Generation, Transmission, Detection, and Application of Vortex Beams 9819900735, 9789819900732

Table of contents :
Preface
Introduction
Contents
About the Author
1 Introduction
1.1 Optical Vortices
1.2 Orbital Angular Momentum
1.2.1 Background and Meaning
1.2.2 Principle of OAM Multiplexing Technology
1.2.3 OAM Multiplexing Communication System Model
1.3 Vortex-Beam Generation
1.3.1 Space-Generation Methods
1.3.2 Fiber-Generation Methods
1.3.3 Comparison of Vortex-Beam Generation Methods
1.4 Transmission Characteristics of Vortex Beams
1.4.1 Atmospheric-Turbulence Effect
1.4.2 Research Methods of Beam-Propagation Characteristics
1.4.3 Transmission Characteristics of Vortex Beams
1.5 Transmission Characteristics of Vortex Beams
1.5.1 Traditional Adaptive-Optics Correction Technology
1.5.2 AO Correction Without a Wavefront Sensor
1.5.3 Vortex Beam Phase Distortion Correction
1.6 Separation and Detection of Vortex Beams
1.6.1 Fork Grating
1.6.2 Interference Characteristics
1.6.3 Diffraction Characteristics
1.6.4 Reconstructed Wavefront
References
2 Vortex-Beam Spatial-Generation Method
2.1 Basic Principle of Vortex Beams
2.2 Types of Vortex Beams
2.2.1 Laguerre–Gaussian Beams
2.2.2 Bessel Beams
2.2.3 Hermite–Gaussian Beams
2.3 Vortex-Beam Generation Methods
2.3.1 Computer-Generated Hologram Method
2.3.2 Mode-Conversion Method
2.3.3 Spiral-Phase-Plate Method
2.3.4 Spatial Light Modulator Method
2.3.5 Optical-Waveguide Device-Conversion Method
2.4 Higher-Order Radial LG Beams
2.5 Generation of Fractional Vortex Beams
2.5.1 Principle of LG-Beam Preparation by the Holographic Method
2.5.2 Experimental Study on the Orbital Angular Momentum of Fractional Laguerre–Gaussian Beams
References
3 Vortex-Beam Generation Using the Optical-Fiber Method
3.1 Introduction
3.2 Optical-Fiber Mode Theory
3.2.1 Wave Equation
3.2.2 Vector Modes in Optical Fiber
3.2.3 Guide-Mode Cutoff and Distance Cutoff
3.2.4 Scalar Modulus Under a Weakly-Conducting Approximation
3.2.5 Analysis of the Principle of Using Optical Fiber to Generate Vortex Light
3.3 Analysis of the Influencing Factors of a Vortex Light Generated by Optical Fiber
3.3.1 Influence of the Incident Wavelength on the Vortex Light
3.3.2 Influence of the Refractive-Index Difference Between the Inside and Outside of the Optical Fiber on the Vortex Light
3.3.3 Influence of the Fiber-Core Radius on the Vortex Light
3.3.4 Effect of the Incident Angle on the Excitation Efficiency of Vortex Light
3.3.5 Effect of Off-Axis Incident Fiber on Vortex Light
3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light
3.4.1 Principle of Using Few-Mode Fibers to Generate Vortex Light
3.4.2 Analysis of the Excitation Efficiency of Vortex Light
3.4.3 Experimental Research
3.4.4 Phase Verification
3.5 Changing the Fiber Structure to Produce Vortex Light
3.5.1 Structural Design
3.5.2 Influence of the Low-Refractive-Index Layer on OAM Mode
References
4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian Beams
4.1 Introduction
4.2 Influence of the Radial Index on the Superposition State of High-Order Radial LG Beams
4.2.1 Interference and Superposition of LG Beams with the Same Topological Charge
4.2.2 Interference and Superposition of LG Beams with the Same Radial Index
4.2.3 Interference Superposition of LG Beams with Arbitrary Radial Indices and Topological Charges
4.3 Influence of the Transmission Distance on the Superposition State of High-Order Radial LG Beams
4.4 Influence of the Beam-Waist Radius on the Superposition State of High-Order Radial LG Beams
4.5 Effect of Off-Axis Parameters on the Superposition State of High-Order Radial LG Beams
4.6 Experiment on a Superimposed High-Order Radial LG Beam
4.6.1 Experimental Device
4.6.2 Hologram Production
4.6.3 Analysis of Results
References
5 Transmission Characteristics of Vortex Beams
5.1 Introduction
5.2 Transmission of LG Beams in Atmospheric Turbulence
5.2.1 Theoretical Analysis
5.2.2 Propagation Characteristics of an LG Beam Passing Through a Slanted Channel of Atmospheric Turbulence
5.3 Transmission of Bessel–Gaussian Beams in Space
5.3.1 Theory of BG-Beam Propagation in Turbulence
5.3.2 Characteristics of a BG Beam Passing Through an Atmospheric-Turbulence Channel
5.4 Research on the Stability of Orbital Angular Momentum in the Slanted Propagation of a Vortex Beam
5.4.1 Comparison of the Light-Intensity Distribution of Vortex Beams
5.4.2 Comparison of the Harmonic Components of a Vortex Beam
References
6 Adaptive-Optics Correction Technology
6.1 Introduction
6.2 Basic Adaptive-Optics Principles
6.2.1 Adaptive-Optics Correction Technology
6.2.2 Shack–Hartmann Algorithm
6.2.3 Phase-Recovery Algorithm
6.2.4 Stochastic Parallel Gradient-Descent Algorithm
6.2.5 Phase-Difference Algorithm
6.3 Wavefront Correction of an OAM Beam After Passing Through Atmospheric Turbulence
6.3.1 Phase-Recovery Algorithm
6.3.2 Stochastic Parallel Gradient-Descent Algorithm
6.3.3 Phase-Difference Algorithm
6.4 Experimental Research
6.4.1 Phase-Recovery Algorithm
6.4.2 Stochastic Parallel Gradient-Descent Algorithm
6.4.3 Phase-Difference Algorithm
References
7 Crosstalk Analysis of an OAM-Multiplexing System Under Atmospheric Turbulence
7.1 Introduction
7.2 Propagation Theory of OAM Beams in Atmospheric Turbulence
7.2.1 Multiphase-Screen Transmission Method
7.2.2 Random Phase-Screen Generation
7.2.3 Crosstalk of an OAM-Multiplexed Beam in Atmospheric Turbulence
7.3 Analysis of the Intensity Phase of an OAM-Multiplexed Beam in Atmospheric Turbulence
7.3.1 Formation of an OAM-Multiplexed Beam
7.3.2 Influence of Different Transmission Conditions on the Light Intensity and Phase
7.4 Spiral-Spectrum Characteristics of OAM-Multiplexed Beams Under Atmospheric Turbulence
7.4.1 OAM-Multiplexed Beam Spiral-Spectrum Theory
7.4.2 Spiral-Spectrum Analysis Under Different Transmission Conditions
7.5 Analysis of the Bit Error Rate of an OAM-Multiplexed Beam Under Atmospheric Turbulence
7.5.1 Bit-Error-Rate Theory of OAM-Multiplexed Beams
7.5.2 Analysis of the Bit Error Rate Under Different Transmission Conditions
7.6 Experiment on the Influence of Atmospheric Turbulence on an OAM-Multiplexed Beam
7.6.1 Principle of the Experiment
7.6.2 Analysis of Results
References
8 Properties of a Superimposed Vortex Beam
8.1 Introduction
8.2 Fabrication of the Vortex-Beam Superposition State Using a Grating Method
8.2.1 Theoretical Analysis
8.2.2 Grating Superposition
8.3 Double OAM Beam Prepared Using the Phase-Superposition Method
8.3.1 Theoretical Analysis
8.3.2 Characteristic Analysis of Superimposed Vortex Beams with Different Topological Charges
8.4 Vortex-Beam Superposition Interference Experiment
8.4.1 Experimental Design
8.4.2 Experiment of the Grating-Superposition Method
8.4.3 Result Analysis of the Grating-Superposition Method
8.4.4 Experiment on the Phase-Superposition Method
8.4.5 Results and Analysis of the Phase-Superposition Method
References
9 Vortex-Beam Detection
9.1 Introduction
9.2 Separate Detection of OAM States Using the Coordinate-Transformation Method
9.2.1 Theoretical Foundations
9.2.2 Superimposed Light-Field Distribution for Different Topological-Charge Numbers
9.2.3 OAM-State Multiplexing System Based on the Coordinate-Transformation Method
9.3 Using a Grating to Detect the Angular Momentum of a Vortex-Light Orbit
9.3.1 Transmission Function of a Grating and Its Representation
9.3.2 Vortex Light Field and Its Diffraction
9.3.3 Phase Correction and the Fan-Out Technique
9.3.4 Periodic-Gradient Grating
9.4 Using Interferometry to Detect the Vortex-Light Phase
9.4.1 Vortex Self-Interference Detection Method
9.5 Measuring the Vortex Optical Phase Using Diffraction Methods
9.5.1 Triangle Diffraction Method
9.5.2 Square-Hole Diffraction Method
9.5.3 Single-Slit-Diffraction Detection Method
9.5.4 Circular-Hole-Diffraction Detection Method
9.6 Summary
References
10 Diffraction Characteristics of a Vortex Beam Passing Through an Optical System
10.1 Diffraction Model of a Vortex Beam Passing Through a Mak-Cass Antenna
10.1.1 Mak-Cass Antenna Structure
10.1.2 Mak-Cass-Antenna Diffraction Model
10.2 Analysis of the Diffraction Characteristics of a Vortex Beam Passing Through a Mak-Cass Antenna Optical System
10.2.1 Diffracted Light-Field Model
10.2.2 Diffraction Spots and Phase Distribution
10.2.3 Spiral-Spectrum Distribution
10.2.4 Transmission Efficiency of the Mak-Cass Antenna
10.3 Analysis of the Diffraction Characteristics of a Vortex Beam Passing Through an Aperture Diaphragm
10.3.1 Theoretical Model of Aperture-Diaphragm Diffraction
10.3.2 Theoretical Diffraction Analysis of a Vortex Beam Passing Through a Diaphragm
10.3.3 Analysis of the Experimental Diffraction Pattern of a Vortex Beam Passing Through an Aperture
10.3.4 Aperture-Diaphragm Detection-Effect Comparison
10.4 Summary
References
11 Propagation Characteristics of a Partially Coherent Vortex-Beam Array in Atmospheric Turbulence
11.1 Beam-Array Overview
11.2 Intensity Distribution of a Radial Partially Coherent Vortex-Beam Array in Atmospheric Turbulence
11.2.1 Mathematical Model of a Radial Partially Coherent Vortex-Beam Array
11.2.2 Cross-Spectral Density Function on an Observation Plane
11.2.3 Expression of the Light Intensity on the Observation Plane
11.2.4 Effect of Light-Source Parameters on the Light-Intensity Characteristics in Non-Kolmogorov Turbulence
11.2.5 Influence Analysis of Radial-Array Parameters
11.2.6 Influence Analysis of Single Partially Coherent Vortex-Beam Parameters
11.3 Influence of Non-Kolmogorov Turbulence Parameters on Light-Intensity Characteristics
11.3.1 Impact Analysis of Non-Kolmogorov Turbulence Intensity
11.3.2 Non-Kolmogorov Turbulence Internal- and External-Scale Influence Analysis
11.4 Summary
References
12 Propagation Characteristics of Scalar Partially Coherent Vortex Beams in Atmospheric Turbulence
12.1 Basic Theory of Laguerre–Gaussian–Schell-Mode Vortex Beams
12.1.1 Laguerre–Gaussian–Schell Beams
12.1.2 Laguerre–Gaussian–Schell Vortex-Beam Model
12.1.3 Propagation Theory of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence
12.2 Phase-Singularity Evolution of Far-Field Laguerre–Gaussian–Schell Vortex Beams
12.2.1 Relationship Between a Phase Singularity and the Topological Charge
12.2.2 Effect of the Transmission Distance on the Phase-Singularity Evolution
12.2.3 Effect of the Correlation Length on the Phase-Singularity Evolution
12.3 Intensity Distribution of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence
12.3.1 Effect of the Atmospheric-Turbulence Intensity on the Light-Intensity Distribution
12.3.2 Influence of the Internal and External Atmospheric-Turbulence Scales on the Light-Intensity Distribution
12.4 Beam Propagation of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence
12.4.1 Analysis of Variation of a Beam Spread with Light-Source Parameters
12.4.2 Analysis of the Beam Spread with Atmospheric-Turbulence Intensity
References
13 Propagation Characteristics of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
13.1 Polarization Theory of Partially Coherent Vector Beams
13.2 Cross-Spectral Density Matrix of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
13.2.1 Intensity and Degree of Polarization
13.2.2 Polarization-Direction Angle
13.3 Polarization-Distribution Degree of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
13.3.1 Influence of the Light-Source Parameters on the Degree of Polarization
13.3.2 Influence of Atmospheric Turbulence on the Polarization Degree
13.3.3 Variation of the Polarization Degree with the Transmission Distance
13.4 Polarization-Direction-Angle Distribution of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
13.4.1 Influence of Atmospheric Turbulence on the Polarization-Direction Angle
13.4.2 Influence of the Transmission Distance on the Polarization-Direction Angle
13.4.3 Polarization-Direction Angle Detection of the Topological Charge
13.4.4 Polarization-Direction-Angle Model of a Far-Field Diffracted Light Field
13.4.5 Results of Using the Polarization-Direction Angle to Detect the Topological Charge
13.4.6 Analysis of the Influence of the Light-Source Parameters on the Detection Effect
13.5 Summary
References
14 Vortex-Beam Information Exchange
14.1 Flexibility of an OAM Vortex-Beam Topological Charge
14.1.1 Conversion of a Single OAM Beam
14.1.2 Conversion of an OAM-Multiplexed Beam
14.2 Principle of OAM Vortex-Beam Channel Reconstruction
14.2.1 OAM-Beam Information Exchange
14.2.2 OAM-Beam Mode Switch
14.3 Demultiplexing an OAM-Multiplexed Vortex Beam
14.4 Experimental Research on the Channel Reconstruction of OAM Vortex Beams
14.4.1 Experimental Research on OAM Information Exchange
14.4.2 Experimental Research on Exchanging Two Information Beams Among Three OAM-Multiplexed Beams
14.4.3 Experimental Research on the Mode Conversion of One Beam in Multiple OAM-Multiplexed Beams
14.4.4 Experimental Research on Exchanging Two OAM Multiplexed Beams with the Same Mode and Different Information
14.4.5 Experimental Research on Deleting/Adding a Beam Pattern in Multiple OAM-Multiplexed Beams
References

Citation preview

Optical Wireless Communication Theory and Technology

Xizheng Ke

Generation, Transmission, Detection, and Application of Vortex Beams

Optical Wireless Communication Theory and Technology Series Editor Xizheng Ke, School of Automation and Information Engineering, Xi’an University of Technology, Xi’an, Shaanxi, China

The book series Optical Wireless Communication Theory and Technology aims to introduce the key technologies and applications adopted in optical wireless communication to researchers of communication engineering, optical engineering and other related majors. The individual book volumes in the series are thematic. The goal of each volume is to give readers a comprehensive overview of how the theory and technology in a certain optical wireless communication area can be known. As a collection, the series provides valuable resources to a wide audience in academia, the communication engineering research community and anyone else who are looking to expand their knowledge of optical communication.

Xizheng Ke

Generation, Transmission, Detection, and Application of Vortex Beams

Xizheng Ke School of Automation and Information Engineering Xi’an University of Technology Xi’an, Shaanxi, China

ISSN 2731-5967 ISSN 2731-5975 (electronic) Optical Wireless Communication Theory and Technology ISBN 978-981-99-0073-2 ISBN 978-981-99-0074-9 (eBook) https://doi.org/10.1007/978-981-99-0074-9 Jointly published with Science Press The print edition is not for sale in China mainland. Customers from China mainland please order the print book from: Science Press. © Science Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The rapid development of the Internet industry has put forward higher requirements for the communication industry. In particular, high speed will be a mandatory requirement in future developments of the communication industry. With the emergence of fields such as massive data transmission, cloud computing, and artificial intelligence, the channel capacity provided by traditional communication methods is significantly limited. To increase the channel capacity in communication systems, a vortex-beam carrying orbital angular momentum has been incorporated in a new multiplexing method. This method can solve the problem of rate and channel capacity in multiplexed communication from the root. To realize orbital-angular-momentum multiplexing communication, it is necessary to achieve breakthroughs in the corresponding key technologies: generation of orbital-angular-momentum vortex beams, channel coding, free-space transmission characteristics, and separation and detection of beams at the receiving end. The contents of this book are divided into five parts. The first part introduces the generation of vortex beams carrying orbital angular momentum; a series of methods for generating vortex beams are also described in detail. Chapter 3 introduces space generation and fiber generation methods. Chapter 4 introduces the generation and superposition properties of high-order radial-exponential Laguerre-Gaussian beams. The second part introduces the transmission characteristics of vortex beams, mainly Laguerre-Gaussian and Bessel-Gaussian beams, in atmospheric turbulence, and analyzes the effects of atmospheric turbulence and light-source parameters on beams. The effects of light intensity distribution and helical spectral properties are discussed in Chap. 5. Chapter 6 introduces research on phase distortion correction of vortex beams using adaptive optics technology. Chapter 7 introduces crosstalk characteristics of multiplexed vortex beams in turbulent atmospheric environments. The third part introduces a detection method for vortex beams. Chapter 8 introduces the superposition state of a vortex beam that can detect its topological charge. Chapter 9 introduces vortex beams from the perspective of interference, diffraction, grating, and detection characteristics. Chapter 10 introduces the diffraction characteristics of vortex beams after passing through a Maksutov-Cassegrain antenna and studies how

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to improve the emission efficiency. The fourth part introduces the propagation characteristics of partially coherent vortex beams in free space. Chapter 11 introduces the light intensity evolution of partially coherent radial vortex-beam arrays after transmission in turbulent flows and discusses the influence of turbulence parameters on their transmission characteristics. Chapter 12 introduces the influence of atmospheric turbulence on the intensity distribution and beam expansion of LaguerreGaussian vortex beams. Chapter 13 introduces partially coherent vector vortex beams and analyzes the effects of atmospheric turbulence parameters on electromagnetic Gaussian-Shell vortex-beam polarization and the effect of the polarization direction angle. Finally, Chap. 14 in the fifth part introduces vortex-beam information exchange and channel reconstruction. This book is the result of collective research by the Optoelectronic Engineering Technology Research Center of Xi’an University of Technology. Wang Jiao, Xu Junyu, Xue Pu, Wang Chaozhen, Ge Tian, Wang Xiayao, Ning Chuan, Shi Xinyu, Zhao Jie, Xie Yanchen, Chen Yun, and many other graduate students participated in this study. The revision work of this book also contributed to the research on the subject. In addition, in the process of writing this monography, a great deal of academic texts were consulted. I would like to express my gratitude to all the authors. Their work has inspired and helped me. I thank them for their enthusiasm in scientific research. I would like to express my gratitude to Shaanxi Provincial Key Industry Innovation Chain Project (2017ZDCXL-GY-06-01), which funded the publication of this book. The present book is a summary of research on vortex beams. Owing to the author’s level of knowledge, there are unavoidable omissions in the book. Criticisms and corrections from readers are welcome. Xi’an, China May 2022

Xizheng Ke

Introduction

High speed will be a mandatory requirement in the future of the communication industry. With the emergence of fields such as massive data transmission, cloud computing, and artificial intelligence, the channel capacity provided by traditional communication methods is significantly limited. To increase the channel capacity in communication systems, a vortex-beam carrying orbital angular momentum has been incorporated in a new multiplexing method. This method can solve the problem of channel capacity in multiplexed communication from the root. This book focuses on the key technologies supporting orbital-angular-momentum multiplexing communication: generation, transmission, detection, and application of vortex beams. A series of methods for generating vortex beams are described and compared in detail. Laguerre-Gaussian and Bessel-Gaussian beams are taken as examples to introduce the transport properties of vortex beams in atmospheric turbulence. We show that superposition of vortex-beam state, interference, diffraction, and grating can realize the detection of the topological charge of vortex beams. We also introduce the application of vortex beams in optical communication and the transmission characteristics of partially coherent vortex beams in atmospheric turbulence. Finally, we describe vortex-beam information exchange and channel reconstruction. This book can be used as a teaching book for senior undergraduates and graduate students in majors such as communication and optics in colleges and universities, and as a reference book for researchers, engineers, and technicians.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Background and Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Principle of OAM Multiplexing Technology . . . . . . . . . . 1.2.3 OAM Multiplexing Communication System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vortex-Beam Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Space-Generation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fiber-Generation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Comparison of Vortex-Beam Generation Methods . . . . . 1.4 Transmission Characteristics of Vortex Beams . . . . . . . . . . . . . . . . 1.4.1 Atmospheric-Turbulence Effect . . . . . . . . . . . . . . . . . . . . . 1.4.2 Research Methods of Beam-Propagation Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Transmission Characteristics of Vortex Beams . . . . . . . . 1.5 Transmission Characteristics of Vortex Beams . . . . . . . . . . . . . . . . 1.5.1 Traditional Adaptive-Optics Correction Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 AO Correction Without a Wavefront Sensor . . . . . . . . . . 1.5.3 Vortex Beam Phase Distortion Correction . . . . . . . . . . . . 1.6 Separation and Detection of Vortex Beams . . . . . . . . . . . . . . . . . . . 1.6.1 Fork Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Interference Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Diffraction Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Reconstructed Wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex-Beam Spatial-Generation Method . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Principle of Vortex Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Types of Vortex Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2.1 Laguerre–Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Bessel Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Hermite–Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Vortex-Beam Generation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Computer-Generated Hologram Method . . . . . . . . . . . . . 2.3.2 Mode-Conversion Method . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Spiral-Phase-Plate Method . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Spatial Light Modulator Method . . . . . . . . . . . . . . . . . . . . 2.3.5 Optical-Waveguide Device-Conversion Method . . . . . . . 2.4 Higher-Order Radial LG Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Generation of Fractional Vortex Beams . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Principle of LG-Beam Preparation by the Holographic Method . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Experimental Study on the Orbital Angular Momentum of Fractional Laguerre–Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 53 54 55 56 58 59 61 62 63 66

Vortex-Beam Generation Using the Optical-Fiber Method . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Optical-Fiber Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Vector Modes in Optical Fiber . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Guide-Mode Cutoff and Distance Cutoff . . . . . . . . . . . . . 3.2.4 Scalar Modulus Under a Weakly-Conducting Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Analysis of the Principle of Using Optical Fiber to Generate Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Analysis of the Influencing Factors of a Vortex Light Generated by Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Influence of the Incident Wavelength on the Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Influence of the Refractive-Index Difference Between the Inside and Outside of the Optical Fiber on the Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Influence of the Fiber-Core Radius on the Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Effect of the Incident Angle on the Excitation Efficiency of Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Effect of Off-Axis Incident Fiber on Vortex Light . . . . . 3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Principle of Using Few-Mode Fibers to Generate Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 76 76 77 81

66

69 73

84 86 89 89

90 92 94 96 97 97

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xi

3.4.2

Analysis of the Excitation Efficiency of Vortex Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Experimental Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Phase Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Changing the Fiber Structure to Produce Vortex Light . . . . . . . . . 3.5.1 Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Influence of the Low-Refractive-Index Layer on OAM Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

Superposition Characteristics of High-Order Radial Laguerre–Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Influence of the Radial Index on the Superposition State of High-Order Radial LG Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Interference and Superposition of LG Beams with the Same Topological Charge . . . . . . . . . . . . . . . . . . 4.2.2 Interference and Superposition of LG Beams with the Same Radial Index . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Interference Superposition of LG Beams with Arbitrary Radial Indices and Topological Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Influence of the Transmission Distance on the Superposition State of High-Order Radial LG Beams . . . . . . . . . . . . . . . . . . . . . . 4.4 Influence of the Beam-Waist Radius on the Superposition State of High-Order Radial LG Beams . . . . . . . . . . . . . . . . . . . . . . 4.5 Effect of Off-Axis Parameters on the Superposition State of High-Order Radial LG Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Experiment on a Superimposed High-Order Radial LG Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Experimental Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Hologram Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission Characteristics of Vortex Beams . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Transmission of LG Beams in Atmospheric Turbulence . . . . . . . . 5.2.1 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Propagation Characteristics of an LG Beam Passing Through a Slanted Channel of Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Transmission of Bessel–Gaussian Beams in Space . . . . . . . . . . . . 5.3.1 Theory of BG-Beam Propagation in Turbulence . . . . . . . 5.3.2 Characteristics of a BG Beam Passing Through an Atmospheric-Turbulence Channel . . . . . . . . . . . . . . . .

99 100 103 105 105 107 109 111 111 112 113 115

119 120 122 123 125 125 126 127 133 135 135 136 136

140 147 147 150

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5.4

Research on the Stability of Orbital Angular Momentum in the Slanted Propagation of a Vortex Beam . . . . . . . . . . . . . . . . . 5.4.1 Comparison of the Light-Intensity Distribution of Vortex Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Comparison of the Harmonic Components of a Vortex Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

Adaptive-Optics Correction Technology . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Basic Adaptive-Optics Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Adaptive-Optics Correction Technology . . . . . . . . . . . . . . 6.2.2 Shack–Hartmann Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Phase-Recovery Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Stochastic Parallel Gradient-Descent Algorithm . . . . . . . 6.2.5 Phase-Difference Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Wavefront Correction of an OAM Beam After Passing Through Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Phase-Recovery Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Stochastic Parallel Gradient-Descent Algorithm . . . . . . . 6.3.3 Phase-Difference Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Experimental Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Phase-Recovery Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Stochastic Parallel Gradient-Descent Algorithm . . . . . . . 6.4.3 Phase-Difference Algorithm . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crosstalk Analysis of an OAM-Multiplexing System Under Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Propagation Theory of OAM Beams in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Multiphase-Screen Transmission Method . . . . . . . . . . . . . 7.2.2 Random Phase-Screen Generation . . . . . . . . . . . . . . . . . . . 7.2.3 Crosstalk of an OAM-Multiplexed Beam in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Analysis of the Intensity Phase of an OAM-Multiplexed Beam in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Formation of an OAM-Multiplexed Beam . . . . . . . . . . . . 7.3.2 Influence of Different Transmission Conditions on the Light Intensity and Phase . . . . . . . . . . . . . . . . . . . . 7.4 Spiral-Spectrum Characteristics of OAM-Multiplexed Beams Under Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . 7.4.1 OAM-Multiplexed Beam Spiral-Spectrum Theory . . . . . 7.4.2 Spiral-Spectrum Analysis Under Different Transmission Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 157 163 165 165 166 166 168 169 172 175 179 179 182 185 189 189 192 196 202 205 205 208 208 209 210 211 211 213 215 216 217

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Analysis of the Bit Error Rate of an OAM-Multiplexed Beam Under Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Bit-Error-Rate Theory of OAM-Multiplexed Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Analysis of the Bit Error Rate Under Different Transmission Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Experiment on the Influence of Atmospheric Turbulence on an OAM-Multiplexed Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Principle of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

7.5

8

9

219 220 221 223 223 224 226

Properties of a Superimposed Vortex Beam . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Fabrication of the Vortex-Beam Superposition State Using a Grating Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Grating Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Double OAM Beam Prepared Using the Phase-Superposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Characteristic Analysis of Superimposed Vortex Beams with Different Topological Charges . . . . . . . . . . . 8.4 Vortex-Beam Superposition Interference Experiment . . . . . . . . . . 8.4.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Experiment of the Grating-Superposition Method . . . . . 8.4.3 Result Analysis of the Grating-Superposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Experiment on the Phase-Superposition Method . . . . . . . 8.4.5 Results and Analysis of the Phase-Superposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 229

Vortex-Beam Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Separate Detection of OAM States Using the Coordinate-Transformation Method . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Superimposed Light-Field Distribution for Different Topological-Charge Numbers . . . . . . . . . . . 9.2.3 OAM-State Multiplexing System Based on the Coordinate-Transformation Method . . . . . . . . . . . 9.3 Using a Grating to Detect the Angular Momentum of a Vortex-Light Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Transmission Function of a Grating and Its Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 251

230 230 231 232 232 235 239 239 240 243 244 248 248

252 252 254 256 258 258

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9.3.2 Vortex Light Field and Its Diffraction . . . . . . . . . . . . . . . . 9.3.3 Phase Correction and the Fan-Out Technique . . . . . . . . . 9.3.4 Periodic-Gradient Grating . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Using Interferometry to Detect the Vortex-Light Phase . . . . . . . . . 9.4.1 Vortex Self-Interference Detection Method . . . . . . . . . . . 9.5 Measuring the Vortex Optical Phase Using Diffraction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Triangle Diffraction Method . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Square-Hole Diffraction Method . . . . . . . . . . . . . . . . . . . . 9.5.3 Single-Slit-Diffraction Detection Method . . . . . . . . . . . . . 9.5.4 Circular-Hole-Diffraction Detection Method . . . . . . . . . . 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Diffraction Characteristics of a Vortex Beam Passing Through an Optical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Diffraction Model of a Vortex Beam Passing Through a Mak-Cass Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Mak-Cass Antenna Structure . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Mak-Cass-Antenna Diffraction Model . . . . . . . . . . . . . . . 10.2 Analysis of the Diffraction Characteristics of a Vortex Beam Passing Through a Mak-Cass Antenna Optical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Diffracted Light-Field Model . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Diffraction Spots and Phase Distribution . . . . . . . . . . . . . 10.2.3 Spiral-Spectrum Distribution . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Transmission Efficiency of the Mak-Cass Antenna . . . . . 10.3 Analysis of the Diffraction Characteristics of a Vortex Beam Passing Through an Aperture Diaphragm . . . . . . . . . . . . . . . 10.3.1 Theoretical Model of Aperture-Diaphragm Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Theoretical Diffraction Analysis of a Vortex Beam Passing Through a Diaphragm . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Analysis of the Experimental Diffraction Pattern of a Vortex Beam Passing Through an Aperture . . . . . . . 10.3.4 Aperture-Diaphragm Detection-Effect Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

260 261 262 265 265 268 268 270 271 273 274 275 277 278 278 279

281 281 283 287 289 290 291 293 297 299 300 300

11 Propagation Characteristics of a Partially Coherent Vortex-Beam Array in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . 303 11.1 Beam-Array Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.2 Intensity Distribution of a Radial Partially Coherent Vortex-Beam Array in Atmospheric Turbulence . . . . . . . . . . . . . . . 306

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11.2.1 Mathematical Model of a Radial Partially Coherent Vortex-Beam Array . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Cross-Spectral Density Function on an Observation Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Expression of the Light Intensity on the Observation Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Effect of Light-Source Parameters on the Light-Intensity Characteristics in Non-Kolmogorov Turbulence . . . . . . . . . . . . . . . . . . . . 11.2.5 Influence Analysis of Radial-Array Parameters . . . . . . . . 11.2.6 Influence Analysis of Single Partially Coherent Vortex-Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Influence of Non-Kolmogorov Turbulence Parameters on Light-Intensity Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Impact Analysis of Non-Kolmogorov Turbulence Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Non-Kolmogorov Turbulence Internaland External-Scale Influence Analysis . . . . . . . . . . . . . . . 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Propagation Characteristics of Scalar Partially Coherent Vortex Beams in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . 12.1 Basic Theory of Laguerre–Gaussian–Schell-Mode Vortex Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Laguerre–Gaussian–Schell Beams . . . . . . . . . . . . . . . . . . . 12.1.2 Laguerre–Gaussian–Schell Vortex-Beam Model . . . . . . . 12.1.3 Propagation Theory of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . 12.2 Phase-Singularity Evolution of Far-Field Laguerre–Gaussian–Schell Vortex Beams . . . . . . . . . . . . . . . . . . . . 12.2.1 Relationship Between a Phase Singularity and the Topological Charge . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Effect of the Transmission Distance on the Phase-Singularity Evolution . . . . . . . . . . . . . . . . . . 12.2.3 Effect of the Correlation Length on the Phase-Singularity Evolution . . . . . . . . . . . . . . . . . . 12.3 Intensity Distribution of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . 12.3.1 Effect of the Atmospheric-Turbulence Intensity on the Light-Intensity Distribution . . . . . . . . . . . . . . . . . . 12.3.2 Influence of the Internal and External Atmospheric-Turbulence Scales on the Light-Intensity Distribution . . . . . . . . . . . . . . . . . .

xv

306 306 314

315 315 320 324 325 328 329 331 333 333 334 337

337 341 342 344 346 347 348

350

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12.4 Beam Propagation of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Analysis of Variation of a Beam Spread with Light-Source Parameters . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Analysis of the Beam Spread with Atmospheric-Turbulence Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Propagation Characteristics of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . 13.1 Polarization Theory of Partially Coherent Vector Beams . . . . . . . 13.2 Cross-Spectral Density Matrix of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . 13.2.1 Intensity and Degree of Polarization . . . . . . . . . . . . . . . . . 13.2.2 Polarization-Direction Angle . . . . . . . . . . . . . . . . . . . . . . . 13.3 Polarization-Distribution Degree of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence . . . . . . . . . . . . . 13.3.1 Influence of the Light-Source Parameters on the Degree of Polarization . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Influence of Atmospheric Turbulence on the Polarization Degree . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Variation of the Polarization Degree with the Transmission Distance . . . . . . . . . . . . . . . . . . . . . 13.4 Polarization-Direction-Angle Distribution of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence . . . . . 13.4.1 Influence of Atmospheric Turbulence on the Polarization-Direction Angle . . . . . . . . . . . . . . . . . 13.4.2 Influence of the Transmission Distance on the Polarization-Direction Angle . . . . . . . . . . . . . . . . . 13.4.3 Polarization-Direction Angle Detection of the Topological Charge . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.4 Polarization-Direction-Angle Model of a Far-Field Diffracted Light Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.5 Results of Using the Polarization-Direction Angle to Detect the Topological Charge . . . . . . . . . . . . . . . . . . . . 13.4.6 Analysis of the Influence of the Light-Source Parameters on the Detection Effect . . . . . . . . . . . . . . . . . . 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Vortex-Beam Information Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Flexibility of an OAM Vortex-Beam Topological Charge . . . . . . . 14.1.1 Conversion of a Single OAM Beam . . . . . . . . . . . . . . . . . 14.1.2 Conversion of an OAM-Multiplexed Beam . . . . . . . . . . . 14.2 Principle of OAM Vortex-Beam Channel Reconstruction . . . . . . .

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Contents

14.2.1 OAM-Beam Information Exchange . . . . . . . . . . . . . . . . . . 14.2.2 OAM-Beam Mode Switch . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Demultiplexing an OAM-Multiplexed Vortex Beam . . . . . . . . . . . 14.4 Experimental Research on the Channel Reconstruction of OAM Vortex Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Experimental Research on OAM Information Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Experimental Research on Exchanging Two Information Beams Among Three OAM-Multiplexed Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Experimental Research on the Mode Conversion of One Beam in Multiple OAM-Multiplexed Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4 Experimental Research on Exchanging Two OAM Multiplexed Beams with the Same Mode and Different Information . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.5 Experimental Research on Deleting/Adding a Beam Pattern in Multiple OAM-Multiplexed Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Author

Xizheng Ke is Doctor of Science, Second-level Professor, Head of the discipline of information and communication systems of Xi’an University of Technology, and Deputy Director of the Key Laboratory of Military-civilian Joint Construction of Intelligent Cooperative Networks in Shaanxi Province. He is Famous Teacher in Shaanxi Province, Fellow of Chinese Institute of Electronics, Foreign academician of the Russian Academy of Natural Sciences, Director of Chinese Optical Engineering Society, Executive Director of Shaanxi Optical Society. He received a bachelor’s degree from Shaanxi Institute of Technology in 1983 and a doctor of science degree from the University of Chinese Academy of Sciences in 1996. From 1997 to 2002, he conducted post-doctoral research at Xidian University and the Second Artillery Engineering College. Dr. Xizheng Ke is Member of editorial board of Journal of Electronics, Infrared and Laser Engineering, Journal of Electronic Measurement and Instrument, Laser Technology, Applied Optics, Journal of Xi’an University of Technology, Journal of Time and Frequency, and Journal of Atmospheric Science Research. He is National Science and Technology Award Review Expert and Member of the disciplinary review group of the Shaanxi Academic Degrees Committee. In 2000, Dr. Xizheng Ke won the Outstanding Young Scholar Award of the Chinese Academy of Sciences. In 2009, he was awarded the title of “Excellent Science and Technology Commissioner of Guangdong Province by the Ministry of Science and Technology”. In 2015, he was awarded the title of “A Golden Phoenix on Green Poplar” in Yangzhou City. In 2018, he won the China Industry-University-Research Innovation Award, and in 2019, he won the China Industry-University-Research Innovation Achievement Award. Since 2001, he has won 16 provincial and ministerial science and technology awards, including 1 first prize and 5 second prizes. He has obtained more than 20 nationally authorized invention patents, published 9 monographs in Science Press, and published more than 400 academic papers in domestic and foreign journals. More than 30 Ph.D. candidates have been cultivated in his direction.

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Chapter 1

Introduction

1.1 Optical Vortices A vortex is one of the most common phenomena in nature. It is ubiquitous in classical macroscopic systems, such as water, clouds, and cyclones. It also exists in quantum microscopic systems, such as superfluids, superconductors, and Bose– Einstein condensation. A vortex is considered an inherent morphological feature of waves [1]. When people study tidal motion, they pay particular attention to the vortex of the tide. When the tide is in contact with the isotidal line, the tidal peak disappears. This phenomenon demonstrates that there are singularities—that is, optical vortices—in ocean waves (as shown in Fig. 1.1) [2]. Richards and Boivin et al. [3, 4] found that a singular ring is formed at the focal plane of an aspheric lens. Through experimentation, they found that there is an optical vortex at the focal plane of the lens, owing to line rotation. This confirms that optical vortices also exist in light-wave fields. In 1973, Carter et al. [5] used a computer to simulate the characteristics of a singular ring and found that when the light beam was slightly disturbed, a singular ring could be generated or removed. In 1974, Nye et al. [6] discovered the existence of phase singularities in seawater acoustic waves when studying speckle fields, and, for the first time, extended the concept of singularities to the field of electromagnetic waves. In 1981, Baranova et al. [7, 8] discovered randomly distributed optical vortices on a laser spot, and experimentally found that the probability of generating optical vortices in a speckle light field could be measured under certain conditions. However, an optical vortex field with higher-order topological charges could not be generated. In 1992, Swartzlander et al. [9] found an optical-vortex soliton in a self-focusing medium through theoretical and experimental research. They discovered that the optical-vortex soliton interacted with the nonlinear medium during the transmission process. The propagation of the spin made a significant contribution.

© Science Press 2023 X. Ke, Generation, Transmission, Detection, and Application of Vortex Beams, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-0074-9_1

1

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1 Introduction

Fig. 1.1 Whirlpool phenomena in the ocean tide [11]

In 1998, Voitsekhovich et al. [10] studied in detail the characteristics of the number density of phase singularities under certain undulating conditions. They found that the number density of phase singularities has a certain statistical distribution, not a specific value. The statistical distribution is related to the probability distribution of the spatial derivative of the amplitude. In the twenty-first century, owing to the further expansion of research fields involving optical vortices, the understanding of optical vortices has reached a new height. Because a vortex light is a form of wave, it not only has spin angular momentum, but also orbital angular momentum (OAM), owing to the spiral phase structure. This type of OAM-carrying beam is known as an optical vortex. An optical vortex is a unique light field, and its uniqueness is manifested mainly in its special wavefront structure and definite photon OAM. Figure 1.2 shows the spiral wavefront, light intensity, and phase-distribution diagram of an optical vortex field. Through the transmission of photon OAM to atoms, molecules, colloidal particles, and other substances in the optical vortex field, microscopic particles can be manipulated with little contact and without damage. In addition, the vortex beam has a topological charge, which has important potential application value in quantum secure communication and other fields [12].

1.2 Orbital Angular Momentum

3

Fig. 1.2 Optical vortex field. a Spiral phase; b light intensity; and c phase distribution

1.2 Orbital Angular Momentum Compared with traditional optical communication, a beam with orbital angular momentum (OAM) has a new degree of freedom, which gives the OAM multiplexing technology unique advantages in improving a system’s channel capacity and spectrum utilization. Research can provide a more intuitive understanding of OAM-multiplexed beams.

1.2.1 Background and Meaning Wireless optical communication, that is, free-space optical (FSO) communication, is a technology that uses lasers as carriers to transmit data, voice, and image information. Owing to the absorption and scattering of an optical signal by the atmosphere, a light beam transmitted in space is attenuated; the atmospheric-turbulence effect causes the laser spot to drift, flicker, and expand, resulting in a large bit error rate and even interrupted communication [13]. Although the traditional channel-coding method can suppress turbulence, it cannot satisfy the requirements of multiplexing communication in the face of strong turbulence and dense fog. A new technology is required to improve the channel capacity and spectrum utilization. In existing multiplexing technology, resources, such as frequency, time, code patterns, and space, have been brought into full play. Wave is limited in the information modulation format in the free space fiber, so information is not interoperable between free space and multimode fiber networks. There is no interoperability between networks; therefore, it is difficult to fully satisfy the network capacity and communication security. To increase the information-transmission capacity, improve spectrum efficiency, and establish a highly reliable and highly secure communication network, OAM multiplexing technology has attracted widespread attention. Multiplexing communication based on orbital angular momentum has the following advantages [14]:

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1 Introduction

1. Safety: Owing to the uncertain relationship between the topological load and the azimuth angle of the OAM, the OAM state can only be accurately detected when the OAM beam is completely received. Angle tilt and partial reception will spread the power of the transmission mode to other modes, reducing the probability of correctly detecting the transmission OAM state; therefore, OAM optical communication can effectively prevent eavesdropping. 2. Orthogonality: The vortex beams of different OAM modes are inherently orthogonal, which enables the information to be modulated on different vortex beams. The signals transmitted on different OAM channels do not interfere with each other, which improves the reliability of the information transmission. 3. Multi-dimensionality: Because an OAM vortex beam has an infinite number of eigenstates, multiple channels of information can be transmitted on the same spatial path, thereby increasing the dimension of the multiplexed communication. 4. High spectrum-utilization rate: Because vortex-beam multiplexing communication uses OAM to transmit multiplexed information, the spectrum-utilization rate is much higher than that of long-term evolution (LTE), 802.11n, and digital video broadcasting—terrestrial (DVB-T, terrestrial digital television broadcasting). 5. High transmission rate: The transmission rate of OAM multiplexing communication is higher than that of LTE, 802.11n, and DVB-T. Experimental research shows that it can reach the order of terabits. With the continuous deepening of OAM research, vortex-beam multiplexing technology with OAM, as a new multiplexing dimension, has attracted widespread attention in the information-transmission field. Multiplexing technology with OAM vortex beams is one method to increase the information transmission rate and ensure information-transmission safety. This multiplexing uses the infinite number of OAM quantum states (or modes) for the multichannel transmission of information, uses the orthogonality between different OAM modes to modulate information, and finally, loads the information onto the orbital momentum. Two or more vortex beams send the multiplexed transmission. And can be combined with existing technologies such as wavelength division multiplexing (WDM), space-division multiplexing (SDM), polarization-division multiplexing (PDM), and other related technologies, to build an information-exchange communication system to improve communication-network capacity and spectrum efficiency. It is also supplemented by multiple-input multipleoutput (MIMO) equalization techniques and channel coding to reduce the crosstalk caused by atmospheric turbulence. In the following, three aspects of optical communication are introduced: OAM-multiplexed communication in free space, optical fiber, and underwater. The development of OAM wireless optical communication has been rapid. In 2007, OAM-multiplexing technology was first applied to optical communication by Yuan et al. [15], who experimentally used different OAM states to multiplex multiple optical signals in free space. The multiplexing/demultiplexing were achieved by loading a computer-generated hologram (CGH) onto a spatial light modulator (SLM).

1.2 Orbital Angular Momentum

5

In 2010, Awaji [16] experimentally transmitted two multiplexed OAM beams carrying 10 Gbit/s signals. This work was the beginning of OAM multiplexed communication and laid the foundation for a series of subsequent studies. In 2011, Wang et al. [17] realized a 2 Tbit/s data link using two orthogonal OAM modes combined with 25 WDM channels, using Laguerre–Gaussian (LG) beams as optical carriers. In 2012, Wang [18] proposed and demonstrated a new high-speed communication model using spatial light modulators loaded with a spiral phase diagram to realize OAM multiplexing. They combined it with PDM to achieve a free-space optical communication-system capacity of 1369.6 Gbit/s and spectral efficiency of 25.6 bit/s/Hz. Meanwhile, Wang’s breakthrough lies in using the spatial-state scalability of photons to increase the spectral efficiency of the transmission. In other words, they used two sets of eight polarization-multiplexed OAM beams superimposed in concentric rings, each loaded with 80-Gbit/s 16 QAM signals. They finally achieved a communication capacity of 2560 Gbit/s and spectral efficiency of 95.7 bit/s/Hz, which greatly enhanced the transmission rate of the system. A schematic diagram of OAM multiplexing combined with PDM and SDM is shown in Fig. 1.3. Between 2013 and 2015, this study was followed by further breakthroughs by the team in communication-system capacity and spectral efficiency, using OAM multiplexing in combination with existing dimensional resources [19, 20]. They used OAM to multiplex/demultiplex 1008 data channels, 24 OAM modes, or 12 OAM modes combined with two polarization states, each carrying 42 wavelengths, and each transmitting a 100-Gbit/s quadrature phase-shift keying (QPSK) signal for multiplexing. They ultimately achieved a communication capacity of 100.8 Tbit/s. In 2014, the team [21] also experimentally realized a free-space data link with a total transmission capacity of 1.036 Pbit/s, while the spectrum utilization was as high as 112.6 bit/s/Hz, using an orthogonal frequency-division multiplexing (OFDM)-8QAM modulation signal with 26 dual-polarized OAM channels, and covering 368 wavelengths in the C + L band. In 2012, Tamburini et al. [22] conducted a 442 m transmission experiment using wireless-optical-link OAM-mode multiplexing in Venice, Italy. In 2014, Krenn [23] used OAM beams to achieve 3 km wireless optical communication in a strong atmospheric-interference environment in the center of Vienna, Austria. In the same year, Xu [24] used a multiple-input multiple-output (MIMO) adaptive-equalization method to reduce the crosstalk between OAM multiplexing system signals caused by atmospheric turbulence. Huang [25, 26] used 4 × 4 MIMO technology and heterodyne detection to implement a free-space four-channel OAM-mode multiplexing technology, in which each OAM beam carries information at a rate of 20 Gbit/s, which effectively reduces the system error rate. In 2016, Ren [27] experimentally studied the application of MIMO technology to OAM multiplexing systems. The experiment revealed that using space diversity and MIMO equalization could effectively reduce the impact of atmospheric turbulence on OAM optical communications. In 2017, Zhang [28] proposed an acoustic OAM communication technology based on an active transducer array. The principle was to generate a sound vortex field with eight topological charges through a phased array composed of 64 sound sources that radiated signals encoded with composite vortex

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1 Introduction

Fig. 1.3 Polarization multiplexing communication link of OAM photon space state loaded with information [18]

states. Another acoustic phased array was used at the receiving end for reception and demodulation. In 2013, Bozinovic et al. [29] built an OAM fiber-optic communication system using a vortex fiber that enabled a high degree of separation of the simplex vector modes. They used the OAM beam as a new degree of freedom for fiber-optic communication systems. They achieved an information transmission rate of 1.6 Tbit/s in a 1.1 km-long vortex fiber. This study provided an experimental basis for the use of OAM beams for long-range and high-capacity fiber-optic communication. Wang et al. [30] proposed a lower computational-complexity OAM-mode multiplexing communication scheme based on conventional multimode fibers, by taking advantage of the property that multimode fibers have a small effective refractive-index difference between modes within a mode group, but a large effective refractive-index difference between mode groups. A combination of interference-free multiplexing among intermode groups and small-scale MIMO-assisted multiplexing in intra-mode group was

1.2 Orbital Angular Momentum

7

used to greatly reduce the system and algorithm complexity. A multiplexed transmission of 10-Gbaud QPSK signals in six OAM modes over an 8.8 km multimode fiber (MMF) was achieved using only 2 × 2 and 4 × 4 MIMO–digital signal processing (DSP) experiments. The total transmission capacity was 120 Gbit/s, and the optical signal-to-noise ratio (OSNR) cost was less than 2.5 dB at the forward error-correction code (FEC) threshold for all six OAM modes. In 2016, Baghdady et al. [31] from Eric G. Johnson’s team conducted an in-depth study of OAM-delay measurement (DM)-based underwater optical communication for underwater channels, including three channel environments: pure water, pure seawater, and near-coastal seawater. In addition, the team enhanced the communication rate, based on the OAM-DM, combined with polarization multiplexing. Keith et al. [32] studied an underwater OAM-DM (l = ± 8) optical-communication system using GaN laser diodes with a two-way laser output. The experimental results showed that the higher the link-transmission rate, the more obvious the bit-error rate (BER) increase. They then studied an OAM-DM (l = ± 8) underwater opticalcommunication system with a channel-attenuation coefficient from 0.00875/m (pure seawater) to 0.4128/m (near-coastal seawater) by generating a laser using a fiberpigtail laser diode with non-zero switching-keying modulation. The rate reached 3 Gbit/s and the average BER was 2.073 × 10–4 . Finally, they studied the OAM-DM (l = ± 8) optical-communication system under a channel from pure seawater (attenuation coefficient of 0.0425/m) to near-coastal seawater (attenuation coefficient of 0.3853/m), using GaN laser diodes to generate laser light with a communication rate up to 2.5 Gbit/s and an average BER of 2.13 × 10–4 . In 2017, Miller et al. [33] investigated an underwater multidimensional fusion-modulated visible-light communication system, based on dual-polarization multiplexing combined with OAM-DM (l = ± 4, ± 8) with a total system-transmission rate of up to 12 Gbit/s and a BER of 2.06 × 10–4 . Professor Jian Wang and researcher Yifan Zhao [34] first achieved four-way OAM-mode (l = ± 3, ± 6) multicast communication, carrying four 1.5-Gbaud 8-QAM-OFDM signals per channel. In addition, the possibility of higher modulation formats was verified experimentally. The scheme is scalable, and the number of OAM modes can be further increased. An experimental schematic diagram of the OAM-based underwater wireless optical-multicast link is shown in Fig. 1.4. An arbitrary waveform generator (AWG) generates a 1.5 Gbaud signal, which is amplified by an electrical amplifier; then, a 520 nm single-mode pigtail laser diode is applied for direct modulation. The output green light signal is reflected by an SLM loaded with a forked multicast phase diagram, and then passes through a 200 cm × 40 cm × 40 cm rectangular water tank to simulate an underwater environment. At the receiving end, the multicast beam is demodulated by an SLM2 loaded with a specific adjustable fork pattern, and then transmitted to a high-sensitivity silicon avalanche-photodiode detector for photoelectric conversion. It is finally sent to an oscilloscope to measure the results and record the light-intensity diagram with a camera.

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1 Introduction

Fig. 1.4 Experimental schematic diagram of underwater wireless optical multicast link based on OAM [34]

1.2.2 Principle of OAM Multiplexing Technology Existing multiplexing technologies include frequency-division multiplexing, timedivision multiplexing, code-division multiplexing, and space-division multiplexing. These multiplexing technologies have achieved breakthrough developments in their corresponding research fields: • 1G technology is inseparable from frequency-division multiplexing; • 2G technology refers to time-division and code-division multiplexing; thus, opening the era of digital communication; • 3G technology is applied to space-division multiplexing, so that the same carrier frequency can be reused in different directions; • 4G combines orthogonal frequency-division multiplexing and MIMO technology, and has advantages in terms of communication system capacity and spectrum utilization. OAM multiplexing technology essentially utilizes the orthogonality between OAM beams to load multiple signals to be transmitted onto OAM beams with different topological charges for transmission. At the receiving end, the difference in topological charge is used to distinguish the different transmission channels. This multiplexing method can simultaneously achieve multiple independent OAM-beam channels on the same carrier frequency. Studies have shown that a vortex beam carrying OAM can be expanded into an infinite-dimensional Hilbert space; hence, using OAM multiplexing technology on the same carrier frequency can improve the system-transmission performance [35]. This feature provides a new degree of freedom for efficient spectrum usage. Table 1.1 shows the transmission rate and spectrum utilization of the following common communication types: LTE, 802.11n, DVB-T, and OAM multiplexing. As shown in Table 1.1, the spectrum-utilization rate and system-transmission rate of the OAM multiplexing technology are significantly better than those of the other three communication types. The reason for this is that OAM multiplexing technology uses the OAM mode carried by the carrier as the modulation parameter for multiplexing.

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Table 1.1 Comparison of transmission rate and spectrum utilization for common communication types Communication type

OAM

LTE

802.11n

DVB-T

Spectrum utilization

95.5 bit·s−1 Hz−1

16.32 bit·s−1 Hz−1

2.4 bit·s−1 Hz−1

0.55 bit·s−1 Hz−1

Transmission rate

2.56 Tbit·s−1

326.4 Mbit·s−1

144.4 Mbit·s−1

Mbit·s−1

1.2.3 OAM Multiplexing Communication System Model An OAM beam-multiplexing communication system model engaged in a turbulentatmospheric transmission is shown in Fig. 1.5. This is an example of four-channel multiplexing. First, the original input bit stream is modulated by quadrature phaseshift keying (QPSK), and the Gaussian beam generated by the solid-state laser is loaded using optical-modulation technology. At this point, the electrical transmission signal is converted into an optical signal. The Gaussian beam carrying the modulation information is converted into an OAM beam corresponding to the topological charge, using a spatial phase mask. The four vortex beams with different topological charges are multiplexed, and the resulting OAM multiplexed state is transmitted through atmospheric turbulence. The OAM-multiplexed beam is demultiplexed at the receiving end to obtain the four OAM beams. Then convert the vortex-beam into a Gaussian beam. Finally, the QPSK signal loaded onto the Gaussian beam is extracted and demodulated to restore the original bit stream; that is, the optical signal is converted into the original electrical signal. Reverse Gaussian Output Phase Beam Bit Torrent Shift Mask

Phase Original Gaussian Shift LG Bit Torrent Beam Mask Beam Atmospheric Turbulence

l=-2

qpsk

l=-4

qpsk

OAM3

l=-6

qpsk

OAM4

l=-8

qpsk

qpsk

l=2

OAM1

qpsk

l=4

OAM2

qpsk

l=6

qpsk

l=8

Multiplexing

DeMultiplexing

Fig. 1.5 OAM multiplexing communication system model

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1 Introduction

1.3 Vortex-Beam Generation To realize OAM multiplexing communication, the first step is to generate a vortex beam with orbital angular momentum. The most common methods of generating vortex beams can be divided into two types: space generation and fiber generation.

1.3.1 Space-Generation Methods The methods for generating vortex beams using spatial structures mainly include the following: direct generation, mode conversion, spiral phase plate, and computational holography. (1) Direct generation A vortex beam is generated directly by the laser cavity [36]. In experiments, this method has strict requirements for the axial symmetry of the resonant cavity, and it is difficult to obtain a stable beam output. In 1989, Coullet et al. [36] used the laser resonant cavity to directly generate vortex beams; however, the large intracavity losses made it difficult to produce high-quality vortex beams. Subsequently, methods to enhance the beam quality were proposed, such as the ring beam-pumping method [37], centrally damaged cavity mirror method [38], and thermally induced modeaperture method [39]. The aforementioned conventional vortex lasers can only produce a single-mode vortex beam without changing the resonant-cavity parameters. Digital lasers have been proposed to satisfy the output demands of arbitrary laser modes, without changing the resonant-cavity structure. This method applies an SLM to the laser cavity to act as a piece of resonant mirror, a spatial light modulator, and another mirror to form a laser resonant cavity to produce a laser output, while the controlling computer can flexibly generate vortex beams with various properties. The working principle of a digital laser is illustrated in Fig. 1.6. Although this method is simple and easy to operate, it is difficult to simultaneously generate high-power, multimodal, and high-quality vortex beams [40]. In summary, there are many problems with generating OAM beams in a cavity; therefore, an out-of-cavity conversion method is typically used to generate vortex beams. (2) Mode conversion A non-axisymmetric optical system is composed of cylindrical mirrors, and a Hermite–Gaussian (HG) beam without orbital angular momentum is input. It can be converted into a laser beam through a mode converter composed of two cylindrical lenses. A Laguerre–Gaussian (LG) beam is shown in Fig. 1.7. This method was first proposed by Allen in 1993. They also proposed converting LG beams into HG beams [41]. The HG beam can be turned into a vortex beam with orbital angular momentum by introducing a phase factor that varies with the azimuth angle, based on the Hermite–Gaussian beam [42].

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Fig. 1.6 Working principle of digital laser [40]

Fig. 1.7 Beam-mode conversion between an HG (HG01 and HG10 ) beam and an LG beam

The conversion efficiency of the mode-conversion method is high; however, the optical-system structure in the conversion process is relatively complicated. The processing and preparation of the key optical components used in the system are difficult, and it is difficult to control the types and parameters of the generated vortex beams. This method’s occasions for application are restricted. (3) Spiral phase plate A spiral phase plate [43] is a transparent plate whose thickness is proportional to the azimuth of rotation, relative to the center of the plate. The surface structure is similar to that of a rotating table. When a light beam passes through the spiral phase plate, the amount of change in the phase of the transmitted light beam differs because the spiral surface of the phase plate changes the optical path of the transmitted light beam; thus, a phase factor with spiral characteristics can be generated.

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Fig. 1.8 Spiral phase plates with different topological charges

The conversion efficiency of the vortex beam generated by the spiral phase plate method is relatively high; however, the topological charge of the optical vortex generated by this method is not unique: For a certain phase plate, the laser using a specific mode can only produce a specific output. The types and specific parameters of the vortex beam cannot be flexibly controlled, and high-quality phase plates are difficult to prepare. The spiral phase plates (SPPs) corresponding to the different topological-charge numbers are shown in Fig. 1.8. In 2019, Sheng et al. [44] designed a flat-plate SPP to generate vortex beams. It could flexibly adjust the OAM angular quantum number of the generated vortex beam and the height of the SPP, according to the refractive index of the dielectric material. The accuracy of the generated beam increased with an increase in the phase order. To compensate for the shortcomings of the previous SPP method, SPPs can be stacked in multiple layers. The phase plate can be flipped to directly adjust the angular quantum number and sign of the generated vortex beam. This provides a potential solution for conveniently adjusting optical OAM characteristics for practical production and applications. (4) Computational holography The computational-holography method is based on the principle of light interference and diffraction. Computer programming is used to produce the interference pattern of the target and reference lights and obtain the vortex beam. Using computational holography to generate vortex beams is a fast and flexible method that has a wide range of applications. It can be realized using computational holograms and spatial light modulators. A computational hologram can be used to transform a fork-shaped grating into a negative film and directly allow a Gaussian plane wave to pass through the fork-shaped grating. In 1992, Heckenberg used the computer-generating hologram (CGH) method to generate the required diffraction grating pattern to generate a vortex beam [45]. In 2012, Escuti et al. designed a forked polarization grating (FPG) based on forked singularities. It is made of a liquid–crystal material to efficiently generate OAM beams and perform optical conversion between modes. The relative

1.3 Vortex-Beam Generation

13

power between modes can also be controlled by the polarization state. The team also succeeded in making switchable FPGs that electrically switch between OAM generation/conversion and transmission states. This method has a diffraction efficiency higher than 90%, and the holographically fabricated elements are compact, lightweight, and easily optimized for any wavelength except ultraviolet infrared, a wide range of OAM angular quantum numbers, and transparent apertures [46]. The spiral zone plate (SZP) is another binary CGH that produces vortex beams. It was first used for visible-light and soft X-rays. In contrast to forked gratings and spiral phase diagrams, they are generated by superimposing vortex waves with spherical waves. SZPs are more suitable for scanning transmission-electron- microscopy applications than gratings, where the spherical waves are focused or diverged at different positions of the optical axis so that diffracted beams of different orders can never be simultaneously focused on the sample. In 2012, Koh et al. generated a series of converging OAM electron beams with topological-charge numbers up to 90 [47]. Figure 1.9 shows a schematic diagram of vortex-beam generation using different methods. The digital-micromirror device (DMD) method [49] is used to dynamically generate an OAM beam by dynamically loading the CGH to encode the amplitude and phase of the incident beam. The DMD is internally composed of an array of millions of micromirrors, each of which can be converted to “off” or “on” by tilting −12° or + 12°, respectively. The “on”-state mirror reflects light in the desired direction; thus, controlling the beam generation. The DMD is inexpensive and can be switched to generate OAM modes at high speed; however, the diffraction efficiency is not satisfactory. In 2021, Hu et al. [50] used a DMD to generate and

Fig. 1.9 Schematic diagram of vortex beams generated by different methods [48]

14 Fig. 1.10 Vortex beam generated by SLM method

1 Introduction 半波片 激光器

衰减片

偏振片

SLM

孔径

扩束准直

光束分 析仪

characterize complex vector-mode vortex light and used a binary-encoding scheme based on random spatial multiplexing to generate vector vortex beams, such as Laguerre–Gaussian, Ince–Gaussian, Mathieu–Gaussian, and parabolic-Gaussian. In the spatial light modulator (SLM) method, the shape grating is loaded onto the SLM, and the Gaussian plane wave can be directly incident to the SLM. Bo Bin et al. [51] used a reflective SLM to generate a certain type of beam, and conducted an experimental study on the interference between the generated beam and planar light. The results verified that the generated vortex beam had different topological charges, and the resulting energy-conversion efficiency was relatively high. In 2016, Forbes et al. combined SLM for OAM-beam generation and controlled SLM modulation to generate different types and states of OAM beams by modifying and writing a digital hologram in real time [52]. Figure 1.10 shows the experimental schematic diagram of vortex-beam generation using the SLM method. The hologram displayed on the SLM can be flexibly controlled by a computer to change the position, size, and topological charge of the generated optical vortex. It can also dynamically adjust the position of the optical vortex in real time. The DMD method can be achieved by replacing the SLM in the figure with a DMD. (5) Photonic sieve method In 2016, Forbes et al. combined SLM for OAM-beam generation and controlled SLM modulation to generate different types and states of OAM beams by modifying and writing a digital hologram in real time [52]. Figure 1.10 shows the experimental schematic diagram of vortex-beam generation using the SLM method. The hologram displayed on the SLM can be flexibly controlled by a computer to change the position, size, and topological charge of the generated optical vortex. It can also dynamically adjust the position of the optical vortex in real time. The DMD method can be achieved by replacing the SLM in the figure with a DMD (Fig. 1.11). (6) Metasurface method Most traditional OAM beam–generation methods are based on standard optical devices, such as SPP, SLM, and CGH, which are suitable for laboratory experiments; however, they have the disadvantages of large sizes, long working distances, and low optical-control accuracy. Moreover, they are incompatible with modern optical systems with integrated, ultraminiature, and multifunctional flat-panel optics.

1.3 Vortex-Beam Generation

15

Fig. 1.11 Vortex beam generation by photon sieve method [53]

In recent years, micro-sized flat-optical elements based on plasma-excited metasurfaces have become a reality. Nanoscale-structured metasurfaces have been proven to effectively control the amplitude, phase, and polarization state of light in linear optical regions, and can reconfigure the beam wavefront. The metasurface can cause an abrupt phase change in the incident light through scalable artificial atoms, resulting in an OAM beam. A schematic of converting incident left- and right-handed polarized light into an OAM beam using a J-plate in the metasurface approach is shown in Fig. 1.12. In 2017, Devlin et al. used a two-dimensional metasurface consisting of dynamic and geometric phases to generate an OAM beam. This metasurface is a J-plate consisting of rectangular nanoantennas of different sizes. This allows any orthogonal polarization state to be converted to independent OAM modes, compared to previous Q-plates, and can be used with high-power laser beams, overcoming the limitations of SLMs [54]. In 2021, Dorrah et al. designed a total angular momentum (TAM) plate that enabled two main classes of functions. The first type is a polarization-switchable device that switches between two generated vortex beams by changing the incident orthogonal polarization state. The other is valid for any incident polarization state and allows the flexible control of the optical OAM, spin angular momentum (SAM), and polarization, amplitude, and phase along the propagation direction [55].

16

1 Introduction

Fig. 1.12 Schematic diagram of OAM beam generation using J-plate in the super-surface generation method [54]

1.3.2 Fiber-Generation Methods To adapt to the development and application requirements of OAM optical communication systems, scholars have proposed the use of optical fiber to generate vortex beams, mainly using three methods: (a) fiber-coupler conversion [56–60]; (b) photonic crystal optical-fiber conversion [61–63], and (c) optical waveguide device conversion [64–68]. In 2011, Yan et al. [58] created an OAM beam through the mode superposition of a Hermite–Gaussian beam input using four microfibers. Later, the research team improved and replaced the microfiber with a square-core fiber and placed it inside the ring fiber. This improved OAM-generating coupler only needs one input fiber, in terms of structure, which reduces the processing complexity and is better than opticalfiber generation and the traditional OAM-generation method. Its structure is simple and has great potential for promoting OAM information-transmission technology in future optical fiber. However, the waveguide dispersion is large, and the purity of the current OAM mode is low, which makes the high-order OAM mode unstable and sensitive to changes in the wavelength. In 2014, Brunet et al. [59] fabricated a fiber with an air core and a ring refractiveindex profile using the modified chemical-vapor deposition (MCVD) process; they added a material with a lower refractive index than the cladding in the ring region of the ring fiber to obtain better coupling efficiency. Experiments using this new fiber produced stable transmissions of 36 modes. In 2017, Pidishety et al. [60] used an all-fiber mode-selective coupler, consisting of a single-mode fiber and ring-core

1.3 Vortex-Beam Generation

17

Fig. 1.13 Experimental schematic diagram of an OAM beam generated by the all-fiber modeselection coupler [60]

fiber, to excite the OAM beam by direct phase-matched coupling. It produced a final excitation-mode purity of 75%. Figure 1.13 shows the experimental schematic diagram of an all-fiber mode-selective coupler consisting of a single-mode fiber and a ring-core fiber. The solid line represents transmission in the fiber and the dashed line represents transmission in free space. In 2012, Willner designed a new set of OAM converters using photonic crystal fiber (PCF) [61]. This fiber has a positive hexagonal arrangement of cladding air holes, but produces a larger dispersion and mode loss. The basic principle is to convert the mode of an input Hermite–Gaussian beam. After the conversion, a series of vortex eigenmodes is generated. As long as the appropriate vortex eigenmodes are selected for combination and superposition, the desired OAM mode can be generated. We can improve the performance in practical applications by changing the air-hole arrangement, size, and spacing parameters of the PCF cladding, according to different practical needs. This method improves the defects of the previous air-hole arrangement, and can generate and support 10 OAM-mode transmissions [62]. In the same year, Wong et al. [69] reported a spiral PCF mode converter in Science magazine that can generate more OAM modes. When the laser inputs linearly polarized supercontinuum light into the PCF, this type of converter modulates the input light in the azimuth, changes the phase of the input light, and produces OAM vortex light. In 2020, Israk et al. [63] designed a coil-type large bandwidth PCF surrounded by three layers with a larger air core surrounded by three thin layers in the middle. It supports up to 56 OAM modes with gently varying dispersion for transmission, with limiting losses less than 10–8 dB/m for most modes, nonlinear coefficients and numerical apertures less than 4W−1 km−1 and 0.17, respectively, and bandwidths up to 1900 nm, and these excellent optical properties ensure the excellent performance of this fiber. A cross-sectional schematic of this PCF is shown in Fig. 1.14. In addition to the above advantages, the spiral PCF also has low limiting loss, relatively flat dispersion and small nonlinear coefficient. And the spiral PCF generates an OAM topological load that changes regularly with the change in the fiber-structure parameters, which has great advantages for generating more OAM modes. In 2012, Cai Xinlun et al. reported a silicon-integrated OAM vortex-beam transmitter in Science magazine. This generator transmits a Hermite–Gaussian beam in a silicon waveguide, couples it to a ring waveguide, and produces an echo after passing

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1 Introduction

Fig. 1.14 Schematic diagram of cross section of coil-type PCF surrounded by three layers [63]

through the ring waveguide [64]. This compact silicon integrated OAM vortex beam emitter is shown in Fig. 1.15. Because periodic zigzag protrusions appear on the inner wall of the waveguide, the Hermite–Gaussian beam produces a phase difference when propagating in the ring waveguide. This phase difference causes the light-wave vector to change. Finally, vortex lights with different OAM modes are generated above the ring waveguide. This type of converter is not only small in size, but also produces vortex beams with low phase sensitivity and a stable OAM mode. The converter can be integrated on a large scale, and simultaneously generates multiple OAM beams with controllable topological charges. In 2020, Cognée et al. [68] demonstrated that a plasmonic photonic-resonant microdisk cavity addressed by a waveguide could generate a controlled correlated OAM and SAM beam. This approach was implemented using silicon–nitride disks and aluminum-nanorod antennas for the antenna arrays. This good polarization and OAM purity can be used as a benchmark for the polarization-resolvable interferometric Fourier microscopy of a single device. In 2022, Zeng et al. [70] exploited the topologically selective excited Brillouinscattering effect in a helical PCF to optically isolate vortex beams with topologicalcharge numbers 0, 1, and 2 in mode-division multiplexing. In the 200 m-long tripleand hexa-rotationally symmetric spiral PCF, the isolation can be maintained above 22 dB and within 1 dB when the optical-power dynamics vary by 35 dB. In the future, to further improve the vortex optical isolation, it will be possible to reduce the fiber length while fabricating soft-glass spiral PCFs. These produce

1.3 Vortex-Beam Generation

19

Fig. 1.15 Compact Silicon Integrated OAM Vortex beam emitter [64]

Brillouin gains more than 100 times higher than those produced by ordinary silica glass, and simultaneously generate more spiral PCF cores to increase the mode capacity. Figure 1.16 shows a schematic diagram of topologically selective excited Brillouin scattering. When the pump and Stokes lights in excited Brillouin scattering have equal and opposite absolute values of topological charge and spin number, the forward-propagating pump light signal is dissipated by the excited Brillouin scattering, and the backward-propagating signal is not changed. This achieves a unidirectional transmission of the signal light and reduces the reverse loss.

Fig. 1.16 Schematic diagram of topologically selective stimulated Brillouin scattering [70]

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1 Introduction

1.3.3 Comparison of Vortex-Beam Generation Methods The unique phase structure and OAM characteristics of the vortex beam make it suitable for applications in quantum information transmission, particle manipulation, molecular optics, etc. However, these applications rely on the generation of highquality vortex beams, and the above generation methods each have advantages and disadvantages. Table 1.2 presents a comparative analysis of the existing methods for generating vortex beams. Therefore, considering these existing conditions and technologies, seeking an effective method to generate higher-quality vortex beams has also become an urgent problem in this field. Table 1.2 Comparison of vortex-beam generation methods Method

Advantage

Disadvantage

Direct generation

Produced directly in the laser cavity

It is difficult to obtain a stable vortex beam or to generate high-order vortex beams

Mode generation

High conversion efficiency

The optical structure is relatively complicated and the device preparation is difficult; the types and parameters of the vortex beams are difficult to control

Spiral phase plate

High conversion efficiency

The types and parameters of the vortex beam are difficult to control; high-quality phase plates are difficult to prepare

Computational holograpy The position, size, and parameters of the vortex beam can be controlled; Low production cost

Requires a strict optical-path alignment; low diffraction efficiency

DMD method

Low cost; high switching speed

Non-ideal diffraction efficiency

SLM method

Easy to adjust the parameters; high diffraction efficiency

Limited energy threshold

Photonic sieve method

Wide available band and high spatial resolution

Low diffraction efficiency

Metasurface method

High system integration; high conversion efficiency

Difficult operation and equipment preparation

Fiber generation

Facilitated promotion in optical-communication systems; the resulting vortex beam is relatively stable

In experiments, only low-order vortex beams can be realized at present; high-order vortex beams are currently difficult to achieve

1.4 Transmission Characteristics of Vortex Beams

21

1.4 Transmission Characteristics of Vortex Beams 1.4.1 Atmospheric-Turbulence Effect Atmospheric turbulence is a random chaotic medium. When light passes through atmospheric turbulence, its random movement causes fluctuations in the refractive index, leading to a series of turbulence effects, such as beam distortion, coherence weakening, and light-intensity attenuation. These changes are particularly obvious during strong turbulence or long-distance transmissions and severely restrict the development of free-space optical communication. Atmospheric turbulence has the following characteristics: (1) The turbulent motion exhibits irregular random characteristics Atmospheric turbulence is generated by an external force. When the external force increases, the fluid motion state changes from a laminar to a turbulent flow. The motion gradually loses stability and becomes irregular, disorderly, and nonlinear. (2) The turbulence parameters have statistical characteristics Although turbulent motion is irregular, the motion parameters at adjacent spatial points have certain correlation characteristics. Therefore, statistical methods, such as the statistical average method, can be used to estimate and predict turbulence. (3) The turbulence has a sensitive dependence on the initial conditions Lorenz first inferred that the atmosphere is sensitive to initial conditions. Then, Berry used the results of accurate numerical calculations as proof of Professor Lorenz’s inference and found that atmospheric turbulence is also sensitive to its initial conditions [71]. Because atmospheric turbulence causes random fluctuations in the refractive index of light, wavefront distortion and amplitude fluctuations occur when light waves propagate in atmospheric turbulence. This causes light-intensity flicker, beam expansion, spot drift, fluctuations in the angle of arrival, reduced beam coherence, and other atmospheric-turbulence effects.

1.4.2 Research Methods of Beam-Propagation Characteristics In general, research on the transmission characteristics of light waves mainly considers two aspects: beam characteristics and transmission paths, as shown in Fig. 1.17. The study of beam-propagation characteristics is mainly based on the statistical characteristics of the light field. The generalized Huygens–Fresnel principle is the most commonly used research method. This is the basic principle of wave optics and

22

1 Introduction

Light-intensity distribution Light-intensity expansion Coherence characteristics

Flicker Arrival-angle variations

Light-beam characteristics

Research on the Transmission Characteristics of Light Beams

Spot drift

...... Degree of polarization Polarization characteristics

Polarization angle Depolarization characteristics

Horizontal path Transmission path

Upstream path Sloping path Downstream path

Fig. 1.17 Research foundation of beam-propagation characteristics

the theoretical basis for addressing diffraction problems. The main principle is the following [72]: Any unobstructed point on the wavefront can be regarded as a wavelet source with the same frequency (or wavelength) as the incident wave; light vibrations anywhere thereafter are the result of the coherent superposition of these wavelets. Their wavefront represents the wavefront formed by the light waves emitted by the light source at a certain moment. The secondary disturbance center is a point light source, also known as the wavelet source. A schematic diagram of the generalized Huygens–Fresnel principle is shown in Fig. 1.18. Assuming that the light field of the light wave at the transmitting end, z = 0, is denoted as u 0 (r ; 0), when the light wave travels along the z-axis for a distance L, the light wave on the observation plane can be expressed as: { { u(ρ; L) =

d 2 ru 0 (r; 0)ξ (ρ, r).

(1.1)

Equation (1.1) can be regarded as the impulse response of a propagation system. ξ (ρ, r) has reciprocity in atmospheric turbulence and can be expressed as: ξ (ρ, r) = −

[ ] ik ik|ρ − r|2 exp(ik L) · exp + ψ(ρ, r) . 2π L 2L

(1.2)

1.4 Transmission Characteristics of Vortex Beams

23

x

x Observation plane

Source plane r

Turbulence element

p z

y

y L

Fig. 1.18 Schematic diagram of the Huygens–Fresnel principle

where, ξ (ρ, r) is called a complex random perturbation of a turbulent medium and can be written as a random perturbation of magnitude and phase: ψ(ρ, r) = χ + i ς.

(1.3)

Applying (1.2) to (1.1) produces: k exp(ik L) u(ρ; L) = 2πi L

{

] ik|ρ − r|2 dru 0 (r; 0) exp + ψ(ρ, r) . 2L [

(1.4)

Equation (1.4) is called the "Extended Huygens–Fresnel Principle." It describes the wave propagation and scattering in random media, optical imaging, and differential laser-radar signal-to-noise ratio analysis, etc. This field has a wide range of applications [73].

1.4.3 Transmission Characteristics of Vortex Beams When an OAM beam is used to transmit information, it is less affected by atmospheric turbulence [74]. Lukin [75] used numerical-simulation methods to confirm that a vortex beam propagating in atmospheric turbulence caused less expansion than a Gaussian beam. Although vortex beams have some advantages in free-space optical communication, in practice, atmospheric turbulence will inevitably cause changes in the beam intensity and phase, resulting in an increase in the bit error rate and a decrease in the communication capacity [76].

24

1 Introduction

The influence of atmospheric turbulence on vortex beams was studied, in addition to analyzing the turbulence effects (light-intensity fluctuations, phase fluctuations, spot flickers, beam expansion, drift, M2 factor, focusing characteristics, etc.) when vortex beams propagate in atmospheric turbulence. In addition, the change in the OAM eigenstate of the photon was analyzed, and the phase singularity of the vortex beam was evaluated accordingly [77]. Generally, research on the transmission characteristics of vortex beams starts from two aspects. (1) Transmission characteristics of different vortex beams Owing to differences in the spatial structure, there are many types of vortex beams. In recent years, Researchers are interested in the following types of vortex beams: Bessel–Gaussian vector beams [78], Laguerre–Gaussian beams [79–81], elliptical vortex beams [82], multi-order Gauss–Scherr vortex beams [83, 84], partially coherent Laguerre–Gausscher beams [85], flat-top vortex beams [86], partially coherent Hermite–Gausscher beams [87], Off-axis vortex beams [88], standard and simplified vortex beams [89], super-Gaussian geometric modes [90], array vortex beams [91], and vortex cosine hyperbolic Gaussian beams [92]. In 2019, Luo et al. [91] used the generalized Huygens–Fresnel principle and Rytov approximation to derive expressions for the transmission intensity of arbitrary-order vortex beams and their arrays in turbulent flows. The beam expansion and evolution were also numerically simulated to obtain the mean-square beam width for different beam and turbulence parameter variations. It was found that the vortex-beam array was less affected by turbulence than a single-mode vortex beam, and the radial vortexbeam array would finally evolve into a Gaussian beam. In 2020, Yan et al. [90] used the Fresnel diffraction principle and multilayer phase-screen method to establish a transport model of a vortex beam in atmospheric turbulence. They analyzed the far-field optical-intensity distribution of the super-Gaussian and Gaussian vortex beams in turbulence, and found that the variation in the dispersion degree of the aberrations of the two beams, that is, the beam quality, could be obtained under different parameters. The transmission distance has a greater influence on the beam quality of the super-Gaussian vortex beam, whereas the topological-charge number has a greater influence on the beam quality of the Gaussian vortex beam. In 2021, Hyde et al. [93] proposed a twisted space–time vortex beam (spatiotemporal optical vortex, STOV), which has a coherent optical vortex and random twist in coupled space–time dimensions. They derived the mutual-coherence function, linear-momentum density, and angular-momentum density, and simulated the physical synthesis of the STOV beam and its free space under different degrees of coherence. They also simulated the propagation of STOV beams in free space with different degrees of coherence. It was found that the larger the degree of coherence, the smaller the number of twisted-vortex singularities and the slower the beam decayed; the smaller the Fresnel number, the easier it was for the beam to split into combinations of lower-order vortices. In 2022, Hricha et al. [94] studied the focusing characteristics and focal shift of a partially coherent vortex cosine-hyperbolic Gaussian beam (PCvChGB) passing through a lens system. Based on the extended Collins formula, an analytical formula

1.4 Transmission Characteristics of Vortex Beams

25

for PCvChGB transmission through a thin lens was derived. It was concluded that the spatial coherence length, Fresnel number, and beam parameters had a significant effect on the mean intensity distribution and focal shift of the beam-focusing region. In 2018, Miao et al. [95] introduced a new radially polarized beam—the radially polarized multi-cosine Gaussian Schell model (MCGSM) beam—and investigated the statistical properties of the MCSM beam in isotropic non-Kolmogorov turbulence. The optical intensity, spectral density, coherence, polarization, and polarization states were analyzed, and the cross-spectral density-matrix metric formula was derived for the radially polarized MCGSN beams. Finally, it was found that their polarization states have a self-splitting property, and each beam evolves into a radially polarized structure. In 2022, Arora et al. [96] further argued both theoretically and experimentally for the perturbation of a uniform linearly polarized beam to a vector singularity. The amount of radial displacement of the beam center of mass caused by this perturbation can indicate the strength of the perturbation. (2) Transmission characteristics of vortex beams on different paths Early researchers generally considered the horizontal transmission of light beams under atmospheric turbulence. In 2001, the International Telecommunications Union proposed a model of atmospheric-structure constants that varied with altitude, and investigators have gradually begun to notice the slant transmission of light beams in atmospheric turbulence. Filippus [97] analyzed the influence of strong near-ground turbulence on the coherence of OAM entangled photons. The study showed that a moderately strong turbulence obviously influenced the entangled photon pairs. Pu [98–100] systematically studied the special properties of vortex beams (measuring the spiral wavefront, orbital angular momentum, and topological charge). Zhang [101] also studied the relevant characteristics of vortex beams propagating in atmospheric turbulence. And In addition to transmissions in turbulent flows, transmissions in other media have been studied successively. In 2017, Porfirev et al. studied the propagation characteristics of vortex beams in random aerosol media by means of the Huygens–Fresnel diffraction principle and the fast Fourier transform. The experiments were performed using an aerosol generator with an aqueous solution, defined by the characteristic size and the effective refractive index of the medium changes. The vortex-beam stability decreased as the topological-charge number increased. It was also found that, at short distances, vortex beams are mostly less stable than Gaussian beams; however, at long distances, vortex beams are more stable, probably because of their ability to repair themselves after passing through obstacles [102]. In recent years, researchers have gradually focused their attention on the transport properties of ocean turbulence. In 2022, Lazrek et al. [103] derived the mean intensity distribution and beam spreading of a vortex cosine hyperbolic-Gaussian beam (vchGb) in ocean turbulence, based on the extended Huygens–Fresnel diffraction principle and the Rytov method. The results showed that by increasing the mean square temperature, dissipation rate of the temperature to the salinity-fluctuation ratio, and dissipation rate of turbulent kinetic energy per unit mass of seawater, a

26

1 Introduction

Fig. 1.19 Normalized light-intensity distribution of a vchGb transmitted at different distances in ocean turbulence [103]

vchGb can propagate at shorter distances in weak ocean turbulence and transform into a Gaussian-like beam in the far field. The evolution characteristics of a vchGb in ocean turbulence are influenced by the initial beam parameters, such as the decentered parameter b, topological charge M, beam waist, and wavelength. The results of this study will be helpful in providing further references for underwater optical communication and remote-sensing imaging. Figure 1.19 shows the normalized light-intensity distribution of a vchGb with topological charge 1 after transmitting for 0.1, 0.3, 0.6, 0.9, and 1.2 km in ocean turbulence. The first and second rows are the distributions for the decentered parameter b = 0.1 and b = 4, respectively. It can be seen that the different decentered parameters b lead to different beam states. Eventually the beam evolves into a Gaussian-like distribution for b = 0.1 and a flat-top-like distribution for b = 4, as the transmission distance increases.

1.5 Transmission Characteristics of Vortex Beams For an entire OAM multiplexing system, the beam is affected by various linear and nonlinear effects when propagating in the atmosphere. The most important distortion originates from atmospheric turbulence. The OAM state is a spatial-mode distribution. Therefore, the wavefront is inevitably affected by atmospheric turbulence during propagation, causing wavefront distortion [104]. Atmospheric turbulence not only affects a single OAM state, but also causes modal crosstalk between different OAM states [105].

1.5 Transmission Characteristics of Vortex Beams

27

1.5.1 Traditional Adaptive-Optics Correction Technology The adaptive-optics (AO) theory was first proposed by Babcock in 1953. In this theory, a wavefront sensor is used to measure the wavefront, and a wavefront corrector is used to compensate for the distorted wavefront in real time. Under ideal conditions, the distorted wavefront can be restored to the plane wave [106]. Initially, adaptive-optics systems were primarily used in the field of high-resolution imaging in astronomy. In the late 1980s, astronomers developed a new set of adaptive-optics systems named “COME-ON,” which was used in 3.6 m telescopes at the Southern European Observatory, New Zealand, and Chile. The deformable mirror has 19 units [107]. In a free-space optical communication system, to correct the wavefront distortion caused by atmospheric turbulence, it was proposed to use an AO system to compensate for the distorted wavefront [108–110]. The wavefront distortion caused by the transmission of the vortex beam in atmospheric turbulence can be corrected and compensated for by the AO system. Traditional AO technology combines electronics and optics. It can detect and correct the distorted wavefront in real time, so that the optical system has the energy to adapt to changes in its own and external conditions to maintain the best working state and improve the beam quality and the performance of the communication system. As shown in Fig. 1.20, a traditional AO system is usually composed of three basic units [111]: a wavefront-detection unit, a wavefront-control unit, and a wavefrontcorrection unit. The wavefront detector detects the wavefront distortion caused by atmospheric turbulence in real time, and the computer control system calculates the control voltage that needs to be loaded onto the wavefront corrector. The wavefront corrector is used to compensate for the error caused by atmospheric turbulence in real time.

Fig. 1.20 Traditional AO system with a wavefront sensor

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1 Introduction

1.5.2 AO Correction Without a Wavefront Sensor Most AO systems use wavefront sensors to detect wavefront phase distortion, and the wavefront controller generates a control signal corresponding to the detected distortion to drive the wavefront corrector (deformation mirror) to correct the distortion phase. In 2009, Xia Lijun et al. [112] conducted a wavefront-distortion correction experiment for atmospheric optical communication. The experimental results showed that, after the AO system was corrected, a smaller initial optical power could be used to obtain a better communication quality. In 2014, Hashmi et al. [113] conducted a numerical simulation experiment on inter-satellite AO communication in a laboratory. The experiment showed that after closed-loop correction of the AO system, the Strehl ratio of the system could be increased from 0.30 to 0.75. However, the structure of a correction system with a wavefront sensor is complex, and the wavefront sensor will divert part of the light intensity of the system; therefore, researchers have begun to investigate AO correction systems that do not rely on wavefront sensors. As shown in Fig. 1.21, an AO system without wavefront detection establishes a system-performance evaluation function, based on the image-quality information obtained by the imaging detector, and optimizes the evaluation function with an optimization algorithm to correct the distorted wavefront. An AO correction system without a wavefront sensor is composed of three parts: a wavefront controller, wavefront corrector, and imaging detector (charge-coupled device, CCD). A parallel light emitted by the light source is transmitted through atmospheric turbulence to produce a distorted beam with aberrations. The distorted light beam is incident on the deformable mirror. The deformed mirror first corrects the distorted light beam and reflects the residual wavefront distortion to the CCD. The wavefront controller drives an intelligent algorithm to regenerate the control signal of the deformable mirror, according to the system-performance index value collected by the CCD, to realize the distortion. The beam undergoes multiple closed-loop corrections.

Fig. 1.21 Structure diagram of AO correction system without a wavefront sensor

1.5 Transmission Characteristics of Vortex Beams

29

Early optimization algorithms included hill climbing and multiple high-frequency vibration methods, which are limited, owing to their time consumption and highbandwidth requirements. Therefore, it is necessary to find intelligent algorithms that are easy to implement and can calculate control parameters in parallel. The most commonly used algorithms are genetic algorithms, simulated-annealing algorithms, and stochastic parallel gradient-descent (SPGD) algorithms. In 2011, Yang [114] studied the influence of optimization algorithms, such as those listed above, on the system-correction effect and found that the simulated-annealing algorithm required the shortest correction time. In 2012, Wang [115] studied the application of a genetic algorithm in laser shaping. The simulation results showed that the wavefront-sensor correction system based on a genetic algorithm could increase the Strehl ratio of the system from 0.3771 to 0.9049. In 2016, Anzuola [116] studied the ability of the SPGD algorithm and its modal version (M-SPGD) to correct AO systems. The results show that M-SPGD has more advantages than SPGD, including a fast convergence speed.

1.5.3 Vortex Beam Phase Distortion Correction One of the most important characteristics of vortex beams is their ability to carry orbital angular momentum. Theoretically, the topological charge of the vortex beam can be any value, usually an integer or fractional order. Owing to the orthogonal nature of OAM, it is only necessary to select vortex beams with different topological charges to have their orbital angular momentums be different. This produces an infinite-dimensional vortex beam that enables vortex-beam overlay multiplexing. In a free-space optical communication system, the vortex beam is first used as a carrier wave. The information is loaded onto the carrier wave, multiplexed, and then transmitted through the channel to realize OAM multiplexing communication. This OAM multiplexing mode for optical communication significantly improves the channel capacity. Vortex-beam applications in the field of optical communications face the following problems: First, when vortex beams are applied to optical communications, OAM multiplexing technology is mainly used to achieve information-multiplexing transmissions. Research on issues related to OAM multiplexing transmissions is also particularly important, including how to multiplex multiple vortex beams, how to reduce the OAM mode change after OAM multiplexed beams are transmitted through turbulent channels, and how to avoid the generated mode-crosstalk problem between beams when multiplexing. The second problem involves the optical field characteristics of the vortex beam and the change characteristics of the OAM mode after passing through an atmospheric channel. At present, most investigations use numerical simulation methods or shortdistance simulations in the laboratory; however, actual optical communication must be realized in a real atmospheric-turbulence environment. Therefore, it is necessary

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1 Introduction

to study the optical-field characteristics of a vortex beam after transmission in an actual atmospheric environment and the change characteristics of the OAM mode. The third problem involves using AO correction technology to suppress the effect of atmospheric turbulence and reduce the crosstalk effect of multiplexed vortex beams. The wavefront-distortion correction of a vortex beam is divided, depending on whether a wavefront sensor is present in the system. The adaptivecorrection technology can be divided into two categories: with and without wavefront sensors. The classic method uses the Shack–Hartmann (SH) wavefront sensor [117]. The non-wavefront sensor method mainly includes three types: the Gerchberg–Saxton (GS) algorithm [118], the SPGD algorithm [76]and the phase diversity (PD) algorithm[119]. In 2015, Xie et al. [120] proposed a stochastic parallel gradient-descent (SPGD) algorithm based on a Zernike polynomial for correcting multiple distorted OAM beams. The crosstalk between modes can be reduced by 5 dB, and the Zernike polynomial coefficients of the corrected modes can be obtained from the feedback loop. In 2018, Baranek et al. [121] corrected the vortex-beam optical aberration by introducing optimized spiral-phase modulation combined with the GS algorithm, and further combined it with SLM into an adaptive correction system. The results showed that the accuracy and efficiency of the GS-algorithm aberration correction could be significantly improved by using the vortex image spot as the target intensity mode during the iteration process. In 2018, Xu et al. [119] investigated a dual-correction method by combining a liquid–crystal adaptive optics (AO) technique and PD (Priority-driven scheduling) algorithm. They first used the AO correction technique to compensate for the aberrations caused by turbulent disturbances, and then used the PD algorithm to correct for residual wavefront aberrations. Figure 1.22 shows the structure of the dual-correction AO system combining LC-SLMs and the PD algorithm. Compared to AO correction technology with a wavefront sensor, AO correction technology without a wavefront sensor has the advantages of a simple hardware implementation and good adaptability to complex environments, such as light-intensity flicker. Therefore, AO has become increasingly popular.

1.6 Separation and Detection of Vortex Beams OAM state-detection methods can be divided into four main types: 1. Using a fork-shaped diffraction grating to convert a specific OAM light into a Gaussian beam in the diffraction direction; 2. Having the OAM light interfere with the Gaussian beam and distinguish it by the interference patterns of different modes; 3. Passing the OAM light through slits or small holes to produce diffraction patterns to distinguish different modes;

1.6 Separation and Detection of Vortex Beams

31

Fig. 1.22 Structural diagram of a double-correction adaptive-optical correction system based on an LC-SLM combined with the PD algorithm [119]

4. Using optical elements to reconstruct the wavefront of OAM light to make it easy to distinguish.

1.6.1 Fork Grating When a Gaussian beam passes through a cross-shaped grating, a vortex beam is generated. A vortex beam can also be converted into a Gaussian beam after passing through the corresponding cross-shaped grating. Therefore, the topological charge of the vortex beam can be detected, as shown in Fig. 1.23.

1.6.2 Interference Characteristics The superposition and interference of two vortex beams with different topological charges can be used to study their phases and other characteristics. The basis of these studies is the interference of plane waves and vortex light, as well as spherical waves and vortex light. The electric and magnetic fields of a plane wave are on the same plane, and the space occupied by the propagation process is called the wave field [122]. Many

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Fig. 1.23 Use a fork grating to detect a vortex beam

vibration modes are also present in the wave field. The sine-function form is a relatively common vibrational mode. A wave surface exists at any point when the wave propagates. The wavefront in the direction of propagation is called the wavefront or wave surface. The electric-field expression of a plane wave is E 1 = A1 exp(−ikx), and the electric-field expression of a vortex beam is E 2 = A2 exp(ilθ ), where k = 2π/λ. k is the plane wave number, l is the topological charge of the vortex beam, and θ is the phase angle. Let A1 = A2 = A0 and A0 , A1 , A2 be constants; then, the amplitude after superposition is / E = E 1 + E 2 = A0 exp(i2π x λ) + A0 exp(ilθ ).

(1.5)

Using the light-intensity calculation formula, I = E E ∗ , we can obtain: [ / ] I = A20 2 + 2 cos(2π x λ − lθ ) .

(1.6)

From Eq. (1.6), the interference image of the plane wave and vortex beam can be obtained. Figure 1.24 shows the interference images of the plane wave and the vortex light when the topological charges are 0, 1, −1, 2, 0.5, and 1.5, respectively. Figure 1.24a shows a graph of the topological charge with a straight stripe-like pattern, and the following raster images have different degrees of fork-like dislocation. When the topological charge is 0, it can be regarded as an ordinary Gaussian beam. Figure 1.24c, d, e demonstrate that when the topological charge takes different values, the number of bifurcations at the center of the interference image differs, and the number of bifurcations is the same as the topological charge. Figure 1.24b, c show that if the sign of the topology is different, the opening direction of the image is opposite. Of course, in theory, not only integer values but also fractions can be used. The corresponding interference images are shown in the figure. For fractional interference images, half of the fringes in the figure appear laterally misaligned, and the black and white fringes appear misaligned. This shows that the

1.6 Separation and Detection of Vortex Beams

33

(a) l = 0

(b) l = − 1

(c) l =1

(d) l = 2

(e) l = 0.5

(f) l =1.5

Fig. 1.24 Plane-wave interference pattern [122]

plane wave and interference image of the vortex light are produced by the dislocation of straight stripes. When combining the integer order and fractional order, the number of bifurcations is equal to the topological charge; that is, the number of dislocations of the interference fringe is the same as the topological charge. Therefore, the interference image is also called a straight-stripe misaligned grating, or fork grating.

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A spherical wave is a light wave with an equal-phase sphere and equal amplitude. In an ideal situation, the light emitted by an ordinary point light source is spherical. The spherical coordinate equation can be used to discuss the interference between spherical waves and vortex beams [123]: E = E1 + E2 =

/ A0 exp(i2πr λ) + A0 exp(ilθ ). r

(1.7)

If the two beams of light are a spherical wave and a vortex beam, the light intensity of the interference superposition can be expressed as ] [ 2 1 I = E E ∗ = A20 1 + 2 + cos(kr − lθ ) . r r

(1.8)

Similarly, through the computer numerical simulation of Eq. (1.8), the ideal vortex light and spherical wave can be simulated and calculated, and the corresponding interference image can be obtained, as shown in Fig. 1.25. It can be seen intuitively that the interference between the spherical wave and the vortex light is different from that of the plane wave. It has light and dark stripes spirally rotating around a center point, instead of straight stripes with alternating light and dark stripes. The distribution characteristics of the interference fringes are also related to the value of the topological charge, so if its value changes, the spiral shape with different values will change accordingly. The rotating fringe at the center position is consistent with the topological charge value. As shown in Fig. 1.25a, when the topological charge is 1, the spiral stripe produces a rotating stripe from the center point. As shown in Fig. 1.25c, when the topological charge is 2, the spiral stripe is at the singular point. There are two rotating stripes at the position, and the direction of rotation is opposite. By analogy, the relationship between the topological charge and the spiral fringe can be obtained. Similar to the cross-shaped grating, the topological charge is the same as the number of rotating fringes. In addition, the topological charge can not only be an integer, but also a fraction. Similar to plane-wave interference, when the topological charge is a fraction, a dislocation phenomenon occurs. As shown in Fig. 1.25e, when the topological charge is 0.5, the fringes of the interference image of the vortex light and spherical light move half a fringe to the right, which means that, regardless of whether it is an integer or a fraction, it is an interference pattern caused by the dislocation of the fringes.

1.6.3 Diffraction Characteristics Because the vortex beam has an OAM and a spiral phase structure, it is particularly important to study the diffraction phenomenon of the vortex beam passing through the spot. In 2006, Sztul et al. [122] conducted a double-slit interference experiment

1.6 Separation and Detection of Vortex Beams

(a) l = 1

35

(b) l = − 1

(c) l = 2

(d) l = 3

(e) l = 0.5

(f) l = 1.5

Fig. 1.25 Spiral interferograms [122]

using a vortex beam. The double-slit interference fringe of the vortex beam was used to measure the OAM of the vortex beam. In the same year, Soares et al. [124] demonstrated a new measurement of the topological charge of a vortex beam carrying orbital angular momentum, which is the diffraction phenomenon of a triangular aperture. In 2009, Ghai et al. [125] studied the intensity distribution of Laguerre–Gaussian (LG) beams diffracted using a single slit. After the LG beam passes through a single

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1 Introduction

slit, the diffraction fringes break and bend. The direction and degree of the bending are closely related to the topological charge. In 2011, Jixiong [99] conducted a doubleslit interference experiment using an optical vortex, and measured its orbital angular momentum. The light-intensity distribution and helical spectrum of the LG beam after a single-slit diffraction were studied, and the results showed that the spiral spectrum of the LG beam widened after passing through a single slit; thus, it can be used to measure the OAM of the optical vortex. These studies are the result of detecting the OAM of a single mode. In 2014, Zhou [126] developed a dynamic double-slit interference system using the rotation of double slits on an opaque screen, where the beam is scanned while the double-slit output optical power fluctuates between high and low. This property can be used to detect OAM states. This method can measure higher-order OAM with improved robustness. A schematic diagram of the dynamic double-slit interference system is shown in Fig. 1.26. The device consists of an opaque screen with two air slits, a Fourier lens, and a photodetector. The black circle in Fig. 1.26 transmits the incident OAM beam; then, the lens moves the double-slit interference pattern to the center, and the photodetector converts the light intensity to a voltage value. The voltage is minimized when the interference is phase-dissipated and maximized when it is coherent. From Fig. 1.27, the topologicalcharge magnitudes of the incident OAM beam can be obtained, based on the number of wave peaks between the two-phase extinction fringes. In 2018, Acevedo et al. [127] measured OAM by simulation and experimentally using rectangular and pentagonal apertures, and found that a non-equilateral pentagon could detect topological-charge magnitudes and signs up to 20; however, the rectangular aperture could only detect magnitudes. In 2016, Fu et al. [128] extended the detection range of Dammann vortex gratings from −12 to + 12 to −24 to + 24 by synthesizing them with −12 to + 12 order spiralphase diagrams. They also detected single-mode or superposition-state OAMs by the appearance of solid bright spots in the far-field diffraction patterns [121]. Figure 1.28 shows the far-field diffraction pattern corresponding to a synthetic Dammann vortex Fig. 1.26 Schematic diagram of dynamic double-slit interference [126]

1.6 Separation and Detection of Vortex Beams

37

Fig. 1.27 Receiving power curve when topological charge is ±40 a Rectangular coordinates; b Polar coordinates [126]

Fig. 1.28 Far-field diffraction pattern corresponding to synthetic 5 × 5 Daman vortex grating a superimposed state OAM beam; b far-field diffraction pattern observed by CCD1; (c) far-field diffraction pattern observed by CCD2 [128]

grating. The superimposed OAM beam can be detected, based on the appearance of solid bright spots in the beam array. Previous studies were able to detect low-order OAM modes. In 2020, Li [129] designed a gradually-changing period spiral-spoke grating (GCPSSG) that combines a spiral phase diagram with an axicon hologram and a gradually-changing periodphase grating to detect topological-charge numbers up to 160. These methods can detect the topological charge of a vortex beam. In the same year, Li [130] designed a spiral phase grating that could detect the angular indices, as well as the radial indices, by judging the distribution pattern of the far-field spots. Figure 1.29 shows the far-field diffraction results of the LG beam when the radial index is not zero. It can be seen that the light field changes from the original Hermite-like Gaussian spot to a combination of multiple Hermite-like Gaussian spots. The total number of bright spots is ( p + 1) × ( p + |l| + 1); that is, the value of l + p is equal to the number of dark stripes of a single Hermite–Gaussian-like spot, while the value of p + 1 is equal to the number of Hermite–Gaussian-like spots. The orientation of the light field corresponds to the sign of the topological-charge number. This allows the radial indices and topological-charge numbers of LG beams to be measured simultaneously. These studies present some of the results of conventional singlemode (OAM) detection. In the same year, Araujo et al. [131] numerically simulated

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Fig. 1.29 Far-field simulation results of an incident LG beam with a non-zero radial index diffracted by a helical phase grating [130]

femtosecond vortices and vortex beams with noninteger topological charges using triangular-aperture diffraction. In 2016, Brandao et al. [132] analyzed the diffraction pattern of a partially coherent vortex beam passing through a triangular aperture and detected the topological charge of the vortex beam, thereby proving that this method is also applicable to partially coherent beams.

1.6.4 Reconstructed Wavefront In 2010, Lavery et al. [133] proposed an efficient OAM-state detection method; the detection process is shown in Fig. 1.30. This method is based on the principle of static optics. It uses a coordinate transformation to convert a beam with a spiral phase into a beam with a lateral phase gradient. It then focuses different OAM states on different lateral positions through a lens, thereby distinguishing the different states through the lateral position. However, because each OAM state has a certain width in the lateral position, this will cause the light spots of the adjacent lateral positions to overlap. When the overlap area is large, the OAM state cannot be distinguished. Later in 2013, Mirhosseinil et al. [134] improved Berkhout’s results by incorporating a “copying technique” (fan-out) algorithm for the beam to make the separated spot finer and increase the separation efficiency from 77 to 92%. In 2019, Ruffato et al. [135] built on their previous work by integrating two key optical components of the

1.6 Separation and Detection of Vortex Beams

39

coordinate-transformation method, the beam unwrapper and phase corrector, on the same side of a transparent-quartz sheet to implement a compact OAM mode sorter. As shown in Fig. 1.31b, both phase diagrams were prepared in the form of pure phase-diffraction optics using high-resolution electron-beam lithography to etch the phase diagrams onto a thin resistive layer. This method enables the detection of OAM-mode separation from −10 to 10, including single-mode and superimposed states in the non-paraxial condition. This significantly improves the system integration, while reducing alignment difficulties. Finally, the replication of synthetic optics devices by rapid mass-production techniques (nanoimprint lithography) allows for high throughput and low cost, while enabling integration into OAM mode-division multiplexed optical platforms. Gao et al. [136] studied the generation of superposition states and the separation and measurement of superposition states using a Mach–Zehnder (M-Z) interferometer. An M-Z interferometer with Dove prisms on two arms was used to separate the OAM states with odd and even angular quantum numbers, realize the OAMstate measurement of the vortex beam, and provide the corresponding experimental results. The M-Z interferometer with Dove prisms on both arms can separate spiral beams carrying different OAMs. In addition to the above conventional coordinate-transformation methods, some unconventional geometric-transformation methods improve the system performance with slight improvements on the original ones. In 2017, Zhao et al. [137] proposed a coordinate-transformation method for separating OAM modes with radially varying phases, where the OAM topological charge and radially varying phase determine the horizontal and vertical positions of the diffracted spot. Simulation and experimental Transmitting terminal

Receiving terminal

Transmitting channel

Information of light path 1

Modulator

Laser

Optical beam splitter

SLM Optocoupler

Modulator

Free space

SLM2

Lens L1

SLM3

SLM CCD detector

Lens L2

Information of light path N

Fig. 1.30 Schematic diagram of an OAM-state multiplexing scheme based on efficient OAM-state separation method. Note SLM stands for a spatial light modulator, used to generate LG light of different values; SLM2 is used to transform the light field coordinates; SLM3 is used to correct the phase of the light field; L1 is a Fourier conversion lens; L2 is used to focus the light; a CCD detector can convert optical images into digital signals.

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1 Introduction

Fig. 1.31 Schematic diagram of two OAM state log-polar separation systems a Traditional logpolar classification system: separated coaxial unwrapping and phase corrector; b Non-paraxial compact system: two elements integrated on the same surface of a transparent plate with reflective backside [135]

results showed that two to three superimposed OAM modes could be separated, and the separation efficiency was improved, compared to other studies [133, 134]. In 2018, Chen et al. [138] spirally transformed an OAM beam, and the phase diagram corresponding to the OAM beam was represented by a spiral line. Then, the collected spiral line was converted to a parallel line to achieve n-fold spot spreading. Simulations and experiments showed that this method could improve the optical refinement by nearly three times, compared to the log-polar transformation, and could finally separate the high-resolution OAM pattern. Multiplexing communication systems that use different OAM states can significantly increase the information capacity of the system. However, in the existing

References

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OAM-state multiplexing system, the detection of each OAM state requires an independent detection branch, which limits high-speed data. The transmission, effective detection, and separation of OAM-state information in multiplexing systems have become a problem that urgently needs to be solved. Therefore, to better detect and separate OAM states, it is necessary to further improve the efficiency of OAMstate separation. Then, the performance of the entire communication system can be effectively adjusted.

References 1. Yuan XC, Jia P et al (2014) Optical vortices and optical communication with orbital angular momentum. J Shenzhen Univ Sci Eng 31(4):331–346 2. Whewell W (1833) Essay towards a first approximation to a map of cotidal lines. Proc R Soc Lond 3(1):188–190 3. Richard B, Wolf E (1959) Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system. Proc R Soc Lond A 253(1274):358–379 4. Boivin A, Wolf E (1965) Electromagnetic field in the neighborhood of the focus of a coherent beam. Phys Rev Lett 138(6B):1561–1565 5. Carter WH (1973) Anomalies in the field of a Gaussian beam near focus. Opt Commun 7(3):211–218 6. Nye JF, Berry MV (1974) Dislocations in wave trains. Proc R Soc Lond A 336(1605):165–190 7. Baranova NB, Zeldovich BY, Mameav AV et al (1981) Dislocations of the wavefront of a speckle inhomogeneous field. Jetp Lett 33(4):206–210 8. Baranova NB, Zeldovich BY, Mameav AV et al (1982) An investigation of the dislocation density of a wave front in light fields having a speckle structure. Zhurnal Eksperimentalnoi I Teroreticheskoi Fiziki 83(52):1702–1710 9. Swartzlander GA, Law CT (1992) Optical vortex solitons observed in kerr nonlinear media. Phys Rev Lett 69(17):2503–2506 10. Voitsekhovich VV, Kouznetsov D, Moronov DK (1998) Density of turbulence induced phase dislocations. Appl Opt 37(21):4525–4535 11. Science (2014) The best satellite photo of the tide map of the Spanish island. Modern Sci 294(4):36 12. Bouchal Z, Celechovsky R (2004) Mixed vortex states of light as information carriers. New J Phys 6(6):1–15 13. Lv H (2011) Research on the orbital angular momentum of vortex beam for space photon communication. Doctoral Dissertation of Xi’an University of Technology, pp 1–2 14. Lu XH, Huang HQ, Zhao CL et al (2008) Vortex beam and optical vortex. Prog Laser Optoelectron 45(1):50–56 15. Lin J, Yuan XC, Tao SH et al (2007) Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states. Appl Opt 6(21):4680–4685 16. Awaji Y, Wada N, Toda Y (2010) Demonstration of spatial mode division multiplexing using Laguerre-Gaussian mode beam in telecom-wavelength. In: 2010 23rd annual meeting of the IEEE photonics society. Denver, CO, USA 17. Fazal IM, Wang J, Yang JY et al (2011) Demonstration of 2-Tbit/s data link using orthogonal orbital-angular-momentum modes and WDM. In: Frontiers in optics 2011. San Jose, California United States 18. Wang J, Yang JY, Fazal IM et al (2012) Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat Photonics 6(7):488–496 19. Huang H, Xie GD, Yan Y et al (2013) 100 Tbit/s free-space data link using orbital angular momentum mode division multiplexing combined with wavelength division multiplexing. In:

42

20.

21.

22.

23. 24.

25.

26.

27.

28. 29. 30. 31. 32.

33.

34.

35. 36. 37. 38. 39.

1 Introduction 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC). Anaheim, CA, USA, pp 1–3 Huang H, Xie GD, Yan Y et al (2014) 100 Tbit/s free-space data link enabled by threedimensional multiplexing of orbital angular momentum, polarization, and wavelength. Opt Lett 39(2):197–200 Wang J, Li SH, Luo M et al (2014) N-dimentional multiplexing link with 1.036-Pbit/s transmission capacity and 112.6-bit/s/Hz spectral efficiency using OFDM-8QAM signals over 368 WDM pol-muxed 26 OAM modes. In: 2014 the European conference on optical communication. Cannes, France, pp 1–3 Tamagnone M, Craeye C, Perruisseay-Carrier J (2012) Further Comment on Encoding many channels on the same frequency through radio vorticity: first experimental test. New J Phys 14(3):811–815 Krenn M, Fickler R, Fink M et al (2014) Twisted light communication through turbulent air across Vienna. 16(11):1–9 Xu ZD, Gui CC, Li SH et al (2014) Fractional Orbital Angular Momentum (OAM) free-space optical communications with atmospheric turbulence assisted by MIMO equalization. Integr Photonics Res, Silicon Nanophotonics Huang H, Cao YW, Xie GD et al (2014) Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization. Opt Lett 39(15):4360–4363 Huang H, Xie GD, Ren YX et al (2013) 4×4 MIMO equalization to mitigate crosstalk degradation in a four-channel free-space orbital-angular-momentum-multiplexed system using heterodyne detection. In: European conference and exhibition on optical communication. IET, pp 708–710 Ren YX, Wang Z, Xie GD et al (2016) Demonstration of OAM-based MIMO FSO link using spatial diversity and MIMO equalization for turbulence mitigation. In: Optical Fiber Communications Conference and Exhibition (OFC) Shi C, Dubois M, Wang Y et al (2017) High-speed acoustic communication by multiplexing orbital angular momentum. Proc Natl Acad Sci USA 114(28):7250–7253 Bozinovic N, Yue Y, Ren Y et al (2013) Terabit-scale orbital angular momentum mode division multiplexing in fibers. Science 340(6140):1545–1548 Wang A, Zhu L, Wang L et al (2018) Directly using 88 km conventional multi-mode fiber for 6-mode orbital angular momentum multiplexing transmission. Opt Express 26(8):100381 Baghday J, Byrd M, Li WZ et al (2015) Spatial multiplexing for blue lasers for undersea communications. In: Proceedings of SPIE. SPIE, Cardiff Joshua B, Keith M, Kaitlyn M et al (2016) Underwater optical communication link using orbital angular momentum space division multiplexing. In: Oceans 2016-MTS/IEEE. Monterey, CA, USA. IEEE, Monterey Miller JK, Morgan KS, Li W et al (2017) Underwater optical communication link using polarization division multiplexing and orbital angular momentum multiplexing. In: Oceans. Anchor age, AK, USA, IEEE, Anchorage Zhao YF, Xu J, Wang AD et al (2017) Demonstration of data-carrying orbital angular momentum-based underwater wireless optical multicasting link. Opt Express 25(23):28743– 28751 Jonathan L, Barry J, Jacqui R et al (2010) Quantum correlations in optical angle-orbital angular momentum variables. Science 329(5992):662–665 Coullet P, Gil L, Rocca F (1989) Optical vortices. Opt Commun 73(89):403–408 Omatsu T, Miyamoto K, Lee AJ (2017) Wavelength-versatile optical vortex lasers. J Opt 19(12):123002 Kano K, Kozawa Y, Sato S (2012) Generation of purely single transverse mode vortex beam from a He-Ne laser cavity with a spot-defect mirror. Int J Opt 359141 Thomas GM, Minassian A, Sheng X et al (2016) Diode-pumped Alexandrite lasers in Qswitched and cavity-dumped Q-switched operation. Opt Express 24(24):27212–27224

References

43

40. Forbes A (2017) Controlling light’s helicity at the source: orbital angular momentum states from lasers. Philos Trans R Soc A: Math Phys Eng Sci 375(2087):20150436 41. Beihersbergen MW, Allen L, Der VV et al (1993) Astigmatic laser mode converters and transfer of orbital angular momentum. Opt Commun 96(1–3):123–132 42. Mcgloin D, Neil SB, Padgett MJ (1998) Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam. Appl Opt 37(3):469–472 43. Yao AM, Padgett MJ (2011) Orbital angular momentum: origins, behavior and applications. Adv Opt Photonics 3(2):161–204 44. Wu W, Sheng Z, Wu H (2013) Design and application of flat spiral phase plate. Acta Phys Sin 68(5):054102 45. Heckenberg NR, Mchuff R, Smith CP et al (1992) Generation of optical phase singularities by computer-generated holograms. Opt Lett 17(3):221–223 46. Yanming Li JK, Escuti MJ (2012) Orbital angular momentum generation and mode transformation with high efficiency using forked polarization gratings. Appl Opt 51(34):8236–8245 47. Saitoh K, Hasegawa Y, Tanaka N et al (2012) Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates. J Electron Microsc 61:171 48. Bai Y, Lv H, Fu X et al (2022) Vortex beam: generation and detection of orbital angular momentum [Invited]. Chinese Opt Lett 20(1):012601 49. Mirhosseini M, Loaiza OSM, Chen C, Rodenburg B, Malik M, Boyd RW (2013) Rapid generation of light beams carrying orbital angular momentum. Opt Express 21(25):30196– 30203 50. Hu XB, Rosales GC (2022) Generation and characterization of complex vector modes with digital micromirror devices: a tutorial. J Opt 24:034001 51. Bin B, Menkeneimule, Zhao JL et al (2012) Generation of vortex beams with a reflected type phase only LCSLM. J Ptoelectronics·Laser 12, 23(1):74–78 52. Forbes A, Dudley A, McLaren M (2016) Creation and detection of optical modes with spatial light modulators. Adv Opt Photonics 8:200–227 53. Liu R, Li F, Padgett MJ, Phillips DB (2015) Generalized photon sieves:fine control of complex fields with simple pinhole arrays. Optica 2:1028 54. Devlin RC, Ambrosio A, Rubin NA et al (2017) Arbitrary spin-to-otbital angular momentum conversion of light. Science 358(6365):896–901 55. Dorrah AH, Rubin NA et al (2021) Structuring total angular momentum of light along the propagation direction with polarization-controlled meta-optic. Nat Commun 12:6249 56. Wang J, Yang JY, Fazal IM et al (2011) 25.6-bit/s/Hz spectral efficiency using 16-QAM signal over pol-muxed multiple orbital-angular-monmentum modes. In: IEEE Photonics Conference (PHO). Arlington, IEEE, pp 587–588 57. Yan Y, Yang Y, Huang H et al (2012) Efficient generation and multiplexing of optical orbital angular momentum modes in a ring fiber by using multiple coherent inputs. Opt Lett 37(17):3645–3647 58. Yan Y, Wang J, Zhang L et al (2011) Fiber coupler for generating orbital angular momentum modes. Opt Lett 36(21):4269–4271 59. Brunet C, Vaity P, Messaddeq Y et al (2014) Design, fabrication and validation of an OAM fiber supporting 36 states. Opt Express 22(21):26117–26127 60. Pidishety S, Pachava S, Gregg P et al (2017) Orbital angular momentum beam excitation using an all-fiber weakly fused mode selective coupler. Opt Lett 42(21):4347–4350 61. Yue Y, Zhang L, Yan Y et al (2012) Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber. Opt Lett 37(11):1889–1891 62. Yue Y, Yan Y, Ahmed N et al (2012) Mode properties and propagation effects of optical Orbital Angular Momentum (OAM) modes in a ring fiber. IEEE Photonics J 4(2):535–543 63. Israk MF, Razzak MA, Ahmed K et al (2020) Ring-based coil structure photonic crystal fiber for transmission of Orbital Angular Momentum with large bandwidth: Outline, investigation and analysis. Opt Commun 473(15):126003 64. Cai XL, Wang JW, Cai XL et al (2012) Integrated compact optical vortex beam emitters. Science 338(6105):363–366

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65. Markin DM, Solntsev AS, Sukhorykov AA (2013) Generation of orbital angular momentum entangled biphotons in triangular quadratic waveguide arrays. Phys Rev A 87(6):063814 66. Guan BB, Scott RP, Fontaine NK et al (2013) Integrated optical orbital angular momentum mulplexing device using 3-D waveguides and a silica PLC. In: Conference on Lasers and Electro-Optics (CLEO). San Jose, CA, USA 67. Zhang DK, Feng X, Cui KY et al (2013) Generating in-plane optical orbital angular momentum beams with silicon waveguides. IEEE Photonics J 5(2):2201206 68. Cognee KDHLPKA (2020) Generation of pure OAM beams with a single state of polarization by antenna-decorated microdisk resonators. ACS Photonics 7:43 69. Wong GK, Kang MS, Lee HW et al (2012) Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber. Science 337(6093):446–449 70. Zeng X et al (2022) Nonreciprocal vortex isolator via topology-selective stimulated Brillouin scattering. Sci Adv 8(42):eabq6064 71. Huang S (2013) The propagation of beam array through atmosphric turbulence in a slanted path. Master Thesis of Beijing University of Posts and Telecommun-ications, pp 27–28 72. Han J, Liu J (2007) Engineering optics. Xidian University Press, pp 288–305 73. Wei HY (2006) Study on the characteristic of laser beam in the slant path through the atmospheric turbulence. Master Thesis of Xidian University 74. Molina TG, Terriza G, Torres JP et al (2002) Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum. Phys Rev Lett 88(1):013601 75. Lukin VP, Konyaev PA, Sennikov V (2012) Beam spreading of vortex beams propagating in turbulent atmosphere. Appl Opt 51(10):C84–C87 76. Gibson G, Couttial J, Padgett MJ et al (2004) Free-space information transfer using light beams carrying orbital angular momentum. Opt Express 12(22):5448–5456 77. Xiao XL, Wang BY, Cheng SG (2015) Evolution of phase singularities of vortex beams propagating in atmospheric turbulence. J Opt Soc Am A 32(5):837–842 78. Zhu KC, Hou GQ, Li Xu XG et al (2008) Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere. Opt Express 16(26):21315–21320 79. Wang T, Pu JX, Rao LZ (2007) Propagation of partially coherent vortex beams in the turbulent atmosphere. Opt Tech 33(S1):4–6 80. Wang T, Pu JX, Chen ZY (2009) Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere. Opt Commun 282(7):1255–1259 81. Jiang YS, Wang SH, Zhang JH et al (2013) Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence. Opt Commun 303(16):38–41 82. Liu XH, Pu JX (2011) Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence. Opt Express 19(27):26444–26450 83. Wang F, Liang CH, Yuan YS et al (2014) Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment. Opt Express 22(19):23456–23464 84. Tang MM, Zhao DM (2015) Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media. Opt Express 23(25):32766–32776 85. Chen R, Liu L, Zhu SJ et al (2014) Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere. Opt Express 22(2):1871–1883 86. Liu HL, Lu YF, Xiao J et al (2016) Radial phased-locked partially coherent flat-topped vortex beam array in non-Kolmogorov medium. Opt Express 24(17):19695–19712 87. Chen YH, Wang F, Yu JY et al (2016) Vector Hermite-Gaussian correlated Schell-model beam. Opt Express 24(14):15232–15250 88. Ding PF, Pu JX (2012) Change of the off-center Laguerre-Gaussion vortex beam while propagation. Acta Phys Sin 61(06):198–203 89. Xu Y, Tian H, Feng H et al (2016) Propagation factors of standard and elegant LaguerreGaussian beams in non-Kolmogorov turbulence. Optik 127(22):10999–11008 90. Yan JW, Yong KL, Tang SF et al (2020) Comparison of propagation characteristics between super gaussian and gaussian vortex beam in air. Acta Optica Sinica 40(2):17–22

References

45

91. Luo CK, Lu F, Miao ZF et al (2019) Propagation and spreading of radial vortex beam array in atmophere. Acta Optica Sinica 39(6):28–33 92. Hricha Z, Lazrek M, Yaalou M et al (2021) Propagation of vortex cosine-hyperbolic-Gaussian beams in atmospheric turbulence. Opt Quant Electron 53(7):1–15 93. Hyde MW IV (2021) Twisted spatiotemporal optical vortex random fields. IEEE Photonics J 13(2):1–16 94. Hricha Z, Halba EM, Lazrek M et al (2022) Focusing properties and focal shift of partially coherent vortex cosine-hyperbolic-Gaussian beams. J Mod Opt 69(14):779–790 95. Tang M, Zhao D, Li X et al (2018) Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence. Optics Commun 407:392–397 96. Gauri A, Stuti J, Hanuman S et al (2022) Perturbation of V-point polarization singular vector beams. Opt Laser Technol 158:108842 97. Stefanus RF (2010) Decoherence of orbital angular momentum entanglement in a turbulent atmosphere. Physics 10(09):1–4 98. Chen ZY, Chen GW, Zhang GW, Rao LZ et al (2008) Determining the orbital angular momentum of vortex beam by young’s Double-Slit interference experiment. Chin J Lasers 35(7):1063–1067 99. Gao F, Chen BS, Pu JX et al (2011) Intensity distribution and spiral spectra of LaguerreGaussian beam passing through a Single-SlitLaser Optoelectron Prog 48(09):090501–1– 090501–7 100. Liu H, Chen ZY, Pu JX (2009) Detection of the orbital angular momentum of franctional optical vortex beams. J Optoelectron·Laser 20(11):1478–1482 101. Zhang YX, Zhang WH, Qi WH et al (2009) Orbital angular momentum of single photons for optical communication in a slant path atmospheric turbulence. Laser J 30(5):63–65 102. Porfirev A, Kirilenko M, Khonina SN et al (2017) Study of propagation of vortex beams in aerosol optical medium. Appl Opt 56:E5–E15 103. Lazrek M, Hricha Z, Belafhal A (2022) Propagation properties of vortex cosine-hyperbolicGaussian beams through oceanic turbulence. Opt Quantum Electron 54(3):1–14 104. Zou L, Zhao SM, Wang L (2014) The effects of atmospheric turbulence on the orbital angular momentum-multiplexed system. Acta Photonica Sinica 43(9):52–57 105. Zhang L (2014) Study of orbital angular momentum mode crosstalk caused by atmospheric turbulence based on Kolmogorov model in free space optical communication. Acta Photonica Sinica 34(12):s201004 106. Zhou RZ (1996) Adaptive optics. National Defense Industry Press 107. Robert T (2011) Principles of adaptive optics. CRC Press 108. Feng F, White HI, Wilkinson D (2014) Aberration correction for free space optical communications using rectangular Zernike modal wavefront sensing. J Lightwave Technol 32(6):1239–1245 109. Sodnik Z, Armengol JP, Czichy RH et al (2009) Adaptive optics and ESA’s optical ground station. In: Proceedings of spie the international society for optics engineering, pp 7464, 746406 110. Berkefeld T, Soltau D, Czichy R et al (2010) Adaptive optics for satellite-toground laser communication at the 1m Telescope of the ESA Optical Ground Station, Tenerife, Spain. In: Proceedings of spie the international society for optics engineering, pp 7736, 77364 111. Jiang WH (2006) Adaptive optical technology. Chin J Nat 28(1):7–13 112. Xia LJ, Li XF (2010) Transmission wave-front correction of atmospheric optical communication based on adaptive optics technology. J Terahertz Sci Electron Inf Technol 8(3):331–335 113. Hashmi AJ, Ashmi AA, Eftekhar AA, Adibi A et al (2014) Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars. Opt Commun 333(4):120–128 114. Yang HZ, Xin XYLI (2011) Comparison of several stochastic parallel optimization algorithms for adaptive optics system without a wavefront sensor. Opt Laser Technol 43(3):630–635 115. Wang WB, Zhao SA, J, et al (2012) Genetic algorithm used in laser shaping. Laser Infrared 42(10):1115–1119

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116. Anzuola E, Gladysz S, Stein K (2016) Performance of wavefront-sensorless adaptive optics using modal and zonal correction. SPIE Remote Sens 100020J 117. Duan HF, Li E, Wang HY et al (2003) the effect of mode orthogonality on precision of wavefront measurement and correction using hartmann-shack sensor. Acta Optica Sinica 23(9):1143–1148 118. Polsnd SP, Krstajic NA, Knight RD et al (2014) Development of a doubly weighted GerchbergSaxton algorithm for use in multibeam imaging applications. Opt Lett 39(8):2431–2434 119. Xu ZH (2018) Research on high-resolution liquid crystal adaptive optics technique with phase diversity. University of Chinese Academy of Sciences (Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences), pp 32–35 120. Xie G, Ren Y, Huang H et al (2015) Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm. Opt Lett 40(7):1197–1200 121. Baranek M, Behal J, Bouchal Z (2018) Optimal spiral phase modulation in Gerchberg-Saxton algorithm for wavefront reconstruction and correction. In: Thirteenth international conference on correlation optics. SPIE 10612:71–78 122. Wang PP (2013) Investigation on dot-array imaging lidar based on Dammann gratings. Master Thesis of Zhejiang University, pp 12–14 123. Xu JY (2017) The experimental study of important technologies of OAM beam generation and control. Master’s Thesis of Xi’an University of Technology, pp 25–30 124. Sztul HH, Alfano RR (2006) Double slit interference with Laguerre-Gaussian beams. Opt Lett 31(7):999–1001 125. Soares WC, Caetano DP, Fonseca JS et al (2008) Direct determination of light beams’ topological charges using diffraction. In: Conference on lasers and electro-optics, 2008 and 2008 conference on quantum electronics and laser science. CLEO/QELS 2008. IEEE, pp 1–2 126. Ghai DP, Senthilkumaran P, Sirohi RS (2009) Single-slit diffraction of an optical beam with phase singularity. Opt Lasers Eng 35(23):123–126 127. Zhou H, Shi L, Zhang X et al (2014) Dynamic interferometry measurement of orbital angular momentum of light. Opt Lett 39(20):6058–6061 128. Acevedo C, Moreno YT, Guzmán Á et al (2018) Far-field diffraction pattern of an optical light beam with orbital angular momentum through of a rectangular and pentagonal aperture. Optik 164:479-487 129. Fu S, Wang T, Zhang S et al (2016) Integrating 5×5 Dammann gratings to detect orbital angular momentum states of beams with the range of −24 to +24. Appl Opt 55(7):1514–1517 130. Li YX, Han YP, Cui ZW et al (2020) Measuring the topological charge of vortex beams with gradually changing-period spiral spoke grating. IEEE Photonics Technol Lett 32(2):101–104 131. Li YX, Han Y, Cui Z et al (2019) Simultaneous identification of the azimuthal and radial mode indices of Laguerre-Gaussian beams using a spiral phase grating. J Phys D Appl Phys 53(8):085106–085111 132. Arauji ED, Anderson ME (2011) Measuring vortex charge with a triangular aperture. In: CLEO, 2011—laser science to photonic applications. IEEE, pp 1–2 133. Brandao PA, Cavalacnti SB (2016) Topological charge identification of partially coherent light diffracted by a triangular aperture. Phys Lett A 380(47):4013–4017 134. Berkhout GC, Lavery MP, Courtial J et al (2010) Efficient sorting of orbital angular momentum states of light. Phys Rev Lett 105(15):153601 135. Mirhosseini M, Malik M, Shi Z et al (2013) Efficient separation of the orbital angular momentum eigenstates of light. Nat Commun 4(1):1–6 136. Ruffato G, Massari M, Girardi M et al (2019) Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared. Opt Express 27(17):24123–24134 137. Xiao XQ, Qi XQ, Gao CQ (2011) Experimental study of detecting orbital angular momentum states of spiral phase beams. Acta Physica Sinica 60(1):275–279 138. Li C, Zhao S (2017) Efficient separating orbital angular momentum mode with radial varying phase. Photonics Res 5(4):267–270

References

47

139. Vorontsov MA, Sivokon VP (1998) Stochastic parallel-gradient-descent technique for highresolution wave-front phase-distortion correction. J Opt Soc Am A 15(10):2745–2758 140. Wen Y, Chremmos I, Chen Y, et al (2018) Spiral transformation for high-resolution and efficient sorting of optical vortex modes. Phys Rev Lett 120(19):193904

Chapter 2

Vortex-Beam Spatial-Generation Method

Vortex beams have infinite dimensions, in theory. The topological charge can be taken from negative infinity to positive infinity, and as an integer or fraction. The orbital angular momentum (OAM) characteristic also provides a theoretical basis for OAM multiplexing and high-dimensional quantum communication. In this chapter, the basic characteristics of vortex beams are briefly introduced, along with several common vortex beams.

2.1 Basic Principle of Vortex Beams A vortex beam is a type of beam with a special spatial phase. Its phase structure is unique and its phase is continuous. Because its central or axial intensity in the propagation direction is zero, it is also called a dark hollow beam or hollow beam. The vortex beam differs from both the plane wave of an ordinary beam and the spherical shape of a special beam; instead, it presents a spiral shape. with the change in the propagation direction and distance, the phase of the vortex beam is spirally distributed; thus, it is also called a spiral beam. It is has been theoretically verified that vortex beams have orbital angular momentum, and the topological charge in the OAM can be any integer or fraction. Compared with ordinary beams, vortex beams have a continuous helical wavefront. The phase is uncertain in the direction of beam propagation, which is the phase singularity. Because the central light intensity of the beam is zero, it is called a dark empty beam. Its light field expression contains a phase factor exp(ilθ ), where l is the topological charge number The light-field expression can be expressed as [1] E(x, y) = u(r, z) exp(ilθ ) exp(−ikz)

© Science Press 2023 X. Ke, Generation, Transmission, Detection, and Application of Vortex Beams, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-0074-9_2

(2.1)

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Fig. 2.1 Three-dimensional wavefront structure of a vortex beam [1]

(a) I=1

(b) I=2

where u(r, z) represents the optical-field amplitude, exp(ilθ ) is the phase factor, l is the topological charge, and k is the wave number. Equation (2.1) shows that the special properties of the vortex beam are caused by the phase factor, and the phase distribution is determined by z = 0. Compared with plane light, when a vortex beam rotates, the phase changes by 2πl, and the phase distribution can be expressed as [2]: φ(r, θ, z) = lθ + kz

(2.2)

According to Eq. (2.2), the phase wavefront is spiral, the central intensity is zero, and a dark core is formed, which is a phase singularity. After the beam is focused, an annular light-intensity map is obtained. The phase-simulation diagram of the vortex beam during the propagation process is shown in Fig. 2.1 [1]. Modulations by amplitude, frequency, phase, and polarization are properties that can be used by traditional beams. Because the vortex beam is different from ordinary laser beams, it has characteristics that they do not have. These characteristics can be used as important means for carriers to carry information for data transmission and communication. Specifically, a vortex beam has a new degree of freedom called orbital angular momentum. Ordinary carrier waves do not have this characteristic in traditional communication mode. Figure 2.2 shows the light-intensity distribution and phase distribution of vortex beams with orders of 1–4 at the cross section in the transmission direction. As shown in Fig. 2.2, the central light intensity of the vortex beam is zero, and the larger the order is, the larger the central dark spot is. The phase distribution of the vortex beam continuously changes. The variation range is [0, 2πl] where l represents the topological charges. In Fig. 2.2, because the topological charges are 1–4, the phase variation ranges are [0, 2π ], [0, 4π ], [0, 6π ], and [0, 8π ], respectively.

2.2 Types of Vortex Beams

(a) First order (b) Second order (c) Third order

51

(d) Fourth order

Fig. 2.2 Intensity distribution and phase distribution of first through fourth-order vortex beams

2.2 Types of Vortex Beams Researchers have established several physical models to study vortex beams. ∗ beams Common vortex beams include Laguerre–Gaussian (LG) beams [3], T E M01 [4], Bessel beams [5, 6], and Hermite–Gaussian (HG) beams [7].

2.2.1 Laguerre–Gaussian Beams The Laguerre–Gaussian beam is a typical laboratory vortex beam. Under a paraxial approximation, the Helmholtz equation can be expressed as follows: ( ) ∂E 1 ∂2 E 1 ∂ ∂E ρ + 2 =0 + 2ik 2 ρ ∂r ∂r ρ ∂ϕ ∂z

(2.3)

The solution to Eq. (2.3) can be obtained as the Laguerre–Gaussian mode, and the expression of its light field is [1, 8]: / E(r, z) =

]. [ ]. [ 2r 2 P0 r2 2 p! | || Lp exp(ilϕ) exp − 2 π( p + |l|)! ω2 (z) ω (z) ω2 (z)

(2.4)

where p is the radial quantum number, p + 1 is the number of hollow rings on the cross section of the beam, L |l| p is the Laguerre polynomial, l is the angular quantum number, P0 is the transmitting power of the laser, and ω(z) is the waist radius of the beam. A Laguerre–Gaussian beam can be converted into a Hermite–Gaussian beam using the geometric mode-transformation method, and the Laguerre–Gaussian beam

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Fig. 2.3 Light-intensity distribution of different l values

(a) LG0 beam intensity 1

(c) LG0 beam intensity 3

(b) LG0 beam intensity 2

(d) LG0 beam intensity 4

can also be diffracted into a Gaussian beam using the computer-generated hologram (CGH) method. The light-intensity distribution of the LG beam is shown in Figs. 2.3 and 2.4. Figure 2.3 shows the light-intensity distribution for different l values. Figure 2.4 shows the light-intensity distribution for different P values. As can be seen from Fig. 2.3, when p = 0, the larger |l| is, the larger the dark spot in the center of the spot distribution is, and the farther the maximum light intensity of the LG beam is from the beam center. Because p + 1 represents the number of radial nodes, the number of rings in the intensity distribution of the LG beam in Fig. 2.3 increases. There is one more ring in LG 11 than in LG 10 . LG 12 has one more ring than LG 11 and two more rings than LG 10 . As can be seen from Figs. 2.3 and 2.4, when the topological charge number l = 1.5 is the same, the overall spot diameter of the LG beam increases gradually with the increase of radial index p; however, the central bright ring diameter decreases gradually. When the radial index p is the same, the overall spot diameter and the central bright ring diameter of the LG beam gradually increase, with the increase of topological-charge number l. In Eq. (2.4), when the condition p = 0, l /= 0 is met, the Laguerre–Gaussian ∗ light beam. Its light-field expression can be expressed beam is also called a T E M01 as [8]: [ ] E(r, θ, z) = E 0 r exp(ilθ ) exp F2 (z) − r 2 /F1 (z) ,

(2.5)

2.2 Types of Vortex Beams

53

Fig. 2.4 Light-intensity distribution of different P values

LG00 beam intensity

(a)

(c)

LG11 beam intensity

(b)

(d)

LG01 beam intensity

LG21 beam intensity

where E 0 is a real number representing the actual amplitude of the beam; wave number k = 2π/λ, and exp(ilθ ) represents the phase factor. F1 (z) and F2 (z) represent the divergence and phase-movement functions of the beam in space, respectively. ∗ beam can be expressed In the polar coordinate system, the light field of the T E M01 as [8]: (

) / kA 2 r2 / + iπ − E(r, θ, z) = E 0 r exp(ilθ ) exp 2ln , A + 2z/ik z + ik A 2

(2.6)

where A is a constant whose value is related to the initial girdle radius.

2.2.2 Bessel Beams The higher-order Bessel beam does not diffuse in a cross section perpendicular to the direction of light propagation. In 1987, when studying the solution of a wave equation in free space, Durnin et al. [9] found that there was a “special” solution of the wave equation in free space. The beam represented by this “special” solution could be described using the expression of a bessel function, so it was called a Bessel beam.

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When a beam propagates along an axial straight line in free space, the expression of its light field is a particular solution of the Helmholtz equation, which can be expressed as [1, 8] E(r, θ, z, t) =

A Jl (kr r ) exp(ilθ ) exp(ik z z), ω(z)

(2.7)

where ω(z) is the beam-waist width at a propagation distance of z, A is the normalization constant, Jl is the lth-order Bezier function, l is the order or topological charge of the vortex beam, and kr and k z are the components of the wave vector of the beam in the radial and propagation directions, respectively. When α meets the condition 0 < α < ω/c, the intensity distribution of the Bessel beam at the cross section of its propagation direction is expressed as [8] I (r, θ, z) = |E 0 Jm (αr )|2 ,

(2.8)

Equation (2.8) indicates that the transverse light strength of a Bessel beam is almost unchanged, regardless of the transmission distance. In other words, the Bessel beam does not diverge in the direction of propagation, which verifies its non-diffraction property. Theoretically, a Bessel beam can carry an infinite amount of energy, violating the law of energy conservation [1]. To investigate this problem, Gori et al. [10] experimentally produced a non-diffraction Bessel beam, based on an ideal Bessel beam, which laid a foundation for studying the characteristics and applications of Bessel beams.

2.2.3 Hermite–Gaussian Beams Higher-order Hermite–Gaussian beams and fundamental-mode Gaussian beams satisfy a certain relationship; that is, Hermite–Gaussian beams can be obtained by the differential equation of fundamental-mode Gaussian beams. The electromagnetic expression of a Hermite–Gaussian beam is [11] [ ] ] [/ ik −1 E p(r ) = E 0 exp − r. Q −1 ik. r Q r . , r . H · p ε h 2

(2.9)

where E 0 is the amplitude constant, Q −1 e is the complex curvature tensor of the matrix,H p is the Hermite polynomial, and r is the position vector on the beam cross section, its energy density is [11] w = |xε0 | = |u(x, y)|2 ,

(2.10)

2.3 Vortex-Beam Generation Methods

55

where E is the electric-field intensity of the beam, B is the magnetic-induction intensity, and u(x, y) represents the amplitude expression of a high-order Hermite Gaussian beam [11]: u(x, y) = exp(iβ) exp(η)H p (ζ ),

(2.11)

β, η, ζ are all real numbers, and the OAM density of Hermite–Gaussian beams is expressed as [11]: [ ] [ ] ∂u ∗ ∂u ∗ i ωε0 ∗ ∂n ∗ ∂n x(u −u ) −y u −u , jz = 2 ∂y ∂y ∂x ∂x

(2.12)

where ω represents the angular frequency of the beam and u ∗ represents a conjugate. Then, the following [11] can be obtained: [

] ∂a ∂a || ||2 −y up . jz = w0 ε0 x ∂y ∂x

(2.13)

| |2 ]} {[ |u p | = where = exp k[ax 2 + (b + c)x y + dy 2 , a [ 2 ] k 2 − 2 ax + (b + c)x y + dy . The average OAM carried by a single photon of the vortex beam can be obtained as. ˝ hω jz r dφdr dz ˝ . (2.14) Jz = wr dφdr dz where a represents the photon energy and ω is Planck’s constant. It can be seen from the formula that Hermite–Gaussian beams have a higher OAM than Laguerre– Gaussian beams.

2.3 Vortex-Beam Generation Methods There are many methods for generating vortex beams, including the computergenerated hologram method, geometric mode-conversion method, spiral-phase-plate method, and spatial light modulator (SLM) method. The following sections describe the primary methods.

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2.3.1 Computer-Generated Hologram Method In 1992, Heckenberg et al. [2] realized phase coding through the interference fringes between the main light wave and the reference light wave, and obtained a vortex beam by using a hologram generated by the computer. This method can control the topological charge of the generated vortex beam, as shown in Fig. 2.5. The principal process of generating a vortex beam by computer-generated holography (CGH) can generally be divided into the following two steps: First, establish the CGH function according to the actual situation and create the CGH required in the experiment. Second, create an experimental light path to irradiate the reference beam on the hologram to obtain the required target beam. (1) Create the hologram Assuming that the amplitude of the target beam is E(r, z) and the amplitude of the reference beam is R(r, z), the amplitude expressions of the two beams can be expressed as [8] E(r, θ, z) = |E O (r, z)| exp[iϕ(r )].

(2.15)

R(r, θ, z) = |R O (r, z)| exp[i ψ(r )].

(2.16)

When two beams of light are transmitted in the same direction and interference at the transmission distance, the expression of the light-intensity distribution is [8]: I (r, θ, z) = |E(r, θ, z) + R(r, θ, z)|2 = |E(r, θ, z)|2 + |R(r, θ, z)|2 + E(r, θ, z)R ∗ (r, θ, z) + E ∗ (r, θ, z)R(r, θ, z). (2.17) It can be seen from Eq. (2.17) that the first two terms in the equation are independent of the light intensity of the target and reference beams, and the last two terms are related to the amplitude and phase information of the corresponding beam. Through the interference of the two light beams, the phase of the object light wave

Fig. 2.5 Computer-hologram method [12]

2.3 Vortex-Beam Generation Methods

57

contains a dislocation structure and produces interference fringes. The interference fringes of the beam can then be recorded with a special medium to obtain the required hologram, which can be regarded as a fork grating of the vortex beam. (2) Obtain the target beam When the laser beam irradiates the hologram, the interference fringes recorded in the hologram can modulate the wave surface of the laser beam. Thus, the diffracted beam contains optical-vortex information, and two conjugate-phase diffracted beams are obtained. When the reference beam is used as the target beam, the third term is the useful term in the diffracted beam [8]: '

E (r, θ, z) ∝ |R(r, θ, z)|2 E(r, θ, z).

(2.18)

This is equivalent to producing a virtual image of the object light wave. When the conjugate beam of the reference beam is used as the target beam, the fourth term is the useful term in the diffracted beam [8]: '

E (r, θ, z) ∝ |R(r, θ, z)|2 E ∗ (r, θ, z).

(2.19)

It is equivalent to producing a real image of the object light wave. Figure 2.6 shows the optical-path experimental diagram of a vortex beam generated using computer-generated holography [8]. First, a Gaussian beam is emitted by the He–Ne laser and irradiated on the hologram. After being focused by the lens, it is vertically incident on the filter. Finally, experimental results are obtained using a CCD camera. CGH has the advantages of convenient operation, flexible application, fast conversion rate, and a wide application field. Theoretically, this method can accurately generate vortex beams of any order. According to the research needs, a special hologram is created by a computer to produce the required beam. However, this method requires a high-resolution hologram-imaging instrument and computer configuration, and can only produce low-order vortex beams. Fig. 2.6 Device diagram of a vortex beam generated by computer-generated holography He-NeLaser

holographi c grating

Lens

ccd filter

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2 Vortex-Beam Spatial-Generation Method

2.3.2 Mode-Conversion Method The geometric mode-conversion method uses an optical device to change the mode of the laser beam and then produces different modes of outgoing light. This optical device is known as a mode converter. In 1991, Abramochkin et al. [13] successfully used a cylindrical lens to experimentally convert Hermite–Gaussian (HG) beams into Laguerre–Gaussian (LG) beams. In 1993, Beijersbergen et al. [4] further completed the mutual conversion from a high-order Hermite–Gaussian beam to a Laguerre– Gaussian beam using two cylindrical lenses and a mode converter. Many common vortex beams can be generated using the mode-conversion method. During optical transmission in an optical fiber, the research object is represented as an HG beam, and the vortex beam carrying OAM is usually represented in the form of an LG beam. The relationship between the HG beam, A(r), and LG beam, U (r, θ ), is. U (r, θ ) = A(r ) exp(ilθ ).

(2.20)

In Eq. (2.20), r is the radial distance from the central axis of the Gaussian beam, θ is the azimuth, and l is the topological charge. According to Eq. (2.20), the HG beam can be converted into an LG beam by introducing a random phase–factor, exp(ilθ ). The optical purity of the OAM obtained by this conversion method is high; however, its device volume is relatively large. There are generally two ways to generate vortex beams using this method: One is to use a mode converter composed of two cylindrical lenses to convert HG beams and LG beams into each other. The other is to generate the corresponding high-order Bessel beam from a high-order LG beam through a conical prism. Herein, the method of generating a vortex beam using two cylindrical lenses is briefly introduced. An optical path diagram is shown in Fig. 2.7 [1]. The focal lengths of the two cylindrical lenses are f and are symmetrical √ to each other. As shown in Fig. 2.7a, the distance between the two lenses, LG, is 2 f . After any order of beam passes through this device, the phase will change by π/2; that is, the HG beam will be converted into the corresponding LG beam after passing through Fig. 2.7 Conversion diagram of different cylindrical lenses

f

f

f

2f

2f

(a) Lens distance:

f

2f

(b) Lens distance: 2 f

2.3 Vortex-Beam Generation Methods

59

the mode converter. In Fig. 2.7b, the distance between the two lenses is denoted as 2 f . After the beam passes through this device, beams with a phase difference of π can be converted between each other. Because π is the change in phase, this is also called the “π converter.” The beam-conversion efficiency of the geometric mode-conversion method is relatively high, and a highly pure vortex beam can be produced. The disadvantage is that a laser in a laboratory can only output one mode of the HG beam in general, and the HG beam itself is difficult to obtain; therefore, it is difficult to generate LG beams of different modes. The structure of the conversion system is complex and requires high precision when fabricating the device, which is not convenient or flexible in practical applications.

2.3.3 Spiral-Phase-Plate Method A spiral phase plate (SPP) is a transparent optical diffractive element, as shown in Fig. 2.8 [12]. Its external structure is similar to that of a rotating step. After the beam passes through the SPP, the optical path of the outgoing beam changes differently, owing to the different thickness of the SPP; thus, the phase of the outgoing beam is superimposed with a spiral phase factor, exp(ilθ ), where l is the topological charge of the SPP and θ is the rotation azimuth. This results in an optical vortex in the light field. A phase singularity will appear in the center of the outgoing beam, with a spiral wavefront structure. The thickness h of the SPP is directly proportional to the rotation azimuth θ . It can be expressed as [1]:

Fig. 2.8 Ideal spiral phase plate [12]

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2 Vortex-Beam Spatial-Generation Method

h = h0 + hs

θ , 2π

(2.21)

where h 0 is the thickness of the phase-plate base and h s is the thickness of the step. Assuming that the step height h of the SPP changes very little, the light intensity of the outgoing beam can be ignored. It can be considered that, after the laser beam passes through the SPP, it only modulates the phase of the outgoing beam. The relevant phase-delay expression formed after the incident beam passes through the SPP is [1]. .θ (φ,λ) =

] [ 2π (n − n 0 )h s θ + nh 0 , λ 2π

(2.22)

where λ is the wavelength of the incident beam, and n and n 0 are the refractive index of material refractive index and air in the air in SPP, respectively; the number of topological lots of SPP. l = h s (n − n 0 )/λ. It can be seen that the size of .θ is determined by the rotation azimuth, θ . When θ increases from zero to 2π , .θ increases from zero to 2lπ . The thickness of the spiral phase plate of the ideal model increases linearly with the increase in the rotation azimuth; however, in the actual manufacturing process, a multilevel-step spiral phase plate (ML-SPP) is often used. Compared with the ideal SPP, the thickness of the MLSPP no longer changes linearly, but in discrete phase steps. If ML-SPP is composed of N phase steps, then the phase difference between two adjacent orders is 2πl/N . The device diagram for generating a partially coherent vortex beam using an SPP is shown in Fig. 2.9 [14].

Lens Rotating ground glass

ccd

He-Ne Laser f1

f2

Spiral phase plate

Fig. 2.9 Device diagram for generating a partially coherent vortex beam using the spiral-phaseplate method

2.3 Vortex-Beam Generation Methods

61

The He–Ne laser emits a Gaussian beam. After passing through the rotating ground-glass sheet, a partially coherent beam is obtained. The function of the two lenses is to expand and collimate the beam, so that the laser beam is vertically incident on the helical phase plate, which then forms a partially coherent vortex beam. The resulting diagram of the vortex beam is collected using a CCD camera. The spiral-phase-plate method is commonly used for generating vortex beams. This method has a high conversion efficiency and can also convert high-power laser beams. However, its disadvantage is that a spiral phase plate can only generate oneorder vortex beams, in theory. Moreover, it cannot flexibly control the types and specific parameters of the OAM beam. In addition, the manufacturing process and processing technology for spiral phase plates have strict requirements and the cost is relatively high.

2.3.4 Spatial Light Modulator Method The working principle of the spatial light modulator (SLM) is as follows: First, the holographic phase diagram is simulated on a computer, according to the phase information of the initial beam and target beam. Then, after the Fourier transform of the lens, the hologram information is loaded into the computer to form a holographic grating with a reflection mode. Finally, the incident beam is passed through the SLM with phase information to produce the required beam. Figure 2.10 [1] shows the device diagram for generating a vortex beam using an SLM in the laboratory. The beam emitted by the He–Ne laser is changed into a plane beam through the collimation of a beam-expansion system composed of a lens, and then enters the spatial light modulator. The experimental results are collected by a CCD camera, and the vortex beam is obtained.

spatial light modulator

He-Ne Laser Lens

control system Fig. 2.10 Device diagram of a vortex beam generated by the SLM method

ccd

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2 Vortex-Beam Spatial-Generation Method

Compared to computer-generated holography, the vortex beam generated by an SLM has the advantages of high energy-conversion efficiency, high diffraction efficiency, simple experimental device and technology, and easy integration into an optical system. The disadvantage of the SLM is that it cannot handle high-power laser beams.

2.3.5 Optical-Waveguide Device-Conversion Method In 2012, Cai developed a silicon-integrated OAM vortex-beam transmitter, which couples Hermite–Gaussian light into the ring waveguide after transmission in the silicon waveguide, and generates the echo-wall mode through the ring waveguide [15]. Vortex rotations with different OAM modes are generated above the ring waveguide. The phase difference of the Hermite–Gaussian beam propagating in the ring waveguide is due to the periodic sawtooth protrusion on the inner wall of the waveguide, which leads to a change in the optical wave vector. This optical-waveguide device-conversion method can simultaneously produce multiple OAM beams with controllable topological charges. A schematic diagram of a micro-silicon-based vortex-beam emitter is shown in Fig. 2.11. Optical-waveguide devices have a stable performance, low cost, small volume, and easy integration. They have great development potential for realizing the OAM mode using optical-waveguide devices. Fig. 2.11 Schematic diagram of a micro-silicon-based vortex-beam emitter [16]

2.4 Higher-Order Radial LG Beams

63

2.4 Higher-Order Radial LG Beams An LG beam is a representative vortex beam. Its light-field expression in cylindrical coordinates is as follows: / √ ( 2 ) ( 2) 2r 2r |l| 2 p! (−1) p r l ( ) exp − 2 L lp up = ω π( p + |l|)! ω ω ω2 ) ( zr 2 exp(iϕ) (2.23) exp(−ilφ) exp −i z R ω2 where, r, φ and z are cylindrical coordinate parameters; k is the wave constant; z R represents the Rayleigh distance, z R = kω02 /2; ϕ represents the Gouy phase, ϕ = (2 p + |l| + 1) arctan(z/z R ); ω is the beam radius at distance z, ω = [ ]1/2 [ / ] ω0 1 + (z/z R )2 ; I = A20 2 + 2 cos(2π x λ − lθ ) represents the waist radius of the beam; l represents the number of topological charges; and p represents the radial exponent. At p = 0, the beam-intensity distribution is a single ring. When p > 0, the beam-intensity distribution has a multi-ring. LG beams can be generated directly by lasers, but are more commonly generated by mode conversion and computer-generated holography. The mode-conversion method converts Hermite–Gaussian (HG) beams into LG beams using a cylindrical prism. In computer-generated holography, LG beams are generated by the phase modulation of a fundamental-mode Gaussian light or plane wave by an SLM-loaded fork grating. The form of a binary fork grating in cylindrical coordinates is [17]: l

2r φ =n+ cos φ, π .

(2.24)

where n = 0, ±1, ±2, ... and . is the period of the fork grating. The diffraction efficiency of the fork grating can be improved by adjusting parameter . to optimize the phase hologram. In addition, by introducing a blazed grating into the fork grating, the quality of the obtained LG beam can also be improved [18]. The transfer-function formula of a hologram is [17]: T (r, φ) = exp[iδ H (r, φ)],

(2.25)

where δ represents the amplitude of the phase modulation. The formula for the mode H (r, φ) of the hologram is [17]: ( ) 2π 1 mod lφ − r cos φ, 2π , H (r, φ) = 2π .

(2.26)

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2 Vortex-Beam Spatial-Generation Method

where mod(a, b) = a − bint(a/b) is the remainder of a/b. When p = 0, the forkopening direction of the phase hologram depends on a positive or negative topological charge, and the bifurcation number depends on the size of the topological charge. When p > 0, its phase hologram is similar to that of p = 0; however, one or more cyclic dislocations will appear. High-order radial LG beams can be obtained by the phase modulation of plane waves with a spatial light modulator, that is, ϕ(r, φ) = −lφ + π φ(−L lp (2r 2 /ω02 )),

(2.27)

where θ (x) is the unit step function and ω0 is the waist of the outgoing light on the SLM plane. Figure 2.12 shows the computer hologram required for a single high-order radial LG beam. Figure 2.12a shows the fork grating required to generate the LG 33 beam and Fig. 2.12b shows the fork grating required to generate the LG −5 2 beam. The phasemodulation function used in Fig. 2.12a, b is ϕ(r, φ) = −lφ + π φ(−L lp (2r 2 /ω02 )), in which the beam-waist radius is ω0 = 1.0 mm. As shown in Fig. 2.12, when P > 0, one or more circular dislocations appear in the fork grating of the LG beam. To investigate the influence of the waist radius on the structure of the fork grating of a high-order radial LG beam, we present the fork grating of the LG 11 beam for different waist radii. Figures 2.13a, b are fork gratings corresponding to LG 11 beams with waist radii ω0 = 2.0 mmand2.5 mm, respectively. (a)

(b)

Fig. 2.12 Hologram. a Fork grating of LG 33 beam; b Fork grating of LG −5 2 beam

Fig. 2.13 Fork-shaped gratings of LG 11 beam with a ω01 = 2.0 mm; b ω01 = 2.5 mm

(a)

(b)

2.4 Higher-Order Radial LG Beams

65

Comparing Fig. 2.13a, b, it can be seen that the fork-grating diagram corresponding to the high-order radial LG beam with different waist radii is different because the total phase of the high-order radial LG beam is determined by exp(−ilφ) and Laguerre polynomial L lP (2r 2 /ω02 ) [18], while when p /= 0, the Laguerre polynomial is related to ω0 . In addition, compared to the zero-order radial LG beam, the fork grating of the former has ring dislocations, whereas the fork grating of the latter has no ring dislocations. As shown in Fig. 2.13, there is a circular dislocation in the fork-grating diagram with the generated LG 11 beam; the central fringe dislocation opens downward, and there is a fringe at the fringe dislocation. However, when the beam-waist radius is larger, the radius of the circular dislocation increases. Figures 2.14a–d and 2.15a–d are the theoretical and experimental light-intensity distribution diagrams of the LG 10 , LG 30 , LG 11 , and LG 31 beams, respectively. The simulation and experimental parameters are set as follows: wavelength λ = 632.8 nm, transmission distance z = 1.5 m, and beam-waist radius ω0 = 1.0 mm. As shown in Figs. 2.14 and 2.15, when P = 0, the light-intensity distribution of the LG beam is a single ring. When p > 0, the light-intensity distribution of the LG beam is a multi-ring, and the number of bright rings is p + 1. Comparing Fig. 2.14a, b with c and d, when the radial index p is fixed, the diameter of each bright ring of the LG beam increases with the increase in topological charge l. Comparing Fig. 2.14a, b with c and d, when the topological charge l is fixed, the overall spot diameter of the LG beam increases and the diameter of the central bright ring decreases with an increase in the radial index p. The experimental results are consistent with the theoretical simulation results. Fig. 2.14 Theoretical intensity distribution of LG beams: a LG 10 ; b LG 30 ; c LG 11 ; and d LG 31

(a)

(b)

(c)

(d)

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2 Vortex-Beam Spatial-Generation Method

Fig. 2.15 Experimental intensity distribution of LG beams: a LG 10 ; b LG 30 ; c LG 11 ; and d LG 31

(a)

(b)

(c)

(d)

2.5 Generation of Fractional Vortex Beams 2.5.1 Principle of LG-Beam Preparation by the Holographic Method The LG beam is prepared using the holographic method. By simulating the interference between the plane wave and the vortex beam, the hologram is drawn by calculating the light-intensity distribution using a computer. The generated interference pattern is loaded onto the spatial light modulator. The laser collimates the beam into the spatial light modulator through the beam-expansion system, obtains the diffraction-field distribution function of the emitted beam, according to the diffraction equation, and simulates the intensity distribution of the fractional Laguerre– Gaussian beam. In the experiment, the fractional-interference grating obtained by changing the topological charge was loaded onto the spatial light modulator to obtain the Laguerre–Gaussian beam. (1) Hologram preparation A vortex beam propagating along the z-axis is assumed [19]: [ ] r2 ikr 2 − 2 − i (2 p + l + 1)ψ E 1 (r, ρ, z = 0) = exp − 2R w [ 2 ] 2l [ 2 ] 2r p 2r , L lp × exp(−ilϕ)(−1) w2 w2

(2.28)

2.5 Generation of Fractional Vortex Beams

67

where l denotes the order of the vortex beam, ϕ denotes the azimuth, p denotes the radial index, and r denotes the radial distance. The angle between the propagation direction of the other plane wave and the z axis is α. The plane-wave function can be written as E 2 = exp(ikx sin α +ikz cos α) and E 3 = exp(iky sin α + ikz cos α), respectively, assuming that the light waist of the two beams is z = 0, and α = π/3. When two beams interference in this plane, the interference light-intensity distribution is [20] I1 = |E 1 + E 2 |2 = E 12 + E 22 + 2E 1 E 2 cos(lθ − kx sin α).

(2.29)

I2 = |E 1 + E 3 |2 = E 12 + E 32 + 2E 1 E 3 cos(lθ − kx sin α).

(2.30)

Let both E 1 and E 2 be unit amplitudes. According to Eqs. (2.29) and (2.30), interference patterns with topological charges of 0.5 and 1.5 are obtained, as shown in Fig. 2.16.

(a) l = 0.5

(c) l = 0.5

(b) l = 1.5

(d) l = 1.5

Fig. 2.16 Interference patterns of a vortex optical-electric field and an oblique plane wave with different topological charges. a and b show vertical dislocations. c and d show horizontal dislocations

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2 Vortex-Beam Spatial-Generation Method

+

=

Fig. 2.17 Fractional and fractional-superposition patterns of the vortex beams: a Vertical dislocation, l = 0.5; b horizontal dislocation, l = 0.5; and c composite phase grating, l = 0.5

(2) Fabrication principle of the composite holographic grating for fractional vortex-beam arrays Using the superposition of horizontal and vertical complex phase-amplitude gratings, the fractional and fractional-superposition interference patterns of the vortex beams are prepared, as shown in Fig. 2.17. We use a composite-fork grating to generate a beam array with a position discrete phase, which is represented by lθ − kx sin α in Eq. (2.18). Figure 2.18 shows the process of preparing the superimposed grating; that is, the vertical and horizontal dislocation gratings are superimposed to obtain a composite phase grating. The phase expressions of the interference gratings in Fig. 2.18a, b, c are as follows: (L x θ + kx sin α) mod 2π,

(2.31)

(L y θ + ky sin α) mod 2π,

(2.32)

((L x θ + kx sin α) mod 2π ) + ((L y θ + ky sin α) mod 2π ),

(2.33)

SLM Laser

PBS Extender Lens

CCD

Fig. 2.18 Schematic diagram of the experimental device

2.5 Generation of Fractional Vortex Beams

69

where L x is the number of central dislocations in the x direction, L y is the number of central dislocations in the y direction, θ is the angular distribution, sin α determines the angle of the grating from the x-axis α, and mod2π is the period of 2π . (3) Obtaining a complex phase-superimposed grating vortex beam and its array When the laser collimates the superimposed fork grating, the superimposed grating prepared by Eq. (2.32) changes the phase of the beam, and the distribution of the outgoing beam in the diffraction field is [21]: u single

f ar (ρ, θ ) =

+∞ .

An F[u 00 (ρ, θ ) exp(inlθ )] ∗ F[exp(−inkx sin α)],

n=−∞

(2.34) u superposition far ≤ (ρ, θ ) =

+∞ .

[ ] An F u 00 (ρ, θ ) exp(inlθ )

n=−∞

[ ] ∗ F exp(−inkx sin α) exp(−inky sin α)

(2.35)

where An is the Fourier coefficient, F is the two-dimensional Fourier transform, (ρ, θ ) in Eq. (2.34) is the coordinate before the Fourier transform, and (ρ ' , θ ' ) in Eq. (2.35) is the coordinate after the two-dimensional Fourier transform.

2.5.2 Experimental Study on the Orbital Angular Momentum of Fractional Laguerre–Gaussian Beams A schematic diagram of the experimental device is shown in Fig. 2.18. The laser emitted by the He–Ne laser (λ = 632.8 nm) passes through the beam-expansion collimation system composed of a lens, and is incident on the reflective spatial modulator (RL-SLM-R2). By changing the superimposed grating loaded on the spatial light modulator (SLM) that can produce different vortex-beam arrays, we can obtain the integer order with the same topological-charge vortex-beam arrays with different integer orders, but the same fractional order. A different fractional-order superposition filters out stray light. The light-intensity distribution of different orders and their superposition were collected and displayed on a white screen using a charge-coupled device (CCD). In the experimental process, we found that the alignment degree of the central dislocation affected the light-intensity distribution of the beam. Moreover, the different values of the angle α between the plane wave and the z-axis also affected the beam-diffraction efficiency. When α is π/4, the laser can easily aim at the centralfork dislocation; however, the “dark space” phenomenon can hardly be seen in the diffracted first-order spot. When α is π/3, the distance between the gratings

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2 Vortex-Beam Spatial-Generation Method

in the interference grating is obviously reduced. To accurately hit the central-fork dislocation, many fine adjustments are required; however, it is easier to capture the beam-intensity distribution. Under the following conditions: laser wavelength λ = 632.8 nm, transmission distance Z = 10 m, spot radius ω = 0.002 m, and angle θ = π between the plane wave and transmission direction, the diffraction spots of gratings with topological charges of 2, 0.5, and 2.21 are simulated, according to the interference patterns obtained from Eqs. (2.29) and (2.30). The interference gratings with topological charges of 2, 0.5, and 2.21 are loaded onto the spatial light modulator to obtain the diffraction spots of integral and fractional vortex beams, as shown in Fig. 2.19. The comparison between Fig. 2.19a, b, c, d, e, f shows that the experimental results are basically consistent with the theory. In addition, it can be seen that when the topological charge is taken as an integer, the light-intensity distribution of the image is circularly symmetrical. When the topological-charge score is taken, the image is obviously not circularly symmetrical. In fact, it shows a “notch” distribution, and the number of “notch” distributions increases with an increase in the topological-charge score. This special distribution has applications in the particle-trapping field, among others. Figure 2.20a, b show the diffraction patterns of the laser through a fork grating with a transverse dislocation of 0.5 and a vertical dislocation of 0.5, respectively. Then, these two gratings are superimposed, and the same method is used to obtain a 3 × 3 beam array, as shown in Fig. 2.20c, in which eight states carry OAM for loading information and one state (located at the positive center) carries alignment information. Using the above principle, the interference grating superimposed with integer order and integer order, fractional order and fractional order, and the complex-phase binary-amplitude superimposed grating are loaded into the spatial light modulator sequentially. The experimental light spot is shown in Fig. 2.21. Figure 2.21a shows the spot diagram of the superposition of the same integer-order topological charges; that is, when the horizontal and vertical topological charges are 2, the topological charges of the beam array generated after passing through the modulator are −4, −2, −2, 0, 0, +2, +2, and +4. Figure 2.21b also shows the spot diagram of an integer-order topological-charge superposition. Compared with Fig. 2.21a, b, the larger l, the more scattered the spot of the superimposed beam array. Figure 2.21c shows the spot diagram of the superposition of different integer-order topological charges. The horizontal topological charge is 2 and the vertical topological charge is 3. The topological charges of the beam array generated after passing through the modulator are −5, −3, −2, −1, 0, +1, +2, +3, and +5. Comparing the topological charges of the spot array in Fig. 2.21a, c, it is found that when transmitting a beam with information under the same conditions, the spot array in Fig. 2.21c can transmit more information phases and has higher utilization. Figure 2.22a shows the spot diagram of the superposition of the same fractionalorder topological charges; that is, when the horizontal and vertical topological charges are 0.5, the topological charges of the beam array generated by the modulator are: −1, −0.5, −0.5, 0, 0, +0.5, +0.5, and +1. Comparing Fig. 2.22a with Fig. 2.21a,

2.5 Generation of Fractional Vortex Beams

71

(a) l = 1

(b) l = 1

(c) l = 0.5

(d) l = 0.5

(e) l = 1.5

(f) l = 1.5

Fig. 2.19 Simulation diagram and experimental diagram of integer-order and fractional-order grating diffraction

among the eight information phases transmitted, the number of effective information phases is six; however, the light spot in Fig. 2.22a is more complete and brighter than that in Fig. 2.21a. Figure 2.22b shows the spot diagram for the superposition of different fractionalorder topological charges. The horizontal topological charge is 0.5, the vertical topological charge is 2.21, and the modulated beam-array topological charges are:

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2 Vortex-Beam Spatial-Generation Method

+

(a) Horizontal spot l=0.5

=

(b) Vertical spot l=0.5

(c) Composite grating l=0.5

Fig. 2.20 Topological-charge variation principle of a fractional vortex-beam array

(a)l=2×2spot array

(b)l=3×3spot array

(c) l=2×3spot array

Fig. 2.21 Vortex-beam array obtained after loading different integer-order superimposed gratings

(a) l=0.5 × 0.5 spot array

(b) l=0.5 ×1.5 spot array

(c) l=2 ×l= 0.5 spot array

Fig. 2.22 Vortex-beam array obtained after loading different fractional-order superimposed gratings

−2, −2.21, −1, −0.5, 0, +0.5, +1, +2.21, and +2. Compared with Fig. 2.21c, the equal-interval transmission-phase information has more continuity. Figure 2.22c shows the spot diagram of the superposition of fractional and integer-order topological charges. The topological charges of the modulated beam array are −2.5, −2, − 2.21, −0.5, 0, +0.5, +2, +2.21, +2, and +2.5. Through comparisons, it was found that when the fractional-order vortex beam acts as a carrier to carry information, similar to the integer-order vortex beam, the

References

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topological charge carried by the spot array will change with the change in the superposition mode of the modulation grating. The results show that the beam-carrier array obtained by different fractional-order superimposed grating-modulation gratings has the highest information-phase decomposition efficiency when transmitting information. Moreover, equal-interval decomposition is conducive to the recovery of information at the receiver. Using the same experimental equipment and path, the light intensity was stronger than that of the integer-order and other spot arrays, which is conducive to long-distance transmission.

References 1. Chen ZT (2013) Study on the characteristics of vortex beam. Master’s thesis of Yan shan University 2. Heckenberg NR, Mcduff R, Smith CP et al (1992) Generation of optical phase singularities by computer-generated holograms. Opt Lett 17(3):221–223 3. Chen MJ, Xing GL, Zhang YX (2016) Variation of orbital angular momentum transmission characteristics of Laguerre Gaussian beam in weakly turbulent ocean. J Radio Wave Sci 31(4):737–742 4. Marco B, Les A, Veen VD et al (1993) Astigmatic laser mode converters and transfer of orbital angular momentum. Opt Commun 96(1):123–132 5. Guy I, Korwan D (1994) Model of vortices nucleation in a photo refractive phase-conjugate resonator. Opt Lett 41(5):941–950 6. Barry L, Rebecca P, Vladimir T et al (1994) Nonlinear rotation of three dimensional dark spatial solitons in a Gaussian laser beam. Opt Lett 19(22):1816–1818 7. He D, Yan HW, Lv BD et al (2009) Synthetic optical vortices formed by Hermite Gaussian vortex beams and their evolution. China Laser 36(8):2023–2029 8. Li YT (2012) Theoretical and experimental study on vortex beam generation by computergenerated holography. Master’s thesis of Yan Shan University 9. Zhang M, Xu JB, Li J et al (2009) Transverse force of Rayleigh particle in Bessel beam. Intense Laser Part Beam 21(1):135–138 10. Franco G, Giorgior G, Cesare P (1987) Bessel-Gauass beams. Opt Commun 64(6):491–495 11. Xu C (2015) Research on transmission performance of free space optical communication system based on vortex beam. Beijing University of Posts and Telecommunications 12. Lu XX, Huang HQ, Zhao CL et al (2008) Vortex beam and optical vortex. Prog Laser Optoelectron 45(1):50–56 13. Eugeny A, Volostnikov V (1991) Beam transformations and non-transformed beams. Opt Commun 83(1):123–135 14. Hua NM, Chen ZY, Pu JP (2011) Experimental study on partially coherent vortex beams. J Opt 31(11):103–106 15. Cai XL, Wuang JW, Michael S et al (2012) Integrated compact optical vortex beam emitters. Science 338(6105):363–366 16. Zhang DK, Feng X, Cui KY et al (2013) Generating in-plane optical orbital angular momentum beams with silicon waveguides. IEEE Photonics J 5(2):12–17 17. Jochen A, Kishan D, Les A et al (1998) The production of multiringed Laguerrea Gaussian modes by computer-generated holograms. Opt Acta Int J Opt 45(6):1231–1237 18. Naoya M, Taro A, Takashi I et al (2008) Generation of high-quality higher-order LaguerreGaussian beams using liquid-crystal-on-silicon spatial light modulators. J Opt Soc Am A 25(7):1642–1651 19. Kishan AJD, Les A et al (1998) The production of multiringed Laguerreâ Gaussian modes by computer-generated holograms. Opt Acta Int J Opt 45(6);1231–1237

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20. Li YT, Zhu YY, Feng JS et al (2012) Experimental study on vortex beam generated by computer generated holography. National Symposium on optoelectronic technology Beijing. In: Proceedings of the 10th National optoelectronic technology academic exchange, vol 50(15), pp 3907–3911 21. Li F, Gao CQ, Liu YD et al (2012) Experimental study on Laguerre Gaussian beam generated by amplitude grating. J Phys 57(2):860–866

Chapter 3

Vortex-Beam Generation Using the Optical-Fiber Method

A vortex beam generated by the optical-fiber method is composed of vector modes in an optical fiber. The analysis of vector modes in optical fiber is also an analysis of the vortex-beam generation conditions. The fiber-mode theory provides the electromagnetic-field solution for vector modes in optical fiber, and the vortex beam generated by the optical fiber connects the vortex beam with the vector-mode solution.

3.1 Introduction The most commonly used spatial-structural devices for vortex-beam generation methods include spiral phase plates [1], spatial light modulators [2], computer holograms [3], etc. Compared with the vortex beam generated by a spatial-structure device, the purity of the vortex beam generated by an optical fiber will be higher because the vortex beam itself is an intrinsic solution from the optical fiber [4]. In 2009, Krishna et al. [5] placed obliquely incident linearly polarized light onto a single-mode fiber to produce a vector beam. In 2013, Fang et al. [6] found that changing the polarization direction could detect different modes in an optical fiber. In 2016, Zhang et al. [7] used acoustic-wave modulation to generate a second-order vector beam for the refractive index of a few-mode fiber. In recent years, scholars at home and abroad have also attempted to use specially structured optical fibers to generate vortex beams. In 2014, Li et al. [8] proposed an optical-fiber structure with 19 ring cores. The simulation showed that the optical fiber could transmit high-order modes. In 2016, Zhang et al. [9] proposed a method for generating vortex beams using Bragg fiber gratings. In 2017, Wong et al. [10] designed a spirally twisted photonic crystal fiber to excite the vortex mode. Several methods have been proposed for generating vortex beams by changing the structure of the optical fiber; however, because of the complexity of the optical-fiber

© Science Press 2023 X. Ke, Generation, Transmission, Detection, and Application of Vortex Beams, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-0074-9_3

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manufacturing process, most of them remain in the theoretical stage and have not been verified experimentally.

3.2 Optical-Fiber Mode Theory 3.2.1 Wave Equation Maxwell’s equation is expressed as [11] ∂B ∂t

(3.1)

∂D +J ∂t

(3.2)

∇ ×E=− ∇ ×H=

∇ ·D=ρ

(3.3)

∇ ·B=0

(3.4)

where E represents the electric-field intensity, H represents the magnetic-field intensity, D represents the potential shift vector, B represents the magnetic-induction intensity, J represents the current intensity, and ρ represents the charge density; “∇×” represents a curl and “∇” represents a divergence. For optical waveguides, the transmission material is a medium, rather than a conductor, and the medium has no charge or current; thus, ρ = 0 and J = 0 [12]. According to Maxwell’s equation and the relationship between D and E, and B and H, the wave equation can be expressed as [13] ∇ 2 E = εμ0

∂ 2E ∂t 2

(3.5)

∇ 2 H = εμ0

∂ 2H ∂t 2

(3.6)

where, ε is the conductivity and μ0/is the permeability. When the angular frequency / of the light is ω, ∂ ∂t = j ω, ∂ 2 ∂t 2 = −ω is true. The wave equation can be written as [14] ∇2E + k2E = 0

(3.7)

∇2H + k2H = 0

(3.8)

3.2 Optical-Fiber Mode Theory

77

√ where k = ω μ0 ε = nk0 , k 0 is the wavenumber in free space, and n is the refractive / / √ √ index of the medium. k0 = ω μ0 ε0 = 2π/λ, n = ε ε0 = εr . Equations (3.7) and (3.8) are vector Helmholtz equations. Given the boundary conditions, each component of E and H can be solved.

3.2.2 Vector Modes in Optical Fiber Using a rectangular coordinate system to represent the electric and magnetic fields in a cylindrical fiber cannot directly reflect the change in the vector mode in the fiber. In this study, we used a cylindrical coordinate system. The Laplace operator in a cylindrical coordinate system is [14] ∇2 =

( ) ∂ 1 ∂2 1 ∂ ∂2 r + 2 2+ 2 r ∂r ∂r r ∂θ ∂z

(3.9)

We substitute Eq. (3.9) and the E and H components in the cylindrical coordinate system into Eqs. (3.7) and (3.8). The expressions for Er , E θ , and E z and Hr , Hθ , and Hz in an optical fiber in a cylindrical coordinate system can be obtained as [15] [ ] [ ] [ ] [ ] ( 2 ) Ez 1 ∂ Ez 1 ∂ Ez ∂ 2 Ez 2 + + 2 + k −β =0 ∂r 2 Hz r ∂r Hz r ∂θ Hz Hz ( ) ) ∂ Er 1 ∂ 2 Er Eθ ( 1 ∂ 2 ∂ Eθ r + 2 − 2 − β 2 − k 2 n 2 Er = 0 + 2 r ∂r ∂r r ∂θ 2 r ∂θ r ( ) ) ∂ Eθ 1 ∂ 2 Eθ Eθ ( 1 ∂ 2 ∂ Er r + 2 − 2 − β 2 − k 2 n2 Eθ = 0 + 2 2 r ∂r ∂r r ∂θ r ∂θ r

(3.10)

(3.11)

(3.12)

We simply write the E component here; however, the information is also true for the H component. β represents the longitudinal-propagation constant. We solve Eq. (3.10) to obtain the component of the vector mode in the z direction in the optical fiber [16]: . E z (r, θ, z) = exp[−i (βz ∓ μθ )] . Hz (r, θ, z) = exp[−i (βz ∓ μθ )]

(

)

A J ur Jμ (u) μ (a ) r < a A K ur r ≥ a K μ (u) μ a

(3.13)

B J ur Jμ (u) μ (a ) r < a B K ur r ≥ a K μ (u) μ a

(3.14)

(

)

78

3 Vortex-Beam Generation Using the Optical-Fiber Method

where n is the refractive index of the fiber core, u is the transverse-normalization constant, A is the electric-field constant, B is the magnetic-field constant, a is the fiber-core radius, μ is the order of the circumferential modulus, Jμ (u) is the Bessel equation when the transverse-normalization constant is u, and K μ (u) is the modified Bessel equation when the transverse-normalization constant is u. Using the curl expression in a cylindrical coordinate system, Maxwell’s equation can be written into the expression of the cylindrical coordinate system. If the r, θ , and z components on the left and right sides are equal, the component forms of the r and θ directions represented by E z and H z can be obtained [15]: ( ) ∂ Ez ωμ0 ∂ Hz −i β + exp[−i (βz ∓ μθ )] k 2n2 − β 2 ∂r r ∂θ ( ) ∂ Ez μ −i β + i ωμ H = 2 2 0 z exp[−i (βz ∓ μθ )] k n − β2 ∂r r ) ( ∂ Ez ∂ Hz −i exp[−i(βz ∓ μθ )] β Eθ = 2 2 − ωμ 0 k n − β2 ∂r ∂θ ( ) μ −i ∂ Hz iβ ∂ E exp[−i (βz ∓ μθ )] = 2 2 − ωμ z 0 k n − β2 r ∂r ) ( ∂ Hz −i ωε0 n 2 ∂ E z exp[−i(βz ∓ μθ )] β Hr = 2 2 − k n − β2 ∂r r ∂θ ( ) −i ∂ Hz μ = 2 2 β − i ωε0 n 2 E z exp[−i (βz ∓ μθ )] k n − β2 ∂r r ( ) β ∂ Hz −i 2 ∂ Ez + ωε0 n exp[−i(βz ∓ μθ )] Hθ = 2 2 k n − β 2 r ∂θ ∂θ ( ) μ ∂ Ez −i iβ Hz + ωε0 n 2 exp[−i (βz ∓ μθ )] = 2 2 k n − β2 r ∂r Er =

(3.15)

(3.16)

(3.17)

(3.18)

Substituting Eqs. (3.11) and (3.12) into Eqs. (3.15) through (3.18), the expressions for E r , E θ , H r , and H θ can be obtained [15]: [ (u ) (μ )] ⎧ a2 μ Jμ a r μ Jμ' a r ⎪ ⎪ ⎪ −i 2 Aβ − Bωμ0 exp[−i(βz ∓ μθ )]r < a ⎪ ⎨ u a Jμ (u) r Jμ (u) [ Er = (ω ) ( )] ⎪ ⎪ μ K μ ωa r ω K μ' a r a2 ⎪ ⎪ − Bωμ0 exp[−i (βz ∓ μθ )]r ≥ a ⎩ −i 2 Aβ ω a K μ (ω) r K μ (ω) (3.19)

3.2 Optical-Fiber Mode Theory

79

[ (u )] ( ) ⎧ a2 μ Jμ' a r μ Jμ ua r ⎪ ⎪ ⎪ −i − Bωμ0 Aβ exp[−i(βz ∓ μθ )]r < a ⎪ ⎨ u2 r Jμ (u) a Jμ (u) [ Eθ = (ω )] ( ) ⎪ ⎪ μ K μ' a r ω K μ ωa r a2 ⎪ ⎪ − Bωμ0 exp[−i (βz ∓ μθ )]r ≥ a ⎩ −i 2 Aβ ω r K μ (ω) a K μ (ω) (3.20) [ ] ( ) (u ) ⎧ u J r a2 μ Jμ' a r ⎪ 2μ μ a ⎪ ⎪ −i − Bβ n Aωε exp[−i (βz ∓ μθ )]r < a 0 1 ⎪ ⎨ u2 r Jμ (u) a Jμ (u) [ Hr = ( )] (ω ) ' ω 2 ⎪ r K K r ⎪ μ ω a μ μ a ⎪ a 2 ⎪ − Bβ exp[−i (βz ∓ μθ )]r ≥ a ⎩ −i 2 Aωε0 n 1 ω r K μ (ω) a K μ (ω) (3.21) [ ] ( ) ( ) ⎧ K' ωr a2 ω K μ ωa r ⎪ 2μ μ a ⎪ ⎪ −i − Bβ n Aωε exp[−i (βz ∓ μθ )]r < a 0 1 ⎪ ⎨ u2 a K μ (ω) r K μ (ω) [ Hθ = ( )] (ω ) ' ω 2 ⎪ ⎪ ⎪ −i a Aωε n 2 μ K μ a r − Bβ ω K μ a r exp[−i (βz ∓ μθ )]r ≥ a ⎪ ⎩ 0 1 ω2 r K μ (ω) a K μ (ω) (3.22) where ε0 is the conductivity and μ0 is the permeability. When r = a for the optical fiber, the tangential electric and magnetic fields are continuous. Substituting Eqs. (3.19) through (3.22) into the boundary conditions, we obtain [16]: [ ] ' ' ) ( ωμ0 1 Jμ (u) 1 K μ (ω) i μβ 1 1 + A + 2 −B =0 a u2 ω a u Ju (u) ω K u (ω) [ ] ' ' ( ) ωε0 n 21 Jμ (u) n 22 K μ (ω) 1 i μβ 1 =0 + A + +B a u Ju (u) ω K u (ω) a u2 ω2

(3.23)

(3.24)

For the homogeneous equation composed of Eqs. (3.23) and (3.24), if A and B have nonzero solutions, their coefficient determinant should be zero, and the characteristic equation can be derived [16]: [

'

'

1 Jμ (u) 1 K μ (ω) + u Jμ (u) ω K μ (ω)

][

'

'

n 21 Jμ (u) n 22 K μ (ω) + ω K μ (ω) un 22 Jμ (u)

]

[ = μ2

n 21 1 1 + 2 ω n 22 u 2

](

1 1 + 2 u2 ω (3.25)

Equation (3.25) is the characteristic equation of the fiber vector mode or the dispersion equation. The dispersion equation can be used to determine the μ-order mode β value or u value.

)

80

3 Vortex-Beam Generation Using the Optical-Fiber Method

The fiber contains four vector modes: the radial vector beam TE0v , angular vector beam TM0v , mixed-vector polarization beam HEμv , and mixed-vector polarization beam EHμv . μ represents the circumferential-module order and v represents the radial-module order. The results of Eq. (3.25) can be interpreted as follows. The TE0v mode occurs when μ = 0 and Ez = Er = Hθ = 0. The TM0v mode occurs when μ = 0 and Eθ = Hr = Hz . The HEμv mode occurs when μ takes a positive value. The EHμv mode occurs when μ takes a negative value. In the phase factor, when the result is positive z increases, θ decreases, and a clockwise rotation indicates right-handed polarization. when the result is negative, z increases, θ increases, and a counterclockwise rotation indicates left-handed polarodd even and H E μν , ization. EH and HE have odd- and even-mode states, denoted by H E μν odd even and E Hμν and E Hμν , respectively. Figure 3.1 shows the light intensity and polarization of the vector mode in the fiber. As shown in Fig. 3.1, the vector modes corresponding to the same light-intensity distribution as the modes in the fiber may be different. Figures 3.1a, b, d, e show the first-order mode in the fiber. There is no difference in the light-intensity distribution, but there is a certain difference in the polarization. Figure 3.1a shows that the TEmode polarization is distributed in the radial direction. Figure 3.1b shows the angular distribution of the TM-mode polarization. In Fig. 3.1d, e, the polarization difference odd even and H E μν is π/2. Comparing Fig. 3.1c, d, e, f, the polarizationbetween H E μν distribution directions of the odd and even modes of the HE and EH modes are opposite, and the distribution of the light intensity is also different. I

(a)

(b)

1

(c)

0 I

(d)

(e)

(f)

1

0

(a)TM01; (b)TE01; (c)

EH11odd;

(d) HE21

odd

; (e)

HE21even

;(f) EH11

even

Fig. 3.1 Light intensity and polarization distribution of vector modes in optical fiber

3.2 Optical-Fiber Mode Theory

81

3.2.3 Guide-Mode Cutoff and Distance Cutoff When μ = 0, the right side of Eq. (3.25) is 0, and the two solutions are the characteristic equations of TE0v and TM0v , respectively. They can be expressed as [17] '

'

1 Jμ (u) 1 K μ (ω) + =0 u Jμ (u) ω K μ (ω) '

(3.26)

'

n 21 Jμ (u) n 22 K μ (ω) + = 0. ω K μ (ω) un 22 Jμ (u)

(3.27)

When ω → 0, the guided mode is cut off and J0 (u) = 0 can be obtained from the recurrence and asymptote of the modified Bessel function. The Bezier curves are shown in Fig. 3.2. As shown in Fig. 3.2, the Bezier curve oscillates with multiple roots. After the order of the Bessel function is determined, only one root exists for a certain horizontal normalization constant u. The roots of the Bessel function are presented in Table 3.1. The first several roots of J 0 (u) are 2.4084, 5.52021, 8.6537,…, corresponding to the u values at the cutoff of TE01 and TM01 , and TE02 and TM02 , respectively. Because u = v at the cutoff, the normalized frequency is. 1.0

J0(u) J1(u) J2(u) Bessel function J(u)

Fig. 3.2 Relationship between the horizontal-normalization constant u and the Bessel function J(u)

0.5

0.0

-0.5

0

Table 3.1 Roots of the Bessel function

Bessel function

6

12

18

24

Horizontal normalization constant u

J0 (u)

J1 (u)

J2 (u)

First root

2.4048

3.8317

5.1356

Second root

0

5.5201

7.0156

Third root

0

8.6537

10.1735

30

82

3 Vortex-Beam Generation Using the Optical-Fiber Method

/ 2πa n 21 − n 22 > 2.4084. λ

V =

(3.28)

From Eq. (3.25), the characteristic equation of the HEμv module can be written as [15] '

'

Jμ (u) u Jμ' (u)

+

K μ (ω) ωK μ' (ω)

=−

μ μ − 2. u2 ω

(3.29)

We use the recurrence of the Bessel function to simplify Eq. (3.29) to [16] Jμ−1 (u) u Jμ (u)

−

K μ−1 (ω) ωK μ (ω)

= 0.

(3.30)

When μ = 1, Eq. (3.30) is expressed as [15] J0 (u) K (ω) − 0 = 0. u J1 (u) ωK 1 (ω)

(3.31)

Figure 3.2 shows that the first few roots of J 1 (u) are 0, 3.831, 7.0160,…, which correspond to the cutoff frequencies of HE11 , HE12 , and HE13 , respectively. When far from the cutoff condition (ω → ∞), the first few roots of J0 (μ) are 2.4084, 5.52021, 8.6537,…, which correspond to being far from the cutoff frequencies of HE11 , HE12 , and HE13 , respectively. Among all the vector modes in the optical fiber, the cutoff frequency of HE11 is 0, which means that HE11 will transmit in any optical fiber and at any wavelength. This is also called the main mode or fundamental mode in the optical fiber. When μ > 1, Eq. (3.31) is expressed as [17] u Jμ−1 (u) = . Jμ (u) 2(μ − 1)

(3.32)

The solution of Eq. (3.32) corresponds to the cutoff frequency of the HEμv mode in the fiber. Substituting μ and v into Eq. (3.13), the light-intensity distribution of the HE vector mode when μ and v are different is shown in Fig. 3.3. It can be seen from Fig. 3.3 that the halo of Fig. 3.3c is thinner than that in Fig. 3.3b, and the halo of Fig. 3.3b is thinner than that in Fig. 3.3a. Therefore, μ determines the thickness of the vector-mode energy ring and the area of the singular point in the middle of the ring. The larger the μ, the thinner the ring of the vector mode, and the larger the area of the singular point in the middle of the ring. The energy will increase along with the increase in the μ concentration. Figure 3.3d shows one more aperture than is shown in Fig. 3.3a. It can be observed that v determines the radial distribution of the vector mode. The larger v is, the greater the number of rings in the radial direction, and the more energy will spread to the other ring.

3.2 Optical-Fiber Mode Theory

83

Fig. 3.3 μ and v light-intensity distributions of the HE vector mode at different times. a μ = 2, v = 1; b μ = 3, v = 1; c μ = 4, v = 1; d μ = 2, v = 2; e μ= 3, v = 2; f μ = 4, v = 2

From Eq. (3.25), the characteristic equation of the EHμv mode is obtained as follows [17]: '

Jμ (u) u Jμ' (u)

'

+

K μ (ω) ωK μ' (ω)

=

μ μ + 2. u2 ω

(3.33)

The solution of Eq. (3.33) is the same as that for HE mode. We substitute the u value obtained by being far from the cutoff condition into the electric-field Eq. (3.18). The polarization of the vector mode in the fiber is the same as that shown in Fig. 3.3; therefore, it is not described here. For all modes in the fiber, when the normalized frequency of the fiber is V < 2.4048, the TE, TM, and HE modes have not yet appeared, and only the HE11 mode is transmitted in the fiber. Therefore, V =

/ 2πa n 21 − n 22 < 2.4084. λ

(3.34)

Equation (3.34) shows the transmission condition of the step-index single-mode fiber. When the V value of the fiber is less than 2.4084, there is only / one mode in the fiber. The effective refractive index of the optical fiber is N = β k0 , which reflects the change in the longitudinal-propagation constant in the optical fiber. β represents the longitudinal-propagation constant and k 0 is the wavenumber. The relationship between the effective refractive index N of the fiber, normalized constant u solved in the characteristic equation, normalized attenuation coefficient ω, and normalized frequency V is expressed as [15] ) ( u 2 = k02 n 21 − N 2 a 2

(3.35)

84

3 Vortex-Beam Generation Using the Optical-Fiber Method 1.4480

Effective refractive index b/K0

TE01 HE11 HE21 HE41 HE12

TM01 EH11 HE31 EH21

HE11 TE01 TM01 EH11

HE21

HE12

1

2

3

HE31

TE01 HE11

4

5

HE41

1.4472

6

TM01

TE01

TM01

HE21

EH21

n1 0

(b)

Effective refractive index b/K0

n2 (a)

2.8

Normalized frequency V

3.0

3.2

3.4

Normalized frequency V

Fig. 3.4 Curves of the effective refractive index of the mode with normalized frequency. a 0 ~ 3rd mode; b 1st mode

) ( ω2 = N 2 − k02 n 22 a 2

(3.36)

V 2 = u 2 + ω2

(3.37)

Using the characteristic equation of vector mode u and ω, the effective refractive index of the vector mode in the optical fiber can be solved by substituting Eqs. (3.35) and (3.36). The effective refractive index of an optical fiber represents the mode change of the optical fiber when it propagates in the longitudinal direction. Figure 3.4 shows the change curve of the effective refractive index of the mode in the fiber with the normalized frequency. As shown in Fig. 3.4, the number of modes in the fiber gradually increases with an increase in the normalized frequency V. The HE11 mode first appears in the optical fiber. Combined with Table 3.1, it can be observed that HE21 , TE01 , and TM01 appear when V > 2.4084, and four modes can be accommodated in the fiber at this time. When V > 3.8317, EH21 , HE31 , and EH12 appear in the fiber, and the fiber’s mode capacity is seven.

3.2.4 Scalar Modulus Under a Weakly-Conducting Approximation For a pair of μ and v values, each μ value corresponds to a mode that represents the distribution law of its electric field on the cross section and the transmission characteristics of space; this is called LPμ,v mode. The distribution of the transverseelectromagnetic field along the circumference and radius of the LPμ,v mode in the fiber-core area are respectively [17]:

3.2 Optical-Fiber Mode Theory

85

Fig. 3.5 Influence of μ on the maximum value on the circumference of the scalar mode. a μ = 1, v = 1, two-dimensional; b μ = 2, v = 1, two-dimensional; c μ = 3, v = 1, two-dimensional; d μ = 1, v = 1, three-dimensional; e μ = 2, v = 1, three-dimensional; f μ = 3, v = 1, three-dimensional

.= exp(±i μθ ). R(r ) = A Jμ

(u ) r . a

(3.38) (3.39)

where μ is the order of the Bessel function and μ starts from 0. This is related to the circumferential and radial directions of the electromagnetic field. If we decompose Eq. (3.38) using Euler’s formula exp( j μθ ) = cos(μθ ) + j sin(μθ ) in the physical sense the distribution of the electromagnetic field along the circumference comprises two linear-polarization modes. Under a scalar approximation, the polarization states are not distinguished; they are considered to be linear polarizations with the same polarization state everywhere. In addition, the influence of μ on the maximum value on the circumference of the scalar mode is shown in Fig. 3.5. As shown in Fig. 3.5a, d, if μ = 1, cos(μθ ) = cos θ , θ changes by 2π along the circumferential direction, and a pair of maximum values appear along the circumferential direction. As shown in Fig. 3.5b, e, if μ = 2, cos(μθ ) = cos 2θ , θ changes by 4π along the circumferential direction, and there are two pairs of maxima along the circumferential direction. As shown in Fig. 3.5c, f, if μ = 3 andcos(μθ ) = cos3θ , θ changes by 6π along the circumferential direction, and there are three pairs of maxima along the circumferential direction. We can see that the larger μ is, the greater the number of maxima on the circumference, and the number of maxima is equal to μ. Figure 3.6 shows that the change in the electromagnetic field along the radius is related to v under the scalar approximation. As shown in Fig. 3.6a, d, for the LP11 mode, R(r ) = A Jμ (2.4048r/a) when away from the cutoff, and the electromagnetic field has only one maximum value along the radius. As shown in Figs. 3.6b, e,

86

3 Vortex-Beam Generation Using the Optical-Fiber Method

Fig. 3.6 Influence of v on the maximum value in the radius direction of the scalar mode. a μ = 1, v = 1, two-dimensional; b μ = 1, v = 2, two-dimensional; c μ = 1, v = 3, two-dimensional; d μ = 1, v = 1, three-dimensional; e μ = 1, v = 2, three-dimensional; f μ = 3, v = 3, three-dimensional

for the LP12 mode, R(r ) = A Jμ (5.5210r/a), far away from the cutoff, and the electromagnetic field has two extreme values along the radius. As shown in Fig. 3.6c, f, for the LP13 mode, R(r ) = A Jμ (8.6537r/a) when away from the cutoff; when r = 0.6381a, u13 = 8.6537 and when r = 0, R(r) = 1. The electromagnetic field has three extreme values along the radius.

3.2.5 Analysis of the Principle of Using Optical Fiber to Generate Vortex Light The basic modes in an optical fiber include the radial vector beam, TM, angular vector beam, TE, mixed-vector polarization beam, HE, and mixed-vector polarization beam, EH. The orbital angular momentum (OAM) in the optical fiber is composed of the fundamental mode of the optical fiber [18]. In a cylindrical fiber, the electromagnetic field can be decomposed into two parts: radial transmission and angular polarization: ⎧ ⎫ ⎨ f v (φ) ⎬ E(r, φ, z, t) = e(r ) gv (φ) exp(iβz − i ωt). ⎩ ⎭ f v (φ) ⎧ ⎫ ⎨ f v (φ) ⎬ H(r, φ, z, t) = h(r ) gv (φ) exp(iβz − i ωt). ⎩ ⎭ f v (φ) . cos(vφ) even modes f v (φ) = . sin(vφ) odd modes

(3.40)

(3.41)

(3.42)

3.2 Optical-Fiber Mode Theory

87

. gv (φ) =

− sin(vφ) even modes . cos(vφ) odd modes

(3.43)

The odd mode in the optical fiber contains a sine component, and the/even mode contains a cosine component; therefore, there is a phase difference of π 2 between the even and odd / modes. In addition, i E(r, φ, z, t) and E(r, φ, z, t) have a phase with the difference of π 2. The even-mode-vector electric field is superimposed / even-mode-vector electric field with a phase difference of π 2. The equations of f v (φ) and gv (φ) become complex numbers, and the electric-field strength can be expressed as [19] ⎫ ⎧ ⎨ exp(iσ φ) exp(ilφ) ⎬ E(r, φ, z, t) = e − exp(i σ φ) exp(ilφ) exp(iβz − i ωt) ⎭ ⎩ exp(ivφ) ⎫ ⎧ ⎨ exp(i σ φ) exp(ilφ) ⎬ H(r, φ, z, t) = h − exp(i σ φ) exp(ilφ) exp(iβz − i ωt), ⎭ ⎩ exp(ivφ)

(3.44)

(3.45)

where σ = ±1 represents the left-handed or right-handed circularly polarized light, l represents the topological-charge number, and v = l + σ represents the total angular momentum. exp(ilφ) indicates that the light in the optical fiber has both orbital angular momentum and circular-polarization characteristics. First, it shows that the OAM mode can exist in the optical fiber because the OAM mode can be composed of eigenmodes with the following relationship [20]: ± even odd O AM±l,m = H El+1,m ± i H El+1,m .

(3.46)

∓ even odd O AM±l,m = E Hl−1,m ± i E Hl−1,m .

(3.47)

± The superscript O AM±l,m represents the spin state of OAM, “ + ” represents righthanded circular polarization and “−” represents left-handed circular polarization. From Eqs. (3.46) and (3.47), it can be seen that the OAM mode in the fiber exhibits circular polarization. The direction of the spin angular momentum of the HE mode is consistent with that of the orbital angular momentum. The direction of the spin angular momentum in the EH mode is opposite to the direction of the orbital angular momentum [21]. When only the fundamental mode HE11 exists in the fiber, Eq. (3.46) can be expressed as [22] ± even odd O AM±0,1 = H E 11 ± i H E 11 .

(3.48)

88

3 Vortex-Beam Generation Using the Optical-Fiber Method

It can be seen from Eq. (3.48) that the topological charge of the fundamental mode HE11 is 0, which indicates that the fundamental mode HE11 is simply a circularly polarized Bessel beam. In addition to the vector mode in the optical fiber, an LP mode also exists. The LP mode is formed by the superposition of different vector modes. The representation of the LP mode is similar to that of the OAM mode: L Pl,m = H El+1,m ± E Hl−1,m

(3.49)

L Pl,m = H El+1,m ± T Ml−1,m

(3.50)

L Pl,m = H El+1,m ± T El−1,m .

(3.51)

From Eqs. (3.49) to (3.51), it can be seen that LPl,m is a synthesis of HEl+1,m , EHl-1,m , TEl-1,m , and TMl-1,m . Figure 3.7 shows the light intensity and polarization distribution of the LP11 mode. The first row represents the basic mode in the fiber and the second row represents the light intensity and polarization distribution of the synthesized LP11 mode. Because the longitudinal-propagation constant β of the HE mode is different from that of the other modes, the LP mode causes the mode to deviate with the transmission distance. The OAM mode in the optical fiber can also be composed of TE and TM eigensolutions [23]: ∓ O AM±1,m = T M0,m ± i T E 0,m .

(3.52)

The TE and TM modes in the fiber are two different modes. As shown in Fig. 3.4, the effective refractive indices of the TE and TM modes are different; therefore, the longitudinal-propagation constants of the two modes are inconsistent. When the distance is the same, the TE mode propagates faster than the TM mode, and as the propagation distance increases, the orbital angular momentum formed by these two TM01

+

HE21even

-

HE21odd

+

TE01

-

LP11

Fig. 3.7 LP11 mode characteristics. (top) Basic mode. (bottom) Light intensity and polarization distribution

3.3 Analysis of the Influencing Factors of a Vortex Light Generated …

89

modes will deviate from each other. Therefore, the modes that can stably generate vortex light in an optical fiber are the HE and EH modes.

3.3 Analysis of the Influencing Factors of a Vortex Light Generated by Optical Fiber 3.3.1 Influence of the Incident Wavelength on the Vortex Light The core refractive index of the fiber is 1.4677 and the cladding refractive index is 1.4628. To study the first four HE modes in the fiber, the fiber-core radius is set to 10.1 μm. At this time, when the incident wavelength is between 500 and 2000 nm, the V value changes between one and six. The HE mode increases from HE11 to HE41 . According to Eqs. (3.35) and (3.36), there is a relationship between the incident wavelength and the effective refractive index of the fiber. Substituting Eqs. (3.35) and (3.36) into Eq. (3.29), the incident wavelength of the fiber HE mode and the effective refractive index of the fiber can be obtained. Figure 3.8 shows the relationship between the effective refractive index of the HE mode and the incident wavelength. As shown in Fig. 3.8, when the incident wavelength is 1550 nm, only the fundamental mode exists in the fiber; that is, the HE11 mode. When the incident wavelength is 850 nm, a second mode can be supported: HE21 . When the incident wavelength is 632.8 nm, three modes can be supported, HE11 , HE21 , and HE31 . When the incident wavelength is 532 nm, four modes can be supported: HE11 , HE21 , HE31 , and HE41 .

HE11 HE21 HE31 HE41

1.467 Effective refractive index

Fig. 3.8 Relationship between the HE-mode effective refractive index and the incident wavelength

1.466

HE11

532

1.465 632.8

1.464

1.463 500

HE21

HE31 HE41 850

1000

1550

1500

Incident wavelength/nm

2000

90

3 Vortex-Beam Generation Using the Optical-Fiber Method

1.466

Fig. 3.9 Relationship between the EH-mode effective refractive index and the incident wavelength

Effective refractive index

EH11 EH21

1.465

1.464

1.463

532

EH11

EH21 632.8

600

800

1000

1200

Incident wavelength/nm

The core refractive index of the optical fiber is 1.4677 and the cladding refractive index is 1.4628. The core radius of the fiber is 10.1 μm. According to Fig. 3.4a, when the incident wavelength is between 500 and 1200 nm, the EH mode in the fiber can increase from EH11 to EH21 . According to Eqs. (3.35) and (3.36), the incident wavelength of the fiber and the effective refractive index of the fiber are related. Substituting Eqs. (3.35) and (3.36) into Eq. (3.33), we can obtain the EH-mode incident wavelength and effective refractive index. Figure 3.9 shows the relationship between the EH-mode effective refractive index in the fiber and the incident wavelength. As shown in Fig. 3.9, when the incident wavelengths are 1550 nm and 850 nm, there is no EH mode in the fiber. When the incident wavelength is 632.8 nm, EH11 mode appears in the fiber. When the incident light is 532 nm, EH11 and EH21 modes appear in the fiber. The smaller the wavelength of the incident wave in the fiber, the more EH modes are generated.

3.3.2 Influence of the Refractive-Index Difference Between the Inside and Outside of the Optical Fiber on the Vortex Light Substituting Eqs. (3.35) and (3.36) into Eq. (3.29), the relationship between the difference in the refractive indices between the inside and outside of the fiber and the effective refractive index can be obtained. To study the relationship between the first three HE modes and the difference between the inner and outer refractive indices, the incident wavelength of the fiber is set to 1550 nm, the core radius is set to 10.1 μm, the cladding radius is set to 62.5 μm, and the refractive index of the outer layer of the

3.3 Analysis of the Influencing Factors of a Vortex Light Generated …

HE11 HE21 HE31

1.467 Effective refractive index

Fig. 3.10 Relationship between the HE-mode effective refractive index and the difference between the inner and outer refractive indices of the fiber

91

1.466 HE11

1.465

HE21

1.464

HE31

1.463 0

1

2

3

4

Internal and external refractive index difference

-3

x10

fiber is 1.4628. When the refractive-index difference between the inside and outside of the optical fiber increases, the HE mode changes, as shown in Fig. 3.10. It can be seen from Fig. 3.10 that the HE mode in the fiber increases with an increase in the internal and external refractive indices; thus, the effective refractive index also increases. The HE11 mode can exist in the fiber when the refractive-index difference is less than 2.2 × 10–4 . The effective refractive index of the fiber starts to increase from 1.4628, when the refractive-index difference is greater than 1.31 × 10–3 , and the HE21 mode can exist in the fiber. When the refractive-index difference is greater than 3.09 × 10–3 , the effective refractive index of the fiber begins to increase from 1.4268, at which time the HE31 mode can exist in the fiber. As the internal and external refractive indices of the fiber increase, the number of HE modes that can be accommodated in the fiber also increases. However, the difference in the refractive index between the inside and outside of the optical fiber should not exceed 0.36%. If the difference in the refractive index between the inside and outside of the optical fiber is too large, the weakly conductive structure of the optical fiber will be destroyed. In the optical fiber, the EH mode can also produce vortex light. According to Eqs. (3.35) and (3.36), there is a certain relationship between the internal and external refractive-index difference of the optical fiber and the effective refractive index of the optical fiber. Substituting Eqs. (3.35) and (3.36) into Eq. (3.33), the relationship of the internal and external refractive-index difference with the effective refractive index can be obtained. The incident wavelength of the optical fiber is set to 1550 nm, the core radius is set to 12.1 μm, and the outer refractive index is 1.4628. When the refractive-index difference between the inner and outer layers of the optical fiber increases, the EH mode in the optical fiber changes, as shown in Fig. 3.11.

Fig. 3.11 Relationship between the EH-mode effective refractive index and the difference between the internal and external refractive index of the fiber

3 Vortex-Beam Generation Using the Optical-Fiber Method

1.4645 Effective refractive index

92

EH11 EH21

1.4640

EH11

1.4635

EH21

1.4630 2.5

3.0

3.5

4.0

4.5 x10-3

Internal and external refractive index difference

As shown in Fig. 3.11, the EH mode in the fiber increases with an increase in the difference between the internal and external refractive indices; thus, the effective refractive index also increases. The EH11 mode can exist in the fiber when the refractive-index difference is greater than 2.89 × 10–3 . When the refractive-index difference is greater than 3.79 × 10–3 , the effective refractive index of the fiber begins to increase from 1.4628, and the EH21 mode can exist in the fiber. As the internal and external refractive indices of the optical fiber increase, the number of EH modes that can be accommodated in the optical fiber also increases.

3.3.3 Influence of the Fiber-Core Radius on the Vortex Light According to Eqs. (3.35) and (3.36), there is a certain relationship between the fibercore radius and the effective refractive index of the fiber. Substituting Eqs. (3.35) and (3.36) into Eq. (3.29), the relationship between the radius and effective refractive index can be obtained. The common internal and external refractive-index values of Corning Optical Fiber in the United States were used as the internal and external refractive indices of the optical fiber. Hence, we set the core refractive index of the fiber to 1.4677, the cladding refractive index to 1.4628, and the wavelength of the incident light to 1550 nm. Figure 3.12 shows the relationship between the effective refractive index of the HE mode in the fiber and the fiber-core radius. As shown in Fig. 3.12, when the HE mode in the fiber has a core radius of 4.1 μm, only the HE11 mode exists in the fiber. When the core radius is greater than 4.89 μm, the HE11 and HE21 modes occur in the fiber. When the fiber core is greater than 7.98 μm, three modes appear in the fiber: HE11 , HE21 , and HE31 . As the core of the optical fiber increases, the number of HE modes that can exist in the optical fiber also increases.

3.3 Analysis of the Influencing Factors of a Vortex Light Generated …

1.467

Fig. 3.12 Relationship between the HE-mode effective refractive index and the fiber-core radius

HE21 HE41

HE51

1.466 Effective refractive index

HE11 HE31

93

HE11

1.465 HE31 HE21

1.464 HE41 HE51

1.463 0

4

8

12

16

Fiber radius/ μm

We use the same values to investigate the EH-mode characteristics. Figure 3.13 shows the relationship between the effective refractive index of the EH mode in the fiber and the fiber-core radius. As shown in Fig. 3.13, when the EH mode in the fiber has a core radius of 4.1 μm, there is no EH mode in the fiber. When the core radius of the fiber is greater than 7.51 μm, the EH11 mode exists in the fiber. When the fiber core is larger than 10.02 μm, two modes exist, EH11 and EH21 . As the core size increases,

EH11 EH21 EH21

Effective refractive index

1.466

1.465 EH11 EH21

1.464

EH31

1.463 4

6

8

10

Fiber radius/ μm

12

14

Fig. 3.13 Relationship between the EH-mode effective refractive index and the fiber radius

94

3 Vortex-Beam Generation Using the Optical-Fiber Method

more EH modes can exist in the optical fiber. A suitable fiber core can be chosen to accommodate high-order vector modes.

3.3.4 Effect of the Incident Angle on the Excitation Efficiency of Vortex Light At different incident angles, light travels on different paths in the optical fiber; therefore, the time taken to reach the exit end also differs. According to the theory of planar waveguides [11], light with different angles, θ represents different modes. Assuming that the incident light is a Gaussian beam with a spot radius of ωs , the total power of the excitation-mode field is the sum of the powers of each order-mode field [24]: P=

. j

( )1 n 1 ε0 2 . |ai |2 . Pj = 2 μ0 j

(3.53)

ai is the field-strength coefficient of each mode field, n1 is the core refractive index, ε0 is the dielectric constant, and μ0 is the magnetic permeability. The excitation power of each mode field is given by [25] | |2 | .∞ .2π | | | E x ψl Rd Rdφ | 1 | ) ( 2 | | a n 1 ε0 2 0 0 Pl = . .∞ 2 4π μ0 el (R)Rd R

(3.54)

0

⎧ ⎨ el (R) cos lφ ψl = el (R) sin lφ . ⎩ el (R)

(3.55)

where R = r/a and a represents the fiber radius. The Bessel function is used to expand Eq. (3.48) in the cylindrical coordinate system, and the excitation efficiency of each mode field after elimination is [16]: . . . . ωs r0 2l+2 Pl (K 0 nθ ωs )2l (K 0 nθ ωs r0 )2 . =4 exp − p l! ωs2 + r02 ωs2 + r02

(3.56)

where P is the incident light power, n is the refractive index of the incident medium, and r 0 is the mode-field radius of the fiber. The incident light is Gaussian light with a 632.8-nm wavelength, and the optical-fiber parameters are selected from the Corning SMF-28e optical fiber.

3.3 Analysis of the Influencing Factors of a Vortex Light Generated … Fig. 3.14 Relationship between the incident angle θ and the excitation efficiency Pl /P

95 l=0 l=1 l=2

Excitation efficiency Pl/P

1.0 0.8 0.6 0.4 0.2 0.0 -0.2

-0.1

0.0

Incident angle θ /rad

0.1

0.2

The relationship between the incident angle of each mode of the vortex light and the excitation efficiency of the vortex light is shown in Fig. 3.14. Then, by deriving Eq. (3.56), the incident angle of the lth-order mode of the vortex light formed by the optical fiber at the maximum excitation efficiency can be obtained [16]: ) 1( θ = l 2 ωs2 + r02 /K 0 nωs r0 .

(3.57)

As shown in Fig. 3.14, when the incident angle is 0°, that is, when the fiber is incident perpendicularly, only the fundamental mode HE11 exists in the fiber. The excitation efficiency of the other modes in the fiber is 0. As the incident angle increases, the excitation efficiency of the fundamental mode in the fiber decreases and the excitation efficiency of the other modes increases. From Fig. 3.14 and Eq. (3.57), it can be seen that when the incident angle in the fiber reaches 2.607°, the excitation efficiency of the first-order mode is the largest, and when the incident angle reaches 3.465°, the excitation efficiency of the secondorder mode is the highest. When the incident angle ranges from 3.466° to 6.333°, the excitation efficiency of the fundamental mode is lower than that of the other modes. When the incident angle is greater than 6.333°, the energy of the light coupled into the optical fiber is too low, and the light cannot be transmitted in the optical fiber. The reason for this phenomenon is that when light is incident vertically, the light appears as a meridian in the optical fiber, and the only mode in the optical fiber is the HE11 mode. When the light is incident obliquely, it forms a spatial light in the optical fiber. The different incident angles correspond to different modes of vortex light generated by the optical fiber.

96

3 Vortex-Beam Generation Using the Optical-Fiber Method

3.3.5 Effect of Off-Axis Incident Fiber on Vortex Light The incident wavelength, refractive index difference between the inside and outside of the optical fiber, and core radius determine whether a specific mode can exist in an optical fiber. The excitation efficiency of each mode in the fiber is related to the off-axis quantity. When the light is incident perpendicularly, only the meridian fiber exists in the fiber. When the incident is offset, the path traversed through the fiber is different; therefore, the time taken to reach the exit end of the fiber is also different. To form the spatial light in the optical fiber, different offsets correspond to different mode orders of the vortex light. When light is irradiated into the optical fiber off-axis, the light in the optical fiber is unevenly irradiated, and spatial light forms in the optical fiber, generating highorder modes. The excitation efficiency expression in a cylindrical coordinate system is [24]. ( )2l . . . . rd2 4 rd ωs r 0 Pl = exp − 2 . p ⎡[l + 1] ωs ωs2 + r02 ωs + r02

(3.58)

Taking the derivative of the above formula, the relationship between the maximum excitation efficiency of the lth-order mode and the off-axis offset is expressed as [16] ( )1 2 rd = l 1/ 2 ωs2 + r02 / .

(3.59)

Gaussian light is used as the incident light, and the optical-fiber parameters are selected from Corning SMF-28e. Figure 3.15 shows the relationship between the excitation efficiency of each mode and the off-axis offset. l=0 l=1 l=2

1.0 Excitation efficiency Pl/P

Fig. 3.15 Relationship between the excitation efficiency of each mode and the off-axis offset

0.8 0.6 0.4 0.2 0.0 -20

-10

0

10

Off axis offset rd/nm

20

3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light

97

As shown in Fig. 3.15, when the offset is 0°, that is, when the fiber is incident perpendicularly, only the fundamental mode HE11 exists in the fiber, and the excitation efficiency of the other modes in the fiber is 0. As the incident angle increases, the excitation efficiency of the fundamental mode in the fiber decreases, and the excitation efficiency of the other modes increases. When the offset reaches 4 nm, the excitation efficiency of the first-order mode is up to 18.38%. When the offset is 6 nm, the excitation efficiency of the second-order mode is up to 6.77%. When the offset is between 5.6 nm and 10 nm, the excitation efficiency of the fundamental mode is lower than that of the other modes. When the offset is greater than 10 nm, the energy of the light coupled to the fiber is too low.

3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light 3.4.1 Principle of Using Few-Mode Fibers to Generate Vortex Light A few-mode fiber is a fiber with only the fundamental and low-order modes. The detailed parameter values of the few-mode fiber used in the experiment are listed in Table 3.2. For a 632.8 nm wavelength, it can be calculated from Eq. (3.34) that the normalized frequency of the few-mode fiber is 3.591. According to Fig. 3.4a, the normalized frequency at this time is 2.4048 < V < 3.8317, and the modes that can exist in the fiber are the fundamental mode (HE11 ) and the first-order modes (TE01 , TM01 , and HE21 ). According to the theory of optical fibers generating vortex light (such as Sect. 3.2.5), the fundamental mode in an optical fiber forms first-order vortex light. According to Eq. (3.46), we can superimpose the first-order fundamental modes in the fiber to form a vortex light: + even odd O AM11 = H E 21 + i H E 21 .

(3.60)

− O AM11 = T E 01 + i T M01 .

(3.61)

Table 3.2 Parameter values of the few-mode fiber in the experiment Fiber type

n1

n2

Length (m)

Core diameter (μm)

Mode-field diameter (μm)

Corning® HI 1060

1.4628

1.4677

1

5.3

5.9 ± 0.3 @ 980 nm

98

3 Vortex-Beam Generation Using the Optical-Fiber Method

The TE01 and TM01 modes in the few-mode fiber can be superimposed to form a vortex light. As shown in Fig. 3.4b, because TE01 and TM01 have different propagation constants, when they propagate the same distance, the phase changes of the two modes are inconsistent, which causes the modes to move away from each other. The effective refractive indices of the odd and even modes of the HE21 mode are equal; that is, the propagation constants of the two modes are equal. Hence, stable vortex light can be formed and no mode separation occurs, despite the increase in the transmission distance. In the experiment, we used the superposition of the HE21 odd and even modes to form a vortex light. According to the principle of vortex light generated by an optical fiber, the phases of the odd and even modes of HE21 in the optical fiber have a phase difference of π/2. We can convert and superimpose the phase difference by passing HE21 even in the optical fiber through a λ/4 wave plate, as shown in Fig. 3.17. Figure 3.16a shows the polarization change during the superposition process. It can be observed that after HE21 even and HE21 odd are superimposed, the vortex light generated by the few-mode fiber is circularly polarized. Figure 3.16b shows the phase change during the superposition process. It can be observed that before the superposition, HE21 even and HE21 odd have no spiral wavefront, and after the superposition, the phase presents a spiral-phase wavefront. Before and after stacking, the light-intensity distribution of the HE21 mode did not change significantly. The difference between the vortex light and an ordinary hollow beam lies in the helical change of the phase. The light intensity of the HE21 mode does not reflect whether it is a vortex light. It is necessary to further verify whether there is a helical phase to confirm that the light beam generated by the few-mode fiber is vortex light. HE21even

HE21odd

OAM11

1

(a)

+ i

= 0

.

(b)

+ i

= -.

Fig. 3.16 Polarization and phase changes of HE21 even and HE21 odd superimposed light. a Polarization-change diagram; b phase-change diagram

3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light Fig. 3.17 Relationship between the incident angle and excitation efficiency in HI 1060 fiber

l=0 l=1

1.0 Excitation efficiency Pl/P

99

0.8 0.6

l=0

0.4 0.2 l=1

0.0 -0.2

-0.1

0.0

0.1

0.2

Incident angle θ/rad

3.4.2 Analysis of the Excitation Efficiency of Vortex Light In this experiment, the few-mode fiber produced only the fundamental mode and the first-order mode. Therefore, in the experiment, the energy of the fundamental mode can be reduced by changing the incident angle, and the energy of the first-order mode can be increased to obtain the first-order mode. The parameters of the optical fiber take the parameters of the HI 1060 optical fiber. According to the laser-efficiency Eq. (3.56), the relationship between the incident angle and excitation efficiency of the vortex light in the few-mode fiber is shown in Fig. 3.17. As shown in Fig. 3.17, when the incident angle is 0°, that is, when the light is incident perpendicular to the fiber, only the fundamental mode HE11 exists, and the excitation efficiency of the other modes in the fiber is 0. As the incident angle increases, the excitation efficiency of the first-order mode first increases and then decreases. The reason for this phenomenon is that when the light is incident vertically, the light appears as a meridian in the optical fiber, and the only mode in the optical fiber is the HE11 mode. When the light is incident obliquely, it forms a spatial light in the optical fiber, and different incident angles correspond to different modes of vortex light generated by the optical fiber. According to Fig. 3.17 and Eq. (3.57), the excitation efficiency of the first-order vortex light of the few-mode fiber is the largest when the incident angle is 2.895°, and its excitation efficiency is 18.39%. When the incident angle of the few-mode fiber reaches 5.729°, the excitation efficiency of the vortex light is zero because of the attenuation of the energy of the light incident on the fiber. Therefore, in the experiment, when the incident angle is controlled between 2.895° and 5.729°, the basic mode of the first-order vortex light can be obtained in the few-mode fiber.

100

3 Vortex-Beam Generation Using the Optical-Fiber Method

3.4.3 Experimental Research As shown in Fig. 3.18, the light beam is converted into linearly polarized light after passing through the polarizer. The Gaussian beam can be coupled to the polarizationmaintaining fiber through the focusing lens, and the polarization-maintaining fiber is connected to the polarization controller. The polarization controller can realize a continuous change in the polarization state of the Gaussian beam by squeezing the optical fiber. The light beam emitted from the polarization controller can be any polarization. Then, the light beam emitted by the polarization controller is coupled into the fewmode fiber at a certain angle of incidence. A few-mode fiber can convert the incident Gaussian beam into the fundamental mode. An analysis of the experimental principle reveals that when the incident angle is controlled between 2.895° and 5.729°, the light emitted by the few-mode fiber is the first-order mode of the vortex light. The light emitted from the few-mode fiber is divided into two beams by the beam splitter: one beam detects the polarization of the vortex light and the other beam detects the phase of the vortex light. The light beam for the polarization passes through the polarizer, and the polarization angle of the polarizer is rotated to detect the polarization of the light beam. After passing the λ/4 wave plate, the detected light beam is incident on an engraved triangular hole to detect the phase of the vortex light. In this experiment, the first-order vector light is excited by adjusting the angle between the polarization-maintaining fiber and the few-mode fiber. The excitation efficiency of the first-order vortex light of the few-mode fiber is highest when the incident angle is 2.895°, and its excitation efficiency is 18.39%. The spot distribution

Fig. 3.18 Experimental-setup diagram

3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light

(b)

(a)

101

(d)

(c)

Fig. 3.19 Light intensity in first-order mode. a First-order mode simulation; b first-order mode experiment; c LP11 simulation; and d LP11 experiment

observed on the CCD, after a fine adjustment of the incident angle, is shown in Fig. 3.19. Figure 3.19a, c show the facula simulation diagrams of the first-order mode appearing in the optical fiber, and Fig. 3.19b, d show the facula diagrams of the first-order mode observed in the experiment. The light spot in Fig. 3.19b may be TM01 , TE01 , or HE21 even . The three beams have the same spot shape, but different polarization states. Figure 3.19d shows the LP11 pattern formed during the experiment. The reason for the formation of the LP11 pattern is the degeneracy of multiple vector patterns. It can be seen from the experimental results that the intensity distributions of several vector beams in the first-order mode are consistent. According to the principle of vector-mode excitation in a fiber, the polarization mode changes periodically along the length of the fiber. When the incident polarization is circular polarization, HE21 even can be generated in the fiber [25]. In this experiment, a polarization controller was used to adjust the polarization of the incident light to circular, and a polarizer was used to detect the polarization direction of the light emitted by the few-mode fiber. The polarization state of the emitted light could be discriminated by rotating the polarizer. The light-intensity distribution when the polarization direction is changed is shown in Fig. 3.20.

No polariser

(a)

(b)

(c)

(d)

(e)

Fig. 3.20 Light-intensity distribution of HE21 even mode after passing through the polarizer. a HE21 even simulation diagram; b unpolarized HE21 even experimental diagram; c experimental diagram at 90º polarization; d experimental diagram at 135° polarization; e experimental diagram at 180º polarization

102

3 Vortex-Beam Generation Using the Optical-Fiber Method

From Fig. 3.20b, it can be seen that the HE21 even experimental graph is similar to the Fig. 3.20a light-intensity simulation graph. When the polarization is 90º, only light polarized in the vertical direction can pass through HE21 even . The light in other polarization directions cannot pass through the polarizer. Figure 3.20c shows the light-intensity distribution when the polarization is 90º. From the 90º polarization detection result, there is vertical polarization in the detected light spot. When the polarization is 135º, only the light with 135º polarization can pass through HE21 even . Figure 3.20d shows the light-intensity distribution when the polarization is 135º. The light-intensity distribution is perpendicular to the polarization direction. Judging from the 135º polarization-detection result, there is 135º polarization light in the detected spot, and the light intensity is distributed at a position perpendicular to 135º. When the polarization is 180º, only the light with 180º polarization can pass through HE21 even . Figure 3.20e shows the light-intensity distribution when the polarization is 180º. The light-intensity distribution at this time is consistent with the polarization direction. From the 180º polarization-detection result, there is 180º polarization light in the detected light spot, and the light-intensity distribution is at 180º. From the polarization distribution, it can be determined that the detected light in Fig. 3.20b is first-order mode HE21 even . The experiment generated first-order vector mode HE21 even from the optical fiber. According to Eq. (3.60), the generation of vortex light in an optical fiber requires the superposition of the even and odd modes of the first-order vector beam HE21 . In this experiment, the mixed polarized light HE21 even generated in this experiment passed through a λ/4 wave plate, and HE21 even was superimposed with the HE21 even delayed by π/2; that is, HE21 even and HE21 odd were superimposed. Figure 3.21 shows the distribution of the light intensity before passing through the λ/4 wave plate. Figure 3.21a shows the light-intensity distribution diagram before passing through the λ/4 wave plate, and Fig. 3.21b shows the three-dimensional light-intensity distribution. Figure 3.21c shows the X-direction experimental and theoretical light intensities before passing through the λ/4 wave plate, and Fig. 3.21d shows the Y-direction experimental and theoretical light intensities before passing through the λ/4 wave plate. From Fig. 3.21c, d, there are certain errors in the theoretical and experimental data for each picture. The main sources of errors are (1) background light noise and (2) camera exposure. Figure 3.22 shows the light-intensity distribution after passing through the λ/4 wave plate. Figure 3.22a shows the light-intensity distribution diagram after passing through the λ/4 wave plate, and Fig. 3.22b shows the three-dimensional lightintensity distribution. Figure 3.22c shows the X-direction experimental and theoretical light intensities after the λ/4 wave plate, and Fig. 3.22d shows the Y-direction experimental and theoretical light intensities after the λ/4 wave plate. From a comparison of Fig. 3.21c–d and Fig. 3.22c–d, there is no obvious change in the light intensity before and after the λ/4 wave plate; however, the energy is slightly attenuated after passing through the λ/4 wave plate. We use a correlation to express the relationship between the light beam produced by the few-mode fiber and the first-order vortex light. In Fig. 3.21c, the correlation between the X direction and

3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light

(a)

103

(b)

Fig. 3.21 Light-intensity diagram before passing through the λ/4 wave plate. a Two-dimensional light intensity; b three-dimensional light intensity; c X-direction theoretical and experimental results; d Y-direction theoretical and experimental results

the X direction of the first-order vortex light is 88.02%. In Fig. 3.22c, the correlation between the Y direction and the Y direction of the first-order vortex light is 83.46%. It can be seen from the experimental results that the intensity of the vortex light generated by the few-mode fiber conforms to the distribution of the first-order vortex light. The difference between vortex light and ordinary beams is the spiral-phase wavefront. While detecting the intensity of the vortex light, we also need to detect the phase of the vortex light to determine the topological-charge size and the direction of the light beam generated by the few-mode fiber.

3.4.4 Phase Verification The difference between a vortex light and a vector beam is that the vortex light has a spiral-phase wavefront. In the above experiment, the light intensity generated in the few-mode fiber conforms to the first-order vortex light-intensity distribution; however, the phase of the vortex light was not detected, which does not indicate that vortex light was generated in the fiber. After comparing the phase-detection methods of the vortex light, the triangular-diffraction method was selected as the

104

3 Vortex-Beam Generation Using the Optical-Fiber Method

(b)

(a)

Fig. 3.22 Light-intensity diagram after passing through the λ/4 wave plate. a Two-dimensional light intensity; b three-dimensional light intensity; c X-direction theoretical and experimental; (d) Y-direction theoretical and experimental

phase-detection method for the vortex light. The light in Fig. 3.22a is shown in Fig. 3.23, after diffraction through a triangle. Figure 3.23a shows the light spot of the vector mode in the fiber after passing through the λ/4 waveplate, and Fig. 3.23b shows the diffraction spot of Fig. 3.23a after passing through a triangular hole. As shown in Fig. 3.23b, the diffraction-spot distribution is triangular. The light beam without a phase singularity forms a spot with a Gaussian distribution after passing through the triangular hole, and the light beam with a phase singularity forms a triangular distribution spot after being diffracted by the triangle. Fig. 3.23 Diffraction result of the light spot to be inspected passing through a triangular hole. a Light spot to be inspected; b triangular-diffraction result

(a)

(b)

3.5 Changing the Fiber Structure to Produce Vortex Light

105

In the triangle-diffraction pattern, the topological charge of the vortex light is indicated by the number of triangular-side spots minus one. The direction of the apex of the triangle in the triangle pattern determines the positive or negative topological charge of the vortex light [26]. The side of the triangle in Fig. 3.23b is composed of two spots. The topological charge of the vortex light is the number of spots that make up the triangle side, minus one. It can be deduced that the topological charge of the spot in Fig. 3.23a is 1. The triangular diffraction spot is composed of vertical edges and vertices. When the vertices are on the left, the topological charge is positive; when the vertices are on the right, the topological charge is negative. The vertex of the triangle in Fig. 3.23b is on the left; therefore, the topological charge of the spot in Fig. 3.23a is positive. The topological charge l = + 1 of the spot in Fig. 3.23a can be determined using triangular diffraction. In this experiment, the mixed-polarization vector beam HE21 even is generated in the few-mode fiber, and HE21 even and HE21 odd are superimposed to generate the vortex light. The results show that the few-mode fiber can generate a first-order vector mode. After superposition, the first-order vector mode was converted into a first-order vortex beam with a spiral-phase wavefront.

3.5 Changing the Fiber Structure to Produce Vortex Light 3.5.1 Structural Design To effectively degenerate and separate the vector modes, the fiber structure needs to meet the high refractive-index gradient and the high mode field gradient. This can be achieved in a fiber structure with a high refractive-index difference and sharp refractive-index profile. Considering this comprehensively, the fiber structure with an inverted parabolic refractive-index gradient distribution is improved, and a low layer is added between the core and cladding. The refractive-index layer, which increases the refractive-index difference, allows it to contain more modes. Its refractive-index distribution is generally expressed as. ⎧ / ) ( ⎪ ⎨ n 1 1 − 2N Δ r 2 /a 2 0 ≤ r ≤ rcor e n(r ) = n2 rcor e < r ≤ r2 . ⎪ ⎩ n3 r > r2

(3.62)

The blue curve in Fig. 3.24 is the refractive-index profile of the improved inverted parabolic graded-index profile fiber (the red part of the original fiber structure). As shown in the structure diagram in Fig. 3.3. 24, n1 and n2 are the refractive indices of the core center and the low–refractive-index layer, respectively. r core = 3 μm, n1 = 1.4539, n2 = 1.440, and n3 = 1.444. N is the curvature parameter of the

106

3 Vortex-Beam Generation Using the Optical-Fiber Method

1.50

Fig. 3.24 Improved optical-fiber structure refractive-index profile

na=1.494

Refractive index size

1.49 1.48 1.47 1.46 n1

1.45

n3=1.444

1.44 -6

n2=1.440

-4

-2

0

2

Fiber radius /µm

4

6

inverted parabola; let N = -4; the relative refractive-index difference is Δ = (n 21 − n 22 )/2n 21 ; the refractive-index difference between the low-refractive-index layer and the cladding layer is Δ n 2 = n 2 − n 3 . The maximum refractive-index difference appears at the boundary between the core and the low-refractive-index layer: n a = n 1 − (n 1 − n 2 )N and Δ n max = n a − n 2 . As a special case, when N = 0, it is a conventional step-index-distribution fiber structure. The core structure of the optical fiber is a graded refractive-index structure, and the cutoff frequency V of its inverted parabolic core also varies with the function. The formula is. / 2πr n(r )2 − n 22 . (3.63) V (r ) = λ If we set λ = 1550 nm, the number of modes in the fiber gradually increases with an increase in the normalized frequency V. The optical fiber transmits vortex light in the range of the core radius r 2 . The calculation shows that the optical-fiber structure is V = 5.421. According to the root of the Bessel-function curve corresponding to the normalized frequency value, the number of modes that can be accommodated in the designed optical fiber is 12, which can transmit vortex beams with a topological charge of three. From Eq. (3.25), the relationship between the effective refractive index and the wavelength of the accommodating mode in the fiber can be determined. As shown in Fig. 3.25, in the wavelength range, as the wavelength increases, the effective refractive index of the accommodating mode in the fiber decreases. The calculation shows that the effective refractive-index difference of the LP11 mode group {TE01 , HE21 , TM01 } in the wavelength range is 2.1 × 10–4 , the effective refractive-index difference of the LP21 mode group {HE31 , EH11 } in the wavelength range is greater than 3.6 × 10–4 , and the LP31 mode group {EH21 , HE41 } in the wavelength range is greater than 1.1 × 10–3 . The maximum effective refractive-index difference in the wavelength range can be greater than 1.1 × 10–3 .

3.5 Changing the Fiber Structure to Produce Vortex Light

HE11

1.490 Effective refractive index neff

Fig. 3.25 Relationship between the effective refractive index and wavelength

107

TE01 HE21 TM01

1.485

HE31 EH11 EH12 EH21

1.480

HE41

1.475 1.470 1.465 1500

1520

1540

1560

1580

1600

Wavelength λ / nm

The effective refractive-index difference of the LP21 mode group of the fiber structure with a low-refractive-index layer is increased by at least 1 × 10–4 , compared to that of the LP21 mode group of the original fiber structure. An increase in the effective refractive-index difference prevents the HE and EH modes from coupling into the LP mode. The coupling between the modes is reduced, and each mode can be transmitted independently and stably. Thus, this structure exhibits a good pattern-separation effect.

3.5.2 Influence of the Low-Refractive-Index Layer on OAM Mode Because the modes of EHl-1,m and HEl+1,m are both determined by characteristic equations, their intensities and phase distributions are the same, and they correspond to the OAM mode with a topological charge of l. Assuming that wavelength λ = 1550 nm, Fig. 3.26 shows the intensity distribution of a high-order Bessel beam. The phase jumps l times in one cycle, and an l-order vortex beam can be transmitted. The corresponding relationship between the modes that the fiber can accommodate and the mode superposition that can generate an OAM mode with a topological charge of l is shown in Fig. 3.26. The number of guided modes in an optical fiber depends not only on a certain structural parameter, but on the normalized operating frequency of the optical fiber. In the case of constant refractive indices n1 and n2 , the number of optical fiber modes changes with inner diameter r1 and outer diameter r2 . Figure 3.27 shows that when the refractive index of the fiber and r2 remain unchanged, the order of the optical fiber that can transmit vortex light gradually decreases with an increase in r1 , and the order of the transmittable OAM mode decreases with an increase in r1 . The decreasing trend of the order tends to reduce with an increase in r1 .

108

3 Vortex-Beam Generation Using the Optical-Fiber Method (b)

(a)

(c)

(f)

(e)

(d)

Fig. 3.26 Light intensity and phase diagram of the vortex beam. a OAM11 light intensity; b OAM21 light intensity; c OAM31 light intensity; d OAM11 phase; e OAM21 phase; f OAM31 phase

25

Fig. 3.27 Relationship between the number of OAM orders that can be accommodated and the core radius r1 OAM Mode order l

20 15 10 5 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Core radius r1/ μm

When the inner diameter r1 and outer diameter r2 remain unchanged, the refractive index n2 of the low-refractive-index layer is changed to obtain the information in the following figure. It shows the corresponding relationship between the topological charge of the optical fiber that can transmit vortex light and n2 . As n2 decreases, the order of the modes that can transmit the vortex light increases (Fig. 3.28). A new type of fiber structure improves on the original fiber structure with an inverted-parabolic graded refractive-index profile. Calculations and simulations provided the intensity-distribution and phase-distribution diagrams of the OAM mode

References

4.0 OAM Mode order l

Fig. 3.28 Relationship between the transmittable OAM order and the low-refractive-index layer

109

3.5 3.0 2.5 2.0 1.444

1.440

1.436

1.432

Low refractive index layer n2

1.428

of this fiber. This theory proves that this new, improved structure can accommodate high-order OAM modes.

References 1. Guo MJ, Zeng J, Li JH (2016) Generation and Interference of Vortex Beam Based on Spiral Phase Plate. Laser Optoelectronics Progress 9:230–236 2. Jennifer C, David G (2003) Modulated optical vortices. Opt Lett 28(11):872–874 3. Zhang YH (2010) Computer-produced amplitude hologram of vortex beam. Neijiang Technol 31(8):20–21 4. Alexeyev A, Tatiana F, Volyar A et al (1998) Optical vortices and the flow of their angular momentum in a multimode fiber. Semiconductor Phys 1:29–34 5. He YL, Ye HP, Liu JM et al (2017) Order-controllable cylindrical vector vortex beam generation by using spatial light modulator and cascaded metasurfaces. IEEE Photon J 9(5):1–10 6. Fang ZQ, Yao Y, Xia KG et al (2013) Vector mode excitation in few-mode fiber by controlling incident polarization. Opt Commun 294(294):177–181 7. Zhang WD, Huang LG, Wei KY et al (2016) High-order optical vortex generation in a few-mode fiber via cascaded acoustically driven vector mode conversion. Opt Lett 41(21):5082–5087 8. Li SH, Wang J (2014) A compact trench-assisted multi-orbital- angular-momentum multiring fiber for ultrahigh-density space-division multiplexing(19 Rings×22 Modes). Sci Rep 4:3853–3857 9. Zhang XQ, Wang AT, Chen RS et al (2016) Generation and conversion of higher order optical vortices in optical fiber with helical fiber bragg gratings. J Lightwave Technol 34(10):2413– 2418 10. Zhang LX, Wei W, Zhang ZM et al (2017) Propagation properties of vortex beams in a ring photonic crystal fiber. Acta Physica Sinica 11. Katsunari O (2005) Fundamentals of optical waveguides (Second Edition). Opt Photon Counterterrorism Crime Fighting 7486:535–554 12. Chen JH (2002) Discussion on Maxwell’s equations. Phys Eng 12(4):18–20 13. Qiao HL, Wang Y, Chen ZG (2013) Full-vectorial finite-difference analysis of modes in waveguide with arbitrary shap. Acta Physica Sinica 62(7):24–31

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14. Zhang XQ (2016) Research on the generation and control of vortex beam in optical fiber. Doctoral Dissertation of University of Science and Technology of China, Anhui, pp 13–19 15. Sun PJ (2016) Generation of vector vortex beam in optical fiber. Master’s Thesis of Harbin University of Science and Technology, Harbin, pp. 22–27 16. Sun YA (2003) Optical fiber technology: theoretical basis and application. Beijing Institute of Technology Press, Beijing, pp 51–52 17. Wei JC (2017) Theoretical research and design of vortex optical fiber. Master’s Thesis of Beijing Jiaotong University, Beijing, pp 30–35 18. Charles B, Ung B, Pierre B et al (2014) Vector mode analysis of ring-core fibers: design tools for spatial division multiplexing. Lightwave Technol J 32(23):4648–4659 19. Charles B, Leslie R (2016) Optical fibers for the transmission of orbital angular momentum modes. Opt Fiber Technol 31:172–177 20. Pedram D, Fares A, LEE H (2006) Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber. Phys Rev Lett 96(4):043604 21. Siddharth R, Poul K (2013) Optical vortices in fiber. Nanophotonics 2(5–6):455–474 22. Nenad B, Stevenj G, Poul K et al (2012) Control of orbital angular momentum of light with optical fibers. Opt Lett 37(13):2451–2457 23. Siddharth R, Poul K (2009) M YAN, “Generation and propagation of radially polarized beams in optical fibers.” Opt Lett 34(16):2525–2527 24. Gabriel MT, Juan T, Liuis T (2002) Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum. Phys Rev Lett 88(1):013601 25. Nirmal V, Krishna IV (2009) Generation of switchable vector beam with two-mode optical fiber and its characteristics. In: ICOP 2009 international conference on optics and photonics. Ühandigarh, India, p 11 26. De AL, Matthew A (2011) Measuring vortex charge with a triangular aperture. Opt Lett 36(6):787–789 27. Sonja FA, Les A, Mites P (2010) Advances in optical angular momentum. Laser Photonics Rev 2(4):299–313 28. Marco B, Robert S, Les A et al (1992) Multiphoton resonances and Bloch-Siegert shifts observed in a classical two-level system. Phys Rev A 45(3):1810–1815 29. Gibson G, Johannes C, Miles P et al (2004) Free-space information transfer using light beams carrying orbital angular momentum. Opt Express 12(22):5448–5456 30. Lu H, Ke XZ (2009) Research on the beam with orbital angular momentum used in encoding and decoding of optical communication. Acta Optica Sinica 29(2):331–335 31. Guo XL, Ke XZ (2015) Realization of optical phase information encode by using orbital angular momentum of light beam. Chinese J Quantum Electron 32(1):69–76

Chapter 4

Superposition Characteristics of High-Order Radial Laguerre–Gaussian Beams

Laguerre–Gaussian (LG) beams are widely used in orbital angular momentum (OAM) multiplexing communication systems; however, in research, they are currently limited to zero-order radial LG beams (radial index is zero). Higher-order radial LG beams with a radial index greater than zero can also be applied to OAMmultiplexed communication systems to double the channel capacity, communication rate, and frequency-band utilization. The high-order radial LG beam has a higher receiving power than the zero-order radial LG beam because of the limitation of the receiving aperture. Research on the superposition state of high-order radial LG beams has mainly focused on two aspects. One is the influence of the radial index on the superposition state of LG beams. The other is the effects of the transmission distance and beamwaist radius on the superposition of high-order radial LG beams.

4.1 Introduction When LG beams are superimposed coaxially, composite vortex beams with different characteristic distributions, propagation characteristics, and special applications can be formed. Naidoo [1] used an intracavity mode-selection method to coherently superimpose two zero-order radial LG beams with opposite topological charges in a solid-state laser. Huang [2] adopted an extra-cavity conversion method that coaxially superimposed computational holograms on multiple LG beams. Aity [3] studied the self-repairing characteristics of the halo lattice structure formed by the coaxial superposition of LG beams. Li [4, 5] measured topological charge through LG-beam superposition. However, their work is limited to the case of radial exponents, and the superposition state of high-order radial LG beams has rarely been studied. Ando [6] discussed the structural characteristics of the phase singularity of the LGbeam superposition state at the source plane, according to the difference in topological © Science Press 2023 X. Ke, Generation, Transmission, Detection, and Application of Vortex Beams, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-0074-9_4

111

112

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

charge and radial index, and provided general expressions for its light-intensity distribution and phase distribution. However, they did not further discuss the influence of the propagation distance and beam-waist radius on the LG-beam superposition state.

4.2 Influence of the Radial Index on the Superposition State of High-Order Radial LG Beams When the radial exponent of the LG beam is not 0, assuming the phase factor of the LG beam is, the complex amplitude of the beam at the source plane is expressed as [6] / u lp (r, θ )

=

p! 2 (−1) p π ( p + |l|)! ω0

(√

2r ω0

)|l|

( 2 ) ( 2) 2r r exp(−ilθ ), exp − 2 L |l| p ω0 ω02 (4.1)

where ω0 is the beam-waist radius. l is the topological charge; its value range is (−∞, +∞), which can be an integer or a fraction. p is the radial exponent, which is an integer with a value range of (0, +∞); and L lp (x) is the Laguerre polynomial. For convenience, let [7] / A|l| p (ω0 )

=

p! 2 π ( p + |l|)!

(√

2r ω0

)|l|

( 2 )|l| ( 2 ) 2r (−1) p r exp − 2 L |l| , p ω0 ω0 ω02

(4.2)

|l|

where A p (ω0 ) is the dimensionless radial amplitude related to p, |l|, and ω0 . Substituting Eqs. (4.2) into (4.1), we obtain u lp (r, θ ) = A|l| p (ω0 ) exp(−ilθ ).

(4.3)

When the phase factors are exp(-il1 θ ) and exp(-il 2 θ ), the radial exponents are p1 and p2 , and the waist radii of the two LG beams are ω01 and ω02, which are coherently superimposed at the source plane. The electric-field distribution of the double LG-beam superposition state is u two (r, θ ) = A|lp11| (ω01 ) exp(−il2 θ ) + A|lp22| (ω02 ) exp(−il2 θ ).

(4.4)

It can be seen from Eq. (4.4) that utwo (r, θ ) is related to p, |l|, and ω0 ; therefore, we consider the influence of the topological charge, radial index, and beam-waist radius on the superposition state of the double LG beams.

4.2 Influence of the Radial Index on the Superposition State of High-Order …

113

4.2.1 Interference and Superposition of LG Beams with the Same Topological Charge When two LG beams with the same topological charge are superimposed coaxially, the complex amplitude expression of the formed superposition state is [ ] |l| u(r, θ ) = u lp1 (r, θ ) + u lp2 (r, θ ) = A|l| p1 (ω0 ) + A p2 (ω0 ) exp(−ilθ ).

(4.5)

Comparing Eqs. (4.5) and (4.3), it can be seen that the composite vortex beam formed by the superposition of two LG beams with the same topological charge and different radial exponents has a light-intensity distribution similar to that of a single high-order radial LG beam. When two LG beams with the same topological charge are superimposed coaxially, to study the influence of the radial index on their light-intensity distribution and phase-distribution characteristics, we keep the radial index of one LG beam constant and gradually increase the other. Figure 4.1 shows the light-intensity distribution of two LG beams with the same topological charge when superimposed coaxially. The simulation parameters are as follows: topological charge of the two LG beams, l1 = l2 = 3; beam-waist radius, ω01 = ω02 = 1.0 mm.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4.1 Light-intensity distribution of the superimposed LG beams with the same topological charge and different radial exponents. a LG0 3 and LG1 3 beams; b LG0 3 and LG2 3 beams; c LG0 3 and LG3 3 beams; d LG1 3 and LG1 3 beams; e LG1 3 and LG2 3 beams; f superimposed LG1 3 and LG3 3 beams

114

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

As shown in Fig. 4.1, when two LG beams with the same topological charge and different radial exponents are coaxially superimposed, the intensity distribution of the two LG beams becomes bright and ring-shaped. It can be seen from Fig. 4.1a that when LG0 3 and LG1 3 are superimposed, their light-intensity distribution has two bright rings. Figure 4.1b shows that when LG0 3 and LG2 3 are superimposed, their lightintensity distribution has three bright rings. Figure 4.1c shows that when LG0 3 and LG3 3 are superimposed, their light-intensity distribution has four bright rings. Figure 4.1d shows that when LG1 3 and LG1 3 are superimposed, their light-intensity distribution has three bright rings. Figure 4.1e shows that when LG1 3 and LG2 3 are superimposed, their light-intensity distribution has three bright rings. Figure 4.1f shows that when LG1 3 and LG3 3 are superimposed, their light-intensity distribution has four bright rings. In summary, when two LG beams with the same topological charge are superimposed coaxially, their light-intensity distribution appears as a bright ring shape. The number of bright rings σ depends on the radial exponent of the LG beam with the largest radial exponent; that is, max{ p1 , p2 }; specifically, σ = max{ p1 , p2 } + 1. Figure 4.2 shows the phase-distribution diagram of the superimposed double LG beams with the same topological charge, which corresponds to the superposition state in Fig. 4.1. The simulation parameters are as follows: topological charge of the two LG beams, l 1 = l 2 = 3; the beam-waist radius, ω01 = ω02 = 1.0 mm.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4.2 Phase distribution of the superimposed LG beams with the same topological charge and different radial indices. a LG0 3 and LG1 3 beams; b LG0 3 and LG2 3 beams; c LG0 3 and LG3 3 beams; d LG1 3 and LG1 3 beams; e LG1 3 and LG2 3 beams; f superimposed LG1 3 and LG3 3 beams

4.2 Influence of the Radial Index on the Superposition State of High-Order …

115

Figure 4.2a, d have similar phase distributions, and both have two bifurcation layers with three bifurcations in each layer. This is because the LG beams involved in the superposition have a maximum radial index of one. Figure 4.2b, e have similar phase distributions, with three bifurcation layers and three bifurcations in each layer. This is because the LG beams involved in the superposition have a maximum radial index of two. Figure 4.2c, d have similar phase distributions, and both have four bifurcation layers with three bifurcations in each layer. This is because the LG beams involved in the superposition have a maximum radial index of three. By analogy, as long as the maximum radial exponents of a group of double LGbeam superposition states are the same, and the topological charge remains the same, the members of the group have similar phase-distribution structures. The number of bifurcation layers η depends on the maximum radial exponent, max{ p1 , p2 }; specifically, η = max{ p1 , p2 } + 1. The number of bifurcations ε in each layer depends on the topological charge l; specifically, ε = l.

4.2.2 Interference and Superposition of LG Beams with the Same Radial Index When two LG beams with the same radial index and different topological charges are superimposed coaxially, the light-field expression of the superposition state is u(r, θ ) = u lp1 (r, θ ) + u lp2 (r, θ ) = A|lp1 | (ω0 ) exp(−il1 θ ) + A|lp2 | (ω0 ) exp(−il2 θ ). (4.6) When two zero-order radial LG beams with the same topological-charge number and a small difference in charge are superimposed coaxially, their light-intensity distribution is dark and “petal”-like. When two zero-order radial LG beams with the same topological-charge number and a large difference are superimposed coaxially, the light-intensity distribution is a double-bright ring. When two zero-order radial LG beams with opposite topological charges are superimposed coaxially, their lightintensity distribution is bright and “petal”-like. Therefore, when studying the coaxial superposition of two LG beams with the same radial exponent, we consider the following three cases: (1) Topological charges with the same sign that have a small difference in charge; (2) Topological charges with the same sign that have a large difference; (3) Topological-charge numbers that have opposite signs. We consider the first case; that is, the case where the topological charges have the same sign and the difference is small. When two LG beams with the same radial index are superimposed coaxially, the topological charges (l1 = 1, l 2 = 3) of the two LG beams remain unchanged and their radial indices p are gradually increased to 0,

116

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

(a)

(b)

(c)

(d)

Fig. 4.3 Light-intensity distribution of the superimposed LG beams with the same radial index and different topological charges. a LG0 1 and LG0 3 beams; b LG1 1 and LG1 3 beams; c LG2 1 and LG2 3 beams; d superimposed state of LG1 3 and LG3 3 beams

1, 2, and 3 in sequence, as shown in Fig. 4.3. We chose the topological charges of the LG beam as l1 = 1 and l2 = 3, and the beam-waist radii are ω01 = ω02 = 1.0mm. As shown in Fig. 4.3, when two LG beams with the same radial index and a small difference in the topological charge are superimposed coaxially, their light-intensity distribution is different from the superposition state of a zero-order LG beam. It is no longer a dark “petal” shape, but is closer to a brighter ring. As shown in Fig. 4.3a– d, as the radial index increases, the number of bright rings in the light-intensity distribution gradually increases. The spot pattern at the center of the light-intensity distribution transitions from a dark “petal” shape to a bright ring, and the radius of the central bright ring decreases with the increase in radial index. When the topological-charge difference of the two high-order radial LG beams is small and the radial exponents are both p, the sum of the number of “petals” and the number of bright rings of the superposition light-intensity distribution is σ = p + 1. We consider the second case; that is, the case where the topological charges have the same number and a large difference. When two LG beams with the same radial index are superimposed coaxially, the topological charges of the two LG beams remain unchanged, and their radial indices gradually increase to 0, 1, 2, and 3 in sequence, as shown in Fig. 4.4. We selected the topological charges l 1 = 1 and l2 = 30, and the beam-waist radii as ω01 = ω02 = 1.0 mm. (a)

(b)

(c)

Fig. 4.4 Intensity distribution of superimposed LG beams with the same radial index and different topological charges. a LG0 1 and LG0 30 beams; b LG1 1 and LG1 30 beams; c superimposed LG2 1 and LG2 30 beams

4.2 Influence of the Radial Index on the Superposition State of High-Order …

117

Figure 4.4 shows that when two LG beams with the same radial index and a large difference in topological charge are superimposed coaxially, the intensity distribution of the two LG beams is a multi-bright ring; that is, an independent multi-ring structure. The central light intensity is zero and a hollow dark spot is formed. Figure 4.4a shows that when beams LG0 1 and LG0 30 are superimposed, the light-intensity distribution has two bright rings. Figure 4.4b shows that when beams LG1 1 and LG1 30 are superimposed, the light-intensity distribution has four bright rings. Figure 4.4c shows that when beams LG2 1 and LG2 30 are superimposed, the light-intensity distribution has six bright rings. From the above analysis, it can be concluded that the number of bright rings σ depends on the radial exponent p of the LG beam, and satisfies σ = 2( p + 1). By analogy, when the radial indices are both p, there will be 2(p + 1) bright rings in the light-intensity distribution of the LG-beam superimposed state. There are two reasons for this phenomenon. One is that the light intensity distribution of a single LG beam with a radial exponent of p has p + 1 bright rings. The second is that when the topological charge differs greatly, the two LG beams will maintain the bright ring shape of a single beam. Therefore, when two high-order radial LG beams with large topological-charge differences are superimposed coaxially, the light-intensity distribution consists of 2(p + 1) bright rings. Then, it can be extended to the case where multiple (n) LG beams are coaxially superimposed, and the light-intensity distribution is n(p + 1) bright rings. We consider the third case; that is, the case where the topological charges are opposite to each other. When two LG beams with the same radial exponent and opposite topological charges are superimposed coaxially, the light-field expression of the superposition state is u two (r, θ ) = u lp (r, θ ) + u −l p (r, θ ) |−l| = A|l| p (ω0 ) exp(−ilθ ) + A p (ω0 ) exp(−ilθ ).

=

(4.7)

2 A|l| p (ω0 ) cos(lθ )

The light-intensity distribution can be expressed as [8] I = u two (r, θ ) × u ∗two (r, θ ) = |u two (r, θ )|2 .

(4.8)

Combining Eqs. (4.1), (4.7), and (4.8), when two LG beams with the same radial index, opposite topological charges, and the same beam-waist radius are superimposed coaxially, the light-intensity distribution is | |2 I = 2| A p,l | [1 + cos(2lθ )].

(4.9)

To obtain the angular position θv of the maximum light intensity, we let ∂I =0 ∂θ

(4.10)

118 (a)

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian … (b)

(c)

(d)

Fig. 4.5 Light-intensity distribution of superimposed LG beams with opposite topological charges. a LG0 3 and LG0 −3 beams; b LG1 3 and LG1 −3 beams; c LG2 3 and LG2 −3 beams; d superimposed LG3 3 and LG3 −3 beams

Solutions must meet the following criterion: θv =

kπ (k = 0, 1, . . . , 2|l| − 1). l

(4.11)

When two LG beams with the same radial index are superimposed coaxially, the topological charges (l 1 = 3, l 2 = −3) of the two LG beams are kept unchanged and their radial indices are gradually increased; p is 0, 1, 2, and 3 in sequence, as shown in Fig. 4.5. We set the topological charges as l1 = 3 and l 2 = −3 and the beam-waist radii as ω01 = ω02 = 1.0 mm. Figure 4.5a–d show the theoretical light-intensity distribution diagrams of the composite vortex beam coaxially superimposed with LG0 3 and LG0 −3 beams, LG1 3 and LG1 −3 beams, LG2 3 and LG2 −3 beams, and LG3 3 and LG3 −3 beams. Figure 4.5b shows the light-intensity distribution diagram of the composite vortex beam formed by the coaxial superposition of the LG1 3 and LG1 −3 beams. It can be seen that there are 12 bright “petals”: two layers of bright “petals” on the periphery with six in each layer. The angular positions are 0, 1/3π, 2/3π, π, 4/3π, and 5/3π, respectively. Figure 4.5c shows the light-intensity distribution diagram of the composite vortex beam formed by the coaxial superposition of the LG2 3 and LG2 −3 beams. It can be seen that there are 18 bright “petals”: Three layers of bright “petals” on the periphery with six in each layer. The angular positions are 0, 1/3π, 2/3π, π, 4/3π, 5/3π, respectively. The fourth is not listed. By analogy, when two high-order radial LG beams with the same radial index and opposite topological charges are superimposed coaxially, the light-intensity distribution of the generated composite vortex beam has a circularly symmetrical, multi-layered bright “petal” distribution. The light-intensity distribution is p + 1 layers of bright “petals,” where each layer has |l2 − l1 |, the angular position is φv = kπ/l, (k = 0, . . . |l2 − l1 |) and the number of bright “petals” totals ( p + 1)|l2 − l1 |. In addition, as p increases, the overall spot radius of the superimposed high-order radial LG beam gradually increases, whereas the radius of the innermost bright “petal” gradually decreases. Figure 4.6a–d show the theoretical phase-distribution diagrams of the composite vortex beam formed by the coaxial superposition of LG0 3 and LG0 −3 beams, LG1 3 and

4.2 Influence of the Radial Index on the Superposition State of High-Order … (a)

(b)

(c)

119

(d)

Fig. 4.6 Phase distribution of superimposed LG beams with opposite topological charges. a LG0 3 and LG0 −3 beams; b LG1 3 and LG1 −3 beams; c LG2 3 and LG2 −3 beams; d superimposed LG3 3 and LG3 −3 beams

LG1 −3 beams, LG2 3 and LG2 −3 beams, and LG3 3 and LG3 −3 beams, respectively. Figure 4.6a shows the phase-distribution diagram of the superimposed LG0 3 and LG0 −3 beams, starting from the center of the circle, where there are six emission-like isophase lines. Figure 4.6b shows the phase-distribution diagram of the superimposed LG1 3 and LG1 −3 beams. There is a circular cross-section; that is, there are two phase mutations, and each phase mutation appears six times (There are six isophase lines.). Figure 4.6c shows the phase-distribution diagram of the superimposed LG2 3 and LG2 −3 beams. There is a circular cross section with three phase mutations, each of which appears six times (There are six isophase lines.). The fourth is not listed. By analogy, when two high-order radial LG beams with the same p and opposite signs of l are superimposed coaxially, the phase distribution of the resulting composite vortex beam has a p + 1-layer phase mutation, and each layer has 21 phase mutations (There are 2 l equal phase lines.).

4.2.3 Interference Superposition of LG Beams with Arbitrary Radial Indices and Topological Charges When two LG beams with any radial index and topological charge are superimposed coaxially, the complex amplitude expression of the formed superposition state is u two (r, φ) = u lp11 (r, φ) + u lp22 (r, φ).

(4.12)

Its light-intensity and phase distributions are expressed as l I p(two) = u two (r, θ ) × u ∗two (r, θ )

(4.13)

φ lp(two) = arg[u two (r, θ )].

(4.14)

Figure 4.7 shows the light-intensity distribution of the superimposed double LG beam with different radial indices and topological charges. From Fig. 4.7a–d, it can

120 (a)

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian … (b)

(c)

(d)

Fig. 4.7 Light-intensity distribution of superimposed LG beam with arbitrary radial index and arbitrary topological charge. a LG0 3 and LG0 −3 beams; b LG1 3 and LG1 −3 beams; c LG2 3 and LG2 −3 beams; d superimposed LG3 3 and LG3 −3 beams

be seen that the light-intensity distribution of the superimposed double LG beam has four layers, whether it is a bright ring or a “petal” shape. This is because three is the maximum value of the radial index of the coaxially superimposed LG beams.

4.3 Influence of the Transmission Distance on the Superposition State of High-Order Radial LG Beams To study the transmission characteristics of the composite vortex beam formed by coaxial superposition of two high-order radial LG beams, we theoretically simulated the light-intensity distribution characteristics of the composite vortex beam at different distances. For convenience of discussion, we assume that the radial exponents are the same when the high-order radial LG beams are superimposed coaxially. Considering the first case, the LG1 1 and LG1 5 beams are selected as research objects, and the influence of the transmission distance on the “petal”-like composite vortex beams is studied. Figure 4.8 shows the light-intensity distribution diagram of the superimposed high-order radial LG beams with a small topological-charge difference. As shown in Fig. 4.8, with an increase in the transmission distance, the light-intensity distribution of the superposition state of the LG1 1 and LG1 5 beams exhibits a diffraction-broadening phenomenon and rotates in the counterclockwise direction. Considering the second case, we choose the LG1 1 and LG1 30 beams as research objects to study the influence of the transmission distance on a multi-ring-shaped composite vortex beam. Figure 4.9 shows the light-intensity distribution of the superimposed high-order radial LG beams with large topological-charge differences. With an increase in the transmission distance, the light-intensity distribution of the superimposed LG1 1 and LG1 30 beams exhibits diffraction broadening and does not rotate.

4.3 Influence of the Transmission Distance on the Superposition State …

(a)

(b)

121

(c)

Fig. 4.8 The intensity distribution of higher-order radial LG beam superposition states with a small topological-charge difference. a z = 0.5 m; b z = 1.0 m; c z = 1.5 m

(a)

(b)

(c)

Fig. 4.9 The intensity distribution of higher-order radial LG beam superposition states with large topological-charge difference. a z = 0.5 m; b z = 1.0 m; c z = 1.5 m

The multi-annular composite vortex beam still retains a complete halo during the transmission process without affecting the orbital angular momentum. This type of beam is stable. This is because when the topological charges of the LG beams differ significantly, the charges remain unchanged, and their independent transmissions do not affect each other. Considering the third case, we choose the LG1 2 and LG1 −2 beams as research objects to study the influence of the transmission distance on the multi-ring-shaped composite vortex beam. Figure 4.10 shows the light-intensity distribution of the superimposed high-order radial LG beams with opposite topological charges. The calculation parameters are set as follows: beam-waist radius, ω01 = ω02 = 1.0 mm; wavelength, λ = 632.8 nm. Figure 4.10a–c show the light-intensity distribution diagrams of the composite vortex beam coaxially superimposed with the LG1 2 and LG1 −2 beams at different transmission distances. As shown in Fig. 4.10, as the transmission distance increases, the spot of the composite vortex beam gradually expands, which is caused by the diffraction of the beam. At this time, the light-intensity distribution of the composite vortex beam does not rotate because the two high-order radial LG beams with the same radial index and opposite topological charges have opposite phases, and they cancel each other to zero when superimposed on each other. Because no additional phase is generated, angular rotation does not occur.

122

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

(a)

(b)

(c)

Fig. 4.10 Theoretical light-intensity distribution of the composite vortex beam under different transmission distances. a z = 0.5 m; b z = 1.0 m; c z = 1.5 m

4.4 Influence of the Beam-Waist Radius on the Superposition State of High-Order Radial LG Beams To study the influence of different beam-waist radii on the light-intensity distribution of superimposed high-order radial LG beams, we used a composite vortex beam coaxially superimposed with LG1 2 and LG1 −2 beams as the research object. The transmission distance z = 1.5 m and the wavelength λ = 632.8nm. Figure 4.11a–d show the light-intensity distribution characteristics of the composite vortex beam (at this time, ω01 < ω02 ), which keeps the beam-waist radius ω01 constant at 1.0 mm and gradually increases the LG1 −2 beam-waist radius ω02 . Figure 4.11 shows that when ω01 = ω02 , the light-intensity distribution of the composite vortex beam is symmetrical, and there is no spinning “tailing” phenomenon. When ω01 < ω02 , the light-intensity distribution of the composite vortex beam rotates clockwise and the “tailing” phenomenon appears. When the waist-radius difference between the two LG beams is larger, the composite vortex beam formed by their interference and superposition rotates more clearly. This is because the Guy phase of the LG beam is related to the beam-waist radius, and the superposition of two LG beams with different beam-waist radii produces an extra phase and causes an angular rotation. (a)

(b)

(c)

(d)

Fig. 4.11 Intensity distribution of the composite vortex beam under different beam-waist radii ω02 a ω02 = 1.0 mm; b ω02 = 1.1 mm; c ω02 = 1.2 mm; d ω02 = 1.3 mm

4.5 Effect of Off-Axis Parameters on the Superposition State of High-Order … (a)

(b)

(c)

123

(d)

Fig. 4.12 Intensity distribution of the composite vortex beam under different beam-waist radii ω01 . a ω01 = 1.0 mm; b ω01 = 1.1 mm; c ω01 = 1.2 mm; d ω01 = 1.3 mm

Figure 4.12a–d show the light-intensity distribution characteristics of the composite vortex beam (at this time, ω01 ≥ ω02 ) while beam-waist radius ω02 is kept constant at 1.0 mm and beam-waist radius ω01 is gradually increased. As shown in Fig. 4.12, when ω01 = ω02 , the light-intensity distribution of the composite vortex beam is symmetrical, and there is no spinning “tailing” phenomenon. When ω01 > ω02 , the light-intensity distribution of the composite vortex beam rotates counterclockwise, and a clockwise rotation “tailing” appears. When the waist-radius difference between the two LG beams is larger, the composite vortex beam formed by their interference and superposition rotates more clearly. Comparing the simulation results in Figs. 4.11 and 4.12, we can obtain a new method to distinguish the topological-charge sign of high-order radial LG beams. When we observe that the light-intensity distribution of the composite vortex beam rotates clockwise and the "tailing" direction rotates counterclockwise, the topological charge of the beam with a larger waist radius is negative, and the topological charge of the other beam is positive. When we observe that the light-intensity distribution of the composite vortex beam rotates counterclockwise and the “tailing” direction rotates clockwise, the topological charge of the beam with a larger waist radius is positive, and the topological charge of the other beam is negative.

4.5 Effect of Off-Axis Parameters on the Superposition State of High-Order Radial LG Beams When studying the off-axis superposition of two high-order radial LG beams, we only need to consider the relative displacements of the two beams. Assuming that the central phase singularity of one beam is located at the origin, and the central phase singularity of the other beam is located off the origin, the displacements from the x-axis and y-axis are respectively Δx and Δy. Then, the light-field expression after the off-axis superposition of such two high-order radial LG beams is 2 u two (x, y) = u lp11 (x + Δx, y + Δy) + u −l p2 (x, y).

(4.15)

124

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

According to the different signs of Δx and Δy, the LG beam position can be divided into five situations, as shown in Fig. 4.13a–e. Observing Fig. 4.13c, e, when Δy < 0, the center phase singularity of the LG beam moves downward. From Fig. 4.13a, c, when Δy > 0, the center phase singularity of the LG beam moves upward. Observing Fig. 4.13b, c, when Δx < 0, the center phase singularity of the LG beam moves to the right. From Fig. 4.13c, d, when Δx > 0, the center phase singularity of the LG beam moves to the left. To study the effect of off-axis parameters on the light-intensity distribution of the superimposed high-order radial LG beams, we used a composite vortex beam formed by the coaxial superposition of LG1 1 and LG1 −1 beams and LG1 3 and LG1 −3 beams as the research object. Figure 4.14 shows the superposition light-intensity distribution of high-order radial LG beams under different longitudinal offsets Δy. When the longitudinal

Fig. 4.13 LG beam intensity distribution at different positions

4.6 Experiment on a Superimposed High-Order Radial LG Beam

125

Fig. 4.14 Intensity distribution of the composite vortex beam under different longitudinal offsets

offset is Δy = 0, the light-intensity distribution of the superimposed high-order radial LG beam is symmetrical about the origin. When the longitudinal offset is Δy /= 0, the light-intensity distribution of the superimposed high-order radial LG beam is no longer symmetrical about the origin. When Δy < 0, with an increase in |Δy|, the central phase singularity of the superposition state gradually moves downward from the central position. When Δy > 0, with an increase in |Δy|, the central phase singularity of the superposition state gradually moves upward from the central position. Figure 4.15 shows the light-intensity distribution of the superimposed high-order radial LG beam under different lateral offsets Δx. It can be seen from the figure that the light-intensity distribution of the superimposed high-order radial LG beam is symmetrical about the origin when the longitudinal offset is Δx = 0. When the longitudinal offset is Δx /= 0, the light-intensity distribution of the superimposed high-order radial LG beam is no longer symmetrical about the origin. When Δx < 0, with an increase in |Δx|, the central phase singularity of the superposition state gradually moves to the right from the central position. When Δx > 0, with an increase in |Δx|, the central phase singularity of the superposition state gradually moves to the left from the central position.

4.6 Experiment on a Superimposed High-Order Radial LG Beam 4.6.1 Experimental Device A spatial light modulator (SLM) is used to phase-modulate the plane wave to generate a high-order radial LG beam and its superposition state. Figure 4.16 shows a diagram of the experimental device for coaxially superimposing high-order radial LG beams to form a new composite vortex beam. The light source selected in the experiment was

126

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

Fig. 4.15 Intensity distribution of the composite vortex beam under different lateral offsets

SLM

laser

GDF

P

BE

GDF: gradient density filter P: polarizer BE: beam extender A:aperture

A CCD

PC1 PC2

Fig. 4.16 Experimental setup diagram

a He–Ne laser of wavelength λ = 632.8 nm. The gradient density filter (GDF) can reduce the emission power of the laser and the polarizer can filter out the light from other polarization directions to obtain the light of a single polarization direction. In addition, a half-wave plate was used to adjust the polarization direction of the incident light. Double lenses were used to expand the beam and collimate the light source so that the beam width matched the effective area of the SLM.

4.6.2 Hologram Production This section describes the superposition state of two high-order radial LG beams. The coaxial superposition of two high-order radial LG beams is equivalent to the superposition of the corresponding fork gratings [5]. The superimposed hologram– formation process is shown in Fig. 4.17. The plane wave was incident on the SLM loaded with the superimposed hologram for the interference experiment to obtain the composite vortex beam of the interference superposition.

4.6 Experiment on a Superimposed High-Order Radial LG Beam

127

Fig. 4.17 Superimposed grating-hologram formation process

Fig. 4.18 Fork grating with superimposed LG3 3 and LG2 −5 beams

Figure 4.18 shows the superimposed hologram of the LG3 3 and LG2 −5 beams. Because of the different radii of the annular dislocation of the LG3 3 and LG2 −5 beams, when the two fork patterns are superimposed, the hologram will appear more complicated. This will reduce the diffraction efficiency of the fork grating to a certain extent, and then affect the quality of the LG-beam superposition state. Figure 4.19 shows the superimposed hologram corresponding to the superimposed LG1 1 and LG1 −1 beams under different beam-waist radii. From Figs. 4.19a–c, when the beam-waist radii are equal, there is only one ring dislocation in the superimposed hologram. When the beam-waist radii are not equal, the ring-shaped dislocation of the superimposed hologram presents a more complicated structure. This is because the beam-waist radius affects the phase function of the high-order radial LG beam, which, in turn, affects the structure of the superimposed hologram.

4.6.3 Analysis of Results Figure 4.20a–f and 4.21a–f show the theoretical and experimental light-intensity distribution diagrams of the composite vortex beam formed by the coaxial superposition of LG0 1 and LG0 −1 beams, LG1 1 and LG1 −1 beams, LG2 1 and LG2 −1 beams, LG0 3 and LG0 −3 beams, LG1 3 and LG1 −3 beams, and LG2 3 and LG2 −3 beams, respectively. The simulation and experimental parameters are as follows:

128

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

(a)

+

=

(b)

+

=

(c)

+

=

Fig. 4.19 Formation process of superimposed grating holograms under different beam-waist radii. a ω01 = 2.0 mm, ω02 = 2.0 mm; b ω01 = 2.0 mm, ω02 = 2.5 mm; c ω01 = 2.5 mm, ω02 = 2.0 mm

wavelength λ = 632.8 nm, transmission distance z = 1.5 m, beam-waist radius ω01 = ω02 = 1.0 mm. Figure 4.20b shows the experimental light-intensity distribution diagram of the superimposed LG1 1 and LG1 −1 beams. When the center intensity is 0, there are two bright “petals” in the inner layer and two bright “petals” in the outer layer, and they are symmetrically distributed. Figure 4.20e shows the experimental light-intensity distribution diagram of the superimposed LG1 3 and LG1 −3 beams. When the center intensity is 0, there are six symmetrically distributed bright “petals” in the inner layer and six bright “petals” in the outer layer. The rest are not described. As shown in Figs. 4.20 and 4.21, with the increase of P, the number of bright “petals” gradually increases, and the overall spot diameter gradually increases; however, the diameter of the central bright “petals” gradually decreases. The experimental results were consistent with the theoretical analysis results. Figure 4.22a–c and 4.23a–c show the theoretical and experimental light-intensity distribution diagrams of the composite vortex beam formed by the coaxial superposition of the LG1 1 and LG1 −1 beams under different transmission distances.

4.6 Experiment on a Superimposed High-Order Radial LG Beam (a)

(b)

(c)

(d)

(e)

(f)

129

Fig. 4.20 Theoretical intensity distribution of the composite vortex beam. a LG0 1 and LG0 −1 beams; b LG1 1 and LG1 −1 beams; c LG2 1 and LG2 −1 beams; d LG0 3 and LG0 −3 beams; e LG1 3 and LG1 −3 beams; f superimposed LG2 3 and LG2 −3 beams (a)

(b)

(c)

(d)

(e)

(f)

Fig. 4.21 Experimental light-intensity distribution of a composite vortex beam. a LG0 1 and LG0 −1 beams; b LG1 1 and LG1 −1 beams; c LG2 1 and LG2 −1 beams; d LG0 3 and LG0 −3 beams; e LG1 3 and LG1 −3 beams; f superimposed LG2 3 and LG2 −3 beams

Figure 4.22d–f and 4.23d–f show the theoretical and experimental light-intensity distribution diagrams of the composite vortex beam formed by the coaxial superposition of the LG1 3 and LG1 −3 beams under different transmission distances. The simulation and experimental parameters are as follows: transmission distance, z = 0.5 m, 1.0 m, and 1.5 m; beam-waist radius, ω01 = ω02 = 1.0 mm; and wavelength, λ = 632.8 nm. As shown in Figs. 4.22 and 4.23, when the composite vortex beam is transmitted in free space, as the transmission distance increases, the light-intensity distribution of the composite vortex beam formed by coaxial superposition does not rotate. There

130

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

Fig. 4.22 Theoretical light-intensity distribution of the composite vortex beam under different transmission distances. a, d z = 0.5 m; b, e z = 1.0 m; c, f z = 1.5 m

Fig. 4.23 Experimental light-intensity distribution of the composite vortex beam with different transmission distances. a, d z = 0.5 m; b, e z = 1.0 m; c, f z = 1.5 m

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

is a certain degree of diffraction and broadening, and the light intensity gradually weakens. The experimental results are consistent with the numerical calculation results. Figure 4.24a–d, e–h are the theoretical light-intensity distribution diagrams of the superimposed LG1 1 and LG1 −1 beams and the superimposed LG1 3 and LG1 −3 beams under different beam-waist radii ω02 . Figure 4.25a–d, e–h are the experimental lightintensity distribution diagrams of the superimposed LG1 1 and LG1 −1 beams and the superimposed LG1 3 and LG1 −3 beams under different beam-waist radii ω02 . Observing Figs. 4.24 and 4.25, when ω01 = ω02 , the light-intensity distribution of the composite vortex beam is symmetrical and there is no spinning “tailing” phenomenon. When ω01 < ω02 , the light-intensity distribution of the composite vortex beam rotates clockwise and is accompanied by a counterclockwise rotation “smearing.” The larger the beam-waist radius difference, the more obvious the rotation. The experimental results are consistent with the numerical calculation results.

4.6 Experiment on a Superimposed High-Order Radial LG Beam

131

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 4.24 Theoretical light-intensity distribution of the composite vortex beam under different beam-waist radii ω02 . a, e ω02 = 1.0 mm; b, f ω02 = 1.1 mm; c, g ω02 = 1.2 mm; d, h ω02 = 1.3 mm (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 4.25 Experimental light-intensity distribution of the composite vortex beam under different beam-waist radii ω02 . a, e ω02 = 1.0 mm; b, f ω02 = 1.1 mm; c, g ω02 = 1.2 mm; d, h ω02 = 1.3 mm

Figure 4.26a–d, e–h are the theoretical light-intensity distribution diagrams of the superimposed LG1 1 and LG1 −1 beams, and the superimposed LG1 3 and LG1 −3 beams under different beam-waist radii ω01 . Figure 4.27a–d, e–h are the experimental lightintensity distribution diagrams of the superimposed LG1 1 and LG1 −1 beams, and the superimposed LG1 3 and LG1 −3 beams under different beam-waist radii ω01 . From Figs. 4.26 and 4.27, when ω01 = ω02 , the light-intensity distribution of the composite vortex beam is symmetrical, and there is no spinning “tailing” phenomenon. When ω01 > ω02 , the light-intensity distribution of the composite vortex beam rotates counterclockwise and is accompanied by a clockwise “smear.”

132

4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian …

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 4.26 Theoretical light-intensity distribution of the composite vortex beam under different beam waist radii ω01 . a, e ω01 = 1.0 mm; b, f ω01 = 1.1 mm; c, g ω01 = 1.2 mm; d, h ω01 = 1.3 mm (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 4.27 Experimental light-intensity distribution of the composite vortex beam under different beam waist radii ω01 . a, e ω01 = 1.0 mm; b, f ω01 = 1.1 mm; c, g ω01 = 1.2 mm; d, h ω01 = 1.3 mm

The larger the beam-waist radius difference, the more obvious the rotation. The experimental results are consistent with the numerical calculation results.

References

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References 1. Naidoo D, Ait-Ameur K, Brunel M et al (2012) Intra-cavity generation of superpositions of Laguerre–Gaussian beams. Appl Phys B 106(3):683–690 2. Huang SJ, Miao Z, He C et al (2016) Composite vortex beams by coaxial superposition of Laguerre-Gaussian beams. Opt Lasers Eng 78:132–139 3. Vaity P, Singh RP (2011) Self-healing property of optical ring lattice. Opt Lett 36(15):2994–2996 4. Li XZ, Tai YP, Lv FJ et al (2015) Measuring the fractional topological charge of lg beams by using interference intensity analysis. Opt Commun 334(1):235–239 5. Ke XZ, Xu JY (2016) Interference and detection of vortex beams with orbital angular momentum. Chin J Lasers 43(9):192–197 6. Ando T, Matsumoto N, Ohtake Y et al (2010) Structure of optical singularities in coaxial superpositions of Laguerre-Gaussian modes. J Opt Soc Am A 27(12):2602–2612 7. Xiang YY, Luo HL, Wen SC (2010) Intensity and phase rotation of beams in left-handed metamaterials. High Power Laser Particle Beams 22(8):1834–1838 8. Angyita JA, Herrerors J, Djordjevic IB (2014) Coherent multimode OAM superpositions for multidimensional modulation. IEEE Photonics J 6(2):1–11

Chapter 5

Transmission Characteristics of Vortex Beams

This chapter introduces the horizontal and slant propagation theory of Laguerre– Gaussian (LG) and Bessel–Gaussian (BG) beams in atmospheric turbulence, and analyzes their orbital angular momentum (OAM) variation characteristics. The study compares the OAM stability of Laguerre–Gaussian beams and Bessel beams during slanted propagation through the atmosphere. We analyze the influence of the atmospheric refractive-index structure constant, beam wavelength, zenith angle, OAM index, beam radius, and other parameters on the harmonic components of the two types of beams with the transmission distance.

5.1 Introduction When LG beams propagate in atmospheric turbulence, the spatial inhomogeneity will change the photon-wave function, that is, cause changes in the LG mode, leading to changes in the OAM index. In theory, OAM can form an infinite-dimensional Hilbert space. Using this feature, high-dimensional communication can be realized by carrying a variety of different OAM vortex optical-transmission information. The change of OAM during the atmospheric transmission will affect the communication quality; hence, it is particularly important to analyze the change law of OAM. The presence of atmospheric turbulence in free space causes the vortex-beam phase to fluctuate, its OAM to disperse, and adds inter-symbol crosstalk to the communication. This is especially important. This chapter mainly analyzes the changes of the spiral spectrum of Laguerre–Gaussian beams and high-order Bessel– Gaussian beams when propagating in atmospheric turbulence, through the influence of the refractive-index structure constant and the zenith angle.

© Science Press 2023 X. Ke, Generation, Transmission, Detection, and Application of Vortex Beams, Optical Wireless Communication Theory and Technology, https://doi.org/10.1007/978-981-99-0074-9_5

135

136

5 Transmission Characteristics of Vortex Beams

5.2 Transmission of LG Beams in Atmospheric Turbulence 5.2.1 Theoretical Analysis (1) LG beams travel horizontally in the atmosphere An LG beam emitted by a light source propagates in atmospheric turbulence under paraxial-approximation conditions. Assuming that the emission aperture of the LG beam is R, the amplitude of the LG beam at the incident surface z = 0 is [1]. u mp (ρ, ϕ)

A = w0

(√

2ρ w0

)|m| (−1) p L |m| p

(

2ρ 2 w02

)

( exp

−ρ 2 w02

) exp(−imϕ)

(5.1)

Under the Rytov approximation, the light field of the LG beam at z in a turbulent medium can be expressed as [2] i u(r, θ, z) = − exp(ikz) λz ] [ ¨ ik m 2 u p (ρ, ϕ) exp[ψ(ρ, ϕ, r, θ, z)] exp (r − ρ) ρdρdϕ 2z

(5.2)

where ψ(ρ, ϕ, r, θ, z) is a complex phase introduced by atmospheric turbulence. The spatial inhomogeneity caused by atmospheric turbulence will change the photon-wave function, resulting in a change in the LG mode. To analyze the OAM components of the LG beam after turbulent flow, the spiral-spectrum definition can be used to calculate the weight of the total beam energy occupied by each component of the spiral harmonics. Then, the OAM changes and the influencing factors of the LG beam in a turbulent medium can be analyzed. The complex amplitude u(r, θ, z) of the LG-beam light field at z in the turbulent medium can be expanded by the spiral harmonic function exp(ilθ ). +∞ . Let, u(r, θ, z) = √12π al (r, z)exp(ilθ ) where [3] l=−∞

1 al (r, z) = √ 2π

.2π u(r, θ, z)exp(−ilθ )dθ,

(5.3)

0

From Eqs. (5.2) and (5.3), we can obtain. |al (r, z)|2 =

1 2π

.2π .2π 0

0

u(r, θ1 , z) exp(−ilθ1 ) × u ∗ (r, θ2 , z) exp(−ilθ2 )dθ1 dθ2

5.2 Transmission of LG Beams in Atmospheric Turbulence

137

( √ )2|m| [ ( )]2 ) ( 2r 2 2r 1 A2 −2r 2 |m| Lp = exp 2π w 2 (z) w(z) w(z)2 w(z)2 .2π .2π × 0

]> [ exp ψ(r, θ1 , z) + ψ ∗ (r, θ1 , z)