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On the Theory of Thomson Scattering and Reflectometry in a Relativistic Magnetized Plasma

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On the Theory of Thomson Scattering and Reflectometry in a Relativistic Magnetized Plasma 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 <j> versus é diagram indicating rejions of stability for numerical eval­ uation of the Shkarofsky functions by forward, reverse and central recursion 55 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.
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