J DOJ ItOAJQy Ris«-R-663(Em
On the Theory of Thomson Scattering and Reflectometry in a Relativistic Magnetized Plasma
Henrik Bindslev Balliol College, Oxford
Rise National Laboratory, Roskilde, Denmark December 1992 On the Theory of Thomson Scattering and Reflectometry in a Relativistic Magnetized Plasma
Henrik Bindslev Balliol College, Oxford
Department of Engineering Science Parks Road, Oxford.
Trinity term 1992
Risø National Laboratory, Roskilde, Denmark December 1992 Abstract A theoretical model of Thomson scattering in a magnetized plasma, taking spatial dispersion into account, is developed A initio. The resulting expressions allow tlromal motion to be included in the descrip tion of the plasma and remain valid for frequencies of the probing radia tion in the region of w^ and co^ provided the absorption is small. With these expressions the effects of the dielectric properties of magnetized plasmas on the scattering of electromagnetic radiation by density fluctua tions are investigated. Cold, hot and relativistic plasma models are con sidered. Significant relativistic effects, of practical importance for milli meter wave scattering in large Tokamaks, are predicted. The complete expression for the source current of the scattered field is derived in the cold plasma limit by a kinetic approach. This result is at variance with the widely used expression derived from a fluid model of the plasma. It is found that a number of mistakes were made in the traditional fluid deri vation, which explains the differences between earlier results and our results in this limit The refractive indices and the cutoff conditions for electromagnetic waves in plasmas are investigated for cold, hot and relativistic plasma models. Significant relativistic modifications of refractive indices and lo cations of cutoffs are found for frequencies in the range of cou« and (o^ Fully relativistic expressions for locations of the X-mode cutoffs are der ived and new algorithms are given which extend the regime in which the weakly relativistic dielectric tensor can be computed. For X-mode plasma reflectometry it is demonstrated that these effects may shift the location of the reflecting layer by a significant fraction of the minor radius and that the cold model may lead to considerable underestimarjons of the density profile. Relativistic effects predicted for Omode reflectometry are smaller than for X-mode, but not negligible. An algorithm for reconstruction of density profiles which allows a relativistic plasma model to be used is presented.
This thesis is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Oxford
ISBN 87-550-1874-2 ISSN 0106-2840
Grafisk Service, Risø, 1992 Contents
Preface viii
1 Introduction 1
1 Thomson scattering 5
2 Thomson scattering 7
2.1 Introduction 7
2.2 Basic principles of Thomson scattering 8
2.3 Review of previous work on Thomson scattering 12
2.3.1 Early experiments 13
2.3.2 Scattering cross section when w' 3> wpe 13
2.3.C Cross sections and transfer when u>' ss MJ^. 13
2.3.4 Fluctuations and the spectral form factor, S(k,w) 15
2.3.5 Electron temperature and density measurements 18
2.3.6 Turbulent density fluctuation measurements 18
2.3.7 Bulk ion temperature measurements 19
2.3.8 Impurity ion measurements 20
i ii Contents
2.3.9 Magnetic field measurements 20
2.3.10 Fast ion measurements 21
2.4 Outline of the relevant scattering theory 22
3 The Collective Thomson Scattering Diagnostic at JET 26
3.1 Choice of frequency for the JET collective scattering diagnostic . . 26
3.2 Sketch of the JET collective scattering diagnostic 28
3.3 Signal to noise ratio 34
3.4 Operating the diagnostic 36
3.5 Interpretation of data 37
4 Dielectric properties of a relativistic magnetized plasma 39
4.1 Introduction 39
4.2 Review of previous work on dielectric properties of relativistic plasmas 40
4.3 Modelling a plasma with spatial and temporal dispersion 44
4.4 The rclativistic plasma model 46
4.5 The weakly relativistic approximation 47
4.6 Computation of the Shkarofsky functions 49
4.7 Computation of the dielectric tensor 56
4.8 Computation of the dispersion function and its derivatives 57
4.9 Refractive index 58
4.10 Cutoffs CI
4.11 Summary 69 Contents ill
5 Propagation of electromagnetic waves 70
5.1 Introduction 70
5.2 Ray-tracing 71
5.3 Propagation through an inhomogeneous and anisotropic plasma . . 75
5.4 Coherent detection 77
5.5 Power flux in a plasma with spatial dispersion 80
5.6 Mode conversion 87
6 Field due to current sources in plasma 96
6.1 Introduction 96
6.2 Near field 96
6.3 Far field 97
6.4 Far field energy flux 102
7 Bilinear plasma response 105
7.1 Introduction 105
7.2 Source currents for scattered field 107
7.3 Expansion of response in powers of v 116
7.4 Parametrization of distribution function 127
8 Fluctuations 129
5.1 Introduction 129
5.2 Review of methods for calculating S(k,u/) 130
5.3 Electrostatic dressed particle approach 131 iv Contents
S.4 Electromagnetic dressed particle approach 137
9 Theoretical model of Thomson scattering 141
9.1 Introduction 141
9.2 Equation of transfer for a scattering diagnostic 142
9.3 Symmetry of the equation of transfer 149
9.4 Numerical results 161
9.5 Summary 171
II Reflectometry 173
10 Reflectometry 175
10.1 Introduction 175
10.2 General algorithm for density profile reconstruction 176
10.3 Reconstruction of density profiles from simulated data 180
10.4 Summary 187
11 Summary and conclusions 188
A Notation 192
B Fourier-Laplace transformation 195
C Notes on Yoon and Krauss-Varban (1990) 196
Bibliography 197 List of Figures
3.1 Schematic diagram of the fast ion collective scattering diagnostic being developed at JET 29
3.2 Sketch of Vlasov converter 30
3.3 Universal polarizer on the launching side 32
3.4 Universal polarizer on the receiving side 32
3.5 Gyrotron power modulation 35
4.1
4.2 Contour plots o{ the logarithm of the weakly relativistic plasma dispersion function, log(|A|). in the complex ft plane 59
4.3 Refractive index,//, as a function of electron density. 60
4.4 Relativistic CMA diagram 62
4.5 Rclativistic cutoff densities normalized by cold cutoff densities as functions of temperature 64
4.G Relativistic CMA diagram for high temperatures 65
4.7 Temperature required to reduce the R-cutofF frequency to w« as a function of (u-'/j/^Ve — 1) 66
4.8 Weakly relativistic and the fully relativistic predictions of the limit at which the R-cutofF is removed 68
v VI List of Fifflires
5.1 X-mode rays traced using the cold and the weakly relativistic model 74
5.2 Illustration of variables describing a transverse polarization state. . 89
5.3 The Poincaré sphere 90
5.4 Evolution of an arbitrary polarization state on the Poincaré sphere. 92
S.l Spectral density of electron density fluctuations 13S
9.1 Scattering system 151
9.2 Scattering geometry. 163
9.3 (a) Dielectric form factor, (b) coupling term, (c) flux term, (d) and
(e) real and imaginary parts of refractive index, as functions of nc, with parameters relevant for the JET collective Thomson scattering diagnostic when scattering from X-mode to X-mode 164
9.4 Dielectric form factor, (a) cold, (b) relativistic, as functions of the frequency of the scattered radiation, with parameters relevant for the JET collective Thomson scattering diagnostic when scattering from X-mode to X-mode 166
9.5 Relativistic dielectric form factor and real part of refractive index
as functions of ne, with parameters relevant for the JET collective Thomson scattering diagnostic when scattering from X-mode to X-mode 167
96 Relativistic dielectric form factor and real part of refractive index
as functions of ne, with parameters relevant for the JET collective Thomson scattering diagnostic when scattering from 0-mode to 0-mode 168
9.7 Relativistic dielectric form factor and real part of refractive index
as functions of ne, with parameters relevant for the TFTR collective Thomson scattering diagnostic 169
9.S Cold dielectric form factor, (i) based on SlTENKO (1967), (s) present work (see text), as functions of the frequency of the scattered ra diation, with parameters relevant for the JET collective Thomson scattering diagnostic when scattering from X-mode to X-mode.. . 170 Li»r of Figures yii
10.1 Cutoff frequency as a function of major radius 177
10.2 Actual electron density profile anH reconstructed density profiles. . 1S2
10.3 Normalized reconstructed density profiles 184
10.4 Reconstructed density profiles for a range of different temperature profiles 186 Preface
This thesis describes most of my work on the theory of Thomson scattering and reflectometry, which was carried out oyer a three year period from August 19S9 to July 1992 at JET Joint Undertaking. Abingdon, UK, where I was p&r* of the fast ton collective Thomson scattering group on attachment from Risø National Laboratory, Roskilde. Denmark.
Until early this year it was the intention that the present thesis should include ex perimental work on the JET collective Thomson scattering diagnostic, including analysis of experimental data. In anticipation of this, a considerable amount of time was spent developing computer codes for running the diagnostic and for an alyzing the experimental results. Due to events beyond my control this diagnostic will, however, not come into operation before the second half of 1993.
My three years at JET and two years as a D. Phil, student at Oxford University, Department of Engineering Science and Balliol College (October 1990 - present) have been a tremendous experience, and have kept me on a steep learning curve. Many people have contributed to this. I would particularly like to thank Tom Hughes, my daily supervisor at JET, with whom I have spent uncountable hours engaged in enjoyable and enriching discussions and who has contributed signifi cantly towards making my experience at JET a true research apprenticeship. I am also indebted to John Allen, my supervisor at Oxford University, to Vagn Jensen, my supervisor at Risø National Laboratory who first introduced me to plasma physics, and to Alan Costley; all of whom gave me much support and helpful ad vice. Highly speculative discussions with Stephen Smith of Essex University were also an important stimulus in this work. I would like to thank Finn Clark, who spent a year at JET before entering university, for his assistance in developing the computer package FIHRAY (see Section 3.4). Finally I would like to thank Risø National Laboratory and The Danish Research Academy for financial support over the past three years.
Henrik Bindslev 9 August 1992
vin Chapter 1
Introduction
Plasmas are often very complicated with many competing processes, making it difficult to obtain detailed and accurate knowledge of their behaviour. This is particularly true of thermonuclear fusion plasmas where the high temperatures prevent the insertion of any physical probes into the plasma. Central electron temperatures of 12 keV and central densities of lO^m-3 are obtained routinely in plasmas produced in the JET Tokamak.1 Here the diagnosis must rely on in formation derived from the electromagnetic radiation and particles escaping from the plasma, the magnetic fields surrounding the plasma, and the influence of the plasma on the propagation of electromagnetic radiation sent into it from the out side. This does provide for a wide range of diagnostics, but the measurements are usually indirect, the quantity being measured depending in a complicated way on a range of plasma parameters. A comprehensive review of diagnostic methods for fusion plasmas can be found in HUTCHINSON (1987 a).
Many important parameters characterizing the plasma cannot be determined un ambiguously from any single measurement. Instead several quantities arc inferred simultaneously from the measurements provided by a range of diagnostics. Some parameters can be obtained from a single diagnostic but rely on the values of other parameters determined by other techniques. These indirect methods of diagnosing the plasma require a detailed and accurate theory of the processes in the plasma which give rise to the measurable quantities, as well as an understanding of all the processes which may affect the measurement.
In fusion plasmas there are generally significant minority populations of highly en ergetic ions which are important for the dynamics of these plasmas. Populations of fast ions with energies up to several MeV, containing as much as a quarter of the
'Tlf I'-rliiiiral details of the JET machine are given in TllE JET PROJECT (1076). Physics result.« am! new developing nis of JET arr summarized each year in the annual report, the latest of whHi i> JET JOINT ESDERTAKIS«. ANSIAI. REPORT (1990).
1 9 Chapter 1. Introduction
total plasma energy, are frequently produced by ion cyclotron resonance heating in present JET plasirns. Nertral beam injection can also produce significant pop ulations of fast ions with energies up to 140 keV. In the coming deuterium-tritium phase of JET (beginning in 1995) significant populations oi fusion-produced alpha particles are expected. The alphas will have a birth energy of 3.5 MeV and it is expected that most of them will be confined and thermalize in the plasma. The dynamics of the fast ions is important in the study of fusion plasmas for several reasons. To sustain a fusion plasma fast ions must be well confined so that their energy goes into heating the bulk plasma rather than the wall. Since a population of fast ions is a source of free energy it may drive up instabilities [ROSENBLUTH and RUTHERFORD, 1975; TSANG et al., 1981; COPPI and PEGORARO, 1981; CHENG et a/., 19S4; WHITE et al, 19S5; SPONG et a/., 1985; Li et al., 1987; Fu and VAN DAM, 19S9 b] which could lead to enhanced transport and specifically to the loss of the fast ions before they have thermalized. Toroidal Alfvén Eigenmodes ave potentially relevant instabilities to which much attention is presently given [CHENG et al, 19S4; CHENG and CHANCE, 1986; Fu and VAN DAM, 1989 a and 6; Fu and ClIENG, 1990; CHENG, 1990]. On the other hand, it is also believed that populations of fast ions may stabilize certain modes.
Spatially resolved measurements of the velocity distributions of fast ions will give valuable information about the extent to which these ions are confined in the plasma und will help to improve our understanding of the interaction between fast ions and the bulk plasma.
On the basis of theoretical studies two techniques capable of diagnosing fast ions in the MeV range in JET have been identified: (a) detection of escaping neutral he lium atoms generated by double charge exchange between injected beam neutrals and fusion alphas [PETROV, 1990; KuSAMAet al., 1990]; and (b) collective Thom son scattering of an intense probing beam of millimeter waves [WOSKOBOINIKOV, 1985; HUGHES, 19S6; HUTCHINSON, 1987; COSTLEY, 1988].
The main objective of the work described in this thesis was to establish a firm theoretical basis from which the experimental results from the second of these techniques, collective Thomson scattering, can be interpreted. Particular attention was given to the problems encountered when the frequency of the probing radiation is in the range of the plasma frequency and the cyclotron frequency.
An important part of the work was the inclusion of relativistic dielectric effects in the theory of Thomson scattering. From this some experience was gained in relativistic calculations of the dielectric properties of plasmas, which also facil itated the investigation of relativistic effects in reflectometry. The relationship between the electron density profiles and the phase shifts measured as functions of frequency in reflectometry was found to be modified significantly by relativis tic effects, a result which has important consequences for the interpretation of 2
reflectometry data. This work is also included in the thesis.
Part I of the thesis describes the theoretical investigations relevant to the theory of Thomson scattering, while Part II describes the investigation of relativistic effects in reflectometry. The work presented in Part II relies extensively on the investigations of the dielectric properties of a relativistic magnetized plasma given in Part I, Chapter 4.
SI units are used throughout this thesis except that the temperature is considered in units of energy. A list of symbols used widely in this thesis is given in Appendix A. Part I
Thomson scattering
5 Chapter 2
Thomson scattering
2.1 Introduction
When electromagnetic waves propagate through a plasma a small fraction of the radiation is scattered. By launching a beam of radiation into a plasma and collect ing some of the scattered radiation, a range of information about the properties of the plasma may be deduced from the spectral properties of the scattered radi ation. This method of plasma diagnosis has proved to be very powerful both in the ionosphere and in laboratory plasmas.
A new generation of diagnostics based on the scattering of radiation from an intense beam of microwaves and intended to give information about small popula tions of highly energetic ions in fusion grade plasmas, is presently under construc tion at JET and TFTR [CosTLEV ei ai, 19S8, 19S9a and 19S9b; WoSKOV et al., 19SS; MACHUZAK et al., 1990]. These developing diagnostics pose new require ments for the theoretical models of Thomson scattering, where we define Thomson scattering broadly as the scattering of electromagnetic radiation by free charged particles in the plasma. The added requirements have necessitated a thorough review and further developments of the theory of Thomson scattering.
Part I of this thesis describes the present state of the theoretical models of Thom son scattering. Chapter 2 gives a brief introduction to Thomson scattering followed by an overview of previous work on Thomson scattering, both experimental and theoretical.
To interpret the data expected from the scattering diagnostics at JET and TFTR it is necessary that the theoretical models take into account a range of effects whirh have been ignored in the past. These new requirements are discussed and
7 j
S Chapter 2. Thomson scattering
an outline is given of the steps involved in developing such a theory. In Chapter 3 a brief description is given of the fast ion and alpha particle collective Thom son scattering diagnostic presently under construction at JET. In the following five chapters the theoretical basis required for a model of Thomson scattering is developed. Some of the topics dealt with in these chapters have wider applica tions than Thomson scattering. This has had some influence on the organization of the topics. In Chapter 4 the dielectric properties of a nlativisUc plasma are investigated and compared to those predicted with hot and cold plasma models. In Chapter 5 the propagation of electromagnetic waves through an inhomogeneous and anisotropic plasma is investigated and the relation between field strength and power for a broadband electromagnetic field in a spatially dispersive plasma is derived. Expressions for the fields resulting from embedded current sources are deduced in Chapter 6, while Chapter 7 investigates the bilinear plasma response, which is responsible for the scattering of electromagnetic waxes. A discussion of methods for calculating the fluctuations in a plasma is given in Chapter S. The theory developed in the previous five chapters is brought together in Chapter 9 to give a model of Thomson scattering with computable expressions. These expres sions are investigated numerically. Some general symmetry properties of Thomson scattering are also discussed in this chapter.
This completes the treatment of Thomson scattering given in this thesis. It should be stressed that the treatment given here does not pretend to be complete. Deriva tion of a comprehensive theory of scattering by free charges in a plasma is a very large undertaking. An attempt is, however, made at giving a framework on which further developments can build, and to outline some of the problems which still need attention.
2.2 Basic principles of Thomson scattering
Thomson scattering is the scattering of electromagnetic radiation by free charges (ie. not electrons bound in atomic states).
When electromagnetic radiation is incident on a collection of free charges the charges are accelerated by the fields of the incident radiation. Any charge un dergoing acceleration emits electromagnetic radiation. The field strength of the radiation emitted by an accelerated charged particle is simply proportional to the charge of the particle and to the the acceleration it is undergoing. Due to the greater mass of ions compared with electrons the ions are not accelerated as much as the electrons air in the incident field and thus the ions do not reradiate as strongly, or put differently, the scattering cross section of the ions is much smaller than that of the electrons. In most Thomson scattering experiments it is therefore acceptable to ignore the scattering due to ions. 2.2- Basic principles of Thomson scattering 9
If the charge is alone in a vacuum then the radiated field due to the acceleration of the charge is given by the Lienard-Wiechert potentials [LANDAU and LlFSHITZ, 19S7. §63J. When the density of charges is low then the presence of the other charges does not influence the field emitted from each charge significantly and it is an acceptable apprcximation to add up the Lienard-Wiechert potentials of all the charges to find the reradiated (or scattered) electromagnetic field resulting from the acceleration of the charges in the incident field.
The relative phases of the fields emitted by the individual charges depend on the phase of the incident field at the point of each charge. If the charges are distributed uniformly and the wavelength of the incident radiation is large relative to the interparticle distances (such that the phase difference between the incident field at the locations of neighboring particles is negligible) then the reradiated fields cancel each other out. For the waves from the individual charges not to cancel each other there must either be density variations (collective scattering) or the wave length of the incident radiation must be so short that the discreteness of the charges and the randomness of their locations allows the fields to add up incoherently.
At higher densities the presence of other charges modifies the fields emitted by each charge and the summation over Lienard-Wiechert potentials does not describe the reradiated field accurately. More specifically this is the case when the frequency, a.'1, of the probing radiation is in the range of the electron plasma frequency, u;^, where
(2.1)
Horr nt is the election density, qe the electron charge, mt the electron mass and co the vacuum dielectric constant.
When u;1 is in this range it is more helpful to think of the scattering as being due to fluctuations in the dielectric properties of the plasma, where the fluctuations in the dielectric properties are caused by fluctuations in the electron density, among other things.
In either picture a wave with frequency ws and wave vector ks which has been scattered from an incident wave with frequency u>' and wave vector k1 is due to the Fourier component
(k,u.') = (k'-ki,w5-u;i) (2.2) 10 Chapter 2. Thomson scattering
of the fluctuations in the plasma.
For diagnostic purposes the frequency of the probing electromagnetic waves will be sufficiently high that the dielectric properties of the plasma are determined by the density and velocity distribution of the electrons, and by the local fields. It follows that the scattering is caused by fluctuations in the electron distribution and the fields, confirming also in this picture that the scattering by ions is negligible. Both the electron distribution and the fields can, however, under certain conditions be significantly influenced by the behavior of the ions. In a simple picture the ion-induced perturbations may be visualized as the result of each ion dragging a screening cloud of electrons and other perturbations along with it. Continuing in this simple picture, the Doppler shift of the light scattered by the perturbations caused by an ion is directly related to the velocity, V,, of the ion:
u;5 = J + v, • (k5 - It') . (2.3)
By calculating the form and magnitude of the perturbation caused by each ion and the perturbations caused by other phenomena it should thus be possible to extract an estimate of the one-dimensional ion velocity distribution in the direction of k (see equation (2.2)) from the spectrum of scattered light.
The influence of the ion dynamics on the spectrum of scattered electromagnetic waves was first observed experimentally by BOWLES (1958 and 1959) in his iono spheric radar scattering experiments, where the Doppler broadening of the scat tered radiation was found to be consistent with ion, and not electron, thermal velocities.
In an early theoretical paper, which included the ion dynamics in calculations of the electron density, SALPETER (19G0) showed that the extent to which the the ion dynamics dominate the electron density fluctuations depends on the scale length of the perturbation relative to the Debye length,
AD = \/S. (2.4)
Here Te is the electron temperature.
SALPETER introduced ;\ parameter
o = \/kXD , (2.5) 2.2. Basic principles of Thomson scattering H
now referred to as the S«lt -itr parameter. It was shown that for a > 1 the electrons behave collectively and the electron density fluctuations are dominated by the ion dynamics. Scattering in this regime is termed collective scattering. For a < 1 the spectrum of the scattered light reflects the thermal motion of the electrons. Despite the limitations of the theory developed by SALPETER the numerical value of Q remains a good indicator of the extent to which the spectrum of the scattered light is dominated by the ion or by the electron dynamics.
As a framework within which to discuss previous theoretical work a brief outline of the present form of the theory of Thomson scattering is useful. The equation of transfer for a scattering system relates the spectral power density of the scattered radiation at the receiver to the power of the incident radiation and the plasma properties. When only scattering due to fluctuations in the electron density is included in the theory then the equation of transfer can be written in the form
^=FO,Ai«„,%±>G . (2.6) Jm 2TT
The elements in equation (2.6) are
dPs/du>: Spectral density of the accepted power at the receiver.
P': Power of the incident beam.
Ok- Beam overlap [BINDSLEV, 1989]. This is a measure of the extent to which the beam patterns of launcher and receiver overlap.
A0: Vacuum wave length.
re: Classical electron radius.
nt'. Average electron density.
S(k,UJ)I Spectral density of the electron density fluctuations. This term is refered to by many names in the literature e.g. spectral density function and spectral form factor.
G: Dielectric form factor (Geometrical factor). This factor describes the coupling of the incident mode to the scattered mode. G has traditionally been referred to as the Geometrical factor. For millimeter waves the term "geometrical" is somewhat misleading since most of the physics entering this factor has to do with dielectric effects.
Superscripts i and s refer to incident and scattered fields respectively. A detailed derivation of the equation of transfer (2.C) is given in Chapter 9. 12 Chapter 2. Thomson scattering
The spectral density function, S(k,u.'), can be written as
S(k,u) = Se(k,u,) + ££-(kf«) (2.7)
where Se(k,u;) is the thermal fluctuation spectrum of the electrons by themselves and S,(k,u;) are the contributions to the electron fluctuations which are driven by the ions of species i. These terms are usually referred to as the electron and ion fea tures respectively. The ions are frequently split into a relatively small populations of highly energetic ions and one or more large populations of (near) thermalized ions, referred to as the bulk ions. The fluctuations associated with each of these populations are usually separated, the fluctuations due to energetic ions being re ferred to as the fast ion or alpha particle features while the fluctuations associated with the bulk ions are referred to as the bulk ion features. Each of the ion features can be split into two terms, one term describing the dielectric properties of the plasma and the electron response and another term which depends only on the ions. In an unmagnetized plasma this term is simply the one-dimensional velocity distribution of that ion species along k with w/Jt as the argument. Expressions for the spectral density function in the electrostatic limit are given by HUGHES and SMITH (19SS). AAMODT and RUSSELL (1990 and 1992) and Cmu (1991 and 1992) have given expressions for the fluctuations of the electron density as well as fluctuations in other quanties based on a fully electromagnetic description of the plasma.
If, in addition to the scattering by density fluctuations, scattering due to other kinds of fluctuations is included then the equation of transfer may take the form
Qp* i — = P'O, AJAJifn, - £S<">(k,u,)G<"> , (2.8) where the summation is over a range of fluctuating quantities (e.g. density and magnetic field). The dielectric form factors, G^n\ associated with each of the fluctuating quantities are in general different.
2.3 Review of previous work on Thomson scat tering
A review of the subject of Thomson scattering was given in the monograph by SHEFFIELD (1975). 2.3. Review of previous vrock on Thomson scut tering 13
2.3.1 Early experiments
The significance of the ion dynamics in the spectrum of scattered electromagnetic waves was first observed experimentally in ionospheric radar scattering experi ments carried out by BOWLES, 195S and 1959 (see discussion in Section 2.2).
HUGHES (1962) first suggested that Thomson scattering of laser light could be used to measure the electron temperature in laboratory plasmas.
An overview of early work on scattering of radio waves in the ionosphere can be found in BOWLES (1964). EVANS and KATZENSTEIN (1969) gave a review of early work on laser scattering in laboratory plasmas.
2.3.2 Scattering cross section when u/' > u/,*
In most laser scattering experiments and in some of the ionospheric scattering with radiowaves the frequency of the probing radiation is much greater than the plasma and electron cyclotron frequencies. This kind of scattering is frequently referred to as incoherent scattering, not to be confused with the scattering not being collective. For these conditions the scattering cross sections can be found by summing over Lienard-Wiechert potentials [LANDAU and LiFSllITZ, 19S7, §63]. Derivations of the scattering cross sections for these conditions wtre given by many authors, for instance EVANS and KATZENSTEIN (1969). See also HUTCHINSON (1987a).
2.3.3 Cross sections and transfer when J' « u;^
In Thomson scattering the dielectric properties of the plasma must be taken into account when the frequency of the probing radiation is not considerably greater than the electron cyclotron frequency and the plasma frequency. This is usually the case for collective scattering diagnostics, but generally not for laser scatter ing intended for electron temperature and density measurements. It is necessary to consider the propagation of a beam of radiation through an inhomogeneous anisotropic plasma, and the coupling of the incident radiation to the scattered ra diation via fluctuations in the plasma. The latter involves determining the source currents assuming the fluctuations to be known and determining the fields which result from these currents.
The problem of wave propagation in an anisotropic plasma was addressed by LIGHTHH.I. (19G0) and MERCIER (1964). MERCIER also made some initial in vestigations of the propagation of radiation when the plasma is inhomogeneous, 14 Chapter 2. Thomson scattering
using Liouville's equation to describe the evolution of an ensemble of independent light particles in one particle phase space (r,k). BEKEFI (1966) investigated the propagat ion of electromagnetic waves in an anisotropic inhomogeneous phv ma giv ing expressions which are readily applied to the modelling of a collective Thomson scattering diagnostic.
The field recalling from a current distribution embedded in an unmagnetized plasma described by Vlasov's equation was investigated by BIRMINGHAM (1963). The result was applied to the scattering of electromagnetic waves by density fluc tuations, where the source currents resulting from the interaction were obtained on the basis of a cold fluid model.
AKIIIEZER et al. (1962 and 1967) and SlTENKO (1967) gave expressions for the scattering cross sections associated with fluctuations in electron density, electron fluid velocity and electric and magnetic fields in a cold plasma.
The theoretical results derived by AKIIIEZER tt al. (1967) and SlTEXKO (1967) are identical apart from a minor misprint in SlTENKO's equation (11.5). Their results an- much quoted and will therefore be given special attention. AKHIEZER et al. and SlTENKO determined the source currents which drive the scattered field (see S«*ctif>ns 2.4 and 1.2) on the basis of the cold fluid equations. Their derivation is however not correct. The principal mistake lies in an inappropriate manipulation of the f.uid velocity which results in the loss of significant terms. Furthermore, the momentum equation that they use leaves out terms which are significant to the same order as terms that are included. Their results are indeed found to be at variance with the cold plasma limit of the results derived on the basis of a kinetic treatment (see AAMODT and RUSSELL (1992) and Sections 7.2 and 7.3 in this thesis). The differences between Ak'HIEZER et al. and SlTENKO's expression for the source current and the cold plasma limit of the expression derived from the kinetic model are discussed further at the end of Section 7.3.
Expressions for the source current in an unmagnetized plasma were derived by AKHIEZER, AKHIEZER and SlTENKO (1967) in the cold plasma limit on the basis of the kinetic equations. Unfortunately there is a mistake in this derivation which brings their kinetic result into agreement with the result they obtained with the fluid description. For a magnetized plasma only the fluid approach is taken, leading to the same result as that given by SlTENKO (1967).
SlMONIcii (1971) and SlMONICH and YEH (1972) investigated the scattering cross sect ions associated with electron density fluctuations in a cold plasma allowing for the «ni5ofropy of a magnetized plasma. The equation of transfer was given for a scattering system, but the effect of an inhomogencous plasma was not taken into account. 2.3. Review of previous work on Thomson scattering 15
The spatial resolution and wave vector resolution obtainable with a scattering diagnostic using coherent detection was analysed by HOLZHAl'ER and MASSIG {197S).
BRETZ (19S6) investigated the scattering cross section for X-mode radiation prop agating perpendicular to the magnetic field. This investigation was generalized by Hl'GHES and SMITH (19S9) to scattering from either characteristic mode (0 or X) into either mode for a general scattering geometry (ie. no limitations were placed on the directions in which the radiation propagates). The equation of transfer for a single moded (diffraction limited) scattering system was given, allowing for the ef fect of an inhomogeneous plasma. In both these papers the scattering was assumed to be due only to electron density fluctuations and the treatments were based on the cold plasma model. BINDSLEV (1991 a and 6) generalized the treatment given by Hl'GHES and SMITH (19S9) to a weakly relativistic plasma. Expressions for the equation of transfer and scattering cross section were derived assuming an inhomogeneous, anisotropic and spatially as well as temporally dispersive plasma. Predictions based on weakly relativistic, hot and cold plasma models were com pared. The scattering was assumed to be due to fluctuations in the conductivity tensor induced by electron density fluctuations only.
The investigation by AAMODT and RUSSELL (1990 and 1992) included scattering due to fluctuations in the electric field, the magnetic field, the density and the fluid velocity interacting with the same set of quantities relating to the incident field. Their derivation was based on a kinetic description of the plasma. To simplify the problem a number of assumptions were made which essentially limit the applicability of tlu* result to a cold plasma. Expressions were given for the source current of the scattered field and the differential scattering cross-section. (See also the discussion at the end of Section 7.3 below.) Corrections were found to expressions derived on the basis of the cold fluid model and new terms were identified.
2.3.4 Fluctuations and the spectral form factor, 5(k,w)
AKIIIEZER. PROKIIODA and SlTENKO (1958) (submitted March 1957) predicted the collective scattering of electromagnetic waves by fluctuations in the electron density. This was before the publication of the surprising results from BOWLES' ionospheric radar scattering experiments (see Subsection 2.3.1), which prompted a number of authors to investigate the fluctuation spectrum of the electrons taking the interaction with the slower moving ions into account. Most athors restricted attention to the density fluctuations (notable exeptions are AkIIIE/ER r.t al. (19G2 arid 19G7) and SlTENKO (19G7)). This is justified when *•' » u;pe and the plasma is not relativistir as was generally the case for the early experiments. 16 Chapter 2. Thomson scattering
Expre' sions for the spectrum of fluctuations are generally derived by one of two ap proaches: (a) based on the fluctuation dissipation theorem [LANDAU and LlFSHITZ (19S6), §124, 125] and (6) based on solving the full set of kinetic equations with appropriate initial conditions [LlFSHITZ and PlTAEVSKII, 19S1; KLIMONTOVICII, 19S2J.
The first approach, (.z), was adopted by DouGHERTY and FARLEY (1960) while FEJER (1960) and SALPETER (1960) adopted the second approach, (6) (see also the discussion in Section 2.2). These papers, which did not include the effect of a magnetic field, were able satisfactorily to explain BOWLES' experimental results.
HAGFORS (1961), FARLEY et d. (1961) and SALPETER (i961) included the effect of a magnetic field, HAGFORS solving the problem by the kinetic approach. He found that over most of parameter space the presence of a magnetic field only manifests itself for fluctuations with wave vectors close to perpendicular to the magnetic field. The effect here is to modulate the spectrum with the period of the ion cyclotron frequency, provided a ^> 1.
Independent of this work, Ak'HIEZER, AKIIIEZER and SlTENKO (1962) (see also AKUIEZER et a!. (1967)) derived expressions for the fluctuations in density, cur rent, fields and the velocity distributions, all for magnetized equilibrium or quasi equilibrium plasmas. The fluctuation dissipation theorem was used in these deriva tions including the derivation of the fluctuations in the velocity distribution. It is worih noting that this treatment was fully electromagnetic (ie. the electric fields were accounted for by the wave equation rather than Poisson's equation).
RoSENBLUTH and ROSTOKER (1962) addressed the problem of fluctuations in a non-equilibrium plasma. Their approach involved the determination of the two time distribution function from i truncated hierarchy of equations linking different orders of distribution functions [ROSTOKER, I960]. They show that their results have a simple interpretation in terms of dressed •particles and that the results can be obtained by a simple and intuitively appealing approach known as the dressed •particle approach. This approach has since become a popular way of deriving expressions for the fluctuation spe:trum. A particularly clear exposition of the method was given by BERNSTEIN el al. (1964) for the relatively simple case of an unmagnetized plasma.
DuBois and GlLINSKY (1964) made a quantum mechanical investigation of inco herent scattering, taking the effect of Coulomb collisions into account.
In a review of incoherent scattering EVANS and KATZENSTEIN (1969) also reviewed the theory of density fluctuations, deriving expressions for the spectral density of the fluctuations using the dressed particle approach. Most attention was given to the unmagnetized case, though results for a magnetized plasma [HAGFORS (19G1)] 2.3. Review of previous work on Thomson scattering 17
were quoted.
PAVLENKO and PANCHENKO (1990) considered scattering in the presence of a pump wave in lower hybrid range of frequencies. SlTENKO's expression for the source current was used. The fluctu- tions were determined by solving the kinetic equations with the field given in the electrostatic limit.
Up to this point most expressions for the density fluctuations were derived on the basis of the electrostatic approximation (see Chapter S) with the work by AKHIEZER et al. (1962 and 1967) and SlTENKO (1967) being prominent excep tions. Fully electromagnetic expressions for the density fluctuations were derived by AAMODT and RUSSELL (1990 and 1992), and ClIIU (1991 and 1992). They showed that the spectrum of density fluctuations predicted with the electromag netic model can differ significantly from the predictions based on the electrostatic model, particularly in the range of frequencies around the lower hybrid resonance.
In AAMODT and RUSSELL'S treatment all fluctuations are calculated using a full kinetic description of the plasma (approach b) allowing for electromagnetic effects.
ClIIU (1991 and 1992) derives his expression for the density fluctuation spectrum from a generalized dressed particle approach allowing for electromagnetic effects. Particular attention is paid to fluctuations around the slow and the fast branch of the dispersion relation at frequencies above the lower hybrid resonance and the fast magnetosonic wave which propagates at frequencies below the lower hybrid resonance. A considerable enhancement of the power density in the fluctuation spectrum is found around frequencies which satisfy the dispersion relation. Ac cording to CHIU the enhancement around the slow wave appears in the same fre quency range as the enhancement around the lower hybrid resonance found with the electrostatic theory. Although the enhancement is large the bulk ions usually dominate the spectrum in the range of parameters in which the enhancement oc curs, making the extraction of fast ion parameters difficult. The fast magnetosonic wave, on the other hand, propagates in a wide parameter range where the fast ion feature is not obscured by the bulk ions. The fast magnetosonic wave, although it is predominantly longitudinally polarized, has a non-trivial transverse part in ics polarization, which is why it is not picked up by the electrostatic approxima tion. HUGHES (private communication, 1991) has shown that the maximum in the fluctuation spectrum cV.culated by AAMODT and RUSSELL (1990) is also at the frequency of the fast T iagnetosonic wave.
ClIIU speculates that the enhancement aro i..d the fast wave can be used with great benefit in alpha particle and other fast ion collective scattering diagnostics. See also the discussion of a paper by WONG in Subsection 2.3.10 below. 18 Chapter 2. Thomson scattering
2.3.5 Electron temperature and density measurements
The measurement of electron temperature and density by Thomson scattering is a well established diagnostic technique (see e.g. LUHMANN and PEEBLES, 1984; Hutchinson, 19S7 a). Non-collective scattering is required such that the Doppler shift of the scattered radiation reflects the thermal motion of the individual elec trons rather than their collective motion. This is ensured if a = 1/fcAp <§; 1 which for most laboratory plasmas implies the use of lasers. As a result it is generally 1 the case that w > ujpe and hence the scattering cross sections can be obtained by summation of the Lienard-Wiechert potentials resulting from the individual elec trons. For non-relativistic plasmas, only the scattering due to density fluctuations need be considered [Hutchinson, 19S7 a].
2.3.6 Turbulent density fluctuation measurements
The first measurements of low frequency density fluctuations in a Tokamak by collective Thomson scattering were made by MAZZUCATO (1976) on the ATC Tokamak using a 70 GHz, 20 W microwave beam launched into the plasma in 0-mode.
Measurement of microturbulence by collective Thomson scattering of CO2 laser light was demonstrated by SURKO and SLUSHER (1976), also on the ATC Tokamak.
A review of collective Thomson scattering of COj laser light was given by SLUSHER and SURKO (1980). An alternative experimental setup which makes use of light scattered from two intersecting CO2 laser beams was described in SURKO and SLUSHER(19S0).
A Fourier optics approach to understanding far forward scattering was given by EVANS, HELLERMANN and HOLZHAUER (19S2)
A review of available sources in the FIR to millimetre wavelength range and their potential for use in collective Thomson scattering was given by WOSKOBOINIKOW, CoilN and TEMKIN (19S3) (Woskoboinikow is now Woskov).
LUHMANN and PEEBLES (1984) in their review of diagnostics for magnetically confined fusion plasmas gave an overview of the techniques involved in collective Thomson scattering from the far infra red to the millimetre wave region.
In i\ review comparing measurements of microturbulencc with theory LlEWER (19S5) gave an extensive overview of experimental results on density fluctuations obtained with collective Thomson scattering. 2.3. Review of previous work on Thomson scattering 19
Experimental results on the scattering of 137 GHz gyrotron radiation by density fluctuations in the Tara Tandem Mirror axicell were reported by MACHUZAK et al. (19S8). See also MACHUZAK (1990).
The consequences of scattering in a regime intermediate to the Bragg regime and the Raman-Nath regime (that is when k2L/kl « 1 where L is the width of the fluctuation prependicular to k) was discussed by DOYLE and EVANS (1988).
A FIR 1222 //m collective scattering diagnostic with heterodyne detection in stalled on TEXT was described and experimental data presented by BROWER et al. (19S8).
BRETZ et al. (19SS) described a collective scattering diagnostic operating in X- mode at 60 GHz on TFTR. While most scattering diagnostics for studying density fluctuations had been operating in 0-mode, this diagnostic, by operating in X- mode below the electron cyclotron frequency was able to probe fluctuations with longer scalelengths at densities where the O-mode would have been cut off. Initial results were presented by BRETZ, et al (1990 a) and BRETZ, NAZIKIAN and WONG (1990 6).
2.3.7 Bulk ion temperature measurements
Bulk ion measurements have been demonstrated on a number of low temperature laboratory plasma devices [HoLZHAUER, 1977; KASPAREK and HoLZHAUER, 1983 a; LACHAMDRE and DECOSTE 19S5].
Numerical simulations to determine the feasibility of measuring the bulk ion tem perature by collective Thomson scattering in the presence of noise have been car ried out by SHARP et al. (1981), WATTERSON et al. (1981) and SlEGRIST et al. (1982). The performance over a wide range of parameters were investigated. SlEGRIST et al. included measurement of the direction of the magnetic field in their investigation. A mainly analytic investigation of the ion feature is given by GRÉSILLON et al. (1984). More recently a study was carried out by ORSITTO (1990) to establish the feasibility of obtaining information about the bulk ions and impurity ions, including the bulk ion temperature, from the collective Thomson scattering diagnostic at JET.
BEIIN et al. (19S9) reported the first successful measurements of the bulk ion temperature in a Tokamak by collective Thomson scattering. A D20 laser pumped by a powerful C02 laser was used as the source of the probing radiation for the TCA Tokamak. 20 Chapter 2. Thomson scattering
2.3.8 Impurity ion measurements.
Impurity ions may significantly distort the fluctuation spectra. While this can be upsetting for bulk ion measurements this does provide a means of obtaining in formation about the impurities [EVANS and YEOMAN, 1974; BRETZ, 1977; SHARP et al, 19S1; KASPAREK and HOLZHAUER, 1983 a; ORSITTO, 1990 (see Subsection 2.3.7)]
2.3.9 Magnetic field measurements.
Localised measurement of the pitch angle of the magnetic field by Thomson scat tering has been proposed by a number of authors. One approach relies on the fact that the spectrum of scattered radiation is modulated with the period of the elec tron cyclotron frequency when the scattering wave vector is perpendicular to the magnetic field. The depth of the modulation decreases rapidly as k moves away from perpendicular to the magnetic field. BRETZ (1974) proposed measuring the field direction, and from that the current, by scanning the direction of k to find the maximum modulation in the spectrum of scattered radiation. Recently this method has been proposed as a diagnostic for JET [CAROLAN et al. (1990)].
An alternative approach relying on the modification of the ion feature (for defini tion of ion feature see expression (2.7) and subsequent discussion) when k is near perpendicular to the magnetic field was suggested by SlEGRIST et al. (1982) and further explored by HOLZHAUER and KASPAREK (1984). Experimental verification of this approach was given by KASPAREK and HOLZHAUER (1983).
WOSKOV and RlIEE (1992) have recently suggested using the spectral location of the lower hybrid resonance as a way of determining the direction of the magnetic field.
The possibility of localized measurement of magnetic field fluctuations with Thom son scattering is investigated by LEHNER, RAX and ZOU (1989), HAAS and EVANS (1990) and VAHALA, VAHALA and BRETZ (1992 and 1990). They noted that, for propagation perpendicular to the magnetic field, fluctuations in a scalar quantity such as tlv. density cannot scatter an incident characteristic mode (O or X mode) into the other characteristic mode. Perpendicular fluctuations in the magnetic field do. however, scatter from one characteristic mode to the other even when only radiation propagating perpendicular to the magnetic field is considered. This provides a special oportunity to observe the scattering due to fluctuations in the magnetic field which normally would be masked by scattering due to density fluc tuations. An experimental setup is proposed and investigated in some detail by VAHALA, VAHALA and BRETZ (1992). 2.3. Review of previous work on Thomson scattering 21
2.3.10 Fast ion measurements.
The feasibility of using collective scattering of COi laser light for diagnosis of fusion produced alpha particles was investigated by D. P. HUTCHINSON et al. (1985).
In a review of gyrotron performance relative to other millimetre to submillimetre sources WoSKOBOINIKOW (19S6) first suggested the use of gyrotron radiation for diagnosis of alpha particles and other fast ions by collective Thomson scattering (see also WOSKOBOINIKOW, COHN and TEMKIN, 1983).
VAHALA, VAHALA and SlGMAR (1986, 1988) and HUGHES and SMITH (1988) considered the spectrum of density fluctuations in a plasma with a population of fusion produced alpha particles with a classical slow down distribution [GAFFEY, 1976]. Expressions for the fluctuations were used which assumed a magnetized plasma and the electrostatic approximation. The alpha particle contribution to the fluctuation spectra was investigated for a wide range of parameters relevant for JET- and TFTR-like plasma conditions and for diagnostic systems based on CO2 laser light and on gyrotron radiation.
Following preliminary work by HUGHES (19S6), a study was made by HUTCHINSON (1987 b) to determine the feasibility of diagnosing fusion alpha particle populations in large Tokamaks by collective Thomson scattering of gyrotron radiation.
A fast ion collective Thomson scattering diagnostic using 140 GHz gyrotron ra diation was proposed for JET in a report by COSTLEY et al. (19S8). Physics issues relevant to this diagnostic were presented in COSTLEY et al. (19S9 a) and technical aspects discussed in COSTLEY et al. (19S9 b).
A preliminary study of the feasibility of using a 60 GHz, 200 kW gyrotron for diagnosis of fusion alphas in TFTR was carried out by WOSKOV et al. (1988).
A CO2 laser based fusion alpha collective Thomson scattering diagnostic for CIT was discussed by RICHARDS et al. (1988).
An overview of ongoing developments of millimetre to submillimetre collective scattering diagnostics for measurements of fast ion populations in large Tokamaks was given by MACHUZAK et al. (1990).
A design study was made by SllEFER et al. (1990) of a fast ion collective Thomson scattering diagnostic based on a 152 /an free election laser (FEL). A similar FEL based system (A0 = 200//m) was proposed for ITER by COSTLEY et al. (1990 b) while ZllUKOVSKY et al. (1991) proposed a C02 laser based system for ITER. 2? Chapter 2. Thomson scattering
Expressions for the fluctuation spectra resulting from arbitrary fast ion velocity distributions represented by expansions in orthogonal polynomials were derived in the electrostatic approximation, and numerical calculations for various non- equilibrium distributions were given by RllEE et al. (1990).
WONG (1991) has proposed a novel experimental setup intended to use only scat tered radiation from that part of the spectrum which is enhanced due to the proximity of a frequency satisfying the dispersion relation. The technique assumes an isotropic velocity distribution for the fast ions. An array of detectors is placed on a circle which is centred on the line which extends from the emitting antenna. Assuming the probing radiation is high enough that refraction can be neglected then the detectors all pick up scattered radiation which has been scattered through one given angle. The norms of the scattering wave vectors are therefore the same for all the receivers. However, the component of the scattering wave vector parallel to the magnetic field varies from one detector to the next. Since the frequency at which the dispersion relation is satisfied is very sensitive to the parallel component of k it follows the the enhancement will occur at different frequencies in the differ ent detectors and thus the detectors will sample different parts of the spectrum. WONG shows that this setup may in principle yield the information required to determine the fusion produced alpha particle velocity distribution, though more work is clearly needed to establish the viability of this approach.
2.4 Outline of the relevant scattering theory
The modelling of Thomson scattering is traditionally split into two major parts:
(A) the determination of the fluctuations in electron density and other quantities which give rise to scattering, and
(D) relations between incident and scattered fields, given the fluctuations.
Part (.4) typically deals with the spectral density function S(k,w). This subject is discussed briefly in Chapter S. Part (B), which is the main subject of the present thesis, is concerned with scattering cross sections and equations of transfer between launcher and receiver in a scattering diagnostic system. If vacuum propagation is assumed the task is a relatively straightforward summation of Lienard-Wiechcrt fields (see e.g. HUTCHINSON, 19S9). This is, however, only an acceptable ap proximation when «,•' » wy. When this condition is not satisfied the dielectric properties of the plasma must be taken into account. This has been done in the cold plasma approximation by a number of authors [AKHIEZER tt al., 1958, 1962 and 19G7; SlTENKO, 19G7; SlMOMCII. 1971; SlMOMCH and YE1I, 1972; BRETZ, 2.4. Outline of the relevant scattering theory 23
19S7; HUGHES and SMITH, 1989]. Here the theory of dielectric effects on Thomson scattering is extended to hot and relativistic plasmas [BINDSLEV, 1991 a and b].
A major motivation for the present work is the fact that in the millimetre wave collective Thomson scattering diagnostics at JET and TFTR [COSTLEY ti al, 19SS; WOSKOV et al., 19S8] the frequency of the probing radiation is in general not high relative to the electron plasma frequency. This means that a number of simplifying assumptions, which are well satisfied for the majority of laser scattering experiments, are not valid. Most notably, vacuum propagation cannot be assumed. In the work by HUGHES and SMITH (1989), the Thomson scattering cross section and the scattered power accepted by the receiving antenna were investigated for a cold plasma. It was shown that, in the parameter ranges relevant for the planned alpha and fast ion Thomson scattering diagnostic at JET, it is necessary to take the dielectric properties of the plasma into account. The dielectric effects manifest themselves in the term referred to as the dielectric form factor (geometrical factor), G. Relative to predictions based on vacuum propagation, large increases in the scattered power were found for X to X mode scattering, with a singularity at the R-cutoff (ordinary and extraordinary mode are referred to as 0 and X mode respectively). X to X scattering may therefore be an attractive option. At the high temperatures found in large tokamaks, the scattering theory based on the cold plasma model is, however, not reliable in this regime: relativistic effects must be taken into account.
In this thesis the theory of Thomson scattering in a magnetized plasma with spatial dispersion is developed ab initio. Spatial dispersion, which is caused by thermal motion, must be taken into account when describing the plasma by hot or relativistic models. Expressions for the differential scattering cross section and the dielectric form factor (geometrical factor) G are derived.
To obtain an outline of the processes involved in a scattering experiment it is helpful to follow the radiation from the emitting antenna through the plasma to the receiver, identifying the various processes along the way.
a. An incident field is launched into the plasma from an antenna located out side the plasma. The field in the scattering region depends on the antenna properties, and on the plasma properties along the path from the antenna to the scattering region.
b. In the plasma, fluctuations occur. The characteristics of these fluctuations depend on the plasma parameters.
c. The incident field interacts with the fluctuations giving rise to a set of cur rents. 24 Chapter 2. Thomson scattering
d. These currents in turn give rise to an electromagnetic field which is the scattered field.
e. A part of the scattered field propagates from the scattering region to the receiving antenna located outside the plasma. The fraction of the scattered power received by the receiving antenna depends on the plasma properties along the path from the scattering volume to the antenna, and on the prop erties of the antenna.
To solve the scattering problem to the extent that quantitative predictions can be made about the outcome of a scattering experiment, given information about the relevant plasma parameters, each of the processes outlined above must be taken into account. While the complete problem can be split into a number of largely independent subproblems the results of each subproblem is generally the input required for the next subproblem (e.g. the output of process 6, the fluctuations, is an input into process c which gives the source currents for the scattered field). The form of the input required for a particular subproblem will in a number of cases depend on the way in which the subproblem is solved (e.g. which approximations have been made). For this reason it is more convenient to solve the scattering problem in the reverse order of that indicated in the outline above. Due to the similarities between processes a and e it is however most logical to consider these two together. Developing a theoretical model of a Thomson scattering diagnostic system can thus logically be split into the following subproblems to be considered in the order indicated:
1. Given knowledge of the plasma from the plasma edge to the scattering region, and of the antenna properties, determine the relation between the power at the antenna and the field in the scattering region. Field strengths as well as direction and spread of wave vectors must be determined. Due to the reciprocity between operating an antenna as an emitter and as a receiver, and the reciprocity between forward and reverse propagation in a plasma, the treatments for the incident and the scattered fields are essentially identical. This step covers the processes a and e in the outline given above.
2. Given the plasma conditions and a set of currents in the scattering volume, determine the scattered field arising from these currents. This covers d above.
3. Given the plasma conditions, the incident field and the plasma fluctuations in the scattering volume, determine the currents which arise from the inter action between the incident field and the fluctuations. This covers c above. 2A. Outline of the relevant scattering theory 25
4. Given the plasma conditions in the scattering volume, determine the plasma fluctuations in this volume. This covers 6 above.
The subproblems 1 to 4 form the subjects of Chapters 5 to 8 respectively.
While a theoretical model of scattering is most readily formulated as a prediction of the outcome of an experiment given all relevant plasma parameters, the practical objective is of course to estimate some of the parameters from the outcome of the experiment. Even with perfect experimental data, limitations may still exist on the range of plasma parameters which can be inferred because the mapping from experimental data to plasma parameters may not be unique. In practice the quality of the experimental data will place more severe limitations on the range of parameters which can be inferred.
Despite these limitations a model, as outlined above, can still be used to interpret experimental data and obtain useful information. In practice it is assumed that most of the plasma parameters, on which the model depends, are known either from other diagnostics or from reasonable guesses. The remaining parameters are estimated by fitting the predictions of the model to the experimental data. Which parameters are assumed to be known and which are to be estimated depends on the experimental situation, the available information and the sensitivity of the model to the various parameters.
If the objective of the scattering experiment is to diagnose the fluctuations in the plasma then only steps 1 to 3 in the model are required. For diagnosis of fast ions and alpha particles the fluctuations are only an intermediate result and step 4 must be included. Chapter 3
The Collective Thomson Scattering Diagnostic at JET
3.1 Choice of frequency for the JET collective scattering diagnostic
The most important decision to be made when designing a diagnostic system based on collective Thomson scattering is the choice of frequency of the probing radiation. The decision must be based on the following considerations:
(1) Availability of sources. On the basis of various considerations, the most important of which are discussed below, it is found that the probing ra diation must be in the millimeter to far infra red (FIR) range. The very small scattering cross section (received power in the interesting part of the scattered spectrum is typically 15 orders of magnitude down from the in cident power) places considerable demands on the power of the source of the probing radiation, which in practice can only be met at present by gy- rotrons and COi lasers. In the longer term free electron lasers may come into consideration.
(2) Accessibility. The probing radiation must be able to penetrate the plasma to the region of interest. The layers to avoid are, for radiation propr.gating in the ordinary mode, the cutoff at the plasma frequency; for radiation in the extraordinary mode, the right and left hand cutoffs; and for propagation in either mode, the fundamental cyclotron resonance layer and the cyclotron harmonic resonance layers.
26 31- Choice of frequency for the JET collective scattering diagnostic 27
(3) Limited refraction. Even if the region of interest is accessible, the diagnos tic will only be useful if the refraction is sufficiently small so that overlap of launcher and receiver beam patterns can be ensured and the scattering geometry reliably defined.
(4) Suitable scattering geometry. To distinguish the feature due to the alpha parti' from the electron feature in the spectrum of the scattered radiation
for cue alpha particle densities expected in JET plasmas (n0 R; ne/500), the Salpeter parameter, a, must as a rule of thumb be greater than 2 [HUTCHIN SON, D.P., 19S5; HUTCHINSON, I.H., 1987 b]. This requirement limits the range of the angle, 0, between incident and scattered wave vectors, which can be used for a particular choice of probing frequency:
2sin0/2<^Z^ . (3.1) 4rAjr>
A0 is the vacuum wavelength of the probing radiation and /* is the refractive index of the plasma in the scattering region. For a Debye length of A/> = 100/im, which is typical for a fusion plasma, equation (3-1) leads, for COi
laser light (A0 = 10.6/im, u ~ 1), to
C02 : 0 < 0.5° . (3.2)
At small scattering angles poor spatial resolution along the beams and stray radiation entering the receiver are serious problems. The use of 2 mm gyrotron radiation (/* < 1) leads to
2mm : 0 < 105° , (3.3)
which offers much greater choice of scattering geometry with good spatial resolution.
(5) Background plasma emission. In the millimeter wavelength region radia tion from the plasma is dominated by electron cyclotron emission (ECE). To minimize the ECE background the probing frequency can cither be placed below the first harmonic or in the dip between the first and second har monics (in low aspect ratio machines like JET and TFTR the overlap in the emission from neighbouring harmonics is too great at higher harmonics). Alternatively the frequency of the probing radiation can be placed beyond the ECE spectrum where the background is dominated by Brcmsstrahlung and the recombination connnuum.
On the basis of considerations 2-5 above it was found that the optimum wave length of the probing radiation for the JET collective scattering diagnostic is ~ 2S Chapter 3. The Collective Thomson Scattering Diagnostic ait JET
200/im [HUGHES. 19S6J. Suitable sources are however not presently available. Taking availability of sources into account, probing with microwaves at 140 GHz (A« = 2.14mm) was found to be the best choice for the JET diagnostic (COSTLEY, 19SS]. This places the frequency of the probing radiation between the fundamental and the second harmonic of the electron cyclotron frequency.
The use of gyrotrons for fast ion collective scattering diagnostics was first sug gested by WOSKOBOINIKOV, (19S6). A fast ion collective scattering diagnostic is presently also under development at TFTR [WOSKOV, 19SS]- This system will use microwaves at 56 GHz which is below the ECE spectrum.
A recent overview of ongoing work was given by MACHUZAK et ml. (1990).
3.2 Sketch of the JET collective scattering diag nostic
Detailed descriptions of the fast ion collective scattering diagnostic under develop ment at JET are given by COSTLEV et «/. (19SS and 19S9). The main components of the system are shown in Figure 3.1.
The system includes the following components:
Gyrotron. The high RF power required will be delivered by a gyrotron. A gy- rotron works by feeding energy from a beam of accelerated electrons into an electromagnetic field which is resonant in an RF cavity. The coupling is achieved by imposing a static magnetic field in the RF cavity which causes the electrons to gyrate at a frequency that almost matches a resonant fre quency of the cavity. The difference between the two frequencies is necessary in order to convert the kinetic energy of the electrons into RF power. The gyrotron to be used at JET is produced by Varian, USA [FELCH, 19S7]. It operates at 140 GHz and excites the TEuj.i whispering gallery mode. The indices refer to the azimuthal, radial and longitudinal (in cavity) mode num bers. The cavity in the gyrotron is fed by a beam of 80 keV electrons which at a beam current of 17 amperes produces 400 k\V of RF power in pulses of up to 5 seconds duration. Up to 1 MW of RF power can be produced in shorter pulses with a beam current of 35 amperes. It should be possible to change the output frequency from 119 GHz (TE12.2.1)
to 147 GHz (TF.|rt.2.i) i» steps of 7 GHz by varying the static magnetic field, each step changing the azimuthal mode number by one.
Anode voltage modulator. In order to distinguish the scattered signal from the 3.2. Sketch of the JET collective scattering diagnostic
MA tOOkV
Figure 3.1: Schematic diagram of the fast ion collective scattering diagnostic being developed at JET. 30 Chapter 3. The Collective Thomson Scattering Diagnostic at JET
background ECE, which is typically three orders of magnitude more intense than the signal, it is necessary to chop the signal. This is done by switching the electron beam feeding the gyrotron on and off at a frequency of 10-30 kHz. This relatively high frequency (considering the power in the electron beam) is desirable in order to minimize the change in plasma parameters, including the increase in ECE due to heating by the probing beam, inside each chopping cycle.
Vlasov converter Although the whispering gallery mode (WGM), excited in the gyrotron, can propagate in a waveguide with relatively low losses it suffers considerable mode conversion in bends and is not suitable for launching into the plasma. More suitable is the HEn mode which couples well to a free space Gausian beam. Conversion into this mode is achieved with a Vlasov converter [VLASOV, 1974 and 1975] which converts the WGM into a colli- mated linearly polarized beam propagating through free space. This beam is then focused and, after passing through a universal polarizer, launched into the waveguide as a HEn mode. The Vlasov com ei ter is sketched in Figure 3.2.
/ Cylindrical mirror / \
Straight cut \, Waveguide
/ JG92 518/4 Helical cut
Figur'o e 3.2: Sketch of Vlasov converter.
A simple picture of the principle behind the Vlasov converter can be given in the geometrical optics limit. In this limit the WGM can be represented by beams propagating close to the waveguide wall suffering multiple reflections as they wind their way forward in the waveguide. The pitch angle of the 3.2. Sketch of the JET collective scattering diagnostic 3_1
screw that the ray trajectory forms is given by the relation between the wavenumbers parallel and perpendicular to the wave guide. The surfaces of constant phase form screws with the opposite handedness to the ray screws and intersect the rays at right angles. The Vlasov converter consists of two elements, an end piece of waveguide cut in a particular way and a parabolic mirror. The cuts in the waveguide form two lines, one parallel to the axis of the waveguide and the other making a helix with one turn, the ends of the two cuts joining up. The pitch angle of the helical cut is identical to the pitch angle of the ray trajectories inside the waveguide. Now consider a ray. It spirals its way down the waveguide, being bent around by many reflections in the waveguide wall. At some point it passes the straight cut in the waveguide (note that no rays pass the helical cut) and continues on a straight free-space trajectory. The direction in the plane perpendicular to the waveguide axis depends on how far nto the waveguide the ray suffered its last reflection, while the angle to the waveguide axis is equal to the pitch angle. The surfaces of constant phase leaving the cut waveguide form cones with the straight cut as the axis. After reflection in the cylindrical mirror with parabolic cross section and focal line coinciding with the straight cut in the wave guide, the surfaces of constant phase are flat. This normally results in a beam where approximately SO % of the power is in a Gaussian beam (TEMoo mode) which couples very efficiently (approximately 99 %) to the HEn mode in the wave guide, giving an overall efficiency of approximately 80 %. Recent improvements in Vlasov converters are aimed at producing a power distribution which is more nearly Gaussian.
Universal polarizer. To ease the interpretation of the spectrum of scattered ra diation it is desirable to couple both the launched and the received beams to one of the characteristic modes of the plasma. These modes are in general elliptically polarized. The radiation co:ning from the Vlasov converter is lin earis notarized. On the receiver side the mixer only accepts radiation which is linearly polarized along a particular direction. Polarizers capable of con verting linearly polarized radiation into an arbitrary elliptical polarization state are therefore required. On the launch side this is achieved with two plane mirrors arranged as a roof top and a flat corrugated reflector placed above the roof top [JOHNSON, 1977] (see sketch in Figure 3.3). For linear polarization incident on the polarizer the ollipticity of the outgoing polarization is varied by rotating the orientation of the grooves relative to the direction of the polarization that is incident on the grooved reflector, while the orientation of the major axis of the polarization elipse is rotated by rotating the whole polarizer around the axis of the incoming and outgoing radiation. On the receive side the conversion from an arbitrary elliptical polarization is achieved with a rot a table half wave minor and a rotatable quarter wave 32 Chapter 3. The Collective Thomson Scattering Diagnostic at JET
Grooved mirror mnvmfT
Plane mirrors
Figure 3.3: Universal polarizer on the launching side.
Quarter wave mirror
Half wave mirror
Figure 3.4: Universal polarizer on the receiving side. 3.2. Sketch of the JET collective scattering diagnostic 33
mirror arranged as shown in Figure 3.4. The half wave and quarter wave mirrors each consist of a wire grid placed in front of a mirror at a distance of a quarter and an eights of a wavelength respectively. The effect of the quarter wave mirror is to change the ellipticity of the polarization while the half wave mirror rotates the major axis.
Transmission wave guide. The probing radiation is transmitted from the out put of the universal polarizer to the torus through a wide-band corrugated wr..eguide. The transmission line has a diameter of 88.9 mm, is 55 meters long and includes 5 quasi-optical mitre bends. The corrugation and low conductivit of the waveguide attenuate spurious modes while leaving the HEn largely unaffected [BARKLEY, 19SS]. The received scattered radiation is transmitted from the torus to the detection system and tracking system (see Launcher and receiver below) through three (two for the tracking system) similar transmission lines though with diameters of 30 mm for the radiation going to the detection system and 10 mm for the radiation going to the tracking system. Close to the torus the receiver lines for the tracking sys tem are tapered down to fundamental waveguide to reject unwanted modes. In the waveguide going to the receiver unwanted modes are attenuated by corrugation in the waveguide.
Launcher and receiver. The launching and receiving antennas will be similar apart from the addition of two slave antennas on the receiving side. The launcher will be placed at the top and the receiver towards the bottom of the vessel. The directions of the beams can be changed on a shot to shot basis. During a shot the receiver beam automatically tracks the incident beam by comparing the scattered power levels received by the two slave antennas. By changing the directions of the beams the volume of plasma being probed i,an be shifted. The scattering wave vector, k, (see equation (2.2)) can also be changed, allowing the one dimensional velocity distributions along various directions to be investigated.
Detection system. The principal components of the heterodyne detection sys tem are:
Notch filter. This is a filter with high rejection in a very narrow band around the probing frequency, intended to reject stray radiation from the launcher which has been picked up by the receiving antenna. It is necessary to reject or strongly attenuate this stray radiation in order not to saturate the mixer. Local oscillator and mixer. The signal (note that this includes the scat tered signal and the background noise) is mixed with the output from a local oscillator operating at 128 GHz. The mixer is a nonlinear photo electric device where ideally the instantaneous photo current (output) is proportional to the the square of the incident field strength. The 34 Chapter 3. The Collective Thomson Scattering Diagnostic at JET
photo current contains elements which can be attributed to the signal only (ijj), the local oscillator only (t«) and the product of signal and
local oscillator (isi). Since the local oscillator will be much stronger than the signal we generally have i33 •< isi •< i«. i„3 can be ignored and in generally only has a DC and a high frequency component (2 x /L0). i3i contains high and low frequency components where the latter, which is the beat frequency between the signal and the local oscillator, is the signal of interest. This technique of mixing the signal with a local oscil lator having a frequency that is shifted relative to the centre frequency of signal spectrum is called heterodyning. A good mathematical discus sion of the technique can be found in CUMMINS and SwiNNEY (1970). The principal advantages of heterodyning are (a) that the frequency spectrum of the signal is down shifted to frequencies where electronic amplification and detection can be used and (b) that the photo current is proportional to the field of the signal rather than the power of the signal. The benefit of (a) is that a complicated spectral analysis is more readily realized while the benefit of (b) is that detector noise, referred to the antenna, is much reduced. Detector channels. The beat current from the mixer (IF signal) is am plified and (through various power dividers and filters) split into 32 channels, 24 on the low side and 8 on the high side of the centre fre quency. The signal generally decreases with increasing distance from the centre frequency. To compensate for this the bandwidth of each channel is made equal to 1/5 of the displacement of the channel from the centre frequency.
3.3 Signal to noise ratio
At the receiver antenna the spectral power density of the signal, dPs/du, is ap proximately three orders of magnitude lower than the spectral power density of the noise background, dPh/du>. To distinguish the signal from the background the gyrotron power is chopped at a frequency of typically 30 kHz (see Figure 3.5). For each channel the signal is estimated by subtracting the estimate of the back ground noise, obtained when the gyrotron is off, from the estimate of the sum of background and signal, obtained when the gyrotron is on.
Let P* and Ph be the signal and background power that falls inside the bandwidth of a particular channel which has a bandwidth of W. Let E{{P) be the estimate of the parameter P based on the data obtained in the i"1 chopping cycle and let E\{P) be the estimate obtained by averaging over N chopping cycles. Let K{P} and Vjv{P} be the variances of these estimates. As an estimate, £,{PS}, of the signal we have 33. Signal to noise ratio 35
Time
Figure 3.5: Gyrotron power modulation.
b h Et{P*} = E,{P' + P ) - Ei{P ) . (3.4)
The variance, Vr{Ps}, of the estimate of the signal is
V'{P5} = F{PS + Pb} + V{Pb) . (3.5)
Let P be the power falling inside a channel with a band width W and assume that the spectral density varies little over the band width. The variance of the estimate of P based on an observation period r is then given by [CUMMINS, 1970; BLACKMAN 1959]
^J-TTW; • (3'6)
In the following we will neglect the 1 in the denominator of (3.6) which is small compared with Wr in the cases of interest to us. It is in any case dependent on a Gaussian statistic of the radiation and the characteristics of the band filter and integrator [BLACKMAN,1959]. Let Tc be the period of a chopping cycle and let Tsb and t(, be the effective periods of the chopping cycle where the gyrotron is on and off respectively, see Figure 3.5. We then have for the variance of the estimate of the signal
(P» + Pbl2 (ph)t 36 Chapter 3. The CoUecti\-e Thomson Scattering Diagnostic at JET
Vv{n " A-U'r, +1VU^ (38)
Assuming that TA = T^, = T and letting 17 = 2r/Tc be the useful fraction of the chopping period and T = iV"rc be the total integration time, i.e. the time resolution of the measurements, then
v.,,,, . ^ + QW ,3,)
Wl - ^^^ . (3..0,
The post- integration signal to noise ratio, (S/A*), is the ratio of the signal divided by the standard deviation of the estimate of the signal
S P* 1 = —=_ =y^VT (3.11) b 2 b 2V A N/2(J* + P ) + 2(P )
* ^b\A"^ {^«^b)
Typical values of the quantities entering (3.11) are
Ph/P* = 1000 W = 1GHz T = 200mSec n = 1.
With these values we would expect a final signal-to-noise ratio of 7.
3.4 Operating the diagnostic
The directions of the axes of the launcher and receiver beam patterns can be varied on a shot to shot basis, thus allowing the volume of plasma being probed to be 3-5. Interpretation of data 37
shifted and the scattering wave vector k to be varied. The main consequence of changing the modulus of k is to vary the relative contributions of the ion and electron features in the spectrum. The direction of k determines the direction in which the one-dimensional velocity distributions may be determined.
The location of the scattering volume and the modulus and direction of k cannot all be varied independently, but three degrees of freedom do exist. One useful way of defining the scattering geometry is by specifying the desired values of the major radius, RQ, the vertical coordinate of the centre of the scattering volume, z0, and the modulus, k, of k, or alternatively, the angle between k and the magnetic field. From a prediction of the equilibrium (magnetic field and density profile) of the coming shot, the specification of the desired scattering geometry and selection of the characteristic plasma mode that the beams should couple to, the required launch angles can be found by an iterative ray-tracing procedure. Having found the launch directions and the resulting beam trajectories the required polarization for coupling purely to the selected mode is determined. A computer code, FINRAY, has been written to perform this task.
During the shot the direction of the receiver beam is automatically varied in one dimension in order to track the incident beam. The orientation of the receiver antenna is recorded at frequent intervals together with the corresponding spectra of the scattered radiation so that the real scattering geometry at each time slice can be estimated for use in the analysis.
3.5 Interpretation of data
The quantities this diagnostic seeks to measure are a small number of parameters describing the one-dimensional fast ion velocity distribution, the temperature of the bulk ions and the densities of the various ion species relative to the electron density.
After a shot the actual location of the scattering volume and the magnitude and direction of the scattering wave vector are found for each time slice by ray-tracing, using the recorded antenna direction and data on the magnetic field and the den sity profile supplied by other diagnostics. At the scattering point local values of the magnetic field and electron density and temperature are obtained from other diagnostics. The spectral analysis can be carried out in several ways. One proce dure which seems feasible is to find estimates of the parameters by least squares fitting in two stages: (1) The central part of the spectrum of the scattered radi ation is used to determine the bulk ion temperature and relative density. Some information on impurities may also be obtainable [OltsiTTO, 1990]. (2) The rest of the spectrum is then fitted to, to find the parameters describing the fast ions. 38 Chapter 3. The Collective Thomson Scaiieiing Diagnostic at JET
The bulk ion data found in step (1) would be assumed to be given in this step. Chapter 4
Dielectric properties of a relativistic magnetized plasma
4.1 Introduction
Although electron temperatures found in large tokamaks (5-15 keV) remain small relative to the rest mass energy of the electrons (511 keV), recent investigations have revealed significant relativistic shifts in the locations of cutoffs and modifica tions of the refractive index in the regions leading up to these cutoffs [BATCHELOR ct al., 1984; ROBINSON, 1986 a; BINDSLEV, 1991 a, b , c and 1992]. This is caused partly by the fact that the refractive index in the vicinity of a cutoff is quite sensitive to the plasma response. It is, however, also found that the relativistic modifications of the Hermitian part of the dielectric tensor are larger than would be expected from a straightforward comparison of the electron temperature with the electron rest mass energy.
This chapter investigates the relativistic modifications of the dispersion in a plasma in the regimes which are relevant for millimetre wave diagnostics in large tokamaks and in particular for collective Thomson scattering and reflectometry.
The chapter is organized as follows. In Section 4.2 an overview of previous work on relativistic dielectric effects is given. Section 4.3 gives the basic equations gov erning electromagnetic waves in a spatially and temporally dispersive plasma. In Section 4.4 the Telativiftic plasma model is outlined. A computationally convenient expression for the weakly Telativintic dielectric tensor (due to SllKAROFSKY, 198G) is given in Section 4.5. In Section 4.6 new methods are given for evaluating the Shkarofsky functions in regimes which hitherto were inaccessible because of nu merical instabilities. In Section 4.7 a brief discussion is given of the codes used in
39 40 Chapter 4. Dielectric properties of a relativistic magnetized plasma
this work for computing the dielectric tensors derived from various plasma models. The dispersion function, A, is defined in Section 4.S where computationally con venient expressions for A and derivatives of A are also given. Section 4.9 presents numerical evaluations of the refractive indices of ordinary mode (O-mode) and ex traordinary mode (X-mode) based on the cold, the hot and the weakly relativistic plasma models. In Section 4.10 numerical evaluations of a number of quantities relating to the fully relativistic O-mode cutoff and X-mode R cutoff are presented, 2 including the locations of the relativistic cutoffs in an cjce/w versus u^/u/ diagram (CMA diagram).
4.2 Review of previous work on dielectric prop erties of relativistic plasmas
The dielectric properties of a relativistic magnetized plasma were first examined by GALITSKII and MlGDAL (1961) using a quantum mechanical approach (work done in 1951).
The first classical derivation of the dielectric tensor, e, for a relativistic magnetized plasma was given by TRUBNIKOV (1959) (work done in 1956). Two expressions for e were given. The first includes a summation over Bessel functions, each term relating to a harmonic of the electron cyclotron frequency, and leaves the three dimensional integration over momentum space undone. This expression assumes only that the background distribution is homogeneous and stationary. The other expression for e is more compact with only one infinite integration. This expression assumes a background distribution in thermodynamic equilibrium. Despite the apparent simplicity of this expression it is not readily evaluated numerically. It has however served as the starting point of a number of derivations of numerically more tractable expressions for e.
DNESTROVSKII et al. (1964) derived a more readily evaluated expression for the dielectric tensor for propagation perpendicular to the magnetic field which is valid in the weakly relativistic regime where
Starting from TltUBMKOV's result, SHKAROFSKY (1966 a) derived an expression for the weakly relativistic dielectric tensor for propagation at arbitrary angles to the magnetic field. In this paper SllKAROFSKY introduced the functions, Fq, (see expression (4.20)) which have become known as the Skharofsky functions. Effi- 4 2. Review of previous work on dielectric properties of relativistic plasmas 41
cient evaluation of these functions is a crucial step in numerical evaluations of the weakly relativistic dielectric tensor. SlIKAROFSKY (1966 b) used this weakly rela tivistic formulation to investigate the plasma dispersion in the vicinity of electron cyclotron harmonics, concluding that non-relativistic theory was inadequate in these regions.
Tractable expressions for the Shkarofsky functions were derived by AlROLDI and OREFICE (19S2) and further developed by KRIVENSKI and OREFICE (1983) who shewed that the lowest orders of Shkarofsky functions could be expressed in terms of the readily evaluated Fried and Conte dispersion function, Z, and that the Shkarofsky functions satisfy a second order recursion relation, allowing higher orders of Tq to be obtained as well. These are the relations used in this present work to evaluate the Shkarofsky functions (see discussion in Section 4.6). A range of useful relations and approximations for the Shkarofsky functions were given by ROBINSON (19S6 b). KRIVENSKI and OREFICE also gave an expression for the dielectric tensor in a weakly relativistic plasma with a drifting Maxwellian electron velocity distribution.
A comprehensive review of methods for evaluating the dielectric properties of relativistic and weakly relativistic plasmas was given by BORNATICI et al. (1983). This paper includes discussions of the various approximations to the rclativistic dielectric tensor.
The dielectric properties of a fully relativistic magnetized plasma in thermody namic equilibrium were investigated by BATCHELOR, GOLDFINGER and WEITZNER (19S4). A computationally convenient expression for the fully rclativistic dielectric tensor was derived and dispersion curves given for perpendicular propagation in O-mode and X-mode at temperatures ranging up to several hundred keV. It was noted that the O-mode cutoff, the X-mode R-cutoff and the upper hybrid reso nance were shifted by relativistic effects and that the R-cutoff and upper hybrid resonance disappear at high temperatures.
A detailed study of the fully relativistic dielectric tensor was carried out by BOR- NATICI and RUFFINA (19S5). Relations were pointed out between limiting forms of elements in the fully relativistic expressions and various approximate expressions for e.
A particularly elegant derivation of the weakly rclativistic dielectric tensor was given by SlIKAROFSKY (19S6). This paper provides an expression for the weakly relativistic dielectric tensor in terms of an expansion in the finite Larmor radius parameter.
7 K = (k±Pe) (4.2) 42 Chapter 4. Dielectric propert irs of & relativist ic magnetized p/asma
which readily lends itself to numerical evaluations. Here p€ is the mean thermal Larmor radius
pi = T,/(>>.) , (4.3)
and kx is the component of the wave vector which is perpendicular to the magnetic field. SHKAROFSKY'S work is further discussed in sections 4.5, 4.6 and 4.7.
Expressions for the dielectric tensor in a fully relativistic magnetized plasma, as suming a wide range of momentum distributions including loss-cone distributions with field aligned drift and the thermodynamic equilibrium distribution, were de rived by YOON and ClIAN'G (19S9). The expressions involve infinite summations over complicated elements with infinite integrals. An alternative formulation of the dielectric tensor for a fully relativistic magnetized plasma in thermodynamic equi librium was derived by YOON and DAVIDSON (1990). Although the expressions are cast as convenient series and asymptotic expansions in Ac, it is not obvious that the special functions introduced in this formulation are readily evaluated except in the weakly relativistic limit, where they reduce to the Shkarofsky functions. The very simple (and therefore attractive; expression for the weakly relativistic dielectric tensor provided in this paper appears to have a number of important deficiencies. The eTy = —tfT elements are dimensionally wrong. MosER (1991. private communication) investigated the dispersion for 0-mode propagating per pendicular to the magnetic field (which is independent of the e^ = —etz elements) predicted with this form of the dielectric tensor, and found a singular behavior of th^ refractive index around the electron cyclotron resonance. This w*ould be consistent with a hot theory but not with a weakly relativistic one.
YOON and KRAI'SS-YARBAN, 1990 derived an expression for the weakly relativis tic dieWtric tensor for a loss-cone distribution and used the result in a study of growth rates of the X-mode and Z-mode1 gyroharmonic maser instabilities. By reducing a loss-cone parameter to zero YOON and KRAUSS-VARBAN'S result gives an expression for the weakly relativistic dielectric tensor for an electron distribu tion in thermodynamic equilibrium. There arj some misprints in the expressions for the weakly relativistic dielectric tensor provided in this article. With the cor rections given in Appendix C of the present thesis the numerical results obtained with YOON and KRAUSS-VARBAN'S expressions in the 'sotropic limit agree with those obtained with SlIKAROFSKY's expressions.
Expressions for the relativist«- dielectric tensor and various limiting forms, assum ing both isotropic and anisotropic momentum distributions, have been presented by many authors including, in addition to those already mentioned. FlDONK rt al
1 In astrophysics it is common to use the label Z-roode for the extraordinary mode at frequencies between the L-cutoff and the upper hybrid resonance 4.2. Review of previous vrork on dielectric properties of relativistic plasmas 43
(19S6). M.AZZICATO et al (19S7), 0REF1CE (19SS), ZlEBELL (19SS), GRAXATA (1990) and BORNATICI et «/. (1990). For a comprehensive list of references up to 19S3 see BORNATICI et *l. (19S3).
Following the initial investigations by BATCIIELOR, GOLDFINGER and W'EITZNER (19S4) of the relativistic modifications of the dispersion around cutoffs and the dis appearance of the X-mode cutoff-resonance pair (see discussion above), a number of workers liave investigated these phenomena. The low density and high temper ature limit for the disappearance of the R-cutoff in a weakly relativistic plasma was given by PRITCHETT (19S4). ROBINSON (19S6 a) investigated the disper sion in a weakly relati.istic plasma for propagation perpendicular to the magnetic field. Arbitrary distributions as well as isotropic Maxwellian and loss-cone distri butions were assumed. Calculations of the refractive index were presented showing a wiggle (anomalous dispersion) in the dispersion curve for O-mode around the cyclotron frequency and the reconnection of the X-mode branches on either side of the R-cutoff and the upperhybrid resonance, and the eventual elimination of the cutoff-resonance pai*- with increasing temperature. Curves were given displaying the relativistic shift of the R-cutofFin the weakly relativistic approximation. Sim plified expressions for the dielectric tensor and locations of cutoffs in a weakly rela tivistic plasma with an anisotropic velocity distribution were presented by SAZIIIN (19S7) together with plots giving a qualitative indication of the relativistic shifts of cutoffs (see also discussion in Section 4.10 below).
A comprehensive survey of the qualitative nature of the weakly rclativistic dis persion relations for perpendicular propagation, with particular attention given to elertron Bernstein waves, was carried out by ROBINSON (19SS a and 6), paper a dealing with O-mode and paper b with X-mode (see also ROBINSON, 19SS c). The papers give a good overview of previous work on both non-relativistic and rela tivistic Bernstein waves, and show that relativistic effects modify significantly the dispersion of Bernstein wave:; (O-mode Bernstein waves, X-mode Dnestrovskii- Kostomarov waves and X-mode Gross-Bcmstein waves) and of the cold branches (O mode and X-mode) around the electron cyclotron harmonics, as compared with the dispersion obtained with a non-relativistic hot theory. In particular it is found that most o( the Bernstein waves decouple from the cold branches leav ing just small wiggles, also referred to as anomalous dispersio.., in the dispersion curves of the cold branchc-s around the electron cyclotro«. harmonics. The range o.* parameter space where Bernstein waves can exist is also founr' 10 be severely limited by relativistic effects and the Bernstein wave branches are generally found to be strongly damped for both large and small values of the refractive ii.dcx. A good overview of the modes predicted with a hot non-relativistic plasma model was given by Pl'RI. LF.ITF.RER and TrTTF.R (1973 and 1975). 44 Chapter 4. Dielectric properties of a reiativistic magnetized plasma
4.3 Modelling a plasma with spatial and tempo ral dispersion
A medium is temporally dispersive when the current at a point in the medium depends on the past history of the fields at that point. A consequence of temporal dispersion is that for an electromagnetic wave the dielectric properties (e.g. the dielectric tensor) of the medium depend on the frequency of the wave. If the current at a point depends also on the past history of the fields at other points in the medium then the medium is spatially dispersive and the dielectric properties of the medium depend also on the wave vector of the electromagnetic wave. A cold plasma is temporally dispersive but not spatially dispersive. Inclusion of thermal motion in the plasma model, as in the hot and the relativistic plasma models, introduces spatial dispersion.
In a medium with spatial and temporal dispersion Maxwell's equations can be written in the form [LANDAU, LlFSHITZ and PlTAEVSKII, 1984, §103]:
SB VxE = --g-, (4.4,
VxD = ^£o_ + _+jj, (4.5) where
0P(r,t) = jir'(i,t,r\t')-E(r\t')dr'dt' (4.6) dt and j refers to current« which are not attributable to the linear response of the medium to the field in question, j may of course be the source of the field and could for instance be the result of the interaction between fluctuations in the medium and another field.
The rank two tensor, In a stationary and homogeneous medium the kernel of the integral in (4.6) sim plifies: dP(r,Q = j Taking the curl of (4.4) and eliminating B gives the inhomogeneous wave equation: VxVxE + -- + - —= --£ . (4.8) 2 2 2 2 2 c at e0c at e0c ot Fourier-Laplace transforming (see appendix B) yields: /i2 k x (k x E(k,u/)) + e(k,w) • E(k,a/) = j(k,w), (4.9) where \i is the refractive index, /J = kc/u, w the angular frequency, k the wave vector and k the unit wave vector, e is the dielectric tensor, e(k,u,) = I+lf^). (4.10) U>£Q I is the identity tensor a.ul tr(k,ui), the conductivity tensor, is the Fourier trans form of the kernel of the integral in (4.7). It is convenient to introduce also the susceptibility tensor, x(k,u>): = *(k,w)-I. The tensor acting on E on the left hand side in (4.9) will be referred to as the wave tensor, A: A(k,w) = e(k,w) + ft7 {kk -1} . (4.12) With this tensor the inhomogeneous wave equation can be written compactly: AE •--- —j . (4.13) 46 Chapter 4. Dielectric properties of a relativistic magnetized plasma 4.4 The relativistic plasma model The cold plasma model is well known. In this paper the hot plasma model refers to a magnetized plasma with an isotropic Maxwellian velocity distribution for the electrons. In the hot model the dielectric tensor, c, is derived from Maxwell's equations and the non-relativistic Vlasov equation (see e.g. SWANSON, 19S9). In the wlativistic model of a homogeneous, magnetized plasma with an isotropic T^'.Ov-ity distribution in thermodynamic equilibrium the plasma current induced by an electromagnetic field is given by |f = I>;9i/vi/i(p)dP. (4-14) where the perturbation of the momentum distribution of the j'th species, fj, is found from the linearized Vlasov equation, Here B° is the static magnetic field, ff is the unperturbed momentum distribution of the j'th species, Vj = p/ijm3 and 7, = v^l + (p/mjCy . (4.16) The static electric field is assumed to be identically zero. There is always a frame of reference in which the static electric field is zero and it is thus to this reference frame that the present model refers. If this frame is not identical to the laboratory frame we can simply make a Lorentz transformation into the appropriate frame. The unperturbed velocity distribution corresponding to thermodynamic equilib rium is /« = te-H C=^ (4 17) i4 U) '' 4^(m;c)3A'2(0) ' ^ Tt ' ' A"„ is the modified Bessel function of the second kind and order n. This model presupposes that collisions are sufficiently frequent that an equilibrium distribution 4.5. The weakly relativistic approximation 47 is established and maintained, yet still so rare that they can be neglected in the equation (4.15) for the perturbed velocity distribution. The relativistic dielectric tensor is derived from the definitions (4.7) and (4.10) with the plasma current response described by equations (4.14), (4.15) and (4.17). Various expressions for the fully relativistic dielectric tensor have been derived (see Section 4.2). One result, due to TlllIBNIKOV, (1959), is given here: 2 1 c Kz (R) 2 e = I + - m T« ) - T< > (4.18) S IP 3 WWc<.Jv2(Ce) JO \ "IR }• cos f — sin £ 0 T(,) = { sinf cos£ 0 0 0 1 sinH ~ k\ sin i{\ — cos f) fcj.k||£ sin £ T(2) 2 T(=, = J J12 ~k\(\ - cosO *j.it|^(l - cosO T(2) _T(2) ,2c M3 R = \ w« \ wce J \ u/ce ) The coordinate system is oriented such that z is in the direction of the magnetic field, B, and the wave vector, k, lies in the x-z plane, so k\\ = fcz and tj. = kx. Only the plasma current due to the electrons has been included, the extension to more species being obvious. 4.5 The weakly relativistic approximation Fully relativistic expressions for e, including that given in equation (4.18), are difficult to handle numerically. A much more tractable form known as the weakly rclativittic dielectric tensor, derived from TnUBNIKOV's result (4.18) by SlIKAIlOF- SKY (1986), is given below. It is valid for (e > 1 and Ae < 1 (Ae is defined below) and is a very good approximation at the temperatures found in large Tokamaks. 48 Chapter 4. Dielectric properties of a i-elativistic magnetized plasma 2 000oc 0o o / !• »i \ IT C-u>,P « £ij=I~ 19) w- ^r=-oop=o 2 r n ^"p+n+3/2 -iN{p + n)J p+n+3/2 ^^p+n+S/2 WU,'ce M p(p + 2n) ik k c2,, 2 -Mn \(p+nf-2 ± n 2ra + 2p - 1 -Tp+n+3/2 —— (P + n)S +n+5/2 (jJLJce 2 M,1 3 -M23 Ae(^+n+5/2+2V' Jp;n+7/2) j (_l)p(n + p_l/2)! a n n = \N\ , p ~ p!(n+p/2)!(„i n + (p-l)/2)!2» ' /•oo ^ = -jexp(-^2)/ (1 -7'<)-«exp[-^ + ^2/(l -i*)] m d Ta *? d(d>2)m ' (4.21) v = (4.22) = t'M*^) (4-23) For evaluating derivatives of the dielectric tensor, derivatives of the Shkarofsky functions, 7™, with respect to J/>2 are also required. It is readily shown by direct differentiation of expression (4.20) that QJTm J 1 _ _ T-m+l (4.24) d{4>2) ,+1 4.6*. Computation of the Shkarofsky functions 49 Evaluating these expressions numerically is straightforward, though care must be taken to avoid numerical instabilities when computing the Shkarofsky functions, ^"(p, v)- Stable evaluation of these functions is the topic of Section 4.6. The condition £e > 1 is clearly well satisfied in Tokamaks. The requirement that Ae < 1 is needed to ensure convergence of the expansion in Ae. In practice this places an upper limit on the frequency, u < y/Cuce , (4.25) and an upper limit on the perpendicular refractive index, ,x = ^<^^- (4-26) The latter imposes a limit on how close to a resonance the dielectric tensor can be evaluated with this expression since // becomes large near a resonance. 4.6 Computation of the Shkarofsky functions A crucial element in obtaining an accurate and reliable evaluation of the weakly relativistic dielectric tensor is a numerically stable algorithm for evaluating the required Shkarofsky functions, ^"(0, i/')> anc^ their derivatives (see expressions (4.20), (4.21), (4.22) and (4.23)). Recursion relations and independent expressions for the lowest orders of the Shkarof sky functions were given by SHKAROFSKY (1986) and KRIVENSKI and OREFICE (19S3). A forward recursion was suggested. Although this provides a stable al gorithm for a wide range of parameters there are important ranges of parameters where forward recursion leads to catastrophic numerical instabilities. One such range is where 0 < |i/'| «C I. Although a stable algorithm was given for the case where i' = 0, a gap, 0 < |?/'| < 1, remains where neither approach gives satisfac tory results. One consequence of the existence of this unstable region is that ray-tracing based on a weakly relativistic plasma model using this algorithm breaks down for propa gation which is nearly perpendicular to the magnetic field. In this connection it is noteworthy that the weakly relativistic ray-tracing code, TORCH, comes to a halt when the propagation becomes nearly perpendicular to the magnetic field [BONOLI, 50 Chapter 4. Dielectric propetiies of a relativisiic magnetized plasma 1991. private communication].2 This problem was also found with RELRAY (see Section 5.2) which was what prompted the investigation that led to the results described in this section- It will be shown that backward recursion is stable in a large part of the range where forward recursion is unstable, but that an important range remains where neither approach is stable. A third approach, which we will call cenfm/ recursion, turns out to be stable in this remaining region. The three approaches cover the ranges of parameters which are usually of interest. The Shkarofsky functions satify the following relations [SlIKAROFSKY, 1986]. Recursion relation for Shkarofsky functions: 2 2 v .F,+2 + q?„„ - Recursion relation for derivatives of Shkarofsky functions: 2 1 V' ^7+2 + qf?+i ~ Relation between different derivatives of Shkarofsky functions: jr. = jrn-i _ jrn-i (4 2g) Since the recursion relations (4.27) and (4.2S) are of second order, T^ must be specified for two different values of q for each m to select a particular solution to each of the recursion relations. We will refer to the specified values of T™ as initial values. Expressions relating low order Shkarofsky functions to the Fried and Contc dispersion function are given by KRIVENSKI and OREFICE (1983): Z(j>-) + Z(-xl>- •^3/2(0,^) = 2wi • (4-30) We need tr consider also the values of .^i/jtø,^) and ^3/2(^,1^) in the small argument limits: Propagation exactly perpendicular to the magnetic field is not a problem because, as mentioned above, stable algorithms exist for computing the Shkarofsky functions when i> = 0. 46. Computation of the Shkarofsky functions 51 2 p — 0 => <£ ;Fi/2->0 (4.31) 2 tf->0 =» ^3/2^-Z'(-«-^ 2'"M). (4.32) Initial values for the derivatives of the Shkarofsky functions can also be given in terms of the Fried and Conte plasma dispersion function [SHKAROFSKY, 1986]. However, these expressions frequently contain differences between large and almost equal numbers resulting in the loss of accuracy. With the improved recursion methods outlined below !F^~X is determined with sufficient accuracy that the initial values for T^ are best determined with relation (4.29) which rarely results in differences being taken between large almost equal numbers. The set of solutions to the recursion relations (4.27) and (4.28) is covered by the set of series composed of a particular solution to the inhomogeneous recursion relation plus a linear combination of the two linearly independent solutions, f^Kf^K to the homogeneous recursion relation *2fl% + «//S. - Numerical noise, inherent in any numerical calculation, will excite both of the homogeneous solutions at every stage of the recursion. If one or both of these homogeneous solutions have a large growth rate for increasing q then an unwanted homogeneous solution will grow and dominate the solution found with forward recursion. The same is true of backward recursion if one or both homogeneous so lutions have a large growth rate for decreasing q. If the two homogeneous solutions have large growth rates in opposite directions then neither forward nor backward recursion will succeed in finding the particular solution which is compatible with the initial values. In this case a new recursive method, central recursion, turns out to be successful. With forward recursion 7%^ and T^+i are determined from Tq+2 = -^(1-^,+^JF,), (4.34a) starting from the initial values. 52 Chapter 4. Dielectric properties of a relativistic magnetized plasma m In reverse recursion the values F, and Fq representing a solution to the recursion relations are obtained from F* = ^(-1+9^+1 + ^+2), (4.35a) K = ^(-"»JT^+^i + ^Ca). (4"35b) starting with seed values for the two highest orders of q. Each of these solutions contain the sought particular solution plus some contribution from the homoge neous solutions ir,m=^r+fl-/ia)+6-/i6)- (4-36) The two linearly independent homogeneous solutions are obtained by reverse re cursion with the homogeneous relation 4 37 /. = £(9/SI+*2£M) ( - ) starting from two different sets of seeds. The unwanted contribution to Fm from the homogeneous solutions is then deter mined by solving the equations a FT/7 = fr,2 + "mfl fl + bmfl%, (4.38a) Fm = ?5h + fl*/$ + *»/$ • (4.38b) n for am and bm (F[ /2 and T™,^ are the known initial values). Given the values of am and bm one can eithci subtract the unwanted homogeneous contribution from Fm to find F™ or correct the seed values and start a new recursion to determine m a new solution Fg . The latter approach generally converges rapidly towards T™. There are many ways of optimizing this algorithm which would require lengthy and tedious explanations and are therefore omitted. 4.6. Computation of the Shkarofsky functions 53 In practice both forward and reverse recursion are successful even in the presence of homogeneous solutions with moderate growth rates in their respective directions. So if for instance one homogeneous solution has a large growth rate in the for ward direction, making forward recursion unstable, while the other homogeneous solution is moderately growing in the reverse direction then reverse recursion as described above will be successful. With the new method, central recursion, the solution to the recursion relation is found iteratively by a sequence of estimates of the solution, where step n in the iteration provides an estimate F™n of the solution for all values of g obtained from the values of the estimate F™n_j at the previous step n — 1. The iterative procedure is F,+1.„ = ^(l + ^F,,..,-^^,,,.,), (4.39a) F^1" = V^r'+^Vi-^^,.-,). (4.39b) This procedure gives a sequence of estimates of the solution which converge to the correct solution provided the sequence obtained with the homogeneous recursion relation goes to zero for all starting values. This is the case if IV ' * ' < 1 (4.40) 7min •* where q„^n is the lowest order for which the Shkarofsky functions are determined by the iteration. Since the initial values provide the values of the Shkarofsky functions for q = 1/2 and q = 3/2 it follows that q„^n = 5/2 and the condition for convergence becomes |<*2|+|tf2|<1.5. (4.41) Finally it should be noted that for large parts of parameter space all the required derivatives may be found without loss of accuray by successive application of the relation (4.29) leaving just Tq to be determined from the recursion relation. This is particularly advantageous in terms of computing time for reverse and central recursion. To find out when to use which algorithm we need to examine the growth rates, 54 Chapter 4. Dielectric properties of a relativistic magnetized plasma r(.) °l', = :7(7f I (*" = !» 2), (4.42) of the homogeneous solutions, a may be thought of as the growth rate for increas ing q and 1/Q as the growth rate for decreasing q. We note that if the coefficients in the recursion relation vary slowly with q then oi',«oll (4-43) and the approximate value of Q^ can be found from i'2(Q{;))2 + qa^)-<>2 = 0 ; i = 1,2. (4.44) The solutions are «... , -'-^ , (4.45a) «<» = ;f, ,„„. (4.45b) V -q-sfq^TWP ' In the limit - * 1°i1° £: _ „ (4-46) both forward and reverse recursion suffer catastrophic numerical instabilities. Cen tral recursion will be successful provided the condition (4.41) is satisfied. For q —* oo central recursion gives T™ —* 0 which can also be shown directly from the definition of the Shkarofsky functions (4.20). Condition (4.41) does however exclude a large part of the relevant parameter space. This limitation and the failure of both forward and reverse recursion thus places an upper limit on the number of terms that can be included in the summation in expression (4.19) for the weakly dielectric tensor. While this limitation should be kept in mind it is in practice rarely a problem since the summation in (4.19) converges long before this limit. If the Shkarofsky functions for large q were required for parameters where 4.6. Computation of the Shkarofsky functions oo central recursion fails then the low q functions could be computed by forward or reverse recursion and central recursion applied only for large values of q. This would overcome the limitation imposed by (4.40) by increasing q^^. The growth rates for two other limiting cases at moderate values of q are of interest. For large values of the product of p and v'' ( region .4 in Figure 4.1 a) we find that »<» IWI 4 V' oc (4.47) <*) <> 1 * which implies that the growth rates are identical for both homogeneous solutions. When |p| < |v'| (region -4') growth rates are less than 1 and forward recursion is stable while for \o\ < |v| (region ,4r) growth rates are greater than 1 and reverse recursion is stable. Figure 4.1: o versus %' diagram indicating regions of stability for numerical evaluation of the Shkarofsky functions by forward, reverse and central recursion. For small values of the product of p and ip (region D in Figure 4.1) the limiting values of the growth ratrs are 56 Chapter 4. Dielectric properties of a relativist!«: magnetized plasma \ov\ 2 . (4.4S) * 9 For moderate to large values of t>' and small values of $ (region B1) o(,) is mod erate to small, while am is small, making reverse recursion unstable and foward recursion workable. When t> is small and 6 is moderate to large (region Br) then o(,) is large and a<2) moderate to large, making forward recursion unstable and reverse workable. When both p and v are small (region Bc) one finds that one growth rate is large and the other small, making both forward and reverse recur sion unstable. In this case condition (4.41) is satisfied and the central recursion is stable. A more detailed numerical analysis shows that for moderate values of q (q < 10) forward, reverse and central recursion give accurate results in the regions R. F and C respectively where the regions refer to the diagram in Figure 4.1 b. 4.7 Computation of the dielectric tensor Computer codes have been developed, as part of this work, to evaluate dielectric tensors and their derivatives derived from four magnetized plasma models: (a) cold, (b) hot equilibrium, (c) weakly relativistic equilibrium based on SllKAROF- SKY (19S6) and (d) weakly relativistic equilibrium based on YOOX and IvRAUSS- VARBAX (1990). With a number of corrections to YOOX and KRAL'S-VARBAX'S work (see AppendiA C), the two relativistic codes give identical results. In the low temperature limit the rclativistic, the hot and the cold codes all give identical results. For propagation close to perpendicular to the magnetic field, both the real and the imaginary part of the refractive index found with codes (c) and (d) agree accurately with curves given by BATCHELOR, GOLDFIXGER and WFITZXF.R(19S4). For the conditions investigated in Section 4.9 the .summation over powers of Ar in expression (4.19). for the weakly relativistic dielectric tensor, converges rapidly. For these conditions A, is typically smaller than 5-If)"2 and it is generally sufficient to include only terms tip to the fifth power in Ar. 4.S. Computation ofthr dispersion function and its derivatives 57 4.8 Computation of the dispersion function and its derivatives Electromagnetic waves in a homogeneous source-free plasma satisfy the homoge neous wave equation AE = 0 , (4.49) where the wave tensor. A was defined in equation (4.12). Xon-trivia! solutions to (4.40) only exist if A = |A| = 0 . (4.50) This is the dispersion equation. For numerical evaluation it is more convenient to have expressions in component form. The dispersion function. A, may be expressed in component form as A = f^A.iAyA* , (4.51) while the derivative of the dispersion function with respect to a parameter x takes the form f •fc = «(mn .jt-^T^mJAnt , (4.o2) where t,jk is the antisymmetric permutation semi tensor (referred to by some authors as the standard Levi-Civita symbol), 1 if i.j.k form an even permutation of 1, 2. 3 (e.g. 3, 1, 2) U,k -1 if i.j.k form an odd permutation of I, 2, 3 (e.g. 3, 2, 1) (4.53) 0 otherwise and f*k is given by 68 Chapter 4. Dielectric properties of a relativistic magnetized plasma + _ J 1 if i,j, k form an even permutation of 1, 2, 3 ,}k ) 0 otherwise 4.9 Refractive index In a cold plasma an explicit expression for the refractive index n can be derived by solving for /* in (4.50). In hot and relativistic plasmas, where e is a function of fi, (4.50) is a transcendenta'. equation in /i. In this case the refractive indices of the mocies of the plasma are found numerically by searching for the roots of A in the complex n plane. Whereas in the cold plasma at most 3 modes exist, in the hot and relaLivistic plasmas there are ir general an infinity of modes, though most of them are heavily damped. A typical situation at frequencies above the R cutoff is illustrated in Figure 4.2. Ir • ^fiectometry using O-mode radiation the frequency, /, of the probing beam is u.^.icilly below the cyclotron frequency, fc. Cold, hot and weakly relat'vistic calcu lations of the O-mode refractive index as a function of density at / = 30, 60 GHz < fc, Tt = 10 keV and propagation perpendicular to the magnetic field are presented in Figures 1.3 (a) and (b). The cold and the hot curves ire indistinguishable. X-mo^ .eflectometry makes use of waves with frequencies between the fundamen tal and the second hannonic of the electron cyclotron frequency and above the R cutoff frequency. Calculations of the refractive index in this region are presented in Figures 4.3 (c)-(f). Cold, hot and weakly relativistic calculations for Te = 10 keV, / = 100, xlO and 120 GHz and perpendicular propagation are compared in Figures 4.3 (c)-(e). While the hot model produces only a small change relative to the cold model and tends towards the cold model at the R cutoff, the weakly relativistic model predicts a substantial change in the refractive index and the density of the R cutoff is increased considerably. In Figure 4.3 (f) the refractive index is calculated relativistically for four different temperatures to display the temperature dependence of the relativistic effects. The Te = 0.05 keV curve is indistinguishable from the cold curve. The collective Thomson scattering diagi ostic at JET will make use of both 0- niode and X-mode radiation at frequencies between the fundamental and the second harmonic of the electron cyclotron frequency, while the TFTR collective scattering diagnostic will make use of X-mode with the frequency below uicc. Cal culations of the refractive inder relevant for these diagnostics are presented in Figures 9.3, 9.5, 9.G and 9.7. It ;s noteworthy that the relativistic effects are more important than the effects 4.9. Refractive index M Figure 4.2: Contour plots of the logarithm of the weakly relativistic plasma dispersion function, log(|A|), in the complex n plane. The weakly relativistic modes appear as singularities in this plot. Th.? cold O- and X-mode refractive indices are also indicated. Parameters: 19 3 Te = 15 keV, ne = 6.0 • 10 m- , B = 3.0 T, Z(B,k) = 90°, / = 124 GHz. (/c K 84 GHz, /„ « 70 GHz, fR « 123 GHz) Chapter 4. Dielectric properties of a relativistic magnetized pi 1.0 O-mode f=30GH2 0.5 -CoX*Relativisti c hot ^ 0 a) 1.0 O-mode f=60GHz Relativistic 0.5 Cold 0 1.0 X-mode f=100GHz •_i \ >^-~ Relativistic 0.5 0 0 1.0 X-mode f=110GHz Relativistic 0.5 d) 1.0 X-mode f=120GHz Refativistic 0.5 0 e) 1.0 X-mode M20GHz 0.5 Relativistic VkeV=0.05—\ 2.5 Electron density / lO^nr3 4.3: Refractive index, /*, as a function of electron density. Parameters: B = 3.4 T, L (B, k) = 90°. (a)-(e) cold, hot and weakly relativis tic, Te = 10 keV, (a) O-mode, / = 30 GHz, (b) O-mode, / = 60 GHz, (c) X-mode, / = 100 GHz, (d) X-rnods, / = 110 GHz, (e) X-mode, / = 120 GHz. (f) weakly relativistic, Te = 0.05, 5, 10, 15 keV, X-mode, / = 120 GHz 4.10. Cutoffs £1 found with the hot model in all the regimes explored here. 4.10 Cutoffs The locations of the O-mode cutoff and the X-mode R and L cutoffs predicted by a fully relativistic model [BATCHELOR tt al, 19S4; BINDSLEV, 1991 a] are given by 3A'2(C.) = vo = (4.55) 4) C2/ (p772)e-^# k ^ / O cutoff Jo 3A'„(C) '4) = »li = , (4.56) k ^ / R cutoff Jo \ 7 / 7 - (w«/w)2 '<) 3/v2(Ce) = vL = , (4.57) \ / L cutoff ^/°°(i u<*/-u\ p"e"^ 7 / 72 ~ (^ce/w)2 where K2 is the modified Bessel function of the second kind and order 2. Equation (4.55) is identical to equation (43) in BATCHELOR tt al. (19S4) (subse quently referred to as B.G.W.) while (4.56) and (4.57) are derived from B.G.W.'s equations 39 to 41.3 The locations of cutoffs found with the weakly relativistic codes at temperatures below 20 keV agree accurately with the results found with the fully relativistic expressions (4.55), (4.56) and (4.57). The expression for the location of the L cutoff has not been published before and is included for completeness, but here the main attention is given to the locations of the O-mode cutoff and the X- mode R cutoff where the relativistic modifications are most significant. In Figure 4.4 the locations of the O-mode cutoff and the X-mode R cutoff predicted by the 3There are misprints in B.G.W., equations .38 and 39 for X-mode. These should road and (lltyy T try — 0 • 62 Chapter 4. Dielectric properties of a relativistic magnetized plasma fully relativistic model (equations (4.55) and (4.56)) are plotted in an ur„/u; versus 2 2 u ju: diagram (CMA diagram) for Tt = 0.1, 5, 10, 15, 20 keV. JG92 185/6 CO- CD' Figure 4.4: Locations of the R cutoff and O-mode cutoff predicted by the fully relativistic plasma model, plotted in an w^/w versus W^JUJ2 diagram (CMA diagram). Te = 0.1, 5, 10, 15, 20 keV. The Te = 0.1 keV curves are almost identical to the cold curves. As may be seen in Figures 4.3 and 4.4, the largest relative shift occurs for the R cutoff near the cyclotron frequency. Relntivistic dielectric effects should therefore be most noticeable in this region. Relativistic effects increase the density at which a wave is cut off in both O-mode ind X-mode. Cutoff densities calculated with equations (4.55) and (4.56) and normalized by the equivalent cutoff densities found with the cold theory, 4.10. Cutoffs £2 (l±*-) = Vo, (4.5S) ^ntcold' O cutoff \«ecold / R cutoff *• ~ uctlU are plotted as functions of temperature in Figure 4.5. In X-mode the temperature dependence of the cutoff density varies with wce/u;. Curves are given for O-mode and for X-mode with a range of values of u}ce/u. Expanding (4.55) to third power in Te provides a simple expression for the nor malized O-mode cutoff density which ir accurate to five significant digits at tem peratures below 20 keV: ,0 = 1 + 4.S92 • 10- (J?L) - 2.24 . HT« (JL)' + 2.3 • 10" (£)' . (4.60) A simple approximation to the relativistic refractive index for O-mode radiation propagating perpendicular to B is given by fo * \h - -^ • (4-61) V %w At temperatures below 20 keV expression (4.61) with Vo given by (4.60) is generally accurate to two or more significant digits except near resonances, and typically accurate to four significant digits in the region near the cutoff. While Tokamaks, Stellarators and most other laboratory plasmas at present rarely achieve electron temperatures in excess of 20 keV, much higher temperatures are encountered in space plasmas and conceptual designs of fusion reactors fuelled with D-He3 assume electron temperatures in the 50-100 keV range. The locations of the cutoffs predicted with expressions (4.55) and (4.56) in the temperature range up to 100 keV are plotted in a CMA diagram in Figure 4.6. ROMNSON (19S6 a) presented a plot showing the temperature required to reduce the R-cutoff frequency from the cold plasma value to a given value, w/j, as a function of (u;n/u>ce -1) (see his Figure 5). His calculations were based on a weakly relativistic approximation for a Maxwellian distribution. We have repeated his 64 Chapter 4. Dielectric properties of a relativistic magnetized plasma 8 Te/keV Figure 4.5: Cutoff densities predicted by the fully relativistic plasma model and normalized by the equivalent cutoff densities found with the cold theory as functions of temperature. Curves are given for O-mode and for X-mode with a range of values of uce/u. The X-mode curve with wce/u; = 0 is identical to the O-mode curve. 4.10. Cutoffs M. I.O O-moHp mt-nff ^^- 1.4 VkeV=l00^ 80-, 1.2 60^ 40-. R cut-off 20 — 1.0 yc^Z^I °~* f 0.8 — ioo ^\V\NSNV 80 ^v\vSV\ 0.6 - 6oA^v\\V\ 40 NTXXXXN 0.4 — 20-^\VV\ Te/keV=0 -^s?xV\ \ 0.2 \ 0 . I . I \ ^ N 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 co„2 (0 Figure 4.6: Locations of the R cutoff and 0-mode cutoff predicted by the fully relativistic plasma model, plotted in a CMA diagram. Te = 0. 20, 40. GO, SO, 100 keV. 66 Chapter 4. Dielectric properties of a relativistic magnetized plasma 100 JG92 507/1 > Figure 4.7: Temperature required to reduce the R-cutoff frequency to u.'/? as a function of (-.-/»/u;« - 1) for u,> = 0.2,0.3,0.5,0.75,1.0. 4.10. Cutoffs £1 calculations using the fully relativistic expression (4.56). The results are presented in Figure 4.7. The curves in Figure 5 in ROBINSON (19SG a) are calculated up to a temperature of 50 keY. In this range his weakly relativistic curves are in good agreement with the curves in Figure 4.7. except that ROBINSON'S curve for <*.>«/<*;« = 1 appears accidentally to have been tilted. The cold limit of (u.'/}/u/ce — 1) for this curve appears to be slightly below 0.6 whereas a simple calculation shows that the cold limit should be 0.61S. The ordinate of this curve at Te = 50 keV appears to be 0.41 whereas our calculation gives 0.40. As may be seen from Figure 4.4 the R-cutofF only exists for «; > u;ce. This can also be seen directly from expression (4.56) for the cutoff frequency. When j.- < w'„ a pole appears in the integrand of the integral in the denominator of (4.5G). The conseqt ce of this is that the integral acquires an imaginary part, implying that the cutoff condition (k = 0) can no longer be satisfied for real values of the frequency. The disappearance of the R-cutofF when u; — u„ was also noted by PRITCHETT (19S4). In the cold plasma limit the R-cutofF exists right up to the vacuum limit, preventing X-modc radiation, coming from the low field side, from reaching the cyclotron resonance layer. At finite temperatures, however, the R-cutofF terminates at a finite plasma density leaving open the access to the resonance layer for X-mode radiation from the low field side and allowing X-mode cyclotron radiation to escape to the low field side. This could have consequences for Electron Cyclotron Resonance Heating (ECRH) from the high field side in toroidal devices (e.g. Tokamaks) since this may uburn a hole" through the plasma as noted by ROBINSON (19S6). and may facilitate ECRH from the low field side in high temperature or low density devices such as D- He3 fusion reactor. On the basis of the weakly relativistic model it can be shown [PRITCIIETT, 19S4] that the condition for removal of the X-mode R-cutofF is -f < oT • (4-62) From expression (4.50) we find that the fully relativistic condition for removal of the R-cutofF is Curves showing the weakly rclativistic and the fully rclativistic predictions of the 6S Chapter 4. Dielectric properties of a re/ativistic magnetised plasma limit at which the R-cutoff is removed are plotted in Figure 4.8. 0.4 JG92 507/2 0.3 to. Fully relativistic to. 0.2- \ Weakly relativistic No X-mode R-cutoff _j i 100 ykeV Figure 4.8: Weakly relativistic and the fully relativistic predictions of the limit at which the R-cutofF is removed. The R-cutoff is removed in the parameter region below the curves. It is seen that the two models are in close agreement for Te < 20 keV, but the weakly relativistic model noticeably underestimates the density limit for the existence of the R-cutoff at higher temperatures. 4 11. Snniinarv 69 4.11 Summary The dielectric properties of plasmas have been investigated for frequencies in the range of the electron cyclotron frequency and the plasma frequency. Calculations of the Hermitian part of the dielectric tensor, refractive indices and locations of cutoffs based on cold, hot and relativistic plasma models have been compared. While only small differences are found between the cold and the hot models, sub stantial differences in all three quantities are found between the cold and hot pretlictions on the one hand and the relativistic on the other. The differences are 2 larger than a simple comparison of Tt with mtc might suggest. New methods have been presented which allow numerically stable evaluation of higher order Shkarofsky functions in parameter regimes where a simple forward recursion is unstable. As a complement to the fully relativistic expression by B.-vTCIIELOR tt al. for the location of the O-mode cutoff, fully relativistic expres sions for the locations of the X-mode R and L cutoffs are presented. Finally a simple and accurate approximation is given for the relativistic refractive index for perpendicular O-mode propagation. Chapter 5 Propagation of electromagnetic waves 5.1 Introduction In this section the propagation of electromagnetic waves through a stationary in homogeneous magnetized plasma is investigated. The gradients are assumed to be sufficiently small that the WKB approximation is a valid description of the propagation of electromagnetic waves. For a detailed discussion of when this is the case see for instance the discussion in BORNATICI el al. (19S3). The main aims of this section are to derive an expression relating the power emitted from the launching antenna to the field delivered to the scattering region, and also an expression relating the fields generated in the scattering volume to the power received by the detector. Both antennas are of course located outside the plasma. As mentioned previously the two problems are related due to the reciprocity of the propagation of electromagnetic fields. The development in this chapter is split into five sections. Section 5.2 deals with the problem of ray tracing. In Section 5.3 general expressions arc given for the propagation of electromagnetic waves through an inhomogeneous and anisotropic plasma. Section 5.4 deals with the problem of coherent detection of radiation emanating from a source embedded in an inhomogeneous anisotropic plasma. The relation between field strength and power for a broad band electromagnetic field in a spatially dispersive plasma is derived in Section 5.5, while Section 5.G deals with mode conversion in the edge region of the plasma where the plasma density is low. 70 5.2. J?av-rraciii£ 71 5.2 Ray-tracing A classical derivation of the ray equations from the eikonal equation via the Hami'ton-Jacobi equation of motion to Hamilton's equations of motion can be found in LANDAU and LlFSHlTZ (19S7) §53. The result is dr 5L- (5.1a) dt = ' dkJ dk (5.1b) dt ~ dr Here r describes the trajectory of the envelope of a wave package and dr/dt thus represents the direction in which the energy of the wave package travels. If u; and k in equations (5-la) and (5.1b) are multiplied by h a simple quantum mechanical interpretation of the ray equations as the Hamiltonian equations of motion of a light particle ensues. For further discussions of the ray equations in connection with plasmas see for instance BEKEFI (1966) and BORNATICI et al. (19S3). The standard derivation of the ray equations assume that the frequency, u:, and the wave vector, k. are both real. The waves must satisfy the local dispersion relation A(k.-:.B(r).nr(r),re(r)) = 0 . (5.2) For real values of.*.-1 the values of k which satisfy the dispersion relation (5.2) for a plasma are in general complex reflecting the fact that there is in most cases at least a small amount of absorption or emission. If, however. |*.J « |*«.| , (5.3) corresponding to propagation with small losses (or emissions) it is usually accept able the ignore the imaginary part of k and the imaginary parts of the derivatives of the dispersion function which are introduced below. A more detailed discussion 'When solving boundary value problems, such as the stationary emission from an antenna, by Fouri'-r Laplace transforms it is most convenient to assume that u is real and k is complex to reflect the fact that the amplitude of the oscillations docs not vary in time but that it docs vary in space For an initial value problem it can be more convenient to assume that k is real and u is complex, or that both quantities are complex. 72 Chapter 5. Propagation of electromagnetic waves of v/hen this is acceptable is given in BORNATICI et al. (1983). It is common to require that the anti-Hermitian prrt of the dielectric tensor, ea, be small While this ensures that (5.3) is satisfied the requirement that ea be small is more restric tive than the requirement that klm C kRe. In practice the condition (5.3) is not a severe limitation since the trajectory which a beam of electromagnetic radiation takes through the plasma is usually only of interest when the absorption is small. The problem of ray-tracing when (5.3) is not satisfied has been dealt with by FRIEDLAND and BERNSTEIN (1980). The ',rm of the ray equatic s (5.1a) and (5.1b) is not the mort convenient for numerical calculations. To write the equations in a more convenient form we observe that for the dispersion relation (5.2) to be satisfied the dispersion function, A, must remain constant (zero in fact) for simultaneous variations in k and LJ, and in r and a/, 6\(r,k,w ) = 6r— +Sk— + 6u—= 0 . 5.4 or ok OUJ With this relation the partial derivatives of u with respect to r and k can be rewritten in terms of partial derivatives of the dispersion function, allowing the ray equations to be rewritten as dr _ d\/dk ^ - ^* (55b) dt ~ 0A/&*" l5,5b) As discussed above, these equations give a good description of electromagnetic wave propagation when plasma scale lengths are large relative to the wavelength and absorption is negligible. In addition to a cold ray-tracing code a weakly relativistic ray-tracing code, RELRAY, was constructed as part of this work. Both codes are based on integrating the ray equations (5.5a) and (5.5b). To reduce the loss of accuracy in the numeri cal manipulations, analytical expressions (rather than finite differences) were used for the derivatives of A, 5.2. Ray-tracing 73 These derivatives were calculated from analytical expressions for the wave tensor, A, and its derivatives by the relation (4.52). Analytic expresions for the derivatives of the wave tensor require analytic expressions for the derivatives of the dielectric tensor (see expression (4.12) for the definition of the wave tensor). The weakly relativistic ray-tracing code, RELRAY is based on the dielectric tensor described in Section 4.5. It was found that particular care had to be paid to the evaluation of the Shkarofsky functions entering the expression for the weakly rel ativistic dielectric tensor for propagation close to perpendicular to the magnetic field. In this regime evaluation of the Shkarofsky functions by simple forward recursion is numerically unstable and leads to inaccuracies in the dielectric ten sor preventing accurate integration of the ray equations. With the algorithms described in Section 4.G this problem was overcome. RELRAY works reliably for propagation at any angle to the magnetic field and for most parameters, with strong resonance regions (e.g. the fundamental electron cyclotron resonance) be ing an exception. This exception is fundamental in that the ray-tracing equations (5.5a) and (5.5b) implicitly assume that the dispersion relation can be satisfied for real values of k and u> as discussed above. The results of tracing rays in X mode through a toroidal plasma with circular cross section with a cold ray-tracing code and with RELRAY for a number of different temperature profiles are presented in Figure 5.1. The plasma was modelled by: B - B0^-f- (5.6a) "e = ("eo - nei) (l - {r/af) " + nei (5.6b) 2 PT Tc = (T£0-Te,)(l-(r/a) ) + Te,. (5.6c) The rays were launched in the poloidal plane and remain there since no magnetic shear was included in the equilibrium (this was done to simplify the presentation and is not a limitation of the codes;. It is noteworthy that the refraction of the rays is reduced with increasing temperature. 74 Chapter 5. Propagation of electromagnetic \vav JG91 561/1 3.0 4.0 5.0 Major radius (m) Figure 5.1: X- mode rays trared using the cold and the weakly relativistic ii del with a number of different temperature profiles. The rays remain in the plane shown. The plasma used is defined in equa tions (5.6a) to (5.6c) with R0 = 3 m, a = 1.5m, B0 — 3.5 T, ,9 3 17 -3 ne0 = 6.01 • 10 m- , na = 1 • lO ™ , pn = 0.7, Te0 = 1.1, 5.1, 10.1, 15.1 keV, Td = 0.1 keV, pT = 0.7. W/2TT = 140 GHz. 5.3. Propagation through an inhomogeneous and anisotropic plasma id 5.3 Propagation through an inhomogeneous and anisotropic plasma Let / be the energy flowing across a surface, A, per unit area. It should be noted that I is an integral over directions of propagation. The energy flow, in a beam emanating from an antenna with an aperture which is large relative to the wave length, can have a very small spread in directions of propagation. In such a beam /, with .4 perpendicular to the beam, is a convenient way of describing the energy flow. We will refer to / as the intensity in accordance with the definition given by BORN and WOLF (19S7). Let X, the normalized intensity, be the intensity of a beam divided by the total power, P, of that beam: X = I/P (5.7) When describing radiation emanating from sources in the plasma it is convenient to differentiate with respect to direction of propagation. We will write the intensity per unit solid angle of wave vector as d2I/dk. In an anisotropic medium the ray direction, identical to the direction of the group velocity, is in general n >i parallel to the direction of the wave vector. We shall therefore also define an intensity per 2 solid angle of ray direction: d I/dvg. BEKEFI (19G6), amongst others, has shown that in a loss free plasma 2 d I/dvg = constant (5.8) /'?., along a ray trajectory. / dk 2 2 I- (5.9) dva Noting that k (0-//0k)-k vp.k (5.9) may be written as TG Chapter 5. Propagation of electromagnetic waves dk (5.10) Given the power radir.ted per unit solid angle of wave vector the power radiated per unit solid angle of ray direction is found by simple multiplication with the Jacobian, \dk/dvg\. Along any ray, (5.8) gives the following relation between the vacuum intensity and the intensity in the plasma: Pi Pi 1 PI (5.11) dv d dkcL dvgdu /*?„ 9 ^ With the relation (5.10). (5.11) gives PI |k-v,| PI (5.12) dkd*' dkdu The flux tube with cross sectional area 6ap at a point p in the plasma will, at a point r in vacuum have the cross sectional area 6av. For the normalized beam intensity we have Ipfap = IvSnv (5.13) The WKB approximation is generally adequate for describing the transfer of power to and from the scattering volume for the collective Thomson scattering experi ments planned at JET and TFTR. The only time it would break down at JET is when a cutoff is approached. The problems caused by refraction would how ever become prohibitive before the WKB approximation broke down. Ray tracing can therefore be used to trace out flux tubes to give an estimate of where the power goes. The intensity in the tube is given b> the relation (5.13). This ignores diffraction. Methods for dealing with diffraction in inhomogeneous anisotropic plasmas were derived by MAZZUCATO (19S9) on the basis of the theory of inho mogeneous waves by ClIOUDIIARY and FFJ.SEN (1973). The WKB approximation also neglects the fact that some power is reflected in regions with gradients in the refractive index. In a sniftering experiment the on'•• consequence this is likely to have is the loss of some power. For the scattering experiments at JET and TFTR the loss by this mechanism will be small and unlikely to be of any consequence to the performance of the diagnostic. i 5.4. Coherent detection Li. The beam emitted by the antenna will in vacuum have a certain angular spread of wave vectors, ftt. From equations (5.12) and (5.13) one finds that the spread, ftp. in the plasma is related to Q„ as npfap = ^-^- «,&,. (5.14) 5.4 Coherent detection Expressions for the power received by a coherent detector when both the receiver and the source of the radiation are located in vacuum arj well known. In this section we want to find expressions for the power received by a coherent detec tor when the detector is located in vacuum and the source is embedded in an inhomogeneous and anisotropic plasma. We start by f.ndmg the power rect '. from a small source with crosi sectional area kip as seen from the receiver. To this end we first find the radiation at a point v in vacuum resulting from a source at a point p in the plasma and then, at the point v, we use ti..- well-known expression for the étendue of a receiving antenna. Let t' - nower emitted from the source per unit solid angle of ray direction and unu «ngiilar frequency be represented by FPiKu) a3/ &P , (5.15) dvg du: dvg cL; where tap is the cross sectional area of the source perpendicular to the ray direction which connects point p with point v. The cross-sectional areas of the flux tube which connects these two points are fap at p and 6av at v. The receiver would sec. no difference between a source with intensity Ip and cross sectional area tnp located at the point p and one with intensity Iv and area Sav located at v, where the intensities are related as in equation (5.12). The power. P\ received from a source, which radiates uniformly across the beam pattern of the receiver, ran be related to the intensity, /, of the emission from the source through OP* n &i <** ok cb 7S Chapter 5- Propagation of electromagnetic waves where .4 is the cross sectional area of the beam pattern and Q is a constant of proportionality with dimension of solid angie. Often Q is visualized as the cone of radiation from the source which is accepted by the receiving antenna. In vacuum the product QA. which is called the étendue, is equal to the square of the vacuum wave length, A0 [SlEGMAN, 1966): *=*2 (5.17) If the cross sectional area of the source is small relative to the cross section of the beam then tne above result is not applu.tble. Based on results given in chapter 4 of CoLLIN and ZUCKER (1969), an expression similar to (5.16) and (5.17), but applicable for small sources is readily derived [BINDSLEV, 1?S9, Appendix A]. The power received from a small -ource with cross sectional area £u perpendicular to the beam is given by dP* (5.1S) Ok du: Q = Kl (5.19) where I is the normalized beam intensity introduced in ecjuation (5.7). Thus we find that the power received from the point v is given by or _ 2 J±_ ia (5.20) a* ° l dkd* t Using (5.12) and (5.13), (5.20) takes the form OP' i.a Ik • v,| *»/ T ear, (5.21) d~- 0 p /<2 dkd«, At a detector located outside the medium the power, Ps, received from a small sourer embedded in the anisotropic medium is thus given by 5.4. Coherent detection 12 where A0 is the vacuum wave length of the received radiation and X* is the normal ized intensity of the beam pattern of the receiving antenna at the position of the source. Note that in vacuum k* • v*//is2 = 1, reducing (5.22) to the well known vacuum relation. For coherent detection of the field resulting from an extended source the fields at the detector resulting from each point in the source must be added up to give the total field, from which the total power follows. We will assume that the source of s the field is a current distribution, j(r, r), embedded in the plasma. Let {ElM(uJ )}p be that part ~»f the field at the detector which results from the current at the point rp in the plasma. From equation (5.22) and the preceding discussion it follows that {£]|„(w*)}r is s related to j(rp.i-r ) as i {E:j^)}p = Cy/i^)(e- y-j(rp,^)e-'^^ , (5.23) where (e-1)* is a vector which selects that component of the polarization which is accepted by the recci/ing antenna (see discussion in chapter 6). C is a constant of proportionality, which can be inferred from (5.22) and (6.31). C is independent of rp except for a possible phase factor which would appear if neighbouring ray paths had different optical lengths (i.e. if waves emitted from the detector antenna had buckled phase fronts). Assuming that the refractive index of the plasma varies slowly across the detector beam, it follows that the optical lengths of adjacent rays will be equal at planes perpendicular to the rays and no phase factor enters the constant of proportionality. In this case the field nt the detector resulting from a distributed source is E:„ = C(e-iy-j(k*,S) (5.24) where j(k^^) = yv/^v)j(rP'w,)e",(k^'>)f/rp • <5-25) From equation (5.22) and by comparison of expression (5.24) with (5.23) we find that with an extended source the power received by the receiving antenna per unit angular frequency is given by SO Chapter 5. Propa«;. fion of electromagnetic wares dr _ 2k'-y^P(k;U.-) IT A° —rt -,- -— (° 26) where yp(k,tf) dkdL- is the power per unit solid angle of wave vector and unit angular frequency emit ted by the current distribution j(r.i) into the mode accepted by the receiving antenna. j(r,r) is the original current distribution weighted by the square root of the normalized beam intensity of the receiving antenna, j(r.0 = ^(7)j(r,0 . (5.27) 5.5 Power flux in a plasma with spatial disper sion The energy flux associated with propagating modes in a plasma will be derived here. It is well known that in a spatially dispersive medium the energy flux is the sum of the electromagnetic flux, accounted for in the conventional Poynting vector, and a kinetic flux due to correlated movement of particles with the wave. The expression for the kinetic flux associated with a quasi-monochromatic mode has been derived by a number of authors, e.g. BERS (1963 and 1972) and LANDAU, LlFSIUTZ and PITAEVSKH (19S4). Here, however, a broadband field, propagating in all directions, is investigated. To ensure that integrations over frequency and direction of propagation are taken correctly, the expression for the energj- flux in a spatially dispersive medium is developed from Poynting's theorem. The Poynting vector is defined as S = E x H . (5.28) From Maxwell's equations (4.4) and (4.5) and the constitutive equation (4.7) in section 4.3, Poynting's theorem is readily derived: * •5.5- Power mt.v in a plasma with spatiai dispersion SI V S = H VxE-E VxH - -T-w~i'dr~E h~E J • (529) jp is the plasma current induced by E: j,(r. f) = J while j represents externally induced currents. For most of the terms in (5.29) the interpretation is straightforward: the term on the lef hand side is the divergence of the electromagnetic flux, the fLst two on the right account for the change in electrostatic and magnetos tat ic energy while the last on the right accounts for the exchange of energy with external sources. The !?rm E • jp is the exchange of energy between the field and the plasma particles. However, jp is a function of E which complicates the interpretation of this term. In the limit where fitm E • jp can be approximated by three terms which account for dissipation, change in energy and divergence of flux associated with correlated particle movement. It is this latter term, the kinetic flux, which is of particular interest here. For susbsequcnt applications we are interested in the kinetic flux associated with a broad band field consisting of propagating waves which all belong to the same branch but cover a wide range of wave vector directions. The wave vectors asso ciated with these waves for a particular value of the frequency, u;, generally reside on a continuous surface in k space. Each wave is thus identified by the frequency. »4.-. and the direccion, k, of its wave vector. A broad band field of propagating waves covering a range of wave vector directions is conveniently represented by (a detailed discussion of this point is given in Section 6.3) E(r,0 = j~-J J EPJk^K'^'^'iå^ . (5.31) It 2» m 9 where the summation is over modes m and kn = ^m(k, w) is the norm of the wave vector which satisfies the dispersion relation for given values of k and ui. For a field of the type given in equation (5.31) we have S2 Chapter 5. Propagation of electromagnetic waves 532 E(r.r)-jp(r.r) = ^/^'(ki^'iK^-*)^^^) ( ) ^(fc-kp-r-C-,—)«) f^ilLV (3^ dktdvtdktd* \ki-v.i/ \k2-v2y where k = *k , k = k(k,u) The summation ever modes is dropped, implicitly assuming that the flux does not contain cross-mode terms. Let - , _ k2 + kJ u,2 + u>i Ko - r , u.-o - - Integrating (5.32) over time and space (e.g. to find the time-averaged power crossing a surface) the integrand will, due to the harmonic term, vanish except where (k2,u.-2) ss (k|*u/i). In this case expanding tr around (ko,w.'0), retaining only zero and first order terms, is a valid approximation: d dtr(k ,u ) + x0 0 / (u;2-wi)/2 + 0(k2 - kj,w2-u.'i) Substituting (5.33) into (5.32) and adding the complex conjugate expression, mak ing use of the fact that the left hand side is real, we find 5.5. Power flux in a plasma xrith spatinl dispersion S3 J JE(r.t)}p(rJ)drdt v r = j j£{Er(ki,*itfJE?(kz,*2)}drdt IT i- T (5.34) + //vrl £nk,.^,)(i) (^)"*#2.^)} *<" + where £ { •-• ) is the integral operator: The second term on the right hand side of (5.34) is clearly the kinetic flux leaving the volume V. For convenience we define a Poyiitimj tensor (s,,); » MMA_|hi>Æ + MA + IM^ , (5.35) tyo which has the property S,j £"( k|, o.-| )£,( k2. W2) E(k2.^2)x(k;xE'(k,.u:,)) E-(k,,^)x(k2 x_E(k2,u;2)) o + —* 7> (5.36) E(k2,u/2) X H'(k,,u;,) + E'(k,,ur,) X H(k2,W2) From (5.29) and (5.34) it follows that the average power, P, carried across a surface, A, over a period, T, is given by (5.37) S4 Chapter 5. Propagation of electromagnetic weaves where n is a unit vector normal to the surface .4. Integrating over .4 and 7*. (5.3S) £T(ki,--.)£r(k».^) lr^-\ (r^-) <•< f\(n x (k2 - kJ)) iri^z - ^"i) <*i _l In the integration over r. it was assumed that klm <£ (L\k x n|) where L is the dimension of -4 in the direction of (k x h). pa is the position vector of the centre of the surface -4 and r the midpoint in the time interval, £r(^-) is a peaked function with a maximum of T/2~, a width of the order 2« JT and /^ &T(U) (LJ = 1 t\ is defined similarly, only in two dimensions. Carrying out one of the k and one of the u.' integrations, |*|« latkjuxn) -i p -2fcImkp. (5.39) = Tuff L J' |k-v.|2' W 4 Sy + (;)(I))N>"'* • The Jacobian. ilkfij |. stems from the integration of 8\ over k: ?(k) d(K< xfi) d(kR„ x n) (5.40) 3(k) d(K,l) k2 = |n-v,| --* Ik • vf I where L means for fixed u.'. £f'£; is the power spectrum of Ep around the location p and time r. Sub JA,T scripts .4. T indicate that the power spectrum is obtained from knowledge of E(r, t) on the surface .4 and time period T. It is therefore also indicated that the reso- 5.5. Power flu.v in a plasma with spatial dispersion S5 lut ion in the power spectrum is approximately 2~/T in u.» while the resolution of k x li is (2»)2/.4. The resulting resolution in k is d(k) Ak = I A{kRe x n} 9(kR. x n) (2^)2\k-v \ a (5.41) k*tA \h - v9| Inserting (5.40) in (5.39), integrating only over positive frequencies and making use of the fact that the direction of the power flow is parallel to the group velocity (see for instance BERS, 1963), we find that k2 p = r° /c-2*in,k-/>. _l (5.42) (2zfTJ-coj |k iff ' S-(f){tF)M<--» dkdtjj 2 Note that the difference between |A-| and fcR, is of second order in ktm and hence is not included here. The dielectric tensor was substituted for the conductivity tensor in (5.42). Letting -4 extend to infinity in both directions and neglecting damping, we find that the energy flux associated with a mode is given by 0»Pm(fc,U>,T) 2 kl 3 Erm (r) dkdw (2T) |k-v3| (*-(?){*}> (5.43) where the Poynting tensor now takes the simpler form (5.44) {,jh~ 2/iou, For subsequent calculations it is convenient, to introduce the normalized flux, 86 Chapter 5. Propagation of electromagnetic waves »-sIHfHfcFH- (545) With the normalized flux the energy flux takes the form S can be given a simple physical interpretation. Assume two electromagnetic fields, one in a plasma and one in vacuum, both with the same electric field strengths. S is the ratio of the energj' flux of the field in the plasma to the flux of the vacuum field. Alternatively, if the energy fluxes of the two fields are identical then S is the ratio of the square of the vacuum electric field strength to the square of the electric field strength in the plasma. When e = ek we find the following useful relations: k • (é* • S • é) = e0cfi k • é" x (k x é) = — e0cft é* • k x (k x é) = —é'-eé , (5.47) - w£o#£jj _ £o£ kdSjj (5.48) 2 dk ~ /i 2 dk From (5.45), (5.47) and (5.4S) we find that the normalized flux satisfies the fol lowing relation: '•'-M'-ii)-' (5«» 2/ ' dk 'e 5.6. Mode conversion £1 5.6 Mode conversion In this section a simple approximate model, due to DE MARCO and SEGRE (1972), is given which describes the mode conversion that takes place in the plasma edge. The model gives an intuitive understanding of the mechanism by which mode conversion takes place and readily lends itself to computations. Based on this model a new method is given for calculating the optimum vacuum polarization for coupling to a particular characteristic mode in the plasma. In terms of the underlying mechanism it would be mom correct to refer to the mode conversion in the edge region as a lack of mode resilience. To illustrate this point consider the propagation of a linearly polarized electromagnetic wave through a plasma with very low density and a static magnetic field which has a shear along the direction of propagation of the wave. If the density is sufficiently low the wave propagates as if in vacuum and thus does not change its polarization. Assume that the static magnetic field is parallel to the electric field of the wave at point .4 and perpendicular to the electric field at point B. At point A the wave is polarized as the ordinary mode of the plasma while at point B it is polarized as the extraordinary mode. There has been a total conversion of power from the ordinary mode to the extrordinary mode, although relative to a fixed coordinate system the polarization has not changed. When the density becomes sufficiently high then the plasma modes become re silient, that is, the power propagating in a plasma mode remains constant despite changes in the polarization of that mode. If the density in the example given above were sufficiently high, then the electric field vector of the wave would at point B be turned by 90° relative to its direction at A and thus still be in the ordinary mode. Whether the plasma modes are resilient depends both on the local properties of the plasma (determined in the cold plasma limit by the density and the magnetic field) and on the rate of change of the polarization of the plasma modes. Mode resilience or mode conversion can, within certain limits, be investigated by following the evolutions of the Stokes vectors which describe the polarization of the propagating wave and the polarizations of the plasma modes [RAMACHANDRAN and RAMASEHAN, 1961; DE MARCO and SEGRE, 1972]. The main limitation of this approach is that only the transverse part of the polarization can be modelled by a Stokes vector. This limits the applicability of this model to low densities, which fortunately is also where there is the greatest need for a model since this is the regime where mode conversion takes place. Whether mode resilience sets in before the model becomes invalid is of course not assured. SS Chapter 5. Propagation of electroroajgnctic waves In a general dielectric medium the electric field vector of a propagating mode can be written on the form E'(r,0 = Re{Eci(kr"urf)} where E is a complex vector. In the WKB approximation E and k vary slowly with r and t compared with the phase factor. Projected onto the plane perpendicular to k, E' traces out an ellipse with major axis a and minor axis b. We define a coordinate system with z pointing in the same direction as the wave vector, k. If E lies entirely in the (x,y) plane then the polarization is purely transverse and the polarization state is completely described by b/a (a = |a|, 6 = |b|), the angle between a and z and the direction of rotation (see Figure 5.2). Rotation in the positive sense corresponding to (ER, x E,„)-k > 0 is called right hand polarization and rotation in the negative sense is called left hand polarization [BORN and WOLF, 1937]. Any transverse polarization can be represented by a unit Stokes vector cos 2\ cos 2>l' S = cos 2\ sin 2tl' * e (0; v[ (5.50) sin 2\ where tan \ •*0 (5.51) and V' = Z(a,x) (5.52) \ is positive for right handed polarization and negative for left handed polarization. The significance of a, b, \ and 4' is illustrated in Figure 5.2. The locus of the unit Stokes vector resides on the unit Poincaré sphere [BORN and WOLF, 19S7] (see Figure 5.3). The north and south poles of the Poincaré sphere correspond to right and left handed circular polarization respectively, while the equator corresponds to linear polarization with the orientation turning through Figure 5.2: Illustration of variables describing a transverse polarization state. 90 Chapter 5. Propagation of electromagnetic reaves Zt / -2*- Figure 5.3: The Poincaré sphere. 5 6 Mode ctmxrrsion 21 ISO' as the equator is circumscribed. Intermediate points represent general ellip tical polarization states. An important property of the Poincaré sphere is that opposing points represent orthogonal polarization states. Any polarization state can be represented by a linear combination of two linearly independent polarization states, in particular by a linear combination of two orthogonal polarization states. A further property of the Poincaré sphere is that for a general polarization state, S, the fractions of power, pi and pj. in each of the two orthogonal polarization states. Si and S*, *re given by P. = 1+^''ST (5.53a) P2 = 1"^I'S, (5.53b) which has a simple geometrical interpretation (see Figure 5.4). All polarization states with a given ratio between the powers in each of two orthog onal modes sit on a circle which lies in a plane perpendicular to the line connecting the orthogonal modes, and which intersects this line at a distance 2pj from the locus of Si and a distance 2pt from the locus of S2. We will refer to such a circle as an equi-powcr circle. The azimuthal location of a point on the circle depends on the relative phase of the orthogonal modes in the polarization represented by this point. In a homogeneous medium with two characteristic modes with different phase velocities (like a magnetized plasma) the only polarization states which remain unchanged during propagation are those of the characteristic modes. The polar ization state of any other electromagnetic wave changes as the relative phase of the characteristic components of the wave varies. If the power ratios of the char acteristic components remain unchanged then the polarization state moves on an equi-power circle. Assuming the the WKB approximation is valid, RAMACHANDRAN and RAMASE- IIAN (1961) showed that the evolution of the polarization state of an arbitrary transverse wave propagating through an inhomogeneous anisotropic medium is given by the differential equation dS 06 - ^ = ^S,xS (5.54) 92 Chapter o. Propagation of electromagnetic waves Equi-power circle Figure 5.4: Evolution of an arbitrary polarization state on the Poincaré sphere. a.fi. Åforfe conversion JB where 6 is the relative phase between the two characteristic modes Q - P\-*i (5.55) = -(*,-*,)* so 5 Yz = -in - /*») ( •») Expression (5.54) is readily integrated numerically to give the evolution of the polarization state as the wave propagates through the plasma. Expression (5.54) shows that in a homogeneous plasma the polarization state of a propagating wave processes around the line connecting the two characteristic modes at an angular rate given by do/dz (p = 2~ corresponds to one revolution). In an inhomogeneous plasma the polarization states of the characteristic modes also change. In this situation the motion on the Poincare sphere of the polarization state of a propagating wave is processing around a moving centre. In particular if the wave was originally a characteristic mode it will not remain so as the char acteristic polarization moves away. If the rate of precession is much smaller than the rate of change of the characteristic polarization states, then the polarization state of the wave remains unchanged but the distribution of power in each of the two modes changes. This corresponds to the low density case described at the beginning of this section. If on the other hand the rate of precession is large rela tive to the rate of change of the characteristic polarizations then the polarization of the wave will follow the changes of the characteristic polarizations and remain approximately on an equi-power circle (which of course is moving), implying that the ratio of power in the two modes is conserved. This corresponds to the high density case described at the beginning of this section. From the expression for the rate of precession (5.56) we find that the power prop agating in each mode is conserved if l;,2~/<,|u?L > 1 . (5.57) Here L is the scale length for changes in the characteristic polarization, which for a magnetized plasma is mainly determined by the scalelcngth of the magnetic shear, 94 Chapter 5. Propagation ofejectrotnagnctic waves In the limit where /it = /t3 the polarization of a propagating wave does not change and powers in characteristic modes are not conserved. DE MARCO and SEGRE (1972) introduced this method for cold magnetized plas mas, giving explicit expressions for the quantities entering (5.54) for propagation perpendicular to the magnetic field in a cold plasma and giving analytic expres sions for the polarization change in the case where the integrated phase difference o is small. Investigations of large polarization changes by numerical integration of (5.54) were made by SEGRE (197S). Expressions for propagation at an arbitrary angle to the magnetic field and conditions under which the modes are resilient were given by SEGRE (1990). Numerical investigation of the polarization change to be expected for parameters relevant to the JET collective Thomson scattering diag nostic were made by BOYD (1990). These results indicate that mode conversion can be significant (of the order of 10 %) if the low density limit of the polarization of a given characteristic mode is launched. Expression (5.54) can be used as the basis of a new method to determine the optimum vacuum polarization to be launched or accepted by the antennas in or der to couple purely to a single plasma mode in the region of the plasma where the modes are resilient. The optimum vacuum polarization is that vacuum state which evolves in such a way that it coincides with the polarization state of the desired mode when regions of the plasma are reached where n > 1 and the mode therefore is conserved. This vacuum state ran be found by integrating expression (5.54) backwards starting from the desired characteristic state sufficiently deep into the plasma that n > 1 and following the ray trajectory out of the plasma. The integration should of course start at a sufficiently low density that this the ory is valid. Its validity can be checked by determining the extent to which the polarization is transverse. The ray trajectories of O-mode and X-mode starting with parallel wave vectors arc generally not significantly different at the relevant plasma parameters. A number of other methods exist for computing the mode conversion in a plasma. Generally they do not have the simple elegance of the method introduced by DE MARCO and SEGRE (1972). All the approaches arc essentially valid in the same range of parameters although some of the other approaches do allow absorption to be included. Early work on coupling of modes in inhomogeneous plasmas was described among others by BUDDEN (19G6). FlDONE and GRANATA (1971) investigated the mode conversion due to magnetic shear in a slab model. The problem was formulated 5.6. Aforfe conx-erskm & in terms of a set of four coupled equations representing forward and reverse prop agating waves in the ordinary and the extraordinary modes where the coupling coefficients were due to the magnetic shear. The method was applied to the prob lem of mode conversion of electron cyclotron emission (ECE) at the upper hybrid resonance. Bo YD (19S5) applied this approach to find the limiting polarization of ECE leaving the plasma. CRAIG (1976) also considered the problem in a slab geometry, seeking the solution by matching the wave expansions in O-mode and X-mode at each slab junction. AlROLDI, OREFICE and RAUPOM (19SS) general ized Craig's approach, giving a differential formulation. Experimental evidence of mode resilience of ECE radiation was given by HUTCHIN SON (1979). Chapter 6 Field due to current sources in plasma 6.1 Introduction In this chapter expressions are found for the fields which result from a current distribution embedded in a magnetized plasma with spatial as well as temporal dispersion. In Section 6.2 the expression for the near field is found. The far field expression is derived in Section 6.3 and expressions for the energy flux in the far field are given in Section 6.4. 6.2 Near field The fields resulting from a current distribution, j, can be found by solving for E in equation (4.13). which is repeated here for convenience: AE=—j. (6.1) yJCo To this end it is useful to introduce the eigenvalues, Aj, and unit eigenvectors, g;, of the wave tensor. A. Let the columns of the tensor g^ be the unit eigenvectors (index j identifies the j*" eigenvector). Let X,j be a diagonal tensor containing 96 6.3. Far field SZ the eigenvalues: f Ai 0 0 ] kj = { 0 A32 0 I 0 0 A3 j The eigenvalue equation for A can be expressed as A,j gjk = gu A/* • (6.2) Solving for A, A^ = gu Au- glj , (6.3) where 9ij 9jk - hk Note that A is in general not Hermitian and therefore g~* ^ g*{. The inverse of A is K, =9n\k i 9kj_-i , (6.4) where A,'1 0 0 ) ^ = 0 AJ' 0 0 0 A71 Solving for E in (6.1) and inserting (6.4) we obtain 9ii 9jk Jk E = — V (6.5) W£0 J? Xj where there is no implicit summation over repeated indices. 6.3 Far field E is analytic in the four complex arguments (k, u>), except on a number of singular surfaces, which will play an important role in the Fourier inversion of (6.5). 9S Chapter 6. Field due to current sources in plasma Assuming j(r,t) is physically reasonable then j(k,w) will have no singularities for finite values of (k,u;). g,} and g~^ can also be assumed to be analytic for all finite values of (k,u>). The only singularities of E are therefore the poles where one of the eigenvalues, A„, equals zero. When A„ = 0, the eigenvalue equation for g„ (equation 6.2) is identical to the homogeneous field equation Ae = 0 . (6.6) Here é is the polarization vector of the electric field of the mode (k,w). A solu tion to (6.6) only exists when Det {A} = 0, which is the dispersion equation for electromagnetic waves in the medium. This implies that, for values of w and k which satisfy the dispersion equation Det{A(k,w)} = 0, there will be at least one eigenvalue A„(k,u;) which is equal to zero. In this case g„ is identical to é: é = g„, {el}- = ffJ wnen *„ = 0 • (6-7) For the subsequent analysis it is convenient to express k in polar coordinates: k = Jtk (6.8) where v [ kKt < 0 for Re{k • n} < 0 ' n is for the moment an arbitrary unit vector. Note that if the complex conjugate of k had been used in the definition of k then E would not have been an analytic function of k and k. With the definition given in (6.9) E is an analytic function of (k, k), which is important for the subsequent analysis. The surfaces in (k,u>) space, on which E(k,w) is singular, are the surfaces where the dispersion relation, Det{Aa-m,k,u>)} =0 , (6.10) is satisfied. Solving for km in (6.10), fj.3. Far field 22 tn = UM , (6.11) gives the location of the pole in the complex k plane as a function of (k,u>). (6.11) is of course simply a parametric representation of the singular surfaces, with (k,u;) as the free parameters. To define the inverse Fourier transform of (6.5) it is necessary to specify the con tours for the ijj and k integrations. E is the response of a causal system to the driving force j. From the condition that the system response be causal, it fol lows (see Appendix B) that the path of integration in the u plane must pass above all singularities in the finite w plane. For sufficiently large values of the imaginary part, wlm, of u; the integration over k space can be confined to the three-dimensional real space, R3. Let C^ be a contour for the u; integration on which the values of u satisfy this criterion. The inverse transform of E(k, w) can then be written E(r,0 = —i- / / JEikk^y^*-^k2dkdkdu . (6.12) a 2T ck Here C'k = R and k is integrated over the half sphere for which k • n > 0. Since we are dealing with a boundary value problem, as opposed to an initial value problem, it is preferable to integrate over real values of u. For this to be possible the medium must be assumed to have no absolute instabilities (see BERS, 1963 and 1972). Furthermore, when the imaginary part of u is reduced to zero it may happen that one or more of the poles in the k plane cross the real axis. The contour of integration in k, Ck, must then be deformed to keep the poles on the same side of the contour. In order to identify the far field in the direction n from the sources, the contour for the k integration is lifted a distance ifl up from the real axis. The expression for the field then takes the form EP {kmkr ut) d(kmk) E(r,o = j^3J jE J^y - dkdu (6.13) V R 2ir d(k) + pTT^-'*" / / /E(w,k + i/3)e'{kUr-ut)(k + iøf dkdkdu (2T) where the summation includes the poles which are situated above the contour Ck and below i/3. E£,(k,u;) is defined as 100 Chapter 6. Field due to current sources in plasma. -i Res{E(k,u/) in which Res{E(k,u;)rfk; km] is the residue of E(k,u/) at a singular point k = km when the integration is with respect to k: Res{E(k,u;)dfc;itm} = ^T /E(fck,w)dJt . (6.15) 27T2 J The Jacobian, \d(kmk)/d(k)\, of the transformation kmk i-» k, gives the ratio between a differential surface element, S(kmk), on the singular surface and its corresponding element, Six., on the unit sphere. While the Jacobian is defined for complex values of km we will generally only be interested in situations where |Im{fcm) | <; |Re{fcm} |andlm{fcm} is a slowly varying function of (k,w), in which case the physical interpretation of the Jacobian is straightforward. The Jacobian is introduced at this stage to give a definition of Ep which facilitates an intuitive understanding of the physics contained in subsequent expressions. The expression for the Jacobian is 2 d(kmk) k ^- , (6.16) d(k) k-vp where vg is a unit vector normal to the singular surface in k space. Let A(k,w) = Det{A(k,u;)}. On the singular surface, where A = 0, SA = ^-SV + ^6u = 0 . (6.17) ok du For 6u = 0, dA 6k = 0 . (6.18) dk It follows that the vector d.\/dk is normal to the singular surface in complex k space. Dividing by the scalar -dA/du; we find that du/dk is normal to the surface: M „ -av* _ a* ak" aA/aj ait 6.3. Far field 1Q1 and hence that *.llf£ • (6-20) With the assumptions made about k we will generally have that (3w/5k),m : (3w/3k)R. and 6kIm : 6kRt are of the same order as klm : kKt. Thus, ignoring the term Im ||^} • Sklm, which is of second order in ktm, the real part of (6.1S) reads •SkRt = 0 . (6.21) (I)R e (v3)R. (^> (va)Im) is therefore parallel to the group velocity, (du/dk)Rt, which ex plains the notation. At large distances from the source the second integral in (6.13) vanishes, the field there being due only to the poles which represent the propagating modes. In the direction n from the source the expression for the far field thus takes the form nr,t) = -^J f^KiK^y^'-^t^rdkdu; , (6.22) where it must be remembered that km is a function of (k,w). To express E£, in terms of the source currents and the dielectric properties of the medium, (6.5) is inserted in (6.14): v L E m (k, w) = Res \ — dk; km \ ± (6.23) In (6.23) there is implicit summation over the index j but not over v. Assuming that 1/A^ has a simple pole at k = km, R«{j^*;*«} = (fjr I • (6-24) k=k„ 102 Chapter 6. Field due to current sources in plasma Solving for A in (6.2): K = g^i Ai>9i» • (625) ; Differentiat ng with respect to k and making use of the facts that Ay gj*\k=k = 0 1 and g" Ay I =0 (the latter is easily seen by substituting in (6.3) for Au), we lk—km find ~dk = g" ~W9>» • (626) Summation is implicit, except over v. From (4.12), § = ¥^*}+l • ™ With (6.24), (6.26) and (6.27) and making use of (6.6) and (6.7), the expression for E£, takes the form Ep =é k7(!"/JL , (6-28) m _, f-2e de\ ^oC •(— + dk) where k = km. In the subsequent development the index m will be dropped. 6.4 Far field energy flux The energy flux associated with the field Ep(k,w) is analysed in 5.5 for modes where \klm\ < JfcRe|. A finite observation time T is assumed. It is shown that the power, 03P/dkcb, carried by a mode (k,u>) per unit angular frequency and per unit solid angle of k. is given by 6.4. Far Geld energy fiux 103 yP(k,u;) 2 JL-2 i r i = ,» ^ 3 - €0cS^\E 'E?\ , v(6.29) 3k5L' (2*) |k-v,| T[ JT ' • i|(«.-c?)W)H where u> takes only positive values. S is the energy flux, associated with a field in the plasma, normalized by the fiux of a vacuum field with identical amplitude. S^, referred to as the Poynting ttnsor (see equation 5.44), is a rank three tensor which is Hermitian in the indices ij and has the property that its bihnear product with the electric field vector is equal to the real part of the Poynting vector: S0E*£J=Re{ExH) P P p [E 'E ]T/T is the power spectrum of E in the observation period which has a duration T. Superscript h indicates the Hermitian part of the tensor concerned. It should be noted that (de]h deh With expression (6.28) for Ep the power spectrum takes the form 1 l £P-£PI = £i! Jr_ (6.30) wt°r '{~+dk)e\ Inserting (6.30) in (6.29) we obtain 1 2|/'|2|k • v,| S £ 1 ^P(k,u;) e" j (6.31) dkcb (2*)V \ * ok) e| 104 Chapter 6. Field due to current sources in plasma To reiterate: this is the energy fluxo f the far fieldpe r unit solid angle of wave vector and unit angular frequency, resulting from a set of source currents, j, embedded in a spatially dispersive medium. Making use of the results from the previous chapter we find from equations 6.31, 5.22 and 5.24 that of the electromagnetic field produced by an extended current distribution in the plasma, the total power per unit angular frequency received by a coherent detector outside the plasma which is coupled to a single plasma mode is given by {k 6 R} . (6.32) &" ~ .. ,3 I, (~2t de\ . Chapter 7 Bilinear plasma response 7.1 Introduction The dielectric properties of a plasma are non-linear as may be seen from the fact that the basic set of equations governing the electrodynamics of a collisionless plasma, Vlasov's and Maxwell's equations, are non-linear in the dynamic variables E, B and /. The non-linear part can usually be neglected, leaving a linear set of equations which lead, in the familiar way, to the linear dielectric response of a plasma characterized by the dielectric tensor. A fundamental property of linear systems is that separate solutions or waves do not interact. In a non-linear system waves do mix and set up new waves. The solutions to the linearized system are of course not rigorous solutions to the full non-linear system but the distortions of these solutions are small when the non-linear part of the response is weak. In this case the linear solutions form a suitable base for solving the non-linear system, the non-linear terms giving rise to small additional parts to the solutions. To set out the mathematical treatment appropriate to this problem, consider the linear operator Li and a set of solutions A"/, t'• = 1, 2, 3, ..., to the homogeneous equation UX\ = 0 . (7.1) Since Li is linear it follows that any linear combination of solutions is also a solution to the homogeneous equation with L\\ L,X, = 0 ; Xi^aiX]. (7.2) 105 106 Chapter T. Bilinear plasma response Summation over repeated indices is implicit. Consider now the nonlinear equation I,A7 + ^/mn.Y;A-;=0 (7.3) where A/mn is a bilinear operator which for convenience we will assume to be symmetric in m,n. 7 is a parameter which does not enter the operators. If 77 = 0, then equation (7.3) is identical to (7.1) and X[ = A*/ would be a solution to (7.3). A formal solution to equation (7.3) is given by X! = limX(B), (7.4a) n) ,) ,) L,X} = -J?MmnAt- At- , (7.4b) L,A-,(0) = 0, (7.4c) when T] is sufficiently small that the recursion relation (7.4b) is convergent. Here the upper index (n) indicates that the function X\ is the result of the n'th iteration of expression (7.4b). Writing X\ as A'/ = A, + Z, , (7.5) we find that Zt = lim 2j"> , (7.6a) n) 1 1 1 L,ZJ = -r?Mmn (XmXn + 2AmZ<»- > + Zjr ^«"- *) , (7.6b) Z\0) = 0. (7.6c) From equations (7.6a) to (7.0c) it is readily seen that to first order in v, £/ is given by 7.2. Source currents for scattered field IQI LtZt = -T,MmnXmXH ; Z,-»0 far ^-»0. (7.7) The effect of the small bilinear term nMmn is to add Z| to the linear solution, A'j. Of the set of solutions to the inhomogeneous equation (7.7), Zi is the particular solution which vanishes when tj -* 0. In a more physical picture we can regard the inhomogeneous part of equation (7.7) as the source which drives Zj. It is noteworthy that the source term is bilinear in the linear solution A"/, which is due to the fact that only the first order term in n was included. This is the first Born approximation. If higher orders of 17 were included, higher orders in A*j would also appear. This is the mathematical framework within which we will describe the scattering of electromagnetic waves by free electrons in a plasma. The incident field interacts with the fluctuations in the plasma through the non-linear terms. This interaction results in a set of currents which drive the scattered field. Quadratic terms in the incident field and quadratic terms in the fluctuations also give rise to new fields. Although these fields are not due to scattering they are in principle observable and will be included in the scattered field to start with. It will, however, be shown that these additional fields arc either negligibly small or appear at frequencies outside the range of interest. 7.2 Source currents for scattered field To identify the currents which drive the scattered field we start with the basic set of equations describing the electrodynamics of a collisionless plasma. The treat ment is limited to one species of charged particles, the electrons, the extension to additional species being obvious. E and B are the total electric and magnetic fields while / is the velocity distribution of the electrons including all perturbations. The evolution of the velocity distribution is described by Vlasov's equation: |+V.|+,(E+VXB).|=0. (7.8) The current associated with / is given by: = while the behavior of the electric and magnetic fields are given by Maxwell's equa tions: 1 dE VxB = ^-^-+WJ (7-10) OB VxE = --£-. (7.11) This is the complete set of equations describing the electrodynamics of a plasma, neglecting collisions. It is convenient to replace equation (7.10) with the inho- mogeneous wave equation, obtained by taking the curl of (7.11) and eliminating B: 1 d*E dj Vx(VxE) + ?aF = -/<0^ (712) Each of the dynamic variables can be separated into a largt term representing the mean and a perturbation associated for example with fluctuations and waves. The mean is understood as a spatial and temporal average over distances and times which are large compared with the characteristic distances and times of the perturbations. The mean terms will be identified by an upper index 0 and the perturbations by an upper index 1. / = r + Z' (7.13a) B = B° + B' (7.13b) E = E1. (7.13c) It is assumed that no static electric field is present (see discussion of this point in Section 4.4). The perturbation terms can be divided into terms associated with the incident field, marked by upper index i; with the fluctuations, marked by upper index 6; and with an additional field resulting from the nonlinear interaction. We label this field with an upper index s although, as mentioned earlier, it contains more than what is usually regarded as the scattered field. 7_g_ Somre currents for scattered nefrf Ufi /• = f + f + r (7-Ha) B' = B' + ff + B* (7.14b) E' = E*+ £: + £•. (7.14c) Assuming that the mean momentum distribution is stationary (df°/dt = 0) and homogeneous (dp/dr — 0) leads to the following zero order term in equation (7-S): (vxtf)-^=0. (7.15) dp Equation (7.15) is satisfied if /° = /°(P|hPA) (7-16) where p[| and px are the components of the momentum parallel and perpendicular toB°. Subtracting (7.15) from (T.S) leaves terms which are of first and second order in the small perturbativc terms. dp Of1 „ OP , , dp -l-+v.-2r+, -rfE'+vxB')-^. If f° is isotropic (i.e. /° depends only on the norm of p) then (vxB')~ = (vxB')p^ = 0 (7.18) and the first term on the right hand side of equation (7.17) simplifies to 110 Chapter T. Bilinear piasma response - \\> note that equation (7.17) has the same form as equation (7.3). The distribu tions associated with the fluctuations and the incident field we define as solutions to the linearized Ylasov equation df* df* « df* . « df° dp dp „ dp • Of l + vl+^xB0)-! = -^ + vxB')^. (7.20) The electric and magnetic fields are calculated from Maxwell's equations with the currents given by the associated velocity distribution. With these definitions the velocity distribution associated with the scattered field, which contains all nonlinear effects, is to the first Born approximation given by (see equation (7.7)) dP dp „ dp ^_ + v.^+9(vxB»)-^ (7.21) = -,(F + vxB*)^ dp -,( + B<>). (§£ + §£), while the electric and magnetic fields are calculated from Maxwell's equations with the currents, j\ given by j* = f/v/'(p)rfp. (7-22) In equation (7.21) the terms on the left and the first term on the right arc linear in the dynamic variabli-s while the last term on the right is bilinear. The scattered field is the result of the coupling described by this bilinear term and would not exist without it. 7.2. Source currents for scattered field 111 To identify which parts of the bilinear term are significant for the observed scat tered field we note that the set of equations governing the perturbed dynamic variables can be written LiZfat) + MitA'i(r,r)Xfc(r,r) = 0 (7.23) where Z{ is a vector containing the dynamic variables of the scattered field, A,- is a vector containing the sum of the dynamic variables of the incident field and the fluctuations, £, is a linenr operator and Mjk is a bilinear operator, symmetric in j and k. Fourier-Laplac1 transforming (see Appendix B) the set of equations (7.23) we get LiZifcu) + T^MjkJxA* ~ k> -w'Wk'V)dk'du' = 0. (7.24) The bilinear term contains elements of the form XlX', X'XS and XSXS. When identifying which elements can satisfy the requirements on the matching of wavevectors and of frequency implicit in equation (7.24), it must be kept in mind that Xi(r,t) is real and thus X;(k,w) = A"(-k, -w). The incident field is assumed to be within a very narrow frequency band while the fluctuations cover a broad band. The scattered field is observed in a range of frequencies around the frequency, u>', of the incident field, the width of the range being smaller than w'. The bilinear element A'1 A'1, gives rise to source terms at (k,w) = (0.0) and (k,u>) = (2k1,2w') and thus does not contribute to the scattered field in the frequency range of interest. The element X'XS satisfies the k and u matching conditions when (k*,u/) = (k* - k\u° - J) and when (ks,us) = (ks + kV' + w1). The latter fluctuations at the higher frequency can usually be ignored compared with the lower frequency fluctuations [Hutchinson, 1987]. The contribution from the element XSXS is generally much smaller than the con tribution from A'1 A* and can therefore be ignored. The velocity distribution associated with the scattered field thus satisfies 112 Chapter 7. Bilinear plasma response f+vÆ + q{yxB>^l _ _,(E. + vxB-).§£ (7.25) dfs -q(E' + vxB').|dp - -q{ES + VXBf)^ dp The first part of the integration of equation (7.25) is essentially identical to the integration of the linearized Vlasov equation required for the derivation of the linear dielectric response of a plasma. In general two approaches have been taken to tackle this problem. In the western literature the Vlasov equation is tradition ally integrated by the method of characteristics [CARRIER and PEARSON, 1976], commonly referred to in plasma physics as integrating over unperturbed orbits [SWANSON, 1989]. AAMODT and RUSSELL (1992), in their derivation of expres sions for the scattering by free electrons in a plasma, integrate the Vlasov equation by integrating over the Green's function to the homogeneous part of the Vlasov equation (left hand side of equation 7.25). This approach is in fact identical to the method of characteristics. In the Russian literature the problem is usually tackled by first carrying out a Fourier-Laplace transform over space and time. This reduces the Vlasov equation to an ordinary differential equation in the az- imuthal angle in momentum space which is readily integrated [TRUBNIKOV, 1959; LlFSHITZ and PlTAEVSKII, 19S1]. This is the approach we have adopted here. Fourier-Laplace transforming equation (7.25) gives s (-n/ + iv • k )f + q(\ xB°)^f- = ucQ (7.26) dp where u:c is the electron cyclotron frequency, "c = ^ (7-27) m. and Q = Q° + Q° (7.28) 7.2. Source currents for scattered field 112 Q°(k%u/S) = -V + vxBs)^ (7.29a) uc op Q°^ - -hk?Iqfc*v**y% (T29b) + (E5 + vxBs) ^-\dkl The dependence on wave vector and frequency has not been shown explicitly for quantities with upper indicies i, s or S in order to reduce the lengths of the equa tions. The following convention is applied when dealing with Fourier-Laplace transforms: X' = .Y'fkW) (7.30a) A's = A"(kW) (7.30b) Xs = JT^k'-kV-u;'). (7.30c) B', B" and B4 can of course be replaced by E\ Es and E* respectively through B" = -^k° x E° ; a = i, s, S . (7.31) To integrate (7.2G) we switch to polar coordinates for p: PiP\\,Px, pj. = costø)kj. + sintø)(B0xkj.) (7.33) ø = P||Xpi- (7.34) In polar coordinates the gradient takes the form T* "T~PH + "^—PJ- + —"TTv • (7.35) dp dpi, " dp± p±d«T 114 Chapter 7. Bilinear plasma response and the last term on the left hand side of (7.26) simplifies to 9(vxB°).^ = ,(vxB»)-4l (7-36) dp pj. d = -qvxB?0 ±dp PJ. d dfs = U'd Changing to polar coordinates in (7.26) we thus obtain V uc uic ) The general solution to equations of the form -dF~ + ~d7~f{x'y)-h{x'y) is /(x,y) = e-g[l*] J* e9{x''y)h{x',y) dx', where c is an integration constant to be determined from boundary conditions. The general solution to (7.37) is therefore if, ci(o*'+0sin*')Q/ J/\ ^ if / where we have introduced the variables a = V||fc» " , (7.38) 0 - U±*l. (7.39) 7.2. Source currents for scattered Held H£ Substituting the integration variable $' for r where s 3 C / = c-" ""* / ~ eH-°T+0™{ The integration constant, c, must be chosen such that /* is periodic in ^ over 2TT. This is accomplished if c = oo or c = —oo. The integral must further be convergent for positive values of the imaginary part of u; (see Appendix B) which leads to c = —oo, so The right hand side of equation (7.40) consists of three terms, one for each of the terms in Q (see equation (7.28)). The first term 5 OT ipi n{ T) (/ )° = c-tf «»* p e- {e ' *- Q°{ s - iU*zoX* • E = q J v(f)° dp . (7.42) Solving for % m equation (7.42) with (/*)° given by (7.41) yields the relativistic susceptibility tensor and hence the relativistic dielectric tensor, e,j = 6{j + \ij- The source currents, j", for the scattered fields result from the two remaining terms in equation (7.40) y = qJyfdp. (7.43) where ?sin ioT f = c- * r e- {e'^-^Q'tø - T)} dr . (7.44) 116 Chapter 7. Bilinear plasma, response We can now write down the wave equation, including source currents, which the scattered field satisfies: (/is)2ks x (k5 x Es) + e» • Es = —j" . (7.45) Given the incident field and complete knowledge of/*, equations (7.43) and (7.44) can in principle be integrated and the scattered field found. To make the problem numerically tractable and facilitate interpretation of experimental data some as sumptions may have to be made about the nature of /' and fs in order to reduce the number of parameters required to describe the experimental situation. Two approaches to simplifying the problem are presented in the following two sections. 7.3 Expansion of response in powers of v In this section it will be assumed that the distributions /' and fs are significant only for low velocities, or more specifically, that the following conditions are sat isfied: v - < 1 (7.46) c (7.47) -^ < 1 . (7.48) s takes on all integer values (Z is the set of all integers, positive and negative). These assumptions are identical to those made by AAMODT and RUSSELL (1991) in their derivation of expressions for the scattering of electromagnetic waves by free electrons in a plasma. The approach adopted here is, however, entirely different to that taken by AAMODT and RUSSELL, and thus complements their derivation. Care is taken to make explicit all important steps in the derivation and expose the points in the derivation where the assumptions (7.46) to (7.48) are required. With the assumptions made above, the operator relating the current j* to the distributions and fields of the incident field and the fluctuations can be represented by an expansion to lowest order in velocity, j" will then no longer depend on the full detail of the distributions but only on zeroth and first order moment? of the distributions. 7.3- Expansion of response in powers ofv 117 We will refer to the terms on the left hand sides of equations (7.46), (7.47) and (7.4S) as the small terms. To expand the operator for j* in v we start by expanding equation (7.44) in powers of velocity. To this end it is convenient to cast (7.44) in a slightly different form. We note that the part enclosed in braces in equation (7.44) is periodic in r over 2jr. This part can therefore be expanded in a Fourier series tf***-*>Q"( ø - T) = £ e"Tf- j ' e'Ø^-^Q'tø - T>-"' dr'. (7.49) Inserting (7.49) in (7.44), integrating over r and replacing r' with r (7.50) 3rzLi{a-s)2xJo Here we have made use of the fact that a,m < 0, from which it follows that ,-HO-S)T c-.(c-,)r dT _ 1 Jo -i(a-s) i(a - s) Since rukf. - (w» + suc) a — s = w. ,s then for V\\kf, 5 (7.51) i(o — .s) u;* + swc \ w + 5u;c / For /? = Vx.k\/ijjc c"5,in(*-T)»5l + i,/?sin(^-T) + ... . (7.52) Inserting (7.51) and (7.52) in (7.50) and retaining only first order terms in v, ns Chapter 7. Bilinear plasma response r(P»,PJ.,*) = (1-tfsM) £ -T^(l + -^rM (7-53) 2 l_ r ', T Al + iØsintø - T))Q"(^ - T)e"" dr . 2K Jo The relationship between a momentum distribution and its associated velocity distribution is dp f(v\\,v±, dp (7.55) dv -(HP) and me is the rest mass of an electron. To first order in the small terms we thus have for the current: ^-JJjl-Hdv^fJdT (7.56) f 2Trme , f^, u>* + su:c J J J v± O"(tu,t»i,0 - T)v(r||,vj.,^)c"* f|l H vj. sin(ø — T) =• v± sin Change of variables to J" = 27r ,=^>c, w* + suc J J J ux Jo T „, Wl,<6')R(r) • v(V||,vx,4>')e-" 7.3. Expansion of response in powers ofv 119 Note that V(t>||, Vj., where cos r — sin r 0 R(r) = < sinr COST 0 (7.57) 0 0 1 On integrating over r, = ( T i Y. ~rr— / / / —dv]\dv±d4>'Q' {vll,v±,.4>') (,) 1 + -—Ji t;,, + ^ Wj. sintø') R -v(V||,i>x,') ( i*i 2 ^R< >(^)-v(rn ,f±,^)) wc w here R(,) = T- I*" R(r)c-UT dr = I 2Wo v 2 0 0 260s and 120 Chapter 7. Bilinear plasma response R(2) = — [' R(r)sintø' + T)e~isT dr 2~ Jo sin 0 + < 0 0 0 •{{SutP-S-u*-*) Summing over s gives J° = (la) (,6) ( f 1 + $± v± sb(*')] T + 4«||T - !& vxTW(A • v(t,„, „x, 4f) \ L ^ J c / where i -in i - fi2 i - fi2 x 1 + ft2 -t2« 0 (i-n2)2 (i-n2)2 1 (1 - Q2)2 (1-ft2)2 0 0 1 .3. Expansion of response in powers ofv 121 X«2) uu -I m = £ —ft R . f sin^' -cos^' 0 1 - i cos ^' sin & 0 > " I 0 0 0 J 1 /sin^ + 2iftcos^\ 1 /cos^-2tftsin^\ 2 V 1 - 4ft2 ) 2 { 1 - 4ft2 j 1 /,cos^,-2iftsin^\ 1^ /sinff +2iftcos^ 2tftcos^' \ 2\ l-4ft2 / 2 V l-4ft2 J sin^jHftco5_^\ ( l-ftJ J anc (7.58) Changing variables to v = v(v||,t'x,^'): <» q r = —l dvQ°(v) where 122 Chapter 7. Biliaea: plasma response (2i) l? T = piT » 1 f v> ~v' ° -I 0 0 Oj 1 (v, + 2iftvA 1 /rT-2iflv,\ 2V 1-4JP / 2V l-4fl2 7 1 /ty - 2tftv,\ 1 /r,+2tftvr\ 2V 1-4Q2 / 2\ 1-4JP 7 • (^) J For convenience we introduce the following set of tensors: 1 0 0 ) (010) [000] p(i) _ ! (2) (3) o i o I , p =-ioo, P = l o o o (7-59\ . ) oooj [ o o o J ( o o i j In terms of this set. the tensors entering the expressions for j* take the form Io T< > = _i—-pO) _ _^L_p(2) + pP) T (7.60) 1-fi2 1-n2 1 + m l2 (3> T<"> = j?, .r - ^_ p + pw (7-61) (i-n2)2 (i - ft2)2 and 2 T(2») l-2fl 2 ijk 2 2 (7-62) M 4fi l-4fl i 2f22 M 4Q2 + é"l-4fl2 .3. Expansion of response in powers ofv 123 Writing (7.59) in component form with implicit summation over repeated indices: X = — / ^ihl*'" fair + tffr) <7«> ([,+fr-Hu,+^",-f^')"' wrher e F = E+vx B Equation (7.63) contains integrals of the form -/*£ rfv Integrating by parts with respect to r, and noting that we find Og '-&< These integrals contain terms of the form vjV%, = SnTJ^ + vrfr dt'i = saTj;\+v,T Partial integration of equation (7.63) gives +5 M'"+«H - ^ |4?'+»SI} •») • Intr«Klucing the tensors ur? A-«') = _ Jl£_T<'-) (7.65) " (u^)2 " t,-w; and -iff - -j£?[^ [wf'+wi*-#• -Tl <7M> 4^tun) <7-67> equation (7.64) can be written Integrating over v and retaining only terms up to first order in v/c, 7.3. Expansion of response in powers ofx 125 -iuj'so 1 dJ x ] E + (7,69) 3i = (27y J *' 1 '' ^ ' "^ ) s i s + Xl! elmn(v mBi + v mB n) + X$\tøÉi+*iÉ[)] where n° = jf^rfvrM (7.70a) xJ» = f^dvvjnv). (7.70b) Note that qev is the current carried by the electrons. From the definitions (7.60) and (7.65) we find that Xjj is the cold plasma susceptibility tensor, X}jY(')} =_ Xo • (7-71) It is important to note that both the tensors X\j and X^l are evaluated with the wave vector and the frequency of the scattered field. The first and second right hand terms in expression (7.69) are identical to those found by AAMODT and RUSSELL (1992). For the remainder of the terms AAMODT and RUSSELL introduce a number of additional simplifications, making it difficult to compare the third right hand term in (7.69) with their result. Equation (7.69) is the cold plasma limit expression for the source current derived on the basis of the kinetic plasma model. We will compare this result to the expression for the source current derived by AKHIEZERc* al. (1967) and SlTENKO (1967) on the basis of the cold fluid equations. Their expression for the source current of the scattered field in a magnetized plasma reads 126 Chapter 7. Bilinear plasma response X 1 3 (772) " ' "^ \ »n-0 - ^c^' ^'^ - Here u is the fluid velocity u° = — (7.73a) n n = n° + ni + n* + ns . (7.73b) Note that Gaussian units are used in (7.72). The rectangular brackets in (7.72) are missing in SlTENKO's expression (11.5): this is clearly a misprint. (It has nevertheless been reproduced in a number of more recent articles.) In the derivation of (7.72) AKH1EZER et al. and SiTENKO ignore the difference between — and n n° + n° which results in the loss of second order terms. Since scattering is due to second order terms this is not acceptable. The momentum equation from which they start (see SiTENKO (1967) equation (11.1)) also leaves out second order terms. As a consequence (7.72) is not the correct expression for the source current in the cold plasma limit. Since this result has nonetheless been quoted widely we shall briefly compare the two forms of the most significant term. The first term in (7.72) represents scattering due to density fluctuations. This term is identical to the first half of the first term in equation (7.69) apart from the important substitution of ui'x' for u'x", which means that in this term AKHIEZER ct al. and SiTENKO incorrectly use the plasma conductivity associated with the incident field rather than that associated with the scattered field. The consequence of this substitution is explored numerically in Figure 9.8. With the aid of the relations between the various perturbations associated with the incident field and the electric field strength of the incident field, .4. Parametrization of distribution function 121 B\ = ;W#, (7.74) w —3 I = -^xtø, (7-75) n' = -^kt^El, (7.76) 9 the source current for the scattered field can be written as * - ^(5?/*'*' (777) s i s "^0 i „S X,t" - XlJtjmk——XmiBk 2 + [XijCjkmtn K , V( )*\ -« The expression for the source current, jff, can be written in a number of other ways, for instance by the use of Maxwell's equations, and for specific applications various further simplifying assumptions may be in order. For the purpose of generating a general routine for numerical evaluation of the scattering cross-section or equation of transfer for a scattering system the form given in equation (7.77) is convenient. This completes the derivation of the expression for the current which results from the interaction between an incident field and fluctuations in the plasma, under the assumptions made in equations (7.46) to (7.48). 7.4 Parametrization of distribution function In this section the problem will be simplified by assuming that the velocity distri bution, including perturbations due to the incident field and the fluctuations, can be approximated by some function /(«i,n2,..., at,..., p) depending on a discrete set of parameters, a,, 128 Chapter 7. Bilinear plasma response i /'(r,<,p) = /°(p) + / (r,i,p) + /*(r,',P) = /(«i,a2,.--,ai,...,p). (7.78) Since the perturbations are functions of space and time the parameters, a^, must also depend on space and time. Let the equilibrium values of the parameters be denoted by a°. We then have /°(p) = /K,«$,.-.,«?,. ...P). (7.79) The perturbations, /' and f6, are small compared with /°. /' can therefore be approximated by a Taylor expansion to first order in a; around the equilibrium values, a°, giving f = <(r,t)^, (7.80) f = «?(r'<)f£, (7-81) where a\ and a\ are the deviations of fl; from a° due to the incident field and the fluctuations respectively. Fourier-Laplace transforming equations (7.80) and (7.81), inserting the expres sions for /' and fs into expression (7.29b) for Q", and this in turn into expression (7.40) for /* we find, by comparison with expressions (7.41) and (7.42) that the source current for the scattered field is given by s j*(k> ) = -tu>e0^j^ J (al(k* -k\S -^mk^J) (7.82) + a;(kl,wi)E'(k,-k\w»-wi)) dk'dJ The difficulties in this approach lie in deriving the susceptibility tensor for a suf ficiently general velocity distribution. On the other hand, expressions for the susceptibility tensor available in the literature can be used directly to explore the scattering due to fluctuations in the parameters which have been included in these expressions. Note that fluctuations in the magnetic field, which are of course linked to fluctuations in the current, are included in E* (see equation 7.31). Chapter 8 Fluctuations 8.1 Introduction Since the author has not contributed anything new to the theory of fluctuations this chapter serves only to outline some of the methods and results presently used in collective Thomson scattering with particular emphasis on the diagnosis of fusion alpha particle populations and other fast ions. In actual experiments it is usually the spectral density of the scattered power, dPs/8u;', that is measured. In most experiments dP'/du' is linearly related to the spectral density, S(k,u/), of the fluctuations of one or more quantities. The expectation value of the spectral density of the fluctuations is the Fourier transform of the ensemble average of the auto-correlation function of that quantity. For instance for the electron density we have 5ne(k,^) = A-^ - (S.l) where (n\) (k,uj) is the Fourier-Laplace transform of the density autocorrelation function (nl)(r,t)=/ lim ^ j^nt{r + v',t + t')ne(r',t')dr'dt\ . (8.2) Here Q denotes ensemble average. 129 130 Chapter 8. Fluctuations For a stationary and homogeneous system satisfying the ergodic hypothesis ' av eraging over space and time is identical to averaging over the ensemble. In this case the expression for (n*) (r, i) simplifies to (nl)(r,t) = (ne(r + r',r-M')ne(r\r')) . (8.3) 8.2 Review of methods for calculating 5(k,u;) A number of methods have been employed to derive expressions for the spectral density of fluctuations. Particular interest has centred around the spectrum of the electron density fluctuations since fluctuations in this quantity are the dominant cause of scattered radiation in most laser scattering experiments as well as in the early ionospheric radar scattering. With increased interest in scattering of radi ation with frequencies closer to the plasma frequency and the electron cyclotron frequency, scattering due to fluctuations in other quantities, most notably fluid ve locity and magnetic field, must also be considered [AAMODT and RUSSELL, 1992] (see also Section 7.3). In equilibrium or quasi-equilibrium (e.g. Ti ^ Te) plasmas the fluctuation dissi pation theorem provides a powerful tool for determining the power spectrum of the fluctuations in the dynamic variables of the plasma [SlTENKO, 1967] including fluctuations in the velocity distribution [AKHIEZER et o/., 19C7]. This approach relies on the entropy of the system being expandable in the dynamic variables around a local maximum, hence the requirement that the system be at least in a quasi-equilibrium [LANDAU and LlFSHITZ, 1986]. Laboratory plasmas are frequently not in equilibrium or quasi-equilibrium states; certainly a fusion plasma with a continuous source of fast non-thermal ions is not. If the state is also non-stationary then the problem of finding expressions for the spectrum of fluctuations becomes virtually impossible to solve [LlFSHITZ and PlTAEVSKIl, 1981]. In many cases of interest including fusion plasmas it is however an acceptable approximation to assume that the state is stationary even if it is not in equilibrium. A stationary non-equilibrium state may be maintained if the system has steady sources and sinks of, for instance, energy and particles. Though it is possible to apply an amended version of the fluctuation dissipation theorem to a stationary non-equilibrium system [SlTENKO, 1982] this approach is not common, and a number of other methods are more widely applied. 'The ergodic hypothesis slates that the system will, irrespective of its initial condition, in the course of time come arbitrarily close to any point, in phase space and the time spent in a differential element of phase space is proportional to the system distribution function in that element. 8.3. Electrostatic dressed particle approach 131 One approach is to find the temporal evolution of the system from a given set of initial conditions by means of the kinetic equations. This provides the product of the system variables at the initial time and some later time for a particular set of initial conditions. The auto-correlation function is found by averaging over initial conditions where the probability of a particular configuration of particles in the initial state is given by the distribution function. This method is described in LlFSHITZ E- M. and PlTAEVSKII (19S1) and has been applied to the problem of fluctuations relevant for scattering by a number of workers including AAMODT and RUSSELL (1992). In the most fundamental approach the two-space-time-point correlation function is obtained directly from Liouville's equation for the distribution of the ensemble of N particle systems [ROSENBLUTH and ROSTOKER, 1962; SlTENKO, 1982]. A coupled chain of equations results which is truncated by an expansion in a set of small quantities which account for the discreteness of the plasma (these parame ters go to zero in the continuum approximation). The physics contained in this is that the importance of the multi-particle correlations vanishes if the system is observed for sufficiently long time. ROSENBLUTH and ROSTOKER (1962) used this method to derive the spectral density of the electron density fluctuations in a stationary non-equilibrium plasma (see also SlTENKO, 1982). Both unmagne- tized and magnetized plasmas were considered. The self-consistent electric fields associated with the fluctuations were accounted for by Poisson's equation rather than the two rotational Maxwell equations. This approximation is referred to as the electrostatic approximation. Having derived an expression for the fluctuation spectrum by this fundamental method, ROSENBLUTH and ROSTOKER derived the same result by the dressed test particle approach. The physics involved in this method is readily appreciated and gives an intuitive understanding of several of the phenomena encountered in the fluctuation spectra. For this reason the dressed particle approach has become very popular in the literature. It has recently been extended by ClffU (1991) to allow the self-consistent field to be fully electromagnetic, i.e. the field is accounted for by the two rotational Maxwell equations (resulting in the wave equation) rather than Poisson's equation. 8.3 Electrostatic dressed particle approach Because of the insight the dressed particle approach affords and the valuable results it, has yielded the dressed particle approach will be outlined here, both in the electrostatic approximation and, in the next section, in the full electromagnetic form. 132 Chapter 8. Fluctuations If the locations and movements of the particles in the plasma were uncorrelated the fluctuations in density would be due entirely to the discreteness of the particles. The field from each particle does however affect the motion of all the other par ticles. In the dressed particle approach the effect of this field is accounted for by treating the rest of the plasma as a continuum, allowing the one-particle distribu tion function, which is governed by the Vlasov equation, to be used in calculating the plasma response to each particle. In this picture each discrete particle is sur rounded by a shielding continuum cloud, which extends approximately a Debye length from the particle. Clearly one condition for this model to be adequate is that there be many charged particles in a Debye sphere (in a JET plasma there are typically of the order of 108 electrons in a Debye sphere). In the electrostatic approximation the screening cloud surrounding one test particle is accounted for by the linearized Vlasov equation and Poisson's equation: 1 ^ + v.^ + -^(vxBV% = —V* -^, (8.4) at or ma ov ma dv 1 A* = --qb6(r - r,(*)) - 7- 5>- [flu dv . (8.5) Here f\bl is the perturbation to the velocity distribution of the ath species caused by a test particle, (b, /), of species b following a helical trajectory 17(f) which is unperturbed by the presence of the rest of the plasma particles. /° is the unperturbed distribution of the ath species and B° is the static magnetic field. Solving equations (8.4) and (8.5) for f*bl and integrating over velocity space yields an expression for the density of particles of the ath species in the screening cloud surrounding the test particle, (b,l) [RosENBLUTH and RosTOKER (1962); SHEFFIELD (1975)]. Instead of starting from the Vlasov equation, as is done in most expositions of the dressed particle approach, we can make use of the fact that the dielectric properties of a plasma have now been studied extensively as a subject of its own, and we can therefore build on the results of these studies. This was the starting point adopted by HUTCHINSON (1987 a) in his presentation of the subject, which was limited to unmagnetized plasmas. Here we generalize his presentation to a magnetized plasma. One advantage of presenting the derivation in this form is that it makes it more clear what physics is included and what is left out of the electrostatic approximation and facilitates comparison with the derivation of the electromagnetic result, given in the next section. The dielectric displacement, D^, surrounding a test charge, pb\ satisfies the relation 8.3. Electrostatic dressed particle approach 133 pt,(r,*) = V-Dw(r,t). (8.6) Fourier-Laplace transforming (8.6) we can introduce the susceptibility of the plasma and the electric field, Ew, induced by the test charge: Pki(k,u) = z'k-Dw(k,w) = iJtt0k-e-Ew(k,u;). (8.7) In general it is not possible to find the electric field, E«, from equation (8.7). If, however, it is assumed that E6/ is electrostatic, so that EM(r,0 = -V*H(r,0, (8.8) then the transform of E&j takes the form EM(k,w) = k£&r(k,w), (8.9) E&*(k,u) = -ifc*w(k,w)T which allows the field to be determined from equation (8.7): Pi U Eir(k,u)=.i f> \. (8.10) ik£0K • e • k The charge displacement induced in a particular species, a, by this field can now be found by first finding the current induced in this species, j„w(k,w) = -?w£0X-E6((k,w) a tal = -?WoX -k^/ (l<,W), (8.11) and then applying the charge continuity equation, 134 Chapter 8. Fluctuations pabi(k,Lj) = —k-joW(k,w) = -.^k-xa-k£Hk,«). (8.12) Inserting the expression for the electrostatic field (8.10) we find the charge density, pabi of species a in the screening cloud surrounding the test charge / of species 6, X k poW(k,u;) = -V °'- p»(k,u;) . (8.13) kc k By introducing the charge number, Z„, of the nth species (—1 for electrons), we can rewrite this expression in terms of particle densities2 n.«(k,W) = -&QEiiS)^ft(klW) (8.14) e(k,w) Za where \a is the electrostatic susceptibility associated with the ath species \a = kx°k, (8.15) and £ = ! + £*<., (8.16) a is the electrostatic permittivity, while is the Fourier-Laplace transform of 6(r — i"/(i)). The fluctuations in the electron density can now be found as the sum of the fluctuations associated with the independent movements of the set of discrete 2This expression corresponds to one term in the summations in SHEFFIELD (1975), equation (8.2.10). S.3. Electrostatic dressed particle approach 135 electrons and the electron components of the screening clouds surrounding all the charged particles in the plasma, including the electrons. The result is3 S(k,W) = Se(k,u/) + £S,(k,u,) (8.17) where the electron feature is given by 5«(k,w) = (8.18) £ nt and the various ion features take the form S,(k,w) = *1 n (8.19) n. £ { Here TZ represents the spectral density associated with the uncorrelated motion of the particles of the species a. For an isotropic Maxwellian distribution the electrostatic susceptibility can be written X« = <*l H exp(-A„)i,(A0)1 + /=- while the spectrum associated with the uncorrelated motion can be expressed as a 2^ ( (u>-kw) \ £ exp(-A0)J,(Aa)exp - (8.21) Here Z is the Fried and Conte plasma dispersion function [FRIED and CoNTE, 1961], // is the modified Bessel function of the first kind and order / and Aa = I va = i/— , wco = \ *>ca f V ma ™a 3These results have been derived by many workers, e.g. SHEFFIELD, 1975. The particular forms for \ and U given here are due to HUGHES and SMITH, 1988. 136 Chapter 8. Fluctuations kv. " V ™.£o at, for the electrons, is the well known Salpeter parameter. The Larmor radii of very energetic ions can be considerably larger than the scat tering volume. An heuristic argument has been put forward [SHEFFIELD, 1975] that in this case the influence of the magnetic field on the motion of the fast ions can be neglected. The argument is that since only a fraction of the orbit of such an ion is seen in the scattering volume then the fact that it is following a helical trajectory rather than a straight line does not affect what goes on in the scattering volume. A stringent proof would, however, require that the difference be determined between the fluctuations predicted with and without the fast ions being influenced by the presence of a magnetic field and then showing that the limits on resolution in k space and in frequency resulting from the finite size of the scattering volume and the finite observation time implies that the scattering diagnostic is not able to resolve this difference. If this is the case then the fast ions can be treated as unmagnetized, meaning that the static magnetic field is ignored in the Ylasov equation for these ions. This leads to a considerable simplification of the expressions for the susceptibility. \, and the fluctuation spectrum of the uncorrelated motion, TZ. For an unmagnetized population, a, the electrostatic susceptibility is given by „.(^yrfS • *, (s.22) \ k J J-v, du\\ ti|| - us/k where the integration contour goes below the pole at U|| = w/k. Here UJJ is the component of v parallel to k (not to be confused with ry which is parallel to B°). For specific velocity distributions the integral may be carried out analytically but numerical evaluation for an arbitrary distribution presents no problem. For an unmagnetized population 11 reduces to no = ZS&foMk) (8.23) where /||„(w||) is the one-dimensional velocity distribution along k /||.>(»I|) = Jfc(y)6(v • k - «||)dv . (8.24) 8.4. Electromagnetic dressed particle approach 137 An example of the spectral density function for fluctuations in the electron density is shown in figure S.l. The dip in the thermal ion feature is readily understood on the basis of the dressed particle picture; the bulk ions respond slower than the electrons due to their greater mass, but at low velocities they still have time to respond and thus contribute significantly to the shielding, thereby replacing electrons. The electrons generally move much faster than the ions so the shielding of each electron is almost entirely due to the deficit of one electron in the Debye sphere surrounding it. This explains why the electron feature is small when a, = l/i"AD > 1: in this case the wavelength of the fluctuation is much greater than the width of the shielding cloud and the electron appears to coincide with the cloud representing the absence of an electron. 8.4 Electromagnetic dressed particle approach The dressed particle approach has recently been extended by ClIIU (1991) to in clude an electromagnetic treatment of the self-consistent field. Let j* be the current associated with the uncorrelated motion of the particles of sjHTics 6. They induce an electric field. E*, in the plasma given by E*(k.u;) = A'^k.«) - j*(k,u/). (8.25) This field in turn drives a current, x" ' E* in the ath species. The total current, J" for the nth species is thus » Summation over repeated lower index is implicit. Thr statistical independence of the particles associated with the current j* yields the following expression for the spectral density of the fluctuations in j* [ClIIU, 1001] , / (jl»(k.-;)>; (k'.* )> = (8-27) s 1 (2ff) ftV (k + k')rf(* + J)SW £ //»(vW« - %•„ " /««*)C?°Cf', 13S Chapter 6*. Fluctuations 13M 1»J 140.0 140.2 140.4 •røaGHz I 142.0 Figure 8.1: Spectral density, 5(k*—k',w* -w'), of electron density fluctuations and individual features of 5 as functions of w\ (a) shows the total spectral density, 5, as well all the features which are due to the bulk deuterium and tritium populations, SD and Sj\ the energetic fusion alpha particles, S0; and the electrons, St. (b) shows S and S0. Parameters: 0-mode to O-mode scattering, u-72ff = 140 GHz, Z(k\k*) = 20°, Z(k,B°) = 75°, B° = 3.4 3 ,9 3 tesla, nt = 8 • lO'^m" , Tt = 12 keV, n0 = nT = 3.95 10 m- , ,7 3 7b = TT = 10 keV, n0 = 5 - 10 m- . The alpha particles have a classical slow-down distribution with a birth energy of 3.4 MeV and a critical velocity of 0.09 times the thermal velocity of the electrons. Curves were calculated using TOMSCAT [HUGHES and SMITH, 19SS] 8.4. Electromagnetic dressed particle approach 139 w.her e a C« > = é,!p*-J,(kj.pa) + éy^J!(klPa) + é,nJ,(k±Pa) , pa = ^ . (8.28) K±Pa « Wca From this expression and expression (8.26) the fluctuation spectrum of the total current, J°, of each species can be found: (j>j°,) (k,«) = Ofi?) (k,«) + {xti*j!)(xh*}*Y E(AH) (k.w). (8.29) Expressions are readily obtained for the fluctuations in all electromagnetic and fluid variables by applying the operators relating the variables to each other and making use of expressions (8.27) and (8.29). ClHU applied his findings to obtain an expression for the density fluctuations. The result follows from (8.29) and the continuity equation (n°n°) (k,u) = ^f (J*J*,) (k,w) . (8.30) Here we give expressions for fluctuations in some of the other relevant quantities. Fluctuations in the electric field are obtained from (8.25) and (8.27)4 (£,£,,) (M) = \?A#Z(ji&)fcu) (8-31) 6 and fluctuations in magnetic field can be found from (8.31) and (7.31) giving (BtB,,) (k,U>) = ^»Wfcj. {EkEki) (k>w) # (g>32) Cross correlations between different terms are found by simply applying the appro priate relation for each term. For instance, the cross correlation between current and electric field would be 4The fluctuations in E can also be obtained from (8.29) in which case the vacuum wave operator must be used. 140 Chapter S. Fluctuations b (J°E,,) (k, u/) = £(*-* + X°,*jZ)Ky (jlj k.) (k,w) . (8.33) The inverse wave operator, A-1, is singular when A = 0, that is when the dispersion relation i.c .atisfied. This leads to singularities in the fluctuation spectrum at values of k and u; which satisfy the dispersion relation. If the wave is damped, implying that k or u> must have an imaginary part to satisfy A = 0, then the singularity is replaced by a Gaussian bump whith a height and width which depends on the damping. In the electrostatic case the dispersion relation is £ = 0. Increases in the fluctuation level are also predicted in this approximation for values of k and u which satisfy the dispersion relation, e = 0, as may be seen from expressions (8.18) and (8.19). In the parameter range which is of interest for the diagnosis of fast ions the elec trostatic dispersion relation is satisfied at the lower hybrid resonance. The electro magnetic dispersion relation is, however, satisfied for two branches, the fast and the slow magnetosonic waves. For large values of k the slow branch is well approximated by the resonance found with the electrostatic approximation. The electrostatic approximation does not, however, approximate any part of the fast wave branch. ClUU explains this as being due to the fact that the fast wave is intrinsically electromagnetic. Despite the electromagnetic nature of the fast wave the density fluctuations associated with it are still substantial because of a significant longitudinal polarization. Chapter 9 Theoretical model of Thomson scattering 9.1 Introduction The equation of transfer for a scattering system relates the spectral density of the scattered power which is received by the receiver to the incident power and the properties of the plasma including the nature of the fluctuation:: in the region of the plasma where the radiation is scattered. Investigations by AAMODT and RUSSELL (1992) have shown that in the cold plasma limit the interaction between the incident radiation and the electron den sity fluctuations are the main source of scattered radiation for most regions of parameter space, though regions exist where fluctuations in other quantities must be included. In this chapter the equation of transfer for a scattering diagnostic is derived and investigated. Only scattering due to fluctuations in the electron density will be included though the required steps for including scattering due to other quantities will be pointed out. The expressions given here will also apply to plasmas where weakly relativistic effects are important. The equation of transfer is derived in Section 9.2. The derivation builds on the developments given in the previous four chapters and may be seen as an example of the application of the expressions derived in those chapters. In Section 9.3 the symmetry properties of equations of transfer for scattering systems are investigated and discussed. Section 9.4 presents numerical evaluations of the quantities entering the equation of transfer derived in Section 9.2. 141 142 Chapter 9. Theoretical model of Thomson scattering 9.2 Equation of transfer for a scattering diagnos tic We will assume that the observed scattered radiation is scattered in a region which is sufficiently homogeneous that the results derived in Chapters 6 and 7 can be applied. From expression (7.82) we find that the current resulting from the interaction of an incident field with fluctuations in the electron density is given by j"(kW) = _,-w»£o|£ JL. |n«(k. _ k'?u;* _u/jE^kVJdk'ck/ , (9.1) s where n e are the fluctuations in the electron density. Since the susceptibility tensor is linear in the electron density the derivative of the susceptibility tensor with respect to nt is simply given by £ - * . (9.2) Let the incident field be a monochromatic beam: EXr, 0 = ^^ (E'-'éVl**-^ + (£''?)'c-'l»r-JtA . (9.3) The total power P' in this beam is F = ^5'|FT (9.4) The Fourier-Laplace transform of the incident field is E'Ck1, J) = i;E'ié')/li{k-ki)6{u - J) + Tr(E'iéi)'Jli(k + )ii)6{u; + J). (9.5) Inserting expression (9.5) in (9.1) we find 9.2. Equation of transfer for a scattering diagnostic 143 J'(kW) = oC0 (2*I3*' i*"* I ^I(k " kl)n'(kS " k>U,> ~ M?) A + (fTé1)* y ^(k + kOn^k* - k, ws + w!) The last term on the right hand side of equation (9.6) represents the effect of density fluctuations at approximately twice the probing frequency and generally propagating faster than the phase velocity of the probing radiation. Such density fluctuations are unlikely to be significant as noted by HUTCHINSON (1987 a) in his treatment of scattering based on vacuum propagation. Neglecting the last term in (9.6) and expressing the convolution over k as a Fourier transform of a product we find r(ks,u;s) = =7^E''X* • é' Jy/T(y)ni(r,u;s -^e-^-^'dr . (9.7) Inserting expression (9.7) for the current in expression (6.31) would give an ex pression for the field which is scattered from the incident field by fluctuations in the electron density. Of the power emitted by an extended current distribution in the plasma, the spectral density of the power received by the receiver is given by expression (5.26). As discussed in Section 5.3, the quantity dkdui entering expression (5.26) is the scattered field which would result from a current distribution which was identical to the actual current distribution weighted by the square root of the normalized beam intensity, y/T*, of the receiver. The effect of the finite width of the beam pattern of the receiving beam is therefore to weight the current distribution entering the expression for the scattered field by vl*. These results are summed up in expression (6.32) which gives the received power per unit angular frequency in terms of the current distribution. Assuming that the response length of the plasma is short compared with the width of the beam pattern1 we have 'The response length of the plasma, rr, is defined here as the maximum distance, |r|, over which o-(r,<) (sec equation 4.7) is appreciably different from zero. The response length is related to the 144 Chapter 9. Theoretical model of Thomson scattering J"(kW) = -7^E"X* - é'nftk« - k\u/» - J) , (9.S) where nJ(r,W) = N/li(r)I*(r)n«(r,a;). (9.9) The square root of the normalized intensity, s/J, plays the same role here as the "weight function", w, in the paper by HOLZHAUER and MASSIG (1978). The current enters expression (6.32) for the received scattered field in the term | (e_1)s • j"|2. Inserting (9.S) for the current this term takes the form s |(e-') -r(kv)f 1 2 nt 'V ) -x--* •|2 = ^\E>?0bTS^)\{^y-x*-é\ (9.10) Here k = ks-k' , u = u'-j , 0b is the beam overlap [BINDSLEV, 19S9]: 0b = jf(r,t)T{r,t)dr (9.11) and S(k,w) is the spectral density function: 5(k,w) = (9.12) ncObT susceptibility tensor as llxll • dk I Requiring that rp be much smaller than the beamwidth ensures that \* can be regarded as constant in the convolution with \/l*(k*). 9.2. Equation of transfer for a scattering diagnostic 145 Inserting (9.10) into expression (6.32) for the received scattered power, and making use of equation (9.4) to express |E"|2 in terms of the incident power, gives 2 2 s dP* P' Ob S(kM) K \V • v;i S* ("')'1 («•')' • X • é' (9.13) (^„.,4-.(z*+l) e With the classical electron radius, re, and the electron plasma frequency, w,«, q2 r2 1 r = - 2 • —- = 2 4 2 (l 9 141} ' 4^c0mec ' u* (4;r) c n (9.13) can be written ^ = J*0»Aj^ne3^G , (9.15) where the dielectric form factor takes the form V P G = T7 7 ' „„ - (9-16) 'IM-ISMT Equation (9.15) gives the power received in a scattering system, given the incident power, the beam patterns and the nature of the density fluctuations. If from a statistical knowledge of the density fluctuations we wanted a prediction of the received power the best estimate would be obtained by using the ensemble average for the spectral density function, (|n*(k,w)|2) ntUbl This definition of the spectral density function is identical to the definition given by SHEFFIELD (1975) when the beams have rectangular cross sections and uniform intensities. If the anti-Hcrmitian part of the dielectric tensor can be ignored, i.e. 146 Chapter 9. Theoretical model of Thomson scattering e = eA , it follows that e"1 = é* and we have (see Section 5.5): é*-(e-~)-é = , With this relation between the flux and the term stemming from the residue, expression (9-16) can be written in the compact and almost symmetrical form G=5§; (919) where the coupling term, C, is given by s 2 w'w'(u> ) , _ , „i|2 <=^p fp !<*•>• **-f <*»> and the S terms are the normalized fluxes given in equation (5.45). In the limit of no spatial dispersion, the expression for the geometrical factor, G, reduces to that given by HUGHES and SMITH (1989), except for a factor u'x* which incorrectly appears as w'x' in that paper. This point is discussed in Sections 7.3, 9.3 and 9.4. In the low density limit the term C tends towards |(é")s-é'|2, while S tends towards unity and consequently a (7-|(é-)'-é''| for ne->0, (9.21) which is the expression found for the dielectric form factor when vacuum propa gation is assumed. The geometrical interpretation of expression (9.21) is the root of the name "geometrical factor". 9.2. Equation of transfer for a scattering diagnostic 147 The differential scattering cross section, cPE/SkSu;, is often given as an interme diate result. From (5.22) and (9.15) it is readily seen that dkdu; Ag ' 2^ |k*-v»| v ' In the limit of no spatial dispersion (9.22) is identical to the expression given by HUGHES and SMITH (19S9) apart from the substitution of a factor w'x* for u'x* as mentioned above. In order to simplify the results for a weakly relativistic plasma we have so far limited the treatment to scattering due to density fluctuations. We will now write down the equation of transfer, including all the terms that contribute to the scat tering, for a plasma where conditions (7.46). (7.47) and (7.48) are satisfied, which implies a cold plasma. The derivation, which makes use of expression (7.77) for the source current, is essentially identical to the derivation given above for the weakly relativistic case. We will therefore only give a few key steps in the derivation. From expression (7.77) for the source current and expression (9.3) for the incident field, and throwing away the fluctuation terms with the sum frequency, we find s 2 (l(éT-JT) = (§)Vf W(\n e\ ) + BkB'k,{BlBl,) (9.23) 5 s -rWk.{vtA) + Skrk, (E kE k>) + 2Re{7v'e:,(n^f,) + JtfWv (n^.,) s s + B& (BtEl) + UkS'k, (v ekE k,) } Here Ar = ie'iYxW, (9-24) Bt = -U^xh^^xLe], (9.25) 14S Chapter 9. Theoretical model of Thomson scattering Uk = (e?r (xW>»em»r£ + X™\} ej, (9.26) f = -W (x^ix}, + ^S^,) ej - (9.27) The complete equation of transfer in a cold plasma can now be written as dP* 1 2 — = P' 0b X0\>Qr c nt — £ S(k, «)-* S(k,u;)nn = X-frl, (9.29) etc., and the dielectric form factors are defined as G*ø = |£ (9-30) S'S' where (9.31) pe pe _ U^K)* ""pe wpe etc. Although the complete equation of transfer for scattering in a cold plasma is fairly extensive it has been attempted to present it in a systematic manner which to gether with the tensorial formulation should facilitate the writing of codes. All 9.3. Symmetry of the equation of transfer 149 previous derivations (e.g. SlTENKO, 1967 and AAMODT and RUSSELL, 1992) have introduced approximations earlier in the derivations. This is undoubtedly accept able in many situations but there is always the risk that some physical phenomenon is lost in the process. With the complete expression given here the significance of any terms that are left out of a computation can be quantified. A code that includes all terms will have the advantage of being valid in the widest possible range of parameters. The computation time is not likely to be a major problem. Finally it should be pointed out that none of the expressions derived in this section assume that the difference in frequency between incident and scattered radiation is small. 9.3 Symmetry of the equation of transfer Before proceeding to numerical evaluations of the dielectric form factor and related quantities we will first investigate some of the symmetry properties of the equation of transfer. When solutions to a problem can be shown to have certain properties, e.g. symme tries, these known properties may help in the search for solutions to the problem and provide convenient checks on the solutions. This is particularly helpful when solving problems which require a certain amount of simplification to be tractable. The equation of transfer for a scattering system is very nearly symmetrical in incident and scattered field quantities, that is, the equation relating the incident power to the received scattered power is virtually unchanged if the direction of power flow is reversed. This near symmetry has given rise to some speculation as to whether the equation of transfer should in fact be completely symmetrical, the slight asymmetry being due to a simplification introduced in the derivation. In Appendix D of BINDSLEV (1991a) an attempt was made to show that such a symmetry exists. A crucial step in the derivation involved the kernel &r(r,r,r',<') describing the current j(r,f) resulting from an electric field E(r\f') interacting with fluctuations j(rJ) = jfitr(r,ty,t') -E^'J^dr'dt'. (9.32) The symmetry of the equation of transfer was shown to exist if the kernel satisfies ArV,«,r,i') = fa'(r,«,r\<'). (9.33) 150 Chapter 9. Theoretical model of Thomson scattering where &rT(r',f,r.i') refers to a plasma where the magnetic field is reversed and the time evolution of the fluctuations is also reversed, that is {/'(v.lcwttt = /*(-v,k,-u,.) (9.34) Onsager's principle was invoked to establish the identity (9.33). The proof of Onsager's principle given in LANDAU k LlFSHITZ (19S6), §120, assumes kinetic coefficients which do not depend on time. Since it is not obvious to the author that Onsager's principle holds for kinetic coefficients which do depend on time the proof of the complete symmetry of the equation of transfer for a scattering system given in Appendix D of BINDSLEV (1991a) must be regarded as unsafe. The equation of transfer does however possess a certain degree of symmetry as will be shown below. We will also explore the nature of the asymmetric term, both for a general bilinear medium and for a plasma described by the Vlasov equation. To explore the consequences of the reciprocity relations for a scattering system consider two antennas a and 6 pointing towards a plasma with fluctuations. The system is illustrated in Figure 9.1. In forward transmission antenna a acts as the emitter and 6 as the receiver while in reverse transmission 6 is the emitter and « the receiver. The reverse transmission is assumed to take place in a conjugate plasma which we define as a plasma where the external magnetic field is reversed and the time evolution of the plasma fluctuations is also reversed (see equation 9.34). It will later be shown that the symmetries of the equation of transfer hold without reversal of the external field provided the antennas are reset to accept or emit the same characteristic morle as was emitted or accepted for the opposite direction of transmission. The antennas are fed by transmission lines. In a conventional coaxial transmission line TEM modes propagate. These modes are fully characterized by the voltage, V, between the two conductors and the current, /, flowing in each of the conductors. V and / are related by the impedance, Z, of the transmission line V = ZI. (9.35) The antennas may also be fed by wave guides, as is the case for the JET scat tering diagnostic. The modes propagating in wave guides generally have more complicated spatial distributions of electric and magnetic fields. For the present purpose the wave guides can however be represented by transmission lines where a propagating mode is characterized by a voltage, V, and a current. / proportional to the electric and the magnetic field strengths respectively [COLIJN & ZlX'KER, 9.3. Symmetry of the equation of transfer 151 Transmission Transmission. line 1 line 2 Antenna « Figure 9.1: Scattering system. 152 Chapter 9. Theoretical model of Thomson scattering 1909. Chapter 4]. V and / are chosen such that the energy flow associated with the propagating mode is given by l/2Re{lT*}. We can now- define the following quantities: l'u. /'*: Voltage and current of mode propagating in transmission line 1 towards antenna a. This is the field which feeds a when a acts as an emitter. I'1*. /'*: Voltage and current of mode propagating in transmission line 1 away from antenna a. This b the field accepted by antenna a when b is emitting. I'2*. Izs: Voltage and current of mode propagating in transmission line 2 towards antenna 6. This is the field which feeds b when b acts as an emitter. \**\ f"1: Voltage and current of mode propagating in transmission line 2 away from antenna 6. This is the field accepted by antenna 6 when a is emitting. ? E " T E*: Electric fields in plasma emitted by antennas a and b respectively. j.-h j3». Currents in plasma resulting from the interaction of the fluctuations in the plasma with the fields emitted by antennas a and b respectively. The reciprocity relation has been shown to hold in a generalized form in a magne tized plasma with spatial dispersion (see e.g. discussions by GlNZDURG, 1970 and Bl DDKX. 19S5). It states that J EU-. r) x H'(u.% r) - E'(*T r) x H(u.-, r) rfr (9.36) = j[ EV-. r) • j(-.%r) - E(~% r) - JV, r) rfr where S is the surface hounding the volume V. The unprimed fields result from the source currents j(»\r) while the primed fields arc driven by j'(«;, r) and refer to a plasma where the externally imposed magnetic field is reversed. Let V be the volume which extends to infinity and excludes the volumes occupied by the antennas, transmission lines, generators and receiving electronics. The surface, 5, bounding this volume consists of three parts: one extending to infinity and two surrounding each of the antenna systems. Integrating equation (9.36) over this volmnr and surface yields V,V")/,*V,) + l'',''^.;a)J,,V,) - c*h (9.37) V^U-VVV V;V)/*V) = C,kfl (9.38) 93. Symmetry of the equation of transfer 153 C°* = /E3"(r,w*)-jk'(r,w*) Cba = Jti*{r,uh)>ja(r,u>b)dr. (9.40) For the following discussion it is convenient to express Cab and Cba in terms of spatial Fourier transforms of the fields and currents. Cab = -^/E3a(k°,u;a)-f(-k0,u;a)c/ka (9.41) (2*) 1 C*6a = -±-jE:ib'(-kb,u,b)-y(kb,J)dkb. (9.42) (2*> As in expression (9.36) the primed quantities in equations (9.37) to (9.42) refer to a plasma where the externally imposed magnetic field, B°, is reversed. Reversing B° does not affect the trajectory of a beam nor does it affect the intensity distribution across the beam [BUDDEN, 19S5]. This invariance follows from the symmetry of the refractive index surface of a characteristic mode (0 or X). The polarization associated with a mode does, however, change when B° is reversed. To find the effect that reversal of B° has on the polarization of a mode w«> note that by Onsager's principle it follows that 0 ev(B ) = eJt(-B°) (9.43) and hence that 0 A.^B ) = A;,(-B°). (9.44) The unit vector é' describing the polarization of the electric field in the plasma with B° reversed thus satisfies A,,(B°)e; = 0 (9.45) from which it follows that (see equations 6.3 and 6.7) 154 Chapter 9. Theoretical model of Thomson scattering é' = c-1 . (9.46) When the anti-Hermitian part of the dielectric tensor can be neglected the relation (9.46) takes the form é' = iF1 •= é* . (9.47) In this limit the only effect reversing B° has on beams of radiation is to reverse the sense of rotation of the field vectors associated with each of the characteristic modes (0 or X). From these results it follows that the voltages and currents representing the fields in the transmission lines are not affected by reversal of B° provided the antenna systems include polarization filters which are reset such that the antennas couple to the same mode or distibution of modes after reversal of B°. Assuming that polarization filters are included and adjusted to couple to the same distribution of modes when B° is reversed, equations (9.37) and (9.3S) also hold when the transmission line voltages and currents all assume the same direction for B°: K^^j/'Vj + rVK'V) = cab (9.48) 1/2WV) + V2a{J)I2V) = Cba. (9.49) To obtain expressions for Cab and Cba in terms of unprimed quantities we must take account of the change of polarization under reversal of B°. It is convenient to introduce the following notation E(k,w) = Eé (9.50) E'(k,w) = EP* = E*(k,w). (9.51) With this notation and from the discussion given above the expression (9.53) for Cba is obtained directly from equation (9.42). To find the expression for Cab we first express E3a in terms of Ev. All of the elements in the expression for Cab then relate to the plasma with B° reversed. Since Cah is a scalar it is invariant under 9.3. Symmetry of the equation of transfer 155 rotation of the spatial coordinates. Rotating the coordinate system to reverse the direction of B° then gives the expression (9.52) for Cab. C°b = ^/E3at(_k»,u;a).jfc(ka,u;a)cfk0 (9.52) ba 3h b a b b 1 C = -^-jE H-k ,^)-i (k 1u; )dk '. (9.53) Expressions (9.52) and (9.53) assume that the antennas each couple to a single mode. If this is not the case then the expressions for Cab and Cba consist of a sum mation over terms of the type given in (9.52) and (9.53), covering all combinations of the modes in the two fields. The received power is proportional to the emitted power which is expressed by the linear relations: Vlb = Z12/26 (9.54) 21 10 yla = 2 / . (9.55) The currents and voltages in the transmission lines are related by V2a = Z22/2a, (9.57) where a = a, b. Multiplying equations (9.48) and (9.49) by Cba and Cah respec tively, equating the left hand sides of the two new equations and expressing the voltages and currents entering into the resulting equation in terms of Ila and Pb by means of equations (9.54) to (9.57) yields CbaZU = CabZ7l (9i5g) Let Pna be the power associated with the field n,a,(n= 1,2; a = a, b) propagating in the transmission lines 150 Chapter 9. Theoretical model of Thomson scattering Pna = hle{Vna(Inay} . (9.59) With the use of equations (9.54) to (9.58) the received power can be related to the scattered power for both forward and reverse transmission: plb _ TjCai|2p2& (96Q) 2 la p2a = r|C*°| P . (9.61) where T=Re{^7}Re{JL}|Z|* (9.62) and Z12 _ Z21 (9.63) Z ~ CV ~ O ' T depends on the geometry of the system, the properties of the plasma and, most importantly for the present discussion, on the characteristics of both the incident field and that part of the scattered field which is accepted by the receiver. T is, however, symmetric in incident and scattered field quantities, and thus appears in the same form in the equations of transfer for both forward and reverse transmis sion. If |Co6| were identical to |C'6a| then the equation of transfer for a scattering system would be completely symmetric in incident and scattered fields, and the fraction of power coupled from emitter to receiver would be independent of the direction of transmission. This is, however, not generally the case. The dielectric response of a collisionless, homogeneous and stationary plasma is almost completely linear, with a small bilinear coupling between modes, where modes are defined broadly as solutions to the linear system. The bilinear cou pling gives rise to currents which act as source currents for other modes (see also discussion in Chapter 6). The most general form of bilinear coupling is given by b c Jt(r,t) = jQ,jk(rJ,r\t',r"j")X J(r'1t')X k(r",t")dr'dt'dr"dt" (9.64) 9.3. Symmetry of the equation of transfer 157 where Q is the kernel describing the coupling and A* is a vector containing all the dynamic variables characterizing a mode (e.g. fields and velocity distribution). When the plasma is stationary and homogeneous then the kernel simplifies so that the expression for the source current takes the form b i"(r,«) = jQ,Mr-r',t~t'y-r",t'-t")X j(r',t')Xl(r'',t")dr'dt'dr''dt" . (9.65) Fourier-Laplace transformation yields a b b b b b j°(k ,u;°) = j^4 JQ,}k(k\u\k\J)X>(k ,u; )Xl(k° - k ,u* -u )dk dJ . (9.66) Inserting (9.66) in expressions (9.52) and (9.53) we find expressions for the coef ficients Cab and Cba in a general stationary homogeneous medium with a small bilinear response: ab a a h C = —^ [Éf \k', ba b b a C = -^ J Ef(k ,J)Q,lk{-^^ X^ ) (9-68) X3a(ka,u!a)Xs(-kb - ka,ub - ua)dkadkbdua . A collisionless, homogeneous and stationary plasma is described by the Vlasov equation and Maxwell's equations. In such a plasma the bilinear response is due to the terms EV,o^ + E<(M,.^, (M.> where all quantities arc evaluated at the same point in space and time. This implies 7 that Q,jk{r,t,r',t',v",t") = 0 when (r ,*') ^ (r",<"). Since the equilibrium plasma is assumed to be homogenrous and stationary it follows that Q depends only on 15S Chapter 9. Theoretical model of Thomson scattering the differences (r — r',< — f'). The bilinear response in a collisionless, homogeneous and stationary plasma thus takes the form j?(r,0 = JQukiT- r',t - t')A^(r',t')A^(r',Orfr't/t'. (9-70) Fourier-Laplace transformation gives a 6 a J,°(k ,u:°) = j±y4 y (?ot(k»,u;»).Yj(k*,u; )^(k - k\u;» - u:')cflc' which upon insertion in expressions (9.52) and (9.53) gives the expressions for Cb and Cba in a stationary homogeneous Vlasov plasma: ab at o 0 o C = ~y J £f (k ,u; )Qufc(-k ,u;«) (9.72) Å'3fc(k\u;6)A'*(-ka - k6,w° -w6)r/k0 ^ = J£y JE?hW)Qiik(-kW) (9.73) A'3a(k°,w°)A'{(-k6 -k°,w6 -u>°)rfka(fk6 The factors, Cab and Cba, are the inner products of the source current with the receiver field, where the receiver field is defined as the field that would emerge from the receiving antenna if it were driven as an emitter. From the expression for the relativistic dielectric tensor (4.18) it is seen that e(k,u/) depends only on square terms of k. This implies that A(k,u/) = A(-k,u>), from which it follows that the complex unit electric field vector of the receiver field is identical to the complex unit field vector, és, of that part of the scattered field which is accepted by the receiver, the only difference between the two fields being the directions of the wave vectors, which are anti-parallel. The forward and reverse equations of transfer, (9.60) and (9.61) together with the expressions for the coefficients Cab and Cia display a number of significant 9.3. Symmetry- of the equation of transfer 159 symmetries. First of all the equations of transfer, apart from the factors Cai and Cha, are symmetric with respect to quantities referring to the incident field and the received part of the scattered field. The equation of transfer for scattering by density fluctuations in a general Vlasov plasma (9-15) has just this property. It is symmetric in (i) and (s) except in the spectral density, S(k = ks — k',w = w* — w'), and in the coupling term, C oc p*(é5)* - x* • é'|2, appearing in the dielectric form factor, G. 5(k,ur)C is, to a symmetric multiplicative factor, identical to the inner product of che source current and the receiver field. S(k,u/) is in general asymmetric, appearing as 5(k. — u?) in the reverse equation of transfer which, however, is identical to S(k,u/) if the time evolution of the fluctuations are reversed (see equation 9.34). Also the complete equation of transfer for a cold plasma (9.2S) is symmetric down to the inner product of source current and receiver field which is represented by the terms £oj3 So3Ga3. The result obtained by HUGHES AND SMITH (19S9) also satisfies this symmetry, the difference between their result and the cold plasma limit of expression (9.15) appearing in the inner product of the source current and the receiver field. We will now focus our attention on the properties of this inner product. From expressions (9.67) and (9.6S) it is evident that the inner products are in general not symmetric in incident and scattered fields, although the form of ex pressions (9.67) and (9.6S) does not exclude the possibility. There is, however sym metry in the inner products between the incident field quantities (e.g. /'(v),E') on the one hand and the fluctuation quantities (e.g. /*(v), E4) on the other. This is of course because the source current has this symmetry (see expression (9.65)) and the receiver field depends only on the receiver and the unperturbed plasma. This symmetry is evident in both expression (7.69) for the source current in a cold plasma and expression (7.82) for the source current in a general plasma. The source current expression (7.77) is also symmetric in (i) and {6), although this is not so evident due to the rewriting of all incident field quantities in terms of the incident electric field. In the complete cold equation of transfer (9.28) the symmetry of (i) and (6) in the source current is retained but for the reason stated above this is not seen so readily. In the general equation of transfer for scattering due to density fluctuations (9.15) this symmetry is, however, not retained because the term n'EJ is not included. So here the symmetry of (i) and (6) in the inner product is lost due to a simplification of the expression. In the derivation of the scattering cross section given by A>UfODT and RUSSELL (1992) and in the derivation of the equation of transfer given here the inner prod uct of the receiver field with the source current, for scattering due to density fluctuations only, results in the term 160 Chapter 9. Theoretical model of Thomson scatterixig (e ) -X •« n » while the incorrect derivation given by AKHIEZER et al. (1967) and SiTENKO (1967) (see discussion in Section 7.3) results in the term (éT-tfé'n'. AKIUEZER et o/.'s result does satisfy the symmetry between (i) and (6) although it is not explicit in the form presented. For instance, the last term in expression (7.72) is the mirror image of the term representing scattering due to density fluctuations and their expression for the inner product thus includes the terms (éTx'é^+léTx'é'rr', while our approach did yield the mirror image term explicitly, giving (éT-X*-éln* + (é")--x"'é'ni. Since both sets of terms have the form required by expressions (9.67) and (9.68) it would not have been possible to decide between them on this basis. To understand where the difference originated from we observe that in the kinetic approach the source current associated with scattering by density fluctuations takes the form j*(r,0 = Jtr(r - r',t - t') • Ei(Tf,t')ns(rf,t^dr'dt', (9.74) which upon Fourier-Laplace transformation results in j«(k\u;>) = y(r,t) = ns(r,t)Jtr(r-T\t - t') • E\r\t')dr'dt' (9.76) which upon Fourier-Laplace transformation results in j"(kW) = j~-4 / thus replacing u:*x* by u'x' compared with the kinetic result. Quantitatively this difference may be quite significant in the vicinity of cutoffs or resonances or when the difference between the frequencies of the incident and scattered radiation is large. This is illustrated in Figure 9.S at the end of the next section. 9.4 Numerical results Equation (9.15) is the equation describing the power transfer in a scattering sys tem. The quantities which depend on the properties of the plasma are the beam overlap. Ok, the spectral density function, S(k,u;) and the dielectric form factor, G. The beam overlap was investigated by BINDSLEV (19S9) while the spectral density function has been the subject of many investigations, e.g. HUGHES and SMITH (19SS). In this thesis an important new result is the generalized expression for the dielectric form factor which allows for spatial dispersion. This allows the dielectric tensor and derivatives thereof, which enter into the expression for the dielectric form factor, G, to be evaluated not only on the basis of the cold plasma model but also using hot or relativistic models. Computer codes have been developed, as part of this work, to evaluate G with dielectric tensors and their derivatives derived from the four magnetized plasma models discussed in Section 4.7: (a) cold, (b) hot equilibrium, (c) weakly rel ativistic equilibrium based on SlIKAROFSKY (1986) and (d) weakly rclativistic equilibrium based on YOON and KRAUSS-VARBAN (1990). In the range of electron temperatures investigated, Te = 0 to 18 keV, G and its constituent parts ^(•-r-v-i'. P V 162 Chapter 9. Theoretical model of Thomson scattering '-ilKrøWH and (-•(-æ-) appear to depend on the norm of the anti-Hermitian part of the dielectric tensor, only to second order. (The author has been able to show this result analytically for the last term but not for the other terms.) It may therefore be concluded that the effect of e° is negligible except in the vicinity of resonances. We will thus discuss numerical results in terms of equation (9.19). Parameter space is clearly very large and a comprehensive survey of it is outside the scope of this thesis. Here we will only present some illustrative examples with parameters relevant to the planned scattering experiments at JET [COSTLEY et a/., 19SS, 1989 a, b] and at TFTR [WOSKOV et of., 1988]. To describe the geometry of the scattering let ™ (k,xB)(k, xB) ,rt„.oi »*/-^-B • ~»- fcxB|k»B| ,9-'8) and cos0 = ki-)c, , cos^ = k-B . (9.79) The scattering geometry is illustrated in Figure 9.2. The angles tl\, y, and \ are the angles used by BRETZ (1987) and HUGHES and SMITH (19S9) to describe the scattering geometry. We shall only present data 9.4. Xnmerical results J£3 B Figure 9.2: Scattering geometry. 164 Chapter 0. Theoretical model of Thomson scattering JGStJMg O O ~to >ca o3n0 2 4 6 8 10 Electron density /1019m3 Figure 9.3: (a) Dielectric form factor, (b) coupling term, (c) flux term, (d) and (e) real and imaginary parts of refractive index, as functions of ne. Parameters: X to X scattering, u;' = w* = 2« • 140 GHz, 9 = 30°, ! « 7T°, \ % 15°), B = 3.4 T, Te = 12 keV. 9.4. .VumeriVaf re>u/rs ISSt resulting from geometries where i; = ISO" —v\- With this symmetry the scattering geometry is fully described by $ and p. The plots given in Figure 9.3 are calculated with parameters relevant for scat tering in JET (parameters are given in the figure caption). The curves show the dielectric form factor, the coupling term, C, (equation 9.20), the normalized flux, 5. (equation 5.45) and the refractive index, p, as functions of electron density. $ and (i for incident and scattered fields are identical due to the symmetry in the direction of propagation relative to the magnetic field. Both incident and scattered fields are in the extraordinary mode (X to X scattering). The frequency of the radiation is higher than the cyclotron frequency, so the R-cutoff determines the maximum density to which the radiation can propagate. The effect of the R-cutoff is clearly visible in the plots. While the hot plasma predictions tend towards the cold plasma predictions at the R-cutoff. the relativistic plasma model produces a shift in the R-cutoff towards higher densities. This shift can be attributed to the relativistic mass increase of the electrons (sec also the discussion in Chapter 4). Although the hot plasma model does produce a change in predictions relative to the cold model, in this regime much more substantial effects are found with the rclativistic model. It is noteworthy that a nonvanishmg imaginary part to the refractive index is found with the relativistic model and not with the hot model. This absorption is attributable to the relativistic smearing of the cyclotron absorption. In Figure 9.4, cold and relativistic versions of the dielectric form factor are plotted l5 -3 against *.-* for a range of electron densities around ne = 6.5- I0 m . It is evident that as the R-cutoff is approached the shapes of the curves become increasingly sensitive to the electron density. The reliability of the analysis of scattered radiation for diagnostic purposes de pends, among many factors, on the accuracy of the model. Figures 9.3 and 9.4 clearly illustrate the need for a relativjstic model. Another factor of importance to the reliability of the analysis is the sensitivity of the spectrum of scattered radi ation to various plasma parameters. Sensitivity to quantities which the diagnostic seeks to measure is beneficial while sensitivity to other quantities such as the elec tron density reduces the reliability of the analysis. As the R-cutoff is approached the dielectric form factor and hence the spectrum of scattered radiation for X to X scattering becomes increasingly sensitive to the electron density and other pa rameters, making reliable analysis impossible in the vicinity of the R-cutoff. The practical consequence of the relativistic shift of the R-entoff is therefore to increase the upper limit of the density range in which reliable measurements can be made with X to X scattering. At any given electron density, the reduced sensitivity found with the relativistic model, which is illustrated in Figure 9.4, implies that the reliability of the analysis will be better than expected from the cold plasma 166 Chapter 9. Theoretical model of Thomson scattering 136 138 140 142 144 ' ©c/2*GHz Figure 9.4: Dielectric form factor, (a) cold, (b) relativistic, as functions of w*. Parameters: as in Figure 9.3 except that ne = 6.2, 6.4. 6.6, 6.8 - 10,9m-3. 9.4. Numerical results 121 predictions. Figure 9.5: Relativistic dielectric form factor and real part of refractive index as functions of ne. Parameters: as in Figure 9.3 except that Te = 0.05, 5, 10, 15 keV. In Figure 9.5 the dielectric form factor and the real part of the refractive index are plotted against electron density for a range of electron temperatures. The rest of the parameters are the same as in Figure 9.3. Only relativistic curves are plotted, the Te = 50 eV curve being indistinguishable from the equivalent cold plasma curve. The dependence of the R-cutoff on temperature is evident. For 0 to X and X to 0 scattering the differences between cold and relativistic predictions are similarly dominated by the shift in the R-cutoff. Figure 9.6 shows plots for 0 to O scattering with the other parameters identical to those for Figure 9.5. Radiation in the ordinary mode is cut off at the plasma frequency. Again a relativistic shift in the cutoff frequency is found. Attention is drawn to the different scale for the density. At densities found in JET the differences between the cold and relativistic predictions are of little practical im portance. Figure 9.7 shows plots similar to Figure 9.5 with parameters relevant for the scat tering experiment planned at TFTR [WOSKOV et al, 1988]. Here the frequency of the probing radiation is below the cyclotron frequency. This implies that the 168 Chapter 9. Theoretical model of Thomson scattering JG91.308/3 10 Te/keV=0.05 •MS 15 §5 5 .2 E 1.0 =£0.DC, 5 Te/keV=0.05- 0 8 12 16 20 24 Electron density /1019nr3 Figure 9.6: Relativistic dielectric form factor and real part of refractive index as functions of ne. Parameters: 0 to O scattering, otherwise as in Figure 9.5 Numerical results JG91.»'->o. i O •c o (0 2 Te/keV=0.05 ee t E )ie l 1 Q o 0 1.0 CC =i0.5 Te/keV=0.05 51015 —i i i 4 6 8 10 12 14 Electron density /1019nr3 Figure 9.7: Relativistic dielectric form factor and real part of refractive index as functions of ne. Parameters: X to X scattering, u' = w* = 2TT • 56 GHz, 8 = 30°, 4> = S0°, (tf « 92.6°, V'5 * 87.4°, * « 29.6°), B = 5.0 T, T. = 0.05,5,10,15 keV. 170 Chapter 9. Theoretical model of Thomson scattering radiation, which is in the extraordinary mode, is cut off at the L-cutoff. Though a small relativistic shift of the L-cutoff is observed, relativistic effects in the dielectric form factor appear to be negligible for the TFTR parameters. JG92.507/4 200 \ Cold \\ \s i\ O Dielectri c for m facto r 3 O I i i i i 136 138 140 142 144 ©s/27tGHz Figure 9.8: Cold dielectric form factor, (i) SlTENKO's form, (s) our form (see text), as functions of u;s. ,9 -3 Parameters: as in Figure 9.3 except that ne = 6.5 • 10 m . In Figure 9.8 we compare the dielectric form factor in a cold plasma predicted with our expressions (9.19) and (9.20) with the predictions obtained with SlTENKO's form where u9\* is replaced by u>'x' in the expression for the coupling term, C. It is evident that the difference between the results obtained with the two models can be of practical importance. 9.5. Summarv 171 9.5 Summary A theory of scattering has been developed which takes the dielectric effects of the plasma into account and, as a new element, allows for spatial dispersion. Thermal motion results in spatial dispersion. This new expression is therefore required when hot or relativistic effects are included in the dielectric properties of the plasma. The symmetry properties of the equation of transfer for a scattering system have been investigated in detail. The equation of transfer was found to be symmetrical in incident and scattered field variables except for the term representing the inner product of the receiver field pattern and the source current for the scattered field. The source current was found to be symmetrical in the quantities (e.g. fields and perturbations to the velocity distribution) referring to the incident field and the fluctuations respectively. Earlier derivations of the expression for the source current based on the fluid description [AklULZKR et n.L, 1967; SlTENKO, 1967] were found to be incorrect and the results are at variance with our results in the cold plasma limit. In the term representing scattering due to density fluctuations a factor w*x* appearing in our expression for the source current was replaced by u/'x' in the incorrect fluid result. Significant relativistic effects, of practical importance for the planned collective scattering diagnostic at JET, have been found for the advantageous X to X scat tering. Due to the relativistic shift of the R-cutoff to higher densities, reliable analysis of radiation scattered from X mode to X mode appears feasible in an important density range which would not have been considered possible on the basis of the cold plasma predictions. For O to O scattering in JET the relativistic effects appear to be of no importance to the signal level in O-mode. However, it is important, even for O to O scattering, to stay clear of the R-cutoff, preferably at densities above it, in order to minimize or eliminate spurious signals from X to X scattering. This requirement accentuates the importance of an accurate knowledge of the location of the R-cutoff. For the collective scattering diagnostic under development at TFTR [WOSKOV, 19SS] no relativistic effects of any importance are predicted. Part II Reflectometry 173 Chapter 10 Reflectometry 10.1 Introduction Reflectometry is based on measurements of the phase change which a probing beam undergoes while propagating through the plasma from a launching antenna to a reflecting layer at a cutoff and back to a receiving antenna [COSTLEV, 19S6J. In Tokamaks and other plasma devices it is provine to be a useful method for diagnosing the electron density [COSTLEV, 1991]. Broad band and multichannel narrow band rcflcctomctry have become reliable means of diagnosing the elec tron density profile [PREXTICE et ai, 1990; DOVLE et al, 1990; MA.NSO et al., 1991; SlPS 1991]. while correlation reflcctometry offers the possibility of localized measurement of density fluctuations [CRIPWELL et al., 19S9; MAZZUCATO and NAZIMAN . 1991; Zou t.t al., 1991; CRIPWELL et al., 1991]. Dual mode reflectom etry may provide a means of measuring the magnetic field profile [PAVLICIIENKO and Sk'fBENKO, 19S9; COSTLEY ct ai, 1990 a]. It was demonstrated in Chapter 4 that there are significant relativistic modifica tions of the dispersion in a plasma in the regimes which arc relevant for reflec tometry. This chapter explores the consequences these modifications have on the interpretation of rcnVcfomctry data. Methods for analyzing reflectometry data with a relativistic plasma model are given. This chapter is arranged as follows. A general algorithm for reconstruction of electron density profiles from reflectometry data is derived in Section 10.2. In Section 10.3 simulated broad band reflcctometry data arc analysed with the cold and the weakly relativistic models. Consequences for existing and possible future diagnostics are discussed. The results are summarized in Section 10.4. 175 176 Chapter 10. Reåecttnnetry To explore the consequences of the relativistic modifications for reflectometry, plasmas are assumed to have the following profiles of magnetic field. B, electron density. ne and electron temperature, Te: B = J^Ik- (10.1a) "« = (ne0-ne,)(l-(r/ii)*)*'+n€, (10.1b) J, = iTa-TJ^-Wa)1)" + T*. (10.1c) We will refer to these \n~; • as actual profiles. (The reconstructed profiles in troduced in Section 10.2 are not assumed to have this nor any other parametric form.) Cutoff frequencies calculated on the basis of this model with equations (4.55) and (4.56) as functions of major radius, R — Iio + rt are given in Figures 10.1 (a)-(c) for a range of central temperatures, T^t. and for values of density and magnetic field which are typical of JET plasmas. It is clear that in both modes the cutoff point may, in the central region, be shifted by a significant fraction of the minor radius. 10.2 General algorithm for density profile recon struction It is evident that the relativistic modification of the refractive index and in par ticular the shift of the cutoff density will change the relation between the density profile and the phase change which a probing wave undergoes in the plasma. An algorithm for reconstruction of the density profile from reflectometric data which is valid for both O-mode and X-mode in the relativistic model is therefore required. One such algorithm is derived here. It is of course also valid for the cold model. If is assumed that the phase shift which the probing wave undergoes in the plasma is 2(u;/c)'P(w) — ~/2 where u? is the frequency of the probing wave, c is the speed of light and *(u;) is the optical distance from the plasma edge to the cutoff layer [GfNZM'RG, 1970, §30], fidr . (10.2) Jr. »is*!«') 10-2. General algorithm for density profile .cconstruction 177 2.0 2.4 2.8 3.2 3.6 4.0 Major radus/m 1 3 ! 2.4 2.6 3.2 2.0 2.4 2.8 3.2 3.6 4.0 Major radius'm Major radius/m Figure 10.1: Cutoff frequency as a function of major radius. Plasma profiles are defined in equations (10.1b)-( 10.1a). Parameters: i?0 = 3 m, a = 3 1.2 m, pn = 0.5, ntl = 1 -lO^m" , Te0 = 0.1, 5.1,10.1,15.1 keV,pr ,;) 3 = 1, Tel = 100 eV, (a) O-mode, B0 = 3.4 T, ne0 = 3.01 • 10 m- , ,s> 3 (b) X-mode, B0 = 2.8 T, ne0 = 5.01 • 10 m- , (c) X-mode, 19 3 B0 = 3.4 T, ne0 = 3.01 • 10 m- . ITS Chapter 10. Refiectometry z is analytic in /i2 at pi = 0 (cf. equation (4.19)). From this it follows that A is analytic in /i2 at /i = O and hence 2 ^ = 2fid\/dfi = 0 at p = 0T (10.3) On while in general — ± 0 at /« = 0. (10.4) a,« Since A is analytic in A", where A" equals B, nt or Tt (cf. equation (4.19)), it follows from (10.4) that dX dX V = ~ l (105) OX d\ldf K ' tends to a finite limit as ;i -» 0, while dp -dA/dX (10.6) OX ~ d\/dn docs not. Let fi(x) be the refractive index at the distance x from the cutoff. Expanding fi2(x) around the cutoff we find to lowest order in x 2 (d? 0B Of dnt 0? &Tt\ "(T)={OB -o;+0^17+mix-)x (10J) Let 6$ be the integral of // from the cutoff out to a distance A = ,l{ )(,r + + A/ (108) ™ l " = iiBH ir^ wt^ 3 - The second equality is valid to lowest order in A. From equations (10.7) and (10.8) wc get an expression for dne/dx in the vicinity of the cutoff which together with the expression for A, implicit in (10.7), forms the basis for the inversion algorithm: 10.2- General algorithm for density profile reconstruction 179 On, /2/i3 dfdB dfdT. (10-9) dr \Z6* OB dr 0Tt dr Z6* (1010) In equations (10.9) and (10.10) /* is the refractive index at the distance A from the cutoff, while the gradients, which vary slowly, can be evaluated anywhere in the vicinity of the cutoff. The assumptions about analyticity made in the derivation clearly also hold for the cold model. Given the spatial profiles of the magnetic field and of the temperature, an iterative procedure for reconstructing the density profiles is readily derived from expressions (10.9) and (10.10). We shall assume that experimental data on the phase shift function are available at discrete frequencies, u?,. Waves at these frequencies are cut off at points, r,. with densities n,. We shall further assume that the density profile has been reconstructed from the edge to the (i — l)th cutoff point, using the experimental data at frequencies up to and including the (i — l)th frequency. The distance from r^ to r, can then be estimated using expression (10.10) where Sty is obtained from the experimental data at frequency u.-, and that part of the density profile which has already been reconstructed. The density at the i th cutoff point can be estimated with expression (10.9). The explicit form of this iterative algorithm is r„ = « (10.11a) »o = n,„...rf>-o,i*>,To) ; £.-i = £(r,_,); T,_, = Te(r,_, 110.11b) u.'i = u:0 + i£u/ (10.11c) 2 2 (/< ), = li [ui,ni.l,Bi.l,Ti.t) (lO.lld) = W I? T \dx) (|Y)( ''"'-'' -1' --0; X=ne,B,Te (10.11c) *i = f fi(ui,r)dr ; fi(u>h;•_,) = /,(u>„ n„ 2?„ T,) (lO.llf) »-• = *,-*,; *i= f liri(ui,r)dr (lO.llg) ISO Chapter 10. Reflectometry /(VX _ /V\ ag(r.-t) _ / V\ dT(ri-i)\ (dn\ _ \Z8* \dBJ, <* \9Tc) Or J t t (lO.llh) [dr)t - /V\ A, = ^i (10.11!) r, = r.^-Ai (10.11J) "' = "-,_(^) Ai ' (10.11k) n, is the reconstructed electron density at the minor radius rj (r, is negative on the inside of the plasma). Having determined r^ the density, n,, could of course also be determined from equation (4.55) or (4.56), or the equivalent cold equation. This provides a convenient means of checking the reconstruction. If the group delay time, •-IT)- (10.12) is available instead of the phase function then expressions (lO.llf) and (lO.llg) in the algorithm arc replaced by ^ = (u;,-u;,.,)l— J -jf_ (ri(ui,r)-ri{ui-ur))dr. (10.13) 10.3 Reconstruction of density profiles from sim ulated data To simulate rcflectometric data the phase function ^(UJ), as given in equation (10.2), was calculated relativistically for a range of plasmas with profiles given by equations (10.lb)-( 10.1a). Density profiles were then reconstructed from ^(w) with the algorithm given above using the cold and the relativistic plasma model. The reconstructed density profiles are identical to the actual density profiles when the reconstruction is based on the relativistic plasma model. This demonstrates 10.3. Reconstruction of density profiles from simulated data 1S1 that the above algorithm is numerically stable and accurate, and that reconstruc tion based on a relativistic plasma model is feasible. For O-mode reflectometry, reconstruction based on the simple approximation to the relativistic refractive index, given in equations (4.60) and (4.61), also results in accurately reconstructed density profiles. For the parameters used in Figure 10.2 (a) the reconstructed profiles, obtained with this simple approximation, were all accurate to three or more significant digits. When the reconstruction is based on the cold plasma model the reconstructed density profiles underestimate the actual density profiles, by a considerable amount in X-mode and by a smaller, though still significant, amount in O-mode. Examples of density profiles reconstructed with the cold model are given in Figures 10.2 (a)- (c), 10.3 and 10.4. The plasma parameters used for the calculations presented in Figures 10.2 (a)-(c) are typical of JET plasmas and identical to those used for the graphs in Figures 10.1. Figure 10.2 (a) shows reconstructed density profiles obtained for O-mode. To judge the significance of the relativistic effects for the JET O-mode multi channel reflectometer [PRENTICE, 1990] the relativistic shifts of profiles may be compared with the errors that enter the measurement from other sources. To this end it is convenient to separate the density profile into three zones: the edge with very steep gradients (f^/n » 1/«); the confinement region with moderate gradi ents (f^r/n ~ 1/«) and densities which are significant fractions of the peak density; and the central region with small gradients (f^/n Errors not related to relativistic effects manifest themselves as an uncertainty in the radial location of the cutoff point associated with a given frequency. The cutoff density associated with a frequency is not changed by these errors. Non-relativistic errors in the density profile are therefore conveniently expressed in terms of radial uncertainties. The profile shifts caused by relativistic effects are mainly due to changes in cutoff density associated with a given frequency, though shifts in the location of the cutoff point do also contribute. For the purpose of this comparison it is helpful, in the regions with non-vanishing density gradients, to think of the relativistic shifts in terms of radial displacements. At the edge the errors from non-relativistic sources can be brought down to about 4 cm [SlPS, 1992, private communication]. The relativistic shift at the edge is negligible because the temperature is low and even if it is not, the relativistic effects do not shift the radial location of a sufficiently high step in density. Only the height of the step and the densities on either side of the step would be affected. In the confinement region the non-relativistic errors can be brought down to 6 cm 1S2 Chapter 1U. Keriectometry 3.4 3.8 42 Major radws/m c) o I 2.0 2.4 2.8 3.2 3.6 4.0 2.4 2.8 3.2 Major radius/m Major radius/m Figure 10.2: Actual electron density profile and reconstructed density profiles derived using the cold plasma model for analysing phase functions, ^(w), obtained with the relativistic model. Plasma profiles are defined in equations (10.1b)-( 10.1a). Parameters: (identical to those used in Figure 10.1) i?0 = 3 m, a = 1.2 m, pn = 0.5, nel = ,7 3 1 • 10 irT , Te0 = 0.1, 5.1, 10.1, 15.1 keV, p7 = 1, Te, = 100 ,9 3 eV, (a) O-mode, B0 = 3.4 T, ne0 = 3.01 • 10 m" , (b) X-mode, ,9 3 B0 = 2.8 T, ne0 = 5.01 • 10 nT , (c) X-mode, B0 = 3.4 T, 19 -3 ne0 = 3.01-10 m . For X-mode the maximum probing frequency and hence the maximum depth to which the density profiles can be reconstructed is limited by absorption in the outer region of the plasma at the second harmonic of the cyclotron frequency. 10.3. Reconstruction of density profiles from simulated data 183 or less [SlPST 1992, private communication]. This is comparable to the rclativistic shifts and it appears that under some conditions the relativistic effects could be the dominant source of errors in this region. In the central region the non-relativistic errors become very large and details of the profile are seldom available here. It is however possible to give accurate information about the peak density at the points in time when a channel changes from reflection to transmission or the reverse. The non-relativistic errors on these data are negligible. Correcting for the relativistic effects increases the predicted peak density by typically 5 to 8 % depending on the temperature (see Figure 4.5 and equations (4.5S) and (4.60)). It must be concluded that a data analysis which takes the relativistic effects into account is likely to improve significantly the accuracy of the JET O-mode mul tichannel reflectometer. The codes for analyzing data from this diagnostic will therefore be updated in the near future to take relativistic effects into account. Figures 10.2 (b) and (c) show reconstructed density profiles obtained from X- mode. An X-mode reflectometer for profile measurements at JET would clearly require a relativistic data analysis. To estimate the relativistic effects at various values of n,o and Bo it is useful to note that the graphs of the reconstructed densities normalized by nt0 are invariant under changes of ne0 and B0 which keep n(0/Bl constant: Mr) «»J^,^,W|. HON) N is the operator representing the relation between the reconstructed density, h{r), and the actual plasma parameters, ij is a convenient dimensionless parameter : which is proportional to ne0/-Bo The invariance of n(r)/nt0 under changes in plasma parameters which leave 7, ne(r)/nt0, B{r)/B0 and Xe(r) unchanged, as expressed in equation (10.14), is read ily shown to follow from the fact that the dependence of £ on u;p, wc and w can be written (see equations (4.18) and (4.19)) £ = e(^,-,/i,Te) . (10.16) 1S4 Chapter 10. Reåectometiy ^^ OJ c* S I 0.4 ni 0.2 0 2/3 10 4/3 Figure 10.3: As Figure 10.2 but with parameters: Ro/a = 3, pn = 0.5, n^/na = 1/100, TJ = 0.90, 0.60, 0.30, 0.10, 0.07, 0.05, 0.033, T* = 10.1 keV, pr = 1, T€i = 100 eV, X-mode. For r\ - 0.90 and 0.60 the maximum probing frequency and hence the maximum depth to which the density profiles can be reconstructed is limited by ab sorption in the outer region of the plasma at the second harmonic of the cyclotron frequency. For r\ = 0.033 the depth is limited by numerical noise in the reconstruction. 10.3. Reconstruction of density profiles from sjawimied d»t» 1S5 In O-mode the reconstructed density profiles depend very little on the magnetic field and hence on 17, except for the limitation that may be imposed by cyclotron absorption on the spatial region over which the density profile can be probed. The graphs in Figure 10.2 (a) therefore give a good estimate of the relativistic effects in O-mode for a wide range of plasma conditions. In X-mode the relativistic effects vary significantly with 17. Density profiles recon structed with the cold model and normalized by n^ are plotted in Figure 10.3. Here the central temperature, T^, is kept constant at 10 keV while 17 is varied. With the major radius normalized by RQ these curves cover a wide range of plas mas. For the cases covered in Figure 10.3 the actual central density is at least 20 % greater than the peak reconstructed density. The depths to which the pro files with 17 = 0.90 and 0.60 can be reconstructed are limited by second harmonic cyclotron absorption in the outer region of the plasma. The depths to which the profiles could be probed were R/R0 = 1.04 for n = 0.90 and R/RQ = 0.S1 for 17 = 0.60. The cold reconstruction overestimates this depth. In the cold plasma limit the X-mode radiation required to probe to the plasma centre and beyond will pass through the second harmonic cyclotron absorption layer if ">%=(l+«/W (101° where a/Ro is the inverse aspect ratio. For the plasmas considered in Figure 10.3, a/Ro = 1/3 giving i)o = 0.75. The relativistic mass increase lowers the frequency at which cyclotron absorption sets in. This is counteracted by the rclativistic lowering of the cutoff frequency. Equation (10.17) generally gives a good indication of when the centre of the plasma may be probed with X-mode, even when relativistic effects arc taken into account. At large values of u^/u:* the location of the cutoff is a weak function of density. In the cold plasma approximation and assuming that B oc l/R a change, SR, in the location of the R cutoff results in a change, 6nt, in the cutoff density given by -£ • =hf • (1018) This means that at low densities and high fields the reconstruction of density profiles from X-mode rcflectometry becomes prone to noise. Some numerical noise is evident in the »/ = 0.07, 0.05 and 0.0033 curves in Figure 10.3. In Figure 10.4 reconstructed densities obtained for plasmas with a range of tem perature profile shapes are given. Pj (see equation (10.1c)) was varied while all 1S6 Chapter 10. RdJcctometry 2/3 t.0 4/3 Major radws/R. Figure 10.4: As Figure 10.2 but with parameters: Ro/a = 3, pn = 0.5, riti/n^ = 1/100. ^ = 0.5, Trf, = 10.1 keV, pr = 1, 2,3,4, Te, = 100 eV, X- mode. The maximum probing frequency and hence the maximum depth to which the density profiles can be reconstructed is limited by absorption in the outer region of the plasma at the second harmonic of the cyclotron frequency. 104. Summan- 187 other parameters were kept constant. Increasing Pr peaks up the temperature profile. It is evident that reconstruct ion based on the cold model underestimates the central density by an amount which is almost completely independent of the peaking of the temperature profile. The graphs in Figure 10.4 further suggest that the depression in the cold reconstructed density profile at a given point depends on the temperature in the vicinity of that point and very little on the temperature in the rest of the plasma. 10.4 Summary Cold And relativistic predictions for refleetometry have been compared. It is found that relativistic effects are of practical importance for X-mode reflet totnet ry in large Tokamaks. because (a) cold analysis leads to a considerable underestimation of the electron density profile and (b) the location of the cutoff may be shifted by a significant fraction of the minor radius, (a) implies that for density profile measurements using X-mode reflectometry (an attractive option for ITER) the data must be analysed with a relativistic plasma model, (b) has consequences for the determination of where fluctuations observed with correlation reflectometry are situated in the plasma. While the relativistic modifications found in 0-mode are smaller than in X-mode they may still have to be taken into account, except near the plasma edge. For typical JET parameters the relativistic shifts of density profiles derived from O- modc reflertometery can be of the same order as or larger than the estimated non- rclativistic errors on density profiles obtained with the JET 0-mode multichannel reflectometer. This diagnostic will therefore benefit from a data analysis which takes relativistic effects into account. A code for rclativistic reconstruction of the electron density profile from the phase shift function has been written and tested with simulated reflectometric data. Simple and accurate approximations have been found for the rclativistic 0-mode refractive index and for the relation between density and frequency at the O-mode cutoff. These approximations permit accurate reconstruction of density profiles from simulated 0-mode reflectometry. Chapter 11 Summary and conclusions The main motivation for the work presented in this thesis was the need to review and improve the theoretical understanding of the plasma processes involved in collective Thomson scattering in preparation for the fusion alpha particle and energetic ion diagnostics presently being developed at JET and TFTR. Most previous Thomson scattering diagnostics have made use of probing radiation with a frequency much greater than the electron plasma frequency. For such experiments it is an acceptable approximation to neglect the refraction in the plasma and to express the scattered field as a summation of the Lienard-Wiechert potentials stemming from the acceleration of the individual charges in the incident field. The theory covering such experiments is well developed. In the collective Thomson scattering diagnostics being developed at JET and TFTR the frequency of the probing radiation is in the range of the electron plasma frequency and as a consequence the dielectric properties of the plasma must be taken into account, both when modelling the propagation of radiation to and from the scattering volume and when finding expressions for the scattered field in terms of the incident field and the fluctuations in the plasma. Though considerable progress has been made, the task of modelling Thomson scattering taking com plete account of the dielectric properties of the plasma is formidable and much work still needs to be done. Following a review of previous work on Thomson scattering a theoretical model relevant for a collective Thomson scattering diagnostic was developed ab initio. Various simplifying assumptions were made: further work will be needed to reduce the limitations imposed by these assumptions. Some of the methods presently used in calculating the fluctuations which result from the presence of energetic ion populations are reviewed. Although the form of 188 i£2 this presentation is slightly different from tbosr given elsewhere no new material is presented on the theory of fluctuations. The main contributions of this work to the theory of Thomson scattering are: • The inclusion of spatial dispersion in the derivations of the equation of trans fer for a scattering diagnostic and the expression for the scattering cross section, which b given as an intermediate result. This allows the dielectric properties of the plasma to be described by models, such as the hot or the relatiristic models, which take thermal motion into account. • The exploration of reiativistic dielectric effects in scattering due to electron density fluctuations, and comparisons with predictions based on cold and hot plasma models. While there is little difference between the predictions based on the cold and the hot models, it is found that reiativistic effects can modify the scattering significantly. From this it is concluded that the dominant cause of the modifications found with the reiativistic model are due to the reiativistic mass increase, which is absent in the hot model. The reiativistic effects are of practical consequence to the JET collective Thomson scattering diagnostic for scattering from X-mode to X-mode, but they are not important for O-mode to 0-mode scattering at JET nor for the X-mode to X-mode scattering at TFTR. • The derivation of the complete expression in the cold plasma limit for the source current which drives the scattered field. This derivation is based on a kinetic description of the plasma and confirms some of the results obtained by AAMODT and RussFLL (1992) by a similar approach. Our result is at variance with the widely quoted expressions derived by AKHIEZER ct ml. (196?) and SlTF.NKO (1967) by the fluid approach. Errors were found in their derivation which explains the difference between our results. The present results show that in the cold plasma limit the scattering of electromagnetic radiation in a plasma is due to fluctuations in electron density and current, and in the electric and magnetic fields. With these results the theory of Thomson scattering in a cold plasma, for given fluctuations, and taking full account of dielectric effects, is essentially complete. On the basis of the present work some areas of the theory of Thomson scattering which merit further attention can be identified. They include the extension of the present reiativistic expressions to include fluctuations in other quantities besides the electron density, and an investigation of whether a rclativistic treatment of the electron response affects the calculations of fluctuations in electron density and other quantities. The investigations of the rclativistic dielectric effects in Thomson scattering in volved a detailed study of the dielectric properties of plasmas for frequencies in the 190 Chapter 11. Summary and conclusion* range of the electron plasma and cyclotron frequencies and for plasma conditions relevant for JET. While the predictions based on the cold and the hot plasma models differ only slightly, except in regions dose to resonances, and converge at the cutoffs, the predictions based on the relativtstic plasma model differ signifi cantly from those based on the cold and hot plasma models. These differences are particularly pronounced in the vicinity of cutoffs, which are shifted by reUtiristic effects. The differences are larger than a simple comparison of temperature with the electron rest mass energy might suggest. The contributions of this work to the study of reiativistic properties of plasmas include: • The derivation of fidly reiativistic expressions for the locations (in parameter space) of the X-mode R-cutoff and L-cutoff. (A fully reiativistic expression for the location of the O-tnode cutoff was already given by BATCHELOR, GOLDFINGER and WEITZXER (19S4)). These expressions were used among other things to calculate curves for the locations of the R-cutoff and O-mode cutoff which were presented in CMA diagrams. • The derivation of new algorithms for evaluating the Shfcarofsky functions, which enter the expressions for the weakly reiativistic dielectric tensor. These algorithms allow the Shkarofsky functions to be evaluated for parameters for which previously published methods are numerically unstable. A practical consequence of this result is to make possible the tracing of rays with the weakly reiativistic plasma model for propagation at all angles relative to the static magnetic field. Hitherto the reiativistic ray tracing codes broke down when attempting to trace rays which propagate nearly perpendicular to the static magnetic field. • The derivation of a simple expression for the reiativistic refractive index for perpendicular O-mode propagation. This expression is a good approxima tion except near resonances. With the significant reiativistic modifications of the dielectric properties found in certain regions of parameter space it was to be expected that diagnostics operating in these regions would be affected. Many diagnostics a* well as other plasma applications rely on the propagation of electromagnetic radiation and some require ray tracing to be carried out as part of the analysis of the experimental situation. With this in mind it was demonstrated that the trajectory of a ray can depend significantly on reiativistic effects even at the relatively low temperatures found in Toknmaks. Because the refractive index is modified most in the regions near cutoffs and the locations of cutoffs are shifted it is evident that reflectomctry may be affected by 121 reUtivistic effects. This was investigated for the first time here and the study forms the second part of this thesis. To study refiectometry with a relativistic model it was necessary to develop a new general algorithm for inversion of refiectometry data to give information about the density profile. The main conclusions of this study are: • For density profile measurements with X-mode refiectometry relying on re flection at the R-cutoff layer a rrlatiristk analysis is required. A cold anal ysis would lead to substantial under-estimations of the density profile. • For density profile measurements with O-mode refiectometry the relativistic effects are smaller than for X-mode but still significant. It is concluded that the JET O-mode multi-channel refiectometer will benefit significantly from a relativistic data analysis. The simple approximation to the relativistic O-moie refractive index, mentioned above, makes this modification of the analysis codes relatively straightforward and results in only a small increase in the computation time required compared with codes base on a cold plasma model. • The distance between the locations of a cutoff layer predicted with a cold plasma model and with a irlativist'c plasma model can differ by a significant fraction of the minor radius. This can be significant for studies of fluctuations with correlation rrflcctometry It is most likely to be significant when use is made of X-mode reflecting at the R-cutoff, but may also be significant when O-mode is used. Appendix A Notation This is a list of the variables which are used widely in this thesis. The numbers on the right refer to the numbers of the equations where the variables are defined or enter in an obvious context. Additional variables are defined locally, and some of the variables listed here are redefined locally. Vectors and tensors are printed in bold face while the components of vectors and tensors, like all other scalars, are printed without bold face. A° Anti-Hermitian part of the tensor A. Ah Hermitian part of the tensor A. A' A, referring to the incident field. Afm Imaginary part of A. ARr Real part of A. A* A, referring to the scattered field. A Unit vector. A' Complex conjugate of A. Ae A, referring to electrons. Ax Component of A perpendicular to B. >l|| Component of A parallel to B. c Vacuum speed of light. (4.8) é Unit electric field vector. (6.6) /(v) Velocity distribution, /(p) Momentum distribution. g Unit eigenvector of A. (6.2) j Current. (4.5) k Wave vector. (B.l) k Complex norm of the wave vector. (6.9) 192 m k Unit wave vector (6.8) m Particle mass. n Particle density. P Momentum. 9 Particle charge. re Classical electron radius. (9.14) i Time. V Velocity. *s Unit group velocity vector. (6.20) B Magnetic flux density. (4.4) E Electric field strength. (4.4) Ep Electric field scrength in polar Fourier space. (6.5) G Dielectric form factor. (9.16) I Beam intensity. (5.7) I Identity tensor. A'n Modified Bessel function of the second kind and order n. (9.11) ob Beam overlap. P Power. (2.6) P Dielectric polarization. (4.5) S Poynting vector. (5.28) s., Poynting tensor. (5.44) 5(k,w) Spectral density function. (9.17) r Temperature (in units of energy). r Time period. c Coupling term. (9.20) £ Amplitude of monochromatic electric field. (9.3) T Shkarofsky function. (4.20) i Normalized beam intensity. (5.7) s Normalized energy flux. (5.45) a Salpeter parameter. (2.5) 7 Relativistic gamma factor. (4.16) e Dielectric tensor. (4.10) £o Vacuum permittivity. (4.6) C Relativistic temperature parameter. (4.1) 0 Scattering angle. (9.79) X Eigenvalue of A. (6.2) X Finite Larmor radius parameter. (4.2) Ao Vacuum wave length. (5.17) AD Debye length. (2.4) /' Refractive index. (4.13) /^r.y Ray refractive index. (5.9) 194 Appendix A. Notation o-'(r,ty,t) Kernel of conductivity operator. (4.6) f A Determinant of the wave tensor. (6.17) A Wave tensor. (4.12) S Differential scattering cross section. (9.22) Appendix B Fourier-Laplace transformation In this paper a spatial Fourier transform and temporal Laplace transform are used: A(r,t) = j^J^ jAik^e^-^dkdu (B.l) A(k,u) = J°° JA{r,t)e-i{kT-"l)drdt . (B.2) The Fourier-Laplace transform is defined for values of 7 large enough for the inte gral (B.2) to exist. In the application, use will be made of the analytic continuation of the transform. 195 Appendix C Notes on Yoon and Krauss-Varban (1990) Yoon and Krauss-Varban (1990) (referred to below as Y & K) give expressions for the dielectric tensor elements of a weakly relativistic plasma with a loss cone distribution. Setting the "loss-cone index", /, equal to zero reduces the loss-cone to a Maxwellian distribution. Their result with / = 0 and the corrections given below formed the basis of one of the relativistic codes used here. Y k K use the Shkarofsky function, F, (Shkarofsky, 1966) which is calculated by means of the relation between F and the plasma dispersion function (Fried and Conte, 1961) given by Krivenski and Orefice (1983). With ifr and In Y & K's expression for M™y, (m — 1) should be replaced by (m + 1) and in their l expression for M£, C'^,_,(m — 1) should for m = 1 be replaced by C m. 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Fully relativistic expressions for Title and authors locations of the X-mode cutoffs are derived On the Theory of Thomson Scattering and Re and new algorithms are given which extend flectometry in a Relativistic Magnetized Plas the regime in which the weakly relativistic die ma lectric tensor can be computed. For X-mode plasma reflectometry it is demonstrated that Henrik Bindslev these effects may shift the location of the re flecting layer by a significant fraction of the minor radius and that the cold model may lead ISBN ISSN to considerable underestimations of the den 87-550-1874-2 0106-2840 sity profile. Relativistic effects predicted for O- mode reflectometry are smaller than for X- Dept. or group Dale moie, but not negligible. An algorithm for re Optics and Fluid Dynamics Department December 1992 construction of density profiles which allows a relativistic plasma model to be used is presen Croups own reg. number(s) Project/contract No(s) ted. Descriptors INIS/EDB COLD PLASMA; DIELECTRIC PROPERTIES; ELECTROMAGNE Pages Tables Illustrations References TIC RADIATION; RUCTUATIONS; HOT PLASMA. JET TOKA- 206 30 186 MAK; MAGNETIC HELDS; MATHEMATICAL MODELS. PLAS MA DENSITY; PLASMA DIAGNOSTICS; REFLECTIVITY; RE FRACTIVE INDEX; RELATIVISTIC PLASMA; THOMSON SCAT- \bstract (Max. 2000 characters) TER1NG; WAVE PROPAGATION A theoretical model of Thomson scattering in a Available on request from Risø Library, Rise National Laboratory, magnetized plasma, taking spatial dispersion (Risø Bibliotek. Forskningscenter Rise), PO Bo* 49. DK-4000 Roskilde, Denmark. into account, is developed ab initio. The result Telephone *45 42 371212, ext 2268/2269 ing expressions allow thermal motion to be in Telex 43116 Telefax *45 42 36 06 09. cluded in the description of the plasma and remain valid for frequencies of the probing ra diation in the region of o™ and (oce, provided the absorption is small. With these expressions the effects of the dielectric properties of mag netized plasmas on the scattering of electro magnetic radiation by density fluctuations are investigated. Cold, hot and relativistic plasma models are considered. Significant relativistic effects, of practical importance for millimeter wave scattering in large Tokamaks, are predic ted. The complete expression for the source current of the scattered field is derived in the cold plasma limit by a kinetic approach. This result is at variance with the widely used ex pression derived from a fluid model of the plasma. It is found that a number of mistakes were made in the traditional fluid derivation, which explains the differences between earlier results and our results in this limit. The refractive indices and the cutoff condi tions for electromagnetic waves in plasmas are investigated for cold, hot and relativistic plas ma models. Significant relativistic modifica tions of refractive indices and locations of cut offs are found for frequencies in the range of Available on request from: Risø Library Risø National Laboratory,