Terahertz Metrology [1 ed.] 9781608077779, 9781608077762

Citation preview

Terahertz Metrology

6502_Book.indb 1

11/11/14 10:21 AM

For a listing of recent titles in the Artech House Microwave Library, turn to the back of this book.

6502_Book.indb 2

11/11/14 10:21 AM

Terahertz Metrology Mira Naftaly Editor

artechhouse.com

6502_Book.indb 3

11/11/14 10:21 AM

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress British Library Cataloguing in Publication Data A catalog record for this book is available from the British Library. ISBN-13: 978-1-60807-776-2 Cover design by John Gomes © 2015 Artech House All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. 10 9 8 7 6 5 4 3 2 1

6502_Book.indb 4

11/11/14 10:21 AM

To the memory of my parents

6502_Book.indb 5

11/11/14 10:21 AM

6502_Book.indb 6

11/11/14 10:21 AM

Contents

Foreword xv Introduction xvii

1

Terahertz Time-Domain Spectrometers

1

1.1  Pulsed Terahertz Time-Domain Spectometers 1 1.1.1  Principles of Operation 1 1.1.2  Photoconductive Emitters and Detectors 4 1.1.3  Optical Rectification 9 1.1.4  Electro-Optic Detection 11 1.1.5  Terahertz Air-Based Coherent Detection 14 1.1.6  Cherenkov (Tilted Wavefront) Emitters and Detectors 16 1.2  Continuous-Wave Terahertz Time-Domain Spectrometer and Microwave Photonics 17 1.2.1  Principles and Operation 17 1.2.2  Types of Continuous-Wave Terahertz Emitters and Detectors 19 vii

6502_Book.indb 7

11/11/14 10:21 AM

viii

Terahertz Metrology

1.3  Time-Domain Spectrometer Configurations 20 1.3.1 Transmission 20 1.3.2 Reflection 21 1.3.3  Attenuated Total Reflection 23 1.3.4  Asynchronous Optical Sampling 24 1.3.5  Substrate Lenses 25 1.4  Commercial Systems 27 References 29 2

Parameter Extraction in Time-Domain Spectrometers 33

2.1 Introduction 33 2.2  Material Properties and Wave Propagation 34 2.2.1  Complex Refractive Index and Complex Permittivity 34 2.2.2  Plane Wave Propagation Across an Interface— Fresnel Equations 36 2.2.3  Plane Wave Propagation Through a Dielectric Slab 37 2.2.4  Fabry-Pérot Effect 40 2.3  Extracting Optical Parameters from Terahertz Time-Domain Spectrometers Data 41 2.3.1  Transmission-mode Terahertz Time-Domain Spectrometers: Approximation 41 2.3.2  Transmission-Mode Terahertz Time-Domain Spectrometers: Exact Solution 43 2.3.3  Reflection-Mode Terahertz Time-Domain Spectrometers: Single Reflection 46 2.3.4  Reflection-Mode Terahertz Time-Domain Spectrometers: Double Reflection 48 2.3.5  Attenuated Total Reflection 51 2.4  Preprocessing and Postprocessing of Data 53 2.4.1  Signal Averaging in the Time and Frequency Domains 53 2.4.2  Signal Denoising 55 2.4.3  Phase Unwrapping 57 2.5  Sample Requirements 59 2.5.1  Sample Thickness 59

6502_Book.indb 8

11/11/14 10:21 AM

Contentsix



2.5.2  Lateral Dimensions of Samples 61 2.5.3  Loss: Absorption, Scattering, Diffraction 62 References 66 3

Metrology for Time-Domain Spectrometers

69

3.1  Performance Parameters: Sensitivity, Resolution, Precision, Accuracy, Dynamic Range, and Signal-to-Noise Ratio 69 3.2  Dynamic Range and Signal-to-Noise Ratio of Terahertz Time-Domain Spectrometers 72 3.3  System Performance Limits Due to Dynamic Range and Signal-to-Noise Ratio 77 3.3.1  Frequency Resolution 77 3.3.2  Measurement Bandwidth 79 3.4  Sources of Random Noise 81 3.4.1 Amplitude 81 3.4.2 Phase 82 3.5  Systematic Errors 83 3.5.1  Alignment Errors 83 3.5.2  Focal Plane Defocusing 84 3.5.3  Delay Line Errors 86 3.6  Terahertz Beam Profile 87 References 89 4

Evaluation of Uncertainty in Time-Domain Spectroscopy 91

4.1 Introduction 91 4.2  Method of Uncertainty Evaluation 92 4.2.1  General Definitions 92 4.2.2  Evaluation of Random and Systematic Errors 94 4.2.3  Propagation and Combination of Measurement Uncertainty 95 4.2.4  Measurement with Resolution Limit 96 4.3  Sources of Uncertainty in Terahertz Time-Domain Spectroscopy 97 4.3.1  Random and Systematic Errors in Amplitude 98

6502_Book.indb 9

11/11/14 10:21 AM

x

Terahertz Metrology

4.3.2  Random and Systematic Errors in Sample Thickness 101 4.3.3  Random and Systematic Errors in Sample Alignment 103 4.3.4  Systematic Error in the Approximated Transfer Function 105 4.3.5  Systematic Error Due to Reflections 107 4.3.6  Systematic Error in Physical Constants 108 4.3.7  Uncertainty in Optical Constants: A Combination of Variances 109 4.4  Practical Implementation 110 References 113 5

Metrology for Fourier Transform Spectrometers 115

5.1  Spectrometer Configurations 115 5.1.1 Michelson 116 5.1.2 Mach-Zender 116 5.1.3  Polarized Fourier Transform Spectrometer (Martin-Pupplett Interferometer) 117 5.1.4  Modes of Operation: Step and Integrate or Fast Scanning 117 5.1.5  Moving Stage 119 5.1.6  General Fourier Transform Spectrometer Advantages 119 5.2  Fourier Transform Spectrometer Metrology 120 5.2.1  Encoders, Synchronizing, and Time-Stamping 120 5.2.2  Instrument Line Function: A Spectral Resolution Limitation 121 5.2.3 Apodization 122 5.2.4  Cosine and Sine Terms and Phase Correction Techniques 122 5.2.5  Achievable Spectral Resolution (Commercial and State of the Art) 125 5.3  Sample Measurement Parameter Recovery 125 5.3.1  Thin Sample Approximation: Lossless Case 126 5.3.2  Dispersive Medium 130 5.3.3  Thick Sample Analysis (or Noncollimated Measurement) 130

6502_Book.indb 10

11/11/14 10:21 AM

Contentsxi



5.3.4  Absorption Coefficients 131 5.4  Sources of Noise and Uncertainties 133 5.4.1  Time-Domain Errors 133 5.4.2  (Software) Noise Reduction Techniques 136 References 138 6

Terahertz Spectrometer Calibration

141

6.1  Why Calibrate Terahertz Spectrometers? 141 6.2  Frequency Calibration 143 6.2.1 Gases 143 6.2.2 Solids 145 6.2.3  Resonant Mesh Filters 149 6.2.4 Etalons 150 6.2.5  Summary of Frequency Calibration Standards 154 6.3  Amplitude Linearity Calibration 154 6.4  Reflection Standards 161 References 162 7

Terahertz Imaging

165

7.1 Introduction 165 7.2  Image Resolution of a Terahertz Imaging System 165 7.3  Data Analysis in the Far Field 167 7.4  Imaging Thin Layers 170 7.5  Spatial Resolution Tests 176 7.6  Near-Field Imaging 178 7.7  Chemical Imaging 179 7.8  Focal Plane Array Microbolometer Cameras 180 7.9 Conclusions 182 References 182 8

Metrology for Vector Network Analyzers

8.1  Vector Network Analyzers 8.1.1  The Role of a Network Analyzer 8.1.2  Scattering Parameters 8.1.3  Vector Network Analyzer Systems 8.1.4  Extender Heads

6502_Book.indb 11

185 186 186 187 196 200

11/11/14 10:21 AM

xii

Terahertz Metrology

8.1.5  Good Practice Tips 202 8.2  Metallic Waveguides 204 8.2.1  Basic Properties 204 8.2.2 Dispersion 207 8.2.3  Standardized Sizes and Frequency Ranges 209 8.2.4 Flanges/Interfaces 212 8.2.5  Good Practice Tips 216 8.3  Calibration Standards and Methods 218 8.3.1  Calibration: General Principles 218 8.3.2  Types of Calibration Standard 223 8.3.3  One-Port Calibration Methods 226 8.3.4  Two-Port Calibration Methods 228 8.3.5  Good Practice Tips 234 8.4  Errors and Uncertainties 236 8.4.1  Main Sources of Measurement Error 236 8.4.2  Typical Sizes of Errors 238 8.4.3  Connection Repeatability 242 8.4.4  System Verification 244 8.4.5  Good Practice Tips 246 8.5  Concluding Remarks 247 References 248 9

Terahertz Optics

9.1  Terahertz Optical Materials 9.1.1  Transparency at Terahertz Frequencies 9.1.2  Inorganic Crystals for Terahertz Optics 9.1.3  Polymers for Terahertz Optics 9.2  Metal Reflectivities 9.3  Gaussian Optics 9.3.1  Gaussian Beams 9.3.2  Parabolic Mirrors 9.4  Dichroic Optical Elements 9.4.1  Terahertz Transparent/Visible Opaque 9.4.2  Terahertz Opaque/Visible Transparent 9.4.3  Terahertz Reflective/Visible Transparent 9.4.4  Terahertz Transparent/Visible Reflective 9.5  Wire-Grid Polarizers 9.6  Antireflection Coatings

6502_Book.indb 12

251 251 251 255 258 262 265 265 267 269 269 270 271 271 273 275

11/11/14 10:21 AM

Contentsxiii



9.7  Photonic Crystal Fibers and Terahertz Waveguides 277 References 278 10

Terahertz Laser Sources

283

10.1  10.2  10.3  10.4  10.5 

Optically Pumped Gas Lasers 283 P-Germanium Laser 285 Quantum Cascade Lasers 287 Pulsed Difference Frequency Generation 289 High-Resolution Frequency Measurements Using Continuous-Wave Difference Frequency Generation 293 10.6  Optical Parametric Oscillators 294 10.7  Free Electron Lasers 296 References 298 11

Electronic Sources of Terahertz Radiation and Terahertz Detectors

303

11.1  Introduction and Scope 303 11.2  Electronic Sources of Terahertz Radiation: Introduction and Terminology 304 11.3  Vacuum Electronic Sources 307 11.3.1  Backward Wave Oscillators 307 11.3.2  Extended Interaction Klystrons 308 11.4  Solid-State Devices 308 11.4.1  Gunn Diodes and IMPATT Oscillators and Amplifiers 308 11.4.2  Transistor Amplifiers 310 11.5  Frequency Multiplied Sources 311 11.6  Source Characteristics Summary 313 11.7  Detectors of Terahertz Radiation: Introduction and Terminology 315 11.8  Incoherent Detectors 317 11.8.1  Golay Cell 317 11.8.2  Pyroelectric Detectors 318 11.8.3  Room-Temperature Bolometers 319 11.8.4  Cryogenic Bolometers 320

6502_Book.indb 13

11/11/14 10:21 AM

xiv

Terahertz Metrology

11.8.5  Photoconductive Detectors 322 11.8.6  Semiconductor Diode Rectifiers 322 11.8.7 Thermopiles 324 11.8.8  Incoherent Detector Summary 324 11.9  Coherent Detectors: Introduction 326 11.9.1  Schottky Diode Mixers 327 11.9.2  Frequency Measurement 329 11.10  Detector Summary 330 11.11  Laboratory Practice 330 11.12  Emerging Technologies 332 11.13 Summary 332 Acknowledgments 333 References 333 A

Frequency and Wavelength Unit Conversion

335

B

Tips on Fourier Transform

337

B.1  Number of Points Required for Fast Fourier Transform 337 B.2  Sampling Interval and Undersampling 337 B.3  Time-Domain Peak Width and Spectral Bandwidth 338 B.4  Time-Domain Echoes and Spectral Oscillations 339 List of Contributors

341

About the Editor

343

Index 345

6502_Book.indb 14

11/11/14 10:21 AM

Foreword In 1800, the musician and astronomer William Herschel discovered the existence of the infrared spectrum by observing that a thermometer showed a rise in temperature when placed beyond the red component of sunlight that had been separated by a glass prism. In due course, instrumentation was developed to explore further the properties of infrared light, but it was not until the turn of the twentieth century, when H. Rubens and E. F. Nichols published their significant paper, “Heat Rays of Great Wavelength,” that terahertz frequency science and its associated technology were born. Subsequent developments from the newly emerging field of radio communications ensured that, in due course, electromagnetic waves could be generated that were of even greater wavelength than those that had been previously studied by Rubens and Nichols using essentially optical instrumentation and methods. Even at this early stage, it became apparent that there were two distinct approaches to the generation, manipulation, and detection of terahertz frequency radiation: the optical route and the electronic route. Consequently, there would be a gap in technical capability arising from the paucity of sources of radiation in the region where the performance of generating devices began to fall off, for very different physical reasons. It is only in recent years that reasonably successful attempts at closure of this gap have been made, through discoveries such as quantum cascade lasers, time-domain spectroscopy, new electronic materials, and advances in semiconductor processing. In parallel with these remarkable developments, there has been a growing realization that investigation of the xv

6502_Book.indb 15

11/11/14 10:21 AM

xvi

Terahertz Metrology

terahertz frequency properties of matter can provide a whole new insight into the workings of nature, leading to diverse applications in laboratory, observatory, clinic, office, or industrial settings. As with all scientific investigations, it is essential that measurements are standardized: they must be made using mutually agreed upon units that are traceable and convenient and the measurements themselves must be repeatable and undertaken with known precision, accuracy, resolution, and sensitivity. Moreover, there needs to be a mature understanding of the limitations of measurement imposed by noise and the peculiarities of instrumentation, and great caution must be exercised in interpretation, with due regard paid to the methods by which results are extracted from the actual raw data provided by the experimental apparatus. The terahertz science and technology community has, perhaps, been somewhat tardy in recognizing the fundamental importance of metrology in its endeavors, and maybe on occasions has made confusing or even overexaggerated claims about what has actually been detected, imaged, and presented in spectroscopy and sensing studies. The situation has also been complicated by the two different traditions (the optical and electronic alluded to earlier) that have continued since the beginning of the twentieth century. As examples of such obfuscation or lack of rigor, one might consider the long-standing use of both dB.m–1 and cm–1 to describe material absorption, the less-than-careful attention paid to measurements of incident power, the frequent disregard of the optical properties of terahertz frequency beams with multifrequency components, or the artifacts arising from incorrect transformation of the time-domain data. As technology in this region emerges from its Cinderella status and becomes more mainstream, it is essential that such errors and omissions are regarded as unacceptable and that measurements are made with standardized and traceable units, so that there are no longer any concerns or question over published data and images. This book attempts to address these matters. It has been written by several well-known authorities in the field, working in academia or industry or at standards laboratories across the world. It should prove valuable to those engaged in scientific and technical studies of the terahertz frequency properties of materials, devices, and systems from astrophysics through materials science to biology and medicine. Martyn Chamberlain Emeritus Professor, Department of Physics Durham University, Durham, United Kingdom October 2014

6502_Book.indb 16

11/11/14 10:21 AM

Introduction Dear reader, if you are reading this book, then you are already aware of the utility of terahertz measurements and their numerous and multifarious applications in science and in industry. The growing popularity of terahertz is evidenced by the near-exponential rise in the number of publications referring to it (Figure I.1), and any list of its uses would be both incomplete and redundant. Metrology is a necessary adjunct to all fields of experimental science and all types of engineering measurements. It quantifies accuracy, resolution, and uncertainty and establishes the methodologies most conducive to obtaining high accuracy with low uncertainty. All mature fields of measurement have highly evolved metrology with international supervisory bodies and agreed standards. The area of terahertz measurements is still fairly new and developing rapidly, with new techniques and instrumentation arising frequently and gaining acceptance. It is only in the last few years that issues of metrology have begun to be considered and addressed. Terahertz metrology is the primary focus of this book; and in this it differs from other books in this field. This book is aimed at practitioners who wish to perform terahertz measurements, to choose the most appropriate instruments and techniques for their requirements, and to have confidence in the validity of their results. It has two goals. The first goal is to highlight metrological aspects of terahertz measurement techniques, to offer tips on good practice, and to point out the pitfalls. The second goal is to provide a practical, straightforward, easy-toimplement guide to terahertz instruments, devices, components, and methods. xvii

6502_Book.indb 17

11/11/14 10:21 AM

Terahertz Metrology

Number of publications mentioning "terahertz"

xviii

10000

1000

100

1980

1990

2000

2010

Y ear

Figure I.1  The yearly number of publications mentioning terahertz. (Data from Google Scholar. The value for 2014 was estimated by doubling the number for June.)

For that reason, complex processes are explained in greatly simplified terms, with references to publications providing greater detail. The bulk of the book is devoted to time-domain spectroscopy. This is because time-domain spectroscopy is the premier technique for terahertz measurements, massively dominating the field, and indeed whose invention in the late 1980s sparked the growth of terahertz activities (see Figure I.1). Timedomain spectroscopy won its success due to its high dynamic range and its ability to detect both amplitude and phase of the transmitted radiation, translating into highly accurate measurements of absorption and refractive index. Chapter 1 describes the operation of terahertz time-domain spectroscopy systems, both pulsed and continuous-wave, their major and most commonly used components, and their most widely adopted configurations. Chapter 2 details the calculations required to extract the optical parameters of the material tested (i.e., its complex dielectric constant) from the recorded timedomain data, and discusses sample requirements. Chapter 3 focuses on the metrology of time-domain spectroscopy measurements, the limits of its performance, and the sources of noise and systematic errors. Chapter 4 is also a metrology chapter, explaining in detail the evaluation of uncertainties in the optical constants obtained by time-domain spectroscopy and listing their most common sources. Chapter 5 is devoted to the other major technique of terahertz measurements, Fourier transform spectroscopy, which operates over a larger bandwidth

6502_Book.indb 18

11/11/14 10:21 AM



Introductionxix

than time-domain spectroscopy and is capable of higher frequency resolution. However, it detects only optical intensity, and its dynamic range is significantly smaller than that of time-domain spectroscopy. In the mid-infrared, down to about 15 THz, where time-domain spectroscopy does not reach, Fourier transform spectroscopy reigns supreme. However, time-domain spectroscopy wins out at frequencies where it is available. Fourier transform spectroscopy is a far older and more established technique than time-domain spectroscopy, its instrumentation is far more evolved, and its metrology is better developed. For that reason, unlike time-domain spectroscopy, where many practitioners, especially research groups, use bespoke laboratory-assembled systems, Fourier transform spectroscopy is mostly carried out using commercial instruments. Chapter 6 is another metrological chapter, describing calibration techniques for terahertz spectrometers, most of which are applicable to both timedomain spectroscopy and Fourier transform spectroscopy, as well as to other types of free-space techniques. Chapter 7 addresses measurement aspects of terahertz imaging, including imaging techniques and data analysis and is again devoted mostly to timedomain spectroscopy as the major imaging modality. Chapter 8 stands out in the book, being devoted to a radically different terahertz measurement modality: vector network analyzer. Vector network analyzers and terahertz spectrometers (e.g., time-domain and Fourier transform) occupy different and separate spheres of terahertz measurements, in that vector network analyzers are configured for device characterization, whereas spectrometers measure material properties. Indeed, time-domain and Fourier transform belong to the general family of optical spectrometers, whereas vector network analyzers are more closely related to electronic instruments such as impedance meters. Moreover, vector network analyzer operation is based on hollow rectangular metallic waveguides, whereas time-domain, Fourier transform, and other spectrometers operate in free space. Some overlaps exist; vector network analyzers have been occasionally used as a platform for free-space terahertz spectroscopy. Conversely, electro-optic sampling uses a time-domain setup to characterize device response. However, in the main, vector network analyzer and spectrometer measurements remain distinct, divided by both areas of application and measurement methodologies. Moreover, intercomparison experiments between a vector network analyzer and a time-domain spectrometer are difficult and limited in scope. Vector network analyzers are set to grow in importance in terahertz measurements, as more and faster devices (e.g., terahertz transistors) are adopted by the industry. The last three chapters are short guides to terahertz components and devices. Chapter 9 discusses terahertz optics, focusing on the practical aspects

6502_Book.indb 19

11/11/14 10:21 AM

xx

Terahertz Metrology

and availability of various optical elements. Chapter 10 is a summary of the main types of laser-based terahertz sources and their operation and capabilities. These may be employed, for example, in types of measurement requiring a narrowband source, either single-frequency or tunable. Chapter 11 lists the most widely used and important electronic terahertz sources and detectors, explaining their operation, summarizing their capabilities, and offering practical usage tips.

6502_Book.indb 20

11/11/14 10:21 AM

1 Terahertz Time-Domain Spectrometers Mira Naftaly

1.1  Pulsed Terahertz Time-Domain Spectometers 1.1.1  Principles of Operation Terahertz time-domain spectroscopy is unique among spectroscopic techniques in that it employs coherent detection to measure directly the field amplitude and phase of the optical wave. As a consequence, it provides a straightforward and unambiguous determination of both attenuation and phase shift resulting from beam interaction with the material studied. Pulsed terahertz time-domain spectroscopy also has two other nearlyunique features. First, the spectral bandwidth is contained in a single-cycle optical pulse. The resulting terahertz beam is therefore both coherent and broadband. Second, a time-domain spectrometer operates in a closed-loop pump-probe configuration whereby the pump and probe beams are derived from the same laser source and the detector is gated by the combined presence of terahertz and probe pulses. This makes possible the high signal-to-noise ratio and the dynamic range of these systems [1]. Pulsed terahertz time-domain spectrometer systems are activated by ultrafast lasers with a pulse length typically shorter than 100 fs. These are 1

6502_Book.indb 1

11/11/14 10:21 AM

2

Terahertz Metrology

mode-locked lasers with broad gain spectra, because the pulse length is inversely related to the bandwidth. One of the most common pump sources for a terahertz time-domain spectrometer is the Ti-sapphire, which has the center wavelength at around 800 nm and is the laser capable of the shortest pulse length of 2.5w in the collimated beam. An additional factor that must be taken into account when using a focused beam is its Rayleigh length. As discussed in Chapter 1, this is the distance over which the beam waist expands by a factor of 2. If the sample thickness is comparable to the Rayleigh length of the focused beam, its lateral dimensions must likewise be doubled. However, the size requirement is mitigated by refractive index of the sample material: both beam waist and its Rayleigh length are reduced proportionally with the refractive index. 2.5.3  Loss: Absorption, Scattering, Diffraction As mentioned above, measurable transmission loss, expressed as extinction coefficient (κ ) or absorption coefficient (α ), may in fact encompass absorption,

6502_Book.indb 62

11/11/14 10:21 AM



Parameter Extraction in Time-Domain Spectrometers63

scattering and/or diffraction in the sample material. In studying material properties, it is important to attempt to distinguish between these effects, for whereas absorption is an intrinsic property of the material, scattering and diffraction arise from medium-scale material structure. The effects of scattering and diffraction have been reported and analyzed in a number of publications [25–30] with the aim to differentiate between these and absorption. Resonant absorption peaks can be unambiguously identified because they are associated with corresponding Lorentzian dispersion in the refractive index via the Kramers-Krönig relationship; an example is shown in Figure 2.12. Regularly structured materials may produce transmission spectra containing loss features arising from diffraction [25, 26]. An example of such is presented in Figure 2.13 which plots the attenuation spectrum of a textile knitted from liquid-crystal polymer, together with its normalized phase difference (which corresponds to refractive index in a solid sample). It is seen that the behavior of the phase associated with diffraction has a different character from that expected in resonant absorption. In addition, diffraction effects tend to be polarization-sensitive. This, however, is not an unambiguous indicator of diffraction, as many birefrigent crystals also have polarization-dependent absorption. A more definitive means of identifying diffraction is by measuring different thicknesses of the material; whereas the absorption coefficient will remain constant, diffraction effects will be thickness-dependent. To sum,

Figure 2.12  Resonant absorption and corresponding anomalous dispersion in cystine.

6502_Book.indb 63

11/11/14 10:21 AM

64

Terahertz Metrology

Figure 2.13  The loss spectra and fractional phase difference (Δ ϕλ /2 π , corresponding to refractive index) in orthogonal and parallel orientations of a Vectran textile knitted using yarn made from liquid-crystal polymer (produced by Kuraray Ltd.). The attenuation features are due to diffraction.

when examining structured materials, diffraction should always be considered as a possible contributor to the attenuation spectrum, as such steps should be taken to differentiate it from absorption. It is unfortunately the case that non-resonance absorption cannot be reliably differentiated from scattering loss. In both cases the loss coefficient α typically increases as a power of frequency, i.e.:

α = Bf β,  2 < β < 4

(2.59)

This is depicted in Figure 2.14, which compares the loss spectra and refractive indices of silica glass and boron nitride ceramic. In glass, the loss is due to absorption by charge defects, whereas in this sample of hot-pressed boron nitride (HBN) it is caused by scattering. The loss behavior in both cases is very similar. It should be noted that the size of porosity structures acting as scattering centers in HBN is of the order of 10 μ m (i.e., much smaller than the wavelength of terahertz radiation). The rate of increase of the loss coefficient with frequency depends on the size, density and geometry of the scattering centers, as well as on the refractive index of the material. As a general rule, if the

6502_Book.indb 64

11/11/14 10:21 AM



Parameter Extraction in Time-Domain Spectrometers65

Figure 2.14  Loss spectra and refractive indices of silica glass and ceramic boron nitride (HBC grade, ordinary ray). In SiO 2 glass loss is due to absorption; in BN ceramic it is due to scattering.

sample material is visually opaque and appears to be grainy (rather than glassy), scattering should be presumed to contribute significantly to the measured loss. Scattering is of particular concern when examining materials prepared as pressed pellets containing a mixture of powders—for example, many organic materials are commonly diluted with HDPE or PTFE powders to reduce their absorption. In such cases scattering centers may be formed not only by individual grains, but also by mixing inhomogeneities in the prepared pellet. Scattering effects in this type of sample are particularly significant because they are used to investigate absorption spectra containing multiple features whose profiles are important for substance identification. This is because, when spectral features are present alongside scattering, their profile is distorted and they appear broader and less well defined [27–29]. Surface scattering, when the material cannot be adequately polished, may be readily identified and accounted for by measuring samples of different thickness. If the frequency-dependent loss is then plotted as a function of thickness and a linear fit calculated, the absorption coefficient will be its slope and surface loss its offset. Similarly to bulk scattering, significant frequencydependent loss can be caused by sub-wavelength size roughness.

6502_Book.indb 65

11/11/14 10:21 AM

66

Terahertz Metrology

References [1]

Klein, M. V., Optics, New York: John Wiley & Sons, 1970.

[2]

Grischkowsky, D., et al. “Far-Infrared Time-Domain Spectroscopy with Terahertz Beams of Dielectrics and Semiconductors,” Journal of the Optical Society of America B: Optical Physics, Vol. 7, No. 10, 1990, pp. 2006–2015.

[3]

Duvillaret, L., F. Garet, and J.-L. Coutaz, “A reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE Journal of Selected Topics in Quantum Electronics, Vol. 2, No. 3, 1996, pp. 739–746.

[4]

Duvillaret, L., F. Garet, and J.-L. Coutaz, “Highly Precise Determination of Optical Constants and Sample Thickness in Terahertz Time-Domain Spectroscopy,” Applied Optics, Vol. 38, No. 2, 1999, pp. 409–415.

[5]

Dorney, T., R. Baraniuk, and D. Mittleman, “Material Parameter Estimation with Terahertz Time-Domain Spectroscopy,” Journal of the Optical Society of America A: Optics, Image Science, and Vision, Vol. 18, No. 7, 2001, pp. 1562–1571.

[6]

Scheller, M., C. Jansen, and M. Koch, “Analyzing Sub-100-μm Samples with Transmission Terahertz Time Domain Spectroscopy,” Optics Communications, Vol. 282, No. 7, 2009, pp. 1304–1306.

[7]

Howells, S. C., and L. A. Schlie, “Transient Terahertz Reflection Spectroscopy of Undoped InSb from 0.1 to 1.1 THz,” Applied Physics Letters, Vol. 69, No. 4, 1996, pp. 550–552.

[8]

Auston, D. H., and K. P. Cheung, “Coherent Time-Domain Far-Infrared Spectroscopy,” Journal of the Optical Society of America B: Optical Physics, Vol. 2, No. 4, 1985, pp. 606–612.

[9]

Hashimshony, D., et al., “Characterization of the Electrical Properties and Thickness of Thin Epitaxial Semiconductor Layers by THz Reflection Spectroscopy,” Journal of Applied Physics, Vol. 90, No. 11, 2001, pp. 5778–5781.

[10] Gatesman, A. J., R. H. Giles, and J. Waldman, “High-Precision Reflectometer for Submillimeter wavelengths,” Journal of the Optical Society of America B: Optical Physics, Vol. 12, No. 2, 1995, pp. 212–219. [11]

Jepsen, P. U., and B. M. Fischer, “Dynamic Range in Terahertz Time-Domain Transmission and Reflection Spectroscopy,” Optics Letters, Vol. 30, No. 1, 2005, pp. 29–31.

[12] Jeon, T.-I., and D. Grischkowsky, “Characterization of Optically Dense, Doped Semiconductors by Reflection THz Time Domain Spectroscopy,” Applied Physics Letters, Vol. 72, No. 23, 1998, pp. 3032–3034. [13] Thrane, L., et al., “THz Reflection Spectroscopy of Liquid Water,” Chemical Physics Letters, Vol. 240, No. 4, 1995, pp. 330–333.

6502_Book.indb 66

11/11/14 10:21 AM



Parameter Extraction in Time-Domain Spectrometers67

[14] Rønne, C., et al., “Investigation of the Temperature Dependence of Dielectric Relaxation in Liquid Water by THz Reflection Spectroscopy and Molecular Dynamics Simulation,” Journal of Chemical Physics, Vol. 107, No. 14, 1997, pp. 5319–5331. [15]

Jepsen, P. U., U. Møller, and H. Merbold, “Investigation of Aqueous Alcohol and Sugar Solutions with Reflection Terahertz Time-Domain Spectroscopy,” Optics Express, Vol. 15, No. 22, 2007, pp. 14 717–14 737.

[16] Nagai, M., et al, “Terahertz Time-Domain Attenuated Total Reflection Spectroscopy in Water and Biological Solution,” International Journal of Infrared and Millimeter Waves, Vol. 27, No. 4, 2006, pp. 505–515. [17] Møller, U., et al., “Terahertz Reflection Spectroscopy of Debye Relaxation in Polar Liquids,” JOSA B, Vol. 26, No. 9, 2009, pp. A113–A125. [18] Fischer, B. M., M. Hoffmann, and P. U. Jepsen, “Dynamic Range and Numerical Error Propagation in Terahertz Time-Domain Spectroscopy,” in Optical Terahertz Science and Technology, Technical Digest (CD). Optical Society of America, 2005, paper TuD1. [19] Ferguson, B., and D. Abbott, “Wavelet De-Noising of Optical Terahertz Pulse Imaging Data,” Fluctuation and Noise Letters, Vol. 1, No. 2, 2001, pp. L65–L69. [20] Ferguson, B., and D. Abbott, “De-Noising Techniques for Terahertz Responses of Biological Samples,” Microelectronics Journal, Vol. 32, No. 12, 2001, pp. 943–953. [21] Pupeza, I., R. Wilk, and M. Koch, “Highly Accurate Optical Material Parameter Determination with THz Time Domain Spectroscopy,” Optics Express, Vol. 15, No. 7, 2007, pp. 4335–4350. [22] Jin, Y.-S., G.-J. Kim, and S.-G. Jeon, “Terahertz Dielectric Properties of Polymers,” Journal of the Korean Physical Society, Vol. 49, No. 2, 2006, pp. 513–517. [23] Withayachumnankul, W., et al., “Limitation in Thin-Film Sensing with TransmissionMode Terahertz Time-Domain Spectroscopy,” Optics Express, Vol. 22, No. 1, 2014, pp. 972–986. [24] Withayachumnankul, W., B. M. Fischer, and D. Abbott, “Material Thickness Optimization for Transmission-Mode Terahertz Time-Domain Spectroscopy,” Optics Express, Vol. 16, No. 10, 2008, pp. 7382–7396. [25] Fletcher, J., et al., “Propagation of Terahertz Radiation Through Random Structures: An Alternative Theoretical Approach and Experimental Validation,” Journal of Applied Physics, Vol. 101, No. 1, 2007, p. 013102. [26] Thúberge, F., et al, “Spectral Artifacts from Non-Uniform Samples Analyzed by Terahertz Time-Domain Spectroscopy,” Optics Express, Vol. 17, No. 13, 2009, pp. 10 841–10 848. [27] Bandyopadhyay, A., et al., “Effects of Scattering on THz Spectra of Granular Solids,” International Journal of Infrared and Millimeter Waves, Vol. 28, No. 11, 2007, pp. 969–978.

6502_Book.indb 67

11/11/14 10:21 AM

68

Terahertz Metrology

[28] Franz, M., B. M. Fischer, and M. Walther, “The Christiansen Effect in Terahertz TimeDomain Spectra of Coarse-Grained Powders,” Applied Physics Letters, Vol. 92, No. 2, 2008, p. 021107. [29] Shen, Y. C., P. F. Taday, and M. Pepper, “Elimination of Scattering Effects in Spectral Measurement of Granulated Materials Using Terahertz Pulsed Spectroscopy,” Applied Physics Letters, Vol. 92, No. 5, 2008, p. 051103. [30] Naftaly, M., J. Leist, and R. Dudley, “Investigation of Ceramic Boron Nitride by Terahertz Time-Domain Spectroscopy,” Journal of the European Ceramic Society, Vol. 30, No. 12, 2010, pp. 2691–2697.

6502_Book.indb 68

11/11/14 10:21 AM

3 Metrology for Time-Domain Spectrometers Mira Naftaly 3.1  Performance Parameters: Sensitivity, Resolution, Precision, Accuracy, Dynamic Range, and Signal-to-Noise Ratio To quantify system performance, it is necessary to define the performance parameters and to determine their values. Metrological performance parameters include: sensitivity and resolution, precision and accuracy, dynamic range, and signal-to-noise ratio [1]. Sensitivity and resolution are related, in that sensitivity refers to the minimum measurable value, whereas resolution refers to the minimum measurable difference between two values. Precision is the numerical value of measurement reproducibility. The three quantities of sensitivity, resolution, and precision can be defined in terms of the measurement uncertainty, which is discussed in more detail in Chapter 4. Experimental values are determined with associated uncertainty (see Chapter 4), as x ± ∆x, where x represents the mean value x=

N

1 ∑ x (3.1) N i=1 i 69

6502_Book.indb 69

11/11/14 10:21 AM

70

Terahertz Metrology

and Δx is the measurement uncertainty given by the standard deviation of the measured value x, calculated as: ∆x = s =

N

2 1 xi − x ) (3.2) ( ∑ N i=1

where N is the number of measurements. This is Type-A uncertainty, derived directly from the measured data. If the distribution of measured values is normal (Gaussian), then 68.2% of data points will lie in the range of x ± ∆x. Type-B uncertainty applies to derived values and is calculated using standard methods of error propagation from known uncertainties in the data and system operation. Both types of uncertainty are discussed more fully in Chapter 4. In evaluating experimental uncertainties, it is important to bear in mind that more often than not the uncertainty is dependent on the value of the parameter being measured [i.e., s = f (x ) ]. This is certainly true of terahertz measurements, and in particular time-domain spectrometer. For example, the uncertainty in the signal amplitude tends to increase with amplitude, as we shall see next. In some types of measurement it is also customary to define the noise floor, which is the standard deviation when the mean signal equals zero [i.e., NF = s(x = 0) ]. Sensitivity, or the minimum measurable value, can be understood as being roughly equal to twice the noise floor, or twice the uncertainty for a small signal [Figure 3.1(a)]. Similarly, resolution, as the minimum measurable difference between two values, is roughly equal to twice the uncertainty of those values [Figure 3.1(b)]. Precision is the reproducibility of the measurement, and as such can also be interpreted as being represented by the uncertainty [Figure 3.1(c)]. That is because reproducibility can be viewed as the probability that a given data point will lie within one standard deviation (i.e., uncertainty) of the mean value, which can be calculated for any distribution of the measured data. Sensitivity, resolution, and precision are all determined internally to the experimental system, by performing repeated measurements and analysing the data to determine its mean, standard deviation, and type of distribution. In contrast, accuracy is evaluated by reference to an externally known quantity. Accuracy is defined as the difference between the true value and the value measured by the experimental system. As such, it must be determined

6502_Book.indb 70

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers71

Figure 3.1(a–c)  Schematic depiction of metrological performance parameters: sensitivity, noise floor, resolution, precision, and accuracy. Top: sequentially measured data points. Bottom: data distribution.

6502_Book.indb 71

11/11/14 10:21 AM

72

Terahertz Metrology

by calibrating the system against a known standard or calibration artefact [Figure 3.1(c)]. For a measurement to be considered correct it is necessary that the measured and true values agree within error, in other words, that the true value lies within one standard deviation of the mean data (i.e., xmeas − s < xtrue < xmeas + s ). This requires that accuracy have a lower value than precision [i.e., accuracy must be encompassed within precision, as seen in Figure 3.1(c)]. The dynamic range (DR) and signal-to-noise ratio (SNR) of a system are also defined internally to the system, based on the analysis of the measured data. However, unlike the parameters discussed above, DR and SNR are generally used to refer to the directly measured signal, and not to any derived quantities. DR is the ratio of the maximum absolute value of the measurable signal to the noise floor:



DR =

xmax (3.3) NF

whereas SNR is the ratio of the absolute mean signal to its uncertainty:

SNR =

x (3.4) ∆x

SNR and DR reflect different and complementary aspects of the system performance. SNR indicates the minimum detectable signal change and therefore is related to sensitivity and resolution. DR describes the maximum quantifiable signal change and indicates the range of values that can be measured by the system. Clearly, DR must always be significantly larger than SNR for meaningful measurement to be possible, preferably DR > 10 SNR.

3.2  Dynamic Range and Signal-to-Noise Ratio of Terahertz Time-Domain Spectrometers As discussed in Chapter 2, in time-domain spectrometers data are acquired in the time domain, while spectroscopic analysis is commonly carried out in the frequency domain, derived from the time domain data via the Fourier transform. Consequently, the SNR and DR of a time-domain spectrometer may be evaluated either with respect to the original time-domain trace or with respect to the calculated frequency-domain spectrum. In terahertz time-domain spectrometers, these produce very different results, and there

6502_Book.indb 72

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers73

is no simple analytical relationship between the values of SNR and DR of the time-domain data and those of the Fourier transform-derived spectrum. Figure 3.2(a) shows part of a typical time-domain trace, which in this case is a mean of 16 runs. Also plotted is the standard deviation of the data, which varies with the time delay, and is largest where the signal is strongest. Indeed, it has been observed empirically that the standard deviation of amplitude in the time domain is roughly proportional to the absolute value of the amplitude. Figure 3.2(a) also shows the SNR calculated according to (3.4), which is seen to fluctuate strongly and irregularly from point to point, to the extent that it provides no meaningful information as to the system performance. Note that the scatter of data obtained from multiple traces is larger than the noise observed at a fixed delay position. This is demonstrated in Figure 3.2(b) which compares the scatter in the maximum amplitude of 16 traces with the noise measured at a fixed delay position: it is seen that the standard deviation is approximately twice as large for multiple traces as for static measurement. This may be due in part to the mechanical noise and positional jitter caused by the movement of the delay stage. In contrast to SNR, the dynamic range in the time domain meaningfully represents system performance. To calculate the DR, the noise floor, which is the detector noise in the absence of a terahertz signal, is determined by setting the delay so as to record the residual noise signal prior to the arrival of the terahertz pulse. Clearly, the time dependence of the DR will follow the absolute value of the signal, as seen in Figure 3.2. In this example, the maximum value of the DR is at the apex of the negative peak, where it is 1,100, while the SNR is 115. The DR and SNR of spectra obtained via the Fourier transform from the time-domain data are evaluated using a similar procedure. First, it is important to remember that, as explained in Chapter 3, for purposes of noise reduction, multiple measurement traces must be averaged in the time domain, not in the frequency domain (Figure 3.3). The frequency-dependent DR of the spectrum can be evaluated simply as the ratio of amplitude at a given frequency to the noise floor. As seen in Figure 3.3, in time-domain spectral data, the noise floor is clearly identifiable as the frequency-invariable signal minimum. Also shown in Figure 3.3 is the standard deviation of the spectral amplitude. Similarly to the case of signal amplitude, the standard deviation is roughly proportional to the amplitude itself. Note that the noise floor can also be defined as the level where the signal meets the standard deviation. This relationship was used to derive a simple and very useful method for estimating the standard deviation of absorption coefficients (Type-B

6502_Book.indb 73

11/11/14 10:21 AM

74

Terahertz Metrology

Figure 3.2  (a) Left ordinate: Part of a typical time-domain trace (mean of 16 runs), and the standard deviation of the data. Right ordinate: DR and SNR. (b) Comparison between scatter in the data obtained from 16 traces and data noise measured at a fixed delay position. The standard deviation is approximately twice larger for multiple traces than for the static measurement.

6502_Book.indb 74

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers75

Figure 3.3  Terahertz time-domain spectrometer source spectrum derived from the time domain data in Figure 3.2, showing the noise floor, the maximum dynamic range, and the standard deviation.

uncertainty) from the standard deviation of spectral amplitude σ (ν ) (Type A uncertainty). The standard deviation of spectral amplitude is assumed to be approximately given by [2]:

σ (ν ) = C[E(ν ) − h] + h (3.5)

where E(ν ) is the amplitude, h is the noise floor, and C is an empirical proportionality constant. From (3.5) and (3.2), the standard deviation of the absorption coefficient σ α of a sample of thickness d can be obtained using standard rules of error propagation [3], which give: 1



2 ⎤2 s sample 1 ⎡ s 2 ⎥ (3.6) s a = ⎢ ref + 2 2 d ⎢⎣ Eref Esample ⎥⎦

Clearly, the spectral profile of the DR will be identical to that of the source spectrum, as seen in Figure 3.4. In contrast, the spectral SNR must be calculated as the ratio of mean amplitude to its standard deviation, where to calculate the standard deviation of spectral data, the Fourier transform has to be applied individually to each time-domain trace. In Figure 3.4, the

6502_Book.indb 75

11/11/14 10:21 AM

76

Terahertz Metrology

Figure 3.4  The DR and SNR of the spectrum shown in Figure 3.3.

maximum DR is 1,800 while the maximum SNR is 200. Note again that neither the DR nor the SNR of spectral amplitude is directly related to those of the time-domain data. It is a widely accepted custom to quote the maximum value as the DR of a time-domain spectrometer. This approach is justifiable to a degree, because the great majority of time-domain spectrometers produce similar spectral profiles, and the frequency dependence of the DR follows that of the source spectrum. Less justifiable is the confusion of DR and SNR and the resulting practice of quoting SNR as the ratio of signal and the mean (RMS) noise in the absence of signal (which is the definition of the DR). An important consequence of the typical values of the SNR and DR is that the large DR allows the examination of strongly attenuating samples, while the much lower SNR limits the accuracy and amplitude resolution of these measurements. Because the standard deviation is roughly proportional to the signal throughout a large part of the spectrum, the SNR is relatively flat over those frequencies. In the time-domain spectrometer shown in Figures 3.2 through 3.4, this range is between 0.2 and 2 THz (which will, of course, vary among different systems). The constant amplitude uncertainty provided by the flat SNR can be of use in spectral measurements. In contrast, the frequency-dependent DR limits the usable spectral bandwidth for lossy samples, as discussed next.

6502_Book.indb 76

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers77

3.3  System Performance Limits Due to Dynamic Range and Signal-to-Noise Ratio 3.3.1  Frequency Resolution In a pulsed terahertz time-domain spectrometer, the frequency resolution of the measured spectrum is ultimately limited by the system DR. In contrast, the resolution of continuous-wave time-domain spectrometer (and generally, frequency-mixing) systems is limited only by the linewidths of the two lasers and by the accuracy, resolution, and stability of their wavelength control; in consequence, their frequency resolution tends to be higher than those of pulsed systems. This section will focus on pulsed time-domain spectrometers; high-resolution frequency measurements using continuous-wave difference frequency generation will be discussed in Chapter 10. Clearly, the frequency interval of the spectral data sets the lower bound on the spectral resolution (i.e., the minimum measurable difference between two spectral points). As the outcome of Fourier transform, in both the timedomain spectrometer and the Fourier transform spectrometer, the data frequency interval is given by c/2L, where L is the distance length of the delay sweep. In a noise-free system with unlimited available delay, the maximum resolution of a pulsed time-domain spectrometer is set by the repetition rate of the pump laser (typically around 80 MHz for a mode-locked Ti-sapphire). However, in the presence of noise, the achievable frequency interval is much lower, being limited by the DR of the system in the time domain. That is because the signal amplitude diminishes with increasing delay from the main pulse, and therefore so does the DR, eventually approaching unity, as seen in Figure 3.2. From that point onwards, scanning to longer delay spans adds no further data, because the signal is drowned in noise. Thus, the usable delay span is bounded by the DR of the system. Indeed, a study of the dependence of the time-domain spectrometer frequency resolution on noise concluded that in a typical system the maximum achievable resolution is of the order of 1 GHz [4]. In common experimental practice, it is advisable to limit the delay scan length to the region where DR remains >2. The apparent frequency resolution of a time-domain spectrometer may be improved by zero-padding (i.e., extending the time-domain data set by supplementing it with a string of zeros). This is a particularly common practice in systems employing fast shakers with a restricted delay range. As the frequency interval equals c/2L, increasing L reduces the interval by forcing the Fourier transform to generate more data points in the frequency domain, thus increasing the apparent resolution. Clearly, the additional data points

6502_Book.indb 77

11/11/14 10:21 AM

78

Terahertz Metrology

arising from a string of zeros carry no additional information. Nevertheless, provided that the measured spectrum lacks narrow-frequency features, the method can be useful, producing spectra that are significantly more accurate than low-resolution ones, because sufficient information is contained in the main time-domain pulse. Due to the time-frequency relationship embodied in the Fourier transform, narrow spectral features require extended time data to be determined accurately. However, broader features can be derived accurately from shorter time spans. Zero padding utilizes the information contained in the data time span by forcing the transform to generate additional frequency points that trace the spectral features with improved fidelity. An example of zero-padding and its effects is shown in Figure 3.5, which plots absorption of atmospheric water vapor between 1.06 and 1.20 THz obtained with high resolution, low resolution, and zero padding. Water vapor has numerous narrow absorption lines, many of them multiplets, which are too narrow and closely spaced to be clearly resolved even by the long scan spectrum with the data interval of 2.5 GHz. It is seen that all points in the unpadded short spectrum coincide with those of the padded spectrum, while the latter also contains additional points that fill out and refine the spectral profile. Nevertheless, the padded spectrum fails to reproduce the fine detail of the long-scan spectrum.

Figure 3.5  Part of absorption spectrum of atmospheric water vapor obtained using a 60-mm high-resolution scan (2.5 GHz), a 10-mm low-resolution scan (15 GHz), and a 10-mm scan with zero padding to 60 mm.

6502_Book.indb 78

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers79

3.3.2  Measurement Bandwidth We have seen that the dynamic range of a time-domain spectrometer in the time domain limits its frequency resolution. Likewise, in the frequency domain, the dynamic range places limits on the bandwidth, maximum measurable absorption, and usable sample thickness. One of the most important papers dealing with the dynamic range of terahertz time-domain spectrometers and its implications on the bandwidth was published by Jepsen and Fischer [5]. They pointed out that the maximum absorption coefficient α max that is measurable in transmission is determined by the DR via the relationship:



⎡ E ⎡ 4n ⎤ 4n ⎤ = 2ln⎢DR amax d = 2ln⎢ max 2 ⎥ (n +1)2 ⎥⎦ (3.7) ⎣ ⎣ Emin (n +1) ⎦

where d is the sample thickness and n is its refractive index. Emax is the reference field and Emin is the noise floor. That is because, as shown in Chapter 2, the absorption coefficient is calculated from the ratio of the reference and sample spectra using an equation of the form of (3.7). Clearly, the minimum measurable sample spectrum cannot be below the noise floor, giving the maximum measurable absorption coefficient as presented in (3.7). Because the dynamic range of a spectrum produced by a time-domain spectrometer falls with frequency (Figure 3.4) while absorption typically increases, (3.7) sets an upper bound on the effective measurement bandwidth. Figure 3.6 demonstrates the mechanism using model data. The DR curve in Figure 3.6 is typical of a time-domain spectrometer, such as seen in Figure 3.4, and α max is calculated from (3.7) assuming d = 1 and n = 2. The values and profiles of α 1, α 2, and α 3 are of a form commonly found in absorbing materials; the available measurement bandwidth is bounded by the intercept of α max and α i. It is seen that for α 1 > α 2 > α 3 the falling curve of α max limits the bandwidth such that BW1 < BW2 < BW3. Figure 3.7 is an experimental demonstration of (3.7) and the model presented in Figure 3.6. The absorption coefficients of two different material samples were measured, and are plotted together with respective α max calculated from (3.7). It is seen that at frequencies above the intersection of α = α max the absorption spectrum becomes very noisy and decreases in line with the α max curve. In interpreting absorption data, this is a visual indicator that the dynamic range of the system has been exceeded and that the data from that point onwards are no longer valid or meaningful.

6502_Book.indb 79

11/11/14 10:21 AM

80

Terahertz Metrology

Figure 3.6  The relationship between DR and measurement bandwidth explained using model data. The DR curve is typical of a time-domain spectrometer. α max is calculated from (3.7) assuming d = 1 and n = 2. The values and profiles of α 1, α 2 , and α 3 are of a form commonly found in absorbing materials. For α 1 > α 2 > α 3 the falling curve of α max limits the bandwidth such that BW1 < BW2 < BW3.

Figure 3.7  Experimental demonstration of (3.7). α max is calculated for two samples: fused silica (SiO 2 , d = 3 mm, n = 2.0) and ceramic BIN77 (50BN:50AlN, d = 2 mm, n = 2.5); α max is larger for the thinner sample. It is seen that absorption cannot be measured beyond α max , and that this limits the measurement bandwidth.

6502_Book.indb 80

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers81

Note that in (3.7) the maximum measurable absorption is inversely proportional to the sample thickness. The optimal sample thickness to be used in time-domain spectrometer measurements was discussed in more detail in Chapter 2.

3.4  Sources of Random Noise 3.4.1 Amplitude As we have seen, random amplitude noise is a critical parameter determining the performance of a terahertz time-domain spectrometer and limiting its frequency resolution and measurement bandwidth. Unsurprisingly, sources of noise and noise reduction have been a major concern in time-domain spectrometer design. Mechanical sources of amplitude noise include vibrations, air currents in the beam paths, particulates in the air along the beam paths, and thermal deformations. Commercial systems minimize all these by employing rigid structures and enclosed beam paths, while bench-top modular systems must rely on good laboratory conditions and careful design to achieve desired performance. Amplitude noise that arises from pulse-to-pulse variations in the pump laser intensity can often be reduced by increasing the modulation frequency. This is because laser noise commonly has a large 1/f component, the effect of which can be reduced by performing measurements at a higher frequency. However, even when all sources of electronic and mechanical noise have been minimized, the residual amplitude noise in time-domain spectrometers tends to be of the order of 1% of signal intensity. This was seen to be the case by Hübers et al. [6] who have tested different time-domain spectrometers employing both photoconductive and electro-optic detectors. Incidentally, this study confirmed once again that amplitude noise is proportional to the absolute value of amplitude. Laser amplitude fluctuations, beam pointing deflections, and detector noise were all considered as possible noise sources, but were found to be too small to account for the observed noise levels [6–8]. Moreover, it was seen that the random errors in amplitude and phase spectra of a time-domain spectrometer are independent of each other and therefore are unlikely to be attributable to a common cause [6, 8]. As discussed in Chapter 2, a large majority of time-domain spectrometers employ biased photoconductive emitters as terahertz sources. It has been proposed that amplitude noise in such systems is due to the intrinsic properties of photoconductive emitters, and a detailed analysis identified thermal noise and generation-recombination noise as the major components [9].

6502_Book.indb 81

11/11/14 10:21 AM

82

Terahertz Metrology

Frequency-dependent thermal noise of a photoconductive terahertz source has the form [9]:

ST (ω ) =

4kTR (3.8) R + ω 2 L2 2

where T is the temperature, R is the emitter resistance, and L is its equivalent inductance. Therefore, an emitter with high resistivity will be less noisy. In addition, cooling the emitter will also improve its noise performance. Indeed, it has long been experimental practice for emitter mounts to be designed as efficient heat sinks. The generation-recombination noise has the form [9]:



SN (ω ) =

4 g 0 t 2 (3.9) 1 + ω 2t 2

where g 0 is the electron generation rate and τ is their recombination lifetime. The electron generation rate is determined by the pump laser intensity and must be as high as possible to maximize the emitted terahertz power (see Chapter 2). A low-noise emitter therefore requires a short recombination lifetime. An example of photoconductive emitter material with high resistivity and short carrier lifetime is low-temperature grown gallium arsenide (LT-GaAs), which is in widespread use in time-domain spectrometer systems. The validity of this analysis was demonstrated by Hou et al. [9], who compared the performance of emitters fabricated on semi-insulating GaAs (SI-GaAs) and oxygen-implanted GaAs (GaAs:O). The carrier lifetime of GaAs:O was reduced by a factor of 103 compared with SI-GaAs and was similar to that achieved in LT-GaAs, and it also had somewhat higher resistivity. In tests measuring amplitude noise at the peak maximum of the time-domain signal, the GaAs:O emitter had an SNR up to 3.7 times higher than the SI-GaAs emitter. It is claimed that, compared with LT-GaAs, the oxygen-implanted GaAs:O can be fabricated far more easily, cheaply, and reproducibly. 3.4.2 Phase Phase noise in time-domain spectrometers can be interpreted as meaning two different aspects of noise behavior. The first refers to the phase jitter of the modulated terahertz signal as observed by the detector and arises from imperfect synchronicity between the terahertz signal and detector triggering. This type of phase noise is rarely a problem in time-domain spectrometers employing

6502_Book.indb 82

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers83

biased photoconductive emitters, because such emitters have their bias modulated electronically, allowing phase noise to be eliminated by electronic means. However, systems using optical rectification may rely on mechanical modulation, such as optical choppers, and be subject to phase noise arising from their operation. For example, a free-spinning chopper blade will produce slight random variations in the duration, rise time, and fall time of the modulated signal. Moreover, vibration and air currents caused by a mechanical chopper often contribute to increased amplitude noise as well. For all those reasons, and because electronic modulation can be much faster than mechanical, it is preferable to employ electronic or optical means of modulation whenever possible. The second type of phase noise refers to timing jitter of the terahertz signal and is due to imperfect delay line registration. This will be discussed in Section 3.5.3.

3.5  Systematic Errors 3.5.1  Alignment Errors Errors in system alignment can cause a reduction of the terahertz signal; more seriously, they can cause spectral distortions that may lead to erroneous measurements. Naturally, commercial systems undergo rigorous alignment during manufacture. However, many terahertz time-domain spectrometers in use today are assembled on a bench-top from modular components. Attention to accurate alignment is therefore of utmost importance. For example, a study of the imaging properties of parabolic mirrors and the effects of small deviations from perfect alignment concluded that alignment errors of the order of 10 µrad can cause significant image distortions [10]. Visibly observable signs of misalignment include reduced signal amplitude, distortions of the time-domain profile of the terahertz signal, and altered spectral profile. Figure 3.8 shows an example of this effect caused by an angular misalignment of the order of a few milliradians in each of the parabolic mirrors (see Chapter 1 for mirror configuration). It is seen that misalignment not only reduces the signal, but also distorts the profile of the time-domain trace, leading to corresponding distortions in the spectral profile and a reduction in the available bandwidth. Minor errors in mirror position and orientation tend to affect higher frequencies more strongly, typically causing a steeper frequency fall-off in the spectral profile. Misalignment can also lead to a nonlinearity in the amplitude response, producing particularly severe consequences for the accuracy of measured spectral parameters. This will be discussed in detail in Chapter 6.

6502_Book.indb 83

11/11/14 10:21 AM

84

Terahertz Metrology

Figure 3.8  Effect of misalignment of the order of a few milliradians in each of the parabolic mirrors on (a) the observed time-domain signal and (b) spectrum of a terahertz time-domain spectrometer.

3.5.2  Focal Plane Defocusing When an optically thick material sample is placed in the focal plane of the terahertz beam, its wavefront may be disturbed. A typical arrangement used for measuring optical parameters of materials was shown in Figure 1.9, where

6502_Book.indb 84

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers85

the sample under test is placed in the focused beam between two central parabolic mirrors. This is widespread practice, in part because it allows for the examination of samples with relatively small lateral dimensions, whereas, in contrast, collimated beams in time-domain spectrometers commonly have large diameters (typically 50 to 75 mm). However, it is contrary to the accepted design of every other kind of spectrometer, from ultraviolet to far-infrared, which places objects under test in the imaging plane or in a collimated beam (see Chapter 5). If the beam wavefront is substantially distorted by the presence of a dielectric object at its waist, it will disturb beam propagation and especially its refocusing and terahertz-probe overlap at the detector. This, in turn, will affect spectral measurements, in particular causing significant errors in the refractive index. This effect is related to the Gouy shift [11], and has been examined specifically in relation to errors in time-domain spectrometer measurements [11, 12]. Figure 3.9 shows an example of refractive index error observed when using a focused beam, as opposed to a collimated beam, to measure the refractive index of an optically thick sample. Such errors become more significant in samples with large optical thickness: here the material is 5-mm-thick pyrolitic boron nitride with a mean ordinary refractive index no = 2.292. Therefore, it is advisable to design a terahertz time-domain spectrometer so as to afford a measurement position in the collimated part of the beam, such as for example

Figure 3.9  Measured refractive index of a 5-mm-thick pyrolitic boron nitride (pBN) plate using collimated and focused terahertz beams.

6502_Book.indb 85

11/11/14 10:21 AM

86

Terahertz Metrology

Figure 3.10  Positional noise in time-domain traces due to errors in the delay stage registration (here stage position is deliberately offset to demonstrate the effect). The solid black line represents the mean of nine traces. The inset shows the effect of trace averaging on the resultant spectrum.

in Figure 1.9. When focused beam must be used, especially for optically thick samples, the possibility of measurement errors should be borne in mind. 3.5.3  Delay Line Errors As discussed in Chapter 1, a crucial mechanical component in the great majority of terahertz time-domain spectrometers in current use is the delay line. Delay positioning errors can translate into noise and errors in the spectral data via several mechanisms. Systematic positioning error (i.e., incorrect registration of the positional scale) will cause a proportional error in the frequency scale via the Fourier transform. Such an error is easily identified and rectified by frequency calibration, as discussed in Chapter 6. In contrast, random variations in delay registration will appear as amplitude noise in the time-domain trace. In this case, averaging multiple timedomain traces to reduce noise produces a broader mean pulse shape, which therefore leads to a narrower calculated spectrum. Figure 3.10 demonstrates the mechanism.

6502_Book.indb 86

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers87

It may be expected that every type of delay mechanism will be subject to a different from of error [13]. In systems that suffer from variable delay positioning, it is advisable to exercise great care in applying signal averaging and to resort often to frequency calibration. A technique for applying frequency calibration to each individual trace will be described in Chapter 6.

3.6  Terahertz Beam Profile The diameter and profile of terahertz beams at various points in a time-domain spectrometer may need to be measured. This may be done for several reasons: as part of system characterization, to verify and optimize system alignment, to check the spot size for imaging or for studying small samples, and to observe effects of optical components or test pieces inserted in the beam path. A commonly employed method of beam profile characterization is the knife-edge technique, where a thin edge (e.g., razor blade) is inserted in the beam at the test point and is translated across the beam, gradually obscuring it, while the terahertz signal is recorded at consecutive knife-edge positions. The procedure is depicted in Figure 3.11. The maximum of the time domain peak may be plotted against the edge position, and for a perfect Gaussian beam the dependence will be the error function. Alternatively, spectral profiles

Figure 3.11  (a) A schematic depiction of the knife-edge technique for beam measurement. (b) The dependence of signal on the edge position for a perfect Gaussian beam (the error function).

6502_Book.indb 87

11/11/14 10:21 AM

88

Terahertz Metrology

may be calculated, and amplitudes at selected frequencies plotted against each edge position, again resulting in an error function whose width decreases with frequency. However, in practice, the beam profile is unlikely to be perfectly Gaussian. Experimental systems are liable to minor misalignments, flaws in the optical components, aperture effects, and other types of defects that influence beam propagation and profile. As a result, real terahertz beams are likely to have distorted profiles, circular asymmetry, and even local irregularities. Because a knife-edge test yields a signal integrated over the beam area, it is incapable of resolving and identifying the details of the beam profile. An alternative solution is to raster scan a pin-hole over the beam area, producing a two-dimensional map of amplitude, which may also be frequency resolved. An example is shown in Figure 3.12, which plots the maximum of the time-domain peak obtained by scanning a 3 mm aperture over the collimated part of the terahertz beam in a time-domain spectrometer [14]. The imaged beam profile clearly deviates from circular symmetry and contains distinct local irregularities. Although aperture scanning is more time-consuming and laborious than a knife-edge test, it is also far more informative and illuminating.

Figure 3.12  Terahertz beam profile imaged using aperture scanning, obtained by plotting the maximum of the time-domain peak.

6502_Book.indb 88

11/11/14 10:21 AM



Metrology for Time-Domain Spectrometers89

References [1]

Bendat, J. S., and S. G. Piersol, Random Data Analysis and Measurement Procedures, 3rd ed., New York: John Wiley & Sons, 2000.

[2]

Tripathi, S. R., et al., “Practical Method to Estimate the Standard Deviation in Absorption Coefficients Measured with THz Time-Domain Spectroscopy,” Opt. Commun., Vol. 283, 2010, pp. 2488–2491.

[3]

Bevington, P. R., and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, New York: McGraw-Hill, 2003.

[4]

Mickan, S., et al., “The Limit of Spectral Resolution in THz Time-Domain Spectroscopy,” Proc. SPIE, Vol. 5277, 2004.

[5]

Jepsen, P. U., and B. M. Fischer, “Dynamic Range in Terahertz Time-Domain Transmission and Reflection Spectroscopy,” Opt. Lett., Vol. 30, 2005, pp. 29–31.

[6]

Hübers, H. -W., et al., “Terahertz Spectroscopy: System and Sensitivity Considerations,” IEEE Trans. on Terahertz Sci. Tech., Vol. 1, 2011, pp. 321–331.

[7]

Takeda, M., et al., “Exploration of the Origin of Random Error in Spectrum Intensity Measured with THz-TDS,” 35th International Conference on Infrared Millimeter and Terahertz Waves (IRMMW-THz), 2010.

[8]

Hiromoto, N., et al., “Study on Random Errors in THz Signal and Optical Constants Observed with THz Time-Domain Spectroscopy,” 35th International Conference on Infrared Millimeter and Terahertz Waves (IRMMW-THz), 2010.

[9]

Hou, L., W. Shi, and S. Chen, “Noise Analysis and Optimization of Terahertz Photoconductive Emitters,” IEEE J. Selected Topics Quantum Electronics, Vol. 19, 2013, p. 8401305.

[10] Howard, J. E., “Imaging Properties of Off-Axis Parabolic Mirrors,” Appl. Opt., Vol. 18, 1979, pp. 2717–2722. [11]

Kužel, P., et al., “Gouy Shift Correction for Highly Accurate Refractive Index Retrieval in Time-Domain Terahertz Spectroscopy,” Opt. Exp., Vol. 18, 2010, pp. 15338–15348.

[12] Choi, D., et al., “Focused Beam Effect on Measuring Precise Optical Parameters of Liquid Water with Terahertz Time Domain Spectroscopy,” 2011 36th International Conference on Infrared, Millimeter and Terahertz Waves (IRMMW-THz), October 2011, pp. 1–2, 2–7. [13] Cohen, N., et al., “Experimental Signature of Registration Noise in Pulsed Terahertz Systems,” Fluctuation Noise Lett., Vol. 6, 2006, pp. L77–L84. [14] Molloy, J. F., M. Naftaly, and R. A. Dudley, “Characterization of Terahertz Beam Profile and Propagation,” IEEE J. Select. Topics Quantum Electron., Vol. 19, 2013, p. 8401508.

6502_Book.indb 89

11/11/14 10:21 AM

6502_Book.indb 90

11/11/14 10:21 AM

4 Evaluation of Uncertainty in Time-Domain Spectroscopy Withawat Withayachumnankul, Bernd M. Fischer, and Derek Abbott 4.1 Introduction Components and techniques have been developed such that a terahertz timedomain spectrometer can largely overcome intrinsic problems from thermal background radiation and atmospheric absorption. A terahertz time-domain spectroscopy waveform transmitted through a material sample is typically rich in information, because its shape is altered by the material’s characteristic frequency response. Sample and reference waveforms, once converted by Fourier transform into the frequency domain, can be processed to extract the frequency-dependent optical constants of a material by means of a reliable parameter extraction method (see Chapter 2 for more details). Nevertheless, even if one assumes that the parameter extraction process is nearly perfect, the operation of the hardware is far from flawless. Namely, measurements of signals and associated parameters still contain errors that affect the quality of the extracted optical constants. Several sources of random and systematic errors exist throughout the measurement process. As discussed in Chapter 2, these sources are, for instance, signal noise, sample misalignment, 91

6502_Book.indb 91

11/11/14 10:21 AM

92

Terahertz Metrology

and thickness measurement variation. Thus, for a reliable measurement, evaluation of uncertainty is critical in the optimization of measurement accuracy. The uncertainty analysis to be discussed in this chapter is a combination of the analytical models for significant error sources, and is applicable to transmission-mode Terahertz time-domain spectroscopy [1]. Some merits of the uncertainty model discussed in this chapter are as follows: the model reduces the time spent in determining the measurement uncertainty, which previously was carried out using a lengthy Monte Carlo method; the model allows the evaluation and comparison of more than one source of error, rather than the noise in the signal alone; the model offers a standard in the evaluation of uncertainty in the optical constants obtained from terahertz time-domain spectroscopy, and thus permits assessment of and comparison among results from different measurements; an overall uncertainty determined from the model can be used in discrimination of spectral features from artifacts, as any peak that has a magnitude, relative to the baseline, lower than the uncertainty level, can be labeled as an artifact; and through the model, a methodical optimization of the measurement parameters is possible.

4.2  Method of Uncertainty Evaluation This section introduces basics of uncertainty evaluation, including common definitions, evaluation of random and systematic errors, combining measurement uncertainties, and resolution limit in measurement. 4.2.1  General Definitions A general aim of the evaluation of uncertainty is to establish a quantity that exhaustively quantifies ambiguity present in a measurement [2]. An implication of the uncertainty is that it can reasonably localize the true value of a measurand with respect to the mean value, although this statement is ambiguous as the true value is, by nature, inaccessible. Nevertheless, no matter which viewpoint is held, an appropriate measurement result is always accompanied by an uncertainty, which is useful in the justification of the experiment and in comparison among measurements. Evaluation of measurement uncertainty can be categorized into two types, depending on the source of information. According to the International Vocabulary of Basic and General Terms in Metrology (VIM) [3], Type A evaluation of uncertainty involves a direct statistical analysis of measurements under

6502_Book.indb 92

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 93

repeatability conditions, whereas Type B involves obtaining a measurement uncertainty via other means, such as a published value or a deduction from personal experience. In this work, evaluation of measurement uncertainty in terahertz time-domain spectroscopy concerns both types of evaluations. Before further information about methodology for an evaluation of uncertainty is discussed, some important metrological terms are introduced. Consider Figure 4.1, where the variable X is subject to measurement. The true value or measurand x 0 is sought by the observer. In fact, this value cannot be accessed due to the lack of information about the systematic error, which constantly biases the measurement result in one direction. What is obtainable with a single measurement is an estimation xl, which deviates from the true value by both the systematic and random errors. The average value of N statisN tically independent measurements gives an arithmetic mean x = ∑ xl /N . l=1

An expectation, μ x, which is free from random error, would be achieved in the limit if the number of measurements becomes infinite.

Figure 4.1  Some definitions and relations used in evaluation of uncertainty. The systematic error biases the expectation from the true value, whereas the random error added to the expectation yields the estimation. The confidence interval localizes the expectation with regard to the arithmetic mean, and the uncertainty expands the interval to localize the true value. This graph is related to Type A evaluation of measurement uncertainty, where the evaluation is based on a statistical analysis obtained from repeatable measurements. See Section 4.2.2 for the descriptions of confidence interval and uncertainty. (Modified from [4].)

6502_Book.indb 93

11/11/14 10:21 AM

.

94

Terahertz Metrology

4.2.2  Evaluation of Random and Systematic Errors A random error, as mentioned earlier, is the difference between an estimation, xl, and the expectation, μ x. The evaluation of random error is to establish a confidence interval that can localize the expectation μ x, based on the data obtainable from a set of estimations. The Guide to the Expression of Uncertainty in Measurement (GUM) [5] suggests that a confidence interval follows



x−

kP k sx ≤ mx ≤ x + P sx (4.1) N N

where N is the number of measurements, kP is the coverage factor, and sx is the empirical standard deviation of X. The coverage factor kP = 1 defines a standard measurement uncertainty, whereas a higher value defines an expanded measurement uncertainty. Determining the coverage factor kp from a specified level of confidence p requires knowledge of the distribution of measurement results (see Annex G in [5] for more information). The empirical variance of X is defined as



s2x =

N

2 1 xl − x ) (4.2) ( ∑ N −1 l=1

The empirical standard deviation equals the positive square root of the empirical variance. Measurement uncertainty must account for both the random error and the systematic error (i.e., the uncertainty must be able to localize the true value with respect to the arithmetic mean). The aforementioned systematic error is the difference between the expectation μ x and the true value x 0, or f x = μ x − x 0. Thus, the uncertainty, localizing the true value, is an expansion of the confidence interval that localizes the expectation. Figure 4.1 elucidates these relations. The evaluation of systematic error is subjective. Given that a rectangular probability distribution represents a systematic error, the uncertainty suggested by the GUM is a geometric combination between the standard deviations of the random and systematic errors, or



6502_Book.indb 94

ux = k P

f2 s2x + s,x (4.3) N 3

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 95

where f s,x / 3 is the standard deviation of a rectangular probability distribution that covers [−fs,x, fs,x], which embraces an unknown f x. However, as a possible alternative to the GUM, Grabe assumes a constant systematic error defined over a specific time period [4, 6, 7]. Hence, the analysis of the systematic error is separated from that of the random error, and the uncertainty is an arithmetic combination between the confidence interval and the worst-case term,



ux = k P

sx + f s,x (4.4) N

In either case, a final measurement result is reported in terms of the arithmetic mean and its uncertainty, or x ± ux . Further evaluation of systematic errors in terahertz time-domain spectroscopy measurement exploits either geometric or arithmetic combination. The selection is based on the nature of the systematic error. If the systematic error is definitive (i.e., it becomes the systematic deviation), the worst-case scenario is adopted. Otherwise, a rectangular distribution and a geometric combination satisfy the analysis. 4.2.3  Propagation and Combination of Measurement Uncertainty Often, a measurand Φ is a function of many input quantities, X1, X 2, … , X M, and Φ(X1, X 2, … , X M) is called a measurement function. In this case the evaluation of uncertainty needs augmentation to account for the propagation and combination of input uncertainties. Possible means of the evaluation can be categorized into two types, numerical or analytical. An analytical evaluation supported by the GUM provides an explicit solution to the propagation and combination of uncertainty. Yet it requires the linearity assumption of a measurement function in the vicinity of interest. However, a numerical evaluation, in particular, the Monte Carlo method [8, 9], has an advantage over its analytical counterpart in that it readily supports the propagation of uncertainty through a nonlinear function [10]. Nevertheless, a drawback of the Monte Carlo method is that it is time-consuming and, therefore, not flexible. An additional Monte Carlo drawback is that no analytical expression is available, and thus this prohibits further in-depth analysis of the measurement. With regard to the analytical evaluation, the propagation of variance, or, equivalently, the law of propagation of uncertainty, for the measurement function Φ is given by

6502_Book.indb 95

11/11/14 10:21 AM

96

Terahertz Metrology

sf2

M −1 M ⎛ ∂f ⎞2 2 ∂f ∂f = ∑⎜ s s (4.5) ⎟ sxi + 2 ∑ ∑ ∂xi ⎠ ∂xi ∂x j xi x j i=1 j=i+1 i=1 ⎝ M

The sensitivity coefficient ∂f/∂xi is calculated at the arithmetic means of the input quantities. This expression also applies to the propagation of the uncertainty of a systematic error that takes on a rectangular distribution. For the propagation of a systematic error that assumes the worst-case scenario, the law of error propagation is m

f s,f = ∑



i=1

∂f f ∂xi s,xi

; − f s,x ≤ f x ≤ f s,x (4.6) i i i

Both (4.5) and (4.6) are the first-order approximation, and hence valid for a linear or approximately linear measurement function in the region of interest. The effects of random and systematic errors could be combined either geometrically or arithmetically, as discussed in Section 4.2.2. For example, the combined uncertainty for a measurement function with two input quantities, in case of the geometric combination, is [7]



u f

=

kP N

2 f 2 ⎞ ⎛ ∂f ⎞2 ⎛ 2 f s, y ⎞ ∂f ⎛ ∂f ⎞ ⎜ ⎟ + s + s ⎜⎜ s2x + s,x ⎟⎟ + 2 ⎜ ⎟ ⎜ ⎟ 3 ⎠ 3 ⎟ (4.7) ∂x ⎝ ∂ y ⎠ xy ⎝ ∂ y ⎠ ⎜ y ⎝ ⎝ ⎠

2 ⎛

( ) ∂f ∂x

( )

and in the case of the arithmetic combination it becomes [7] u f



=

kP N +

2 ⎛ ∂f ⎞2 ∂f 2 ∂f ⎛ ∂f ⎞ sx + 2 ⎜ ⎟ sxy + ⎜ ⎟ s2y ∂x ∂x ⎝ ∂ y ⎠ ⎝ ∂ y ⎠

( )

( )

∂f ∂f f + f ∂x s,x ∂ y s, y

(4.8)

where uf is the uncertainty for a result produced by function Φ(X, Y ). 4.2.4  Measurement with Resolution Limit A resolution limit in a measuring device can give rise to a systematic error. Occasionally, this error is significant in comparison with a random error in an

6502_Book.indb 96

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 97

observation. It is thus suggested that in such cases this limit be incorporated into an expression of uncertainty [2]. Consider the case that a device with its resolution of δ gives an estimation xl of a quantity X, without any other error. An estimation of the same measurand obtained from a device with an infinitesimal resolution could be any value in between xl ± δ /2 and, in practice, xl + Δ. The adjusted term Δ, with a constraint ⎪Δ⎪ ≤ δ /2, is introduced to compensate for the imperfect resolution. Obviously, the probability distribution associated with the adjusted term Δ is rectangular with the variance of δ 2/12. This variance therefore contributes to the total variance of X or, mathematically [2],



s2x +

d2 (4.9) 12

where s2x is a variance due to random error. In practice, the resolution parameter, δ , is an attribute of a measuring device, and therefore the treatment of resolution limit is a Type-B evaluation of uncertainty.

4.3  Sources of Uncertainty in Terahertz Time-Domain Spectroscopy Many sources of error appear in a terahertz time-domain spectroscopy measurement and parameter extraction process. Significant sources of error are shown in Figure 4.2, where they are listed along with the parameter extraction process and accompanied by their class (random or systematic). In addition to noise, the sample signal also contains reflections, which, if not dealt with appropriately, cause a systematic error. The error in the amplitude from several measurements manifests itself as a variance (or deviation). It propagates down the parameter extraction process, through to the Fourier transform and deconvolution stages, producing the variance in the magnitude and phase of the estimated transfer function. The parameter extraction process requires knowledge of the sample thickness, sample alignment, and air refractive index, each of which have a degree of uncertainty. This step introduces the variances to the estimation. Furthermore, an approximation to the model transfer function gives rise to a systematic error. At the output, all these variances accumulate and contribute to the uncertainty in the extracted optical constants. It is noteworthy that some sources of systematic error can be identified and quantified by calibration, as discussed in Chapter 6.

6502_Book.indb 97

11/11/14 10:21 AM

98

Terahertz Metrology

Figure 4.2  Sources of error in terahertz time-domain spectroscopy measurement. The sources of error in the dashed boxes occur in both the terahertz time-domain spectroscopy measurement and the parameter extraction process. The errors produced by these sources are classified as either random or systematic. They cause the variances and deviations, which propagate down the process, and eventually contribute to the uncertainty in the extracted optical constants.

Sections 4.3.1 to 4.3.6 provide an analysis for each source of error in detail, including a connection between these errors and the variance in the optical constants. The combination of all variances to produce the uncertainty in the optical constants is given in Section 4.3.7. 4.3.1  Random and Systematic Errors in Amplitude The terahertz amplitude is prone to variation induced by many sources of random and systematic errors. As mentioned earlier, the sources of random error include laser intensity fluctuation, optical and electronic noise, and jitter in the delay stage, whereas the sources of systematic error include registration error, mechanical drift, and so forth. The variation in the amplitude may embrace the effects from inhomogeneity in a sample or among samples, if the sample is displaced or replaced with nominally identical samples during several measurements. What is considered here is the amplitude variance model, which combines all these errors and assumes a normal probability distribution. This treatment is valid although systematic error is involved, as the systematic error drifts over time and thus cannot be tackled by the method proposed by [7],

6502_Book.indb 98

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 99

which requires a constant systematic error. The amplitude variance is often statistically obtained from a number of repeated measurements, and thus is regarded as a Type-A evaluation. Regarding the natural difference between two types of error in the terahertz amplitude, a random error occurs in a relatively short time scale, in contrast to a systematic error that can be observed only when the measurement time span is long enough. In addition, the amplitude drift due to the systematic error changes in one direction over time. Due to these facts, recording and averaging signals over a long time span increase the amplitude variation. This scenario is demonstrated in Figure 4.3, in which the drift is larger for a succeeding measurement. Therefore, measuring a terahertz signal is a compromise between the random and systematic errors Ii.e., recording over a short time can avoid the drift, but not random error, while recording over a longer period averages out the random error but accentuates the problem of drift).

Figure 4.3  Amplitude variance of measured time-domain signals. The amplitude variance is plotted against the time and the number of measurements. Any two succeeding measurements are separated by approximately 40 seconds. As the number of measurements grows, the variance dramatically increases. The inset shows the arithmetic mean of the 60 measurements. Interestingly, the two peaks in the variance occur at 11.8 and 12.7 ps, whereas the negative and positive peaks in the mean signal are at 11 and 12.4 ps, respectively. The result is most probably dominated by delay-line registration and mechanical drift.

6502_Book.indb 99

11/11/14 10:21 AM

100

Terahertz Metrology

Given the amplitude variances of the time-domain reference and sample signals, denoted by s2E (k) and s2E (k), respectively, the amplitude-related ref

sam

variances in the optical constants, derived from (2.23a) and (2.23b), read as 2 sn,E (ω ) =



⎫ 2 ⎧ Aref (ω ) ⎪ c ⎪ Asam (ω ) + ⎨ ⎬ ωl ⎪ E (ω ) 4 E (ω ) 4 ⎪ ⎩ sam ⎭ ref

( )

sκ2 ,E (ω ) =



(4.10a)

2 ⎧ B (ω ) c ⎪ Bsam (ω ) + ref ⎨ ωl ⎪ E (ω ) 4 E (ω ) 4 ⎩ sam ref 2 ⎛ n(ω ) − n0 ⎞2 sn,E (ω ) ⎫⎪ +⎜ ⎬ ⎟ 2 ⎝ n(ω ) + n0 ⎠ n (ω ) ⎭⎪

( )

(4.10b)

where









Asam (w) = ∑ℑ2 ⎡⎣ Esam (w)exp( jwkt)⎤⎦ s2E (k) k

(4.11a)

sam

Aref (w) = ∑ℑ2 ⎡⎣ Eref (w)exp( jwkt)⎤⎦ s2E (k) (4.11b) k

ref

Bsam (w) = ∑ℜ2 ⎡⎣ Esam (w)exp( jwkt)⎤⎦ s2E (k) (4.11c) k

sam

Bref (w) = ∑ℜ2 ⎡⎣ Eref (w)exp( jwkt)⎤⎦ s2E (k) (4.11d) ref k

Here k is the temporal index and τ is the sampling interval, and thus k τ is the time. The summation is carried out over the time duration of a recorded terahertz signal. In (4.10) and (4.11), all the parameters are calculated at their mean value. Typically, it is possible to assume that s2E (k) ≈ s2E (k) . ref

sam

In (4.10), the square of the thickness, l 2, is a major factor. Increasing the thickness will seemingly decrease the variance in the optical constants. A physical reason behind this is that, for a very thin sample, the system might not be sensitive enough to detect a small change in the amplitude and phase,

6502_Book.indb 100

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 101

which is masked by noise. A thicker sample allows terahertz radiation to interact more with the material, causing a larger change in signal. However, this competes with the fact that ⎪Esam(ω )⎪ ∝ exp(−l) and thus increasing l will lower the amplitude of a sample signal and lift the overall variance. Equation (4.10) combines the effects from both the reference and sample signals. For flexibility in some applications, the effects from the two can be separated. Thereby, the variances in the optical constants due to the variance in the sample signal are 2 sn,E (w) = sam

sk2 ,E (w) = sam



2

( ) c wl

Asam (w) Esam (w)

4

(4.12a)

2 2 ⎫ 2 ⎧ c ⎪ Bsam (w) ⎛ n(w) − n0 ⎞ sn,Esam (w) ⎪ + ⎨ ⎬ wl ⎪ E (w) 4 ⎜⎝ n(w) + n0 ⎟⎠ n2 (w) ⎪ (4.12b) ⎩ sam ⎭

( )

Likewise, the variances in the optical constants due to the variance in a reference signal are 2 sn,E (w) = ref

sk2 ,E (w) = ref

2

( wlc )

Aref (w) Eref (w)

4

(4.13a)

2 2 ⎫ 2 ⎧ c ⎪ Bref (w) ⎛ n(w) − n0 ⎞ sn,Eref (w) ⎪ + ⎨ ⎬ (4.13b) wl ⎪ E (w) 4 ⎜⎝ n(w) + n0 ⎟⎠ n2 (w) ⎪ ⎩ ref ⎭

( )

Separation of the effects from the reference and sample allows evaluation of the uncertainty where the numbers of measurements for the sample and reference signals are not equal. This separation scheme will be used later in Section 4.3.7, when variances from all sources are combined to yield the overall uncertainty. 4.3.2  Random and Systematic Errors in Sample Thickness One parameter that has an influence on the extracted optical constants is the propagation distance of a terahertz beam inside a sample. The propagation distance equals the sample thickness, when the angle of incidence of the beam

6502_Book.indb 101

11/11/14 10:21 AM

102

Terahertz Metrology

is normal to the sample surfaces. The variance associated with this thickness is partially due to a random error in thickness measurement, which may be subject to, for example, the mechanical pressure exerted during thickness measurement, the rigidity of a sample, and so forth. Errors in thickness can also occur due to a change in properties of the sample, for example, a sample of biological tissue can shrink during the experiment due to dehydration or a cryogenically frozen sample can have a different thickness to that measured at room temperature. Another common source for error in sample thickness is that the sample is not being plane-parallel, unless it is professionally optically polished. This implies that different areas of the sample may produce different results. In addition to the random error, another critical factor contributing to the variance in the thickness is the resolution of a measuring device, deemed systematic error. These two types of error and their impact on the optical constants are evaluated separately, in the following discussion. Random Error in Sample Thickness

Generally, a random error occurring in sample thickness measurements has a normal distribution around a mean value. Given the sample thickness variance s2l caused by this error, by referring to the measurement functions in (2.23a) and (2.23b), the thickness-related variances of the optical constants are ⎡ n(w) − n0 ⎤2 2 2 sn,l (w) = ⎢ ⎥ sl (4.14a) l ⎣





⎡ k(w) ⎤2 2 sk2 ,l (w) = ⎢ s l ⎥ l ⎣

⎦

⎦

⎡ c ⎛ n(w) − n0 ⎞⎤2 2 + ⎢ ⎜ ⎟⎥ sn,l (w) (4.14b) ⎣ n(w)wl ⎝ n(w) + n0 ⎠⎦

where s 2l is typically determined from the statistical distribution of several observations, and thus the evaluation is denoted as Type A. Equations (4.14a) and (4.14b) indicate that, with no limit, increasing the sample thickness results in a decrease of the variance of the optical constants. However, a thicker sample also results in a weaker sample signal, which even2 (w) and sk2 ,E (w) in (4.10). tually gives rise to sn,E Systematic Error in Sample Thickness (Resolution Limit)

The resolution of a common thickness measuring device, such as a ­mi­crometer or calliper, is relatively limited. This introduces a systematic error to the thickness measurement. As a result, readout values are influenced

6502_Book.indb 102

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 103

by the combination of both random and systematic errors. According to the analysis of the resolution limit in Section 4.2.4, the variance in the thickness induced by the resolution limit is δ 2l /12, where δ l is the resolution of a measuring device. The propagation functions, which link this variance to the variances in the optical constants, are consistent with those in (4.14). Thus, ⎡ n(w) − n0 ⎤2 dl2 2 sn,d (w) = ⎢ ⎥ 12 (4.15a) l ⎣ ⎦





⎡ k(w) ⎤2 dl2 sk2 ,d (w) = ⎢ l ⎥ 12 ⎣

⎦

⎡ c ⎛ n(w) − n0 ⎞⎤2 2 + ⎢ ⎜ ⎟⎥ sn,d (w) (4.15b) ⎣ n(w)wl ⎝ n(w) + n0 ⎠⎦

Because δ l is obtained from a published value, the evaluation in (4.15) is regarded as Type B. Similar to the thickness-related variances, the variances here decrease as the thickness increases unless the noise in the signal is considered. 4.3.3  Random and Systematic Errors in Sample Alignment When the angle of incidence of terahertz on a sample slab is not normal to the surfaces, the transfer function becomes complicated. Specifically, overly tilting the sample will result in a complex propagation geometry, a deviated beam direction, and a lower terahertz energy focused onto a detector. To avoid these complications, the angle of incidence is typically assumed to be normal to the sample’s surface, so that a simple transfer function can be adopted. However, with manual placement and adjustment of the sample, it is likely that a small misalignment can occur. The small misalignment of a sample causes a longer propagation path of terahertz inside the sample, as illustrated in Figure 4.4. A change in the propagation distance consequently affects the estimated optical constants of the sample. The alignment error, therefore, needs to be taken into account in the evaluation of uncertainty in optical constants. The type of this error is dependent on experimental practice. If the sample is moved in among several measurements, the error is random. However, if the sample is fixed throughout measurements, the error from the sample alignment is systematic. Despite the possible difference in practice, in this work, the error in the sample alignment is considered systematic. The worst-case scenario can bound the error arising from either case.

6502_Book.indb 103

11/11/14 10:21 AM

104

Terahertz Metrology

Figure 4.4  Tilted sample in terahertz beam path. This exaggerated figure illustrates a small tilt angle from the normal, which might occur due to misalignment of the sample. The terahertz path inside the sample, l θ, is longer than the sample thickness, l, by the factor of 1/cos θ t . The refraction angle, θ t , is related to the incident angle (and the tilting angle), θ i , through Snell’s law, n sin θ t = n 0 sin θ i , but for a small tilting angle, θ t ≈ θ i .

According to Figure 4.4, the propagation distance inside a sample, l θ, is a function of the sample thickness, l, and the refraction angle, θ t, or lq =



l (4.16) cosqt

By assuming that the angle of refraction deviates in a small interval [−f θ, f θ] and has its arithmetic mean at the origin, the deviation in the propagation distance is thus ⎛ 1 ⎞ fl = l ⎜ − 1⎟ (4.17) ⎝ cos f q ⎠



Note that the numerical evaluation of an error propagation, as in (4.17), is allowed by the GUM. Given the deviation in the propagation distance, f l, the alignment-related deviations of the optical constants, derived from (2.23a) and (2.23b), are f n,q (w) =



6502_Book.indb 104

fk ,q (w) =

n(w) − n0 fl l

(4.18a)

n(w) − n0 k(w) c fl + ⋅ ⋅ f (w) (4.18b) l n(w)wl n(w) + n0 n,q

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 105

Substituting f l from (4.17) gives





⎛ 1 ⎞ f n,q (w) = ⎡⎣ n(w) − n0 ⎤⎦ ⎜ − 1⎟ cos f ⎝ ⎠ q

(4.19a)

n(w) − n0 ⎛ 1 ⎞ c fk ,q (w) = k(w)⎜ − 1⎟ + ⋅ ⋅ f (w) (4.19b) cos f n(w)wl n(w) + n0 n,q ⎝ ⎠ q

From (4.19), the deviation in the refractive index due to the sample alignment is independent of the sample thickness, whereas the deviation in the extinction coefficient can reduce for a thicker sample. It is noteworthy that these errors are much more significant for samples with high refractive index and extinction coefficient. 4.3.4  Systematic Error in the Approximated Transfer Function As discussed in Chapter 2, estimation of the optical constants can be based on an approximated transfer function. This approximation certainly gives rise to an error in the estimated optical constants. However, unlike any other error in the measurement process, the systematic error arising from an approximated transfer function is recognizable and quantifiable. Furthermore, it can be completely removed from the optical constants, if an accurate technique for material parameter extraction, such as that given in [11–13], is employed. However, a precise approach involves a complicated iterative calculation, and most researchers trade off this complexity with a small error from the approximation. Here a treatment of the approximation is offered to evaluate the error and to assist the selection of an appropriate approach in determining the optical constants. In this section, for clarification, the exact transfer function in (2.20) is referred to as Hexact(ω ), and its approximation in (2.21) is referred to as Happx(ω ). The phase difference between the approximated and exact transfer functions is f∠H (w) = ∠H appx (w) − ∠H exact (w)



6502_Book.indb 105

⎧⎪ ˆ ⎫ 4 n(w)n0 ⎪ = −arg ⎨ 2 ⎬ ⎩⎪ ⎡⎣nˆ(w) + n0 ⎤⎦ ⎭⎪

(4.20)

11/11/14 10:21 AM

106

Terahertz Metrology

In a similar way, the magnitude difference between the two functions is f ln H (w) = ln H appx (w) − ln H exact (w) = ln

4n(w)n0

2

⎡⎣n(w) + n0 ⎤⎦

− ln

4 nˆ(w)n0

⎡⎣nˆ(w) + n0 ⎤⎦2

2 n(w) ⎡ nˆ(w) + n0 ⎤ = ln ⎢ ⎥ nˆ(w) ⎣ n(w) + n0 ⎦



(4.21)



It can be seen that if κ ≈ 0, which makes nˆ(w) ≈ n(w), then f∠H and f ln⎪H⎪ become zero. Derived from the measurement function in (2.23a), the effect of the phase difference on the refractive index deviation is f n,H (w) =



c f (w) wl ∠H

(4.22)

Likewise, derived from (2.23b), the effect of the approximated transfer function on the deviation of the extinction coefficient is



fk ,H (w) =

⎤ c ⎡ 1 n(w) − n0 f ln H + ⋅ f n,H (w) ⎥ ⎢ wl ⎣ n(w) n(w) + n0 ⎦

(4.23)

Obviously, the thickness, l, of a sample is an important factor in both fn,H (ω ) and fκ,H (ω ). A thicker sample implies a lower contribution to the deviation of the optical constants from the transfer function approximation. A physical explanation is that a thick sample enhances the interaction between terahertz waves and the bulk material, as indicated by the exponential terms in (2.20). This enhanced interaction dominates the transfer function, and eclipses the effect from the complex Fresnel transmission coefficients. Here the values of n(ω ) and κ (ω ) are estimated on the basis of a simplified transfer function. Substitution of the approximated values of n(ω ) and κ (ω ) into (4.22( and (4.23) can determine the approximated deviations, fn,H and fκ,H. These deviations are not correction factors for the approximated optical constants, but are rather used to demonstrate the magnitude of the difference between the approximated and the exact values.

6502_Book.indb 106

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 107

4.3.5  Systematic Error Due to Reflections In the measurement of a sample, particularly in the case of parallel and smooth surfaces, the reflections at air-sample interfaces always occur, resulting in reflected pulses in a recorded signal. These reflections, or the Fabry-Pérot effect, can be removed to some extent from the signal by some means prior to the parameter estimation. Otherwise, the transfer function in (2.21) must incorporate the Fabry-Pérot effect, as follows:

HFP(ω ) = FP(ω )H(ω ) (4.24)

where



⎫−1 ⎧⎪ ⎡ nˆ(w) − n ⎤2 ⎡ wl ⎤⎪ 0 FP(w) = ⎨1− ⎢ ⎥ ⋅ exp⎢−2 jnˆ(w) ⎥⎬ (4.25) ˆ c ⎦⎭⎪ ⎣ ⎩⎪ ⎣ n(w) + n0 ⎦

In this case, the effect must be dealt with during the parameter estimation process by an iterative method such as that of [11]. However, a simple extraction method that ignores the Fabry-Pérot effect is often preferred. The effect then propagates to the extracted optical constants, where it manifests itself as a systematic error. In response to that, this section proposes an analytical model that can trace the propagation of a Fabry-Pérot effect, now a systematic error, from the spectrum to the optical constants. Quantification of this error can show how large its contribution is toward the optical constants. In addition, the estimated error from the Fabry-Pérot effect has merit in that it can be used to discriminate real absorption features from oscillatory artifacts. If the peak amplitude (i.e., the absorption spectrum subtracted by its base-line), is lower than the estimated Fabry-Pérot oscillation, then the peak is not of importance and can be regarded as an artifact. In spite of that, in most cases, the Fabry-Pérot effect can be recognized directly by its regular sinusoidal oscillations. Recall that now there are two expressions of the transfer function: one is with the Fabry-Pérot term, HFP(ω ), and the other is an approximation, H(ω ). The phase difference between the two transfer functions is f∠FP (w) = ∠H (w) − ∠H FP (w)

6502_Book.indb 107

= −arg {FP(w)}



(4.26)

11/11/14 10:21 AM

108

Terahertz Metrology

In a similar way, the magnitude difference between the two functions is f ln FP (w) = ln H (w) − ln H FP (w) = −ln FP(w)



(4.27)

Derived from the measurement function in (2.23a), the effect of reflections on the refractive index deviation is f n,FP (w) =



c f (w) (4.28) wl ∠FP

Likewise, derived from (2.23b), the effect of reflections on the deviation of the extinction coefficient is



fk ,FP (w) =

⎤ c ⎡ 1 n(w) − n0 ⋅ f n,FP (w) ⎥ (4.29) ⎢ f ln FP + wl ⎣ n(w) n(w) + n0 ⎦

Again, the sample thickness plays an important role in scaling the deviation caused by the Fabry-Pérot effect. A longer propagation path within a sample results in a lower deviation of the estimated optical constants. An explanation of this observation is that a longer terahertz path length in a sample leads to a reduction in the amplitude of reflected pulses in an exponential manner. The reduced amplitude of reflections makes the approximation more reasonable. In our analysis, the values of n(ω ) and κ (ω ) are estimated without considering the Fabry-Pérot effect. Substitution of the approximated values, n(ω ) and κ (ω ), into (4.28) and (4.29) can determine the approximated deviations fn,FP and fκ,FP, but not the actual deviations. Thus, fn,FP and fκ,FP are not correction factors for the optical constants. 4.3.6  Systematic Error in Physical Constants The refractive index of air is slightly larger than unity and is dependent on the temperature and pressure. The value at 0.89 THz can be estimated from [14]



6502_Book.indb 108

n0,exact = 1+

86.26(5748 + T ) p ⋅10−6 T2

(4.30)

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 109

where p is the partial pressure of water vapor in millimeters of mercury (mmHg) and T is the temperature in Kelvin. At the temperature of 298.15K (25°C) the saturated vapor pressure is 23.76 mmHg—this yields an index offset of 1.4 × 10 –4. Nevertheless, the value of unity for air is always adopted instead of this exact calculation in the estimation of the optical constants for the sake of simplicity. Thus, this approximation causes a systematic error, where the sign and magnitude of the variation is known a priori. The worst-case analysis is adopted in tracing the propagation of this error to the output optical constants. From the measurement function in (2.23a), the refractive index deviation due to the air-index deviation is fn,n0(ω ) = ⎪fn0⎪ (4.31)



where f n 0 = n 0 − n 0,exact. Furthermore, from (2.23b), the deviation in the extinction coefficient is



fk ,n (w) = 0

c n(w) − n0 ⋅ f wl n(w)n0 n0 (4.32)

The relation is straightforward. 4.3.7  Uncertainty in Optical Constants: A Combination of Variances As shown in Sections 4.3.1 to 4.3.6, many sources of error contribute to the variance of the measured optical constants. This section suggests a combination of variances caused by these sources to produce the uncertainty in the optical constants. The combination could be either arithmetic or geometric, dependent on the type of source. The information relating to the combining of variances can be found in Section 4.2.3. The combined uncertainties for the refractive index and extinction coefficient are estimated by addition of the variances and deviations derived so far, or un (w) = kP

2 sn,E

sam

NE

sam



6502_Book.indb 109

+

2 sn,E

ref

NE

ref

+

2 sn,l + s2 N l n,d

+ f n,q + f n,H + f n,FP + f n

0

(4.33a)

11/11/14 10:21 AM

110

Terahertz Metrology

uk (w) = kP

sk2 ,E

sam

NE

+

sam



sk2 ,E

ref

NE

ref

+

sk2 ,l 2 +s N l k ,d

+ fk ,q + fk ,H + fk ,FP + fk ,n

0

(4.33b)

where the coverage factor kP = 1 is for the standard uncertainty and kP > 1 for an expanded uncertainty; NEsam and NEref are the numbers of measurements for the sample and reference signals, respectively; and Nl is the number of measurements for the sample thickness. Because the sources of error are uncorrelated, no covariance terms appear in the formulas. It is advised that when the measurement uncertainty is reported, the coverage factor, kP, and all the components used to determine the uncertainty be listed, along with their evaluating method (i.e., Type A or Type B evalu2 2 2 , s{n,k , ation) [5]. Typically, but not always, s{n,k },Esam },Eref and s{n,k },l are Type A, or statistical observations, whereas the rest of the components are Type B. The calculation of uncertainty presented in this section follows a recommendation of the GUM in that the uncertainty is directly derivable from the contributing sources of error, and it is directly transferable to other measurands, to which the optical constants are relevant. For example, transferring from the uncertainty in the extinction coefficient to that in the absorption coefficient is via ua = (2w/c)uk . The uncertainty model enables further investigation for dominant sources of error in the system, and also enables optimization of the measurement. A parametric sensitivity analysis can also be performed with these equations. However, it should be remembered that the uncertainty model is based on a linear approximation. This low-order approximation is valid in the case where the sources of error have their variation limited to a small vicinity.

4.4  Practical Implementation The analytical models for the propagation of variance, developed in the earlier sections, are implemented with a set of terahertz measurements to demonstrate the functionality. The measurements are carried out with a lactose sample by a free-space transmission terahertz spectrometer. The terahertz time-domain spectroscopy system in use employs photoconductive antennas (PCA) at the transmitter and receiver. The pump laser is a mode-locked Ti:sapphire laser with a pulse duration of 15 fs and a repetition rate of 80 MHz. This generates the terahertz pulse with a FWHM of 0.4 ps,

6502_Book.indb 110

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 111

and its bandwidth spans from 0.1 to 3 THz. The time constant for the lock-in amplifier is set to 30 ms. The surrounding atmosphere is purged with nitrogen to eliminate the effects of water vapor. The lactose sample is prepared by mixing 25 mg of α -lactose monohydrate with HDPE powder and pressing the mixture using a hydraulic press into a solid disk with a diameter of 13 mm and a thickness of 1.85 mm. The sample is placed at the focal plane between two off-axis parabolic mirrors. These mirrors have a focal length of 100 mm and the collimated beam incident on the first mirror has a diameter of 35 mm. According to the theory of Gaussian beam optics [15], the depth of focus (i.e., twice the Rayleigh length), is 2 mm for the 3 THz wavelength. Thus, the sample thickness of 1.85 mm is thinner than the depth of focus of the highest-frequency component. In addition, the largest waist diameter of the beam is 11 mm for the 0.1 THz component, smaller than sample’s diameter, and thus does not lead to edge diffraction. The reference and sample signals are measured alternately to assure that the drift in the signal amplitude does not influence the result. The reference and sample signals are both measured 10 times. The time between two consecutive measurements is 6 minutes on average. Figure 4.5 shows the mean values of the reference and sample signals, along with their standard deviations. No reflections are observed in the sample signal.

Figure 4.5  Average signals and standard deviations for reference and lactose. The reference and lactose signals, each, are averaged over 10 measurements. The signals have a temporal resolution of 0.0167 ps and a total duration of 34.16 ps. The inset shows the spectra of the reference and sample.

6502_Book.indb 111

11/11/14 10:21 AM

112

Terahertz Metrology

Measured by a micrometer with a resolution δ = 1 μ m, the pellet of lactose has an average thickness of 1.85 mm in the propagation direction, and the standard deviation of the thickness from 10 measurements is sl = 5 μ m. Let us suppose as a worst-case that the tilting angle of the lactose sample during the measurements has a rectangular distribution around the origin, bounded by f θ = ±2°. Throughout the measurement, the ambient temperature is approximately 25°C and the humidity is 60%—this corresponds to the saturated vapor pressure of 23.76 mmHg and the partial pressure of 14.26 mmHg. According to (4.30), the refractive index of air is ≈ 1.0001. Shown in Figure 4.6 are the optical constants of the lactose/HDPE pellet, n and κ , their standard deviations, sn and sκ, their deviations, fn and fκ, and the combined uncertainties, un and uκ, plotted on a logarithmic scale. The optical constants are determined from a pair of the averaged reference and sample signals, using the measurement functions in (2.23a) and (2.23b). The standard deviations, the deviations, and the uncertainties are evaluated by using the proposed analytical models. For comparison, the standard deviations of n and κ due to the amplitude variation, or sn,E and sκ,E , are also evaluated numerically from the 10 profiles of their respective values, available from 10 pairs of the reference and sample signals. The analytical and numerical evaluations appear to provide comparable values of sκ,E or sn,E . The slight mismatch is likely caused by the first-order approximation in the analytical model. The refractive index appears constant at n ≈ 1.46, but actually varies slightly with the frequency. The extinction coefficient, however, is strongly dependent on the frequency, and varies in between 0.001 and 0.01; two absorption peaks at 0.53 and 1.37 THz reproduce those reported in [16]. The variation in the terahertz amplitude gives rise to sn,E and sκ,E equally. Because the extinction coefficient is lower than the index of refraction by two orders of magnitude or more, the extinction coefficient is thus significantly affected by sκ,E . Interestingly, the standard deviation in n caused by the thickness variance, or sn,l, is higher than the standard deviation caused by the amplitude variance sn,E . The deviations from the limited thickness resolution, sn,δ and sκ,δ, from the tilting angle, fn,θ and f k,θ, and from the offset in refractive index, fn0 and fκ,n0, are less than the optical constants’ levels by four orders of magnitude, and are deemed insignificant. The transfer function approximation causes a significant impact in the case of the extinction coefficient, because at low frequencies the value of fκ,H is close to the value of sκ,E . Note that no deviation from the reflections is evaluated here, as the reflections are not present in the signal. The uncertainties un and uκ are evaluated with the coverage factor kP of 1. It can be seen that the uncertainties are dominated by the effects from the amplitude variation, sn,E or sκ,E . The values of un and uκ become larger at

6502_Book.indb 112

11/11/14 10:21 AM



Evaluation of Uncertainty in Time-Domain Spectroscopy 113

Figure 4.6  Uncertainty for lactose measurement. The combined uncertainties in the optical constants are plotted in comparison to the mean values of the optical constants and the standard deviations introduced by various sources of error. The combined uncertainty is calculated with the coverage factor kP = 1. Both figures share the same vertical scale. In (a), the refractive index of the lactose/HDPE pellet is approximately 1.46, compared to its combined uncertainty of 10 –3 . The major sources contributing to the combined uncertainty are signal noise and thickness uncertainty. In (b), the extinction coefficient is in the order of 10 –3 , compared to its combined uncertainty in the order of 10 –4 . The major source contributing to the combined uncertainty is signal noise. The arrowheads indicate the low-frequency resonances of α -lactose at 0.53 and 1.37 THz.

higher frequencies, where the magnitude of the sample and reference spectra is low. The tendency of the uncertainties with respect to the spectral position is similar to that of the results reported in [17], in which a similar terahertz time-domain spectroscopy system is used in characterization of some dielectric materials. To sum, the largest contributions to uncertainty arise from the amplitude and thickness uncertainties, whereas contributions from the thickness resolution, the titling angle, and the index offset are deemed negligible.

References [1]

6502_Book.indb 113

Withayachumnankul, W., et al., “Uncertainty in Terahertz Time-Domain Spectroscopy Measurements,” Journal of the Optical Society of America B: Optical Physics, Vol. 25, No. 6, 2008, pp. 1059–1072.

11/11/14 10:21 AM

114

Terahertz Metrology

[2]

Lira, I. H., and W. Wöger, “The Evaluation of Standard Uncertainty in the Presence of Limited Resolution of Indicating Devices,” Measurement Science and Technology, Vol. 8, 1997, pp. 441–443.

[3]

International Vocabulary of Basic and General Terms in Metrology (VIM), ISO, 2004.

[4]

Grabe, M., Measurement Uncertainties in Science and Technology, New York: Springer, 2005.

[5]

Guide to the Expression of Uncertainty in Measurement (GUM), 1st ed., ISO, 1993.

[6]

Grabe, M., “Principles of Metrological Statistics,” Metrologia, Vol. 23, 1987, pp. 213–219.

[7]

Grabe, M., “Estimation of Measurement Uncertainties—An Alternative to the ISO Guide,” Metrologia, Vol. 38, 2001, pp. 97–106.

[8]

Weise, K., and H. Zhang, “Uncertainty Treatment in Monte Carlo Simulation,” Journal of Physics A: Mathematical and General, Vol. 30, 1997, pp. 5971–5980.

[9]

Joint Committee for Guides in Metrology, Evaluation of Measurement Data—Supplement 1 to the Guide to the Expression of Uncertainty in Measurement—Propagation of Distributions Using a Monte Carlo Method, 2006.

[10] Cox, M., and P. Harris, “Up a GUM Tree,” Counting on IT—The UK’s National Measurement Laboratory Newsletter, Vol. 8, 1999, pp. 4–5. [11]

Duvillaret, L., F. Garet, and J.-L. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE Journal of Selected Topics in Quantum Electronics, Vol. 2, No. 3, 1996, pp. 739–746.

[12] Duvillaret, L., F. Garet, and J.-L. Coutaz “Highly Precise Determination of Optical Constants and Sample Thickness in Terahertz Time-Domain Spectroscopy,” Applied Optics, Vol. 38, No. 2, 1999, pp. 409–415. [13] Dorney, T., R. Baraniuk, and D. Mittleman, “Material Parameter Estimation with Terahertz Time-Domain Spectroscopy,” Journal of the Optical Society of America A: Optics, Image Science, and Vision, Vol. 18, no. 7, 2001, pp. 1562–1571. [14] Chamberlain, J. E., F. D. Findlay, and H. A. Gebbie, “Refractive Index of Air at 0.337mm Wave-length,” Nature, Vol. 206, No. 4987, 1965, pp. 886–887. [15] Saleh, B. E. A., and M. C. Teich, Fundamentals of Photonics, Ch. 3: Beam Optics, ser. Wiley Series in Pure and Applied Optics, New York: John Wiley & Sons, 1991, pp. 80–107. [16] Fischer, B. M., et al., “Chemical Recognition in Terahertz Time-Domain Spectroscopy and Imaging,” Semiconductor Science and Technology, Vol. 20, 2005, pp. S246–S253. [17] Haring Bolívar, P., et al., “Measurement of the Dielectric Constant and Loss Tangent of High Dielectric-Constant Materials at Terahertz Frequencies,” IEEE Transactions on Microwave Theory and Techniques, Vol. 51, No. 4, 2003, pp. 1062–1066.

6502_Book.indb 114

11/11/14 10:21 AM

5 Metrology for Fourier Transform Spectrometers Giorgio Savini Fourier transform spectrometers have been employed since the beginning of the previous century and can boast a large heritage of usage with many examples of applications from industry to space instrumentation. A complete analysis of Fourier transform spectrometer designs and their performance and issues requires more than one textbook. In this chapter, an attempt to characterize the main features that define the performance of a Fourier transform spectrometer and its metrology will be addressed, with sufficient references to more detailed sources for more in-depth discussions.

5.1  Spectrometer Configurations Far from attempting a detailed review of the various architectures of possible Fourier transform spectrometers, it is necessary to highlight some of the main characteristics or differences of the various spectrometers, for comments on the metrology and uncertainties to apply. Hereafter, a brief description of some of the most common and frequent architectures that are widely used in industry and academic research are described, with a more complete review of spectrometer architectures to be found elsewhere, and further details on design are available in [1, 2]. 115

6502_Book.indb 115

11/11/14 10:21 AM

116

Terahertz Metrology

5.1.1 Michelson The Michelson spectrometer design (Figure 5.1(a) in its essential form) has two flat mirrors and two superimposed input/output ports. Old manual designs would allow moving one of the mirrors with a spring screw to control the optical path delay, equal to twice the mechanical shift of the mirror. Most designs reserve one port for the input light, while the other would have a small telescope attached to increase the area of light collection. This two-port configuration will inherently perform Fourier spectroscopy of the difference of the two ports. In the majority of cases this is expected, and the light source is substantially brighter to neglect the contribution of the detector. However, when this is not true, and especially in detailed calibrations, the narcissus contribution of the detector port (in analogy to a telescope detector seeing itself) must be taken into account [3, 4]. 5.1.2 Mach-Zender This highly configurable design figure of interferometer unfolds the light path, which is covered twice in the two arms of the interferometer, by rotating beam divider and reflecting mirrors at 45° and recombining the beams at a second angled beam combiner [Figure 5.1(b)]. This configuration has therefore two separate input and output ports, which are ideal for differential measurements of two sources placed at the two input ports. At the same time, this design allows collecting all output light by recording both positive and negative interferograms. In both these arrangements, the beam divider splits the incoming light from the two ports by as close to 50:50 as possible, while introducing the 180° phase shift for one particular input-output, which also yields conservation of energy.

Figure 5.1  Left: Michelson configuration. Right: Mach-Zender configuration.

6502_Book.indb 116

11/11/14 10:21 AM



Metrology for Fourier Transform Spectrometers117

5.1.3  Polarized Fourier Transform Spectrometer (Martin-Pupplett Interferometer) This design is analogous to the Michelson and in its basic architecture [5] makes use of three polarizers. Here the folded design will be described, which is more commonly employed, but similar to the Mach-Zender, the design can be unfolded. The first polarizer fully polarizes the input light, which is then incident on the second polarized beam-divider that is critically placed with its plane at 45°, but with the polarizing axis oriented so as to present a 45° projected angle to the incoming beam. The input signal is thus very efficiently split in a 50/50 ratio (depending on the quality of the polarizer) in the two arms of the spectrometer. The two beams are then redirected back towards the beam splitter with a 180° phase shift (it is actually a 360° phase shift with the extra 180° caused by the flip of the reference frame of propagation) produced by roof-top mirrors that allow the two beams to be transmitted where previously reflected and vice versa. When the two half signals are recombined (one with an added delay equal to twice the mechanical path difference), a generic elliptical polarization state is obtained that is then modulated through the third and final output polarizer. This polarizer can be (in the reference frame following the direction of propagation of the photon) parallel to the first one, to obtain total constructive interference at the zero-path difference, or perpendicular, for a nulling destructive interference. The detailed algebra related to this simple but elegant interference process can be found in [6]. 5.1.4  Modes of Operation: Step and Integrate or Fast Scanning All of the above Fourier transform spectrometer configurations can be adopted in two substantially different modes of operation, linked to the way in which data is acquired. The first can be intuitively referred to as step and integrate (SaI), and is employed in the majority of cases where duration of the measurement is not an issue and a high signal to noise is required. The second, often referred to as continuous or fast scanning (FS), is used when the overall measurement time is critical and/or the measurement has sufficient S/N to integrate over the shortest necessary time interval to achieve the desired spectral resolution compatible with the detector time constant. Let us define the main parameters that characterize the performance of a Fourier transform spectrometer. The detecting system consists of either a single detector or an array of detectors which will have an integration time constant τ and a sampling interval Δt, defined as the inverse of the sample acquisition frequency ν acq. As the spectrometer mirror is moved by a fixed amount or continuously, signal sampling will correspond to an optical sampling

6502_Book.indb 117

11/11/14 10:21 AM

118

Terahertz Metrology

Figure 5.2  A Martin-Puplett Fourier transform spectrometer configuration. The source is collimated via a parabolic mirror and enters the spectrometer through a first polarizer. A second polarizer, angled to present a projected 45° angle, splits the incoming beam in the two arms of the spectrometer, the fixed one and the moving one (on the right). The beam is then shifted by 180° thanks to the roof-top mirrors and the change of frame of reference, to recombine at the beam combiner with the appropriate phase delay for each frequency. It finally exits the spectrometer through a third polarizer which modulates the elliptical states.

Δx often referred to as optical path difference. This identifies, via the Nyquist sampling theorem, the highest frequency that can be unambiguously resolved after applying Fourier transform:

sN ≤

1 2∆x (5.1)

defined as wavenumber in units of cm–1. Frequencies above this will still be detected (unless appropriate spectral filtering is in place), as this depends ultimately on the characteristics of the detector system, but they will be aliased into the resolved spectral range. This can also be used to the advantage of the measurement. If the system is spectrally sensitive within a frequency band [σ 1,σ 2], then the optical sampling will be given by the Shannon sampling theorem ∆x ≤

1 (5.2) 2( s 2 − s1 )

to have the spectral band appropriately aliased.

6502_Book.indb 118

11/11/14 10:21 AM



Metrology for Fourier Transform Spectrometers119

The spectral resolution will be determined mainly by the maximum optical path difference M of the system equal to twice the mechanical distance traveled L by the scanning mirror from the position of maximum coherent interference or zero path difference:

∆s ≤

A (5.3) 4L

where the A coefficient depends on the apodization chosen (see Section 5.2.3). Techniques exist that can be used to avoid being constrained by the above conditions, which can be found in [7]. Finally, from (5.1) and (5.3), the spectral resolving power can be obtained as

R=

sN (5.4) ∆s

5.1.5  Moving Stage The common element to all the above designs is the scanning stage. This can hold a single mirror for the Michelson as a roof-top or a more complex multimirror stage depending on the design. In a Martin-Puplett configuration (as for the polarized Fourier transform spectrometer), a variety of mirror configurations can be chosen, each with different advantages and disadvantages. Table 1 in [2] shows the implications of using different optics in the interferometer arm and how they impact on potential misalignments. In addition to the optical alignment of the mirrors, there are other factors that can affect the quality of the spectra (detailed in Section 5.4), but can be generally linked to the motion drive and the bearings employed. The most common table-top system consists of a linear motor that moves the mirror along the axis aligned to the optical path. Pivot systems [8] are equally popular (especially in airborne or satellites where the overall center of mass of the system is an issue). The simplification of the mechanism allowed by a pivot and a push-pull motor is paid by the arc motion of the mirrors, which can introduce minor degradation of the instrument line function. 5.1.6  General Fourier Transform Spectrometer Advantages There are two known advantages of using a Fourier transform spectrometer when compared to a monochromator or dispersive spectrometers [9]. The first

6502_Book.indb 119

11/11/14 10:21 AM

120

Terahertz Metrology

is known as the throughput or Jacquinot advantage consisting in the fact that it is not limited by a slit aperture, the throughput depends only on the size of the collimated beam entering the spectrometer. The second advantage identified by Felgett is the multiplexing occurring due to the simultaneous integration of all the wavelengths received by the detector which in turn improves the signal-to-noise ratio.

5.2  Fourier Transform Spectrometer Metrology The quality of the data produced by a Fourier transform spectrometer relies heavily on the metrology associated with the knowledge of the position of the scanning mirror, combined with that of the data points in time. The interconnected basic parameters like the ones introduced in (5.1) to (5.4) define the limits of its capability (without taking into account any system issues and imperfections). Having fixed a Maximum optical scan length L = 2M, and assuming fast scan mode of operation, a mechanical speed v of the stage, and a data acquisition frequency f, we can recover the range of unaliased operation of the system (5.1) and its spectral resolving power (5.2) and (5.4). However, there are a number of issues that require attention to ensure that no basic errors are made. 5.2.1  Encoders, Synchronizing, and Time-Stamping Stage motors are often controlled in either open or closed loop (the latter if encoders are present on the stage). Open loop stages are less reliable as they depend on the assumption that the knowledge of the speed imparted to the stage is exact, whereas the presence of an optical encoder with feedback to the controller allows correction to be made for nonlinear motions in the stage drive. In the first case, as the speed of a stage will not be uniform when analyzed in sufficient detail, a data acquisition system that relies on a fixed frequency will inevitably sample the interferogram at unequal steps. When Fourier transformed, such errors conspire in mixing the frequency components of the spectrum diluting its resolving power. Another solution is to link the encoder directly to the acquisition system, so that the latter acquires the data point at the desired positions flagged by the encoder. Nominal use of an encoder that has no feedback loop is done by timestamping. In this popular approach, both the data and the encoder of the mirror

6502_Book.indb 120

11/11/14 10:21 AM



Metrology for Fourier Transform Spectrometers121

position are each recorded with a time value at which it was read. Providing that the two clocks of the systems being interrogated are synchronized, then the two can be recombined. 5.2.2  Instrument Line Function: A Spectral Resolution Limitation As discussed, the resolution of the spectrometer is related to the maximum optical path difference that can be imposed on the input beam. The product of the infinite waveforms entering the Fourier transform spectrometer and the finite interval over which they are sampled (optical path difference) is converted by Fourier transform to the convolution of the spectrum of the input beam with the Fourier transform of the rectangular window function of finite size. If no alteration is made to the finite set of data, this will be a sinc function, the width of which is dictated by the size of the window function by which the interferogram is multiplied. This sinc function is an inherent limitation of the Fourier nature of the length of the data used, and is referred to as the theoretical instrument line function. An additional modification to the instrument line function that needs to be taken into account, because it affects its shape and width, is the acceptance angle of the Fourier transform spectrometer port or off-axis range. This effect is a consequence of the more general issue of an off-axis beam. This can be due to misalignments in the optical system, or can be an inherent systematic error to be taken into account arising from the array nature of the detector system. This is a well-known effect and has been treated extensively in [10–12]. We can omit the source arrangement and the collimating geometry, and we can generally assume that the collimated beam entering the spectrometer has a divergence angle that is symmetrical with respect to the Fourier transform spectrometer axis and uniform in distribution with an angle range [−θ ,θ ]. Given the same mirror position, the actual maximum optical path difference achieved by the most off-axis of rays will be L¢ =



2M (5.5) cosq

and the difference in optical path



6502_Book.indb 121

∆ D = L¢ − L = L

q2 1 1 − cosq −1 = L ≅L ≅ Lq2 /2 (5.6) cosq cosq 2 − q2

11/11/14 10:21 AM

122

Terahertz Metrology

One can either use (5.5) to define the relevant worst-case sinc function (which can be used as a rule of thumb on the best resolution available), or perform a more detailed calculation with a weighted average of the radiation pattern S(θ ):



ILF = 2L

ILF¢ =

+q

sin( y) ; y

y = 2psL (5.7a)

2L sin( 2psL/cosq ) dq (5.7b) 2psL/cosq

∫ S(q) cosq

−q

One can also use (5.5) to find the maximum θ allowed in a system if a given spectral resolution is required, by imposing that the two resolvable wavelengths be in opposite phases at L′ to obtain θ 2 ≤ 1/R. 5.2.3 Apodization The presence of negative side-lobes and distinct ringing features caused by the sinc ILF can be acceptable for some applications, but not for others. To avoid this, one can multiply the interferogram by an apodization function which is generically defined as a function equal to 1 at ZPD (as this point contains the total power information) and 0 at the edges of the sampled region at maximum optical path difference (with different options for the effective weight as a function of path difference). Many studies have been conducted [13, 14] on the effects of different apodization functions on the instrument line function. Here we report the most practical or commonly used for specific applications. As can be seen in Table 5.1, there is a trade-off in using particular functions. Generally, control of side-lobes comes at a cost in loss of spectral resolution due to the increase of instrument line function postprocessing full width at half-maximum. 5.2.4  Cosine and Sine Terms and Phase Correction Techniques In an ideal instrument, as the mirror crosses the zero path difference position, it would replicate the interferogram symmetrically on both sides due to the mathematical equivalence of the interferogram as a function of optical

6502_Book.indb 122

11/11/14 10:21 AM

6502_Book.indb 123

2

exp(−x2 /2s2 )

⎛ x2 ⎞ ⎜⎝ 1− L2 ⎟⎠

a ≤ x 2

g(s2 )

1

1

1.067

f 2 (a)

f1(a)

1.90416

0.5

1

1.77179

1.20671

0.23 → 0

1

0.21 < f 3 (a) > k2 (see Chapter 2); in this model, zero absorption (i.e., k2 = 0) is assumed. Because the total signal is constant (no absorption), when the signal reflected from the air-material interface increases with n2, the signal reflected from the substrate decreases proportionately. The model presented in Figure 7.4 takes into account only a single reflected pulse; in reality, multiple reflections occur, as shown in

Figure 7.4  Modeled reflected pulse from the layer structure in Figure 7.3, assuming a single reflection. The reflected signal at 0.00 mm is from the air-material interface; the larger peak at 0.10 mm is from the material-substrate interface.

6502_Book.indb 171

11/11/14 10:21 AM

172

Terahertz Metrology

Figure 7.5. The modeled reflected signal in Figure 7.5 demonstrates another important feature, namely, the phase change that is highlighted by (7.7). For n1 < n2, rmaterial > 1 (i.e., there is no phase change in the return signal), whereas for n1 > n2, rmaterial < 1 (i.e., the phase of the return signal is flipped by π ). To compare the model with experimental results, Figure 7.6(a) shows the raw terahertz waveform reflected from a 200 µm thick piece of polyvinyl chloride (PVC). PVC is a polar polymer with strong terahertz absorption. The terahertz signal is complex with two positive peaks and two negative peaks resulting from reflections from the front and back surfaces of the sample. Figure

Figure 7.5  (a) Multiple reflections within the structure shown in Figure 7.3. (b) P1 is the signal from the front surface reflection, P2 is from the substrate interface, and the other signal peaks are due to multiple reflections. The phase of the peaks depends on the sign of the reflection coefficient (7.7).

6502_Book.indb 172

11/11/14 10:21 AM



Terahertz Imaging173

Figure 7.6  The raw terahertz time-domain data obtained from a 200-µm-thick piece of PVC. (a) A time-domain waveform and (b) a B-scan.

7.6(b) shows a b-scan through the PVC film. The optical delay is related to the depth into the sample; the grayscale represents the terahertz field amplitude, with white being positive and black being negative. In order to remove the confusion of multiple peaks shown Figure 7.6(a), filters were applied to generate an impulse function, as discussed above. To study the effects of these filter functions, Figure 7.7 shows the effects of increasing the high-frequency cutoff on the waveform presented in Figure 7.6(a). For low values of HF, around 12 to 16, there is ringing in the waveform due to the edge effects of the filter, which

6502_Book.indb 173

11/11/14 10:21 AM

174

Terahertz Metrology

Figure 7.7  The effect of increasing the high-frequency cutoff on the resulting waveform presented in Figure 7.6(a). The low-frequency filter is constant at 500. The consecutive traces are offset for clarity.

diminishes with increasing HF. However, when the value of HF rises too high, the waveform begins to broaden due to the filtering out of the high frequency components [see Figures 7.2(a, b)]. However, increasing the LF filter has less effect on the waveform; although when the value is too low, side-lobes appear

Figure 7.8  The effect of increasing the low-frequency cutoff on the resulting waveform shown in Figure 7.6(a). The high-frequency filter is constant at 20. The consecutive traces are offset for clarity.

6502_Book.indb 174

11/11/14 10:21 AM



Terahertz Imaging175

in the waveform, as seen in Figure 7.8. The setting of the filter parameters lies largely within the judgement of the user, being a trade-off between noise and signal in the data. Comparing the b-scans shown in Figures 7.9 and 7.6, a clear improvement is seen in the quality of the image, allowing easy assignments to be made of the different peaks in the terahertz waveform.

Figure 7.9  (a) The time-domain waveform from a 200-µm-thick piece of PVC resulting from the application of HF and LF filters. (b) The image shows the corresponding B-scan. Significant improvement in the image is seen as compared with Figure 7.6.

6502_Book.indb 175

11/11/14 10:21 AM

176

Terahertz Metrology

7.5  Spatial Resolution Tests To investigate the effects of filter parameters, a U.S. Air Force (USAF) imaging target was tested using reflection imaging. Figure 7.10 demonstrates that increasing the HF filter leads to a reduction in the spatial resolution of the image, due to the loss of the high-frequency components of the terahertz pulse. A similar result was observed in Figure 7.7, with the loss of resolution in the waveform. In contrast, varying the LF filter (Figure 7.11) has little effect on the resolving power of the system. However, the contrast of the image is improved with increased LF, due to the suppression of the dc background of the image. A terahertz image may also be analyzed in the frequency domain, by calculating the signal spectrum at each point and displaying the images obtained

Figure 7.10  The HF filter varies from 12 to 60. As the high-frequency cutoff increases, the spatial resolution of the image decreases due to the loss of the high-frequency components of the pulse [see Figure 7.2(b)].

6502_Book.indb 176

11/11/14 10:21 AM



Terahertz Imaging177

Figure 7.11  The LH filter varies from 100 to 1,000 (HF fixed at 20). There are small changes to the resolving power of the terahertz optics. However, there is a small but noticeable improvement in the contrast with increased LH value.

at different frequencies. This is shown in Figure 7.12, which uses the USAF test piece, with images displayed at frequencies from 0.1 to 2.5 THz. At the lower frequencies, below 0.5 THz or so, the lines are not resolved and cannot be seen. The lines become visible above 1 THz and are clearly resolved at 2.5 THz. However, at this frequency, the reduced signal-to-noise ratio leads to image noise and consequent blurring. It is useful to compare the theoretical spot size against the achievable spatial resolution. This is shown in Figure 7.13. Each of the traces is a signal amplitude line scan from Figure 7.12. The line densities and widths are listed at the top, with the calculated spot sizes shown on the right. The optics used in the experiment have an effective focal length of 50 mm and an effective aperture of 50 mm, giving f/1 focusing conditions, provided that the mirror is fully illuminated by the terahertz beam. Figure 7.13 also points out the issue of oversampling the intended target. These results highlight the fact that care should be taken when defining the spatial resolution of a system, as the values depend on the measurement conditions and on the analysis algorithms used. It should also be borne in mind that in absorbing materials, different frequencies will penetrate to different depths into the sample. Generally, lower-frequency components will

6502_Book.indb 177

11/11/14 10:21 AM

178

Terahertz Metrology

Figure 7.12  Images of the USAF test piece at different terahertz frequencies. At the lowest frequency of 0.1 THz, the features are not resolved, whereas they are clearly resolved at 1.0 THz and above. In the final image at 2.5 THz, the lines are well defined; however, the reduced signal-to-noise ratio causes blurring.

penetrate deeper than higher frequencies. Thus, the absorption characteristics of the sample can also affect image analysis.

7.6  Near-Field Imaging One of the most effective methods of overcoming the limits of spatial resolution as defined in (7.1) is by employing near-field techniques, which have recently been reported to achieve a feature resolution of better than 200 nm

6502_Book.indb 178

11/11/14 10:21 AM

Terahertz Imaging179



[9]. One of the first applications combining terahertz techniques with nearfield methodologies was demonstrated by van der Valk and Planken [22], who used an electro-optic crystal (GaP) in combination with a metallic tip. The interaction of the crystal, the tip, and the terahertz pulse generates a nearfield image, achieving a spot size of about 20 µm. However, the use of this technique is restricted by the necessity for the interactions to occur near the surface of the electro-optic crystal, thus limiting sample accessibility. A more advanced technique has been recently developed, using aperture-less scattering scanning near-field optical microscopy, known as s-SNOM [23, 24]. In optical s-SNOM, light is focused onto the tip of an atomic force microscope (AFM), where resonant enhancement of scattered light takes place and can then be detected. Similarly, in a terahertz SNOM system, the terahertz pulse is focused onto the tip of an AFM, which is made to oscillate at a frequency Ω. The tip-sample distance is then:

g(t) = g 0 + g1(V )cos(Ωt) (7.8)

where t is the time, and g 1(V ) is the dither amplitude of the needle. The scattered terahertz signal is given by: ∞



Escattered (t) = E0 + ∑ Em cos(mΩt) (7.9) m=1

where E 0 is the background signal which is suppressed by extracting the signal at higher harmonics, m, by using lock-in techniques [25]. The scattered signal is highly dependent on the tip-sample distance. Moon et al. [26] demonstrated the power of this technique by imaging across the edge of a gold film at different harmonic frequencies. The line scans across the film showed resolution of 265 nm, 215 nm, and 185 nm at the first, second, and third harmonics, respectively. The achieved resolution was therefore over three orders of magnitude smaller than the wavelength of terahertz radiation used.

7.7  Chemical Imaging Chemical imaging is one of the most complex challenges of terahertz technology. Koda et al. [27] first demonstrated the technique by imaging envelopes containing illicit drugs. Transmission spectroscopy on sealed envelopes that were raster-scanned across the terahertz beam was performed using a bright

6502_Book.indb 179

11/11/14 10:21 AM

180

Terahertz Metrology

Figure 7.13  Line scans across the horizontal lines shown in Figure 7.12. The line density and line widths are listed at the top. On the right is shown the approximate spot size for each frequency assuming f/1 optics. Consecutive traces are offset for clarity.

tunable terahertz source, a terahertz OPO (see Chapter 10), and an Si-bolometer or a pyroelectric detector (see Chapter 11), so that for each point on the envelope a full terahertz absorption spectrum was acquired. The technique was able to identify methamphetamine and 3.4-methylenedioxy-N-methylamphetamine (MDMA) drugs that were concealed from view. Recently, a group at NNT Corporation used a broadband terahertz time-domain system, again in transmission, to map the distribution of a pharmaceutical cocrystal in a polyethylene matrix [28]. However, for applications where it is not possible to transmit a terahertz signal through the sample, reflection measurements must be used instead. One of the first examples of this was carried out by Shen et al. [29], who successfully demonstrated recovery of chemical information from the sample. It is important to note that the information carried by the reflected signal is no longer fully dependent on the absorption of the sample material, but on its refractive index [30].

7.8  Focal Plane Array Microbolometer Cameras Techniques discussed in this chapter so far have focused on point measurements where a terahertz image or, rather, maps are achieved by raster scanning, either by scanning the terahertz beam or by moving the object. This greatly limits

6502_Book.indb 180

11/11/14 10:21 AM



Terahertz Imaging181

the speed of image acquisition; as in the visible or near-infrared, it is highly desirable to have a 2-D sensor array. The first example of such 2-D imaging in the terahertz region was demonstrated by Lee and Hu [31] using a commercial infrared focal plane array (FPA) that had been optimized for night vision use in the region of 21 to 40 THz (14 to 7.5 microns). The camera was modified by using a germanium lens and HDPE filters to absorb visible light. The terahertz source on this occasion was provided by a room-temperature methanol (CH3OH) vapor laser pumped by a carbon dioxide (CO2) laser, generating about 10 mW at 2.52 THz (118.8 µm) (see Chapter 10). Lee and Hu trialed the system by imaging a small razor blade hidden in a high-density polyethelyne (HDPE) bag, as shown in Figure 7.14. Subsequently, the authors used a

Figure 7.14  The first demonstration of a real-time terahertz camera. (a) Optical image of the razor blade; (b) optical image of the razor blade hidden in an HDPE envelope; (c)– (e) the equivalent THz images. (Reproduced with permission from [31].)

6502_Book.indb 181

11/11/14 10:21 AM

182

Terahertz Metrology

4.3 THz quantum cascade laser (QCL) as a source in a similar system [31], but again did not employ an optimized imaging device. Other groups have started to develop optimized real-time terahertz cameras. The most advanced of these is an uncooled 320 × 240 focal plane array coupled with a commercial QCL system operating at 3.75 THz, developed by the group headed by Oda at NEC, Japan. A 160 × 120 video rate FPA operating at 450 GHz was demonstrated by the group at INO, Canada, who used a tunable CO2 laser to illuminate various hidden samples [32] at different frequencies.

7.9 Conclusions Terahertz applications continue to grow a great rate, as mentioned earlier. Two areas for future expansion of terahertz applications should be highlighted, namely, near-field imaging and focal plane arrays. The first of these extends terahertz imaging beyond its diffraction limit and offers subwavelength resolution. The second will speed up image acquisition rates, although it may be at the expense of reduced spectral resolution. Whatever imaging technique is employed, signal processing is and will remain an important part of the data acquisition process. This chapter illustrates the effects of applying de-noising filters to the waveform and to the acquired image, while also noting that in using time-domain spectrometer attention should be given to the bandwidth of the terahertz pulse. In particular, removing the high-frequency component of the pulse has very marked effects on the ability to recover fine features in the image. Finally, an understanding of the terahertz refractive index of the materials studied is required, as this provides a rich source of information about the sample.

References [1]

Hu, B. B., and M. C. Nuss, “Imaging with Terahertz Waves,” Opt. Lett., Vol. 20, No. 16, 1995, pp. 1716–1718.

[2]

Wallace, V. P., et al., “Terahertz Pulsed Imaging of Basal Cell Carcinoma Ex Vivo and In Vivo,” Br. J. Dermatol., Vol. 151, No. 2, 2004, pp. 424–432.

[3]

Woodward, R. M., et al., “Terahertz Pulse Imaging in Reflection Geometry of Human Skin Cancer and Skin Tissue,” Phys. Med. Biol., Vol. 47, No. 21, 2002, pp. 3853–3863.

[4]

Woodward, R. M., et al., “Terahertz Pulse Imaging of Ex Vivo Basal Cell Carcinoma,” J. Invest. Dermatol., Vol. 120, No. 1, 2003, pp. 72–78.

6502_Book.indb 182

11/11/14 10:21 AM



Terahertz Imaging183

[5]

Churchley, D., et al., “Terahertz Pulsed Imaging Study to Assess Remineralization of Artificial Caries Lesions,” J. Biomed. Opt., Vol. 16, No. 2, 2001, pp. 026001–026001-8.

[6]

Ho, L., et al., “Analysis of Sustained-Release Tablet Film Coats Using Terahertz Pulsed Imaging,” J. Control Release, Vol. 119, No. 3, 2007, pp. 253–261.

[7]

Ho, L., et al., “Terahertz Pulsed Imaging as an Analytical Tool for Sustained-Release Tablet Film Coating,” Eur. J. Pharm. Biopharm., Vol. 71, No. 1, 2009, pp. 117–123.

[8]

Ho, L., et al., “Monitoring the Film Coating Unit Operation and Predicting Drug Dissolution Using Terahertz Pulsed Imaging,” J. Pharm. Sci., Vol. 98, No. 12, 2009, pp. 4866–4876.

[9]

Fukunaga, K., et al., “Terahertz Analysis of an East Asian Historical Mural Painting,” Journal of the European Optical Society, Vol. 5, 2010.

[10] Seco-Martorell, C., et al., “Goya’s Artwork Imaging with Terahertz Waves,” Optics Express, Vol. 21, No. 1, 2013, pp. 17800–17805. [11] Skryl, A., et al., “Terahertz Time-Domain Imaging of Hidden Defects in Wooden Artworks: Application to a Russian Icon Painting,” Appl. Opt., Vol. 53, 2014, pp. 1033–1038. [12] Wietzke, S., et al., “Terahertz Imaging: A New Non-Destructive Technique for the Quality Control of Plastic Weld Joints,” Journal of the European Optical Society, Vol. 2, 2007, p. 07013. [13] Duling, I., and D. Zimbars, “Terahertz Imaging: Revealing Hidden Defects,” Nature Photonics, Vol. 3, 2009, pp. 630–632. [14] Katletz, S., et al., “Efficient Terahertz En-Face Imaging,” Opt. Exp., Vol. 19, 2011, pp. 23042–23053. [15]

Karpowicz, N., et al., “Comparison Between Pulsed Terahertz Time-Domain Imaging and Continuous Wave Terahertz Imaging,” Semicond. Sci. Technol., Vol. 20, 2005, pp. S293–S299.

[16] Airy, G. B., “On the Diffraction of an Object-Glass with Circular Aperture,” Transactions of the Cambridge Philosophical Society, Vol. 5, No. 283, 1835. [17] Shen, Y. C., and P. F. Taday “Development and Application of Terahertz Pulsed Imaging for Nondestructive Inspection of Pharmaceutical Tablet,” IEEE J. Selected Topics in Quantum Electronics, Vol. 14, No. 2, 2008, pp. 407–415. [18] Parrot, E., et al., “Terahertz Pulsed Imaging In Vivo: Measurements and Processing Methods,” J. Biomed. Optics, Vol. 16, No. 10, 2011, p. 106010. [19] Chen, Y., S. Huang, and E. Pickwell-MacPherson, “Frequency-Wavelet Domain Deconvolution for Terahertz Reflection Imaging and Spectroscopy,” Optics Express, Vol. 18, No. 2, 2010, pp. 1177–1190. [20] Bergeron, A., et al., “Components, Concepts and Technologies for Useful Video Rate TH Imaging,” Proc. of SPIE, Millimetre Wave and Terahertz Sensors and Technology V, 2012, p. 85440C-1.

6502_Book.indb 183

11/11/14 10:21 AM

184

Terahertz Metrology

[21] Burford, N. M., et al., “Terahertz Imaging for Nondestructive Evaluation of Packaged Power Electronic Devices,” International Journal of Emerging Technology and Advanced Engineering, Vol. 4, No. 1, 2014, pp. 395–401. [22] Van der Valk, N. C. J., and P. C. M. Planken. “Electro-Optic Detection of Subwavelength Terahertz Spot Sizes in the Near Field of a Metal Tip,” Applied Physics Letters, Vol. 81, No. 9, 2002, pp. 1558–1560. [23] Hillenbrand, R., T. Taubner, and F. Keilmann, “Phonon-Enhanced Light–Matter Interaction at the Nanometre Scale,” Nature, Vol. 418, No. 6894, 2002, pp. 159–162. [24] Hillenbrand, R., and F. Keilmann, “Complex Optical Constants on a Subwavelength Scale,” Physical Review Letters, Vol. 85, No. 14, 2000, pp. 3029–3032. [25] Moon, K., et al., “Terahertz Near-Field Microscope: Analysis and Measurements of Scattering Signals,” IEEE Trans. on Terahertz Science and Technology, Vol. 1, No. 1, 2011, pp. 164–168. [26] Moon, K., et al., “Quantitative Coherent Scattering Spectra in Apertureless Terahertz Pulse Near-Field Microscopes,” Applied Physics Letters, Vol. 101, No. 1, 2012, p. 011109. [27] Kawase, K., et al., “Non-Destructive Terahertz Imaging of Illicit Drugs Using Spectral Fingerprints,” Optics Express, Vol. 11, No. 20, 2003, pp. 2549–2554. [28] Charron, D. M., et al., “Chemical Mapping of Pharmaceutical Cocrystals Using Terahertz Spectroscopic Imaging,” Analytical Chemistry, Vol. 85, No. 4, 2013, pp. 1980–1984. [29] Shen, Y. C., et al., “Detection and Identification of Explosives Using Terahertz Pulsed Spectroscopic Imaging,” Applied Physics Letters, Vol. 86, No. 24, 2005, p. 241116. [30] Shen, Y. C., et al., “Chemical Mapping Using Reflection Terahertz Pulsed Imaging,” Semicond. Sci. Technol., Vol. 20, 2005, pp. S254–S257. [31] Lee, A. W. M., and Q. Hu, “Real-Time Continuous Wave Terahertz Imaging by Use of a Microbolometer FPA,” Optics Letters, Vol. 30, No. 19, 2005, p. 2563. [32] Lee, A. W. M., et al., “Real-Time Imaging Using a 4.3 THz Quantum Cascade Laser and a 320 × 240 Microbolomter Focal-Plane Array,” IEEE Photonics Technology Letters, Vol. 18, No. 13, 2006, pp. 1415–1417.

6502_Book.indb 184

11/11/14 10:21 AM

8 Metrology for Vector Network Analyzers Roland G. Clarke and Nick M. Ridler The vector network analyzer (VNA) has been a feature of radio-frequency (RF) and microwave measurements since the 1960s. Early instruments were often physically large, requiring plenty of manual adjustment to obtain good measurements, and they were generally limited to frequencies well below 20 GHz. However, rapid advances in digital electronics and signal processing, combined with significant progress in microwave signal sources and components, has seen the development of today’s compact, highly automated vector network analyzers. The vector network analyzer is now widely used to measure passive and active microwave components and to characterize high-frequency materials. During the past 20 years, a range of extension modules have been developed by various manufacturers that effectively increase the measurement capability of vector network analyzers to much higher frequencies [1–3]. Stateof-the-art network analyzers equipped with suitable high-frequency extension modules are now capable of measurements at 1 THz. It is these systems that are likely to be found in a terahertz measurement laboratory. In this chapter, we will introduce the typical, commercially available, vector network analyzer equipment adapted for millimeter-wave, submillimeterwave, and terahertz measurements (i.e., used at frequencies ranging from 100 GHz to 1 THz, or thereabouts). Other vector network analyzer systems are in 185

6502_Book.indb 185

11/12/14 8:41 AM

186

Terahertz Metrology

use in some laboratories that adopt a different arrangement or measurement technique (see, for example, [4, 5]). These other vector network analyzer systems will not be discussed further in this chapter. We shall begin by explaining what the vector network analyzer actually measures and then move on to discuss some of the metrology issues that need to be considered by the laboratory practitioner.

8.1  Vector Network Analyzers 8.1.1  The Role of a Network Analyzer As the name suggests, the network analyzer is an instrument that provides information about the scattering of signals1 incident upon a network. In this context, a network is any circuit, device, or, more generally, any physical artifact, to which electromagnetic signals may be applied through one or more ports. The notion of a port is significant because the underlying basis of network analysis assumes that, regardless of what the network consists of, it is connected to the outside world by means of a well-defined system for guiding the incident and emerging signals. In the case of millimeter-wave, submillimeter-wave, and terahertz frequencies, this guiding structure is most likely to be a metallic waveguide (see the following section for a more detailed discussion of metallic waveguides). The network that is to be measured is usually referred to as the device-under-test (DUT). Because the ports are physically realized by a guiding structure, both an impedance (characteristic of the guiding system) and a precise location (i.e., plane) are associated with each network port. When an electromagnetic wave propagates within a guiding structure, the ratio of electric (voltage) and magnetic (current) portions of the propagating disturbance is a property of the structure itself (geometry, materials, and so forth). This ratio is considered to be a characteristic impedance (denoted as Z0) associated with the guiding structure at a given frequency of operation [6]. If, at some point, the propagating wave encounters a different impedance, some (possibly all) of the wave energy is reflected. We may think of this process as a scattering of the incident signal, and it may be shown that the amount of reflection (the reflection coefficient) is directly related to the relationship between the two impedances. For the purpose of measurements, the impedance associated with each port is often termed the reference impedance, and is often (although not necessarily) the same for each port of a given network. 1

We use the term “signal” here to mean measurement signal, which is ordinarily an unmodulated, pure sinusoid, rather than an information-carrying, aperiodic signal.

6502_Book.indb 186

11/12/14 8:41 AM



Metrology for Vector Network Analyzers187

For some types of guiding structure (for example, a coaxial cable) it is relatively straightforward to define this impedance, especially when a transverse electromagnetic wave is propagating in a low-loss transmission line. In the case of metallic waveguides, the characteristic impedance is not so readily defined. However, in practice this does not matter too much, providing that it is recognized that any discontinuity in the electric/magnetic ratio encountered by the propagating wave will generally lead to reflections. It is not necessary to know (or assign) a numerical value to the characteristic impedance to make meaningful measurements of the reflection of signals incident on a given network.2 From a measurement perspective, the physical location of the port, the reference plane, is also important. This is because the incident and emerging signals are traveling waves, and their phase relationships will be a function of position. By measuring at different locations along a waveguide, different phase relationships will be observed. For convenience, the reference planes for practical measurements are usually taken to be at the junctions between two waveguide sections. By applying continuous, sinusoidal, electrical stimulus signals to the network (DUT), the vector network analyzer measures the incident and emerging signals at each port. The results are presented as ratios of emerging signals to incident signals. Thus, the role of the network analyzer is to measure the scattering of signals incident on each port of the DUT. The vector network analyzer is capable of measuring both the magnitude and the relative phase of these signals.3 It is important to realize that the magnitude and phase relationships of the incident and emerging signals are generally frequency-dependent, and therefore the vector network analyzer usually measures the scattering of signals as a function of frequency. 8.1.2  Scattering Parameters The notion of reflection coefficient can be extended if we consider the alternative ways in which the energy in a guided electromagnetic signal may be distributed, or scattered.

2 There are various ways to define impedance in a metallic waveguide. These generally differ by a constant term, and because it is ratios of impedance that determine reflections, the choice of definition is not important. 3 By contrast, a scalar analyzer only measures the magnitudes of the incident and emerging signals.

6502_Book.indb 187

11/12/14 8:41 AM

188

Terahertz Metrology

Figure 8.1  (a) A one-port network; and (b) a two-port network. Incident energy scatters by reflection, internal absorption or by transmission to another port.

First, let us consider a network with only a single port [see Figure 8.1(a)]. The energy associated with an incident signal might be completely absorbed by the network, either by thermal dissipation or by radiation.4 Certainly, this must be the case if there is no steady-state5 impedance discontinuity experienced by the incident wave as it enters the network port [7]. Alternatively, the energy may be partially or wholly reflected back to the original source. By energy conservation, the sum of the reflected and absorbed energies must be equal to the original incident energy. We may also think in terms of time-averaged rates of energy transfer, or average power. In which case, the above statement also holds true, and the sum of the average reflected and absorbed signal powers must be equal to the average of the incident signal power. This idea can be extended to two-ports [see Figure 8.1(b)], and indeed, n-ports. For a two-port network, there are three possible paths for the incident 4 Internal dissipation is indistinguishable from radiating the energy away, as far as this network’s input port is concerned, and so both may be simply treated as absorption. 5 The use of the term steady-state is necessary because in some networks, once the continuous, sinusoidal electromagnetic signals have settled to a final state, an impedance discontinuity that might have been encountered by an initial disturbance may effectively disappear. This is the basis of impedance transformers (for example, those that rely on quarter-wavelength lines) [7].

6502_Book.indb 188

11/12/14 8:41 AM



Metrology for Vector Network Analyzers189

signal power. Once again, the incident signal might be absorbed or reflected by the network, but it might equally well be transmitted through the network and emerge from the second port. Thus, a transmission coefficient may also be considered, which is related to the ratio of the signal emerging from the second port to the signal incident at the first port. Again, energy conservation requires that the sum of the average reflected, absorbed and transmitted powers be equal to the average incident signal power. By considering the incident and emerging signals from each port of a network, we can now define a set of scattering parameters (or scattering coefficients). In Figure 8.2, the incident signals on each port are denoted as a1 and a2, while the emerging signals are b1 and b2. The ratio of the signals b1 and a1 gives the scattering coefficient for a signal that is both incident on Port 1 and also emerging from Port 1. This, of course, simply represents the reflection of a signal arriving at Port 1, providing that there are no other incident signals on the network which could potentially contribute to the signal emerging from Port 1. Therefore, this scattering parameter is defined only when a2 is zero. If we denote this parameter as S11, then we can write



S11 =

b1 a1

(8.1) a2 =0

The first subscript for the scattering parameter designates the port from which a signal emerges, while the second subscript designates the port on which the signal is incident. By taking similar ratios of other combinations of the a and b signals, four scattering parameters may be defined for a two-port network. These parameters describe the reflections for each port and the transmission between each port. It is not necessary to have a separate coefficient for absorption, since

Figure 8.2  A general two-port network, with incident signals (an ) and emerging signals (bn ).

6502_Book.indb 189

11/12/14 8:41 AM

190

Terahertz Metrology

the information about reflection and transmission alone conveys a complete description of the scattering of incident signals. Such scattering parameters are often simply called S-parameters and as such, they are fundamental to vector network analyzer measurements. Table 8.1 shows these scattering parameters and their corresponding physical interpretation. The individual scattering parameters are defined under the conditions that a signal is incident upon one port at a time. However, providing the network is linear, the emerging signals may consist of a sum of contributions from the incident signals at each port. We can therefore write two linear equations relating the emerging signals, the scattering coefficients and the incident signals:

b1 = S11a1 + S12a2 (8.2)



b2 = S21a1 + S22a2 (8.3) These equations may also be written in matrix form, ⎡b1 ⎤ ⎡ S11 ⎢b ⎥ = ⎢ S ⎣ 2 ⎦ ⎣ 21



S12 ⎤ ⎡a1 ⎤ (8.4) ⎥ S22 ⎦ ⎢⎣a2 ⎥⎦

Table 8.1 Physical Meaning of Two-Port Scattering Parameters Scattering Parameter S11

Relationship to a and b Signals S11 =

Physical Meaning

b1 a1 a =0

Reflection coefficient at Port 1

b1 a2 a =0

Transmission coefficient from Port 2 to Port 1

b2 a1 a =0

Transmission coefficient from Port 1 to Port 2

b2 a2 a =0

Reflection coefficient at Port 2

2

S12

S12 =

1

S21

S 21 =

2

S22

S 22 =

1

6502_Book.indb 190

11/12/14 8:41 AM



Metrology for Vector Network Analyzers191

For a network with two or more ports, the scattering parameters are often referred to as the scattering matrix, or S-matrix. We must now look more closely at the way in which the a and b signals are defined. First, the a and b signals represent traveling electromagnetic waves with both magnitude and relative phase. Therefore, they must be treated as complex (phasor) quantities, and the scattering parameters will also be complex quantities. The magnitudes of the a and b signals are defined in terms of the squareroot of power, rather than voltage (E field), current (H field) or simply power. This may seem confusing at first, but there are good reasons for adopting this convention. One of the reasons is that the scattering parameters should provide an immediate insight into the behavior of a network. Consider a network that is reciprocal, such that the ratio of power transmitted from Port 1 to Port 2 is identical to the ratio of power transmitted from Port 2 to Port 1. Intuitively, we might expect the forward and reverse transmission scattering coefficients to be the same. However, if the reference impedance is different for the two ports, only those ratios related directly to power would be the same. The ratios of voltages or currents, for example, would be different in each direction. However, if the propagating voltages and currents are normalized to the square root of the reference impedance, we obtain the desired results. Dividing voltages by the square-root of impedance yields a parameter with the dimensions of the square-root of power; similarly, multiplying a current by the square root of impedance also yields a parameter in terms of the square-root of power. Therefore, the choice of square-root of power (rather than simply power) for the a and b signals arises from the fact that the scattering coefficients are the same, whether derived from voltages or currents, and also because the scattering matrix is symmetrical for reciprocal networks. A further advantage is that the reflection scattering parameters are equivalent to the voltage and current reflection coefficients, since the normalizing factors cancel out. Additionally, if the reference impedances are the same at each port (which is usually the case), the scattering parameters also provide the voltage and current transmission coefficients. These advantages may seem somewhat redundant for metallic waveguides where it is not convenient to think in terms of voltages and currents, but scattering parameters are widely used at lower frequencies where this is both possible and often done. In microwave engineering literature, the complex voltage reflection coefficient is usually denoted by Γ, and is equivalent to the Snn scattering parameter (where n is the port number).

6502_Book.indb 191

11/12/14 8:41 AM

192

Terahertz Metrology

We can now summarize some important (and useful) points concerning scattering parameters: • • •

•

•

•

The Snn parameter is equivalent to the complex voltage reflection coefficient at Port n. The Smn parameter is equivalent to the voltage transmission coefficient from Port n to Port m. As the an and bn signals have the dimensions of the square-root of power, then ⎪an⎪2 and ⎪bn⎪2 are the incident and emerging powers, so that ⎪Snn⎪2 and ⎪Smn⎪2 are the reflection and transmission coefficients for power. For a passive network (a network that does not add any power), the maximum magnitude for reflection coefficients is unity, that is, ⎪Snn⎪ ≤ 1. Similarly, the magnitude for transmission coefficients cannot exceed unity, so we also have ⎪Smn⎪ ≤ 1. From energy conservation, the total scattered power in a passive network cannot exceed the incident power, and therefore ⎪S11⎪2 + ⎪S21⎪2 ≤ 1. If the network does not absorb (dissipate, radiate, and so forth) any of the power, then the network is said to be lossless, and it follows that ⎪S11⎪2 + ⎪S21⎪2 = 1.

Display Formats

Network analyzers display measurement results in the form of scattering parameters, usually as a function of frequency. They may also display other quantities which are derived directly from the scattering parameters. It is worth looking carefully at the vector network analyzer’s display to gain familiarity with the various measurement formats available. Vector network analyzers are generally designed with engineers in mind. Consequently, it is likely that the default measurement format is the S11 parameter, displayed in decibels (dB). This is a logarithmic-magnitude6 format (LogMag), and since the decibel unit is defined for power ratios, this is simply, S11(dB) = 10log10⎪S11⎪2 (8.5)

or equivalently,

S11(dB) = 20log10⎪S11⎪ (8.6)

6

Decibels are based on common (base 10) logarithms.

6502_Book.indb 192

11/12/14 8:41 AM

Metrology for Vector Network Analyzers193



To interpret the display, the user needs to remember that a linear 1:1 reflection will be 0 dB, and a zero reflection will be −∞ dB. In practice, noise and other limitations mean that true zero reflections are not observed, and typically a nominal zero reflection will appear as a negative value in decibels somewhere between −50 dB and −100 dB. This corresponds to a reflected power ratio of between 1 × 10 –5 and 1 × 10 –10. The vector network analyzer user may select other formats for the complex scattering parameters, including linear magnitude and phase (in degrees), and linear real and imaginary components. These are normally displayed on rectangular graphs, as a function of the test frequency. The phase display may seem a little perplexing at first. It should be noted that the scale is usually in degrees, with positive angles above a zero reference and negative angles below it. The key to interpreting this display format is to remember that the points +180° and −180° are coincident. Any measurement trace depicting a continuous, linear change of phase, as a function of frequency, will appear to have sharp discontinuities where the phase progresses between ±180°. The linear complex scattering parameters may also be presented on a polar plot where the measurement trace indicates the scattering data as a function of frequency. The default scale is usually chosen such that the outer circumference corresponds to a unit circle. Thus, a maximum reflection coefficient for a passive DUT will produce a trace (as a function of frequency) that lies on this circle. An impedance grid may be superimposed on the polar reflection scale, in which case we have the famous Smith chart (popular with RF and microwave engineers) which is described in many texts [8]. The impedances are related to the characteristic impedance, Z0, of the ports (the measurement reference impedance). Where Z0 may be meaningfully assigned a value, the Smith chart impedances are de-normalized to give the actual impedance values associated with the network. In rectangular waveguide, where the characteristic impedance is not easy to define, it is common practice to assign the value of 1 to Z0. This simply means that any impedances displayed are really normalized impedances, relative to the waveguide impedance. Figure 8.3 shows a selection of typical vector network analyzer measurements of a one-port device (in this example, the device is a 94 GHz antenna connected by a short length of rectangular waveguide7). Each display format is derived from the same primary measurement data, but presented in different ways. Notice that the phase of the reflection coefficient reduces (almost 7

An antenna is a one-port network as far as the VNA measurement is concerned.

6502_Book.indb 193

11/12/14 8:41 AM

194

Terahertz Metrology

Figure 8.3  Selection of measurement formats available from a typical vector network analyzer: (a) linear magnitude; (b) log-magnitude (reflection coefficient in decibels); (c) phase of the reflection coefficient; and (d) Smith chart.

l­ inearly) as the measurement frequency increases, and the transition from −180° to +180° does not indicate any significant discontinuity in the measured phase. Notice too, that the reflection coefficient (S11) is a minimum at the frequency where the antenna is designed to radiate the energy away efficiently.

6502_Book.indb 194

11/12/14 8:41 AM



Metrology for Vector Network Analyzers195

Figure 8.3 Continued.

Most vector network analyzers allow the user to add markers to display the trace value at particular frequencies, and offer many other data processing/comparison functions. The measurement graphs in Figure 8.3 have been kept relatively simple for clarity. Other parameters may be derived from the scattering parameter data, and are usually available to view directly on the vector network analyzer. For

6502_Book.indb 195

11/12/14 8:41 AM

196

Terahertz Metrology

example, there is a one-to-one correspondence between the reflection coefficient (S11, S22) and the standing-wave ratio (SWR) [9], and most vector network analyzers will display reflection data in this format. One final data format available to the vector network analyzer user is worthy of mention here. The propagation delay [10] through a two-port network may also be determined from the rate of phase change as a function of frequency, for the steady-state transmission coefficients (S21, S12). The vector network analyzer computes the delay time by differentiating the phase measurement with respect to frequency. 8.1.3  Vector Network Analyzer Systems It will be instructive to first consider the essential hardware features of the common vector network analyzers used to measure scattering parameters from a few kilohertz to frequencies in the region of 50 GHz (an RF/microwave vector network analyzer). Figure 8.4 shows a block diagram of a typical two-port vector network analyzer. This is a somewhat simplified view of the major components and system diagram of a two-port8 vector network analyzer, and of course, there are many variations found on commercial products. However, the basic operation is essentially the same. A signal source operating in a continuous-wave fashion is used to supply the test signal. This test signal is routed to one of the two test ports,9 usually by means of an electronically controlled switch. This means that, ordinarily, only one test-port receives a stimulus signal at any given time (although the vector network analyzer will normally switch the test signal quite rapidly between the test ports). This is consistent with the definition of scattering parameters mentioned earlier, where each parameter is obtained under the condition that only one incident (a) signal is present at a given time. Along the path between the signal source and the test port, there are signal separation devices. These can take various forms, but most commonly they are implemented with components known as directional couplers. Effectively, these components separate the signals on the basis of direction of travel. Their 8

Some modern VNAs are multiport instruments; our discussion here is confined to the more common two-port VNA. 9 The term ‘port’ is used for both the network or device being tested and also the VNA’s measurement terminals. The VNA has “test ports,” to which the ports of the DUT network are connected. As with the DUT, the VNA test ports can be realised with any electromagnetic guiding structure, (for example, co-axial cables or metallic waveguides) although clearly these need to be of the same type on both to facilitate connections. At microwave frequencies, ‘adapters’ are sometimes used to convert from one guiding structure to another.

6502_Book.indb 196

11/12/14 8:41 AM



Metrology for Vector Network Analyzers197

Figure 8.4  Block diagram of a typical two-port vector network analyzer. The test signal proceeds through an electronically controlled switch to a signal separation device, and then to the vector network analyzer test port (measurement reference plane). Reflected signals and signals transmitted through to the other port will travel back toward the switch/signal source. Ideally, both the switch and the signal source circuits are designed such that no re-reflection of these signals will occur.

role is quite simple: to tap off a proportion of the outgoing (a) and returning (b) test signals to be measured within the vector network analyzer. This architecture results in signals within the vector network analyzer that are proportionally related to the signals incident on and emerging from the DUT. At a given moment, there should only be one incident-related signal, (either a1 or a2). However, depending on the DUT, there may be two emerging-related signals (b1 and b2) present. Ratios of these internal signals form the basis of the scattering parameter measurements, but it should be noted that there will be inevitable amplitude and phase errors introduced by

6502_Book.indb 197

11/12/14 8:41 AM

198

Terahertz Metrology

losses, internal mismatches, imperfect directivity of the directional couplers, and arbitrary path lengths (phase shifts) in the vector network analyzer system. However, this is not a major problem, as these errors are systematic. Providing they can be determined (i.e., quantified) by a suitable calibration process, the measurements may be mathematically corrected to account for them. Since the internal signals (related to the external a and b signals) are of the same high frequency as the measurement stimulus, they cannot be processed directly. The solution is to downconvert them to a much lower frequency. This is achieved by the use of frequency mixers that combine (i.e., multiply) the RF/microwave test signals with a second signal source [a local oscillator (LO)] to produce an intermediate frequency (IF).10 This process is known as heterodyning, and is essentially the same method used in good quality radio receivers to downconvert incoming signals to more convenient frequencies prior to demodulation. Multiplying two sinusoids of different frequencies yields two new signals whose frequencies will be at the sum and difference of the frequencies of the original signals.11 By careful choice of the LO frequency and then selecting the difference frequency from the mixer output, the IF signal will be at a much lower frequency than that of the original measurement signal. Crucially, these IF signals preserve both the amplitude proportionality and the phase relationships of the original high-frequency signals. The vector network analyzer digitizes the IF signals, and after appropriate manipulation (signal processing), displays the results. Older network analyzer systems often had the measurement signal sources, directional coupler assemblies (known as the test set), IF/digitizer and display/processor units, and separate instruments connected by a daunting maze of cables. Such setups are still commonly found in laboratories, although with more modern vector network analyzers, the entire system is housed within a single instrument box. Millimeter-Wave Systems

For millimeter-wave, submillimeter-wave, and terahertz vector network analyzers, the basic architecture is essentially the same. The main difference the user 10

Some VNAs have only one high-quality signal source, which is used to provide the measurement stimulus. The ‘local oscillator’ functionality is achieved through the use of ‘samplers’ which mix the test signals with harmonically-rich waveforms to obtain the necessary IF signals. Conversely, more modern VNAs, and especially systems which make use of ‘extender heads’ contain two highquality signal sources. These may be standalone instruments in their own right, but are more often integrated into the VNA ‘box’. 11 This easily follows from the trigonometric expansion of sinAsinB which yields terms of cos(A+B) and cos(A-B). Usually, only the difference term is useful.

6502_Book.indb 198

11/12/14 8:41 AM



Metrology for Vector Network Analyzers199

will notice is that some of the key components are housed in separate modules (known as extender modules or extender heads). These are usually intended to be used in conjunction with a standard microwave vector network analyzer, and thus a complete system typically consists of a microwave vector network analyzer, and one or two extender heads. Sometimes, the extender heads are designed and manufactured by a third-party company in close cooperation with the vector network analyzer manufacturer. Figure 8.5 shows the typical hardware configuration for a vector network analyzer with extender modules. There may be additional external units to supply dc power to the extender heads which are not shown here. Also, some vector network analyzer systems may require an external controller to manage the exchange of RF/LO and IF signals between the vector network analyzer and the extender heads.

Figure 8.5  Signal paths of a two-port vector network analyzer with extender heads. The vector network analyzer still contains the signal separation devices and frequency mixers depicted in Figure 8.4 but these are not in use when the vector network analyzer is used with the extender heads. Some vector network analyzers supply the RF, LO test signals via the front panel, while some systems may require additional controller modules which connect the vector network analyzer and the extender heads.

6502_Book.indb 199

11/12/14 8:41 AM

200

Terahertz Metrology

8.1.4  Extender Heads Figure 8.5 shows that the vector network analyzer system typically supplies the extender heads with a radio-frequency (RF) signal and a local oscillator (LO) signal. These signals are usually in the microwave portion of the frequency spectrum, typically a few tens of gigahertz. For millimeter-wave, submillimeter-wave, and terahertz systems, the actual stimulus test signal is generated within the extender head by a process of harmonic multiplying. The technique relies on driving a nonlinear semiconductor device (usually a Schottky diode) with a moderately high-power RF signal. The nonlinearity in the target device leads to the generation of many harmonics (integer multiples) of the RF signal. The power level at each harmonic frequency diminishes as the harmonic number increases, and at the desired output frequency, the power is usually very low compared to the RF input signal. Table 8.2 shows RF frequencies, harmonic numbers, and test-port output powers for a variety of common measurement frequency ranges. These are typical values; individual vector network analyzer systems may have different values, depending on the manufacturer. Also, many manufacturers offer more than one extender head version for each frequency range in order to accommodate different vector network analyzer RF frequency capabilities. The absolute value of the test port power is not important12 because scattering parameters are formed by ratios of incident and emerging signals. There is an implicit assumption here that the network being measured responds linearly to the test signal. If the network response is not linear, interpreting the measurement results requires additional care. The extender heads also contain the directional signal separation components (couplers). The process of down converting the coupled test signals to more convenient frequencies is similar to that employed in the microwave vector network analyzer. However, rather than mix the coupled signals with the LO signal directly, a harmonic of the LO signal is used (in a manner similar to the harmonic multiplying above). By driving a suitable nonlinear semiconductor device with the LO signal, the test signal will mix (multiply) with the harmonics of the LO frequency. One of these harmonics will lead to an IF signal in a convenient frequency range that may be processed in the same manner as that used in the conventional microwave vector network 12

Apart from signal-to-noise ratio considerations, the absolute value of the test signal plays no part in the measurement, providing the DUT is linear, so that the scattering response is not a function of the power level.

6502_Book.indb 200

11/12/14 8:41 AM

Metrology for Vector Network Analyzers201



Table 8.2 Typical RF Frequencies Used to Drive the Harmonic Multipliers and Corresponding Output Power* Desired Test Frequencies

Input RF Frequency (GHz)

110–170 GHz

27–42

140–220 GHz 220–325 GHz 325–500 GHz

Harmonic Number

Test-Port Output Power

4

1 mW

24–37

6

0.3 mW

18–27

12

0.1 mW

27–42

12

20 μ W

500–750 GHz

28–42

18

3 μW

750 GHz–1.1 THz

21–31

36

0.3 μ W

*Individual systems may follow the same operating principle, but use different harmonic numbers, and the actual output power may vary considerably.

analyzer. The LO mixer circuits are included as part of the extender head. In this manner, the millimeter-wave, submillimeter-wave or terahertz signals are all confined to the extender head and DUT. The RF, LO, and the downconverted signals are all connected to the vector network analyzer. The user does not normally need to be concerned with selecting the correct frequencies for the RF and LO signals; this is done automatically in the vector network analyzer, although the user may be required to enter the appropriate harmonic numbers in the vector network analyzer configuration.13 Transmission/Reflection Options

A complete system for two-port measurements will include two identical extender heads. However, the extender heads are by far the most expensive part of the system, and most manufacturers offer a reduced-cost alternative. This is accomplished by reducing the complexity of one of the extender heads by omitting the components required to measure reflection. This leaves the user with a full transmission/reflection head (a1 and b1 signals) and a transmissiononly head (just a b2 signal). The result is a system that can measure S11 and S21, but in order to measure the reverse parameters (S22 and S12 ) the DUT must 13

Many modern vector network analyzers will have this information either preloaded or easily loaded via an instrument state file supplied on suitable media by the extender head manufacturer.

6502_Book.indb 201

11/12/14 8:41 AM

202

Terahertz Metrology

Figure 8.6  A lower-cost option is implemented when the second extender head is only capable of receiving signals. Forward transmission and reflection can be measured, but the DUT must be reversed to measure reverse reflection and transmission.

be physically reversed.14 This is sometimes referred to as a one-path, two-port system (see Figure 8.6). It is also entirely possible to have a system with only one transmission/ reflection extender head. Despite the terminology, such a system can only be used for reflection measurements. The extender head would be capable of measuring transmission if it were used in a system with a second extender head. The labeling really stems from the fact that measuring reflection requires a1 and b1 signals, whereas to measure transmission requires only the b signal at the second port. The vector network analyzer user needs to consider carefully the practical arrangement of the extender heads and the DUT (see Figure 8.7). The block diagram of the connection scheme might seem straightforward, but the physical reality often presents some challenges. For millimeter-wave to terahertz frequencies, the extender heads are likely to connect to the DUT via rigid rectangular metallic waveguides. This implies that the orientation, spacing and alignment of the heads must be carefully thought out. In the next section, we shall discuss some of the properties of metallic waveguides and their interconnections. 8.1.5  Good Practice Tips The following are good practice tips for making VNA measurements: •

Tip 1: Plan the measurements from the perspective of physical connections before attempting to calibrate or even switch on the vector network analyzer. Check the orientation of the DUT waveguides, and

14

The reduced capability of the second extender head also has implications for the calibration process, and some methods of calibration are not available in these systems.

6502_Book.indb 202

11/12/14 8:41 AM



Metrology for Vector Network Analyzers203

Figure 8.7  (a) Vector network analyzer with extender heads. Flexible cables are used to connect the RF and LO harmonic multiplier/mixer drive signals, and return the IF output signals to the vector network analyzer. Also visible are dc power modules that are required to power the extender heads. (b) Close-up of DUT between the extender head measurement reference planes. Here, the DUT is actually an additional length of precision waveguide.

verify that it is possible to actually make the necessary connections. The extender heads may need to be placed on suitable mounts, or perhaps rotated, or supported on stands. It is a good idea to reduce the mechanical strain on the waveguides as much as possible by supporting the weight of the extender heads, while allowing for fine movement of the heads for alignment. Some fairly elaborate air-cushion mounts

6502_Book.indb 203

11/12/14 8:41 AM

204

Terahertz Metrology

•

•

•

and sliding rail systems have been used in some laboratories, although the authors have found that blocks of inexpensive packing foam can perform very well. The use of good quality laboratory jacks, possibly with precision adjustments (micrometers) should also be considered. Tip 2: Avoid, as much as possible, moving the extender heads once a calibration and measurement activity has been commenced. Even small amounts of flexing on the cables that connect the RF, LO, and downconverted test signals to the vector network analyzer can lead to phase shifts, affecting the integrity of the calibration and ultimately reduced quality of measurements. Some movement is often necessary, but the user should try to minimize it as much as possible. Tip 3: Use additional waveguide sections to extend the physical location of the test ports. This will not only make the physical connections more practical, but it also serves to protect the original waveguide test ports from expensive damage. Adding lengths of additional waveguide will not significantly affect the measurement accuracy (the effects of the extra waveguide length are fully accounted for in the calibration process). However, at higher frequencies, some appreciable loss will be present in the waveguide; the main impact of this loss will be to reduce the available dynamic range of the measurement. Tip 4: Measurements of waveguide devices with more than two ports are possible, even with a two-port vector network analyzer system. The scattering parameters for n ports are acquired by measuring pairs of ports in turn. Under these circumstances, it is preferable to terminate ports that are not being measured with reflectionless (i.e., matched) terminations. This ensures that all incident signals are zero, apart from the port being stimulated by the vector network analyzer. However, measuring n-port devices does present practical challenges. The position and orientation of the extender heads, and the amount of head movement required just to make the necessary connections, are often problematic and requires considerable prior thought.

8.2  Metallic Waveguides 8.2.1  Basic Properties Most of today’s vector network analyzers that operate at frequencies above 110 GHz have test ports that are realized using rectangular metallic waveguide. These test ports are where the devices to be tested (e.g., electronic components) are connected. On a two-port vector network analyzer, a one-port device (such

6502_Book.indb 204

11/12/14 8:41 AM



Metrology for Vector Network Analyzers205

Figure 8.8  A rectangular metallic waveguide of width, a, and height, b.

as a termination) needs only to be connected to one of the two test ports. For two-port devices (such as attenuators and amplifiers), these need to be connected to both of the vector network analyzer’s test ports. Rectangular metallic waveguide (or waveguide, for short) has a long history of usage that dates back to the early-to-middle part of the twentieth century. In those early days, waveguides were often used on high power systems (e.g., for radar applications) operating at microwave frequencies (i.e., at frequencies ranging from around 1 to 30 GHz). The waveguides were found to be very suitable for building these types of systems because the waveguides exhibited relatively low electromagnetic losses (i.e., low attenuation). The waveguides were also able to handle the high power levels that were often used for the radar applications. Finally, it was relatively easy to join the waveguides together without causing any significant loss of signal (i.e., the junctions where the waveguides were joined generated very low levels of electromagnetic reflection). A potential drawback with this type of waveguide is that, for most normal applications, it has an inherently narrow bandwidth (for mono-mode propagation15). This was not a problem for the early microwave systems that tended to use only a very narrow range of frequencies (or sometimes only a single frequency was used). The size of waveguide was therefore chosen to accommodate the required operating frequency (or frequencies). Figure 8.8 shows a diagram of a length of waveguide. The important mechanical dimensions of the waveguide are the width, a, and height, b. Most waveguides that are used these days have a width-to-height aspect ratio of 2:1 (i.e., the width of the waveguide aperture is twice the height of the aperture). 15

In general, these types of waveguide can operate over broad bandwidths. However, multiple electromagnetic modes can simultaneously be present over these broad bandwidths and this makes the performance of the waveguide difficult to predict. Conventional waveguide uses therefore usually restrict the bandwidth to ensure only a single mode of propagation is present, hence the name mono-mode propagation.

6502_Book.indb 205

11/12/14 8:41 AM

206

Terahertz Metrology

The bandwidth for mono-mode propagation for a given waveguide size is one octave. Below this frequency range, no electromagnetic energy can propagate in the waveguide. There is a well-defined frequency below which the waveguide stops propagating energy, and this is called the cutoff frequency. The cutoff frequency, fc, is determined by the width (i.e., the broad wall dimension) of the waveguide, a, as follows:



fc =

c 1 × er 2a (8.7)

where c is the speed of electromagnetic waves in vacuum (defined as 299,792,458 m/s) and ε r is the relative permittivity of the material filling the waveguide. In most cases, the waveguide is filled with air (ε r ≈ 1), and so the cutoff frequency is given by:

fc =

299 792 458 (8.8) 2a

Therefore, the frequency range for mono-mode propagation in waveguide is, in principle, from fc to 2fc. However, in practice, the operational bandwidth is taken as less than an octave. This is to avoid problems that occur at frequencies close to fc and 2fc. At frequencies close to fc, the dispersion (see Section 8.2.2) and loss rapidly increases; at frequencies close to 2fc, there is an increased likelihood that higher-order modes will start to propagate and so mono-mode propagation inside the waveguide can no longer be assured. To avoid these problems, a reduced operational bandwidth is usually given for a specific waveguide size. Generally, the suggested minimum frequency, fmin, and maximum frequency, fmax, for a given waveguide size (as defined by the waveguide aperture width, a) is given by:

fmin = 1.25 × fc (8.9)



fmax = 1.9 × fc (8.10)

The basic waveguide properties described above (i.e., low attenuation, high power handling capability, low reflection of waveguide connections) have enabled waveguide to be used successfully at microwave frequencies for many years. In more recent years, there has been an increased interest in the commercial use of frequencies in the millimeter-wave (30 GHz to 300 GHz) and submillimeter-wave (300 GHz to 3 THz) regions. At these higher

6502_Book.indb 206

11/12/14 8:41 AM



Metrology for Vector Network Analyzers207

frequencies, waveguide is still seen as an attractive transmission line for guiding electromagnetic signals. This is despite the attenuation being higher at these frequencies and the power handling capability being lower. In addition, the interfaces used to join two pieces of waveguide also become significantly more reflective (i.e., a significant amount of the signal being transmitted can be reflected by a slight change in cross-sectional geometry caused by two joining waveguide interfaces). We will further discuss these terahertz waveguide sizes, and their interfaces, in the sections that follow, but first we will describe another interesting feature affecting all waveguide sizes: the property known as dispersion. 8.2.2 Dispersion When a sinusoidal electromagnetic signal travels in free space, the free space wavelength, λ , is related to the frequency, f, as follows: λ =



c (8.11) f

where, as before, c is the speed of electromagnetic waves in vacuum (defined as 299,792,458 m/s). However, for a sinusoidal electromagnetic signal traveling inside a rectangular metallic waveguide, the wavelength (often called the guide wavelength) is considerably longer that the wavelength in free space. In fact, the difference between the guide wavelength and the free space wavelength increases as the frequency decreases. When the frequency reaches the cutoff frequency (below which no electromagnetic signal will propagate), the guide wavelength becomes infinite. The guide wavelength, λ g, is given by: lg =



l ⎛ l⎞ 1−⎜ ⎟ ⎝ lc ⎠

2

(8.12)

where λ c is the waveguide cutoff wavelength, λ c = 2a. Equation (8.12) shows that when the free space wavelength reaches the cutoff wavelength (i.e., when the frequency reaches the cutoff frequency), λ /λ c = 1 and so the denominator on the right side goes to zero and so λ g becomes infinite.

6502_Book.indb 207

11/12/14 8:41 AM

208

Terahertz Metrology

Figure 8.9 shows a plot of the guide wavelength and the free space wavelength versus frequency for a standardized waveguide size that is used at frequencies from 500 to 750 GHz. The difference between the free space and guide wavelengths is referred to as the dispersion. The guide wavelength is a nonlinear function of frequency, and this causes the phase velocity at lower frequencies to be faster than it is at higher frequencies. This effect is known as dispersion. Figure 8.9 clearly shows that the dispersion effect increases as the frequency decreases. The size of the waveguide used to calculate the wavelengths shown in Figure 8.9 is a standardized waveguide size (see Section 8.2.3). This waveguide size has an aperture width of, a = 0.38 mm. The cutoff frequency for this waveguide [using (8.8)] is therefore fc ≈ 394 GHz and the recommended minimum and maximum frequencies [using (8.9) and (8.10)] are fmin ≈ 493 GHz and fmax ≈ 749 GHz. In practice, the minimum and maximum suggested operating frequencies for this particular waveguide size are usually rounded to be 500 GHz and 750 GHz, respectively. Figure 8.9 shows that the dispersion increases rapidly below the recommended minimum frequency of 500 GHz and this is one of the reasons why it is not advisable to use this waveguide size below approximately 500 GHz. Although the dispersion is much lower at frequencies close to 800 GHz, the recommended maximum frequency is set to 750 GHz to ensure that the waveguide will only accommodate mono-mode propagation. (Note that the calculated frequency for the onset of higher order modes for this waveguide size is 2fc ≈ 788 GHz.)

Figure 8.9  Free-space wavelength and guide wavelength for a waveguide with width, a = 0.38 mm.

6502_Book.indb 208

11/12/14 8:41 AM



Metrology for Vector Network Analyzers209

The dispersion characteristics for other waveguide sizes show similar behavior to those shown in Figure 8.9. So, in general, for any waveguide size, the suggested useful frequency range can be established as typically 1.25fc to 1.9fc, where fc is determined from the width of the waveguide aperture [using (8.8)]. 8.2.3  Standardized Sizes and Frequency Ranges The above general properties of waveguides imply that any size of waveguide can be used, the choice of waveguide size being governed by which frequencies are of interest. However, in practice, there are many advantages with using a relatively small set of waveguides sizes. One such advantage is that these sizes could be used by others and so it makes it easy to share and exchange components due to the compatibility of their sizes. This has been recognized for many years by users of waveguide at microwave and millimeter-wave frequencies. In fact, agreed standardized sizes of waveguides have been in use at frequencies up to 330 GHz for many years [11, 12]. In recent years, a similar standardization activity has taken place addressing waveguide sizes suitable for the submillimeter-wave/terahertz frequency ranges (i.e., waveguides sizes used for frequencies above 330 GHz). This standardization activity has been led by the IEEE and has resulted in a new standard: IEEE Std. 1785.1-2012 [13]. This standard provides sizes that are based on the same approach that was used to define the sizes used at the lower microwave and millimeter-wave frequencies [11, 12]. This has resulted in two contiguous interleaved series (containing no gaps or overlaps in the frequencies covered by each series). In addition, the new IEEE standard defines the waveguide widths using metric units (i.e., micrometers, μ m). Table 8.3 shows the new standardized waveguide sizes and frequency ranges, where a and b are the waveguide aperture width and height, respectively; fmin and fmax are the suggested minimum and maximum frequencies, respectively; and fc is the cutoff frequency. The names of the waveguide sizes (WM-2540, WM-2032, and so forth) follow a new naming convention that has been developed for these waveguide bands. Because the sizes are defined in terms of metric units, the letters WM are used to indicate that the size refers to waveguide using metric dimensions. These letters are then followed by a number that indicates the size (in micrometers, μ m) of the waveguide width (i.e., broad wall) dimension. The two contiguous interleaved series in Table 8.3 are as follows: (1) WM-710, WM-470, WM-310, and so forth; and (2) WM-570, WM-380, WM250, and so forth. Both series can be extended to higher frequencies (if needed), as follows:

6502_Book.indb 209

11/12/14 8:41 AM

210

Terahertz Metrology

1. Use the waveguide size WM-710 and smaller sizes in Table 8.3. 2. Divide the mechanical dimensions by 10. 3. Multiple the frequencies by 10. 4. Rename the waveguide accordingly. For example, the next two sizes in the series (derived from WM-710 and 570) are shown in Table 8.4. The waveguide sizes WM-2540 to WM-864 in Table 8.3 overlap with sizes given in earlier standards (e.g., the WR sizes given in MIL standard MIL-DTL-85/3C) [11]. In this overlap region, the WM sizes have been set to be the same as the earlier WR sizes (that were specified using Imperial units, that is, inch and mil). This equivalence is shown in Table 8.5.

Table 8.3 Frequency Bands and Waveguide Dimensions Used in IEEE Std. 1785.1-2012 Waveguide Name

a (μm)

b (μm)

WM-2540

2,540

1,270

59.014

75

110

WM-2032

2,032

1,016

73.767

90

140

fc (GHz)

90.790

fmin (GHz)

fmax (GHz)

WM-1651

1,651

825.5

110

170

WM-1295

1,295

647.5

115.75

140

220

WM-1092

1,092

546

137.27

170

260

WM-864

864

432

173.49

220

330

WM-710

710

355

211.12

260

400

WM-570

570

285

262.97

330

500

WM-470

470

235

318.93

400

600

WM-380

380

190

394.46

500

750

WM-310

310

155

483.53

600

900

WM-250

250

125

599.58

750

1,100

WM-200

200

100

749.48

900

1,400

WM-164

164

82

WM-130

130

65

913.99

WM-106

106

53

1,414.1

1,700

2,600

WM-86

86

43

1,743.0

2,200

3,300

1,153.0

1,100

1,700

1,400

2,200

Source: [13].

6502_Book.indb 210

11/12/14 8:41 AM

Metrology for Vector Network Analyzers211



Table 8.4 Extended Frequency Bands and Waveguide Dimensions Using IEEE Std. 1785.1-2012 Name

a (μm)

b (μm)

fc (GHz)

fmin (GHz)

fmax (GHz)

WM-71

71

35.5

2111.2

2,600

4,000

WM-57

57

28.5

2629.7

3,300

5,000

Source: [13].

Table 8.5 Comparison Between New IEEE Standard [13] and the Earlier MIL Standard [11] Waveguide Names MIL Name

New IEEE Name

WR-10

WM-2540

fmin (GHz)

fmax (GHz)

75

110

WR-08

WM-2032

90

140

WR-06

WM-1651

110

170

WR-05

WM-1295

140

220

WR-04

WM-1092

170

260

WR-03

WM-864

220

330

Finally, Table 8.6 shows a comparison between the new IEEE waveguide names and names resulting from a previous attempt to extend the MIL standard naming convention to these submillimeter-wave frequencies. These extended MIL names (WR-2.8, WR-2.2, and so forth) [14] are still often found describing waveguides that are used at these frequencies. Table 8.6 Comparison Between New IEEE Standard [13] and Extended MIL [14] Waveguide Names Extended MIL Name

New IEEE Name

WR-2.8

WM-710

260

400

WR-2.2

WM-570

330

500

WR-1.9

WM-470

400

600

WR-1.5

WM-380

500

750

WR-1.2

WM-310

600

900

WR-1.0

WM-250

750

1,100

6502_Book.indb 211

fmin (GHz)

fmax (GHz)

11/12/14 8:41 AM

212

Terahertz Metrology

8.2.4 Flanges/Interfaces Having discussed the sizes of waveguide that are used for propagating the electromagnetic signals, an equally important aspect is the method used to join together (or interconnect) sections of waveguide. As with the waveguide sizes discussed in the previous section, there is a long history of usage of interconnect mechanisms for waveguides used at microwave and millimeter-wave frequencies. The waveguide interconnect mechanism that enables sections of waveguide to be joined together is often called the flange or interface. A variety of different shapes and sizes of flange are used at microwave frequencies. However, for millimeter-wave frequencies, there is one flange design that is used much more than any other. This is the UG-387 design, as shown in Figure 8.10. This design has been in existence for many years and has been standardized in the MIL-DTL-3922/67E standard [15]. At the centre of the waveguide flange is the waveguide aperture. The waveguide aperture shown in Figure 8.3 is relatively large (3.8 mm × 1.9 mm) and so is used at low millimeter-wave frequencies (50 GHz to 75 GHz). The flange also features four relatively large, threaded, holes (shown at the North, South, East, and West positions in Figure 8.10). Screws are inserted into these holes to enable two flanges to be bolted together and tightened. The method for aligning two flanges that are being connected together is to use alignment holes and alignment pins (as identified in Figure 8.10). These alignment pins are usually permanently fitted to the waveguide flange. The two alignment pins on one of the waveguide flanges fit into the two alignment holes on the other waveguide flange, and vice versa. Thus, a total of four alignment pins (two per flange) fit into a total of four alignment holes (two per flange). The positions and tolerances of these holes and pins translate into providing the alignment for the apertures of the two waveguides being joined together. However, the alignment achieved using conventional UG-387 flanges is often not sufficient for many present day applications. This is particularly

Figure 8.10  A conventional UG-387 flange, used extensively at millimeter-wave frequencies.

6502_Book.indb 212

11/12/14 8:41 AM



Metrology for Vector Network Analyzers213

the case for applications at high millimeter-wave frequencies (i.e., above 110 GHz). This is because, as frequency increases, the required size of the waveguide aperture decreases (as discussed in Section 8.2.3). Therefore, any tolerance and positional error in the alignment pins and holes on the waveguide flanges will have a proportionally larger impact on the achieved alignment of the waveguide apertures, when these apertures are of a smaller size (i.e., for higher-frequency applications). The above problem has led to the introduction of a number of precision UG-387 flange types. The most popular of these precision UG-387 flanges is shown in Figure 8.11. The main design change with this flange, compared with the conventional UG-387 flange, is the introduction of two additional alignment holes, immediately above and below the waveguide aperture. These inner holes are machined to a tighter tolerance than the outer pins and holes found on the conventional UG-387 flange. Additionally, separate detachable dowel pins are inserted into these inner holes during connection, resulting in a significant improvement in the alignment of the waveguide apertures. However, as the required frequency extends still further into the submillimeter-wave and terahertz frequency ranges, the precision UG-387 is no longer adequate (in terms of providing a waveguide connection with relatively low reflection loss) for many applications. This has led to new flange designs being developed specifically for applications at these frequencies. As with the recent work on standardizing waveguide aperture sizes (in IEEE Std. 1785.12012 [13]) above 110 GHz, work is now underway to standardize these new flange designs so that the flanges can be manufactured to agreed dimensions and tolerances. The three new designs that are being standardized are described below. All three designs can provide acceptable connection performance for many applications at submillimeter-wave and terahertz frequencies. Only brief descriptions are given here, as these designs are still in the process of being finalized. Once the designs have been finalized, they will be published in IEEE Std. 1785.2 [16].

Figure 8.11  Precision UG-387 flange, showing the two additional inner dowel holes situated immediately above and below the waveguide aperture.

6502_Book.indb 213

11/12/14 8:41 AM

214

Terahertz Metrology

Precision Dowel Design

To the naked eye, a Precision Dowel flange looks the same as the precision UG-387 design (Figure 8.11). There are two design features that distinguish the Precision Dowel design from the precision UG-387 design: 1. The diameter, position, and tolerance of the two inner dowel holes are more tightly defined. 2. The two dowel pins that fit into the inner dowel holes are of different diameters: • The larger diameter dowel pin provides planar alignment of the waveguide apertures. • The smaller diameter dowel pin provides angular alignment of the waveguide apertures. The two dowel pins are marked so that they can be clearly identified. Ring-Centered Design

As with the Precision Dowel flange, the Ring-Centered flange also looks (to the naked eye) the same as the precision UG-387 design (as shown in Figure 8.11). The main design feature with the ring-centered design is the use of a detachable precision centering ring to align the waveguide flanges (rather than the inner dowel pins that are used with the Precision Dowel design). Figure 8.12 shows photos of the ring-centered design, before and after inserting the centering ring. Plug and Jack Design

This design comes as two parts: a Plug flange and a Jack flange. (Other names that could be used to describe the two parts are “plug and socket” or “male

Figure 8.12  Ring-centered flange design, showing (a) flange before attaching centering ring, and (b) flange with centering ring attached. (Photo courtesy of NRAO).

6502_Book.indb 214

11/12/14 8:41 AM



Metrology for Vector Network Analyzers215

and female.”) As with the Precision Dowel flange and the Ring-Centered flange, the Plug flange also looks (to the naked eye) the same as the precision UG-387 design (shown in Figure 8.11). The Plug and Jack flanges are shown in Figure 8.13. A connection is usually made between two waveguides when one waveguide has a Plug flange and the other has a Jack flange. A Plug version can be connected to another Plug flange as long as inner dowels (or pins) are used during the connection. However, a Jack version must not be connected to another Jack version unless an adapter is inserted between the two Jack interfaces, as this will result in a gap at the interface. The alignment of two waveguides, connected using the Plug and Jack flanges, is achieved by the precision fit between the concentric mating mechanism of the Plug and Jack sections of the two flanges. All three flange types (Precision Dowel, Ring-Centered, and Plug and Jack) are mechanically compatible with each other (and compatible with the earlier UG-387 and precision UG-387 designs). However, to achieve the best electrical performance for a given flange connection, it is necessary to only join together similar flange types, that is, two Precision Dowel flanges, two RingCentered flanges, or a Plug and a Jack flange. Typical performance values (in terms of worst-case reflection coefficient), for these flange type pairings used in three different waveguide sizes, are shown in Table 8.7.

Figure 8.13  Plug and Jack flange design, showing the Plug version (on the left) and the Jack version (on the right). (Photo courtesy of OML, Inc.)

6502_Book.indb 215

11/12/14 8:41 AM

216

Terahertz Metrology Table 8.7 Worst-Case Reflection Coefficients for the Three New Flange Types in Selected Waveguide Sizes Worst-Case Reflection Coefficient (dB)

Waveguide Name

Minimum Frequency (GHz)

WM-570

330

Maximum Frequency (GHz) 500

Precision Dowel Flange

RingCentered Flange

Plug and Jack Flange

−32

−36

−38

WM-380

500

750

−26

−29

−31

WM-250

750

1,100

−19

−22

−24

8.2.5  Good Practice Tips Although rectangular metallic waveguide is generally a very mechanically robust transmission line type, there are some good practice tips that should be followed to ensure that waveguide components remain in good working order and that waveguide connections are reliable and of a good quality. •

•

6502_Book.indb 216

Tip 1: When connecting two waveguides fitted with flanges (such as the types described in Section 8.2.4), make sure that the alignment mechanisms are not subjected to any mechanical stress as the flange faces are brought together. For example, for flanges that use dowel pins for alignment, make sure the dowel pins move easily into the alignment holes and that they are not causing any significant friction. The flanges should first be brought together using the alignment pins before any screws are used to tighten the connection. Tip 2: Use a torque driver to help ensure that screws are tightened to a uniform torque. Although different waveguide manufacturers are likely to specify different values of torque for the waveguide connections, it is a good idea to make connections with repeatable torque (i.e., using a torque driver). Figure 8.14(a) shows a typical torque driver that is used for tightening waveguide flange screws to a specified torque. Figure 8.14(b) shows typical screws that are used for making waveguide connections. Note that the screw has a hexagonal hole into which the torque driver fits. The torque driver has a rounded end so that it can fit into the screw’s hexagonal hole from a variety of different angles. This can be very useful if the flange screws are not easily accessible, due to other obstacles that might be in the way.

11/12/14 8:42 AM

Metrology for Vector Network Analyzers217



Figure 8.14  (a) A typical torque driver used to tighten waveguide flange screws; (b) typical waveguide flange screws (with hexagonal screw holes). •

6502_Book.indb 217

Tip 3: When tightening waveguide flange screws, tighten each screw gradually and in stages. One method is to gradually tighten screws that are on opposite sides of the flange face, for example, tighten the screws at north and south, then tighten the screws at east and west, and then repeat the process until all screws have reached the required tightness. This ensures that even pressure is applied to the two waveguide

11/12/14 8:42 AM

218

Terahertz Metrology

•

•

•

flange mating faces as they are brought together by the tightening of the screws. Never just tighten one screw fully before some tightening of the other screws has taken place; this can cause the flange faces to cock and so the waveguides are not joined together very evenly. Tip 4: Support waveguide components, using your hand, as they are connected together, both during the alignment and screw-tightening stages. This prevents the dowel pins and/or screws from artificially binding during the connection process. Tip 5: Use x-, y-, z-position adjusters (e.g., controlled by micrometers) for aligning bulky items, such as the vector network analyzer extender heads or heavy components. As with Tip 4, this prevents the flange dowel pins and screws from binding artificially during connection. Tip 6: Avoid excessive handling of certain items as this can lead to thermal changes. Since waveguide is made of metal, waveguide devices make very good conductors of heat. For example, holding a waveguide load can cause heat to be conducted inside the waveguide and onto the resistive element used to form the load. An increase in temperature can cause the reflection of the load to change significantly.

8.3  Calibration Standards and Methods 8.3.1  Calibration: General Principles In the foregoing introduction to vector network analyzers, we stated that any measurement of incident and emerging signals will be subject to inevitable systematic errors. For example, the devices used to separate the incident and emerging signals may not be perfect, such that the observed ratio of a and b signals will be in error (a directivity error). Some of the signals which are reflected from the DUT back to the originating test port, or by transmission through the DUT to a different test port, may be rereflected from the vector network analyzer (a source-match error). There will also be attenuation and phase shifts in the various signal paths within the vector network analyzer. To further complicate matters, these errors will be different for each frequency across the range of interest. Fortunately, all of these systematic errors can be grouped together into a relatively simple error model. A brief survey of literature on vector network analyzer calibration will probably give the impression that the topic concerns a large amount of tedious mathematics [17]. While this may be true to an extent, the underlying principles are quite straightforward. A simple illustration may help:

6502_Book.indb 218

11/12/14 8:42 AM



Metrology for Vector Network Analyzers219

Consider a handheld resistance meter of the type found in many laboratories. This sort of instrument is typically supplied with a pair of test leads, through which a small current is passed. The resistance of whatever device is placed between the test leads is determined by measuring the current that flows when a small test voltage is applied, using the well-known relationship of Ohm’s law. However, if the test leads themselves have some resistance, the meter will measure the sum of the test lead resistances plus the DUT resistance. For example, if the leads have a resistance of 1 ohm each, and the DUT is a 10 ohm resistance, the meter will display a reading of 12 ohms (an error of 20%). The solution is quite straightforward. If we first measure a resistance (a calibration standard) for which we already know (with confidence) what the correct measurement result should be, we can easily determine the combined test lead resistance. An ideal choice of calibration standard in this example would be a short-circuit because it is easy to implement, simply by connecting the test leads together. We can also have a high level of confidence in what the correct result should be (i.e., 0 ohms). Any nonzero reading obtained from the calibration step may be subtracted from all subsequent measurements to correct the results. In the case of the vector network analyzer, there are many systematic errors, but these may also be combined into a simplified error model. For a one-port vector network analyzer measurement, this model takes the form of an additional two-port network (sometimes called an error-box network) that intervenes between the desired measurement reference plane, and the actual measurements made within the vector network analyzer. Such a model represents all the possible systematic errors, including arbitrary amplitude and phase errors, imperfect coupler directivity, and internal mismatches. Figure 8.15 illustrates this concept.

Figure 8.15  For a one-port measurement, the systematic errors may be combined into an additional error box, a two-port network which comes between the DUT and the actual measurement. This additional network accounts for all the systematic errors: imperfect directional signal separation (directivity), imperfect proportionality (tracking), and unwanted rereflections from the vector network analyzer (source match).

6502_Book.indb 219

11/12/14 8:42 AM

220

Terahertz Metrology

This intervening two-port network can be fully described by a set of scattering parameters in the same way that any real two-port network can be. If these error scattering parameters can be found, then it is possible to express the unknown one-port DUT scattering parameter (i.e., the reflection coefficient, S11) in terms of the error parameters and the internal, uncorrected measurement value. ¢ , S21 ¢ , S12 ¢ and S22 ¢ are the scattering parameters for the intervenIf S11 ing error network, S11 is the true reflection coefficient of the DUT, and w is the uncorrected internal measurement, then it is easy to show from (8.2) and (8.3) that



¢ + w = S11

¢ S12 ¢ S11 S21 (8.13) ¢ 1 − S22

from which S11 =



(

¢ w − S11

)

¢ w − S11 ¢ + S21 ¢ S12 ¢ S22

(8.14)

¢ , S21 ¢ , S12 ¢ and S22 ¢ ) are first determined, Thus, if the error coefficients ( S11 the vector network analyzer can mathematically correct the subsequent measurements.16 Although there are four error parameters, it may be seen from (8.13) and (8.14) that it is unnecessary to determine S′21 and S′12 individually, and hence there are only three independent parameters to find. In the one-port error model, the term S′11 corresponds to a directivity error (which should ideally be zero), and the term S′22 corresponds to a mismatched source impedance (which again, should ideally be zero). The product of S′21 S′12 corresponds to the various attenuation and phase shifts in the signal paths (sometimes called tracking errors); ideally, S′21 S′12 should be equal to exactly one (i.e., no attenuation or phase shift). In practice, none of these ideal characteristics are found in real vector network analyzer hardware, and thus a calibration process is an essential step to achieving good measurements. In a manner similar to the resistance meter example above, measuring three calibration standards (with known reflection properties) will permit the error coefficients to be found by solving a system of three linear equations, based on (8.13). 16

It is worth noting that (8.13) and (8.14) appear in the literature in a variety of different forms; however, the underlying equations are exactly the same.

6502_Book.indb 220

11/12/14 8:42 AM



Metrology for Vector Network Analyzers221

The reader may wonder why the process of calibration (i.e., finding the error coefficients) cannot be undertaken once by the manufacturer, with the values stored internally in the vector network analyzer. The primary reason for not doing this is that the user is free to extend the signal path (by means of additional waveguides, and so forth) to any arbitrary location. Calibrating locally in this way ensures that all of the systematic errors, including the extended signal path, are included in the error coefficients. Also, the calibration process effectively determines the reference plane (see Section 8.1.1) for the measurement. As a result of adopting this philosophy, a locally performed calibration is essential to obtain accurate, and indeed meaningful, vector network analyzer measurements. Two-Port Errors

The error model for two-port measurements may be developed from the one-port case. However, additional error coefficients are included so that the two-port error model comprises more than just a pair of one-port models. For example, the tracking error for a transmission measurement (say, S21) will be different from the tracking error for a reflection measurement. There may also be a signal leakage path between the two test ports (crosstalk), which will affect measurements of high-isolation networks (DUTs where the true value of S21 or S12 is very small). Calibrating for two-port measurements will generally require both reflection (one-port) and transmission (two-port) calibration standards. From the vector network analyzer user’s perspective, the process of actually performing a calibration is not particularly onerous. With the exception of a couple of small steps discussed shortly, the vector network analyzer user is not required to undertake any of the tedious mathematical calculations, which are always accomplished automatically within the vector network analyzer firmware. All that is required of the user is to physically connect the appropriate calibration standards, in turn, to the vector network analyzer test ports. Other Sources of Error

Before we consider practical calibration techniques, it is worth noting some limitations of the calibration process. In addition to systematic errors, measurements are also subject to random errors (such as noise in the measurement circuits, limited repeatability of waveguide connections, and so forth). Calibration procedures cannot provide any correction for random errors. However, the vector network analyzer user can take steps to reduce the impact of random error sources. For example, measurement data will be subject

6502_Book.indb 221

11/12/14 8:42 AM

222

Terahertz Metrology

to noise (trace noise).17 This can be reduced by averaging successive measurements, and most vector network analyzers allow the user to select an averaging factor. Large values for the averaging factor will inevitably slow down the measurement time, although moderate averaging will have little detrimental effect on performance.18 While averaging can reduce the effect of noise, the amount of noise entering the measurement circuits can also be reduced by limiting the bandwidth of the downconverted IF signal. For optimal reduction of noise, both averaging and IF bandwidth reduction can be used. Other random errors (such as flange connection repeatability) cannot be easily reduced, but some knowledge of their statistical impact can help to quantify the typical uncertainty of any calibrated measurements (see Section 8.4). Dealing with systematic errors through calibration assumes that the systematic errors themselves are constant and do not change with time, or at least, not within the time it takes to perform the desired measurements. Unfortunately, real measurements are beset by drift in the systematic errors. There are two main causes of drift, both of which can be mitigated with careful practice. The first cause of drift is physical movement of any part of the signal path. The components within the vector network analyzer itself, and also within the millimeter-wave extender heads, are unlikely to move. The weak link in terms of physical movement is the flexible cables between the vector network analyzer and the extender heads. Movement of these cables leads to small changes in attenuation and phase shift, and as these parameters are part of the systematic errors, any changes which take place after calibration will degrade the measurement accuracy. It is therefore important to minimize the amount of movement of the extender heads once the calibration process has been undertaken. A second cause of drift is thermal fluctuation of either the measurement signal path or the electronic circuits within the vector network analyzer and the extender heads. Changes in temperature will produce small changes in physical 17 Noise is unavoidably added to the measurement data, mainly in the receiver circuitry. It appears at all measurement signal levels as trace noise. The noise level also determines the minimum signal that can be detected, which governs the dynamic range of the measurement. 18 Some vector network analyzers permit the user to choose between trace averaging and point averaging. Trace averaging will calculate and display a new trace average after measuring each complete set of frequency points in the current range. Changing the DUT leads to the data trace appearing to settle (slowly) toward the new measurement, as the influence of the old data is progressively reduced through the averaging process. This behavior can be undesirable, especially with large averaging factors. In point averaging, the vector network analyzer measures each frequency point multiple times (according to the averaging factor) and updates the trace data at that point, before moving to the next frequency point.

6502_Book.indb 222

11/12/14 8:42 AM



Metrology for Vector Network Analyzers223

dimensions through linear expansion. These can be surprisingly significant at submillimeter-wave and terahertz frequencies. The effects of thermal drift can be reduced by allowing the vector network analyzer equipment to thermally stabilize before use. In practice, this can mean having the equipment switched on for several hours before calibration and measurements take place. It is also important to maintain a stable ambient temperature. The flexible cables used to connect the extender heads to the vector network analyzer, and the waveguides used to connect the extender heads to the DUT, are more exposed to ambient thermal fluctuation than other parts of the signal path. Ideally, measurements should take place in a temperature-controlled laboratory to minimize the possibility of thermal changes to these parts of the signal path. 8.3.2  Types of Calibration Standard Since the calibration process has a significant influence on the integrity of vector network analyzer measurements, the choice of calibration standards is very important. The quality of the final measurements will strongly depend on how accurately the complex (magnitude and phase) reflection and transmission properties of the calibration standards are known. In theory, the actual values for the reflection or transmission properties of calibration standards are arbitrary, providing that we know their characteristics and that they are sufficiently different from each other. (The latter requirement arises from the fact that the solution to the error correction equations will be ill-conditioned if the values are too closely spaced.) A vector network analyzer equipped with extender heads is likely to be supplied with a calibration kit, containing a selection of calibration standards. Clearly, the vector network analyzer must know the reflection and transmission properties of the standards to solve for the error coefficients. In most cases, it is not necessary for the user to enter the characteristics of individual calibration standards into the vector network analyzer. This data is either preloaded into the vector network analyzer (for commonly used commercial calibration kits) or it is supplied on a USB memory device (or other medium) from the calibration kit manufacturer. Occasionally, it may be necessary to manually enter the properties associated with user-fabricated calibration kits, and some types of thru/reflect/line (TRL) calibration kits. To understand why certain items are chosen as calibration standards, we must consider how easy (or difficult) it is to be confident of the associated reflection or transmission properties. In practice, this is more readily achieved with some devices than others. For example, in metallic waveguide, we can predict (with reasonable confidence) the electrical properties of a short-circuit

6502_Book.indb 223

11/12/14 8:42 AM

224

Terahertz Metrology

from basic physical principles. (A short-circuit made from a conducting metallic plate, placed across the waveguide aperture, constitutes an electric wall, from which a total reflection must be produced.) Calculable Reflection and Transmission Standards

The complex reflection coefficient for any waveguide termination may be calculated by recognizing that the amount of reflection will depend on the relationship between the termination impedance and the characteristic impedance of the waveguide structure. If ZT is the impedance of the termination and Z0 is the characteristic impedance of the waveguide, then the reflection coefficient at the termination, ΓT (i.e., the scattering parameter, S11) is given by:



Γ T = S11 =

ZT − Z0 (8.15) ZT + Z0

The impedance of the short-circuit is zero, because the electric field tangential to a perfect conductor must be zero. Consequently, regardless of how the waveguide impedance is defined, the value for the reflection coefficient for a short-circuit termination is −1 (unity magnitude, with relative phase of the reflected and incident waves of 180°). Thus, a short-circuit makes a reliable and easy-to-implement calibration standard. In a waveguide calibration kit, the short-circuit is usually supplied as a waveguide flange without any waveguide aperture, plus some convenient means to hold the device (see Figure 8.16). This is also known as a flush short-circuit (or just flush short). A short length of waveguide (of the same aperture as that used for the test ports) will produce an electrical delay (i.e., a phase-shift, according to the guided wavelength). By application of equations similar to (8.12), this phase shift can be calculated19 with a high degree of confidence. Thus, a short length of waveguide is also a useful calibration standard. Calibration kits often include one or more short lengths of waveguide, often called shims or offset shims. A short-circuit and a short length of waveguide may also be combined to form an offset short-circuit. Again, this device will have calculable electrical properties (a total reflection, with a modified phase shift). Some waveguide calibration kits have offset short-circuits manufactured as a single device. These standards may look similar to the flush short 19 The phase shift will be different for each measurement frequency, and must therefore be calculated (by the VNA) at each frequency point for the calibration/measurement.

6502_Book.indb 224

11/12/14 8:42 AM



Metrology for Vector Network Analyzers225

circuits, but there is a waveguide aperture on the surface of the flange face, leading to a short, rectangular recess that implements the offset short circuit. Open-ended (i.e., open-circuited) waveguides are not usually used for calibration. This is because a portion of the signal energy will radiate away from the open end of the guide. Some of this radiated energy may subsequently be reflected back toward the waveguide from nearby objects, in a manner which cannot be predicted. This makes it difficult to have confidence in the expected reflection response, and thus is unsuited to a calibration standard. However, some recent experimental calibration methods have been proposed that do utilize an open-ended waveguide as a calibration standard [18]. These techniques have not yet been widely adopted, mainly due to the practical requirement to ensure a reflection-less environment in front of the open waveguide. They do have one advantage, which is that problems with flange misalignments that affect offset short-circuits do not arise. Assumed Good Reflection and Transmission Standards

Sometimes it is necessary to simply assume that a particular calibration standard is good. An example here might be an energy-absorbing termination (a matched Figure 8.16  Common calibration load). This type of standard should produce standards: (a) flush short-circuit; (b) offset shim; and (c) matched load. no reflections and it is designed to absorb all of the incident energy. For the purposes of calibration, we usually assume that this is the case. The matched loads provided in a waveguide calibration kit are generally good but not perfect. Here, good typically means that the magnitude of the true reflection coefficient is likely to be in the region of 0.01 (at best) to around

6502_Book.indb 225

11/12/14 8:42 AM

226

Terahertz Metrology

0.03. For this reason, it is sometimes necessary to exercise caution when using matched loads as part of a calibration procedure (see the following discussion on calibration techniques). Table 8.8 summarizes the most common items found in a waveguide calibration kit. We must now consider the options available to the vector network analyzer metrologist for calibrating the vector network analyzer using these standards. 8.3.3  One-Port Calibration Methods As discussed in Section 8.3.1, for a one-port calibration, we require three known terminations (sometimes called a three known loads calibration). The possible calibration strategies are therefore quite limited. The most common Table 8.8 Typical Standards Likely to be Found in a Waveguide Calibration Kit* Typical Quantity

Standard

Comment

1 or 2

Flush short-circuits

Easy to implement in waveguide, high confidence in the reflection coefficient. Usually two are supplied as these are relatively easy to manufacture.

1 or 2

Shims (very short lengths of waveguide); where two shims are supplied, they will be different lengths.

For many calibration methods, the precise length needs to be known.

1 or 2

Offset short-circuits

Optional. Some calibration kits have a combined offset shim plus short-circuit, manufactured as a single device.

2

Matched loads (or sometimes, low-reflecting loads)

Must normally assume that the reflection coefficient is zero, unless using a calibra­ tion method that calls for a low-reflecting load (where the precise reflection value does not need to be known).

1

Verification lines: a length of waveguide, usually around 25 mm long. The length should be precisely known at a standard laboratory temperature.

Not used in the calibration process, but often used to verify the success of a calibration.

*Other items may be found in specific calibration kits, but these are the main calibration standards used.

6502_Book.indb 226

11/12/14 8:42 AM



Metrology for Vector Network Analyzers227

technique is the short/offset-short/load method. This is essentially a waveguide equivalent to the short/open/load (SOL) method used for coaxial calibrations at microwave frequencies. As observed earlier, open-ended waveguides are not commonly used at millimeter-wave frequencies and above. However, as with the lower-frequency short/open/load calibrations, the short and the offsetshort in the waveguide SOL20 calibration provide two strong reflections but with different phases. It is important that, over the frequency range concerned, the length of the offset is such that the phase-shift produced is not the same as that obtained by a flush short-circuit (180°).21 Ordinarily, the length of the shims in the calibration kit are chosen such that either this will not happen or, by using two different shims, at least one will produce sufficient phase difference at every frequency in the band of interest. Most vector network analyzers calculate the actual phase shift for each frequency point by first computing the delay at each frequency, using the cutoff frequency for the guide and the free-space delay. Therefore, it is important that the calibration shims are properly specified in terms of the minimum frequency, which must normally be equal to the cutoff frequency for the waveguide size (8.8), and the free-space delay.22 A variation on this type of calibration calls for an additional measurement of an offset load. This approach is predicated on the fact that, very often, the electrical properties of the offset (i.e., a short length of waveguide with negligible attenuation and calculable delay) are more reliably known than the reflection properties of the load. Also, the load is usually not a perfect (i.e., reflection-less) load. In fact, for the offset-load calibration algorithm to work, it is actually preferable if the load is not absolutely perfect. To perform this type of calibration, a length of offset waveguide is required (in the same manner as 20 The acronym SOL traditionally stands for short/open/load, although for calibration in waveguide, the open-circuit standard is substituted (without loss of accuracy) with an offset short-circuit. Usually, SOL is still used to describe this type of calibration approach, although some manufacturers may use different acronyms, such as SOSL or SSL. 21 The recommended minimum phase difference between the reflection coefficients of two standards is usually 20°. 22 This information is normally supplied with the calibration kit, and may already be installed in the vector network analyzer calibration kit definitions. It is prudent to check that the vector network analyzer has the correct values for minimum frequency and free-space delay for each standard (where appropriate) prior to starting a calibration. Note that some modern vector network analyzers now associate the minimum frequency with the test port waveguide size, rather than the individual calibration standards. The user can still specify a minimum frequency for the standards, but it is only used to inform the vector network analyzer of the preferred lower frequency at which the individual standards are to be used, rather than specify the cutoff frequency for the waveguide size.

6502_Book.indb 227

11/12/14 8:42 AM

228

Terahertz Metrology

the offset-short). The vector network analyzer measures the reflections from the load standards, with and without the offset, and then mathematically manipulates the results to determine a single, more accurate measurement corresponding to an ideal reflection-less load. Again, it is important that the phase shift produced by the offset line does not lead to phases that are coincident with the flush load. An alternative to the offset load calibration standard is a sliding load. These devices are commonly used at microwave frequencies where the load element is mounted on a coaxial, sliding assembly. By making multiple measurements at different positions, the reflection coefficient of a slightly imperfect matched load traces a circle in the complex reflection coefficient plane. The center of this circle corresponds to an ideal, reflection-less load. In metallic waveguides, the sliding load is formed by adding a micrometer adjustment to the energy absorbing element, such that its position (i.e., phase) may be continuously adjusted with respect to the test port. Sliding loads are not easy to use (at any frequency) because a good calibration requires several, well-spaced measurements. However, in the hands of a skilled practitioner, sliding loads can be used to achieve a good quality calibration. It is actually possible to calibrate a one-port vector network analyzer without using a matched load. Provided that two different offset lengths are available, a flush short plus two different offset-shorts may be used. However, for most purposes, a successful one-port calibration is achieved using the short/offset-short/load method and this is generally the recommended strategy. Once calibrated using one of the methods described above, the vector network analyzer may then be used to make error-corrected, one-port measurements. The measured data will be mathematically corrected for all systematic magnitude and phase errors, and the magnitude and phase of the reflection measurement will be relative to the reference plane established at the test-ports by the calibration standards. 8.3.4  Two-Port Calibration Methods For two-port calibrations, there are more options available. The most common strategies are members of either the short/offset-short/load/thru (SOLT)23 family of calibration methods, or the thru/reflect/line (TRL) category [19]. 23

At microwave frequencies (in coaxial line) this method is normally implemented as short/open/ load/thru.

6502_Book.indb 228

11/12/14 8:42 AM



Metrology for Vector Network Analyzers229

SOLT

This calibration comprises a one-port calibration performed at each test-port, plus additional measurements for a thru (through) connection between the two test ports. The additional measurements are needed to allow the vector network analyzer to solve for the transmission tracking error terms. Ordinarily, the thru standard is implemented by simply connecting the waveguide test ports together. Effectively, this forms a zero-length path, with a predictable (i.e., zero) delay and perfect (i.e., lossless) transmission between the ports. Strictly speaking, the thru does not need to be zero length, but the transmission delay does need to be precisely known. Connecting the test-port reference planes together (yielding a zero-length path) makes this requirement easy to satisfy. Some modern vector network analyzers offer a development of the SOLT method, which uses an unknown thru, where the precise characteristics of the thru standard do not need to be known [20]. At first, this may appear to violate the basic principle of calibration methods discussed so far (the need to fully know the transmission/reflection properties of the calibration standards), but the method works by applying some elegant mathematics to the raw measurement data to solve for the required error coefficients. In practice, some facts do need to be known about the unknown thru: it must be known that the thru is reciprocal (i.e., S21 = S12) and the user should be able to estimate the delay (or nominal phase shift). These are generally quite easy criteria to satisfy. When the unknown thru method is used, the calibration algorithm is often called short/offset-short/load/reciprocal thru (SOLR). TRL

The second category of two-port calibrations is based on the thru/reflect/line (TRL) technique [19]. The TRL calibration algorithm operates in an entirely different manner from that of SOLT. Its main feature is that it cleverly avoids the need to know as much accurate information about the electrical properties of the calibration standards, when compared to SOLT. TRL makes use of the fact that a two-port calibration is really a matter of establishing a calibration in an arbitrary wave-guiding structure (transmission medium), and the reflection/transmission behavior of the DUT is relative to this medium. Thus, it is not actually necessary to know everything about the calibration standards, providing it is possible to use a section of the reference wave-guiding structure as part of the calibration. In this way, TRL has been called a self-calibration algorithm.

6502_Book.indb 229

11/12/14 8:42 AM

230

Terahertz Metrology

For the TRL method, the thru is usually formed by a direct connection of the test ports. The delay of the thru standard must be known accurately, but for a zero-path length, this is easy to satisfy, in the same manner as for SOLT. The thru standard is also used to establish the test-port reference planes for the measurement, which are defined to be at the center of the thru path. The reflect standard can be any reflecting device; all that needs to be known is the approximate phase of the reflection coefficient (to within ±90°) and that it produces a reasonably strong reflection. The precise magnitude of the reflection coefficient does not need to be known. In practice, the most commonly used reflect standard is the flush short-circuit. The line standard is simply a short length of waveguide (for example, an offset shim). The delay/phase-shift produced by the line standard does not need to be known precisely (again, ±90° is sufficient). However, there is one very important proviso, which is that the phase-shift produced by the line standard must be different from the thru standard, and the phase difference must not be close to either 0° or 180°.24 It is normally recommended that the phase shift lies between 20° and 160°. For lower frequencies, this is relatively easy to satisfy in metallic waveguide. A typical waveguide band covers an upper/lower frequency ratio of approximately 1.5:1.25 An offset line (shim) that is a quarter-wavelength (based on the guide wavelength) at the geometric mean of this band will have a phase shift of approximately 55° at the lowest frequency of the band and approximately 120° at the upper frequency of the band. As an example, consider the waveguide band covering 110 GHz to 170 GHz (WR-06/WM-1651), for which the geometric mean frequency is 136.75 GHz. The cutoff wavelength is found directly from the broad-wall dimension of the guide, as λ c = 2a = 3.302 mm, and the free-space wavelength at 136.75 GHz is 2.192 mm (using c = 299,792,458 m/s and assuming ε r ≈ 1). Using (8.12), we find that the guide wavelength at 136.75 GHz is lg =



l ⎛ l⎞ 1−⎜ ⎟ ⎝ lc ⎠

2

= 2.932 mm (8.16)

24

The calibration algorithm fails (numerically) if the phase shift is 0° or 180°. The frequencies which are deemed to be the upper and lower useable frequencies for a given waveguide size are rounded to convenient values. This means that the ratio of upper/lower frequency varies slightly among the common waveguide sizes. 25

6502_Book.indb 230

11/12/14 8:42 AM



Metrology for Vector Network Analyzers231

Thus, a quarter-wavelength shim at this frequency will be 2.932/4 = 0.733 mm, producing a phase-shift of 90°. The guided wavelengths at 110 GHz and 170 GHz will be 4.822 mm and 2.085 mm, respectively. The corresponding phase shifts are therefore 54.7° and 126.6°.26 If the required shim length for a quarter-wavelength at the geometric mean of 110 GHz to 170 GHz is only 0.733 mm, we may begin to perceive a possible problem with this approach at higher frequencies. For the WM-380 waveguide size (500 to 750 GHz), the equivalent shim would need to be approximately 160 μ m in length. Such short lengths mean that a λ /4 offset waveguide is too thin to be considered practical. Effectively, such shims would be thin metallic foils with the waveguide flange details impressed, but they would be much too fragile to be considered reliable calibration standards. The solution is to use longer lines which are more physically robust. The penalty that this incurs is that the bandwidth over which each shim satisfies the TRL phase-shift criteria will be reduced, and therefore more than one line is needed to cover an entire waveguide band. Often, the lines are chosen to be 3λ /4 at the centre of two smaller frequencies ranges. Longer lines may be used if the frequency range is split into several suitable subbands [21]. For example, calibrating over the entire frequency range for WR-03/ WM-864, 220 GHz to 325 GHz, would require an impractical single line standard of around 362 μ m. If this band is divided into two subbands (for the purpose of calibration), then lines of 1.290 mm and 1.000 mm will provide phases of nominally 270° in the center of two subbands (220 GHz to 274 GHz and 246 GHz to 325 GHz). Overlapping bands are chosen to ensure that all frequencies in the band may be calibrated with lines which remain well clear of the phase-shifts (0°, 180°) where the calibration algorithm does not work. LRL

A variant of the TRL method is known as line/reflect/line (LRL) [19]. For LRL, the first line takes the place of the TRL thru standard, and therefore (just as with TRL) the phase shift (delay) of this line must be accurately known. The second line need only be different from the first by the required 20° to 160°, 26

The value of the actual phase-shift for a given length of waveguide at a specific frequency may be found by several means. As discussed in 8.3.3., VNAs usually achieve this by first computing delay at each frequency, for which the cut-off frequency and the free-space delay are required, rather than the cut-off wavelength and the free-space wavelength. Whichever method is used, the precise value of free-space propagation is always required, including the effect of relative permittivity of the medium filling the guide (usually air). For lines which are multiple wavelengths long, these calculations often become numerically sensitive, and care must be taken to avoid a loss of accuracy through insufficient numerical precision.

6502_Book.indb 231

11/12/14 8:42 AM

232

Terahertz Metrology

and therefore a quarter-wavelength difference in the two lines is sufficient to meet this criterion.27 As for TRL, the delay/phase-shift of the second line only needs to be specified to within ±90°. Using LRL means that only two shims of reasonable (i.e., practical) thickness need to be used, and these will be sufficient to calibrate across an entire waveguide band. However, care must be taken to specify the calibration standards. For LRL calibration, the vector network analyzer must compute the phase shift for the first line standard, using the cutoff frequency (minimum frequency) and the free-space delay. Thus, LRL suffers from the fact that the calibration accuracy depends on the accuracy with which the properties of the first line can be specified. The advantage of TRL is that a zero path length thru connection has a well-defined delay of exactly zero. For higher millimeterwave frequencies, TRL with multiple lines may offer better results than LRL. Some Further Comments on TRL/LRL Calibration

For TRL and LRL calibration, the measurement reference plane may be established using the first line standard or by using the reflect standard. For TRL, it is preferable to use the thru (the first line standard) as it is well known. If the first line standard is used in LRL, the measurement reference plane is established at the center of the line, and therefore it relies on the accuracy with which this line is known. The alternative is to use the reflect standard to set the measurement reference plane. In this case, the reflection phase of the reflect standard must be accurately known. Ordinarily, a flush short-circuit is used for the reflect standard, and therefore it is possible to specify the reflection phase with confidence. Often, the calibration kit definition simply requires the reflect standard to be defined as a short circuit; the reflection phase is then easily determined by the vector network analyzer. For LRL calibration, it is generally preferable to use the reflect standard to set the measurement reference plane. In addition to specifying the minimum frequency for the offset shims (line standards), a maximum frequency may also be specified/required by some vector network analyzers. Where this is the case, the maximum frequency is used to instruct the calibration algorithm to avoid using a particular standard at unsuitable frequencies, where the phase shift may be approaching 0° or 180°. (As before, the minimum frequency must correspond to the cutoff frequency for the waveguide size, and is used by the vector network analyzer to compute delay/phase-shift.) 27

It can be seen that TRL is essentially a special case of LRL, where the first line (which must always be known) has a zero length.

6502_Book.indb 232

11/12/14 8:42 AM



Metrology for Vector Network Analyzers233

TRL and LRL are two-port calibration techniques. As such, they require the vector network analyzer to be equipped with two transmission-reflection extender heads. However, once completed, TRL/LRL calibrations may be used to perform one-port measurements. Depending on the availability and quality of calibration standards for one-port calibration, a TRL/LRL calibration may well be a preferred option, even for one-port measurements. Selecting a Two-Port Calibration

The reader may well ask what factors should ultimately determine the choice of two-port calibration approach for a given measurement task. In practice, this usually comes down to availability of suitable calibration artifacts and whether or not a particular strategy is practical. In general, TRL (and its variants) will yield more accurate results. This is due to the fact that the standards do not have to be as precisely known compared with SOLT for the algorithm to successfully compute the error terms. Thus, at frequencies up to the region of 400 GHz, TRL (or LRL) is usually the best option. Beyond this, and at frequencies approaching 1 THz, it is often found that the line standards become difficult to manufacture. Multiple longer lines may be used (as mentioned above) but attenuation, and the fact that more steps are required in the calibration process, can mean that TRL/ LRL methods are less attractive. Some manufacturers of terahertz calibration kits have returned to SOLT methods for calibration at terahertz frequencies. For all waveguide-based calibrations, the vector network analyzer user may also be required to enter (or confirm) a value for the characteristic impedance, Z0. In view of the difficulty of defining characteristic impedance for waveguide transmission lines, it is common practice to simply assign the value of Z 0 to 1. This means that any vector network analyzer display of actual impedance values (computed from the reflection measurements of the DUT) are effectively normalized values, relative to the transmission impedance of the waveguide. The value entered for Z0 does not affect the calibration in any way; it simply determines the scaling factor for the display of DUT impedance. Isolation (Calibration for Cross-Talk Errors)

All two-port calibration techniques include an additional, optional stage of the calibration procedure called isolation. The purpose of this step is to quantify the amount of signal leakage between the two ports (i.e., crosstalk). If some of the test signal can leak between the two test ports during a measurement, this amounts to an alternative signal path for transmission measurements. If the DUT has a very low transmission coefficient (low S21 or S12), this alternative signal path could become significant, or even dominate the measurement.

6502_Book.indb 233

11/12/14 8:42 AM

234

Terahertz Metrology

However, in rectangular waveguide, signal leakage is usually very small, and for most vector network analyzers, the signal leakage is less than 1 × 10 –5. Unless the DUT is expected to have a very low transmission characteristic, it is not necessary to conduct the isolation part of the calibration procedure. Indeed, many vector network analyzers invite the user to omit the isolation part of a two-port calibration process (in which case this error source is assumed to be zero). Table 8.9 summarizes the most common methods of calibrating a vector network analyzer for millimeter-wave, submillimeter-wave, and terahertz measurements. The process of removing systematic errors through calibration cannot be regarded as perfect. Some residual errors will always remain. Later in this chapter, we will look at the issue of verification (assessing the outcome of an attempt to calibrate the vector network analyzer) and quantifying the uncertainty associated with the measurements. 8.3.5  Good Practice Tips The following are some good practice tips for VNA callibrations: •

•

6502_Book.indb 234

Tip 1: A vector network analyzer should always be allowed to warm up before use, especially when high-quality measurements are needed. As a general rule, a vector network analyzer with millimeter-wave/ submillimeter-wave or terahertz extender heads should be switched on for at least a few hours before use in order to thermally stabilize. Failing to do this will result in noticeable drift in the measurement trace, particularly if the laboratory temperature is cool ( 0 dB). Tip 6: As mentioned previously, avoid excessive handling of certain items as this can lead to thermal changes. Metallic waveguide is a very good conductor of heat and so holding a waveguide device, such as a calibration load, can cause heat to be conducted inside the waveguide and onto the low-reflecting resistive element used to form the load. An increase in temperature can cause the reflection of the load element to change significantly and hence degrade the quality of the vector network analyzer calibration.

8.4  Errors and Uncertainties 8.4.1  Main Sources of Measurement Error Errors in vector network analyzer measurements are usually caused by imperfections in the system hardware. (When we talk about errors here, we are ignoring any blunders or incompetency on the part of the equipment operator.)

6502_Book.indb 236

11/12/14 8:42 AM



Metrology for Vector Network Analyzers237

These errors include the vector network analyzer and any accessories, as well as components such as the calibration standards and the devices under test. These imperfections can give rise to either random or systematic errors, or sometimes both. We will discuss these sources of error next. Waveguide Transmission Lines

The waveguides that are used as part of the overall measurement setup will never be perfect. For example, for a waveguide aperture, the width, a, and height, b, of the aperture will not be exactly the correct size. In addition, the corners of the rectangular aperture will not be perfect right angles. The lack of perfection in the waveguide flange alignment mechanisms (e.g., using dowel pins and holes) will cause some amount of misalignment, both linear misalignment in the a and/or b directions and angular misalignment. This misalignment will cause systematic errors, due to the alignment mechanisms (dowel pins and holes) not being exactly in the correct place, and random errors, due to the inevitable dimensional tolerances on these alignment mechanisms (e.g., the diameters of the dowel pins and holes). These waveguide dimensional errors will affect the performance of the vector network analyzer test ports (i.e., the waveguides on the extender heads attached to the vector network analyzer), the calibration standards, and, the devices under test. Vector Network Analyzer Hardware

The sources of error in the vector network analyzer hardware are due to electrical noise, nonlinearity in the vector network analyzer’s detectors, and isolation/crosstalk between the vector network analyzer test ports. Electrical noise induces random errors into all vector network analyzer measurements. This can be further divided into trace noise and detector noise. Trace noise can be reduced by applying numerical averaging, that is, by setting the vector network analyzer to take multiple measurements at the time that the vector network analyzer captures the data. Detector noise can be reduced by selecting a narrow intermediate frequency (IF) bandwidth (e.g., 100 Hz or less). Nonlinearity in the vector network analyzer’s detectors will cause an error in the response of the vector network analyzer that varies as a function of measured signal level. For example, the error will increase as the amount of attenuation being measured increases. This nonlinearity error, L, is usually specified in terms of L dB/dB. For example, if L = 0.01 dB/dB, then the nonlinearity error when measuring a 20-dB attenuator will be 0.01 × 20 = 0.2 dB. Isolation/crosstalk between the vector network analyzer’s test ports means that not all of the test signal supplied by the vector network analyzer actually passes through the device under test. Instead, some of the signal is picked up

6502_Book.indb 237

11/12/14 8:42 AM

238

Terahertz Metrology

directly by the vector network analyzer’s receiver detectors without passing through the device under test. This can cause problems, especially when the vector network analyzer is measuring low signal levels (e.g., for devices with high attenuation). Calibration Standards

The assumptions made about the properties of calibration standards will never be exactly true. For example, it is common to assume that the matched load standard really is matched (i.e., produces no reflected signal). In practice, there will always be some signal reflected by the matched load and so the assumption that this reflected signal is actually zero will induce an error into the calibration process. This calibration error will then induce an error in all measurements made with respect to this calibration (using this matched load standard). Similarly, if an offset device is used during calibration (e.g., an offset short-circuit standard), then the amount of offset (i.e., the length of line inherent in the standard) will not be known perfectly. The error in the knowledge about this length will induce an error into the calibration process. As before, this calibration error will then induce an error in all measurements made with respect to this calibration (using this offset short-circuit standard). 8.4.2  Typical Sizes of Errors Much work has been done, over many years, to investigate waveguide transmission line errors due to waveguide dimensional tolerances. These days, a convenient method of evaluating such errors is by using electromagnetic simulation software. A report showing the results produced by one such simulator is available at: http://www.gb.nrao.edu/electronics/edtn/edtn215.pdf [22]. This report shows calculations of mismatch due to tolerances on aperture height and width dimensions and aperture corner radii. The report also shows calculations of mismatch due to both linear and angular flange misalignments. The report by Kerr [22] shows that, for a good-quality waveguide, the most significant sources of mismatch are likely to be due to tolerances on the height and width dimensions of the waveguide aperture as well as linear flange misalignments (in both the a and b directions). Approximate analytical formulas for these effects have been given in [23]. For example, the mismatch (in terms of the linear magnitude of the reflection coefficient, ⎪Γ⎪) due to a height error, δ b, in a waveguide aperture of height, b, can be approximated by:



6502_Book.indb 238

Γ =

db (8.17) 2b

11/12/14 8:42 AM

Metrology for Vector Network Analyzers239



Similarly, the linear magnitude of the reflection coefficient, ⎪Γ⎪, due to a width error, δ a, in a waveguide aperture of width, a, can be approximated by: 1 ⎛ lg ⎞ da (8.18) Γ = ⎜ ⎟ 8⎝ a ⎠ a 2



where λ g is the guide wavelength. We can therefore use (8.17) and (8.18) to estimate the mismatch (in terms of the worst-case reflection coefficient) caused by different waveguide aperture height and width errors. Table 8.10 shows this information (in decibels) for aperture height errors, ⎪δ b⎪, of 1 μ m, 2 μ m, 5 μ m, and 10 μ m. Similarly, Table 8.11 shows similar information (in decibels) for aperture width errors, ⎪δ a⎪, of the same size. Table 8.10 Estimated Worst-Case Reflection Coefficient Due to Errors in Waveguide Aperture Height of 1 μ m, 2 μ m, 5 μ m, and 10 μ m Minimum Waveguide Frequency Name (GHz)

Maximum Frequency (GHz)

Worst-Case Reflection Coefficient (dB) ⎪δ b⎪ = 1 μm

⎪δ b⎪ = 2 μm

⎪δ b⎪ = 5 μm

⎪δ b⎪ = 10 μm

WM-570

330

500

−54

−49

−41

−35

WM-380

500

750

−51

−45

−37

−31

WM-250

750

1,100

−48

−42

−34

−28

Source: [23].

Table 8.11 Estimated Worst-Case Reflection Coefficient Due to Errors in Waveguide Aperture Width of 1 μ m, 2 μ m, 5 μ m, and 10 μ m Worst-Case Reflection Coefficient (dB)

Minimum Waveguide Frequency Name (GHz)

Maximum Frequency (GHz)

⎪δa⎪ = 1 μm

⎪δa⎪ = 2 μm

⎪δa⎪ = 5 μm

⎪δa⎪ = 10 μm

WM-570

330

500

−56

−50

−42

−36

WM-380

500

750

−53

−47

−39

−33

WM-250

750

1,100

−49

−43

−35

−29

Source: [23].

6502_Book.indb 239

11/12/14 8:42 AM

240

Terahertz Metrology

The errors in the vector network analyzer hardware (noise, nonlinearity, isolation/crosstalk) vary significantly depending on the type of vector network analyzer and the vector network analyzer’s operating conditions (including frequency range). Errors due to noise can be reduced by using numerical averaging and a narrow IF bandwidth. Effects due to noise can often be taken into account at the same time as other random effects are evaluated (e.g., the errors due to flange connection repeatability; see Section 8.4.3). Nonlinearity errors can usually be evaluated by measuring a series of known values of attenuation. At lower frequencies, a calibrated step attenuator is often used to provide a series of increasing values of attenuation. Measurement of these attenuation steps helps to establish if there is an error that increases proportionally to the level of attenuation being measured. At the present time, there are no internationally agreed methods for evaluating nonlinearity errors in vector network analyzers operating at submillimeter-wave frequencies. However, work is ongoing in this area at national measurement institutes (such as NIST in the United States, NPL in the United Kingdom, and PTB in Germany), and so these institutes should be consulted for the most up-to-date information about this topic. Isolation/crosstalk errors can be evaluated by measuring the transmission (i.e., ⎪S21⎪ and ⎪S12⎪) when both vector network analyzer ports are terminated with low reflecting loads. In an ideal world, no signal should be passing from port 1 to port 2 (as no connection has been made between ports 1 and 2 and so no signal path has been made available for transmission between the ports). However, in practice, some signal coming from the vector network analyzer’s source circuitry can be picked up by the vector network analyzer’s receiver circuitry due to leakage paths between source and receiver. Figure 8.17 shows some typical results of the measured transmission for a vector network analyzer operating from 750 GHz to 1,100 GHz and terminated with two low reflecting loads. On this occasion, the vector network analyzer’s IF bandwidth was set to 30 Hz. This helps to reduce any effects due to noise on the observed isolation/crosstalk. The amount of isolation/crosstalk in a vector network analyzer system is often dependent on the operating frequency; there is usually more crosstalk present at very high frequencies compared with lower frequency ranges. For example, Figure 8.18 shows the measured crosstalk/isolation for a vector network analyzer operating from 140 to 220 GHz, where the isolation/crosstalk is seen to be typically within the range −70 dB to −90 dB. This can be compared with the isolation/crosstalk shown in Figure 8.17, where the vector network analyzer is operating from 750 GHz to 110 GHz and the measured crosstalk/isolation is seen to be typically from −40 dB to −60 dB.

6502_Book.indb 240

11/12/14 8:42 AM



Metrology for Vector Network Analyzers241

Figure 8.17  Typical forward (⎪S21⎪, solid line) and reverse (⎪S12⎪, dashed line) isolation/ crosstalk for a vector network analyzer operating in the 750 to 1,100 GHz waveguide band.

Figure 8.18  Typical forward (⎪S21⎪, solid line) and reverse (⎪S12⎪, dashed line) isolation/ crosstalk for a vector network analyzer operating in the 140 to 220 GHz waveguide band.

6502_Book.indb 241

11/12/14 8:42 AM

242

Terahertz Metrology

8.4.3  Connection Repeatability As mentioned in Section 8.2.4, waveguide flanges can be a significant source of errors in vector network analyzer measurements. These errors can be both systematic in nature (e.g., due to a consistent misalignment between the two connecting waveguides) or random (e.g., due to dimensional tolerances on the alignment mechanisms and other random effects). The random errors give rise to measurements that do not repeat exactly each time the waveguide flange connection is disconnected and reconnected. Random errors due to connection repeatability can be assessed by repeatedly disconnecting and reconnecting a device a number of times, and analysing the results in terms of the variability in the results (i.e., using statistical techniques). Some typical repeatability assessments using two vector network analyzers (one operating in the 140 to 220 GHz band and another operating in the 750 to 1,100 GHz band) are shown in Tables 8.12 and 8.13, respectively. These tables give (at the minimum, middle, and maximum frequencies for each waveguide band) the experimental standard deviation in the magnitude of the reflection coefficient, s(⎪Γ⎪), determined from repeated disconnects/ reconnects of two devices: a near-matched load and an offset short-circuit. For the assessment shown here, 12 disconnect/reconnects of each device were used to calculate s(⎪Γ⎪) according to the following formula:



s( Γ ) =

2 n 1 Γ − Γ (8.19) ∑ k n − 1 k=1

where n is the number of repeat measurements (in our case, n = 12), Γk (k = 1 to 12) are the repeat measurements of the complex-valued reflection coefficient and Γ is the complex-valued mean reflection coefficient:



Γ=

1 n 1 n Γ R + j ∑ k=1Γ I (8.20) ∑ k=1 k k n n

where ΓR and ΓI are the real and imaginary components of the reflection coefficient, respectively. During the repeatability assessments shown here, the flanges on the devices under test were maintained with the same orientation with respect to the vector network analyzer test port flanges (i.e., the devices were deliberately not inverted as this would introduce an additional source of variability in the measurements). This repeatability assessment therefore only responds to random effects such as the tolerances on the diameters of the dowel pins

6502_Book.indb 242

11/12/14 8:42 AM

Metrology for Vector Network Analyzers243



Table 8.12 Repeatability Assessments in Terms of s(⎪Γ⎪) for the WR-05 Waveguide Band Frequency (GHz)

Near-Matched Load

Offset Short-Circuit

140

0.003

0.006

180

0.002

0.007

220

0.002

0.010

Table 8.13 Repeatability Assessments in Terms of s(⎪Γ⎪) for the WM-250 Waveguide Band Frequency (GHz)

Near-Matched Load

Offset Short-Circuit

750

0.012

0.136

900

0.009

0.093

1,100

0.011

0.078

and holes used to align the waveguides. If the devices are inverted during the repeatability assessment, then the effects to due incorrect position of the dowel pins will also have an impact in the observed repeatability. In general, including flange inversion in the repeatability assessment will increase the amount of variability (and hence the observed standard deviations) in the measurements. This will be particularly true at the very high frequencies (e.g., towards 1 THz and above). Tables 8.12 and 8.13 show that, for the same type of device (i.e., either matched load or short-circuit) the s(⎪Γ⎪) values at the lower frequencies (140 to 220 GHz) are smaller than at the higher frequencies (750 GHz to 1,100 GHz). This decrease in repeatability at the higher frequencies is most likely due to the waveguide aperture at these frequencies being much smaller than the waveguide aperture for the lower frequencies, while the dimensional tolerance errors affecting the flange alignment mechanism for both waveguide sizes are effectively the same. A similar flange dimensional misalignment error will therefore have a much greater impact on the smaller waveguide aperture compared to the larger waveguide aperture. Tables 8.12 and 8.13 also show that, for a given waveguide size (either WR-05, 140 GHz to 220 GHz, or WM-250, 750 GHz to 1100 GHz), the s(⎪Γ⎪) values are smaller for the near-matched load compared with the offset short-circuit. This shows that the measurement repeatability is also related to the size (i.e., magnitude) of the signal being measured: for the near-matched

6502_Book.indb 243

11/12/14 8:42 AM

244

Terahertz Metrology

load, the amount of reflected signal is very low (i.e., close to zero); and for the offset short-circuit, the amount of reflected signal is very high (close to 100% reflected signal). Finally, it is worth noting that the standard deviation calculation shown in (8.19) does not show separately the variation in the real and imaginary components of the complex-valued reflection coefficient. Rather, the equation merges the variability of the two components into one. To examine separately the variation in the real and imaginary components, the calculation should be performed separately on these two components. Advice on performing such a calculation has been given in [24]. 8.4.4  System Verification After calibrating the vector network analyzer system, it is important to perform some kind of verification on the overall system performance to check that assumptions made during the calibration process (e.g., about the system and/or the calibration standards) were correct. For example, that the amount of reflected signal provided by the near-matched load was sufficiently small. Until recently, it has been very difficult to independently verify the performance of a vector network analyzer at these high millimeter-wave and submillimeter-wave frequencies. This is because devices providing known values (against which the vector network analyzer can be checked) have not been readily available. At lower frequencies, manufacturers provide verification kits for the vector network analyser, but at the time of this writing, such kits were not readily available at the very high frequencies. However, some progress has recently been made to address these problems. A new form of standard has been developed that comprises a straight section of waveguide that is orientated during connection such that the waveguide aperture is at right angles to the waveguide apertures on the vector network analyzer test ports. This has been described elsewhere in [25, 26]. This cross-connected waveguide (or cross-guide, for short) forms a section of waveguide that is effectively below cutoff and so its loss can be predicted from electromagnetic theory (e.g., using 3-D electromagnetic simulation software). A cross-guide device can be connected to the vector network analyzer after calibration to verify the vector network analyzer’s performance, by comparing the vector network analyzer results with values predicted by simulation software or with measured values supplied by another laboratory (such as a standards laboratory). The level of agreement is established by comparing the difference between the two sets of results with the expected uncertainty in the results. For example, Figure 8.19 shows the measured and modeled values

6502_Book.indb 244

11/12/14 8:42 AM



Metrology for Vector Network Analyzers245

of transmission coefficient for two different lengths of cross-guide in WR-05 (140 GHz to 220 GHz): 0.54 mm and 1.47 mm. Figure 8.20 illustrates the concept of a cross-connected waveguide and Figure 8.21 illustrates the connection strategy for these cross-guide lines.

Figure 8.19  Measured (solid line) and modeled (dashed line) transmission coefficient values for a 0.54 mm length of WR-05 cross-guide and a 1.47 mm length of WR-05 cross-guide.

Figure 8.20  Illustration showing the concept of a cross-connected waveguide or cross-guide.

6502_Book.indb 245

11/12/14 8:42 AM

246

Terahertz Metrology

Figure 8.21  (a) A conventional waveguide test port; and (b) a waveguide line orientated as a cross-connected waveguide (so that its aperture is rotated 90° with respect to the test port aperture).

8.4.5  Good Practice Tips The following are good practice tips for verifying/assuring a vector network analysis: •

•

•

•

6502_Book.indb 246

Tip 1: Devices used for verification purposes should be connected with the same degree of care and attention as the calibration standards and devices under test. A poor connection of a verification device could lead to the conclusion that the vector network analyzer is not operating correctly when actually it is the verification device that is not operating as it should (due to the poor connection). Tip 2: Always ensure waveguide flange faces are clean. This can be established by visual inspection; a handheld magnifier can be used to help with the inspection of the flange faces. Any dirt can be removed using a cotton swop dipped in a suitable cleaning fluid [i.e., isopropanol (IPA)]. Tip 3: Keep documented records of the performance of the vector network analyzer, for example, periodic measurements (or plots) of the vector network analyzer’s isolation/crosstalk. These records can be used on subsequent occasions to show that the vector network analyzer is performing as it did previously. Tip 4: If possible, try to obtain some dimensional data on the critical waveguides used with the system. For example, the aperture

11/12/14 8:42 AM



Metrology for Vector Network Analyzers247

dimensions (height and width) of the calibration standards and the vector network analyzer extender head test ports (i.e., the calibration reference planes). Check the measured dimensions against values given in the standards [13].

8.5  Concluding Remarks This chapter has given an overview of the vector network analyzer as used at millimeter and submillimeter wavelengths. Emphasis has been given on using these vector network analyzers for making measurements in rectangular metallic waveguide. This is because the extender heads that are used with these vector network analyzers at these frequencies use this type of waveguide to provide the test ports. A description of the operation of the vector network analyzer and its hardware has been given along with an overview of the properties of metallic waveguides used at these frequencies. Information has also been given concerning the use of standards and calibration procedures that are required for the vector network analyzer to make meaningful measurements. An uncalibrated vector network analyzer is rarely of any use in the majority of measurement applications, whereas a calibrated vector network analyzer that has used appropriate standards and calibration routines is capable of making very accurate and very reliable measurements. Some information has been given on understanding the errors and uncertainties associated with vector network analyzer measurements. Ultimately, in the fullness of time, vector network analyzer measurements made at these frequencies should be made traceable to the International System of Units (SI). Although some work has now begun in this area [27], there is still much more that needs to be done before vector network analyzer measurements can be considered traceable at all these frequencies. Finally, it is worth noting that measurements using vector network analyzers can also be made in other transmission media (e.g., in planar circuit, on-wafer environments) at millimeter and submillimeter wavelengths. This is achieved by attaching adaptors (e.g., waveguide to coplanar waveguide onwafer probes) to the waveguide ports of the vector network analyzer extender heads. However, this topic is outside the scope of the material covered by this chapter. At the time of this writing, measurements using on-wafer probes, at these frequencies, are a subject that is still in its infancy. More work needs to be done by researchers and manufacturers before this technology can be taken up and implemented by the on-wafer user community.

6502_Book.indb 247

11/12/14 8:42 AM

248

Terahertz Metrology

References [1]

OML, Inc., www.omlinc.com.

[2]

Virginia Diodes, Inc., www.vadiodes.com.

[3]

Rohde and Schwarz, www.rohde-schwarz.com.

[4]

Goy, P., M. Gross, and J. M. Raimond, “A New 8-450 GHz Millimetre Vector Network Analyser,” 23rd General Assembly of URSI (Union Int. Radio Sci.), Prague, Czechoslovakia, August 1990.

[5]

Goy, P., M. Gross, and J. M. Raimond, “8-1000 GHz Vector Network Analyser,” 15th Int. Conf. on Infrared and Millimeter Waves (IRMMW), Orlando, FL, December 1990.

[6]

Collier, R. J., Transmission Lines, Cambridge, U.K.: Cambridge University Press, 2013.

[7]

Kearns, D. M., and R. W. Beatty, Basic Theory of Waveguide Junctions and Introductory Microwave Network Analysis, London, U.K.: Pergamon Press, 1967.

[8]

Pozar, D. M., Microwave Engineering, New York: John Wiley & Sons, 1998.

[9]

Collin, R. E., Foundations for Microwave Engineering, New York: McGraw-Hill, 1992.

[10] Agilent Technologies, “Network Analyzer Basics,” Application Note 5965-7917E, Agilent Technologies Inc., 2004. [11]

MIL-DTL-85/3C, “Waveguide, Rigid, Rectangular (Millimeter Wavelength),” February 4, 2002.

[12] IEC 60153-2, “Hollow Metallic Waveguides. Part 2: Relevant Specifications for Ordinary Rectangular Waveguides,” 2nd ed., 1974. [13] IEEE Std. 1785.1-2012, “IEEE Standard for Rectangular Metallic Waveguides and Their Interfaces for Frequencies of 110 GHz and Above—Part 1: Frequency Bands and Waveguide Dimensions,” 2013. [14] Hesler, J. L., et al., “Recommendations for Waveguide Interfaces to 1 THz,” Proc. 18th Int. Symp. on Space THz Tech., Pasadena, CA, March 2007. [15]

MIL-DTL-3922/67Ew/Amendment1, “Flanges, Waveguide (Contact) (Round, 4 Hole) (Millimeter),” May 2014.

[16] IEEE Std. 1785.2, “IEEE Standard for Rectangular Metallic Waveguides and Their Interfaces for Frequencies of 110 GHz and Above—Part 2: Waveguide Interfaces,” in preparation. [17] Marks, R. B., “Formulations of the Basic Vector Network Analyzer Error Model Including Switch Terms,” Proc. 50th ARFTG Microwave Measurement Conference, December 1997, pp. 115–126. [18] Liu, Z., and R. M. Weikle II, “A Reflectometer Calibration Method Resistant to Waveguide Flange Misalignment,” IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-54, No. 6, June 2006, pp. 2447–2452.

6502_Book.indb 248

11/12/14 8:42 AM



Metrology for Vector Network Analyzers249

[19] Rumiantsev, A., and N. M. Ridler, “VNA Calibration,” IEEE Microwave Magazine, Vol. 9, No. 3, June 2008, pp. 86–99. [20] Ferrero, A., and U. Pisani, “Two-Port Network Analyzer Calibration Using an Unknown ‘Thru’,” IEEE Microwave Guided Wave Letters, Vol. 2, No. 12, 1992, pp. 505–507. [21] Ridler, N. M., “Choosing Line Lengths for Calibrating Waveguide Vector Network Analysers at Millimetre and Sub-Millimetre Wavelengths,” NPL Report TQE 5, National Physical Laboratory, March 2009. [22] Kerr, A. R., “Mismatch Caused by Waveguide Tolerances, Corner Radii, and Flange Misalignment,” Electronics Division Technical Note No 215, National Radio Astronomy Observatory, Charlottesville, VA, December 2009. http://www.gb.nrao.edu/electronics/edtn/edtn215.pdf. [23] Bannister, D. J., E. J. Griffin, and T. E. Hodgetts, “On the Dimensional Tolerances of Rectangular Waveguide for Reflectometry at Millimetric Wavelengths,” NPL Report DES 95, National Physical Laboratory, Teddington, UK, September 1989. [24] Ridler, N. M., and R. G. Clarke, “Investigating Connection Repeatability of Waveguide Devices at Frequencies from 750 GHz to 1.1 THz,” Proc. 82nd ARFTG Microwave Measurement Conference, Columbus, OH, November 20–21, 2013, pp. 87–99. [25] Ridler, N. M., and M. J. Salter, “Cross-Connected Waveguide Lines as Standards for Millimeter- and Submillimeter-Wave Vector Network Analyzers,” Proc. 81st ARFTG Microwave Measurement Conference, Seattle, WA, June 7, 2013. [26] Schrader, T., et al., “Verification of Scattering Parameter Measurements in Waveguides Up to 325 GHz Including Highly-Reflective Devices,” Adv. Radio Sci., Vol. 9, 2011, pp. 9–17. [27] Ridler, N. M., et al., “The Trace Is on Measurements: Developing Traceability for S-Parameter Measurements at Millimeter and Submillimeter Wavelengths,” IEEE Microwave Magazine, Vol. 14, No. 7, November/December 2013, pp. 67–74.

6502_Book.indb 249

11/12/14 8:42 AM

6502_Book.indb 250

11/12/14 8:42 AM

9 Terahertz Optics Mira Naftaly 9.1  Terahertz Optical Materials 9.1.1  Transparency at Terahertz Frequencies Terahertz measurements and applications in general are greatly inconvenienced by the fact that very few of the highly transmissive materials employed for the visible and near-infrared optics are equally transparent in the terahertz band. Silica-based glasses, in particular, absorb strongly at frequencies above ~0.5 THz. Material absorption at terahertz frequencies arises from several mechanisms, including free carrier absorption, vibrational lattice modes, dielectric relaxations, and disorder in amorphous materials. Free Carrier Absorption

Frequency-dependent absorption by free carriers may be generally described by the Drude model of carrier transport, which gives the complex conductivity as [1, 2]:



s (w ) =

s0 s0 s 0 wtc (9.1) = 2 2 +i 1 − iwtc 1 + w tc 1 + w 2 tc2

251

6502_Book.indb 251

11/11/14 10:22 AM

252

Terahertz Metrology

where σ 0 is the dc conductivity, τ c is the carrier relaxation time, and ω is the electromagnetic field frequency. The complex dielectric constant then becomes: e = e¢ + ie ≤= e∞ + i



is 0 s(w) = e∞ + w w(1 − iwtc )

s0 ⎤ ⎡ s 0 tc ⎤ ⎡ = ⎢ e∞ − 2 2 ⎥ + i⎢ 2 2 ⎥ 1 + w tc ⎦ ⎣ w(1 + w tc ) ⎦ ⎣

(9.2)



where ε ∞ is the intrinsic dielectric constant of the material (real). The refractive index and absorption coefficient due to the Drude mechanism can be calculated from (2.5) and (2.8); their typical form is shown in Figure 9.1. For materials whose conductivities are >1 Ω cm, the peak of absorption usually lies between 0.1 and 0.5 THz. It is seen that the absorption coefficient and refractive index arising from free carriers vary with frequency, both increasing strongly at lower frequencies with the dc conductivity σ 0. For that reason, materials used as transmissive elements in terahertz systems are required to be high resistivity (e.g., float-zone silicon). The free-carrier mechanism is the only type of terahertz absorption that falls with frequency, and its presence is easily recognizable by its characteristic spectral profile.

Figure 9.1  The typical form of absorption coefficient and refractive index arising from the Drude mechanism of free-carrier absorption.

6502_Book.indb 252

11/11/14 10:22 AM



Terahertz Optics253

Absorption by Lattice Modes (Phonons)

Resonant absorption of radiation occurs when the frequency of the incident terahertz wave matches that of vibrational modes (phonons) of the lattice. Only optical phonons in crystals can participate in this process, so it is absent in elemental solids. Phonon absorption in crystalline materials can manifest as well-defined spectral features, such as those seen in Figures 2.12, 4.5, 6.6, and 6.7. Some materials possess very intense phonon resonances clustered in broad frequency bands, termed Reststrahlen bands, where strong absorption renders the material opaque, with a consequent reflectivity close to unity [see (2.6)]. This phenomenon may occur in either crystalline or amorphous materials. A Reststrahlen band also affects the refractive index outside its region, in that it signals the onset of ionic polarizability. According to the Clausius-Mossotti equation, the real dielectric constant ε ′ is related to material polarizability, p, by [1, 2]:



e¢ − 1 Np = (9.3) e¢ + 2 3e0

where N is the number of atoms or molecules per unit volume and ε 0 is the permittivity of free space. From (9.3) it is seen that the real dielectric constant, and therefore the refractive index, increases with polarizability. At frequencies above the Reststrahlen band, only electrons contribute to polarizability, whereas at frequencies below it, there is an additional component of ionic polarizability, which increases the combined value of p and hence raises the refractive index [1]. As a consequence, in materials that possess a Reststrahlen band, the refractive index is higher at terahertz frequencies than it is in the visible. Resonant absorption is determined by the molecular and crystal structure of the material. Generally, lattice modes and, in particular, Reststrahlen bands tend to occur in the higher-frequency part of the terahertz range and in the mid-infrared. Solids with tightly bound light atoms have higher Restsrahlen frequencies than those with weakly bound heavy atoms (this is because phonon modes may be viewed as analogous to vibrational modes of a network of masses interconnected by springs). Dielectric Relaxations in Polar Materials

When an oscillating electromagnetic field interacts with a material that has polarizable bonds, it causes charge separation and creates dipoles which oscillate

6502_Book.indb 253

11/11/14 10:22 AM

254

Terahertz Metrology

in response to the field. These dipole oscillations are unhindered at low frequencies, allowing the material to be transparent. At higher frequencies, however, the dipole motions are impeded by friction in the material, in consequence of which their response is delayed relative to the field, manifesting as absorption. The Debye model describes the frequency behavior of the dielectric constant in terms of the response time τ d of the dipoles, called relaxation time [1]:



⎡ e(0) − e(∞) ⎤ ⎡ ( e(0) − e(∞)) wt d ⎤ +i ε = ⎢ e(∞) + ⎥ (9.4) 1 + w 2 t d2 ⎥⎦ ⎢⎣ 1 + w 2 t d2 ⎦ ⎣

where ε (0) and ε (∞) are the static and high-frequency dielectric constants, respectively. In more strongly polar materials, the difference [ε (0) − ε (∞)] is larger. For example, in nonpolar high-resistivity silicon ε (0) ≅ ε (∞) = 11.7, whereas in strongly polar water ε (0) = 80 and ε (∞) = 1.8. Figure 9.2 shows a typical frequency dependence of the absorption coefficient and refractive index in the terahertz range, as exhibited by polar materials. Absorption is seen to rise with frequency, whereas refractive index falls; the slopes of the curves are steeper for larger [ε (0) − ε (∞)] and for longer τ d . Absorption is low when ωτ d 50

Yes

Z-cut sapphire

3.1

1.0

8.7

>50

Yes

Silicon carbide (6H-SiC)

3.1

~3

Yes

Diamond

2.38

~1 0.09

~1 0.12

0.26

Yes

and nonpolar, making possible high terahertz transparency extending to >10 THz [13–15]. HDPE has a low degree of chain branching, resulting in a higher density than LDPE where branching is high. Both HDPE and LDPE are part crystalline and part amorphous, giving them the characteristic milky-white semitranslucent appearance in the visible; the degree of crystallinity is higher in HDPE (70% to 90%) than in LDPE (40% to 60%). Both HDPE and LDPE have a pronounced lattice mode at 2.2 THz, which is more sharply defined in HDPE (see Figure 6.7). Polyethylene is one of the most commonly used plastics and is available in a wide variety of shapes and forms. It can be easily machined to a desired profile and can be polished to a near-optical finish. Polytetrafluorethylene

Polytetrafluorethylene (PTFE, also known as Teflon) has the chemical formula (C2F4)n and may be regarded as a form of polyethylene where all hydrogen atoms have been substituted by fluorine [12]. PTFE is highly crystalline, resulting in a bright-white appearance in the visible. Fluorine bonds are stronger than hydrogen ones and more polar, shifting the lattice mode to 6.1 THz, compared with 2.2 THz in polyethylene, and giving rise to significant terahertz absorption at higher frequencies, thus reducing the transparency range [13, 14, 16, 17]. PTFE is available in a wide variety of forms, and can be machined to a desired profile. However, it is relatively soft and deformable and cannot be polished to an optical finish.

6502_Book.indb 259

11/11/14 10:22 AM

260

Terahertz Metrology

Cyclic Olefin Copolymers

Cyclic olefin copolymers (COC) are cyclic hydrocarbons copolymerized with other types of organic molecules (e.g., ethylene [12]), which are produced by a small number of manufacturers under proprietary trade names (e.g., TOPAS, Zeonor, Picarin). They are polymer glasses, and have high optical clarity and surface finish. COC materials for terahertz optics are designed to have high transparency up to 10 THz or more, and a refractive index that is similar at terahertz and in the visible [14, 17, 18], to aid optical alignment and make possible accurate copropagation of optical and terahertz beams. COC terahertz optics are available in limited ranges, and they are more expensive than other polymers. Polypropylene

Polypropylene consists of hydrocarbon chains with CH3 side-groups and has a formula (C3H6)n [12]; it has moderate crystallinity and is translucent in the visible. Because all its bonds are covalent and either C-C or C-H, it is nonpolar and, away from absorption bands, has moderately good terahertz transparency up to 10 THz and beyond [15, 16]. However, it suffers from several broad absorption bands at frequencies above 2 THz; moreover, commercial grades commonly have additives which strongly increase its terahertz absorption. Poly(4-Methyl-1-Pentene) (TPX)

Poly(4-methyl-1-pentene) (PMP), commonly known as TPX, has a chemical formula (C6H12)n and, like polypropylene, consists of hydrocarbon chains with attached pentene side groups [12]. It is a polymer glass with good optical clarity and surface finish, which may have a slight yellow or gray cast depending on the grade. It is nonpolar and has good terahertz transparency extending beyond 10 THz with no absorption features and negligible dispersion [13, 15, 18]. As in COC materials, the terahertz refractive index of PMP is close to its value in the visible. PMP is hard, resists deformation, and can be manufactured as flat plates or lenses. Due to its excellent properties, PMP is widely used in terahertz optics and for making sample cells and mounts. It is available as sheets from a number of manufacturers; however, different grades may have slightly different optical properties. PMP lenses are commercially available from suppliers of terahertz optics. Paraffin

Paraffins (waxes) are similar to polyethylene in that they are alkanes whose general chemical formula consists of CnH2n+2 units [12]. However, their carbon

6502_Book.indb 260

11/11/14 10:22 AM



Terahertz Optics261

chains are much shorter: tens of atoms, as compared to tens of thousands in a polymer. Paraffins are therefore nonpolar materials with good terahertz transparency. In some applications or types of measurement it is necessary to use an optical contact fluid, or to suspend test particles in a transparent liquid or jelly, or occasionally to mount a sample in a rigid matrix material. Liquid paraffin, petroleum jelly, and paraffin wax all may serve very well for these purposes. The length of hydrocarbon chains forming the material determines its density and consistency at room temperature: paraffin wax has chains of 20 to 40 carbon atoms, whereas liquid paraffin has chains of 6 to 16 carbon atoms, and petroleum jelly is a mixture of shorter and longer chains. Paraffin wax and petroleum jelly are both partially crystalline, indicated by their translucent appearance in the visible. Figure 9.5 shows absorption coefficients and refractive indices of paraffin liquid and wax. The lattice mode at 2.2 THz is evident in the wax spectrum as a broad shallow band. Table 9.2 summarizes the terahertz transmission properties of the most common low-loss polymers. The data is derived from the published literature, especially [13–18], in many cases together with author’s own measurements. It should be noted that a large degree of variation exists in the published data, especially with respect to absorption, which may be in part attributable to differences in the fabrication processes, additives, and so forth. The values given in Table 9.2 represent the author’s best judgement.

Figure 9.5  Absorption coefficients and refractive indices of paraffin liquid and wax.

6502_Book.indb 261

11/11/14 10:22 AM

262

Terahertz Metrology Table 9.2 Optical Properties of Terahertz Transparent Polymers Terahertz Refractive Index (mean)

Absorption Absorption Absorption at 1 THz at 3 THz at 10 THz Transparency (cm –1) (cm –1) (cm –1) in the Visible

Low-density polyethylene

1.51

0.2

1.6

~2

No

High-density polyethylene

1.53

0.2

1.6

~3

No

Polytetrafluor­ 1.43 ethylene

0.5

2.8

>50

No

Cyclic olefin copolymer

1.52 to 1.53

0.2

0.8

~2

Yes

Polypropylene

1.52

0.3

~1.5

~3.5

Yes

PMP (TPX)— polymethylpentene

1.46

0.3

0.8

~2.5

Yes

Paraffin liquid

1.47

0.5

1.7

NA

Yes

Paraffin wax

1.49

0.8

4.2

NA

No

Polymer

9.2  Metal Reflectivities Metallic mirrors are arguably the most crucial components of any terahertz system, particularly so because lenses are problematic (see Section 9.1) and dielectric mirrors (as in the visible) do not exist. However, reflectivities of metals are higher at terahertz frequencies than in the visible, approaching unity. Metallic mirrors, therefore, provide adequate performance for all terahertz applications, including highly demanding ones such as laser mirrors. They also have the advantages of being broadband with only weak frequency dependence, and polarization insensitivity. The general expression for reflection coefficient at a material/air interface is [1]:



R=

(n − 1)2 + k2 (9.6) (n + 1)2 + k2

where n and k are the real and imaginary refractive indices (see Chapter 2 for the relationship between n, k and the dielectric constant ε ′+ i ε ″). For a

6502_Book.indb 262

11/11/14 10:22 AM



Terahertz Optics263

dielectric where k 100W [32, 33]. Continuous-wave operation has been achieved using a Yb-YAG disc laser and an OPO cavity with tight focusing and high finesse [34]. The possibility of a true terahertz-resonant optical parametric oscillator was demonstrated by Schaar et al. [35]. This was a doubly-resonant OPO with a coupled-cavity configuration, where the primary OPO generated near-infrared signal and idler using a mode-locked Nd-YAG laser and a periodically-poled LiNbO3 crystal, while the secondary OPO used a periodically-poled GaAs crystal to generate difference-frequency terahertz. This device was tunable between 0.4 and 3.5 THz, and produced up to 1 mW of average terahertz power.

10.7  Free Electron Lasers A free electron laser (FEL) uses as its lasing medium a beam of relativistic electrons produced by an accelerator, usually a linear accelerator (linac). Free electron lasers are therefore very large, very expensive installations, and only a few terahertz FEL facilities exist around the world. However, these are an immense experimental resource, because they are capable of generating terahertz optical powers many orders of magnitude higher than any other source. An example of such a facility is seen in Figure 10.8, which shows an artist’s impression of the FEL ALICE (Accelerators and Lasers in Combined Experiments) at Daresbury, United Kingdom. An updated worldwide list of free electron laser facilities is available at http://sbfel3.ucsb .edu/www/fel_table.html.

6502_Book.indb 296

11/11/14 10:22 AM



Terahertz Laser Sources297

Figure 10.8  A scaled artist’s impression of the FEL ALICE at STFC Daresbury Laboratory, United Kingdom, http://www.stfc.ac.uk/ASTeC/Programmes/Alice/35997. aspx. (Courtesy STFC Daresbury.)

A free electron laser consists of an undulator (or wiggler) contained in an optical cavity, as depicted schematically in Figure 10.9. The undulator is an assembly of alternating magnets creating a periodic magnetic field orthogonal to the direction of the electron beam. This imposes an oscillatory motion on the electrons, causing them to emit radiation at the wavelength equal to the wiggler period [36, 37]. However, because the electrons are relativistic, in their inertial frame the period length of the wiggler is contracted, so that the emitted radiation is shifted to shorter wavelengths, given by:



l=

lw (1 + K 2 ) (10.5) 2g

where λ w is the wiggler period and γ is the relativistic Lorentz factor determined by the electron velocity v:



g =

1 (10.6) 1 − v2 /c 2

The factor K is called the wiggler parameter and is a dimensionless quantity that describes the relationship between the wiggler period and the magnetic field B:



K =

eBlw (10.7) 2pme c

where e and me are the electron charge and mass. The wiggler parameter is usually close to unity, so that λ ≈ λ w/γ . The emitted wavelength can be tuned by varying the electron energy (and thus velocity). As an example of

6502_Book.indb 297

11/11/14 10:22 AM

298

Terahertz Metrology

Figure 10.9  Schematic depiction of a free electron laser.

the energies involved, the undulator period λ w is typically around 1 cm, so to generate radiation at 3 THz the Lorentz factor has to be γ = 100. This requires electrons to travel at v = 0.94c and have the energy of 5 MeV. Fine frequency tuning can also be achieved by varying the strength of the magnetic field and thus the value of K. Enclosing the undulator in an optical cavity resonant with the emitted frequency amplifies the spontaneous emission and generates lasing. A linac produces short electron bunches whose spatial extent is much shorter than the undulator period. As a result, all the electrons in a bunch wiggle in phase and emit radiation coherently. If the cavity round-trip time matches the bunch repetition rate of the linac, the emitted radiation has the highest peak power and shortest pulse length. Free electron lasers produce terahertz radiation with peak power up to several megawatts and average powers of hundreds of watts, with broad tunability into tens of terahertz. As a result, they have been employed in a variety of experiments requiring high brightness sources, such as nonlinear effects and detection of very low-yield processes [38]. A particularly interesting application is real-time imaging using detector arrays [38]. They are also especially well suited to studies on biological systems [39]: due to their high peak power, they can penetrate aqueous samples, while the short pulse length and controllable repetition rate make it possible to avoid thermal effects.

References [1]

Douglas, N. G., Millimetre and Submillimetre Wavelength Lasers, Springer Series in Optical Science, New York: Springer-Verlag, 1989.

[2]

Jacobsson, S., “Optically Pumped Far Infrared Lasers,” Infrared Phys., Vol. 29, 1989, pp. 853–874.

6502_Book.indb 298

11/11/14 10:22 AM



Terahertz Laser Sources299

[3]

Rothman, L., et al., “The HITRAN2012 Molecular Spectroscopic Database,” Journal of Quantitative Spectroscopy and Radiative Transfer, 2013, http://hitran.iao.ru/.

[4]

Inguscio, M., et al., “A Review of Frequency Measurements of Optically Pumped Lasers from 0.1 to 8 THz,” J. Appl. Phys., Vol. 60, 1986, pp. R161–R191.

[5]

Bründermann, E., H. -W. Hübers, and M. F. Kimmit, Terahertz Techniques, New York: Springer, 2012, Ch. 4.

[6]

Bründermann, E., et al., “Double Acceptor Doped Ge: A New Medium for InterValence-Band Lasers,” Appl. Phys. Lett., Vol. 68, 1996, pp. 3075–3077.

[7]

Hübers, H. -W., S. G. Pavlov, and V. N. Shastin, “Terahertz Lasers Based on Germanium and Silicon,” Semiconductor Sci. Tech., Vol. 20, 2005, p. S211.

[8]

Bergner, A., et al., “New p-Ge THz Laser Spectrometer for the Study of Solutions: THz Absorption Spectroscopy of Water,” Rev. Sci. Instr., Vol. 76, 2005, p. 063110.

[9]

Sharma, V., et al., “From Solvated Ions to Ion-Pairing: A THz Study of Lanthanum(III) Hydration,” Phys. Chem. Chem. Phys., Vol. 15, 2013, pp. 8383–8391.

[10] Williams, B. S., “Terahertz Quantum-Cascade Lasers,” Nature Photonics, Vol. 1, 2007, pp. 517–525. [11] Scalari, G., et al., “THz and Sub-THz Quantum Cascade Lasers,” Laser & Photon. Rev., Vol. 3, 2009, pp. 45–66. [12] Kumar, S., “Recent Progress in Terahertz Quantum Cascade Lasers,” IEEE J. Select. Topics Quantum Electron., Vol. 17, 2011, pp. 38–47. [13] Vitiello, M. S., and A. Tredicucci, “Tunable Emission in THz Quantum Cascade Lasers,” IEEE Trans. on Terahertz Sci. Technol., Vol. 1, 2011, pp. 76–84. [14] Chassagneux, Y., et al., “Limiting Factors to the Temperature Performance of THz Quantum Cascade Lasers Based on the Resonant-Phonon Depopulation Scheme,” IEEE Trans. on Terahertz Sci. Technol., Vol. 2, 2012, pp. 83–92. [15] Sutherland, R. L., Handbook of Nonlinear Optics, New York: Marcel Dekker, 2003. [16] Takeya, K., et al., “Wide Spectrum Terahertz-Wave Generation from Nonlinear Waveguides,” IEEE J. Sel. Topics Quantum Electron., Vol. 19, 2013, p. 8500212. [17] Ding, Y. J., and W. Shi, “Widely-Tunable, Monochromatic, and High-Power Terahertz Sources and Their Applications,” J. Nonlinear Opt. Phys. Mater., Vol. 12, 2003, pp. 557–585. [18] Shi, W., and Y. J. Ding, “A Monochromatic and High-Power Terahertz Source Tunable in the Ranges of 2.7–38.4 and 58.2–3540 mm for Variety of Potential Applications,” Appl. Phys. Lett., Vol. 84, 2004, pp. 1635–1637. [19] Shi, W., Y. J. Ding, and P. G. Schunemann, “Coherent Terahertz Waves Based on Difference-Frequency Generation in an Annealed Zinc–Germanium Phosphide Crystal: Improvements on Tuning Ranges and Peak Powers,” Opt. Commun., Vol. 233, 2004, pp. 183–189.

6502_Book.indb 299

11/11/14 10:22 AM

300

Terahertz Metrology

[20] Nishizawa, J. -I., et al., “Spectral Measurement of Terahertz Vibrations of Biomolecules Using a GaP,” J. Phys. D: Appl. Phys., Vol. 36, pp. 2958–2961. [21] Taniuchi, T., S. Okada, and H. Nakanishi, “Widely Tunable Terahertz-Wave Generation in an Organic Crystal and Its Spectroscopic Application,” J. Appl. Phys., Vol. 95, 2004, pp. 5984–5988. [22] Jiang, Y., and Y. J. Ding, “Efficient Terahertz Generation from Two Collinearly Propagating CO2 Laser Pulses,” Appl. Phys. Lett., Vol. 92, 2007, p. 091108. [23] Lu, Y., et al., “Efficient and Widely Step-Tunable Terahertz Generation with a DualWavelength CO2 Laser,” Appl. Phys. B, Vol. 103, 2011, pp. 387–390. [24] Vijayraghavan, K., et al., “Broadly Tunable Terahertz Generation in Mid-Infrared Quantum Cascade Lasers,” Nature Communications, Vol. 4, 2013, p. 2021. [25] Lu, Q. Y., et al., “Widely Tuned Room Temperature Terahertz Quantum Cascade Laser Sources Based on Difference-Frequency Generation,” Appl. Phys. Lett., Vol. 101, 2012, p. 251121. [26] Zink, L. R., et al., “High Resolution Far Infrared Spectroscopy,” in Applied Laser Spectroscopy, W. Demtroder and M. Inguscio, (eds.), New York: Plenum Press, 1990, pp. 141–148. [27] Odashima, H., L. R. Zink, and K. M. Evenson, “Tunable Far-Infrared Spectroscopy Extended to 9.1 THz,” Opt. Lett., Vol. 24, 1999, pp. 406–407. [28] Eliet, S., et al., “Doppler Limited Rotational Transitions of OH and SH Radicals Measured by Continuous-Wave Terahertz Photomixing,” J. Molecular Structure, Vol. 1006, 2011, pp. 13–19. [29] Hindle, F., et al., “Recent Developments of an Opto-Electronic THz Spectrometer for High-Resolution Spectroscopy,” Sensors, Vol. 9, 2009, pp. 9039–9057. [30] Hindle, F., et al., “Widely Tunable THz Synthesizer,” Appl. Phys. B, Vol. 104, 2011, pp. 763–768. [31] Malcolm, G. P. A., D. A. Walsh, and M. Chateauneuf, “Non-Linear Optical Generation,” in Physics and Applications of Terahertz Radiation, Springer Series in Optical Sciences, Vol. 173, M. Perenzoni and D. J. Paul, (eds.), New York: Springer, 2014, Chapter 6. [32] Hayashi, S., et al., “High-Power, Single-Longitudinal-Mode Terahertz-Wave Generation Pumped by a Microchip Nd:YAG Laser,” Opt. Express, Vol. 20, 2012, pp. 2881–2886. [33] Minamide, H., T. Ikari, and H. Ito, “Frequency-Agile Terahertz-Wave Parametric Oscillator in a Ring-Cavity Configuration,” Rev. Sci. Instr., Vol. 80, 2009, p. 123104. [34] Kiessling, J., et al., “Pump-Enhanced Optical Parametric Oscillator Generating Continuous Wave Tunable Terahertz Radiation,” Opt. Lett., Vol. 36, 2011, pp. 4374–4376. [35] Schaar, J. E., et al., “Terahertz Sources Based on Intracavity Parametric Down-Conversion in Quasi-Phase-Matched Gallium Arsenide,” IEEE J. Select. Topics Quant. Electron., Vol. 14, 2008, pp. 354–362.

6502_Book.indb 300

11/11/14 10:22 AM



Terahertz Laser Sources301

[36] Murdin, B. N., “Far-Infrared Free-Electron Lasers and Their Applications,” Contemporary Phys., Vol. 50, 2009, pp. 391–406. [37] Barletta, W. A., et al., “Free Electron Lasers: Present Status and Future Challenges,” Nuclear Instr. Methods Phys. Res. A, Vol. 618, 2010, pp. 69–96. [38] Knyazev, B. A., G. N. Kulipanov, and N. A. Vinokurov, “Novosibirsk Terahertz Free Electron Laser: Instrumentation Development and Experimental Achievements,” Meas. Sci. Technol., Vol. 21, 2010, p. 054017. [39] Weightman, P., “Prospects for the Study of Biological Systems with High Power Sources of Terahertz Radiation,” Phys. Biol., Vol. 9, 2012, p. 053001.

6502_Book.indb 301

11/11/14 10:22 AM

6502_Book.indb 302

11/11/14 10:22 AM

11 Electronic Sources of Terahertz Radiation and Terahertz Detectors Peter Huggard 11.1  Introduction and Scope This chapter is directed at presenting electronic terahertz source and detector technologies to the nonspecialized scientist or engineer who want to assimilate quickly what technology is commercially available and what its capabilities and drawbacks are. It is not intended to summarize or even present the state of the art in research: instead, the focus will be on approaches that are available off the shelf or are sold as bespoke products or can be adopted for use in the laboratory or field instrument with minimum specialized knowledge. The reader is assumed to be someone who wants to employ terahertz radiation for a particular purpose or who wants to understand the basics of the field, rather than an existing expert. Armed with this chapter, a search engine, and Internet access, hopefully, the reader will then be able to find quickly potential technology providers and to understand and evaluate more easily individual device data sheets. Further information is available in [1]. The chapter is split into two main sections: electronic terahertz sources and terahertz detectors. Terahertz in this context will be assumed to start at frequencies around 100 GHz and extend upwards until circa 10 THz, or in free space wavelength terms, a range 3 mm ≥ λ ≥ 30 μ m. The approach will 303

6502_Book.indb 303

11/12/14 8:42 AM

304

Terahertz Metrology

be first to present briefly the key parameters used to characterize the sources and detectors. The physical principles behind each source or detector type will be summarized and their characteristics and limitations will be described. Summary tables and figures will be used to assist in the comparison of data, and illustrative references for further information will be listed. Finally, some advice on laboratory practice will be given, and indications of emerging technologies briefly discussed.

11.2  Electronic Sources of Terahertz Radiation: Introduction and Terminology Electronic sources of radiation include solid state devices, usually based on high electron mobility semiconductors, and vacuum tube emitters. In the former, an oscillating current is created in a device whose dimensions are much smaller than the free space wavelength. This alternating current is coupled into a propagating field by means of an antenna structure that has a size comparable to the wavelength. In vacuum tube sources, an electron beam is emitted by a cathode and is electrostatically accelerated. This beam propagates inside the same metal waveguide cavity as the terahertz field. Either the tube geometry and/or an applied magnetic field cause the transfer of kinetic energy from the electron beam into the terahertz field. Some sources are standalone oscillators; others function as amplifiers or frequency converters and so require suitable input signals. Both types of sources exhibit decreasing efficiency as the frequency increases, so that solid-states devices work below about ≈1.5 THz and vacuum tubes work below ≈0.5 THz. These are not hard cutoff limits, but rather indications that other approaches are more viable. Achieving a high oscillating frequency in a semiconductor device requires that the electron transit time is small, and this implies that active layers are very thin. Keeping the parasitic shunt capacitance at an acceptable value means that the device area must be small, and devices are characteristically 1 μ m or less across. This, coupled with the small thickness, limits the output power. Similarly, for vacuum electron tubes, machining the metal waveguide features with sufficiently accuracy and other factors mean that the performance of these oscillators and amplifiers drops sharply above a few hundred gigahertz. Research continues into extending the frequency coverage of both types, in particular, the efficiency and maximum output power at high frequencies. See, for example, [2]. As a third type of electronic source, terahertz emission via electron synchrotron radiation should also be mentioned. This is related to the generation of

6502_Book.indb 304

11/12/14 8:42 AM

Electronic Sources of Terahertz Radiation and Terahertz Detectors305



power in free electron laser, discussed elsewhere in this book. In a synchrotron, when the circulating electron bunch length is shorter than the wavelength, coherent emission occurs, with the power scaling as the square of the number of electrons in the bunch. This creates an intense, short-lived, broadband terahertz pulse at a high repetition rate. The spectrum is continuous in frequency up to a rapid drop, by orders of magnitude in power, above a cutoff point determined by the electron bunch length. The lowest frequency tends to be limited by diffraction and by aperture diameters in the beam pipes. From a user perspective, the disadvantage is that synchrotron beam time is expensive, and access may be limited as optimizing the terahertz emission conflicts with the beam requirements of users at the other end of the electromagnetic spectrum. Some of the parameters to consider when specifying or choosing a source are: •

•

•

•

•

•

•

6502_Book.indb 305

Output power: Is the stated output power available as continuous wave or pulsed? If pulsed, what are the turn-on and turn-off times and the pulse duration? What is the power stability or amplitude modulation noise during the on phase? What is the duty cycle? Is there any radiation emitted during the nominally off time? What is the source lifetime before failure or unacceptable deterioration? Tuning range or spectral coverage: What is the frequency tuning bandwidth? How does the output power vary with tuning? Is any tuning mechanism electronic or mechanical? What are the maximum frequency sweep rate and the linearity of the sweep? Output linewidth: For a given tuning setting, what is the output spectrum? What controls the linewidth? For narrowband sources, what controls the phase noise? What is the level of spurious (e.g., unwanted harmonic) signals? Input signal: Is this required? What are the power level and frequency? What is the interface (e.g., the connector or waveguide size and flange)? Will an antenna, for example, a feedhorn (Figure 11.1), be required? Beam shape: Is the power emitted into free space or fundamental mode metal waveguide so that a Gaussian spatial mode can be obtained by standard methods, or is a more complicated optical system required? Power conversion efficiency and temperature stability: How much heat will need to be removed? What is the accuracy requirement on the operational temperature? Is cooling by convection or forced air current adequate, or is water or cryogenic cooling necessary? Operational temperature: Room temperature or slightly above is preferred. Operating at temperatures below ambient requires a cooling

11/12/14 8:42 AM

306

Terahertz Metrology

Figure 11.1  Example of a corrugated feedhorn antenna for frequencies around 2.5 THz. The top photograph shows the sacrificial mandrel around which the antenna is formed, and the bottom scanning electron micrograph displays the 1-mm diameter aperture of the finished feed horn. The input of the feedhorn is a hollow rectangular waveguide measuring 24 μ m × 96 μ m. (Image courtesy MMT Group, STFC Rutherford Appleton Laboratory.)

•

6502_Book.indb 306

system and, more troublesome, a vacuum enclosure and windows to prevent condensation of atmospheric water vapor. Windows need care in design and material selection to minimize losses and/or to prevent standing waves. Electrical bias and magnetic field requirements: What is the electrical power consumption, in terms of volts and amps? What is the tolerance

11/12/14 8:42 AM

Electronic Sources of Terahertz Radiation and Terahertz Detectors307



•

of the device to transients and electrostatic discharge? Is a high voltage needed, or a strong magnetic field, both of which will increase risk to operators and introduce health and safety requirements (e.g., interlocks)? Lifetime: Especially important for vacuum electronic devices, what is the source lifetime? Is any change in performance with time expected?

11.3  Vacuum Electronic Sources 11.3.1  Backward Wave Oscillators The backward wave oscillator, also known as the carcinotron [1], is the most frequently encountered vacuum electronic continuous-wave source of terahertz radiation in the laboratory. In a backward wave oscillator, an electron beam propagates in a hollow metal waveguide, which has been machined with periodic internal features so that the phase velocity of the terahertz wave and the longitudinal electron velocity are matched. The electrons interact with the axial component of the electromagnetic field and a transfer of kinetic energy serves to amplify the wave. The terahertz field builds from noise: tuning of its frequency over a limited range is possible by varying the accelerating potential applied to the electrons. This is of the order of a few kilovolts, and rises with tube output frequency. Phase-locking techniques can be applied to provide feedback if a narrow line is required. The term “backward” arises as the power density in the wave grows in the opposite direction to the velocity of the electrons, so that the output port is near the cathode. The output signal is usually available to the user as the fundamental mode in a standard size of hollow rectangular waveguide, so that a feedhorn or further waveguide can be used to transport the power to the experiment. Backward wave oscillators have output power and tuning ranges that decline with frequency. Typically, they provide several tens of milliwatts over a ±20% frequency bandwidth at 100 GHz, declining to around 1 mW and ±10% bandwidth at 1.2 THz. Within the tuning range, the power depends strongly on the frequency. The maximum frequency of commercially offered units is currently around 1.4 THz. It should be noted that a longitudinal magnetic field is necessary to confine the electrons. This is generated either by a permanent magnet or an electromagnet, depending on tube design. The requirements for vacuum enclosure, high voltage, magnetic field, and cooling water have size and mass implications for backward wave oscillators. One should reckon on dimension of 50 cm and a mass of tens of kilograms, possibly with a separate high voltage (HV) and electromagnet power supply

6502_Book.indb 307

11/12/14 8:42 AM

308

Terahertz Metrology

and tube units. Either laboratory services or separate chiller units are needed for the cooling circuit. 11.3.2  Extended Interaction Klystrons Other vacuum electron sources and amplifiers of radiation at frequencies up to a few hundred gigahertz are available or under development. These include Orotrons, traveling wave tubes, reflex klystrons, and gyro-traveling wave tubes. Space limitations mean that only one device will be discussed here, the extended interaction klystron (EIK). This shares many features with the backward wave oscillator: electron gun, focusing magnetic field, copropagating electron beam and terahertz wave. In the klystron, the initially uniform electron beam passes through a series of cavities, and the field configuration is such that bunching of the electrons occurs. The bunched electrons transfer kinetic energy to an amplifier, so that the tube functions as a linear amplifier, or if feedback is introduced, an oscillator. The EIK represents a relatively compact high power source, with sufficient lifetime to have been used for a space-borne radar, operating at 0.1 THz, for cloud profiling. Powers available in pulsed mode from EIK amplifiers, operating with a low duty cycle, are kilowatts at 0.1 THz, dropping to tens of watts at 0.25 THz. Power gains are typically 40 dB and 25 dB, respectively. Although these are high, a quick calculation shows that significant input power is required to extract the full output power from the tube. Standard rectangular waveguide interfaces are used for both input and output connections. Continuous-wave devices exhibit similar gains, but deliver lower powers, a maximum of 30W at 0.1 THz, dropping below 10W at 0.2 THz. Tuning bandwidths are of the order ±10%, but the instantaneous bandwidth tends to be below 1 GHz.

11.4  Solid-State Devices 11.4.1  Gunn Diodes and IMPATT Oscillators and Amplifiers Although these semiconductor sources operate on different physical principles, from the user’s perspective they are very similar. Engineered two-terminal semiconductor devices, exploiting the Gunn [3, 4] or impact ionization transit time effects [4], convert power from a low-voltage bias supply into an oscillating electromagnetic field. The GaAs, InP, or Si semiconductor devices are embedded in a resonant cavity, and terahertz power is emitted as the fundamental

6502_Book.indb 308

11/12/14 8:42 AM



Electronic Sources of Terahertz Radiation and Terahertz Detectors309

mode in a standard hollow rectangular waveguide. Either the fundamental oscillation frequency—or its second harmonic—provides the narrowband output. Frequency tuning and optimization of the output power are achieved by mechanically varying the effective size of the cavity, and by changing the bias applied to the device or a separate capacitance in the cavity. This voltage tuning process can also be used to achieve phase locking of the output frequency. Continuous wave Gunn and IMPATT sources are commercially available as fixed frequency and tuneable oscillators and amplifiers. When used as injection-locked amplifiers, waveguide circulators can be used to couple power to and from the device. Waveguide power combining can be used to link two devices in parallel to provide higher powers. Research continues into improving the output power of these devices up to frequencies approaching 0.5 THz [5], as there is a strong decrease in power with frequency, approximately proportional to (frequency)–3: Figure 11.2. At the time of writing this chapter, they were only commercially available at frequencies close to 0.1 THz, where maximum output powers are of the order of 100 mW. These devices are

Figure 11.2  Strong frequency dependence of output powers from a range of Gunn effect devices. Maximum powers of a few hundred milliwatts at 100 GHz drop to less than 1 mW at 500 GHz. (Image courtesy of H. Eisele [6].)

6502_Book.indb 309

11/12/14 8:42 AM

310

Terahertz Metrology

compact, robust, and reliable and operate in air or vacuum with the cavities at room temperature. They are often used as local oscillators for heterodyne receivers and as drivers for the frequency multipliers described next. These use nonlinear processes in semiconductor diodes to provide second or third harmonics of the input frequency at power conversion efficiencies of up to 30%. Pulsed IMPATTs amplifiers and oscillators are capable of providing much higher powers over the same frequency range, a few tens of watts at 100 GHz, dropping to 20W at 140 GHz. Pulse widths are of the order of 100 ns, and repetition rates are tens of kilohertz, so that duty cycles are around 1%. 11.4.2  Transistor Amplifiers In recent years, advances in transistor technology have made MMIC-based amplifiers for low terahertz frequencies a realistic prospect. Sub 100 nm transistor dimensions in, among others, CMOS and InP-based HEMT technologies have yielded impressive performances in the frequency range from 0.1 THz to nearly 0.7 THz [7]. It is expected that the prototype devices reported in the literature will eventually find their way into commercially available components. The current situation in terms of transistor-based power amplifiers is that they are a viable alternative to Gunn and IMPATT devices at frequencies around 100 GHz. Dimensions, power requirements, and output powers are similar: Figure 11.3 shows the external view of a typical device. This approach starts with a narrowband signal around 30 GHz (e.g., from a dielectric resonator oscillator). The signal is frequency multiplied to around 100 GHz, and the available power is increased by a transistor amplifier. This can offer more flexibility than the Gunn oscillator in terms of signal frequency agility. A typical amplifier MMIC contains a few transistors along with impedance matching and bias components. An externally regulated and decoupled low voltage bias is supplied to the transistors. The MMIC is usually packaged in a metal cavity with waveguide ports, and terahertz power is coupled to and from the transmission lines on the chip by waveguide probes. Available output power is usually a compromise with tuning range, and as with Gunn sources and IMPATTs, combining outputs from two and four MMICS can be used to increase the available power. Specific examples of the technology capabilities include narrowband power-combined GaN units yielding 3W just below 0.1 THz, and single devices offering nearly 30 mW in a ±15% bandwidth to 0.13 THz. At the time of this writing, the current commercial availability of transistor amplifiers at higher frequencies was limited, and the intending user is advised to check with manufacturers.

6502_Book.indb 310

11/12/14 8:42 AM



Electronic Sources of Terahertz Radiation and Terahertz Detectors311

Figure 11.3  External appearance of a 0.1- to 0.13-THz transistor amplifier. Saturated output power is close to 30 mW across this band. One of the two standard rectangular waveguide interfaces can be seen on the lower right of the photograph. (Image courtesy of Radiometer Physics GmbH.)

11.5  Frequency Multiplied Sources Frequency multiplication offers a method to transfer the sophisticated signal control available at frequencies below 100 GHz to the terahertz region, that is, narrow line width, amplitude, phase, and frequency modulation, and arbitrary waveform generation (e.g., chirping). Multipliers use a nonlinearity in a semiconductor device to generate harmonics of the instantaneous input signal: the input signal can modulate the resistance or capacitance to form a varistor or a varactor multiplier, respectively [8].1 The devices are typically planar GaAs Schottky diodes or heterostructure barrier varactors. In both cases, the construction principles are similar: the semiconductor devices are mounted on microstrip circuits in metal waveguide cavities. 1

Varactor-based multipliers exploit the nonlinear dependence of the reverse biased diode’s capacitance on the input signal to generate the higher harmonics. Varistor devices similarly depend on a nonlinear behavior of the forward-biased resistance.

6502_Book.indb 311

11/12/14 8:42 AM

312

Terahertz Metrology

Tuning elements and bandpass filters, respectively, match impedances to that of the rectangular waveguides for maximum power transfer and prevent the propagation of undesired harmonics. Construction is achieved by soldering devices to dielectric substrates that have the patterned microstrip circuitry or by preparing self-supporting membranes with metal conducting regions. High power handling is accomplished by using arrays of diodes in combination with high thermal conductivity substrates. Most multipliers operate as either frequency doublers or triplers: generation of a single higher harmonic of a given tone is generally accomplished by cascading two times and/or three times multiplication elements, rather than using a higher order of multiplication in one device. Power conversion efficiencies tend towards a trade-off between bandwidth and efficiency, with frequency doublers more efficient than triplers. Best power conversion performances can be close to 40% in a low-frequency narrowband doubler, with a tripler covering a waveguide band2 typically achieving 10%. The highest power devices can accept an input power above 1W at frequencies below 100 GHz. Efficiency, power handling, and output power fall with frequency, as the diodes get smaller to reduce capacitance and the structures get harder to make to meet tolerances. The effect is that a power close to 1 mW is feasible at 1 THz and circa 10 μ W at 2 THz. Greater efficiency and output power can be obtained by cooling the multipliers to cryogenic temperatures: this is particularly relevant for generating frequencies above 1 THz. Traditionally, ×2 and ×3 multipliers generating local oscillator signals were driven by Gunn diode oscillators providing tens of milliwatts at frequencies around 100 GHz. Phase lock loop techniques can be used to stabilize the frequency of the Gunn diode. Recently, transistor power amplifiers operating around 100 GHz have become available, so it is common to start with a signal of a few tens of gigahertz from a synthesizer or dielectric resonator oscillator and successively amplify and frequency-multiply this until the desired frequency and power level is reached. This offers advantages in frequency sweeping and stability, respectively. Higher orders of multiplication from single devices are used for calibration signals and phase or frequency locking applications. Power levels and efficiency are of less concern here, and such multipliers can be driven directly from synthesizers. 2 Hollow rectangular waveguides have a useful single mode frequency range limited by the cutoff frequency of the fundamental mode and the onset of higher-order modes, the upper limit being about 50% higher than the lower (see also Chapter 8). The coverage of the spectrum from below 0.1 THz to above 2 THz is thus divided into a set of overlapping bands [8] and corresponding agreed waveguide dimensions. Within each band, the appropriate waveguide transmits the fundamental mode with low loss.

6502_Book.indb 312

11/12/14 8:42 AM



Electronic Sources of Terahertz Radiation and Terahertz Detectors313

Figure 11.4  Photograph of a typical frequency multiplier. The SMA connector supplies bias or the input signal. Terahertz power is emitted through the rectangular waveguide port. (Image courtesy MMT Group, STFC Rutherford Appleton Laboratory.)

Multipliers’ size and mass are dominated by the hollow rectangular waveguide interfaces. With a standard UG387/m flange diameter of 19 mm, multipliers are typically housed in a cuboid block of a side ≈ 25 mm (Figure 11.4). This also gives ample space for attachment of a dc bias connector. Arrays of free space multipliers have been studied, but are not a commercial product. Spectral purity of the output signal needs to be determined when using multipliers. Waveguide cutoff can be used to eliminate the second harmonic from the output of a tripler, but care needs to be taken to keep the third (fourth) and higher harmonics from a doubler (tripler) to acceptable levels. Any unwanted tones in the input signal may also combine to produce an output, and input power can leak from the waveguide flanges and reach the detector. Amplitude modulation noise is also a potential problem. Finally, one should be aware of other nonlinear frequency conversion devices, such as upconverters. Often forming part of the component chain used to drive a multiplier, upconverters emit the sum of two input frequencies. This enables the creation of essentially arbitrary terahertz waveforms. For example, a 2 GHz signal from a digital frequency synthesis source can be added to a 30 GHz signal from a dielectric resonator oscillator, and the sum successively amplified and frequency multiplied into the terahertz region.

11.6  Source Characteristics Summary Table 11.1 summarizes the properties of the electronic terahertz sources covered in this short review.

6502_Book.indb 313

11/12/14 8:42 AM

6502_Book.indb 314

Power and Frequency

30W at 0.1 THz, 10W at 0.14 THz

3W at ≈ 0.1 THz

0.1W at 0.1 THz, 1 mW at 1 THz

Pulsed IMPATT

Transistor amplifier

Frequency multiplier

±20 or less

±15 % or less

± 1% or less

2.5

10

10

2.5

0.1W at 0.1 THz, 2 mW at 0.2 THz

Gunn Diode, IMPATT

± 2% or less

20

±20 at 0.1 THz, ±10 at 1.3 2 kW at 0.1 THz, 10W THz at 0.25 THz, pulsed. Continuous-wave powers are lower.

Extended interaction klystron

Size (cm) 50

Tuning Bandwidth (%) ±20 at 0.1 THz, ±10 at 1.3 THz

Backward 50 mW at 0.1 THz, 0.5 μ W wave oscillator at 1.4 THz

Source Type

20g

200 g

200g

50g

5 kg

20 kg

Mass

Needs input signal. Choice of doublers and triplers: trade-off between bandwidth and efficiency.

Needs input signal. Rapidly evolving continuous-wave technology. Low-voltage, ambient operation, currently low frequency only.

100 ns high power pulses. Low duty cycle, low voltage, ambient operation, low frequency only

Standalone continuous-wave amplifier or oscillator, low voltage ambient operation, low frequency only.

Standalone amplifier or oscillator requires HV supply, magnetic fields, and coolant.

Standalone oscillator requires HV supply, magnetic fields, and coolant.

Comment

Table 11.1 Characteristics of Electronic Terahertz Sources

314 Terahertz Metrology

11/12/14 8:42 AM

Electronic Sources of Terahertz Radiation and Terahertz Detectors315



11.7  Detectors of Terahertz Radiation: Introduction and Terminology This treatment of detectors will cover devices used in the laboratory and field to characterize new sources for radar, imaging, process control, and spectroscopy. It will briefly touch on the devices used to characterize molecular emissions and absorptions in the Earth’s atmosphere and will indicate the specialized devices used in low background radio astronomy. Detectors are grouped into incoherent and coherent devices. The terms video, power, and incoherent are interchangeable and indicate that the detector is a power-to-signal transducer. Coherent detectors, also known as heterodyne devices, yield information on the power, frequency spectrum, and phase of the incoming signal. Although there is in principle considerable overlap, as any detector producing a response proportional to the square of the incident field strength can function in both power and heterodyne modes, in practice the Schottky barrier diode is the only component used in both groups. The key incoherent detector parameters include: •

•

•

•

•

6502_Book.indb 315

Spectral coverage or predetection bandwidth: The maximum and minimum frequencies that can be detected and frequency dependence between them. This may be determined by the detection principles or by the mounting structure (e.g., a waveguide mount with its lowfrequency cutoff). Polarization response: The detector may respond predominantly to a single linear or circular polarization to a polarization that rotates with frequency or exhibit no polarization dependence. Responsivity: Specified as the signal per unit incident power, usually quoted in units of V/W. Low responsivity can usually be corrected by following the detector with a low noise preamplifier. Noise equivalent power (NEP): Generally much more important than responsivity, the NEP is the input signal level that gives a response equal to the root mean square noise voltage. It is quoted in units of W/√Hz, as noise generally scales as the square root of the post detection bandwidth. Postdetection bandwidth: This may be equivalently specified as response time. This is determined by the variation in signal amplitude as the source is amplitude-modulated or chopped at different frequencies. The lowest usable frequency may not be dc, as detectors are likely to be ac coupled or to depend on a mechanism that gives no steadystate response. The upper frequency cutoff may be determined by the

11/12/14 8:42 AM

316

Terahertz Metrology

•

•

•

detector’s RC time constant, or the preamplifier gain. Ideally, there is a rectangular frequency response between these two cutoff points, but this may not be the case (e.g., for pyroelectric detectors operating in the more sensitive voltage mode). A uniform frequency response also leads to a more accurate representation when the detector is used to observe short pulses. Here the response time determines the decay after a delta function stimulus. Dynamic range: The ratio between the lowest observable signal, which is set by the NEP, and the onset of saturation, where the response begins to deviate from being linearly proportional to the input power. Dynamic ranges are typically several orders of magnitude and are often quoted in decibels. Operating temperature: Terahertz detectors generally either operate at room temperature or at liquid helium, 4.2 K and below, temperatures. Generally, lower NEPs can be achieved by using cryogenically cooled detectors. This needs to be set against the extra effort in evacuating the detector cryostat; the cost, associated labor, and safety aspects of cryogens; and the need for one or more windows in the signal path. Input beam capability: How big is the sensitive area? Does the detector respond to a single mode of a Gaussian beam? Does it have a waveguide input so that a feedhorn antenna is needed? Does it accept several spatial modes and condense these onto the detector area with a Winston cone?3

One final comment on choosing a detector is that it is possible to have a detector that is too sensitive. It is necessary to consider the radiation that the detector will “see”: this is set by the detector acceptance angle and the temperature and emissivity of objects in the field of view. Inherent temporal fluctuations in the associated emitted thermal radiation set a lower limit on the detector NEP: in this case, the detector is said to be background limited in power (BLIP). Thus, the ultrasensitive bolometers developed for looking at weak signals from the cool universe [9, 10], which operate at millikelvin temperatures and require cooled optical systems, would be overwhelmed and saturated in a laboratory environment, where they are needed to characterize a typically much brighter source and background. A detailed consideration of 3 Winston cones [10] are reflective, nonimaging, optical devices used to collect terahertz radiation and concentrate it onto an incoherent detector. Shaped like an elliptic parabaloid with the tip cut off, the area occupied by a terahertz beam is reduced at the smaller output end at the expense of an increased angular divergence.

6502_Book.indb 316

11/12/14 8:42 AM



Electronic Sources of Terahertz Radiation and Terahertz Detectors317

the background noise limitations and capabilities of thermal and heterodyne detectors, operating at different temperatures and observing different backgrounds, has been made [11].

11.8  Incoherent Detectors 11.8.1  Golay Cell The Golay cell has had a distinguished role in the development of terahertz spectroscopy over more than half a century [12]. Essentially a room temperature bolometer with a sensitive optical sensing mechanism, its uniform broadband response meant that it has been used as a quasi-reference standard in a field where few standards exist. An image of a modern detector is shown in Figure 11.5. Terahertz radiation propagating in free space enters the detector through a window circa 5 mm in diameter. A fraction of the power is absorbed in a thin metal layer. Given the broadband infrared absorbing properties of thin metal films [13], the window material determines the device’s spectral response. Diamond windows cover millimeter-wave to visible frequencies, and single crystal quartz works well below 3 THz and at visible frequencies. High-density polyethylene has very low loss below 1 THz, and the absorption then rises very slowly with frequency, so that the upper limit is close to 20 THz. The thermal energy deposited in the foil heats and raises the pressure of gas in a cell, one wall of which is a flexible mirror. An optical lever system images the

Figure 11.5  Golay cell mounted on a vibration absorbing baseplate. The detector window is the light colored disc, towards the top right of the unit. (Image courtesy Microtech Instruments, Inc.)

6502_Book.indb 317

11/12/14 8:42 AM

318

Terahertz Metrology

shadow of a linear grating onto an inverted replica by reflection in the mirror. Deformation of the flexible mirror moves the image and allows visible or near infrared light to pass to a photo receiver. This arrangement gives an excellent NEP for a room temperature device, approximately 10 –10 W/√Hz. As well as sensitivity and broadband coverage, one further advantage is that the response is not polarization dependent. However, the mechanism reaction time is very slow, with a bandwidth around 10 Hz, so data acquisition times can be long. Although the sensing cell is small, a few centimeters across, the associated optical lever system is rather bulky, so that the device dimensions are approximately 5 × 10 × 15 cm3. A separate low-voltage power supply is also required. Another consideration is that the supply and repair facilities for these devices can be uncertain. At the time of this writing, there were at least two vendors, but this has not always been the case. The units are fragile: mechanical shock or, as this author can testify, too high pulsed laser power can destroy the thin absorbing film. They need to be isolated from vibrations and acoustic noise. Finally, the deformable mirror appears to be prone to aging, and the detectors can be saturated by too much power, so that the linearity of a given Golay cell detector should be verified before use [14]. 11.8.2  Pyroelectric Detectors Pyroelectric detectors are probably the most commonly used room temperature detectors of terahertz radiation. Their operation exploits the temperature dependence of the permanent electric dipole moment exhibited by certain insulating crystals. The most commonly used materials are lithium tantalate (LiTaO3) and deuterated, L-alanine doped, triglycine sulphate (DLaTGS). The crystals are cut into thin discs, ranging in diameter from 1 to 5 mm, and surface electrodes are deposited. These electrodes, or the bulk of the crystal, function as the absorber for the incoming terahertz radiation: when the temperature changes, a potential difference is established between the top and bottom surfaces of the disc. The detector is essentially a capacitor, and the combination of this capacitance and the load resistor determine the speed of response. In use the detector is shunted by the load resistor and buffered by a discrete JFET preamplifier (called a voltage mode) or connected to an operational amplifier where the load resistor provides the feedback for the op-amp: current mode. In both cases careful choice of low noise components is necessary. Voltage mode operation yields the lowest NEP, but the detector responsivity then has a strong modulation frequency dependence. Current mode gives a response that does not depend on frequency, at least up to the

6502_Book.indb 318

11/12/14 8:42 AM



Electronic Sources of Terahertz Radiation and Terahertz Detectors319

point determined by the gain bandwidth of the op-amp. Typical postdetection bandwidths are from a few hertz to 10 kHz, with corresponding NEPs in the 10 –10 to 10 –9 W/√Hz range. The very highest response speeds, with subnanosecond response times, are obtained by dispensing with the semiconductor devices and using the detector to drive a 50Ω load directly. Single detectors are typically packed in metal transistor cans with or without windows, and thus are very compact, with sub 1 cm dimensions. The housing may just serve to protect the detector, or may include a buffer amplifier. The addition of a specifically designed preamplifier to a detector is straightforward. Devices adapted for high average power laser applications are available, with enhanced heat sinking from the crystal to the enclosure. The polarization independent spectral response is governed by the absorber and the choice of window material: some types of devices are known to behave as resonant cavities, which gives a periodic wavelength modulation to the responsivity. This is one drawback compared to the Golay cell; otherwise, the pyroelectric detector is nearly a drop-in replacement with advantages including small size, robustness, and a higher optical damage threshold. Finally, for beam profiling, cameras incorporating two-dimensional arrays of pyroelectric elements are available. Individual element sizes are 25 μ m × 25 μ m or less, and the trade-off in NEP is about one order of magnitude compared to the single elements discussed above. 11.8.3  Room-Temperature Bolometers The most general definition of a bolometer is a radiation absorber attached to a temperature sensor with the combination thermally coupled to a heat sink. Incoming terahertz radiation heats the absorber; electronically monitoring the resulting rise in temperature provides the detection signal. The Golay cell mentioned above is thus a form of bolometer. Optimizing the response means minimizing the heat capacity of the absorber, the thermometer, and the thermal conductance. The detector’s time constant also depends on these quantities [1]. The radiation absorbing and temperature readout functions can be combined, when a material with a large temperature coefficient of resistance is used as the absorber. Alternatively, in a composite bolometer, a thin metal film on a high thermal conductivity dielectric substrate forms the absorber, and a small thermistor monitors the temperature. Absorbers can be made very broadband in frequency terms, and this is a significant advantage of the approach. Room-temperature bolometers are more widely used at super-terahertz frequencies, as the blackbody radiation curves of typical emitters mean that

6502_Book.indb 319

11/12/14 8:42 AM

320

Terahertz Metrology

more incoming radiant power is available to compensate for the modest sensitivity of the detectors, and semiconductor processing approaches mean that two-dimensional arrays with some thousands of elements along a side are practical. For terahertz purposes, larger element sizes with dimensions approaching the free space wavelength are needed. At least one manufacturer offers a vanadium oxide-based two-dimensional array, with about 300 elements on a side, for imaging applications. The time constant is around 10 ms, and so moderately high frame rates are possible in imaging applications. 11.8.4  Cryogenic Bolometers Cryogenic bolometers, based on a number of different physical principles, are widely applied as terahertz detectors in the laboratory. Operation is typically at temperatures of a few Kelvin, and is accomplished by liquid helium or closed cycle mechanical coolers. Wet cryostats usually have capacities for a few liters of liquid nitrogen and helium, and their dimensions are about 0.3m in diameter and 0.5m in height. Radiation enters the vacuum insulated cryostat through a window that has low loss at terahertz frequencies, and is focused onto the detector by a condensing Winston cone (Figure 11.6). The detector is sometimes mounted in an integrating cavity to improve performance. Cooled windows and/or long wavelength transmitting filters are inserted into the beam path to block unwanted infrared radiation: fluctuation in this adds noise to the detected signal and the incoming power can even saturate the detector. Thus, limiting the spectral bandwidth to the minimum required by the measurement may be necessary to ensuring maximum dynamic range. It is noted that the ultrasensitive bolometers developed for radio astronomy are not useful in the laboratory, as they would be saturated by the stray background radiation accompanying any signal, even from a cryogenic source. Cryogenic composite bolometers typically have a metal film absorber deposited onto a thin diamond or silicon nitride substrate. Narrow legs of the same material provide a controlled thermal conductance to the mount. A miniature doped silicon thermistor or a superconducting metal strip held at a temperature corresponding to the midpoint of its resistive transition is used to monitor the temperature of the absorber. A low noise preamplifier and bias circuit is connected to the detector. The system’s speed of response is usually traded off against sensitivity: typically postdetection or modulation bandwidths are a few hundred hertz, and NEPs are in the 10 –12 to 10 –14 W/√Hz range. A faster bolometer uses the free carriers in a piece of doped InSb crystal as the absorber. In this InSb hot electron bolometer (HEB), incident-long wavelength terahertz radiation is absorbed by and heats the electron gas. As

6502_Book.indb 320

11/12/14 8:42 AM



Electronic Sources of Terahertz Radiation and Terahertz Detectors321

Figure 11.6  External view, left, of a liquid helium cooled cryogenic detector cryostat. Terahertz radiation enters through the window situated at the bottom center of the instrument. Right: Internal view of the cold plate of a different cryostat with a linear array of InSb hot electron bolometers coupled to incoming radiation by Winston cones. (Images courtesy of QMC Instruments Ltd.)

the conductivity of this gas is proportional to its temperature, measuring the resistance of the InSb provides the detection signals. Manufacturers variously quote the spectral sensitivity as extending from below 0.1 THz to either 0.5 or 1.5 THz, with a postdetection bandwidth close to 1 MHz, and an NEP below 10 –12 W/√Hz. In terms of external appearance and optical systems, the detectors are very similar to composite cryogenic bolometers. The performance of the InSb HEB can be enhanced if the detector is subject to an applied magnetic field. The free carriers then resonantly couple to the incident terahertz radiation by the cyclotron resonance mechanism. A homogenous magnetic field can be used to increase the detector’s sensitivity in a 0.3 THz wide region between 1.5 and 2.5 THz. Alternatively, a nonuniform field can extend the low frequency responsivity up to about 1.5 THz. The magnetic field is created using permanent magnets inside the detector cryostat. NEP and response times for the cyclotron resonance InSb detector are similar to those of the standard InSb HEB. A third kind of hot electron bolometer is based on the superconductor NbN. A nanometer-scale bridge of superconducting NbN, linking two arms

6502_Book.indb 321

11/12/14 8:42 AM

322

Terahertz Metrology

of a planar antenna, performs the detection. Incident radiation heats the electron gas in this bridge and a readout circuit measures the impedance change of the bridge. A hyperhemispherical contact lens focuses incident terahertz radiation onto the antenna. A range of responsivities and bandwidths are offered; for example, one device covers from 1 to 12 THz, with an NEP below 10 –10 W/√Hz and a postdetection bandwidth of a few gigahertz. Again, the external appearance and cryogenic requirements are similar to the composite bolometers and InSb detectors mentioned earlier. 11.8.5  Photoconductive Detectors A range of cryogenically cooled inorganic semiconductors have also been used as photoconductive terahertz detectors. Impurity levels close to the conduction band are ionized by the terahertz radiation, and the resulting free carriers increase the conductivity of the material, leading to a detection signal. Mounting, electronic, and optical systems for the crystals are similar to those of the composite bolometer described earlier. The range of useful crystals includes ultrapure epitaxial GaAs, and germanium variously doped with gold, zinc, and gallium. The spectral responsivity of all these photoconductors is peaked at a location depending on the dopant. The response curve exhibits a sharp low-frequency cutoff and a gradual decline as the frequency increases. The detector with the lowest lying peak is germanium doped with gallium, Ge(Ga), which has a useful spectral range from 2.4 THz to around 8 THz. Application of uniaxial stress to the germanium crystal can be used to shift the response to lower frequencies, from approximately 1.5 to 6 THz. Response speeds are below 100 kHz, and the NEP is around 10 –12 W/√Hz. Ge(Ga) appears to be the only commercially available photoconductive terahertz detector at the time of this writing. 11.8.6  Semiconductor Diode Rectifiers This term is used to encompass micron and submicron diameter GaAs and InGaAs Schottky diodes that produce an output signal dependent on the instantaneous incident terahertz power. It also includes InAs/GaAlSb diodes. The devices are engineered to have a nonlinear I(V) characteristic, application of an oscillating potential at terahertz frequencies results in a net dc signal. The optimum operating point on the I(V) curve is reached by applying an external forward bias, or in the more recently developed zero bias detectors, is designed to be at the origin. The zero bias detectors are potentially lower

6502_Book.indb 322

11/12/14 8:42 AM



Electronic Sources of Terahertz Radiation and Terahertz Detectors323

noise devices, being free of the shot noise contribution associated with the microampere to milliampere bias current in biased devices. As the GaAs and InGaAs diodes are similar to the devices used in frequency multipliers and Schottky mixers, achieving best performance requires solving the same design challenges: minimizing device capacitance and mounting structure capacitance and inductance, and matching the device impedance to that of the antenna. As the devices are small, response times are very short, and the frequency response typically exceeds that of any subsequent preamplifier. As with the Schottky diode-based terahertz sources discussed earlier, performance rolls off with frequency, with a useful responsivity range for the room temperature technology extending from below 0.1 THz to around 3 THz. Devices currently available exhibit typical postdetection bandwidths from a few hertz to above 1 GHz, with NEPs in the 1 × 10 –12 to 5 × 10 –11 W/√Hz range at 0.1 and 1 THz, respectively. Schottky devices were originally produced as an array of metal anodes on the surface of a small GaAs chip: the end of a ≈100-μ m-diameter wire was chemically sharpened and brought into contact with one of the anodes. The whole was mounted in the corner of a gold plated metal cube which gave directivity to the antenna pattern, or in a waveguide cavity. The corner cube approach has been largely superseded by planar diode technology. In the simplest approach, the diode is fixed across the terminals of a planar metal antenna patterned on a dielectric substrate. As the antenna radiates (receives), predominantly into (from) the substrate, a lens is placed in contact with the reverse of the substrate to help concentrate the incoming beam onto the antenna. Typically the lenses are close to a hemisphere in shape, a few millimeters to 1 cm in diameter, and made of low loss polymer, quartz, or high-resistivity silicon. Wires soldered or epoxied to the antenna convey the output signal to an SMA or similar connector. The frequency and polarization response are determined by the antenna design and device size; typically, the response to a single linear polarization is at least an octave in frequency and sensitivity does not abruptly roll off outside this range. Alternatively, the diode detector can be mounted in a rectangular waveguide cavity. In this case, it is designed to couple to radiation in the fundamental waveguide mode, so the minimum:maximum frequency range ratio will be about 1:1.5; it will be sensitive to a single linear polarization, and a feedhorn antenna will be needed to couple efficiently the radiation from the waveguide into a Gaussian beam in free space. In summary, when detection of a single mode of the radiation field is required at low terahertz frequencies, where the radiation spectrum is narrow and is known beforehand, and perhaps a very wide postdetection bandwidth

6502_Book.indb 323

11/12/14 8:42 AM

324

Terahertz Metrology

is needed, or when robustness and compact size are important, modern roomtemperature diode detectors in either open structure or waveguide form are strong candidates. 11.8.7 Thermopiles Based on what is variously known as the Seebeck, Peltier, or thermoelectric effect, thermopiles convert a thermal gradient created by the absorption of incident radiation into an electrical signal. Devices useful for terahertz radiation consist of a plate with a radiation absorbing coating that is connected to a heat sink by a series array of thermocouples. Thermopile detectors operate at room temperature and can have NEPs approaching those of pyroelectric detectors; the radiation bandwidth is similarly determined by the absorbing properties. Unlike pyroelectric detectors, they provide an output signal in response to a continuous-wave illumination. The main disadvantage is response time; large thermal time constants mean that the bandwidths range from fractions of 1 Hz to a few hertz. For this reason thermopiles are best suited to applications such as source power measurement, where accuracy is more important than speed of response. The detectors are typically available in two types. The first, a large area, 25 mm diameter disc absorber using what is similar to a Peltier cooler operating in reverse as the detector, is specifically designed for laser power measurement. The absorber is exposed to the air. This detector offers high power handling and robustness, at the expense of sensitivity and response time. The second type uses a thin film absorber, with a diameter of a few millimeters, coupled to an array of micromachined thermocouples. The absorber is suspended in a low thermal conductivity gas and radiation enters through a window that is typically a high resistivity silicon plate. The choice of gas affects the responsivity, electrical bandwidth, and NEP. This type is much more sensitive, with a NEP ≈ 10 –9 W/√Hz, but the slow thermal response means that the bandwidth is of the order of a few Hz. The wavelength dependence of the responsivity is determined by the properties of the absorber, and the transmission of the window: standing waves in this can potentially lead to variations in response. As the signal depends on a temperature difference, maintaining the heat sink at a constant temperature or monitoring its drift may be needed. Thermally insulating enclosures and integrated thermistors may be respectively used for this purpose. 11.8.8  Incoherent Detector Summary Table 11.2 summarizes the properties of the incoherent detectors covered.

6502_Book.indb 324

11/12/14 8:42 AM

6502_Book.indb 325

NEP — (W/√Hz)

10 –10

10 –10 to 10 –9

10 –12 to 10 –10

10 –9

10 –12 to 10 –14

10 –12

10 –12

10 –10

10 –12

Detector Type

Golay Cell

Pyroelectric

Semiconductor diode

Thermopile

Composite bolometer

InSb hot electron bolometer, B = 0

InSb hot electron bolometer, B > 0

NbN hot electron bolometer

Photoconductive Ge(Ga) 2.4–8 to 1.5–6

1–12

>0.1–2.5

>0.1–≈1

0.1–30

0.1–10

0.1–3

0.1–10

0.1–10

Coverage (THz)

≈5 × 10 4

≈5 × 10 9

≈5 × 10 5

≈10 6

10 3

1

> 10 9

10 to >10 8

10

50

50

50

50

50

1

2.5

≈1

≈10

4

4

4

4

4

300

300

300

300

Bandwidth (Hz) Size (cm) T (K)

The only cryogenic photoconductor commercially available. Spectral response tuning is by applying stress.

Cryogenic. A range of models offer different spectral coverage, speed and NEP.

Cryogenic. Cyclotron resonance used either to improve sensitivity in a narrow band, or to higher frequencies. Fast speed of response.

Cryogenic, sensitive only at long wavelengths. Fast speed of response.

Cryogenic, broadband, sensitive. Moderate speed of response.

Broadband, slow. Laser power meter versions available.

Robust, compact, and very fast. NEP increases with frequency. Individual devices operate best over 2:1 frequency range.

Robust and compact: modest cost. Responsivity depends on wavelength. Generally needs preamplifier. Two-dimensional arrays for beam profiling also available.

Broadband workhorse, slow, often used as reference, note fragility and possible nonlinearity.

Comment

Table 11.2 Summary of the Characteristics of Commercially Available Incoherent Terahertz Detectors

Electronic Sources of Terahertz Radiation and Terahertz Detectors325

11/12/14 8:42 AM

326

Terahertz Metrology

11.9  Coherent Detectors: Introduction The heterodyne detector parameter set includes spectral coverage, polarization response, dynamic range, operating temperature, and input beam capability. The nature of the detector process is that the incoming signal beats with a stronger local oscillator (LO) signal in a nonlinear device to generate a frequency downconverted output, usually at a frequency below 20 GHz. This signal is then processed using more readily available low-frequency spectrometers, power detectors, and so forth. Coherent detectors tend to respond to a single-mode, single-polarization input signal. Optimising the performance of coherent detectors requires achieving impedance matches of the detector at signal, local oscillator, and downconverted frequencies. This means that the spectral coverage tends to be more limited than incoherent detectors. Specific factors needed to specify a heterodyne detector include the following: •

•

•

•

•

6502_Book.indb 326

Radio frequency (RF) bandwidth: The range of frequencies where the desired performance can be achieved or will not have deteriorated by more than 3 dB. Intermediate frequency (IF) bandwidth: The instantaneous down­ converted output frequency range. Depending on the down conversion mechanism, this can extend from dc or close to dc to several tens of gigahertz. Conversion efficiency: Sometimes expressed as conversion gain or conversion loss, this is the ratio between incoming signal power and the output power at the corresponding downconverted frequency. Local oscillator requirement: The LO power, frequency, tuning range, and interface mechanism (e.g., waveguide and flange) need to be identified. Mixers can be fundamental, with the LO frequency approximately equal to the incoming frequency; or subharmonic, where the LO frequency is a fraction of the signal frequency. In this case, a harmonic of the LO is generated in the mixer device and beats with the incoming power to achieve the downconversion. Sideband response: How does the mixer respond to signals a given frequency above and below the LO frequency, assuming a fundamental mixer? Are these components “folded” on top of each other in the IF output, as in a double sideband (DSB) mixer? This means that there is ambiguity in the downconverted output as regards input frequency and power. Is one component suppressed, as in a single sideband (SSB)

11/12/14 8:42 AM

Electronic Sources of Terahertz Radiation and Terahertz Detectors327



• •

mixer? Or does the device have separate outputs for upper and lower sidebands, in a sideband separating or image rejection mixer? Impedance match at ports: This determines how much signal is reflected, which can lead to standing waves and loss of efficiency. Postdetection IF gain: Detecting very weak signals means that large, that is, tens of decibels, of IF amplification may be required after a mixer. Careful selection of the IF amplifiers’ noise figures, gain, impedance match, susceptibility to interfering signals and temperature stability is necessary to obtain best performance from the receiver.

11.9.1  Schottky Diode Mixers The only readily available technology for coherent detection of signals at low to medium terahertz frequencies uses room-temperature GaAs or InGaAs Schottky diodes (Figure 11.7). Although superconducting devices based on tunnel junction or hot electron bolometer mixers offer superior sensitivity and perhaps operation at a higher frequency, these are not commercial products, but bespoke cryogenic devices developed for specific radio astronomical and cosmological applications. Schottky mixers are internally and externally physically similar to the incoherent diode detectors described above; they usually contain a waveguide cavity and a detector that is sensitive to a single-mode, single-polarization input. An alternative approach mounts the diode and a contacting wire antenna in a corner cube reflector. The assembly usually has three ports: for the RF signal, the LO, and the downconverted mixing product or IF. RF and LO ports are usually hollow rectangular waveguide, and IF signal connections are usually via SMA or their higher-frequency analog connectors. Mixer performance is usually specified in terms of conversion loss. Typically IF powers are a factor of 10 lower than the corresponding RF signal, and significant IF amplification may be necessary to bring the signal to a level where it can be analyzed or detected. Suitable low noise IF amplifiers with bandwidths from a few hundred megahertz to more than 10 GHz are available from several suppliers; the bandwidth should be chosen to match the signal being observed. Provision of a suitable LO signal is key to achieving mixer performance. This may be within the IF bandwidth of the RF for a fundamental mixer, close to half the RF in a subharmonic mixer, or at an integer fraction of the RF in a harmonic mixer. Best performance in terms of conversion loss and system noise is obtained from fundamental mixers, but there is the difficulty of generating adequate LO power, typically milliwatts, at frequencies above

6502_Book.indb 327

11/12/14 8:42 AM

328

Terahertz Metrology

Figure 11.7  Scanning electron micrograph of an air-bridged GaAs terahertz diode subharmonic mixer circuit, which contains two Schottky diodes in an antiparallel configuration. The 1 μ m diameter diodes are located under the circular depressions at the end of the 20 μ m long metal air-bridges. (Courtesy MMT Group, STFC Rutherford Appleton Laboratory.)

0.5 THz. Subharmonic mixers incorporating antiparallel diode pairs are often a practical alternative. If the RF signal strength is high, a harmonic mixer may offer adequate performance. LO signals are derived from synthesizers, Gunn diodes, IMPATTS, transistor amplifiers, and frequency multiplied outputs. Mixer noise is usually expressed in terms of a noise temperature, essentially the temperature of a matched resistive load at a noiseless mixer input that gives the same noise power as the mixer when the resistive load is at 0k. The noise contribution of the first of any IF amplifiers, scaled by the conversion loss of the mixer, needs to be added to that of the mixer. Conversion from noise temperature to NEP is straightforward [11]: at an RF of 1 THz, for a bandwidth of a few gigahertz and a room-temperature environment, a NEP of ≈ 10 –14 W/√Hz is possible with signal averaging for 1 second. Thus mixers offer far superior performance to other room temperature detectors in terms of sensitivity, as well as preserving the spectral and phase content of the RF. The user should be aware that mixers are usually double sideband, so that there is ambiguity as to whether a signal in the IF band comes from an RF that is higher or lower than the LO or a multiple thereof for harmonic mixers. This folding of sidebands onto each other may not be a problem if the

6502_Book.indb 328

11/12/14 8:42 AM

Electronic Sources of Terahertz Radiation and Terahertz Detectors329



Table 11.3 Summary of the Characteristics of Commercially Available Coherent Terahertz Detectors IF Detector NEP Coverage Bandwidth Bandwidth Size Type (W/√Hz) (THz) (%) (GHz) (cm) Schottky mixer