Generation and Application of High Power Microwaves 9781000112375, 9780750304740, 9780750304511, 1000112373

Table of contents :
Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Directors' Preface......Page 10
Contents......Page 14
Introduction to microwave sources......Page 16
Spontaneous and Stimulated Radiation of Classical Electrons......Page 24
High gain free electron laser with waveguide......Page 44
Computer Modelling of Microwave Sources......Page 72
Gyro-Amplifiers......Page 118
Vacuum Microelectronics for Microwave Power Amplifiers......Page 124
Modes and Mode Conversion in Microwave Devices......Page 136
Klystrons and related devices......Page 188
Cyclotron Resonance Effects on a Rectilinear Electron Beam for the Generation of High-Power Microwaves......Page 198
Interaction of Radiation with Plasmas......Page 216
Uses of Intense Microwaves in Tokamaks......Page 234
The Physics of Ion Cyclotron Heating In Tokamaks......Page 252
RF systems for heating and current drive......Page 290
Applications of High-Power Microwave Devices......Page 320
Participants Addresses......Page 340
Index......Page 346

Citation preview

GENERATION AND APPLICATION OF HIGH POWER MICROWAVES Edited by R. A Cairns and A. D. R. Phelps

Generation and Application of High Power Microwaves

Edited by R. A Cairns and A. D. R. Phelps

ISBN 978-0-7503-0451-1

,!7IA7F0-daefbb!

www.crcpress.com  an informa business

Scottish Graduate Series

GENERATION AND APPLICATION OF HIGH POWER MICROWAVES

GENERATION AND APPLICATION OF HIGH POWER MICROWAVES Proceedings of the Forty Eighth Scottish Universities Summer School in Physics, St Andrews, August 1996.

A

NATO

Advanced Study Institute.

Edited by RA Cairns - University of St Andrews A D R Phelps - University of Strathclyde Series Editor P Osborne - University of Edinburgh

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1997 by The Scottish Universities Summer School in Physics CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN-13: 9780750304740 (pbk) 9780750304511 (hbk)

This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

SUSSP Proceedings

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1982 1983 1983 1984 1985 1985 1985 1986 1987 1987 1988

Dispersion Relations Fluctuation, Relaxation and Resonance in Magnetic Systems Polarons and Excitons Strong Interactions and High Energy Physics Nuclear Structure and Electromagnetic Interactions Phonons in Perfect and Imperfect Lattices Particle Interactions at High Energy Methods in Solid State and Superfluid Theory Physics of Hot Plasmas Quantum Optics Hadronic Interactions of Photons and Electrons Atoms and Molecules in Astrophysics Properties of Amorphous Semiconductors Phenomenology of Particles at High Energy The Helium Liquids Non-linear Optics Fundamentals of Quark Models Nuclear Structure Physics Metal Non-metal Transitions in Disordered Solids Laser-Plasma Interactions: 1 Gauge Theories and Experiments at High Energy Magnetism in Solids Laser-Plasma Interactions: 2 Lasers: Physics, Systems and Techniques Quantitative Electron Microscopy Statistical and Particle Physics Fundamental Forces Superstrings and Supergravity Laser-Plasma Interactions: 3 Synchrotron Radiation Localisation and Interaction Computational Physics Astrophysical Plasma Spectroscopy Optical Computing

/continued

v

SUSSP Proceedings (continued)

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

1988 Laser-Plasma Interactions: 4 1989 Physics of the Early Universe 1990 Pattern Recognition and Image Processing 1991 Physics of Nanostructures 1991 High Temperature Superconductivity 1992 Quantitative Microbeam Analysis 1992 Spatial Complexity in Optical Systems 1993 High Energy Phenomenology 1994 Determination of Geophysical Parameters from Space 1994 Simple Quantum Systems 1994 Laser-Plasma Interactions 5: Inertial Confinement Fusion 1995 General Relativity 1995 Laser Sources and Applications 1996 Generation and Application of High Power Microwaves 1997 Physical Processes in the Coastal Zone 1998 Semiconductor Quantum Optoelectronics 1998 Muon Science 1998 Advances in Lasers and Applications

VI

Executive Committee

Professor Alan Cairns

University of St Andrews

Director and Co-Editor

Professor Alan Phelps

University of Strathclyde

Director and Co-Editor

Dr :\Iichael Hooper

University of Strathclyde

Secretary

Professor .Jeff Sanderson

University of St Andrews

Tr-erzsurer

Dr Robin Preston

University of Strathclyde

Social SecretaTy

International Advisory Committee

Dr Paul I3onoli

:\lassadmssct:o In:otitute of Technology, USA

Professor Alan Cairns

University of St Andrews, Scotland

Profos.~or

Alan Phelps

University of Strathclyde, Scotland

Professor

~Ianfred

Thumm

University of Karlsruhe, Germany

Vll

Lecturers

Rudolfo Bonifacio

University of Milan

Marco Brambilla

Max-Planck-lnstitut fiir Plasmaphysik, Garching

Alan Cairns

University of St Andrews

James Eastwood

AEA Technology

Chris Edgcombe

University of Cambridge

Georges Faillon

Thomson Tube Electronique

Nathaniel Fisch

Princeton University

Claude Gormezano

JET Joint Undertaking

Robert Parker

Naval Research Laboratory

Mikhail Petelin

Institute of Applied Physics, Nizhny Novgorod

Alan Phelps

University of Strathclyde

Manfred Thumm

Forschungszentrum Karlsruhe

John Vomvoridis

National Technical University of Athens

Vlll

Directors' Preface

The forty-eighth Scottish Universities Summer School in Physics, a NATO Advanced Study Institute on the subject of Generation and Application of High Power Microwaves, was held in St Andrews in August 1996. We were pleased that we succeeeded in attracting a distinguished group of lecturers, together with a lively group of participants, from 13 countries. The participants attended a wide ranging series of lectures over a period of ten working days and also took advantage of the opportunity to present their own work in a poster session and to take part in both organised and informal discussions. The Directors are grateful to their colleagues on the Organising Committee. As Secretary and Treasurer respectively, Michael Hooper and Jeff Sanderson brought experience of previous Summer Schools and very efficient handling of these vital roles. Another veteran of many Schools, Robin Preston, brought his customary jovial good humour to the role of Social Secretary, making sure, with the aid of his assistants Adrian Cross and Darren McDonald, that everyone was kept entertained in the intervals between doing physics. Finally, Jane Cumberlidge handled the secretarial tasks arising during the School, including organising the duplication of a mountain of lecture notes, with efficiency and good humour. We also wish to acknowledge the assistance provided by the staff of Conference and Group Services at St Andrews University. Especial thanks are due to Jackie Matthews, Lesley Mackie and the rest of the staff at John Burnett Hall for their friendly and efficient service. Our warmest thanks go to the participants, lecturers and all who helped with the organisation for their role in making the School a success, both scientifically and on a social level. Finally, taking on our role as Editors, we would like to thank all the lecturers for their work in producing the material presented in this volume.

Alan Cairns and Alan Phelps St Andrews and Strathclyde, May 1997

ix

Introduction

From the invention of the magnetron in the 1930s, the technology of devices for generating high power microwaves has advanced steadily, both in terms of the power which can be produced and the range of frequencies over which the devices can operate. Early devices, such as the magnetron and klystron, have continued to be developed, while being joined by a plethora of other devices: gyrotrons, free electron lasers, cyclotron autoresonant masers, to name but a few. Commercially available systems can now produce megawatt power levels at frequencies ranging up to several hundred GHz. Such sources have a wide range of applications, including radar, communications, particle accelerators, materials processing and plasma heating. The last of these has been an important driving force in the development of the highest power sources. To heat and drive current in magnetically confined plasmas used in research into controlled nuclear fusion needs multi-megawatt sources with long pulse or continuous operation capabilities. Since high power sources have developed rapidly in the last few years, it seemed to the Directors that a School combining a survey of these developments, together with their application to fusion research, would be timely. The emphasis on applications to fusion research, though not to the complete exclusion of other uses of these sources, is justified not just by the stimulus this application has given to source development, but also by the fact that the physical principles involved have a good deal in common. The basic mechanism of operation of high power microwave sources is the conversion of the energy of a beam of relativistic electrons into electromagnetic wave energy, relying on a resonant interaction between particles and waves to accomplish efficient energy transfer. In a hot plasma the resistivity is low and absorption of a wave by resistive dissipation is weak. Instead, the absorption of energy relies on a resonant wave-particle interaction in some region of the plasma. This is, in many essential respects, the inverse of the process involved in wave generation, and the wave-particle interactions share the same basic theory. While high power microwave devices may all be powered by relativistic electron beam:-;, there are a wide variety of ways in which the conversion of energy into waves may be accomplished in practice, so that a considerable variety of devices has been developed. This book contains an introduction to the basic physics of wave generation and a discussion of computational modelling of the process in realistic device geometries, followed by surveys of the state of the art in some of the more important devices which are in use, or under development. It will be clear from these articles that this is a very active subject, in which progress to higher powers will require all the ingenuity of physicists, engineers and materials scientists.

x

On the plasma heating side, the chapters in the book again range from a basic introduction to the essential physics, to more detailed accounts of the complexities of accurate theoretical modelling of heating and current drive in fusion plasmas, subtle uses of waves to recycle energy from a particles to current drive within a fusion plasma and an account of the design of systems on a large present day tokamak like JET (Joint European Torus) and those projected for ITER (International Tokamak Experimental Reactor). By bringing together contributions from leading research workers in both microwave devices and their applications to fusion, the Editors hope to have produced a volume which will interest researchers in both fields and help them appreciate the similarities in the underlying physics.

xi

Contents Introduction to microwave sources ............................................ 1 Alan Phelps Spontaneous and Stimulated Radiation of Classical Electrons .............. 9 M Petelin High gain free electron laser with waveguide ................................ 29 Rodolfo Bonifacio Computer Modelling of Microwave Sources ................................. 57 James W Eastwood Gyro-Amplifiers .............................................................. 103 Monica Blank Vacuum Microelectronics for Microwave Power Amplifiers ............... 109 M Garven and R K Parker Modes and Mode Conversion in Microwave Devices ...................... 121 Manfred Thumm Klystrons and related devices ............................................... 173 Georges Faillon Cyclotron Resonance Effects on a Rectilinear Electron Beam for the Generation of High-Power Microwaves ..................................... 183 J L Vomvoridis Interaction of Radiation with Plasmas ...................................... 201 RA Cairns Uses of Intense Microwaves in Tokamaks ................................... 219 NJ Fisch The Physics of Ion Cyclotron Heating In Tokamaks ....................... 237 M area Brambilla RF systems for heating and current drive .................................. 275 C Gormezano Applications of High-Power Microwave Devices ........................... 305 Manfred Thumm Participants Addresses ....................................................... 325 Index .......................................................................... 331

xiii

1

Introduction to microwave sources Alan Phelps University of Strathclyde, Glasgow

1

Introduction

In this introductory chapter the aim is to introduce the physics involved in the production of microwaves using free electrons and to discuss some of the types of microwave sources that exist. Later chapters written by international authorities in their fields will focus in more detail on individual types of microwave source and their characteristics. Alan Cairns in his chapter similarly introduces the twin theme of this book, namely the application of high power microwave sources, particularly to plasma heating and that theme is then again treated in greater detail in a series of chapters on the application of these sources. The term "microwave" is taken here as applying to the electromagnetic wave spectrum in the range 1-300GHz, although much of the physical discussion is applicable also outside this range. Microwave sources can be either continuous (CW) or pulsed and "high-power" in this context typically means CW sources with average powers of lO's of kWs to MWs and pulsed sources capable of peak powers from MWs to GWs and beyond. For plasma heating applications the sources presently available are CW, or long-pulsed, with powers of around IMW. Although solid state devices are successful sources of microwaves at lower power levels, for the higher power levels above a few lOOW and certainly for MW power levels, vacuum electronic devices are at present the only really competitive devices in use. The background physics which will be assumed is that normally gained during a first degree in physics or engineering. Electromagnetic fields and waves, some relativity, electron beam physics and collective phenomena of the type encountered in plasma physics are the areas that are needed in understanding the operation of these vacuum electronic microwave sources. Microwaves were generated towards the end of the 19th century in the pioneering experiments of Hertz and a variety of specific microwave sources have been developed during the 20th century. The magnetron was known for two decades before the second

2

Alan Phelps

world war but its development and application accelerated in the early 1940's and the higher power capability of the resonant cavity versions became recognised. The klystron was invented in the 1930's and the traveling wave tube (TWT) and backward wave oscillator (BWO) in the 1940's. The free electron laser (FEL) had roots in the undulator of Motz (1951) and the Ubitron of Phillips (1960) but has only flourished worldwide since the 1970's. The gyrotron's history has been discussed in several places including Flyagin et al (1977), Hirshfield and Granatstein (1977) and Edgcombe (1993). The gyrotron has become the most successful microwave source for application to plasma heating and had some precursors in theoretical papers in the late 1950's and developed through the 1960's and 70's, particularly at the Institute of Applied Physics, Gorky (now Nizhny Novgorod) Russia to become in the 1990's a commercially available source that has reached a relatively mature stage of development. The Doppler upshifted autoresonant relative of the gyrotron, the cyclotron autoresonant maser (CARM) is less well-developed. The autoresonant advantages were noticed by Petelin (1974) ·and developed in the form of a CARM concept by Bratman et al. (1981). The range of vacuum electronic microwave devices has now become large with nearly every potential means of converting electron energy into electromagnetic energy being explored including virtual cathode oscillators (Vircators), magnetically insulated transmission line oscillators (MILO's) and relativistic diffraction grating generators (RDG's).

2

Physics of microwave sources

For an introduction to the subject of microwave sources books are usually more helpful in the first instance than reference to the original primary research papers and therefore in the reference list at the end of this chapter there are several monographs and textbooks. Benford and Swegle (1992), Granatstein and Alexeff (1987), Gaponov-Grekhov and Granatstein (1994), Gilmour (1986) and (1994), Marshall (1985), Luchini and Motz (1990), Freund and Antonsen (1992) and Edgcombe (1993) vary in their emphasis and depth of coverage but are all recommended for further reading. Collections of papers on all types of high power microwave devices have been published in special issues of IEEE Trans. on Plasma Science (1985, 1988, ..... , 1998) and more specifically on gyrotrons in a series of special issues of Int. J. Electron. (1981, 1982, 1984, 1986, 1988a, 1988b) Inherently associated with any understanding of vacuum electronic microwave devices is the underlying physics of electron beam propagation. The books by Miller (1982), Lawson (1988) and Humphries (1990) are relevant in that they treat particularly intense electron beams of the type needed for high power microwave generation. The two books by Pierce (1950) and (1954) are included for their historical significance in the development of microwave tubes, electron guns and Pierce electrodes. The vacuum electronic microwave devices normally depend upon converting electron beam energy into electromagnetic wave energy and therefore high-power electron beams are needed. The amount of current carried by these beams increases with the accelerating voltage, although the relationship is not usually linear. To achieve high power the required accelerating voltage can be ~50kV and so the need for some inclusion of relativistic effects becomes apparent. For those microwave sources where a substantial Doppler upshift of the emitted radiation is desirable the electron energy may need to

Introduction to microwave sources

3

be measured in MeV. A parameter that occurs throughout discussions of microwave sources whenever relativistic effects are important is / where 'Y

= (1 -

(32)-1/2

2)-1/2

= ( 1 - ~2

(2.1)

where (J = v / c and v is the speed of the electron and c is the speed of light. An electron has a total energy 1mc2 and sees a length which is f, in the laboratory rest frame as shortened to €/I in its own frame of reference. This effect is present in all relativistic microwave devices but is particularly relevant to the operation of the FEL. In microwave sources that depend upon the cyclotron motion of the electrons the relativistic dependence of the cyclotron frequency upon electron energy plays an important role in the electron bunching mechanism. The cyclotron angular frequency is: eB w=-. (2.2) 1m Electrons with higher energy have lower cyclotron frequencies and vice versa which leads to a bunching of the electrons in their orbits around the magnetic field and hence leads to the possibility of coherent emission. Free electrons can emit or absorb radiation provided the laws of conservation of energy and momentum are satisfied and also that the electron and photon dispersion properties are satisfied. Some of the types of electron-photon interaction that are possible are bremsstrahlung, Cherenkov and transition radiation interactions. A general physical principle is that all charged particles that suffer a change in their velocity, i.e. an acceleration, will radiate electromagnetic waves. The classical formula for the power P radiated by a single electron experiencing an acceleration x is given by

(2.3) This gives rise to a bremsstrahlung radiation mechanism and since an electron moving in a circular path around a magnetic field is constantly changing the direction of its velocity vector it is continually accelerating and hence liable to radiate. This form of bremsstrahlung is called magnetic bremsstrahlung and is the process by which cyclotron and synchrotron radiation occurs. In Cherenkov interactions electrons that travel faster than the local speed of light will emit radiation. The electrons can not of course exceed c, the speed of light in vacuum, but the speed of light can be slower than c in a medium or in an interaction structure. This is the basis of the interactions that are used in Cherenkov free electron microwave sources. Travelling-wave tubes (TWTs), backward wave oscillators (BWOs) and dielectric lined Cherenkov masers are examples of this type of interaction. A third type of process that can be used to extract energy from an electron beam and couple radiation out is transition radiation. This occurs whenever electrons leave one region and pass into another material where there is a change in the refractive index at the boundary.

.~a~ 112- -.'.I-,

Bz

l (n;c)

r

Wea dX . w aEz -12 Wco ay

12 -

- - - - - - - - w c o = kcoC =

Ex=

.wa~

= A sin ( :7r x) sin ( p; y )t1 Vph· In the longitudinally infinite system such a mismatch between the electron velocity of the wave space harmonic is developed automatically and can be found from the self-consistent system of equations describing the wave propagation and the electron motion. The type of resulting instability depends on the mutual orientation of the electron velocity and the wave group velocity. If these velocities are co-directional, perturbations of both RF field and of electron motion drift in the common direction. If these perturbations compose a wave packet of a limited length, there is a moving reference frame where the packet is growing and expanding, but in the laboratory reference frame after the packet passing any observation point the perturbations are fading; this is a drifting (convective) instability. If the electron velocity and the wave group velocity are counter-directional, the RF field and electron perturbations in the wave packet are propagating in opposite directions; as a result the packet is growing and expanding in such a way that perturbations are increasing monotonically in every point of the laboratory reference frame; a proper name for this effect would be a resident instability. The more traditional term "absolute instability" from the relativistic viewpoint seems misleading: there are moving reference frames where the same process represents an instability of the drifting type.

Radiation of Classical Electrons

23

In Figure 10 the drifting instability takes place at points A and F, the resident instability is realised at points B, C, D, E (and even, with account of the wave packet expansion, at positive group velocities in narrow vicinities of points B and E).

The drifting instability (A and F) is used in forward wave amplifiers: (A) in the travelling wave tube (TWT) at zero space harmonic and (F) in the TWT at +1st space harmonic. The resident instability corresponding to the points C and D is used in the backward wave oscillator (BWO), that corresponding to the point B is used in the (7r-mode generator, and that corresponding to the point Eis used in 27r-mode generators which in the case of oversized cross section are called orotrons.

'///////////////////////////< /

~ 7////7////77/77///////7////l, \

Figure 11. Bremsstrahlung radiation of electrons in cylindrical waveguide

8

Bremsstrahlung radiation

Thus the rectilinear electron beams can radiate only slow waves. as for the fast waves, they can be radiated by electrons, if an external force makes them move with an acceleration (8Ve/8t-:/= 0). For the microwave electronics the most interesting Bremsstrahlung configuration is a cylindrical waveguide pierced with a curve periodic electron beam (Figure 11). Suppose the electrons move along a helical trajectory in a homogeneous magnetic field. From the quantum viewpoint such an electron is an oscillator whose energy £ (see Figure 12) is a continuous function of the translational momentum Pil and a discrete function of the gyrational state of the particle: the vertical distance between two neighbouring Landau curves in Figure 12 is equal to !iwH, where WH = eH0 /mc1 is the cyclotron frequency (gyrofrequency). So, the emission of a photon (transition 0-+A in Figure 12) is followed by the electron energy loss related to changes of both oscillatory and translational motions

(40) One part of energy

D,_£J_

= nnwH

is lost by the electron, because it has jumped down by n Landau steps. Another part

M Petelin

24

'

''

''

''

''

''

''

''

''

''

''

''

'' 0

P11

Figure 12. Quantum diagram of electron m magnetic field and radiation transitions of electrons is lost together with the electron translational momentum. As the losses of the electron energy and of its longitudinal momentum are equal to the energy and longitudinal momentum of the emitted photon

6..£

= hw,

the equation (40) results in

(41)

which describes the resonance between the electron and the electromagnetic wave with account of the Doppler shift. In an elementary radiation event we distinguish three cases:

= 0 transition 0-+B in Figure 12), the radiation is of the Cherenkov type - compare (41) with (la, lb)

• If the electron does not change its gyrational state (n

• If the electron decreases its oscillatory energy (n > 0, transition 0-tA in Figure 12), the resonance condition is of the normal Doppler type . • If the electron increases its oscillatory energy (n < 0, transition 0-tC in Figure 12), the resonance condition is of the anomalous Doppler type.

In the latter case one part of the lost translational energy of the electron is converted into its oscillatory energy and another part into the energy of the photon. The anomalous Doppler radiation can take place, if the wave phase velocity is less not only relative to the velocity of light, but also relative to the translational velocity of electron. As for the normal Doppler radiation (n > 0), it can take place at any wave phase velocities. In the cylindrical waveguide (Figure 11), where the wave dispersion curves are the hyperbolas (Figure 13) (42)

25

Radiation of Classical Electrons

/

0

/

/

/

/

/

/

/

/

/

/

/

/

/

/

//

/

,,./''D

k11

Figure 13. Brillouin diagram for Bremsstrahlung radiation of electrons in cylindrical waveguide

the electron radiates a wave when the corresponding hyperbola (42) intersects with the Doppler resonance line (41). Analysing Figure 13 by analogy with the Cherenkov case (Figure 10), we have • At C and D a drifting (convective) instability used in forward wave amplifiers . • At A and B a resident (absolute) instability used in microwave generators: the point B corresponds to a backward wave generator and the point B to a near-cutoff generator (which is called the gyrotron). The stimulated radiation of the stationary beam of oscillating electrons can be analysed using the same quantum and classical methods as in the Cherenkov case. In particular, in the gyrotron the relativistic dependence of the gyrofrequency w H on the electron energy results in a dependence of the effective beam conductivity on the cyclotron resonance mismatch (Figure 14), which is similar to that shown in Figure 6 for the Cherenkov case. Another practically important method to make electrons move along periodic curved trajectories is the use of periodic static magnetic fields. In the latter case the Doppler resonance condition takes the form (43) analogous to (41). In this last equation n = 27rVj1/d where dis the static magnetic field period. Microwave devices of the sort were named at first ubitrons and later renamed into free electron lasers. Their performance is similar to that of the Cherenkov and the cyclotron resonance devices.

26

M Petelin

Figure 14. Cyclotron resonance absorption line of relativistic electrons

9

Microwave generators and amplifiers

In this section we consider microwave generators and amplifiers with force bunching of electrons. All such devices are featured with the electron bunching of inertial type: the RF field changes the electron energies, which results in the change of their velocities, which in its turn results in the development of electron bunches. However, there are some devices where the RF force shifts electrons from their unperturbed trajectories, but the changes of the electron velocities are negligible. Such a bunching process stops immediately, if the RF field is "switched oft". For example in devices of the magnetron ("M") type devices (Figure 15) electrons move in crossed electric and magnetic fields with velocity V _ EoxHo e - c H6

(44)

Figure 15. Electron-wave interaction in magnetron In the magnetron electrons radiating the synchronous slow wave do not change their velocity which remains equal to cE0 / H 0 : approaching to the anode, the particles give to the wave only their potential energy. If the RF field "disappeared" (e.g., in a non-corrugated drift tube), all electrons would keep the common velocity (44) and, so, distances between them would stay frozen. Thus the electron bunching develops only in the presence of the RF force.

Radiation of Classical Electrons

27

The force bunching results in a relatively simple dependence of the effective conductivity of the stationary beam on the synchronism mismatch: the conductivity is proportional to the space spectrum intensity in the Cherenkov synchronism point (h = w/V.,)contrary to Equations (17, 18). The sign of the conductivity in the M-type devices depends on the transverse structure of the vector RF field near the electron beam; (recall that in the Cherenkov devices with free longitudinal velocity of electrons we need only a non-zero component of the RF electric field). In the magnetron, to make the conductivity negative, it is necessary to corrugate the anode-not the cathode. The force bunching of electrons ( characterised in particular with the "single-hump" conductivity curve-Figure 16) takes place also for the anomalous Doppler cyclotron resonance devices with rectilinear electron beams, for ubitrons with a combination resonance and for some other devices ( most of them exist only in the theory).

a

Ve -V,(O) ph

Figure 16. Electron beam effective conductivity vs synchronism mismatch

10

Sectioned electron devices

Let us return to microwave devices where the longitudinal velocity of electrons is free and where their bunching continues even in drift tubes (in the absence of the RF field). The last effect is used in klystrons, twystrons as well as in similar sectioned devices with periodic curved beams. Note that in the klystron (Figure 17) the interaction of the electron beam with the RF field takes place only in narrow gaps. So the amplifier can be classified as a device based on the (stimulated) transition radiation. However if the RF field in the whole device is expanded into the Fourier integral (17), the stimulated radiation effect may be again ascribed to the Cherenkov components (18) of the RF field spectrum .

•


.B

2

(3.8b)

= 2nmc2

(3.9) which gives for perfect injection on-axis (/3J_ (0)

= 0)

and negligible initial field,

fJ = _ atat

(3.10)

'Y

~

which gives the approximated relation (2.4). By (3.10) we can derive the electron energy equation. From (3.2)

d"( dt

= _e_E.v = __e_8Atat ./3 = _!. 8atat .atat = __!.__ 8 latatJ2 mc 2

J_

mc2

ot

J_

'Y ot

2"(

ot

(3.11)

where, from eqs.(3.7), (3.8),

latat (z, t) 12 (}

a~ - iaw [a(z, t)eiO(z,t)

(k+kw)z-wt

-

c.c.]

+ la(z, t)\ 2

(3.12) (3.13)

35

High gain free electron laser with waveguide is the electron phase in the combined "ponderomotive" (radiation which propagates in the z-direction at the phase velocity w

+

wiggler) field, (3.14)

By substitution of (3.2) into (3.11) and with the paraxial approximation d/dt ~ v d/dz ~ cd/dz, the electron energy equation can be written as 11

k aw iO -d"( = ---(ae + c.c. ) 2 'Y

dz

(3.15)

The r.h.s. of eq.(3.15) is a "ponderomotive" force whose strength is proportional to the electron- radiation coupling (lalawh ex: \E.v_i_I). This force is responsible for electron acceleration (d"f /dz > 0) or deceleration (d"f /dz < 0) and eventually bunching. Note that if a were constant both in modulus and in phase, (3.15) would reduce to d"f/dz ex: cosO, i.e., the pendulum equation (we shall see that d"f/dz ex: d2 0/dz 2 ) (Colson 1977). In order to derive an equation for the phase 0, we start simply differentiating eq. (3.13):

(3.16) Note that the electron phase with respect to the ponderomotive field remains constant, i.e., dO/dz = 0, if k = kw/3,/(1- /3, 1) or (3.17) that is the exact (on-axis) resonance condition (eq. (2.3b)). This relation is obtained also by imposing that Vp = vii' Hence, we get one more interpretation of the resonance condition; namely, an electron has the resonant energy 'Yr (eq. (2.6)) if it is injected with longitudinal velocity equal to the phase velocity of the ponderomotive field. The approximated resonance relation (3.18) follows from (3.17) in the limit 1 - /3, « 1 and in the approximation atot = aw , i.e., neglecting the radiation contribution to the electric transverse velocity like in (2.4). In this approximation, from 1

1

2

1 +a~

2 /3,11 =1-8 --~1--. 72 'Y2 _L

one can derive, in the limit 7 2

»

1 +a~, 1

1 +a~

-~1+-!311 2"(2

(3.19)

Rodolfo Bonifacio

36 which inserted in eq.(3.16) gives

d()_k dz-

l+a~k

w-2T

(3.20)

By introducing the electron resonant energy /r (eq.(2.6)), eq.(3.20) becomes

= kw

d()

dz

(i -,; )

(3.20')

12

However, if the approximation a101 ~ aw is dropped and the exact result (3.12) is used, there are additional contributions to the electron trajectory due to the radiation field and thus corrections to the phase equation (3.20'). The final form is

do dz

3.2

= kw

( 1 - ,; 'Y 2 )

[.iaw (ae ;o - c.c. ) - Ia l2]

+ 2k'Y2

(3.21)

Field equation:

In the second part of the self-consistent scheme for the FEL dynamics we look for the evolution equation of the radiation field, A, as determined by the electron transverse

current Jr The wave equation in the one-dimensional approximation reads

02 1 02 ) 471" ( oz2 - c2 ot2 A(z, t) = -~J.L(z, t)

(3.22)

where J.L is the average over the electron cross section of the transverse density current N

= e I:V.L&(x -

J.L(x, t)

j=l

Xj(t))

(3.23)

By substituting the result (3.10) for the transverse velocity v.L, projecting eq.(3.22) on thee-direction, and using the dimensionless vector potentials (3.7), (3.8), we obtain 1 02 ) . 47re2 N a e-ikwz _ iaei(kz-wt) 02 ( !:i""2 - 2"!i2 a(z, t)e•(kz-wt) = i w &(z - Zj(t)) (3.24) 2 n.L L

uz

c ut

me

j=l

/

where n.L is the transverse electron number density, i.e., the total number of electrons divided by the transverse beam section. Note that the transverse current (the r.h.s. of eq.(3.24)) has a main contribution due to the wiggler, plus a minor contribution due to the radiation field. Now, in the slowly-varying envelope approximation (SVEA)

l~:f « lkal

and

second-order derivatives can be neglected and 2 2 10)aei(kz-wt) . ( -0 2 - 2 oz c ot 2

~

loaf loz « lwal

(o + -10). ae•(kz-wt) ot

2ik oz

c

(3.24a)

(3.24b)

High gain free electron laser with waveguide

37

Furthermore, as the complex variable a is slowly-varying on the scale of the radiation wavelength>., it can be driven only by a transverse current averaged over a volume with longitudinal dimension l 11 several >.'s long. Hence, we average the transverse current also with respect to z, thus obtaining with (3.24a) and (3.13) 1 a ) a= ( -a + --

8z

c 8t

2 n { 1_ 4Jre _ _ __.::!: aw

2k mc l11 2

e-i 8; - ia L -1 } L:-i

Ii

i Ii

(3.25)

Now, introducing the total electron number density, n, and defining the average over a sample of N electrons of any electron dynamically variable f

(!(8, 1))

1 =NL f (8, 1) N

(3.26)

J=l

the wave equation can be written as (3.27) where (3.28) is the plasma frequency. Note that we assume space matching of the radiation and the electron beams (plane-wave and charged sheets in one dimension), whereas for different cross sections a filling factor should be introduced.

3.3

Space charge

We have derived the dynamic eq.(3.27) for the complex field amplitude, and eqs.(3.15), (3.21) for the electron energy and phase. Now, when we write these equations for the j-th electron of the beam, we must add the space-charge force acting on that electron due to the longitudinal self-field created by electron density fluctuations. According to refs. 13,14 the space charge contribution to the energy eq.(3.15) is (

~';) sc = -ik ( : ) 2 ( (e-i8 )e;o, -

c.c.)

(3.15')

Clearly, the space-charge force is appreciable if the electron current is "higher", so that the plasma frequency Wp is "big"; this point will be discussed next. Furthermore, the space charge effect would vanish not only (trivially) if one considers a single electron, but also if the electron phases were homogeneously distributed in such a way that the electron bunching is zero, where the bunching is described just by the quantity in the r.h.s. of (3.15') (OSbSl) (3.29) Note that if we sum the space charge force over the electrons, the result is zero; really it is an internal force with respect to the electron system.

Rodolfo Bonifacio

38

Now the electron dynamics equations (3.15), (3.21) can be rewritten for the j-th electron, with inclusion of space-charge (eq.(3.15')), and in the paraxial approximation

(d/dz::::::: o/oz + (l/v )0/ot: 11

0 1 0) (} (-+-oz v ot

1

11

o lo) 'Y ( -+-oz v ot 1 11

= (v

where

v

3.4

Universal scaling

11

11 )

0

is the "bulk" velocity of the macroscopic electron distribution).

The system of coupled evolution equations (3.27), (3.30), (3.31) can be set in a dimensionless form by introducing the following variables and parameters (Bonifacio 1984, and Bonifacio et al. 1987) :

rj

~ 'YJ p 'Yr

A

---a

z t

2kwpz 2kwpt

p

~(aw~

a

1 +a~ 4p -a2w

w

(1A12

Wp~

'Yr

4 ckw

r/

= ~ IEl 2I 47r) p n"(rmc2

(3.32) 3

In eq.(3.31), the complex field amplitude A is such that plAl 2 gives the ratio between the energy densities of radiation and of the resonant electron beam; z, t are scaled coordinates, in particular, z is a dimensionless length defined in the range 0 ::=:; z ::=:; 47rpNw for 0 ::=:; z ::::; Lw; pis the fundamental FEL parameter (p::::::: 0.1367; 1 J11 3 B-;/ 3 >.Y 3 [SI units], with J the electron current density) and a the space-charge parameter (see eq.(3.33b)) below). By the scaling (3.32), eqs(3.27), (3.39), (3.31) become:

o lo) (} ( -+---= oz v ot 1

(3.33a)

0 lo) ( -+---=

rj

(3.33b)

+ ~ o_] A [~ oz cot

(3.33c)

11

0z

v ot 11

Note that, for fixed wiggler parameters and given initial conditions, the whole system (3.33) depends only on one electron-beam parameter, the FEL parameter p ("universal scaling").

39

High gain free electron laser with waveguide

4

4.1

Steady state: Compton and Raman regimes, linear stability analysis, collective instability and exponential gain Steady state equations and Hamiltonian model

Under a standard transformation of coordinates

z'

=z

(4.1)

the differential operators in the l.h.s. of the FEL equations (3.33) change as follows:

(!+i:t) ({)

18) az+~at

=}

=}

{)

az' az'{) - c"/J1 (1 - -) at'{) (311

II

(4.2a) (4.2b)

Clearly in free space propagation effects can be neglected if the velocity difference between radiation and electrons (slippage) is not appreciable during the interaction in the wiggler. In this case, the derivative with respect to time in (4.2b) can be neglected, so that only the space dependence is left in all equations; one can follow the steady state evolution of the system as it moves along the wiggler z-axis. We now introduce a slightly different scaling which is suitable for the discussion of the collective FEL instability and the high gain regime. Namely, we redefine the electron phase() and the complex field amplitude A in terms of the detuning parameter 6 as follows: 6 = _2.Jr)~

2p

- ,;

,;

(4.3a)

then the scaling is like in (3.32), provided that one performs the simple substitutions (4.3b) , By (4.3a), (4.3b) the steady state FEL equations can be written as: dBi

(4.4a)

dz dfj

(4.4b)

dz

dA

(4.4c)

dz

Note that the detuning parameter 6 appears in the field equation (4.4c). Two constant of motion can be derived from eqs.(4.4). One of them is

(r)

+ IAl 2 = canst.

(4.5)

40

Rodolfo Bonifacio

which in the original variables reads: n('Y)mc2 + IEl 2 /47r = const., that is, energy conservation. However, there is another and phase dependent constant of motion, namely

H (4.6) Actually, a fully canonical treatment for both electrons and radiation leads to a Hamiltonian model of the FEL, including high density effects, which can describe high gain amplifiers operating both in the Compton or in the Raman (Bonifacio et al. 1987) regimes as we shall see next. Notice that if we introduce real and imaginary parts of the field variable, setting (4.7) A= (Bo+ ifo)/v'2& we can write a dimensionless Hamiltonian for (2N + 2) canonically conjugate electron and field variables (Bonifacio et al. 1987)

1

~

-L..J

2p j=l

(r ·+-{fv (r ~ P2r; +- 1 )

J

1 P N

cos

ej ()-

~ sin ej)

OL..J--+ OL..J-j=l

rj

j=l

rj

1 1 )) (0-20 +r 02 )+. 3 , of the root of the dispersion relation (4.13) which rules the exponential amplification versus the detuning parameter D. resonance, and ii) vanishes for D = Dr :::; 2, i.e., the system is unstable (stable) for all detunings D :=:;Dr (D 2 Dr). In the limit p-+ 0, the dispersion relation (4.16) reduces simply to )..3 - D>.2+1 = 0 (4.15) a cubic characteristic equation which is well-known since the theory of travelling-wavetubes (Pierce 1950). In this limit the maximum of >. 3 occurs exactly on resonance, D = 0, and its value is ).. 3 = -./3/2; the critical detuning, or the threshold value for the instability is D = Dr = (27 /4) 113 ~ 1.89. For a given wiggler, the limit p -+ 0 is approached for high electron energy and low current (p . 2 - (a - 2p)>. + 1 + aD

=0

(4.15')

From (4.15'), the maximum growth rate turns out to occur at D ~ fo, whereas for -+ 1 the on-resonance gain vanishes. Hence, space-charge definitely plays a role, and

p

High gain free eiectron laser with waveguide

43

the FEL operates in a different regime, the Raman regime (Bonifacio et al. 1987), in which the electrons interact appreciably not only via the common radiation field, but also directly due to space-charge effects. The electron system can exhibit collective plasma oscillations, i.e., the instability is collective in a stronger sense with respect to the Compton regime. Accordingly, in the electron longitudinal rest frame, the resonance relation is no longer the two- wave Compton relation, w' = w~ (see the relativistic mirror in Sec.2), but is a three-wave relation, w' = w~ - w~ , which describes a stimulated Raman backscattering with the radiation (signal) frequency equal to the difference between the wiggler (pump) and the plasma (idler) frequencies. The scaling (4.3) allows for a description of the passage of the FEL dynamics from the Compton to the Raman regime only by increasing the FEL parameter p. The scaled intensity IAl 2 grows from IAlo « 1 to IAl 2 = 0(1) exhibiting the predicted exponential growth nearly up to the first peak, that is in saturation, where the linear analysis becomes completely invalid. This behaviour has been observed in several high-gain FEL experiments (Orzechowski et al. 1987, 1986). The result IAl 2 = 0(1) at saturation is relevant. Actually, since IAl 2 ex: IEl 2 / pn ex: IEl 2 /n413 (see eq.(3.32) ), where n is the electron density, it follows that at saturation IEl 2 ex: n 413 , namely, the system exhibits a collective behaviour. However, the efficiency of the FEL process, T/ ~ plAl 2 ~ p, is limited to a few percent. One way to avoid saturation and raise both the efficiency and scaling of intensity with the electron density is via a variable-parameter or "tapered" wiggler, as proposed in References 20,23, and experimentally observed (Orzechowsky et al. (1986) and Adinhoffer et al. (1984)). Another way is given by FEL superradiance, where IEl 2 ex: n 2 • The other fact to note is the lethargy: the exponential gain occurs only after a lethargic stage. This stage is the longer, the closer the initial conditions are to the equilibrium state (4.8). It lasts as long as the three modes of the linear analysis interfere, until the divergent mode prevails over the two other modes. On the other hand, if the system is stable (or the wiggler is not long enough), the radiated intensity IAl 2 remains always close to its initial value !Arn. In the following we shall focus on the Compton regime.

5 5.1

Steady-state results in the Compton regime Compton FEL equations and high gain regime

As discussed in the previous sections, for values of p :S 0.01 the system operates in the Compton regime. In this case we can neglect both the space-charge contribution in eq.(4.4b) and the radiative corrections in eqs.(4.4a) and (4.4c); furthermore, we can surely perform the approximation of small relative variations of the electron energy in eqs. (4.4),

11-b)ol«l

1

(1)0

(5.1)

44

Rodolfo Bonifacio

Since rj = ''(j/ p('y) 0 , this condition implies that pfj ~ 1 . Hence, if we define for the j-th electron the variable Pi proportional to the relative energy variation, 1 Pi= fi - p

1 /i - ('Y)o p

('Y)o

(5.2)

eqs. (4.4a-c) reduces to

- ( Ae;tiJ

+ c.c.)

(5.3)

(e-i 0) + it5A The linear stability analysis around the state (4.8) leads from eqs. (5.3) to just the cubic equation (4.15'). In order to better discuss the physics of the Compton regime, and also for future convenience, we can reabsorb the parameter§ from eqs. (5.3). In practice, we go back from the present scaling (4.3) to the original scaling (3.32). The steady-state Compton FEL equations then reads: d(}j

(5.4a)

az

Pi

dpj

az

- ( Aeili;

dA

az

(e-io)

where

+ c.c.)

(5.4b)

=b

(5.4c)

1 'Yi - 'Yr p·---J p 'Yr

(5.5)

with ('Yi - 'Yr)hr « 1 and (}, A, p defined as in (3.32). Clearly, the detuning can be introduced at the level of the initial conditions, and we already know that on resonance the system is unstable in the Compton regime if it starts close to the equilibrium condition (4.8) (which now reads Ao= 0, (e-i8 ) 0 = (Pi)o = 0). We note the following facts. First of all, the universal scaling is here at its best: just no parameters are left in eqs. (5.4). Furthermore, these equations can be derived as Hamilton equations from a ("universal") Hamiltonian

H

=-£ [P7 + j=l

2

i (A*e-i 8J

-

c.c.)]

(5.6)

using the Poisson brackets {B;,Pi} =§ii and {A,A*} = -i/N. Likewise, eqs. (5.3) could be derived from a one-parameter (the detuning ,;z, where >.1(j=1, 2, 3) are the solutions of the cubic eq.(4.15'). With initial conditions close to the unstable state (4.8), e. g. Ao= (M)o =/= 0

(5.9)

where the perturbation to the equilibrium condition (4.8) is a small injected signal. one finds (Bonifacio et al. 1990, and Bonifacio et al. 1992) when looking at the divergent mode:

IAl 2 (z)

~ ~ [4 cosh

2 (

The resonance gain Gres(z) = in Figure 2

'7-z) + Gz) 4 cos

cosh (

'7-z) +

1]

IAI~

(IAl 2 (z) - JAJ6) /IAl6 obtained from eq.

(5.10)

(5.10) is plotted

Rodolfo Bonifacio

46

15

G( z) lO

Figure 2. Linear regime: gain at resonance from Equation (5.10)

Note that for

z«

1, the behavior of the output intensity is

/A/ 2 (z)

::::: ( 1 +

~) /A/~

that is, it remains very close to the initial value

z;::: 1, /A/ 2 (z) diverges exponentially as

/A/ 2 (z)

:::::

IAl6

(5.10')

(lethargy). On the contrary, for

1

g exp( J3:z)/AI~

(5.10")

in agreement with (4.14). The factor 1/9, or 1/3 for /A/, also referred to as "launching loss" term, is due to the initial power distribution among the three modes of the linearized dynamics. Going back to the original variable z, in the exponential gain regime

(5.11)

where

(5.11') is the exponential or unsaturated gain per unit length, in agreement with (4.14'). If the peak power is near the wiggler end, one can define a total unsaturated gain G

= gLw = J3 47rpNw

(5.12)

The high gain regime is such that G > 1. Also, from (5.12) it follows that the gain per wiggler period is of the order of 47r p.

5.3

Marley's small signal gain

With still the initial conditions (5.9), in the stable region, o > or, and in the limit 1, the linear calculation gives for the radiated intensity

z/../J «

/A/ 2

(z) ~ [1 + ~ ( 1 - cos oz - o; sin oz)] /A/~

(5.13)

High gain free electron laser with waveguide

47

In eq. (5.13), IAl 2 is an oscillating function: the hyperbolic functions of eq. (5.10) have become simple trigonometric functions. From eq. (5.13) we can define the smallsignal or interference gain

G(z,c5) =

- [Al 2 IAl 2 (z) [Ai5 °=

where in the original variables (setting /o

4 (

63 1- cosc5z-

c5z

2

(5.14)

= (1)0)

c5z = 47f lo - Ir z Aw 'Yr At the wiggler end (z = Lw = Nw>-w--+

) sinc5z

z=

(5.15)

47rpNw)

, __ N lo - Ir _ uz - 47rp w - - - = Ir

A L.l.

(5.15')

which is a function of the injection energy /o for fixed Nw and /r· Thus we can write G(~)

4(47rpNw)3 J(~)

!(~)

~ (1- cos~ - ~sin~) ~3 2

(5.16)

The small signal gain, proportional to the function !(~), is an asymmetric curve, quite peculiar of a dispersive process. In particular, the small signal gain is zero on resonance, just the opposite result with respect to the high gain regime, where the growth rate is maximum for c5 = 0. On the contrary, here the gain is maximum at ~ ~ 2.5; i.e., a necessary condition to have gain, is to inject electrons with (1) 0 > /r, as anticipated in section 2. A phase space analysis would show that the injection of electrons with ('Y)o >Ir gives an optimum gain, for a given undulator length, if it corresponds to the electron executing on average half- "synchrotron" oscillation in the ponderomotive bucket. On the other hand, for increasing z and thus relaxing the limit z/./6 « 1 , the peak gain not only becomes remarkably higher, but also moves towards smaller detuning values, in agreement with the previous discussion In conclusion, the Matley regime is valid even in the unstable region c5 :::; c5r up to z:::; 1, whereas increasing z, the symmetry is broken and the exponential gain at resonance increases. In order to understand the apparent conflict between the zero small signal gain and the fastest exponential rate both occurring on resonance in the two regimes, one could simply refer to the resonance condition in terms of the equality of the longitudinal electron velocity, 11 with the ponderomotive phase velocity, Vp· In the self-consistent scheme of section 3, eq.(3.14), Vp was evaluated as Vp = w/(k +kw) for a given radiation (and wiggler) field. In that case, in a resonant and randomly phased electron beam, nearly one half of the electrons absorb energy and one half lose energy, with no net gain; in fact, the particles (slightly) bunch around a phase for which there is no coupling with the radiation. However, if the field varies appreciably as in the high gain regime, the ponderomotive phase velocity is modified as 11 ,

w - d¢/dz v ·- ----'-p -

k+kw

(3.14')

Rodolfo Bonifacio

48

where we set A= IAI exp(i). Since d/rlz > 0 (Bonifacio et al. 1986a), the electron beam acts as a dielectric medium which slows down the phase velocity of the pondero~ motive field. Hence, resonant electrons get a longitudinal velocity v > Vp and bunch around a phase corresponding to gain. 11

Going back to the small signal, low gain analysis, the gain function (5.14) can be rewritten in the form G(z, J) = -

:z3

d

(5.14')

2 d(Jz/ 2) sinc2 (Jz/2)

or, at the end of the wiggler,

3 d . 2(~/ 2 ) G( A)=-(47rpNw) '--' 2 d(~/2) smc

(5.16")

Now, from (5.15') and the resonance relation (3.18), ~

~'Y

~w

- = 27rNw- = 7rNw2 'Yr W so that

(5.17) ~w

x=7rNww

(5.18)

By recalling eq. (2.3a), we see that eq. (5.18) expresses the basic result that the small signal gain is proportional to the derivative of the spontaneous spectrum. In fact, in this regime and for quite general undulators, two Madey's theorems hold (Madey (1979)): 1. ('Y - 'Yo) = (1/2)d/d'Yo ((('Y- 'Yofl), namely, there is no gain without energy spread;

2. (('Y- 'Yo) 2 ) oc d2 I(w)/dwd0., in agreement with the previous derivation. Also, it follows that in this regime the "gain" linewidth is of the order of the spontaneous linewidth::::::; l/Nw (sec.2); hence, the electron energy spread must be less than : : : ; l/Nw, and the energy transfer from the electrons to the field, that is the FEL efficiency, is limited to within::::::; l/Nw. This is valid only in the low gain situation z « 1. In the high gain region z > 1, the linewidth is given by ~w/w::::::; p. This can be easily inferred as follows: J : : : ; 'Yi - 'Yr P'Yr

< Dr : : : ; 2

that is

~'Y ~ 'Yr

2p

From the resonance relation (3.18) ~'Y/'Yr = (1/2)(~w/w) so that ~w/w ~ 4p, i.e., the linewidth is of the order of p. We conclude this section by recalling that the FEL dynamics in the Compton regime can be described by means of only three (complex) electron and field collective variables (Bonifacio et al. 1986b). These variables are the complex field amplitude A, the bunching parameter b, and another electron variable which describes electron energy modulation. The evolution equations for these quantities, obtained in suitable approximations, turn out to nicely reproduce the numerical results from the full (2N + 2) eqs.(5.4) even in the saturation regime.

High gain free electron laser with waveguide

6

49

High gain FEL amplifier in a waveguide

Now we consider the FEL process occurring in a waveguide, which is the case of amplifiers operation in the microwave region: see (Orzechowski et al. 1985, Sprangle et al. 1980 and Masud et al. 1987). For sake of simplicity we shall assume a rectangular waveguide, with only one (transverse electric) mode which couples to the electron beam. Hence we shall generalize the FEL equations and the universal scaling to include off-axis propagation in a rectangular waveguide and with a planar wiggler. As an application, we shall discuss the optimization of gain, tuning and slippage with the waveguide (Bonifacio, De Salva Souza (1989b).

6.1

Planar wiggler

Unlike in the previous sections we shall consider a planar wiggler, with geometry such that the electron beam, injected along the z-axis, is subjected to a transverse, periodic magnetostatic field in the y-direction, thus wiggling in the x-z plane. The modifications to be introduced in the previous treatment are well-known (Scharlemann 1989 and Orzekowski et al. 1987). First of all, the undulator parameter, written as aw = eBw/mc 2 kw for a helical wiggler, where Bw is the peak value of the wiggler field, becomes for a planar wiggler

eBw aw= vf2mc2kw

(6.1)

Note that the definitions (2.5) and (6.1) can be unified if one introduces the root-meansquare wiggler field, (Bw)rmS> so that aw= e(Bw)rms/mc2 kw in both cases. Second, in the FEL equations and in the scaling (3.32) one must perform the substitution (6.2a) where dB is the following difference of Bessel functions (Colson 1981) dB= Jo(~) - J1(0

~=

a2 2(1; a~)

(6.2b)

The quantity dB is of the order of one and accounts for a reduction in the electron-field coupling in the presence of a planar wiggler instead of a helical one. Furthermore, in a planar wiggler the motion gets a longitudinal, fast-oscillating component superimposed on the wiggler motion in the (x-z) plane. As a consequence of averaging out this jitter effect, the relative phase of the electron in the ponderomotive field,() (eq.(3.13)), must be referred to the average longitudinal electron motion:

() = (k + kw)Z - wt t = :_ ::::; ~ + +a~) cf3,,

(1 l

c .

2'Y2

(6.3)

With aw and() defined as in eq.(6.1) and (6.2), as we shall understand from now on, the FEL dynamic equations both in the angina! and in the scaled form remain unchanged

50

Rodolfo Bonifacio

with respect to the case of a helical wiggler except for the substitution (6.2a). As an example, the FEL parameter r (eq.(3.32)) with a planar wiggler becomes

(aw )2/3 --

- 1 'Yr

Wp

p--

6.2

(6.4)

4 ckw

FEL equations with a rectangular waveguide

Let the FEL process occur within a rectangular waveguide, whose short (long) dimension is b (a), parallel (orthogonal) to the wiggler field (Bw1(Y). We assume that only one guided mode, the transverse electric T E01 mode, couples with the electron beam. We must start from a full three-dimensional treatment of the field. However, we introduce the following ansatz on the radiation vector potential (compare eq.(3.4)): A(x, t) = -xliil(z, t) sin(k z - wt+ (z, t)) sin(krx.L) 11

(6.5)

where Iii I(z, t) and a and two intersections for 0 < (1 + a!)h2 < a, one for positive and another for negative slippage. Also, from eq. (6.24') it follows that the radiation wavelength A is always greater than without waveguide, and this effect is maximum just at zero slippage (X = 1), where A is doubled with respect to the no-guide case. The total slippage length with a waveguide is ls

= (~g v"

- 1)

Lw

(6.26)

X)

(6.26')

By means of eq.(6.6) and (6.20) one finds easily ls

= LwaX(l -

which generalizes the usual result ls= LwaX = NwA . Figure 6.1 shows also that the slippage length has a maximum as a function of X with a constant, (at X = 1/2), i.e., with respect to the wavelength A. Most important, also the FEL gain can be optimized with respect to the wavelength A for fixed wiggler and waveguide, e.g., by varying the electron energy. Let us consider the steady-state unsaturated gain per unit length g, generalized with the waveguide as (6.27) with kw and the form

p defined in (6.12) and (6.18). Now, we can write the latter parameter in 113 _p-p _ (1 -x)(k-Sbeam) -116

k11 Smode

2

(6.28)

where pis the FEL parameter (6.4) without waveguide (with a planar wiggler). If one sets the waveguide geometrical factor (k/ k11 )(Sbeam/ Smode) equal to one, as we shall assume from now on, the ratio of the gains with and without waveguide is

X) 1

g_kwP_( - - - 1-g kwP 2

5 6

(6.29)

showing a gain reduction due to the waveguide, which is monotonic decreasing function of X. However, if one is interested in the global dependence of the gain g on the wavelength A (i.e., with X ex A variable and a constant as above for tuning and slippage) then, since p ex

JA/ Aw = ~

g ex .jX (1-

x)s/6 = W(X)

2

(6.30)

The curve W(X) is reported in Figure 4. A remarkable result is the existence of a maximum of W(X), which occurs for X = AAw/4b2 = 0.75 . Namely, for a fixed ratio b/ Aw, the gain can be optimized with respect to the radiation wavelength by the choice A ~ 3b2 /Aw·

High gain free electron laser with waveguide

55

0 0.5 1 x 1.5 2 Figure 4. Function W(X), (Equation 6.30), describing the gain per unit length in the waveguide as a function of the radiation wavelength.

References Adinhoffer J A Neil GR, Hess CE, Smith TI, Fornaca SW, Schwettman HA, Phys Rev Lett 52 344, (1984) Baier V N and Mil'shtein A I, Sov Phys Dokl 25 112 (1980), Bernstein I B,and Hirschfield J H, Phys Rev A 20 1661 (1979), Bonifacio R, Pellegrini C, Narducci L M, Opt Comm 50 313 (1984) Bonifacio R, Casagrande F, De Salvo Souza L, Opt Comm 58 259 (1986a) Bonifacio R, Casagrande F, De Salvo Souza L, Phys Rev A33 2836 (1986b) Bonifacio R Casagrande F, Pellegrini C, Opt Comm, 61, 55 (1987) Bonifacio R, Casagrande F, Cerchione G, DeSalvo Souza L, Pierini P, High gain, High power free electron lasers, Editors Bonifacio R, De Salvo Souza L, Pellegrini C, Elsevier, NorthHolland, p 35, (1989a) Bonifacio R, De Salvo Souza L, Nucl Instr Meth A276 394 (1989b) Bonifacio R, Casagrande F, Cerchioni G, De Salvo Souza L, Pierini P and Piovella N, Riv Nuovo Cimento 13 n9 (1990), Bonifacio R, Corsini R, De Salvo L, Pierini P and Piovella N, Riv Nuovo Cimento 15 nll (1992) Colson W B, IEEE J Quant Electr QE-17 1417 (1981) Colson W B, Phys Rev A64 198 (1977) Colson W B, Phys Rev Lett 59 187 (1976) Dattoli G, Marino A, Renieri A and Romanelli F, IEEE J Quantum Electr QE-17 1371 (1981), Deacon DAG, Elias LR, Fairback WM, Matley JM, Ramian G J, Schwettman HA, and Smith T I, Phys Rev Lett 38 892 (1977) Elias L R, Fairback W M, Matley J M, Schwettman H A, and Smith T I, Phys Rev Lett 36 717 (1976) Grover A and Sprangle P, IEEE J Quantum Electr QE-17 1196 (1981), Jackson JD, Classical Electrodynamics, (Wiley, New York, 1975) Kroll N M and McMullin WA, Phys Rev Al 7 300 (1978), Kroll NM, Morton LP, and Rosenbluth MN, IEEE J Quantum Electr QE-17 1436 (1981)

56

Rodolfo Bonifacio

Kroll N M, Morton L P, and Rosenbluth M N, in Jilree Electron Generation of Coherent Radiation, in Physics of Quantum Electronics, edited by Jacobs S, Pilloff H, Sargent M, Scully M, and Spitzer R, Vol 7, chap 5 (Addison-Wesley, Reading, MA, 1980) p 113 Matley JM, J Appl Phys 42 1905 (1971) Matley J M, Nuovo Cimento B50 64 (1979) Masud J, Marshall TC, Schlesinger SP, Yee F G, Fawley WM, Scharlemann ET, Yu SS, Sessler AM, Sternbach E J, Phys Rev Lett 58 763 (1987) Murphy J B, Pellegrini C, Bonifacio R, Opt Comm 53 197 (1985) Orzechowski T J, Anderson B R, Fawley W M, Paul A C, Proznitz D, Scharlemann E T, Yarema S M, Hopkins D B, Sessler A M, and Wurtele J S, Phys Rev Lett 54 889 (1985) Orzechowsky T J, Anderson BR, JC Clark, Fawley WM, Paul AC, Proznitz D, Scharlemann ET, Yarema SM, Hopkins DH, Sessler AM, Wurtele JS, Phys Rev Lett 57 2172 (1986) Orzekowski T J, Scharlemann ET, D Hopkins, Phys Rev A35 2184 (1987) Pierce JR, Traveling Wave Tubes, (Van Nostrand, Princeton, 1950) Scharlemann ET, High gain, High power free electron laser, Ed Bonifacio R, De Salvo Souza L, Pellegrini C, Elsevier, North-Holland, 1989 Shih C C and Yariv A, IEEE J Quantum Electr QE-17 1378 (1981) Sprangle P, Tang C M, Manheimer W M, Phys Rev A 21 302 (1980)

57

Computer Modelling of Microwave Sources James W Eastwood AEA Technology Culham, Abingdon

1

Introduction

This Summer School treats a wide range of high-power microwave tubes, covering the generation of electromagnetic radiation from sub-millimetre to tens of centimetre wavelengths. At the highest frequency are free electron lasers (FELs) and fast-wave devices such as gyrotrons and cyclotron autoresonant masers (CARMs). As wavelengths increase, a variety of devices-Cerenkov masers, backward wave oscillators (BWOs)), travelling wave tubes (TWTs), vircators, klystrons, magnetrons, etc-find favour. The design choices for these tubes depend on many factors, such as frequency, power, pulse length, tunability, mode purity, rep-rating, ruggedness and so forth (Granatstein and Alexeff 1987, Benford and Swegle 1992). The theoretical analysis and computer modelling of certain of these tubes take advantage of device specific approximations (eg Ting et al 1991). Such specific approaches will not be treated in this chapter. The focus here will be to identify and describe generic features and generally applicable techniques, although in doing so reference will be made to specific tubes. Common to all microwave tubes is the interaction of electromagnetic fields with charged particle flow. The generation of microwaves relies on first converting the almost DC electromagnetic energy in the power supply into kinetic and potential energy of ordered electron flow and then transforming in an interaction region the electron energy into microwave energy (cf Figure 2). Different devices exploit different resonant processes to effect this energy conversion. For example, gyrotrons extract kinetic energy from the electron gyromotion perpendicular to the constraining magnetic field. Linear beam devices take kinetic energy from electron flow parallel to the magnetic

58

James W Eastwood

field. Crossed field devices such as the magnetron rely on converting potential energy of the electrons into microwaves. In all cases microwave energy is extracted from the interaction region and is fed via waveguides to the antenna for radiation. In some cases, residual energy in the electron flow is collected and recycled to improve the overall efficiency. Successful tube design relies on getting all the factors in the chain of energy conversion quantitatively correct. Computer modelling now plays an important role in all these steps. Computer simulations of microwave tubes fall broadly into three categories:

Idealised, where simulations using idealised geometries are used to investigate specific mechanisms. Engineering, where simulations are used to investigate a range of design parameters to optimise engineering design before expensive experimental prototypes are built. Interpretive, where modelling reproduces experimentally measured data and is then used to synthesise data inaccessible to direct experiment. Although much of the behaviour of tubes relies on well-established classical electrodynamics, there is no single universal computer code to model all devices. The reason for this is that computer power is limited, and so can handle only a finite range of lengths and timescales; fully three dimensional calculations still tax the most powerful parallel computers. The 'brute-force' approach often leads to computations which take so long to undertake as to be impractical. If a simpler, quicker calculation provides all the information required, then it is to be preferred. The secret of success is to divide-and-conquer. Most information of relevance can be obtained from computations using reduced dimensionality and/or reduced physical models. A large part of the parameter searching needed in tube design is usually performed using parametric, 1-D and 2-D codes, and only in special circumstances does it require fully self-consistent 3-D electrodynamic modelling. Reduced physical models reduce scalelengths to be treated. The range of length scales is determined at one extreme by the shortest scalelength present-for example, the electron gyroradius or debye length, the spacing of permanent periodic magnets, signal wavelength-and at the other by the system size. Similarly, the range of timescales is determined by high frequencies of signals and electron motions and the timescale to reach steady state. If timescales shorter than those of interest are present in the physical equations they cannot be simply ignored, as they often force restrictions on timesteps to maintain stability in numerical calculations. Likewise, short wavelengths cannot be ignored as they can also lead to instabilities through non-physical resonant mode coupling or through nonlinear instabilities. This chapter will touch on all aspects of the use of 'computer experiments' in the design of tubes. We shall identify some of the physical approximations used for different parts of microwave sources, examine numerical methods used to handle these physical approximations, and see examples of how these numerical methods are used. The next section looks in general terms at the nature and role of computer simulation in the design and interpretation of tubes. Section 3 then identifies the components of

Computer Modelling of Microwave Sources

59

a generic tube and summarises the standard computational problems which result. The remainder of the chapter works through these problems in order of increasing complexity. Sections 4 and 5 introduce the numerical methods. Section 6 describes the computations of particle orbits in known electric and magnetic fields. Sections 7-10 work through the range of purely electromagnetic calculations. Sections 11-14 describe the most detailed computer simulation of the interaction between electromagnetic fields and charged particle flow using Particle-in-Cell (PIC) models.

2

The Computer Experiment physical phenomenon mathematical model algebraic approximation numerical algorithms simulation program computer experiment

Figure 1. The steps in developing a simulation model Figure 1 summarises the steps involved in setting up a computer experiment. The starting point is some physical phenomenon. The objective is to obtain physical understanding from the computer experiment. Between these two points we may identify a number of distinct design steps. Each step introduces constraints. The first step is to devise a mathematical model. We are faced with choices of parametrisation for the geometry and for the initial and boundary values. The computer experimenter must know what simplifying assumptions are made in order to identify the range of validity of the equations, and hence of his simulation model. Ordering of equations can lead to changes of causality and to different sets of consistent initial and boundary conditions. Convenient mathematical fictions, such as the cold plasma approximation, may be inaccessible to the numerical model. The mathematical model is usually a set of differential equations. The numerical scheme replaces these equations by discrete algebraic approximations to allow numerical solution on computers. This introduces questions concerning the consequences of finite time step, discrete spatial meshes, element choice, and for particle models, the limited number of particles. Numerical errors may be relatively benign in that they modify

James W Eastwood

60

Tube power supply

gun

focussing field Interaction space

rf Input

rf output

collector

antenna

Figure 2. The generic microwave tube dispersion, enhance diffusion and give small drifts in conserved quantities. Equally well, they can lead to grossly nonphysical behaviour. The onus is always on the computational scientist to demonstrate that the results of the simulation are physically meaningful. Impinging on the choice of discretisation is the question of numerical algorithms. Discretisation replaces continuous variables of the mathematical model by arrays of values, and differential equations by algebraic equations. Unless the tens (or hundreds) of thousands of algebraic equations arising can be rapidly solved on a computer, the proposed simulation becomes impractical. The process of obtaining the mathematical model, discrete approximation and numerical algorithm is usually an interactive search for a good compromise between the quality of the representation and computational cost. Only when a satisfactory choice has been found can the apparatus - the simulation program-be built. As with physical apparatus, a great deal of care is needed in designing and constructing the program to ensure that it behaves in the desired manner. A well-engineered program should be easy to use, easy to read and, when necessary, easy to adapt. It should have a clear modular structure and built-in diagnostics. The analogy with laboratory practice may be taken further; elements of the program should be tested as they are assembled into a preplanned structure. The complete program should, if possible, be tested on known problems to verify both the algorithms and the code. Only when testing and calibration are complete is the simulation program ready for performing computer experiments.

3

The Generic Tube

Figure 2 shows a block diagram of the components of a generic microwave source. As described in the introduction, the purpose of these components is to convert DC electrical power from the power supply to microwave power at the antenna through resonant interactions with the electron flow. The source is divided into four parts:

Power Supply which provides the primary electrical input to generate the electron beam and perhaps also supply currents for coils to generate focusing magnetic fields.

Computer Modelling of Microwave Sources

61

RF Input to drive the tube if it is an amplifier rather than an oscillator. Tube whose role is to convert the primary electric input from the power supply to microwave power.

RF Output which transports microwave energy from the tube and radiates it from the antenna. The tube itself is also divided into three parts : the gun, the interaction space and the collector. Often, but not always, these items have focusing magnetic fields. Sometimes, the collector is absent. In some tubes, the electron source is an integral part of the interaction space. Different parts of the microwave source need different mathematical models in describing their physical processes:

The Power Supply is a low frequency electromagnetic system, built from discrete component capacitors, inductors and resistors feeding power to the tube through some pulse forming network or transmission line. Power supplies are usually described by the coupled set of ordinary differential equations resulting from the application of Kirchoff's Laws. The Electron Gun is also a low frequency component, but requires a more detailed mathematical model to describe its behaviour. Electrons are emitted from cathode surfaces, are accelerated into the interaction region under the combined influence of (usually steady) electric and magnetic fields. The performance of tubes depends sensitively on preparing beams with well defined spatial and velocity distributions. Consequently, gun design often involves complex anode and cathode shapes to fine tune electric fields, and precisely tailored coils to similarly control magnetic field. The computational tasks are to model • electron trajectories in the electric and magnetic fields of the gun • emission of electrons from the cathode • beam space charge and currents • electrostatic fields in complex geometries • magnetostatic fields in complex geometries • thermal loading of mechanical parts (eg grids and beam scrape off) • thermo-mechanical consequences of power loading on mechanical structures.

The Interaction Space is the part of the device where the energy in the charged particle flow is converted to radio frequency (rf) energy. In general, the interaction space requires the consistent treatment of the electron flow and electromagnetic waves, although in linear beam tubes where k · E » lk x El the interaction can be treated electrostatically. The computational challenges in modelling the interaction space include those listed above for the gun region, plus computation of

James W Eastwood

62

• the time evolution of electromagnetic fields in two or three dimensions in the presence of electron flow and wave interaction structures • the coupling to lumped circuit descriptions for rf input and rf output . • diagnostic information such as wave dispersion, gain, power flows, etc. • secondary emission and plasma formation

The Collector, which recovers residual ordered energy from the spent electron beam presents largely the same challenges as the electron gun. Usually, an electrostatic treatment is sufficient, with the only substantial differences from the gun problem being that one has a time varying beam which deposits energy on and may generate secondary electron fluxes from the mechanical structure. Analysis of the effects of beam deposition include thermomechanical problems in the collector and possibly return electron fluxes into the interaction space.

The Output Waveguide and Antenna can be treated using the same electromagnetic solver techniques employed in modelling the tube interaction space. However, it is often computationally advantageous to use special techniques where a full wave solution is not required throughout the space (as is the case where electron flow is not present). A widely used method in waveguides is mode matching (Uher et al 1993), where known modes in waveguide sections are matched across transitions by some fitting procedure. Antennas with steady monochromatic signals can be modelled using surface element methods in the frequency domain. This method of moments (Wang 1991) transforms the Helmholtz equation for the field amplitude into a surface integral formulation, so a computational mesh is required only on the surface of the radiating object.

3.1

Standard Problems

The standard computational tasks arising in modelling the generic microwave tube are: l. Circuit modelling

2. Field calculations

• electrostatics • magnetostatics • electromagnetic eigenproblems • time-dependent electromagnetics • waveguide mode matching • antenna method of moments modelling

Computer Modelling of Microwave Sources 3. Particle calculations

63

• orbits • beam injection • cathode emission • secondary emission

4. Particle-In-Cell (PIC)

• electrostatic PIC • electromagnetic PIC

5. Thermomechanical

• heat flow • thermal expansion • mechanical stresses

These items will be treated in more detail in the remainder of this chapter.

4

Numerical Methods

Considerations of the length and time scale constraints, and limits of finite computer resources lead to the choice of mathematical model. This model is usually a coupled set of differential equations, although it may also involve integral equations and variational principles. The purpose of the numerical method is to approximate the differential equations relating continuous variables by sets of discrete algebraic equations for discrete values.

4.1

Classification of Equations

It is important to classify the equations in order to determine the form of bound-

ary condition required and to help choose the appropriate numerical method. The first distinction, between ordinary differential equations (ODE) and partial differential equations (PDE) may arise from dimensionality or from the choice of formalism.

ODE are divided into initial value (time-dependent on an open domain) and boundary value (time-independent on a closed domain) problems. Examples of an initial value ODE are the sets of equations arising from the application of Kirchoff's Laws to electrical circuits and the equation of motion for particles in a given electromagnetic field. An example of a boundary value ODE is Poisson's equation in one dimension. Less obvious instances of ODE are those obtained by representing hyperbolic PDE by characteristics, using finite element spatial representations in initial value/boundary value problems, or by transformation methods on eigenvalue problems. PDE having time dependence are of two types, hyperbolic or parabolic. Hyperbolic equations have real characteristics, and so may be mapped forwards in time by particle methods. Instances of hyperbolic equations are Vlasov's equation for collisionless charged particle flow and Maxwell's equations for electromagnetic waves. Parabolic

James W Eastwood

64

equations are typified by the heat flow equation. The prototypical time-independent (elliptic) PDE is Poisson's equation; equations of this type arise in electrostatics, magnetostatics and long time (steady state) solutions of parabolic equations. The different physical nature of the hyperbolic and parabolic PDE is revealed by their dispersion relations; the hyperbolic equation has a non-decaying behaviour with time scale proportional to length scale, whilst the parabolic equation gives a decaying solution with time scale proportional to the square of the length scale. An important result which emerges from the characteristic analysis of PDE is the correct form of boundary condition for each type of equation (Richtmyer and Morton 1967). For second order PDE, the appropriate boundary conditions are Cauchy (value and normal derivative given) on an open boundary for hyperbolic equations; Neumann (normal derivative given) or Dirichlet (value given) on an open boundary for parabolic equations; and Neumann or Dirichlet on a closed boundary for elliptic equations. The value or normal derivative boundary condition may be replaced by mixed or by periodic boundary conditions.

4.2

Discretisation

The continuum in space (space-time for time-riependent problems) is replaced by a finite set of values, and the differential equations are approximated by algebraic equations to obtain a system of equations which can be solved using digital computers. There are basically four methods of making the step from the differential continuum to the numerically computable discrete system. 1. Finite difference approximations. 2. Finite element methods. 3. Spectral methods. 4. Particle-in-cell methods (PIC). This distinction is to some extent arbitrary, in that combinations of methods are often used in obtaining simulation models, and that algorithms derived using one method may be identical to those derived using another.

4.3

Finite Differences

The finite difference method (Richtmyer and Morton 1967, Potter 1973) gets discrete equations by replacing • the continuum by values on a lattice of points • the derivatives by value differences • the equations by difference equations on the lattice.

Computer Modelling of Microwave Sources

65

Advantages of the finite difference approximation are that the equations are simple to derive, there are few operations per lattice point, coding is easy in regular geometries, and there is an established body of methods that work. Historically, the finite difference approximation has been the favoured method for electromagnetic and plasma simulations. In almost all time-dependent calculations, finite differences are used for the time coordinate. Limitations of the finite difference approximation are that it is difficult to apply to irregular boundaries and anisotropic media, its numerical dispersion and diffusion may give a poor approximation to the physics of the plasma, and it is more susceptible to nonlinear instabilities than alternative methods. As an example consider the differential equation for the time evolution of the current in an L - R circuit given by Kirchoff's Laws .

dl

dt

= _!:__/

(1)

R

The finite difference approximation to Equation (1) replaces the continuous function of time I(t) by a set of discrete values. For a uniform timestep tit these values 1n, n = 1, 2, 3, ... approximate currents at times tn = ntit. The derivative of a function I evaluated at time t may be defined as

dl

dt

=

lim {/(t + c5t) - I(t - c5t)}

6HO

2c5t

(2)

In the finite difference approximation, the limit is not taken to zero, but to some small quantity tit which gives an acceptable compromise between accuracy and computational cost. The smallness of tit is measured in terms of the characteristic time scale of variation of/. Applying this simple finite difference approximation approximation to Equation (1) gives

(3) Taylor expanding about time tn shows that the difference approximation is consistent with the derivative and is second order accurate:

(4) Higher order approximations are obtained by taking linear combinations of differences over further mesh points. For example, a fourth order approximation is given by

d/ [Jn+l _ 1n-l] dt ~a 2tit

+ (l - a)

[Jn+2 _ Jn-2] 4tit

(5)

where a = 4/3. In principle, Equation (3) can be used to compute the value of Jn+l given values 1n and 1n-1, and repeated application can be used to generate an approximation to

I(t). In practice, it cannot be used as this particular choice of difference equation is an example of an unconditionally unstable scheme. Any small imprecision in the

James W Eastwood

66

calculation will seed an exponentially growing solution which rapidly swamps the exponentially decaying physical root. We shall return to the question of numerical stability in Section 5. Second and further derivatives are treated in the same manner. For instance, the second order accurate approximation to the second derivative is given on a uniform mesh of spacing ~x by d2 = p+l - 2p + 1. Note that the space-time lattice in Equation (20) causes periodicities in the frequency and wavenumber of respectively 27r /At and 27r /Ax. Physically, the only significant wavelengths are those in the Principal (or First Brillouin, in solid state terminology) Zone, lkl < 7r /Ax, and likewise for frequencies lwl < 7r /At. The CFL stability criterion on the time step in this instance is At < Ax/v. The Lax, Lax-Wendroff and Upwind schemes are examples of the second cure, and Backward Euler time differencing is an example of the third (Potter 1973).

5.2

Nonlinear

Even if the time step is sufficiently small for a numerical scheme to be linearly stable, nonlinear instabilities can still lead to failure if the spatial resolution is insufficient. The physical cause of the nonlinear instability is the cascade of energy to short wavelengths caused by nonlinear advection terms. If dissipation is too small, the unresolved short wavelengths couple back into ('alias') long wavelengths, because of the periodicity in wavenumber space caused by the discrete spatial mesh. This coupling can give numerical solutions which blow up in a finite time 1/(t0 - t). This effect has led to erroneous interpretation of some simulation results (Eastwood and Arter 1986). To ensure nonlinear stability requires sufficient physical or numerical dissipation to make the mesh Reynolds Number< 0(1), namely max(Ax/lengthscale) < 0(1)

(21)

A simple illustration of the nonlinear stability is given by the solution of the discrete Burger's equation for wavelength equal to three mesh spacings. Substituting the trial solution u = 21 (t) sin kx into (22)

James W Eastwood

72

yields

2j

= -2J 2 sin 2kx -

2J sin kx.

(23)

For a wavelength equal to three mesh spacings, kx = 27rp/3, p integer, so sin 2kx - sin kx, and Equation (23) reduces to j = J2 - f. This has the solution

J = 1/(1 -

to= log IJo/Uo -

e(t-tol),

l)j.

=

(24)

For Jo < 1, J has the physical exponentially decaying form, but for Jo > 1 the solution becomes singular in finite time t 0 . The stability boundary at Jo = 1 corresponds to the mesh Reynolds number becoming of order unity. PIC methods avoid the non-linear instability by combining the advection and time derivative terms in the lagrangian derivative for the particle trajectories. However, instabilities arising from spatial resolution limitations can occur as non-physical inverse Landau damping through coupling to charge density aliases (Section 11.4).

6

Particle Orbits

The need for particle orbit calculations arises in tracking particles through known fields, and particle integration in PIC models of guns, interaction spaces and collectors. The mathematical problem is to integrate the equations of motion:

dx dt

-=v·

'

dp dt

= F = q(E + v x B)

(25)

= Ji + (p/m0 c) 2

(26)

where in the relativistic case p

= 7m0 v

and 7

Criteria for selecting discrete approximations to Equations (25) and (26) are consistency, accuracy, stability and efficiency: Consistency requires that the difference equation tends to the differential as flt --+ 0. Accuracy requires the deviation of the difference from differential equation to be small for some small flt. Stability is concerned with the propagation and growth of errors. Roundoff and initialisation errors should remain small compared to the solution of the differential equation. Efficiency is concerned with the number of timelevels to be retained and the number of operations required to advance from one step to the next. Application of these criteria when only a small number of trajectories are being followed often leads to the choice of 3rd or 4th order Runge-Kutta scheme. If many orbits are being followed in PIC schemes, then second order accurate leapfrog electric force and implicit Lorentz force integrators are usually favoured because of their lower storage demands (Chap 4, Hockney and Eastwood 1988).

Computer Modelling of Microwave Sources

73

The leapfrog scheme, as its name suggests, 'leap-frogs' positions and velocities to new timelevels. The leapfrog scheme for non-relativistic motion of a charged particle of charge q and mass m in an electric field E is xn - xn-1 vn-1/2 t::..t (27)

=

vn+l/2 - vn-1/2

(28)

1._Ent::..t

m

where xn = x(tn), etc. Given xn-l and vn- 1/ 2 , application of Equation (27) leapfrogs positions over velocities to time tn, and application of Equation (28) leap-frogs velocities over positions to time tn+I/ 2 = (n+ l/2)t::..t. Taylor expanding Equation (27) or Equation (28) about the timelevels of their right hand sides show them to be consistent and O(t::..t 2 ) accurate. The numerical stability of Equations (27) and (28) can be readily evaluated for a harmonic force qEn /m = -f2 2xn. Eliminating v from Equations (27) and (28) and setting xn = x 0 eiwt gives the characteristic equation sin 2(wt::..t/2)

= (nt::..t/2) 2

(29)

imply nt::..t ~ 2 for stability. If nt::..t > 2, w must be complex to satisfy Equation (29), indicating that Equations (27) and (28) would lead to an exponentially growing solution rather than the physical bounded oscillations. The Lorentz force integrator approximates dv -=vxn

(30)

dt

by the second order implicit finite difference approximation vn+l/2 - vn-1/2 = (vn+l/2

+ vn-1/2)

x on t::..t/2

(31)

Equation (31) can be simply rearranged to explicitly give vn+ 1/ 2 in terms of vn- 1/ 2 and nn. Its characteristic equation tan(wt::..t/2)

= ±nt::..t/2

(32)

indicates unconditional stability, with error in the gyrofrequency increasing with t::..t. Replacing nt::..t/2 by tan(f2t::..t/2) in Equation (31) corrects this frequency error. The combined integrator The leapfrog electric and implicit Lorentz force integrator combine straightforwardly. For the relativistic Equations (25) and (26) it is more efficient to perform the momentum integration as a three stage process to avoid recomputing 'Y p

pn-1/2 + qEn !::..t/2

J1 + (p- /moc)

2

(=

J1 + (p+ /m c)

(p+ + p-) x Ot::..t/2'Y p+ + qEnt::..t/2

0

(33) 2)

(34) (35) (36)

Non-cartesian systems (eg polars) introduce centripetal acceleration terms into the momentum equation. These are generally handled by transforming particle coordinate to cartesian at positions xn, accelerating and moving to new position xn+l, and then transforming momentum components back to the non-cartesian system at the new position.

James W Eastwood

74

7

Electrostatics

Electrostatic field solvers are a central part of PIC codes and are used in the modelling of guns, collectors and interaction spaces. Finite difference, finite element and spectral field solvers in one, two and three dimensions find use in microwave tube modelling. To illustrate these, we focus here on the widely used two dimensional (2D) solvers. A rectangular structured mesh or element net is usually favoured for PIC applications, largely because it simplifies the problem of tracking particles and leads to a matrix problem for which rapid elliptic solver (RES) packages are available-see Chapter of Hockney and Eastwood (1988). The drawback of the rectangular brick approach is that complex shaped boundaries are more difficult to fit, and in such cases the more flexible triangular element net may be favoured. The electrostatic problem is to solve '\7 ·EE= p where E = -'V.2¢2 + A1 , .Aac/>a) - (2p1 + P2 + Pa)/12

+ (8B;i)¢2 + (8B;k)¢3 + (8M;;)p1 + (8M;i)P2 + (8M;k)Pa

(55)

-bb;

Similar contributions to rows j and

k are given by coefficients of al/ 8¢2 and f)J / 8¢3 ,

In summary, the finite element matrix equation Equation (54) for the electrostatic problem is generated by

L Clear B 2, Loop through elements (a) compute element matrix contribution (b) assembly element contributions

Computer Modelling of Microwave Sources

79

3. Solve for potential (a) decompose B (b) compute source term b (c) solve for

ef>

Item (3) assumes direct sparse matrix solution. As is the case for the finite difference approximation , iterative methods can also be used.

7.5

3-D Electrostatics

Both finite difference approximation and finite element method methods generalise straightforwardly to three dimensions. Rectangular cells (elements) are replaced by rectangular bricks, triangular elements are replaced by triangular prism and/ or tetrahedral elements.

7.6

Solution of Poisson's Equation

Tube modelling requires efficient solution of Poisson's equation, particularly in PIC calculations where many tens of thousands of solutions are required for each simulation. The methods available for solution depend on details of the permeability, region geometry and method of discretisation. The available methods can be broadly classified as follows (Hockney and Eastwood 1988). Mesh relaxation : initial guess potentials are relaxed to a solution by systematically sweeping the mesh and adjusting values. Matrix methods : the discrete equations are treated as a matrix of linear equations and as solved by variations of standard matrix methods. Rapid elliptic solvers (RES) : special techniques based on the fast Fourier transform and cyclic reduction are used to solve certain classes of problems in O(N log N) operations. The fastest RES usually assume a rectangular domain for the computation. They can be applied in more general shaped regions by embedding the general shaped region in the rectangular region and using the capacitance matrix method for computing surface charges needed to obtain the correct potentials on the general shaped region. More generally applicable RES methods use clever ordering strategies for the equations in the sparse matrix to minimise infill. Where complex shaped regions and nonuniform dielectrics are present these are likely to be the fastest method for computing the potential repeatedly for different source charge densities and/or boundary potentials in a given geometry. The most widely used method for solving the matrix problem is iteration -usually some variant of successive over-relaxation (SOR), multigrid or perhaps preconditioned

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80

conjugate gradient methods. (Strictly, these are matrix methods which find exact solutions in a finite number of steps, but in practice are used as iterative methods.) The attraction of these schemes is the simplicity of programming them although they usually require more computational work to reach a solution than their RES counterparts.

8

Magnetostatics

Magnetostatic calculations are used in the design of beam focusing in the gun, the interaction space and the collector. The computational problems are very similar to those in electrostatics, particularly in two dimensional (eg x, y or r, z) cases. The magnetostatic problem is to find the magnetic field B, where

B = 'V x A = µH;

'V x H = j

(56)

In two dimensions, the only components magnetic field and current density to be considered are those in the ignorable coordinate direction (z in cartesian, ()in r, z polars). As in the electrostatic case, a Green's function formulation can be used for the potential.

8.1

2-D Finite Difference Approximation

In the case of Ampere's law, the finite area approximation to the circulation on the mesh shown in Figure 3 evaluates

(57) as

(58)

where µ'tvH/v =(AN -A0 )/!j,,y, etc. and A are z-component vector potential values at the mesh points.

8.2

2-D Finite Elements

Finite element equations for magnetostatics can be obtained using variational or weighted residual methods in much the same manner as for electrostatics. Here the action integral becomes (59) I= /(l'V x Al 2 /2µ + j · S)dr and variations are taken with respect to the nodal amplitudes of the unknown vector potential.

8.3

3-D Finite Differences

The 3-D extension of the magnetostatic problem requires an equation for each component of vector potential. Fields B and H are stored on a lattice displaced half a cell

Computer Modelling of Microwave Sources

81

Figure 6. In the 3-D finite area method, circulations of vector potential and magnetic intensity are computed around intersecting loops. width in all three coordinate directions from the lattice on which A and j are stored. The finite difference equations are then obtained by evaluating Equation (57) and

(60) on intersecting areas as illustrated in Figure 6.

9

Electromagnetic Eigenmodes

Microwave tubes rely on the resonant interaction between electron flow and electromagnetic fields in the interaction space. An important task in tube design is to tailor resonant modes to optimise the desired interaction at the chosen frequency whilst avoiding other unwanted resonances. The electromagnetic eigenvalue problem is the computational analogue of the laboratory 'cold test', where resonant frequencies and mode structures are computed in the absence of the electron flow. There are two approaches to this; either compute the time evolution of electromagnetic fields and use spectral filtering techniques to identify eigenfrequencies and eigenmodes, or transform the equations for the fields to frequency domain and solve the resulting eigenvalue problem. The time domain solution method is described in Section 10 and the frequency domain is treated here. The internal problem relates to resonant structures and waveguides, whereas the external problem is concerned with antennas.

9.1

The Internal Problem

The frequency domain formulation looks for sinusoidal time variation. Assuming variations"" exp(iwt) for Maxwell's equations in vacuo gives equations for the field harmonic amplitudes: iwE/c2 = V' x B; iwB = -V' x E (61) Eliminating E gives the Helmholtz equation V' 2 B -t- (w/c) 2 B = 0

(62)

James W Eastwood

82

Similar equations can be obtained for harmonics of E, A and ./2 dx' vn f dv '::::' -f

J

A

x+>./2

J f.

vn dv

(86)

where = :L 8(x -x;)8( v -v;) is the distribution of superparticles and the sum is taken over all Np superparticles. In PIC methods, the local space averaging is introduced through assignment to the mesh, which also may be interpreted as the cloud of electrons in the superparticles having finite size. The equations of motion which describe the trajectories of the superparticles are discretised in PIC models using the finite difference approximation as described in Section 6.

11.2

Assignment and interpolation

Equation (85) is most easily approximated by finite differences (cf Section 7). This requires the source charge density to be approximated on the finite difference mesh. Likewise, once the potentials and electric fields are known on the mesh, values of fields must

James W Eastwood

88

be interpolated to particle positions in order to compute accelerations. The prescriptions for these operations are the charge assignment and force interpolation functions. The PIC scheme has a dual representation for charge density, via both the superparticle approximation and the mesh approximation

p(x)::::::: L N,q&(x - x;)::::::: LPpb(x - Xp)

(87)

p

Mapping between these approximations can be obtained using weighted residuals. Multiplying Equation (87) by the charge assignment function Wp(x) and integrating over x gives the expression for the charge, Pp at mesh point p in terms of superparticles: (88) If Wp is chosen to be the 'triangle' function, then Equation (88) leads to the area weighting assignment scheme; where charge from a particle at position x; = Xp + x' Lix (Lix = Xp+i - xp) is assigned to mesh points p and p + 1 according to the prescription;

§pp= N,q(l - x'),

bpp+l

= N,qx'

(89)

Fields at particle positions are obtained by interpolation. To avoid 'self-force' oscillations and to assure momentum conservation, the interpolation functions should be set equal to the assignment function

(90)

11.3

Timestep loop

Collecting the above together gives the steps involved in advancing the PIC calculation by one timestep: start

NP particles, positions

x?,

velocities v~- 1 ! 2

assign solve

(cPp+l - 2¢p + c/Jp-1)/ Lix 2 =Pp/lo

interpolate

Fi = qE(x;) = q Lp Wp(x;)Ep

accelerate

V;

move

n+l _ n X; -X;

step counter

n := n

n+l/2 -_ V;n-1/2

Ep

= -(cPp+l -

c/Jp-1)/2Lix

+ r,u "'·At/m

+ V;n+l/2 At l....l.

+ 1 and go to start.

Counting the number of arithmetic operations shows the work for each of these steps to be proportional to the number of particles, apart from the solve step which is

Computer Modelling of Microwave Sources

89

proportional to Np, where N is the number of mesh points and p is the bandwidth. This gives a total operations count: op count= aNp + /3(N)

(91)

where a = constant and f3 is an almost linear function. If instead, forces were computed by direct pairwise sums using Coulomb's law, the operations count would be proporThis large computational saving is the main reason for using the mesh tional to for force calculation.

NJ.

11.4

Properties

PIC simulation models are amenable to the same dispersion and kinetic analysis used for the corresponding differential system. (Hockney and Eastwood 1988, Birdsall and Langdon 1991). Dispersion for the discrete system differs from the continuous in two principal respects; replacing derivatives by difference introduces periodicity in both w and k (with period lengths 21f / 6.t and 21f / 6.x), and sampling the charge density on a mesh (Equation (88)) introduces non-physical Landau resonances. Both of these factors can lead to numerical instability if 6.t or 6.x is too large. Kinetic analysis treats the PIC particles as finite sized particles. Salient points from these analyses are l. The timestep 6.t must be sufficiently small that wp6.t < 2 to ensure stability of the time integration, where wp is the plasma frequency.

2. Finite spatial mesh effects alter dispersion characteristics, particularly if 6.x > An,where An is the debye length. These changes can be minimised by using correction factors in spectral field equation solvers or by using finite element derivations of discrete equations. 3. Alias mode coupling arising from charge density under-sampling can lead to inverse Landau damping instabilities in cold electron beams. Higher order assignment/interpolation functions weaken these instabilities. 4. Assignment, interpolation and finite difference errors lead to an effective superparticle width a > 6.x. 5. If the number of particles in length a is large("" 10), collision effects remain small.

12

Multidimensional Electrostatic PIC

Multidimensional electrostatic PIC models are used to simulate the interaction of the electron flow with the electric and magnetic fields in the electron gun and collector regions of tubes. They are straightforward generalisations of the 1-D scheme: l. The equation of motion is replaced by its vector form. If applied magnetic fields

are present then the Lorentz force integrator (Section 6) is used.

James W Eastwood

90 4

3 A2

-·--

A1

:x

A3

A4

2

Figure 9. The fraction of charge assigned from a particle at position x to nodes 1-4 1:s proportional to areas A1-A4 in the area weighting scheme. 2. Poisson's equation is replaced by its multidimensional counterparts. The control volume method can be used to obtain the relevant equations in cartesian, polars and other coordinate systems (Section 7). 3. Assignment and interpolation schemes are replaced by the product of 1-D functions. For example, the 2-D area weighting function takes the form

(92) in coordinates (ui, u 2 ). The fraction of charge assigned from a particle position x = (ui, u 2 ) to nodes 1-4 proportional areas Al-A4 as shown in Figure 9, where (u 1 ,u 2 ) are (x,y), (r,8) or (r,z) for cartesian and polars. w(u) is the 1-D assignment function. Complex shaped regions can be treated by embedding in a rectangular region and using the capacitance matrix method on Poisson's equation (Section 7). Particle boundary conditions on the embedded boundary are most readily implemented by using a masking array to tag cells as inside or outside the active computational domain. An alternative approach to complex regions is to build them up by a union of blocks; this method is to be preferred in 3-D, where the extra computations and storage associated with the masked volume of mesh can become a large computational burden.

13 13.1

Electromagnetic PIC Model equations

The appropriate model for the interaction space is the high frequency, high power, Maxwell-Vlasov model. High frequency means that the characteristic periods are less than the transit time of light across the system, and high power means that the relativistic 'Y > 1; these imply that displacement currents and relativistic effects must be taken into account.

Computer Modelling of Microwave Sources

91

The evolution of the fields is described by the time dependent Maxwell equations

oB = ot

oD='VxH-j

-\1 x E

-

ot

(93)

subject to the initial conditions

(94)

\l·D=p where the constitutive relationships relate D to E and B to H

D=EE

(95)

B=µH.

Motion of the electron gas is described by the relativistic Vlasov equation

of P of of -+-·-+F·-=0

(96)

p = F (= q(E + v

(97)

ot

mox

op

whose characteristics are x = v

and p

= /mov

/

=

x B)

J1 + (p/moc)

2

(98)

The model is completed by volume material properties to specify E and µ, and surface models to describe the emission and absorption of both electrons and electromagnetic radiation.

13.2

Timestep loop

The timestep loop of electromagnetic PIC codes is similar to that of electrostatic PIC; the principal differences being that the full Maxwell equations are now advanced using a finite difference time domain or finite element time domain scheme (Section 10), and currents rather than charges are assigned to the nodes of the finite difference approximation or finite element method lattices: start:

n '/\T pos1't'10ns xi, moment a Pin+l/2 , z. = 1, "P

move assign current

jn+l/2

advance fields

nn --+ nn+i, sn+i/2 --+ sn+a/2

interpolate forces n+l/2

--+ n+3/2

update momenta

Pi

step counter

n := n+ I

go to start

P

James W Eastwood

92

The electromagnetic PIC model carries a much richer set of physical interactions that its electrostatic counterpart. It also has more ways in which non-physical behaviour can be realised if care is not exercised. In addition to the possible problems discussed above for the electrostatic case, problems can arise with charge conservation, excessive numerical noise, numerical Cerenkov instability, boundary conditions and geometrical effects. Computational costs can rapidly rise to make some 3-D PIC calculations practicable only if modern massively parallel computers are used efficiently. We address some of these issues further in the remainder of this chapter.

13.3

Charge conservation

The differential Maxwell-Vlasov equation conserves charge; taking the divergence of Ampere's Law and the time derivative of Gauss law gives

~V'·D=-Y'·j=ap

at

at

(99)

The two expressions for the time derivative of V' · D can be shown to be equal from the first moment of Vlasov's equation. A consequence of Equation {99) is that if Gauss's Law is satisfied initially, then the evolution of the fields given by the two time dependent equations will ensure that it is satisfied at later times. In PIC codes, the charge conservation property is not necessarily preserved. Particle positions and currents are updated using the momenta, and the current is used in a discrete approximation to Ampere's Law to update the displacement field D. The new particle positions can be used to compute a mesh charge density, and in general the discrete approximation to Gauss's law

Y'·D-p=c#O

(100)

Three methods have been used to eliminate the charge conservation error c. 1. Poisson's equation is solved at each step to compute the correction required to the longitudinal part of D to make c = 0 (Boris 1971).

2. A 'pseudocurrent' term is added to Ampere's equation proportional to V'c, this diffuses the error away on a timescale determined by the numerical diffusion coefficient (Marder 1987) 3. The discrete field equations, current and charge assignment schemes are chosen to ensure that c = 0 (Eastwood 1991). The third solution to the charge conservation property has the advantages that it does not require the computation or storage of charge density, and requires no solution of Poisson's equation in the timestep loop-consequently it is computationally faster and less memory demanding. In addition, it allows PIC calculations to be run more efficiently on parallel computers. Both finite difference approximation and finite element method based electromagnetic PIC schemes can be defined with the charge conserving property, although it is more straightforward to derive such schemes using the variational finite element method outlined in the next section.

Computer Modelling of Microwave Sources

14 14.1

93

Virtual Particle EM-PIC Physical model

The variational finite element derivation used in the Virtual Particle (VP) method leads to numerical schemes which automatically assure charge conservation, avoid numerical Cerenkov instabilities, provide a simple method for controlling numerical noise and which can handle complex geometries (Eastwood et al 1995). The VP method extends ideas we met earlier. The distribution function is represented by a set of sample points (ie 'superparticles')

J(x, p, t) =

L N,8(x -

x;(t))8(p - p;(t))

(101)

where (x;, p;), i = 1, ... NP, are the coordinates of the NP superparticles, each of mass M = N,m0 and charge Q = N,q, and the Maxwell-Vlasov set may be written in terms of the action integral

I=

I dtdT (

D·E-B·H ) - p¢ + j ·A 2

J

Mc2 ~ -::y;-dt

(102)

General dielectric and magnetic media are added to the model by introducing polarisation, P, and magnetisation, M, terms into the action integral

I = ... +

I dtdT(M . B +

p . E)

(103)

and substituting for M and P from the resulting evolutionary equation using their constitutive relationships. The non-lossy part gives the dielectric function relating D to E, and the lossy parts give the magnetisation and conduction currents, k-

D=tE,

j1 = -Y' x ,6B+O"E

(104)

The magnetisation current term can be used to control numerical noise; taking ,6 to give a decay time of ""' 100 signal periods greatly reduces noise and thence the number of particles needed, whilst having little effect on the the signal. (This may not be true for high Q cavities.) The treatment of field and particle boundary conditions is discussed further below in sections 14.3 and 14.4. The discrete equations are obtained using a variational finite element method, with fields in the general geometry version being represented by their tensor components. Variations with respect to potentials lead to discrete forms of Maxwell's equations and prescriptions for charge and current assignment. Current is assigned from 'virtual particles' placed at points specially interpolated between positions at successive time levels, a procedure which automatically leads to charge conservation. In the following illustration, we use only the lowest order conforming elements. In general curvilinear coordinates (x1,

x2 , x3 ) the Action integral is (105)

James W Eastwood

94

where

E _ _ !!i_ k8xk di=

_ 8Ak

bi_ ijk8Ak E 8xi

(106)

H; = - 1-gijbi

(107)

at '

E0 Vggij Ej,

-

µo,/9

The superparticle current is given by the sum over all particles: I'= Lqp8(x 1 p

14.2

-

x!)8(x 2

-

x;)8(x 3 - x;)xi

(108)

Numerical scheme

Field Equations Treating Action I as a function of the finite element approximations for ¢ and A; then taking variations with respect to the nodal amplitudes gives the element contributions to the discrete approximations of Maxwell's equations. In general, the form of the assembled equations depends on the number of elements and boundaries adjacent to the node in question. However, for an internal node where the element net is topologically equivalent to a cubic lattice, the assembled equations take the familiar form: Bib'= -eiik8iEk, 8;b; = 0 (109) 8idi = eiik8iHk - I', 8;d; = Q

(110)

where the symbol 8 denotes a centred difference. These equations in tensor quantities bi,di, Ek,Hk,li and Q take the same form as the Yee finite difference time domain scheme (Yee 1966) for cartesian field components, but are identical in any coordinate system. This provides great simplification of the simulation program in general geometry. Constitutive Equations Geometrical and material (permeability and permittivity) information appears only in the constitutive relations. The weak approximations to Equation (107), with lumped mass matrices give the simplest explicit expressions for E; and H; of the form: (111)

Elements of the symmetric tensors G5 and GIT are sparse matrices. More general permittivities and permeabilities are handled by replacing Eo and µ 0 by scalar or tensor functions in the computation of G5 and GIT, respectively. Assignment The formulae for current and charge assignment arising from the variational formulation are also coordinate independent. Let the finite element test function approximations to the potentials be ¢ = .oOr Dll-o

Figure 1.3. Normalised waveguide attenuation as a function of waveguide dimensions (Quine 1g68). the cross-section of the waveguide depending on the propagating mode. TEmomode:

Pmax(kW)

TEonmode:

Pmax(kW)

TEmnmode:

TEumode:

=

6.63

X

l0- 7 abm 2 1- (

6.63

X

10- 7 abn2

ab f3mn 2 ko

1- (

~)

~~o)

2

2

2 + (n) 21 [(12.) 2 + (1L) 21 [(m) a 1i. J n m J IE 12

ZoEomEon

Zo

max

IE max

(1.3)

(1.4)

1Emaxl 2

abfJu[G)2+G)2][b2+a2] 2 ko

IEmaxl 2

12

(1.5)

(1.6)

At normal temperature and pressure, a representative figure for Emax which corresponds to the breakdown electric field of dry air is 2.9x 106 V /m. Using this and Equations (1.3) to (1.6) the peak power handling capacity for a rectangular guide is readily determined. Figure l.5a shows the variation of Pmax versus frequency for the

127

Modes and Mode Conversion in Microwave Devices

0.16 E

cg c

0 14

0 12

0

~ 010

~

2 rk c,nm n a cos ncf>

E,

Zh.nmH~

_ J/lnmPnm J' ( Pnmr )e-Jµ"""{ COS n 2 ak c,nm n a sin n

E~

-zh,nmH,.

,. ,{-sinncl> -Jn/lnmJ - - (p,..,.r) - - e ..1 '"m rk~.nm n a ' cos n

f3nm

[kg-(P:mrr2

[kg - ( p: m

zh,nm

n

stnn

10-2

~

;

~

E w

~~ a_

10•3

_T~,,_

- -- - - - - -

TE,,

io-4

1~~---'"--"~~~'-'-~7.....,.,10~.~-=--~~~~7~1~

k,a

Figure 1.8. Peak power handling capability of various modes in circular waveguide as function of waveguide dimensions {Bhartia and Bahl 1984). 1.4.1

Mode conversion in oversized rectangular waveguides

Oversized rectangular waveguides are well suited for many system applications such as lower hybrid plasma current drive and millimetre wave plasma diagnostics and spectroscopy, where the waveguide runs are a few tens of meters. They have low attenuation, reasonable manufacturing cost, simplicity in component construction, and are easy to couple with standard waveguides. For the dominant T Ero mode propagation in standard rectangular waveguide, the transverse dimensions are usually selected such that >.. 0 /2 < a < ).. 0 and b < ).. 0 /2, and typically one might have a~ >.. 0 /./2 and b ~ a/2. Peak power handling capability and the attenuation constant of the rectangular copper waveguide propagating the T Ero mode can be improved by increasing waveguide transverse dimensions (Section 1.2) but now there exists the danger of unintentional mode conversion at tilts (angle ¢>, offsets (S) and cross-sectional dimension changes (~T). Expressions for the corresponding normalised amplitude coupling coefficients C between the T E 10 mode and various parasitic modes in rectangular waveguide are summarised in Table 1.3. Here unit incident power in the T Ero mode and spurious modes terminated in matched loads are assumed. The subscripts t, o and d represent tilt, offset and abrupt dimension change discontinuities. The power coupled to the parasitic modes is equal to C 2 . The knowledge of theoretical coupling coefficients is very useful for component design and in establishing dimensional tolerances.

132

Manfred Thumm

Spurious Modes

Coupling Coefficient C, (Eplane) = -4/2 ( - b ) ( - ) n2 Ao .,,

-·~

(T = b)

(T=a)

l6m 2 ( X-a ) ( - ) (m2-1) o.,,

C0 ( E plane) =

. ( n'TTS) s)sm b ( f2 b S

C, (H plane)=

(T = b)

~·~

(T =a)

m

s)

4m (2 -1 a

2/2 s

. [ sm

Cd(Eplane) = b + Llb (T= b, ilT= Llb)

TEmo modes for even m only n'TTl:lb

( b + Ll b)

]

Composite TE 1,,/TM 1 ,, for even n only

n'!Tilb

b

+ Llb ----.----

2mila/a

Cd (H plane)=

m2 - ( 1

(T=a,AT=Aa)

TE mo for even .m only

Composite TE 1,,/TM 1 ,, for all n

!!!!__

b

C0 (Hplane) =

Composite TE 1 ,,/TM 1,, foroddnonly

+Lla) -

_

l

- T ·---

--

__J__ _ _ _

~---

TEmo for odd m only

2

a

Table 1.3. Coupling coefficients for tilts, offsets and abrupt dimension changes between the T E 10 mode and various parasitic modes in the rectangular waveguides (Bhartia and Bahl 1984). (

. Perturbation Method: Numerical integration of the coupled-wave equations

E

(generalized telegrapher's equations). 2.

E ~ 1 and L, < >. (axisymmetric perturbations) Scattering Matrix Method: Mode matching employing the decomposition of the perturbed waveguide in short homogeneous sections. Inclusion of cutoff-modes (evanescent modes).

3.

f. ~ 1 and L, < >. (non-axisymmetric perturbations) Finite Element (FE) - Finite Difference (FD) - and Finite Integration Method: These methods, though universal, can be used in practice only to design moderately oversized mode converters. --

- -- -- -

r

a

i>.

a

i

Mode Spectrum B

Mode Spectrum A

Figure 2.1. Different types of waveguide perturbations.

2.3

Adiabatic mode converters

Non-degenerate waveguide modes (with different phase constants /31 #- /32) can be converted one to another by a specific adiabatic, that means slow change of the waveguide cross-section or wall impedance. In this case, the coupling parameters C 1 ,;, (note that c1,; are the coupling coefficients) for the couplings to all possible parasitic modes i are small: C1,i

I 2c1, 1

I

= I /31 - /3; « 1

(2.1)

thus the conversion efficiency at the design frequency is T/o = 1 - O(C~,;).

Examples are:

(2.2)

Modes and Mode Conversion in Microwave Devices

137

1. T E 10 (rectangular) - T En (circular) converter

T E 10 (rectangular) - T E 01 (circular) converter

2. Waveguide cross-section tapers: rectangular waveguide, linear dimensions a 1, b1 ---+ a2, b2 circular waveguide, radius a 1 ---+ a2 3. Surface-impedance converters (corrugated waveguide) T En - H En converter T Mn - H E 11 converter 2.3.1

TE 10 to TE 11 and TE10 to TE01 converter

The T E 10 (rectangular) to T En (circular) transducer is a simple device where the rectangular waveguide cross-section with its straight field lines is slowly changed to a circular one with all the field lines ending perpendicular to the wall (Section 1, or lecture 1). The geometries of TE 10 (rectangular) to TE01 (circular) converters are much more complicated. Here we present the principles of three different types of this transducer, the sector-type (King-type, Figure 2.2a) the twin-sector-type (Southworth-type, Figure 2.2b) and the Marie-type mode converter (Figure 2.3). The mode conversion in a linear sector-type transformer and the variation of the cutoff frequencies of the first three modes along a Marie-type transformer are plotted in Figures 2.4a,b, respectively. Herc L is the converter length, k 0 the free-space wavenumber and a the waveguide radius.

Figure 2.2. Rectangular T E 10 to circular T E 01 mode converter (a) sector-type {Kingtype) and (b) twin-type (Southworth-type) {Bhartia and Bahl 1984).

Manfred Thumm

138

-

Prnpective view of tun•ducH "'lnd•el. Pirr1pecfr•• l'iew of 1ution 1 of m~ndrtl.

b,

~$La, c$ ';~l.~

Penpectiwe •iew of uctio" 2 of m1ndrel. -PeupecliYeYiewoluctio"} ol mand

75 0

~ 40 ~sectoon 1--~

H=1

r,,

~1

section2---~sectoon3-'->l

Figure 2.4. (a) Mode conversion (relative power) in a linear sector-type transducer and (b) variation of the cutoff frequencies of the first three modes along a Marie transducer. Abrupt Discontinuities

~L

~ Output D,

Input 0 1

(a) Straight Taper

Output D,

Input 0 1

(bl Variable Taper

Figure 2.5. Waveguide tapers and

(2.4)

140

Manfred Thumm

The corresponding expression for T E 02 - T E 01 coupling in a straight circular T E 01 mode taper with input radius a 1 , where the T E 02 mode is cut-off, and output radius a2 is lei= 0.568a2(a~o~ ai) (2.5) From (2.3) and (2.4), it can be shown that the coupling to the LSE12 mode in an E-plane taper is about 8.5dB stronger than the coupling to the T E 30 tnode in an H-plane taper for comparable dimensions. Solymar also found that the value of the coupling coefficient rapidly decreases with the order of the spurious mode and that the amplitudes of the next higher-order modes are approximately lOdB lower than the values given by (2.3), (2.4) and (2.5). As the coupling coefficients are inversely proportional to the taper length, doubling L reduces mode conversion loss by 6dB. Very low unintentional mode conversion from the propagating mode to. unwanted parasitic modes is obtained by employing relatively short nonlinear tapers with a gradual change of the cone angle. The synthesis of non-linear taper contours for highly overmoded transmission systems has been studied by several workers. A comprehensive summary can be found in the work of Mobius and Thumm (1993). The properties of several gyrotron uptapers are listed in Table 2.2 together with experimental values obtained by measurements of the gyrotron output mode content using wavenumber spectrometers. Mode

TE.n TEo2

TEoJ

TEw.4

Frequency (GHz)

39 70 140 140

Input Diameter (mm)

20.0 12.2 8.5 17.4

Output Diameter (mm)

62.6 63.5 70.0 40.0

Length (mm)

300 570 750 130

Parasitic mode level (dB) Theoretical Experimental Synthesis -27 -20 -20 -19

Analysis -28 -19 -20 -19

-(24±2) ..(15±2) ..(16±2) -{19±2)

Table 2.2. Geometrical parameters and calculated and measured levels of parasitic modes at the output of optimised non-linear gyrotron tapers.

The syntheses employed a near-optimum mode conversion distribution including reconversion and reflection. The synthesised contours were subsequently analysed by numerically solving the generalised telegraphist's equations. The taper input contours are matched to the flare angle of the resonator output. The spurious mode levels predicted by the synthesis and analysis codes were always in good agreement. The high-power experimental results are slightly higher, but this is not surprising since insulation gaps and the RF output window are additional sources of unwanted parasitic modes. The excitation of the parasitic modes T M 10 ,3 and T M10 ,4 together with the T E 10 ,4 transmission loss along the T E 10 ,4 taper and the optimum taper contour are plotted in Figure 2.6. It is evident that the effect of reconversion into the operating T E 10 ,4 mode is quite important. Conversion of the operating gyrotron mode to unwanted spurious modes at discontinuities of an ordinary gyrotron cavity can be reduced substantially by introducing

141

Modes and Mode Conversion in Microwave Devices

R[mm]~-------------, 30 20

10

0

-10

-20

-30

m-

-:I ::!:!.

.... 0

TE10,4

-10 -20

.....

I-

·;;. -30 0

:I !:::. .... N

Cl)

· .....

-1

_.·-·-· ......™103 ™10.4

-40

-2 -3 -4

m-

-u:r ::!:!.

.... 0

!:::.

.... N

Cl)

-5

-50 0

100

50

z(mm)

Figure 2.6. Optimum contour of a 17.4-40.0mm T Ern,4 uptaper together with the excitation of the T M 10 ,3 and T M10,4 modes and the T Ern,4 transmission loss along the taper. smooth transition regions between the individual parts of the open resonators. Cavity and adjacent tapered output waveguide should be optimised as one single unit.

2.3.3

Surface-impedance converters in corrugated waveguide

In cases where the waveguide diameter D = 2a becomes fairly large compared to the free-space wavelength .A 0 , and when the period p of the annular slots is assumed small compared with the guide wavelength, no space harmonics need to be considered and a circumferentially corrugated waveguide wall may be modelled well by an effective anisotropic surface reactance (wall impedance concept: WIC) where the circumferential component is zero Zt/> = 0 and the longitudinal component is finite: Zz = iZ0 Z (for the co-ordinate system, see Figure 2.7). Here Z is a normalised reactance and Z 0 = µ0/Eo is the impedance of free space. For k 0 a » 1 the surface reactance Z may well be approximated as

J

z_w -

tan(k 0 d)

p 1 + tan(k0 d)

2koa

(2.6)

Manfred Thumm

142

-

TE 11

mode converter

-----

Figure 2. 7. Principle of the T En - H En mode converter (cross-sectional view) and polar co-ordinated system (r, ¢, z ). where d and ware the mechanical slot depth and width, respectively. The factor w/p reflects the fact that the finite width of the corrugation teeth reduces the reactance, while the tan(k 0 d) dependence indicates that the annular slots act like short-circuited transmission lines. The longitudinal surface reactance may be reduced further if the slots and teeth are rounded (Clarricoats and Olver 1975). The propagation behaviour of HEmn and EHmn hybrid modes (m > 0) is described by the characteristic (or dispersion) equation for the roots Xmn = Kmna

1

Zkoa

=

Jm(Xmn) x;,n Xmnl:,.(Xmn) 1

[m~ (3;,,n 2

(XmnJ:n(Xmn)) 2 ] lm(Xmn)

( 2 .7)

where Kmn is the radial separation constant (the transverse wavenumber), and f3mn = (k5 - K!n) 1l 2 is the longitudinal propagation constant in the waveguide. lm(X) is a Bessel function of the first kind and order m.

T Eon modes are not affected by the corrugation. The numerical solution of the characteristic Equation (7) for a corrugated waveguide with 2a = 27. 79mm at f = 70GHz is plotted in Figure 2.8. The eigenvalues Kmna of normal H Emn and EHmn hybrid modes are monotonically varying functions of the relative depth d/(>. 0 /4). For d/(>. 0 /4) = 0, 2, 4 ... (Z = 0) the eigenvalues of hybrid modes are equal to the eigenvalues of the modes of the smooth waveguide (it is assumed that there are no ohmic losses in the waveguide). Hybrid modes consist of linear combinations of T Emn and T Mmn field components which are determined by the hybrid factor: Arnn= The balanced hybrid condition, roots

Jm(Xmn) . XmnJ:,.(Xmn)

A2 = 1, (at d/(>. 0 /4)

corresponding to HE modes, and

A= +1, A=

-1,

(2.8)

= 1, 3, 5 ... with Z = oo has two

(2.9)

(2.10)

corresponding to EH modes. Any waveguide possessing a diameter D ~ ).. 0 and a wall with high surface reactance can propagate such a hybrid mode (corrugated, dielectriccoated metal, and dielectric waveguides. Here IAI = oo corresponds to TE modes and IAI = 0 to TM modes.

Modes and Mode Conversion in Microwave Devices

9

-----EH23 HE13

8 "3

llic:

143

EH13

7 6

5 4 3

HE11

2

TE11

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 d/()/4) Figure 2.8. Eigenvalues Xmn of the characteristic equation versus normalised electrical slot depth d/ (>.. 0 / 4) for various modes in a circumferentially corrugated waveguide with inner radius a=13.9mm at 70GHz {k 0 a

= 20.39).

Adiabatic T Eu and H Eu (T Mu to H Eu) mode conversion is achieved in a straight corrugated waveguide section in which the electrical depth of the slots decreases (increases) slowly from an initial value of ).. 0 /2 to a final slot depth of approximately ).. 0 /4 (see Figure 2. 7) . The varying slot depth allows coupling only to higher order modes with identical azimuthal mode number and unwanted mode conversion is associated primarily with the nearby TMu to EH12 (TEu to EHu and TE12 to EH!2) branch. The EH12 hybrid mode has a radiation pattern which is entirely cross-polarised in the 45° plane. If excited it degrades seriously the cross-polarisation performance of the transducer output. Therefore the converter must be long enough to suppress the excitation of this unwanted spurious mode. Computer-aided optimisation of converter length, shape of corrugations and slot depth profile along the converters has been performed employing a scattering matrix code (Thumm 1986). The slot depth changes most slowly at the converter input where coupling to the unwanted mode branches is strongest. Conversion efficiencies in the range of 99% were achieved which was verified by low-power and high-power experiments.

Manfred Thumm

144

2.4

Conversion of degenerate modes

In the case of mode degeneracy, which means that the two coupled modes have the same phase constant (3 = (31 = (32 , we have strong coupling and the coupling parameter C 1,2 » 1, whereas the coupling parameters of these two main modes to other parasitic modes fulfil Cm,n « 1 form= 1, 2 and n f- 1, 2. For constant coupling (c 1,2 (z) = c0 = const) the amplitudes of the two converter modes are given by: 2.12 cos( c0 z) exp( i(3z) i sin(c0 z) exp(i(3z)

(2.11) (2.12)

and the conversion efficiency is in the order of (2.13) Power is periodically coupled from mode 1 to mode 2 and vice versa. This feature is demonstrated in Figure 2.9. Here we shortly describe two types of such converters: • T E 01 to T Mu converter • T Eu and T Mu circular polarizers.

\

"'

\

u

-e ~

"'E

as'--~-'-----~---+

0

I

o~

I

I

rt

2

3rt

T

c,,. L[rod I F'igure 2.9. Normalised wave amplitudes and phase factors versus integrated coupling strength.

2.4.1

TE 0 i-TM11 mode converter

A circular waveguide is bent with an optimised curvature distribution at a proper angle to convert virtually all of the T E 01 power to T Mu, which is polarised perpendicularly

()c

Modes and Mode Conversion in Microwave Devices

145

to the plane of the bend. Power is continuously coupled from T E 01 to T Mn along the converter because T E 01 and T Mn are degenerate modes ( J~ = -J1 ). The conversion angle is given by (} _ P01Ao (2.14) c - 2V2a where p~ 1 = 3.8317 is the first root of the Bessel function J1 . The bend must be long enough to avoid unintentional conversion to parasitic modes, but ought to be as short as possible to minimise ohmic (wall) losses from the high loss TM11 mode. TE01 -TM11 mode converters at 70 and 140GHz with a= 13.9mm were optimised by numerical integration of the forward coupled-wave equations for six coupled modes: TE01 , TMu, TE 11 , TE12, TE21 and TM21 (Kumric and Thumm 1986). Ellipticity coupling (b.m = 2) was included in coupling matrices. Due to multimode coupling, the conversion angle Be depends on the curvature distribution and differs by a small amount from the value calculated with eq. (2.14). In both cases the lowest spurious mode level together with highest power transmission efficiency (shortest arc length) where achieved using a sinusoidal curvature distribution instead of constant curvature. The calculated and experimental data of the two converters are listed in Table 2.3. Frequency

Arc Length

Angle it

70GHz 140GHz

l.31 m 2.52 m

24.0° 12.1°

Transmitted Power Efficiency calculated measured 97.9% (97.6 ± 0.4)% 98.5% 95.2% 95.5% (95.0 ± 1.0)%

Table 2.3. Geometrical data, calculated and measured conversion efficiencies of bent T Ea 1 to T Mu mode converters at 70 and 140GHz.

2.4.2

T E 11 and T M 11 circular polarisers

Mode coupling in elliptically deformed oversized circular waveguides was studied theoretically and experimentally by Thumm and Kumric (1989) in order to optimize T E 11 and T M 11 polarisers for the lMW /70GHz long-pulse (3s) ECRH systems of the W7-AS stellarator at IPP Garching. The polarisation converters essentially consist of smoothwall circular waveguides (inner diameter = 27.79mm) which are gradually squeezed (a(, z) = a 0 [l + b2 (z) sin(2ef>)]). A sine-squared function of the length co-ordinate is used to get an almost elliptical cross-section in the middle and circular cross-sections at both ends [J2 (z) = b!f'ax sin 2 (7rz/ L )]. Starting from linear polarisation oriented 45° with respect to the major axis of the ellipse (Figure 2.10), the magnitude of the two perpendicularly polarised components E~ and E: will be equal when the coupling integral of the deformation satisfies: (2.15) where ca,x(z) is the relevant coupling coefficient and the integral covers the length L of the distorted waveguide section. Since the components and are both oriented midway

Manfred Thumm

146

Figure 2.10. Elliptical deformation a 2 = ao 2 oriented to couple ordinary E~ (vertical) and cross-polarised E~ (horizontal) radial electric field components and the components and with the greatest difference in propagation constants (phase velocities).

E:

E:

between the maxima and minima of the ellipse, they must travel with identical phase velocities (degenerate modes with go phase difference). Thus an incident T Eu or T Mu mode with fixed (linear) polarisation can be converted to circular polarisation and vice versa. The condition (2.15) is equivalent to the statement that there must be a 7r/2 relative phase shift for the independent polarisations and (Figure 2.lO)(Doane 1g86). 0

E:

~

Cl:I

31::

Q

t:l.

-

0.83

1.0

....

E:

1.65

.... ....

0.8

~ Q 0.6

e:

Cl:S 0.4 i:: :§ 0.2 u ~ 0

Horizontal TM11

.... t- .... -+' 0

/

)"':

1.5 0.5 1.0 diameter squeeze wmax (mm)

2.0

Figure 2.11. Measured fractional power in vertical and horizontal T Mu as a function of the maximum diameter squeeze (b.Dmax (70GHz, D=27. 79mm, L=2m, vertical input). Figure 2.11 shows the dependence of the measured power in vertical and horizontal T Mu mode on the maximal diameter squeeze (b..Dmax = 2a 2 ,max at the centre of the deformed waveguide section (z = L/2). Circular polarisation was achieved with (b..D 1 = 0.83mm. To rotate the polarisation by goo requires a doubling of the distortion integral (2.15) to 7r /2. This corresponds to a phase shift of 180° between the components parallel and perpendicular to the major axis of the ellipse. Figure 2.11 shows that the polarisation rotation was found for (b..D 2 = l.65mm. The dependence of the measured power in vertical and horizontal T Eu mode on ( b..Dmax is plotted in Figure 2.12. As expected, because the coupling constant for the T Eu mode is 2.4 times smaller than that for the T Mu mode, circular polarisation was measured at (b..D 1 = i.g8mm. By

Modes and Mode Conversion in Microwave Devices

~

0.8

~

0.6

Q.,

e: Q

1.98

1.0

Q

~ 0.4

Vertical TE11 · Horizontal TE11

1,..-;~-t-

~ 0.2 u

~

0

147

0

0.5 1.0 1.5 diameter squeeze till max (mm)

2.0

Figure 2.12. Measured fractional power in vertical and horizontal T En as a function of the maximum diameter squeeze (!::i.Dmax {'lOGHz, D=27. 79mm, L=2m, vertical input).

rotating the linear input polarisation, arbitrary elliptical polarisation states can be generated.

2.5

Mode converters with periodically perturbed walls

In overmoded circular waveguides a selective transformation of one specific mode with azimuthal mode index m 1 into another mode with azimuthal index m 2 can be achieved by means of a periodic structure of the inner waveguide wall (Figure 2.15):

a(z, .. = 720mm.

because the T M 21 mode is only weakly coupled. The T E 01 mode is also strongly coupled to the unwanted TE 12 mode because b./1(01, 11) is quite close to b./1(01, 12) and c(Ol, 12) > c(Ol, 11). In addition, the TE21 wave is tightly coupled to the generated T Eu mode (large value of c(ll, 21) / b./1(11, 21)). In order to get a conversion efficiency of approximately 93%, it turned out to be necessary to increase the number N of geometrical periods with simple cosine-shaped perturbation to 10 (Thumm 1984). The overall length of this converter is L = 3.12m ((ao = 13.9m, Eu= 0.049, E12 = E21 = O; b./1(01, 11) = 20.15m- 1 ). The results of numerical computations and the geometrical data of various 70GHz = 8) are given in Thumm (1986). The conversion efficiency of a simple cosine-shaped transducer is quite low (TJo = 86.2%) and the levels of parasitic modes are rather high (Figure 2.14a). The multimode calculations show that the conversion efficiency can be improved to T)o= 94.0% by a 1.6% increase in the perturbation period (b./1w = 19.87m- 1 ). This reduces the unwanted T E 12 mode level and causes a continuous re-matching of the required phase difference between the T E 01 and T E 11 modes (Thumm 1984, 1986), resulting in a reduction of the remaining T E 01 mode content from 6.9% to 0.9% (Figure 2.14b). The power levels of undesired modes are further reduced (Figure 2.14c) by superimposing

TE01 to TEu mode transducers with eight geometrical periods (N

Manfred Thumm

152

.... - - ..., .. .,.fl! • 0.10

....·

(al

~

D.10

i o •o

""o.Jo

i

~

0.DI 0.06

- - - - - - - - - - - - - - - -...~,.:.· .. ""'·= ::~

(c)

io.•D

... 0.,0

Figure 2.14. Calculated fractional power in each mode along 70GHz TE01 to TEu converters (a 0 = 13.gmm, 8 geometrical periods) with: (a) simple cosine-shaped perturbation, (b) as (a) but slightly changed geometrical period, and (c) slightly changed geometrical period and superposition of three different, phase matched periodic perturbations.

two additional, phase-matched small perturbations, whose periods correspond to the beat wavelengths between T E 01 and T E 12 (~/3w(Ol, 12) = 24.84m- 1 ) and between T Eu and T E 21 (~/3w(ll, 21) = 9.94m- 1 ). In this case, eight main T E 0 i/T En-periods correspond to ten T E 0 i/T E 12 periods and to four T Eu /T E 21 periods: 8~/Jw(Ol, 11)

=

10~/Jw(Ol, 12)

= 4~/Jw(ll, 21)).

The theoretical conversion efficiency of this improved T E 01 to T Eu mode converter is = 97.4% (including ohmic attenuation of 1.4% in Cu-waveguide). The bandwidth factor for a frequency deviation of 0.3% is 0.997 (0.3% reduction of the conversion efficiency). Measurements with a wavenumber spectrometer give an experimental efficiency of ry0 (exp) = 97.2 ± 0.5 % in excellent agreement with the predicted value. T/o

153

Modes and Mode Conversion in Microwave Devices

0.99 ,, =0.995 x 0.98 x 0.99= 0.965

28 GHz

70 GHz

(b)

(a)

Figure 2.15. (a) Sequence of mode conversionTE02-TE0 1-TEn-HE11 measured at high power (200kW, 1ms) for 28 and 70GHz by thermographic pattern analysers. (b} Principle of periodically perturbed waveguide wall and circumferentially corrugated wall mode converters. The measured efficiencies are given. 2.5.3

Step-type coupling

For moderately oversized waveguides, Vinogradov and Denisov (1990) have developed the principle of very short waveguide mode converters with step-type coupling described by the coupling coefficient

c(z) = {

Co

Os;z El. is possible with fast waves, but irrelevant because of the significant contribution of the axial wave electric field) or any type of EH and HE hybrid waves. This importance of the axial electric fields has been overlooked in some earlier papers, including Iatrou and Vorrivoridis (1991), Vomvoridis and Iatrou (1991) and Vomvoridis and Harnbakis (1991), in which various aspects of the interaction are addressed by keeping the ratios El./Bl. and w/k as parameters independent of each other and the theory is complemented with examples employing TM (or TM-like hybrid) waves: In these papers, the theoretical part is only reliable, when adjusted for TM waves (that is, with El./Bl. = w/k, as has already been done in Section 2, where the quantity D 3 is used from Vomvoridis and Iatrou (1991)). It is noted that the analysis presented so far has had as its objective the presentation of the basic physics of the interaction. For this reason, in developing Equations (2), (8) and (9) a short time t (from the entry time of each individual electron into the system) has been assumed. This has in effect permitted the axial velocity to be a constant, Vz = v0 , that is to omit the effects of the axial magnetic force v l. x Bl., which is perfectly acceptable as long as the transverse velocity, which is initially equal to zero, remains small. For a complete analysis of the interaction, including the calculation of the saturation level and of the non-linear electronic efficiency, this force cannot be ignored. The equations of motion for test electrons under the action of both transverse and axial wave fields have been developed by Frantzeskakis et al (1994), on the assumption that the transverse excursion of each electron remains small enough, so that the linear term suffices to give the transverse variation of the fields. [Also, the same reference defines the helical co- ordinate system e 1 , e 2 , e 3 .] They consist of a system of five (for the three momentum components and the two transverse displacements) non-linear (as is typical of relativistic dynamics) equations (Equations. (3.19) in the reference). Approximate expressions are obtained for the transverse displacements, thus reducing the problem to a system of three equations. This system is seen to possess two constants of the motion, and therefore it is integrable in closed form. At this point the first constant will be used. (The second one will be introduced and used in the following sections.) It reads

(10) where u represents the dimensionless momentum (momentum divided by me), hence 11 3 is its axial component, while n = kc/w is the refractive index (inverse of the dimensionless phase velocity). The term in square brackets represents the effects of the axial electric field. It can be readily seen that this term is much smaller than unity, for any realistic value for the fields and the transverse displacement ri, therefore one can use the shorter expression Ki = wy - u~, which is the standard constant encountered in CRM studies. The significance of Ki can be seen by first noting that exclusively axial velocities have points on the hyperbola "( 2 = l+u~, shown in Figure 1, while the points below this hyperbola have no physical significance. Depending on its initial conditions, an electron with axial velocity will be initially located at a point 0 on this hyperbola. Its subsequent trajectory will depend on the fields acting on it, but in the co-ordinates 'Y - u 3 it will follow a straight line K 1 =constant, with slope equal to d"(/du 3 = £ = w/kc, the phase

191

Cyclotron resonance effects

0.15

';. 0.10

p

>-

~ Q) c:

LU

-0.50

-0.25

2

a.as

0.00

Axial Momentum u

0.25

a.so

3

Figure 1.

Possible paths of a typical electron, starting with no transverse velocity (point 0) and following trajectories consistent with the invariance of K1.

velocity. Since the hyperbola has asymptotes with unit slope, two possibilities exist. (a) For fast waves (when C = w/kc > 1) the electron as it moves along the straight line, continuously increases its energy, while acquiring transverse momentum. Such an evolution is not appropriate for RF generation. (b) On the other hand, if C < 1 (the case of slow waves) the straight line will eventually intersect the hyperbola at some other point. Whether this point will have a higher value of "Y than initially, depends on how Ccompares with the slope of the hyperbola at the initial position 0. This slope is equal to the axial velocity, therefore the electron gains energy, when C > v0 /c (point P 2 ), or when C < 0, while it loses energy (point P 1) when 0 < C < v0 /c, which is precisely the case of the anomalous Doppler shift. Of course, how far from the initial point 0 the electron will eventually arrive, depends on the amplitude of the wave fields (and is described with the aid of the second constant of the motion in the next section): Instead of reaching, say, point Pi, it may not go any further than point F 1 (on the straight line OP 1 , since K 1 is conserved). This allows one to express the electronic efficiency 77 = ("'10 - "Ye)/ ("'10 - 1) as a product of two terms, 17 = 77 1 772 . (It is recalled that the electronic efficiency gives the fraction of the initial beam power lost during the interaction, that is, converted to radiation power. Since the beam power is the product Pb = ("Yo - 1)mc2 lb/ e = Vbh and the beam current is conserved, the electronic efficiency is the ratio of energy differences of the electrons, on the average. In the present interaction however, all electrons follow identical paths, hence the properties of any electron coincide with the average ones.) Postponing the discussion for 772 until the following section, here we discuss the term 7]1 , which is defined by the ratio 771 = bo - "'(p) / ("'10 - 1). With this definition, 'T/J gives the fraction of the initial beam power (or, electron energy) that is available to the interaction, if strong enough wave fields are present to move the electron from its initial position 0 (where "'( = "YP) to position P, the one with the lowest possible energy consistent with the phase velocity (the slope of OP). This efficiency depends only on the phase velocity of the wave (and the initial electron energy) and can be easily

John L Vomvoridis

192 2.0

.,,, = .9

0

1.8 1.6 ('!!

1.4

1.2 1.0 0.0

0.5

t

1.0

Figure 2. Contour plots for the efficiency term 'T/i as function of the beam energy and the wave phase velocity. The vertical axis has also 'T/I = 0.

calculated. It is shown in Figure 2. The value 'T/I = 1 (that is, all energy being available for conversion) is a possibility, which corresponds to the line K 1 = constant crossing the hyperbola at its apex, at which 'Y = l. About the curve 'T/i = 1 in Figure 2, there exists a broad belt of points with really high efficiencies 'T/i, say up to the curves 'T/I = 0.9. High initial beam energies seem to be favourable, since there this belt becomes wider and (more importantly) it shifts to higher phase velocities for the required slow wave. The curve 'T/I = 0 in Figure 2 corresponds to the line K 1 = constant being tangential to the hyperbola in Figure 1 (thus, the beam and the phase velocity are equal). A second contour with 'T/I = 0 coincides with the vertical axis in Figure 2: For zero phase velocity, the lines K 1 = constant are horizontal in Figure 1 and therefore no energy exchange is associated with such waves. (Such waves are nevertheless able to convert the axial momentum to transverse.)

4

Slow-wave CARM

The second constant of the motion can be written in a variety of ways. In their Equation (4.10), Frantzeskakis et al (1994) give this constant for arbitrary hybrid waves, including therefore the effects of the axial electric field, as a relation connecting the electron energy 'Y and the dimensionless momentum component along the transverse magnetic field of the wave, u 2 , in terms of an integral, which can be easily expressed in terms of elementary functions. (The case of TE waves has to be treated separately, since the denominator in this equation cancels out.) In this form, the constant of the motion can be reduced to depend on three parameters, which means that a three-dimensional scan has to be performed to study the trajectories in phase space. Instead of this (which

Cyclotron resonance effects

193

still remains to be done), in this section we present the results in the case of TE waves, drawing from Vomvoridis and Hambakis (1991). (As has been mentioned earlier, in this reference the results are reliable only when £ = n- 1 , as is appropriate for TE waves.) Omitting unnecessary details, the second constant of the motion can be brought in the following compact form (11) where CB and CA are constants proportional to the negative of the magnetostatic field and to the radiation field amplitudes, respectively. In this expression, sin 'l/; is a linear function of 7, such that in Figure 1, sin 1f; = ±1 corresponds respectively to points 0 (the initial energy) and P (the lowest energy consistent with the phase velocity (for K 1 = constant to hold). The angle x gives the direction of the transverse momentum in relation to the transverse radiation fields (x = 7r /2 and x = 3w /2 means co- and anti-parallel to the wave electric field). A two-coordinate constant like K 2 enables one to study the evolution in phase space. For this study, it is of importance to determine the singular points and the critical curves. The critical points contained in Equation (11) are vortices (0-points) and saddles (X-points). [It is reminded that for a constant of the form K(x, y), singular points are those for which simultaneously fJK/fJx = 0, i.e. dy = 0, and fJK/fJy = 0, i.e. dx = 0. The sign of the expression (82C/fJx 2)(82C/fJy 2) - (8 2 C/8x8y) 2 at a singular point indicates its nature: a positive sign means vortex, a negative sign means saddle.] In particular, a vortex is always present on x = 7r , while on x = 0 (= 2w) there are either two vortices and one saddle, or just one vortex. Additional saddles are present at x = 7r /2 and x = 3w /2, with sin 1f; = ±1. Once the saddles are determined, the separatrices (= the trajectories through the saddle) can be drawn. The significance of all this formalism lies in the fact that with minimal effort one can get a pretty good idea on the structure of the trajectories in phase space (e.g. in co-ordinates (x, 1f;) in the present application): The trajectories are closed around a vortex, up to the separatrix, they are deflected from the saddle point, while two distinct trajectories never intersect. Applying these procedures to the constant K 2 , it turns out that in co-ordinates CA, CB there exist seven different domains, each corresponding to different topology for

the phase space trajectories. Ten more topologies correspond to transitions between different domains. (At these transitions, one trajectory passes through two distinct saddles, i.e., two separatrices merge.) Referring to Vomvoridis and Hambakis (1991) for the values of CA, CB which correspond to each topology, we present here the seven main topologies for the possible trajectories, I through VII, shown in Figure 3, along with the transitions VI-III and VII-IV. These transitions occur when the saddles at sin 1f; = ±1 merge, and occur when CB = 0 (i.e. zero magnetostatic field!) and for sufficiently large values of CA (1 . = -

0

(2.27)

co

which is very large near w = 2nci where S is regular. It turns out that this is not only a convenient simplification, but actually gives a much better agreement of Equation (2.26) with the full hot plasma dispersion relation near the fundamental cyclotron resonance (the failure of the complete FLR expansion near w = nci is due to the fact all roots of the disperion relation except the FW have a wavelength much shorter than the ion Larmor radius in this frequency domain). As long as the roots of (2.26) are well separated, the smallest one is just the FW (2.6), with a small correction due to the finite pressure. Near the Alfven resonance (2.9), however, this root is no longer divergent, but has a confluence with a short-wavelength wave. The nature of the new root depends on the value of the ratio 1Pa2f SI = /3; · (me/m;) · J(w/Oci), where f is a numerical factor of order unity (but much larger than unity near the first cyclotron harmonics), and /3; = (w;Jn~J(v;hJc2 ) the normalized ion pressure. a) If the plasma is very tenuous and cold, (2.28) the confluence occurs with a wave whose dispersion relation can be written nj_2

= nj_21 8 = - (n 2 11

-

s·

S) p

(2.29)

If w/k » Vthe this is the cold-plasma shear Alfven wave (SAW), and propagates on the low magnetic field side of the confluence; in the opposite limit it is called 'kinetic Alfven wave' (KAW), and propagates on the high magnetic field side. The third root does not satisfy the condition k;v;hJn~i « 1, and must be discarded. 11

Ion Cyclotron Heating In Tokamaks

263

b) In a typical fusion plasma, however, the opposite condition usually prevails, namely (2.30) In this case, the confluence occurs with the pressure-driven root, whose dispersion relation is n 2 -S ': : :'. --"__ • n.L2 = n2) (2.31) .LB a2 This wave is the first member of the family of electrostatic waves which exist near harmonics of the ion cyclotron frequency, known as Ion Bernstein waves (IBW) [Bernstein 1958); it always propagates on the high magnetic field side of the resonance. In this case SAW or KAW still exist as solutions of the hot plasma dispersion relation, but do not play any role near the Alfven resonance. In other words, condition (2.30) justifies the zero electron inertia approximation, IPI --t oo, which simplifies (2.26) to (2.32) In the following we will consider mainly situations where this approximation is valid.

2.9

First harmonic heating in a single species plasma

In the limit of perpendicular propagation

~2

(J; w 2 w- 2nci

\

A2--t----

x

(2.33)

diverges at w = 20ci. Hence ion FLR effects are always important near the first cyclotron harmonic, although this could not have been predicted from (2.6) alone. The dispersion curves (Figure 9) show a double confluence between the FW and the IBW, separated by a 'frequency gap' in which the two roots of (2.32) are complex conjugate, and the two waves evanescent. The confluences are easily located analytically by searching for the roots of the discriminant of (2.32): 20ci - w I W

_ (3

conf -

1

h}

1±

Jhf..

(2.34)

(h} ': : :'. 1/3 in a deuterium plasma). They both lie on the high-field side of the resonance; the optical thickness of the evanescence layer can be estimated as (2.35) with

nJF taken at the resonance in the cold limit.

To evaluate reflection and transmission through the resonance layer the wave equation must be solved. Because the FLR terms in the dispersion relation depend on the wavevector, however, in a nonuniform configuration they become second order differential operators, whose form is not uniquely determined by the dispersion relation alone.

264

Marco Brambilla

n2

3000

j_

2000

FW

FW

ro/Q . 2.1

1.9

-2000 Fig. 9 - Dispersion curves near the first ion cyclotron harmonic for perpendicular (n11= 0) and oblique (n1F 4) propagation. o? I 0 2 pe ce

=0.5, p =0.02

(fe = Ti = 5.11 ke V), hydrogen plasma.

\

\

5.0%

2.5%

\

0.0%

6000

n2

j_

FW

FW

ro/Q 0.9

0.92

1.02

O.f14 ~

' '- -...

5.0%

2.5%

\400~

Fig.IO - Hydrogen minority in Deuterium plasma,

1.04

0.0% \ 2 /Q 2 =0.5, p =0.02, ro pe ce

Te= Ti= 5.11 keV. Hot -plasma dispersion curves near the ion-ion resonance, perpendicular propagation.

Ion Cyclotron Heating In Tokamaks

265

It turns out [Colestock and Kashuba, 1983] that the correct FLR wave equation in this

case is

(2.36) where R = -S = (2/3)(w~Jn~i), L = -2(w~Jn~J, and ~ 2 can be regarded as constants. The solutions of this self-adjoint equation can be written in the form of Laplace integrals [Gambier and Swanson, 1985], and from their asymptotic behaviour for large X and the radiation conditions the reflection and transmission coefficients can be determined. The result is identical with that of the Budden model, Equation (2.19), with, however, ri2 replacing 'f/l· The 'absorbed' power in this case appears as power transported away from the resonance region by the Bernstein wave, a phenomenon known as (linear) mode conversion. The IBW can be absorbed by electron Landau damping if its parallel phase velocity is sufficiently small; alternatively, it can be absorbed by stochastic acceleration of the ions, which can occur when the wavelength becomes shorter than the ion Larmor radius [Riyopoulos 1986]. If n 11 is not zero, there is a layer of width ~Xcyci (Equation (2.3)) around the cyclotron harmonic resonance where n.L)F is not real (Figure 9 with nu= 4). The evanescence layer is completely washed out and the confluence of the FW and IBW suppressed if ~Xcyci extends beyond the farthest confluence point determined by the minus sign in Equation (2.34). The condition for the validity of Equation (2.36) is therefore Vt hi /]; » lri,11-. c

(2.37)

The l.h. side grows faster than the r.h. side with the plasma temperature, and is moreover proportional to the plasma density, which does not enter in the r.h. side. Since the values of n 11 which can be launched by IC antennas are nearly independent of the plasma size, this condition is more easily satisfied with increasing plasma performance. It has been pointed out [Weynants 1974] that this circumstance might somewhat reduce the efficiency of first harmonic ion heating in a reactor. It turns out, however, that for waves excited from the low field side, which encounter first the cyclotron harmonic, this is only a minor effect. The amount of harmonic cyclotron damping suffered by the FW wave itself while transiting through the first harmonic resonance can be easily estimated in the two limits of negligible and very large Doppler broadening, using once more an equation similar to (2.12), but with Im(>. 2 ) on the r.h. side. An equation which interpolates well between these two extremes is [Brambilla 1993b] ~Px

Px

where

w

7r

81b

= 'f/fb '.:::'. 4/3; nJp ~Rtor 1+8fb 2 I 2 8 - 2nu2vthi c

lb -

fJl

(2.38)

(2.39)

is the parameter which characterizes the transition between the case of quasiperpendicular propagation (8fb « 1) with little harmonic damping and efficient mode conversion, and the case of large Doppler broadening (8fb » 1) in which harmonic cyclotron damping dominates and mode conversion is suppressed. We may also note that in the latter

266

Marco Brambilla

limit the ratio between harmonic and fundamental cyclotron damping in a single species plasma is roughly t.Px(2nci)/t.Px(nci) '.::::' nI/n,~ » 1, as anticipated.

2.10

Hydrogen minority in a Deuterium plasma

The analogy of Equation (2.38) with the two heating regimes in a two-species plasma will have been noted. Indeed, as illustrated by the dispersion curves of Figure 10, minority heating of hydrogen in deuterium goes over continously into the pure n+ case as the H+ concentration tends to zero. The dispersion relation for this scenario for nearly perpendicular propagation can be discussed as in the previous case, taking into account the singularities of both L due to the minority, and a 2 due to the majority. The two confluences between the FW and the IBW are located where (2.40) the last approximation being valid if VH, although small, satisfies VH » f3D· In this limit the farthest confluence asymptotically approaches the position of the cold-plasma ion-ion resonance. Because of the square root in the second term, however, at low minority concentrations the width of the frequency gap due to finite Larmor radius effects easily exceeds substantially the width predicted for the evanescence layer by the cold plasma approximation. Accordingly, the condition to be in the minority heating regime can be written

(2.41) We can add that in this regime the minority is preferentially heated as soon as VH ~ flv, and second harmonic heating of deuterium correspondingly reduced compared to the pure n+ case. The wave equation in the mode-conversion regime is a fourth-order differential equation (since it must describe both the FW and the IBW) which combines the singularities of Equation (2.18) and Equation (2.36). This equation, to our knowledge, has not been solved in closed form. The results of numerical integrations for the reflection, transmission and mode-conversion coefficients, however, can be reproduced accurately using once again the Budden expressions (2.19) and (2.21), with

(2.42) where 771 is given by (2.20) and r/2 by (2.35). The formal analogy of this expression with (2.40) suggests that this might be an exact result, although this has not been proven. As in the pure D+ case, the power described by A± is transported away by the ion Bernstein wave.

2.11

Cold-plasma resonances and mode convers10n

The replacement of a cold-plasma singularity by a mode conversion layer (with or without frequency gap) is a universal characteristic of wave resonances in the absence

Ion Cyclotron Heating In Tokamaks

267

of dissipation when finite pressure effects are taken into account. Thus for nearly perpendicular propagation the confluence of the FW with the pressure-driven wave (or with the shear Alfven wave in very low f3 plasmas) occurs also in the scenarios where the minority resonance does not coincide with the first harmonic of the majority. In this case, however, a-2 is much smaller and the wavelength of the IBW correspondingly shorter, so that the confluence layer is very narrow. In this case any residual damping (ion cyclotron damping of the minority in the tail of the Doppler broadened cyclotron resonance, electron Landau damping, or even collisional absorption) is sufficient to eliminate the confluence; the cold-plasma description of the wave resonance is then perfectly adequate, as the dispersion curves in Figure 5 for oblique propagation clearly illustrate.

2.12

Electron heating

In hot tokamak plasmas damping of the FW on the electrons always competes with ion heating. One can also purportedly avoid damping on the ions when the FW is used for current drive. To estimate the importance of electron damping one must in the first place take into account parallel dispersion in Em rewriting the last line of Equation (2.5) as 2 w 2 Z'( Xe=--. (2.43) lzz '.:::'. - -wpe 2 Xe Xe ) w k vthe In addition, as pointed out by Stix [Stix 1975], it is necessary to take into account the FLR corrections contributed by the electrons to Eyy '.:::'. S - nI(a-2 + 2Te) and Eyz = -Ezy '.:::'. in.L 'nii~e· Expanding the hot plasma dielectric tensor to second order in the electron Larmor radius, and keeping only the leading terms in w/rlce• one finds 11

Te

1

2

2

Vthe = -2 wpe n 2 - 2 Hee C

{

-Xe Z(xe )}

~e

1 2 2 wpe Vthe 2 '( ) -_ -2 -;:;--Xe Z Xe . W>Lce C2

(2.44)

Although not contributing significantly to the FLR dispersion relation, Te and ~e are required to determine correctly the parallel component of the FW electric field and P:bs· From the third line of (2.1) one gets

Ez

= ~.L'Tiipi {Ex - i~eEy}. n.L -

(2.45)

The refractive index of the fast wave satisfies always nI « IPI; moreover, except near the Alfven resonance n1~ = S, IEx/Eyl = O(w/rlci) is of order unity or smaller. If f3e « me/m;, therefore, ~e = O(f3erlce/w) is negligible, and this equation predicts the characteristic inverse dependence of Ez on the density due to screening by the electrons in a cold plasma. Under the more typical condition f3e ~ me/m;, on the other hand, the first term within the bracket in Equation (2.45) is negligible, and one obtains (2.46) In this regime Ez is independent of the plasma density, and proportional instead to the plasma temperature. This residual parallel field is required to maintain charge

Marco Brambilla

268

neutrality. Note that ~e/ P is a real quantity independent of the parallel phase velocity of the wave; Equation (2.46), therefore, is valid even if lx0 el ;

,,

.,.,

_, 0 c

,.;

j u

.!!. ~

~ 0'

~

=

....

UJ

E E ,.;

"'

0

:!.

0

E

c. 0 c.

~

0

c.

.!!.

1'

'

i i :; .g

i: 0

~ ;:"'

.t:

295

~

~

0 u

"' 0

corrugated TE02-bendl90"1

0

UJ

.... corrugated

TE01-bendl90'1

TE01/TE02 mode converter section phase shifter

am straight

TE 02-down-taperi •63. 4mm/o27,6mml

woveguide(~27.8mml

corrugated-vwoll mode filter

corrugated-wall mode fitter

bellows

k- spectrometer

arc detector, gos inlet. RF-monitor

DC-break TE01/TE1l mode converter

gyrotron, 70GHz. 200 kW(cwl

elliptic/circular polarizer

1 t

TEtl/ HE11 mode converter

corrugated

HE11-bendl47'1

corrugated HE 11-up-laper (e 2 7.8mm/0 63.l. mm) bellows

corrugated arc detector, gos inlet barrier window

open-ended corrugated woveguide (antennol

focusing reflector antenna plasma

t

/

/

I

J_

Figure 10. COMPASS ECRH transmission lines

The generators are equipped with conventional electronics which do not present specific problems. Also there are no matching problems. Extensive theoretical and experimental efforts have been made to develop low loss overmoded transmission lines, effort which is described in another course. An example of the complexity of the transmission lines as used in COMPASS is shown in Figure 10. In order to avoid the production of unwanted high order modes all elements of the line are very carefully designed. The required mode at the antenna, generally the HEll mode, closely matches the free space impedance. Less than 2% power is reflected back to the gyrotrons. The substantial effort which has been made both by industry and by laboratories to produce high power tubes and low loss transmission lines in the lOOGHz range of frequencies is now paying

C Gormezano

296

off and, although expensive, reliable ECRH generators are now available for the present generation of fusion devices.

5

RF systems for ITER

The present main parameters ofITER in the current Engineering Design Activity (EDA) phase are the following (for the single null diverted configuration): Major radius Minor radius Plasma elongation Plasma current Toroidal Field MHD safety factor Fusion Power Average wall loading Density Temperature Burn duration

R a

k I

B

q95

Prus Pn (ne) (Te) tburn

=

8.14m 2.80m 1.6 to 1. 75 21 MA (nominal) 5.68T 3.05 1.5 GW (nominal) 1 MW /m 2 (nominal) 1.3 1020 m- 3 (nominal) 10.5 keV (nominal) 1160 sec (nominal)

The main parameters of importance for RF systems in ITER are the nominal values of the magnetic field, the density and the temperature. Due to uncertainties in plasma position control and also due to non-confined alpha particles, a gap of 15cm is foreseen between the last closed magnetic surface and the first wall. Also of importance is the finite number of ports, up to 20, only a fraction of them being available for additional heating, as well as their size, about 2.5m in height and l.5m in width. An order of magnitude of the size of an RF system on ITER is illustrated in Figure 11 where a possible LHCD system is sketched. Note the scale. Auxiliary power systems on ITER have to be designed to perform the following functions: 1. Heating, which further subdivides into: (a) providing sufficient power across the separatrix to access H-mode confinement, (b) increasing the temperature to ignition, (c) supporting driven burn scenarios if confinement proves to be inadequate, (d) maintaining adequate temperature during the current termination phase when the density exceeds the Greenwald limit. 2. Driving on-axis current (0.5MA to IMA) to provide the seed current for high bootstrap current steady-state tokamak discharges. 3. Driving off-axis current (3MA) to maintain the current profile needed to access high performance plasmas in steady state tokamak discharges. 4. Stabilising MHD instabilities either by local heating or current drive (rotating modes) or by inducing sufficient plasma rotation (locked modes). 5. Pre-ionising the plasma to ease breakdown conditions and allow a good start-up.

RF systems for heating and current drive

-~·,~·'"-~

•

!

L.....--\

297

------+-

11!7d0

---7

Figure 11. Sketch of possible addtional heating system {LHCD) on ITER showing (1) back vacuum vessel, (2) cryopump, (3) vacuum windows, (4) vacuum windowa, (5) hyperguide, (6) grill mouth, (7) and (8) transmission lines, (9) klystrons. Distances in centimetres. Typically, the required power is lOOMW for 1000 sec. It is now recognised that no single additional heating system is capable of fulfilling all these tasks and that two or more systems will have to be used. For the time being, all main additional systemsNeutral Beam Injection, ICRH, ECRH and LHCD-are still being considered and design efforts are pursued along these lines. These designs are still evolving together with the main design of ITER and only a snapshot of the status of the present design and of the problems encountered will be given here. Proposals have to be formalised by the end of 1998, for the final Engineering Design Activity (EDA) report of ITER. The challenges that the RF systems will have to face for ITER are formidable, as for every subsystem in ITER, and can be summarised as follows: • The need for antennas, in particular for ICRH and LHCD, to be plasma facing components: disruption forces, alpha particle losses, radiation, neutron flux, etc. • Steady state operation: active cooling of radiating structures and high efficiency operation of the generator. • High neutron yield: antennas being located in a vacuum port will have to provide their own shielding structure to block most of the neutrons as will the blanket in ITER. This will have the consequence that all insulators might suffer from neutrons in particular in the presence of DC or RF fields and that line of sight has to be blocked. • Remote handling: all systems have to be capable of being removed hands off. It is unlikely that repairs will be done in the same way. • High degree of reliability and availability: number of pulses for conditioning, tuning, matching, etc. , will likely be kept at a bare minimum. Intelligent control

C Gormezano

298

electronics systems will have to be extensively used. RF system machines which are already using Tritium operation (JET, TFTR) are facing similar problems, but can only be considered as test beds as far as ITER needs are concerned.

5.1

ICRH system for ITER

The unique features of ICRF, in ITER conditions are: (a) the absence of density limit allowing heating of the plasma centre whatever the density, (b) the possibility of direct ion heating (up to 60%) which provides additional alpha heating power during the crucial phase of the path to ignition, all other heating methods (including Neutral Beam Injection) heating electrons preferentially, and (c) the decoupling of the heating and current drive functions by phase and frequency control. The first two features, absence of density limit and direct ion heating, are of obvious importance. The main focus of design effort has centred on the antenna. Blanket antenna were initially proposed allowing large area antennas to be installed with the advantage of having low voltage handling for a given power and being able to operate in a large frequency range. Due to the complexity of the ITER blanket and the associated remote handling, as well as its evolving design, the design has been re-directed toward the concept of a compact launcher, easily accessible and maintainable, based on conventional coupling straps grouped in 4x2 array modules. Four ports would be required to couple 50MW. The anticipated voltage is 38kV at 60MHz (possibly up to 50kV in dipole phasing) for the coupling as estimated for an ITER burning plasma located at 15cm away from the antenna with a 2cm density gradient scrape-off layer. The frequency range, 40 to 70MHz, allows compliance with the required magnetic field range using second harmonic tritium resonance. Current Drive in this frequency range will not be done in optimum conditions but still will be adequate for ITER needs (central current only). A sketch of such an antenna is shown in Figure 12. The straps are fed by coaxial extensions inserted in the neutron shield. A key issue is to produce an effective system in conditions of varying plasma loads as discussed above. The ITER team presently favours a resonant structure with a variable tuning element, such as a stub, in vacuum. Each array is composed of four modules each including two current straps and one Faraday shield. This plasma facing component is considered to be made of high yield strength copper with Beryllium coating and water cooled at high pressure. A screen-less array is also being considered, in view of the recent successful operation in medium size tokamaks (ASDEX, TEXTOR). Other issues which are being considered are: • development of all metal vacuum transmission coaxial lines, including support of the inner conductor; • the double vacuum window will be located far away from the vessel to reduce neutron irradiation and to ease maintenance; • wide band matching systems; • operation at 2MW per strap (total power of the plant: 64MW);

RF systems for heating and current drive

299

( 19]~)

0

...,0

100

1500

50

1900

I 196 • ) (2 ..5 is about 3x10 7 . Thus, the required radiated power to detect the cm-sized piece of space debris is about 20MW. This is larger than the power of any single 35GHz microwave tube with lOOµs pulse length. Since 35GHz gyro-klystrons with an output power of 700kW have been developed by Russian industry (Table 2), these sources can be coherently combined and fed into a large phased-array antenna. Since the motivation for an amplifier is coherent power combining and not simpler data processing, an amplifier is the only choice for the system. It would take about 30 such gyroklystrons to provide enough power for the system. A single gyroklystron, or possibly gyrotron, would provide enough power to detect a piece of debris with a RCS of about 30cm2 . With a combination of mechanical and electronic steering, the radar could both scan and revisit targets.

5.3

High-power nanosecond radar

In conventional radars, range resolution is obtained with the use of a long pulse chirped in frequency and then compressed by the receiver, for instance, with the use of a matched filter. In almost every case, while the range resolution is c/ 6,.f, there are temporal sidelobes that mix up the signals from nearby range cells. This problem can be especially serious if one is attempting to resolve a low-cross-section target in close proximity with a high-cross-section target, such as a buoy near a ship. Furthermore, pulsed Doppler radars typically have a problem with blind speed when trying to resolve the velocity of a target. Finally, for a low-peak power system, with a not too large ratio of inter-pulse separation to pulse time, the dead time can also become an operating constraint. One possible way to get around some of these difficulties is to use a very-short-pulse high-power radar without pulse compression using a relativistic BWO. The features of such a system, developed at the IAP and to LHCE in Russia are summarised in Table 5. The radar target was a small aircraft, which had a total RCS of about a square meter. The range from the target to the radar device was 50km. The typical received radar signals are shown in Figure 8 after the signals were processed so as to reduce the stationary clutter by 28dB. This nanosecond radar system has demonstrated: • Long target detection range: a helicopter of 2m 2 RCS was reliably tracked up to 105km over a wood covered terrain (theoretical range = 150km) . • High range resolution (important for target identification): sample reflectors as well as details of the helicopter and boats have been seen separately, if distanced by more than lm . • Effective moving target indication (MTI) performance: MTI algorithms provide up to 30dB suppression of echoes from immovable targets and 10~20dB suppression of echoes from the wind-surged vegetation.

Manfred Thumm

320

Microwave generator frequency radiated pulse power pulse duration repetition rate

10 GHz 0.35 GW 7 ns 150 Hz

amplitude instability frequency instability

1% 0.1 % 107 pulses

resource

Antenna radiation-reception beam width 3.5°

Receiver frequency band noise factor

210 MHz 7 dB

A numerical signal processing sub-system is used to perform the moving target indication(MTI).

Table 5. Parameters of high-power nanosecond radar system (Gapanov-Grekhov and Granatstein 1994).

(a)

island

___ "-_,if1~·