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Microwave plasma diagnostics Shankowski, Allison Eugene 1968

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MICROWAVE PLASMA DIAGNOSTICS ALLISON EUGENE SHANKOWSKI B.Sc., U n i v e r s i t y o f . A l b e r t a , 1963. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF' DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the required standard Research Supervisor Members of the Committee, Head of the Department THE UNIVERSITY OF BRITISH COLUMBIA December, 1968 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by hits r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada Department ABSTRACT "" . T h i s t h e s i s i s c o n c e r n e d m a i n l y w i t h d e v e l o p i n g a c c u r a t e m i c r o w a v e m e t h o d s f o r d e t e r m i n i n g e l e c t r o n - d e n s i t y d i s t r i b u t i o n s i n t r a n s i e n t p l a s m a c o l u m n s . Two new m e t h o d s h a v e b e e n d e v e l o p e d a n d t h e s e h a v e b e e n e v a l u a t e d by c o m p a r i n g w i t h o t h e r m e t h o d s . T h e f i r s t i s a m i c r o w a v e r e f r a c t i o n t e c h n i q u e w h i c h d e -p e n d s o n m u l t i p l e r e f l e c t i o n s o f a n o b l i q u e l y - i n c i d e n t m i c r o w a v e .beam b e t w e e n t h e p l a s m a c r i t i c a l l a y e r a n d t h e w a l l s o f t h e d i s -c h a r g e t u b e . - I t o f f e r s s e v e r a l a d v a n t a g e s o v e r p r e v i o u s t e c h n i q u e s u s i n g m i c r o w a v e r e f r a c t i o n ; i t i s s i m p l e r a n d f a s t e r t o i m p l e m e n t , a n d y i e l d s a c c u r a t e a n d m o r e r e l i a b l e r e s u l t s . S e v e r a l m e t h o d s f o r r e c o n s t r u c t i n g p r o f i l e s f r o m r e f r a c t i o n d a t a a r e . d e s c r i b e d . The s e c o n d m e t h o d d e p e n d s o n r e f l e c t i o n m e a s u r e m e n t s a t n o r m a l i n c i d e n c e , a n d i s s u i t a b l e f o r p l a s m a s i n w h i c h t h e maximum p l a s m a f r e q u e n c y i s g r e a t e r t h a n t h e a p p l i e d f r e q u e n c y . T h i s m e t h o d i s b a s e d o n m e a s u r i n g t h e r a t e o f c h a n g e i n p h a s e o f t h e r e f l e c t e d s i g n a l d u e t o D o p p l e r s h i f t i n f r e q u e n c y p r o d u c e d b y m o t i o n o f "ohe p l a s m a c r i t i c a l l a y e r . I t h a s t h e a d v a n t a g e t h a t t h e r e q u i r e d d a t a c a n b e o b t a i n e d f r o m r e l a t i v e p h a s e m e a s u r e m e n t s w h i c h a r e l e s s s u b -j e c t t o e x p e r i m e n t a l e r r o r t h a n a b s o l u t e p h a s e m e a s u r e m e n t s . S e v e r a l m e t h o d s o f r e c o n s t r u c t i n g p r o f i l e s a r e d e v e l o p e d . A p p r o x i m a t e p r o -f i l e s c a n b e d e t e r m i n e d f r o m d a t a a t t w o f r e q u e n c i e s o n l y , w h i l e m o r e d e t a i l e d p r o f i l e s i n v o l v e a p p l y i n g a s t e p - b y - s t e p p r o c e d u r e t o d a t a o b t a i n e d a t s e v e r a l f r e q u e n c i e s . . The s e n s i t i v i t y o f t h e m e t h o d t o c e r t a i n e x p e r i m e n t a l e r r o r s i s i n v e s t i g a t e d a n d t h e v a l i -d i t y o f r a y t h e o r y i n t h e n o r m a l - i n c i d e n c e a p p l i c a t i o n i s e x a m i n e d . A f e w r e l a t e d m e a s u r e m e n t s w e r e c a r r i e d o u t i n o r d e r t o e x a m i n e t h e v a l i d i t y o f some a s s u m p t i o n s made i n c o n n e c t i o n w i t h t h e m i c r o w a v e m e t h o d s f o r d e t e r m i n i n g p r o f i l e s . The e f f e c t o f p o l a r i -z a t i o n , t h e e f f e c t o f a l t e r i n g t h e d i s c h a r g e - c u r r e n t w a v e f o r m , a n d t h e ' p o s s i b i l i t y o f a d e n s i t y g r a d i e n t i n t h e a x i a l d i r e c t i o n w e r e i n v e s t i g a t e d . L a n g m u i r d o u b l e p r o b e s w e r e u s e d i n a f e w c a s e s t o o b t a i n a n i n d e p e n d e n t m e a s u r e m e n t o f t h e e l e c t r o n - d e n s i t y d i s t r i b u t i o n . U s i n g t h e m i c r o w a v e r e f r a c t i o n t e c h n i q u e , e x t e n s i v e m e a -s u r e m e n t s w e r e c a r r i e d o u t i n a n a f t e r g l o w p l a s m a f o r v a r i o u s d i s -c h a r g e c o n d i t i o n s . A f e w i n t e r e s t i n g c h a r a c t e r i s t i c s w e r e o b s e r v e d , a n d a p o s s i b l e r e a s o n f o r t h e o b s e r v e d s h a p e o f e l e c t r o n - d e n s i t y p r o -f i l e s i s s u g g e s t e d . i i i TABLE OF CONTENTS page LIST "OF ILLUSTRATIONS v i i LIST OF SYMBOLS . x ACKNOWLEDGEMENTS .' r x i i i 1. INTRODUCTION 1 2. MICROWAVE REFRACTION TECHNIQUES FOR MEASURING ELECTRON-DENSITY PROFILES I N A LINEAR'DISCHARGE 5 2.1 I n t r o d u c t i o n . . . . . . . 5 2.2 D e s c r i p t i o n o f t h e Method 6 2.3 M o d i f i e d Methods o f Measurements 9 2.3.1 Lens Method 9 2.3.2 M u l t i p l e - R e f l e c t i o n Method 14 2.4 E x p e r i m e n t a l S e t u p . . . . 16 2.4.1 D i s c h a r g e A p p a r a t u s . . . . 16 2.4.2 M i c r o w a v e Setup 19 2.4.3 M e a s u r i n g P r o c e d u r e 22 2.5 E x p e r i m e n t a l R e s u l t s 23 2.5.1 P r e l i m i n a r y Measurements 23 2.5.2 R e s u l t s O b t a i n e d U s i n g M u l t i p l e -R e f l e c t i o n Method 24 2.6 Summary 27 3. RECONSTRUCTION OF ELECTRON-DENSITY PROFILES FROM MICROWAVE- REFRACTION DATA 29 3.1 R a y - T h e o r y A n a l y s i s 29 3.2 R e c o n s t r u c t i o n P r o c e d u r e s 31 3.2.1 U s i n g a P r o f i l e M o d e l 31 3.2.2 A n a l y t i c a l S o l u t i o n . . . . 38 3.2.3 S t e p - b y - S t e p Method U s i n g P i e c e w i s e -L i n e a r A p p r o x i m a t i o n 42 i v page 3.2.4 S t e p - T o y - S t e p M e t h o d U s i n g R e l a t i v e M e a s u r e m e n t s o f " D " 43 .3.3' E v a l u a t i o n o f R e c o n s t r u c t i o n M e t h o d s 48 3.4 E x p e r i m e n t a l P r o f i l e s 55 3. 5 Summary • .- 64 4. D E T E R M I N A T I O N OF E L E C T R O N - D E N S I T Y P R O F I L E S FROM MEASUREMENTS AT NORMAL I N C I D E N C E . 65 4.1 I n t r o d u c t i o n 65 4- 2 D e s c r i p t i o n o f M e t h o d 67 4.3 R e c o n s t r u c t i o n o f P r o f i l e s 72 4^3.1 U s i n g a P r o f i l e M o d e l 73 4.3.2 S t e p - b y - S t e p M e t h o d U s i n g P i e c e w i s e -L i n e a r A p p r o x i m a t i o n 77 4.3.3 U s i n g a M o d i f i e d P i e c e w i s e - L i n e a r A p p r o x i m a t i o n . . 80 4.4 A c c u r a c y 85 4.5 E x p e r i m e n t s 93 4.5-1 M e a s u r e m e n t s 93 4- 5.2 P r a c t i c a l C o n s i d e r a t i o n s 97 4.6 A n a l y s i s o f E x p e r i m e n t a l R e s u l t s . . . . . 101 4.7 V a l i d i t y o f R a y T h e o r y a n d E f f e c t o f C u r v a t u r e o n P h a s e M e a s u r e m e n t s 113 4.8 Summary 116 5. OTHER MEASUREMENTS 119 5.1 I n t r o d u c t i o n 119 5.2 R e f r a c t i o n M e a s u r e m e n t s a t V a r i o u s P r e s s u r e s a n d D i s c h a r g e V o l t a g e s . . 119 5- 3 L a n g m u i r • P r o b e S t u d i e s 125 5- 3.1 E x p e r i m e n t a l S e t u p 128 5.3.2 R e s u l t s 129 5.4 C o m p a r i s o n o f R e s u l t s O b t a i n e d b y U s i n g P a r a l l e l .and P e r p e n d i c u l a r P o l a r i z a t i o n 132 v 5-5 Effect of Altering Current Waveform on Afterglow Measurements 134 5.'6. Test for Uniformity of Electron Density i n A x i a l Direction of Plasma Column 137 5 .7 Measurements During the Current-Flow Period 140 6. CONCLUSIONS ' 145 APPENDIX I Calculation of "dD/dt" for,-a Piecewise-Linear P r o f i l e 148 APPENDIX I I Calculation of "K" for a Piecewise-Linear P r o f i l e 151 APPENDIX I I I Calculation of "K" for a Modified Piecewise-Linear P r o f i l e 154 APPENDIX IV Evaluation of "u" and "K" for Assumed Pr o f i l e s ; 157 APPENDIX V Refraction Data Obtained Under-Various Dis-charge Conditions I64 REFERENCES . 182 v i LIST OF ILLUSTRATIONS Figure • Page 2 . 1 Ray Path i n Inhomogeneous Plasma • 7 2 .2 Proposed Operation of Lens Method i n a Decaying Plasma 10 2 .3 Waveform of Signal Received by Lens Antenna 13 . 2 .4 Multiple Reflections and Received Waveform in a Decaying Plasma 15 • 2 . 5 Discharge Apparatus 17 2 . 6 Microwave Apparatus 20 2.7 "t/D" Diagram 25 2 . 8 Signals Received by A^ and A! 26 2 . 9 "t/D" Diagram 26 ' 3 . 1 Assumed P r o f i l e Model 33 3.2 Solution Curves for P r o f i l e Model, n Q / 0 36 3 . 3 Solution Curves for P r o f i l e Model, x 39 3 . 4 Piecewise-Linear Representation 45 3 . 5 Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using an Assumed P r o f i l e Model 50 3.6 Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using an Assumed P r o f i l e Model 52 3 . 7 Comparison of Actual and Reconstructed P r o f i l e s . 54 3 . 8 Comparison of Actual and Reconstructed P r o f i l e s . 56 3 . 9 "t/D" Diagram Obtained by Multiple-Reflection Method 57 3 . 1 0 Dependence of D on 0^ 58 3 .11 Electron-Density P r o f i l e s at t=3 • 50ms 59 3 . 12 Electron-Density P r o f i l e s at t=3.75ms 60 3 . 1 3 A n a l y t i c a l Representation of Experimental Data Using Least-Square Polynomial Approximation..... 62 v i i 3.14 P r o f i l e s Reconstructed 'by Using Piecewise-Linear Method 63 4.1 Hypothetical Electron-Density P r o f i l e 69 4.2 K as a Function of the Parameter "a" for the-•Profile n/n = (x/R) a 76 ' m x 1 4-3 Piecewise-Linear Approximation. . ."  79 4 . 4 K as a Function of the Parameter M for the P r o f i l e n(x) = Ax 2 + Bx 84 4.5 Comparison of Actual and Reconstructed P r o f i l e s Obtained by using a P r o f i l e Model 87 4 . 6 Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using Piecewise-Linear Approximation 90 4.7 Effect of P r o f i l e Shape on Size of F i r s t Seg-ment of Piecewise-Linear Approximation 92 4 . 8 Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using Modified Piecewise-Linear Approximation 94 4.9 Microwave Apparatus 96 4.10 Reflected and Transmitted Microwave Signals 98 4.11 Doubly-Reflected Wave. 100 4.12 'F as a Function of Time 103 4.13 F as a Function of Time at Several Frequencies.. 105 4 .14 Variation of Maximum Electron Density with Time Variation of 0 with Time, at 26.5&Hz 106 4 .15 Electron-Density P r o f i l e s at t=4.00ms. 108 4.16 Electron-Density P r o f i l e s Determined by Using Modified Piecewise-Linear Representation 109 4 .17 Comparison of P r o f i l e s from Normal-Incidence and Oblique-Incidence Measurements 110 4.18 Effect of Experimental Error on Reconstructed P r o f i l e 112 4.19 Phase Error Due to Ray Theory and Effect of Curvature of Wave-Theory Result 115 5.1 Electron-Density P r o f i l e s i n an Argon After-glow Plasma 122 v i i i 5.2 Normalized Electron-Density P r o f i l e s 124 5 . 3 Effect of Variable Ionic Mobility on P r o f i l e .Shape , 126 5.4 Electron-Density P r o f i l e s Determined from Probe Measurements..-..' • •• ••• 130 5 . 5 Waveforms of Received Signal at Two P o l a r i -zations 133 5 . 6 Operation of Spark-Gap Switch 136. 5 .7 Waveforms of Reflected Signal at Various Positions along the Discharge Tube . • 139. 5 . 8 Waveforms Obtained During Current-Elow Period.. 141 5-9 Waveforms Obtained During-Current-Elow and Afterglow Periods 141 I. 1 Piecewise-Linear Representation 148 I I . 1 Piecewise-Linear P r o f i l e 152 I H . 1 ' Modified Piecewise-Linear P r o f i l e 154 IV. 1 "K" as a Function of C r i t i c a l Density for Various Electron-Density P r o f i l e s 163 V. l "t/P" Diagram for Argon, Q = 30° 165 V .2 "t/P" Diagram, Q± = 40° 169 V .3 "t/P" Diagram, 9 ± = 50° . . 173 V .4 "t/P" Diagram, Q± = 6 0 ° 177 V .5 "t/P" Diagram for Nitrogen, 9 . = 40° 181 i x LIST OP SYMBOLS The following l i s t contains those symbols that have been used consistently throughout the thesis and gives the page on which the symbol f i r s t appears. In a few cases, p a r t i c u l a r l y i n deriva-tions, some symbols have been used to represent different quantities at different times and places. In each case, however, the symbol has been defined where introduced to avoid misinterpretation of i t s meaning. SYMBOL QUANTITY PAGE a Parameter, used with p r o f i l e model.... 32 A Surface area of Langmuir probe 128 p ° * c Phase v e l o c i t y of electromagnetic wave i n free-space 70 D Distance between entrant and emergent rays at oblique incidence 6 e Electronic charge 1 ' E(M) Complete e l l i p t i c i n t e g r a l , second kind 16 E(0|m) Incomplete e l l i p t i c i n t e g r a l , second kind 1 5 9 f Frequency . 70 F Frequency of amplitude v a r i a t i o n i n interference pattern of received microwave signal 70 F(0|m) Incomplete e l l i p t i c i n t e g r a l , f i r s t kind 160 h Depth of penetration at oblique incidence 6 i Net current flowing i n double-probe c i r c u i t 128 J (x) Bessel function of f i r s t kind, zero order 121 o k Boltzmann constant 128 K Factor r e l a t i n g phase s h i f t i n plasma to "free-space" quantities 69 K(M) Complete e l l i p t i c i n t e g r a l , f i r s t kind 16 m Density gradient : 44 m' Apparent density gradient 78 m Electron mass . 1 e m. .. Ion mass 128 1 M Parameter, determined from, experimental data... 83 n Electron-number density 1 n C r i t i c a l density at normal incidence '. 6 c o 0 n c C r i t i c a l density at oblique incidence 6 n Maximum density, at axis of plasma column 3-1 n Q Electron density at free-space/plasma boundary. 32 P Pressure i n discharge tube 123 r Radial distance from plasma axis... 121 R Radius of plasma column 31 5 Sin(9 i) 40 t Time, measured from the instant at which breakdown occurs 23 T g Electron temperature 128 u Velocity of plane located at c r i t i c a l density.. 70 V^ Potential applied across double probes 128 V g Characteristic energy of electrons.*.. 123 x c Spatial coordinate of c r i t i c a l layer 68 x Spatial coordinate of free-space/plasma boundary • 32 a Characteristic decay-time of electron density during the afterglow period...- 46 (3q Free-space phase coef f i c i e n t 69 6 Phase coefficient of plasma medium 68 P P(a) Gamma function 34 e Q P e r m i t t i v i t y of free space 1 Relative p e r m i t t i v i t y 2 Index of refraction 32 x i 9. A n g l e o f i n c i d e n c e • 6 \ F r e e - s p a c e w a v e l e n g t h 114 o •• —6 \x P r e f i x , m i c r o - , d e n o t i n g 10 16 / L ( a ) C o m b i n a t i o n o f g a m m a f u n c t i o n s 34 Yip ' P r o b e s a t u r a t i o n c u r r e n t , p e a k - t o - p e a k 128 0 P h a s e a n g l e 68 co A n g u l a r f r e q u e n c y 2 w A n g u l a r p l a s m a f r e q u e n c y 1 x i i ACKNOWLEDGEMENTS • I wish to express my sincere gratitude to Dr. M.M.Z. Kharadly, the supervisor of th i s project, for his continuous guidance and encouragement throughout the course of this work. I am grateful to the National Research Council of Canada for supporting this work through Grant A -3344 and Block Term Grant A - 6 8 . Also, I wish to acknowledge the award of a B.C. Hydro and Power Authority Graduate Scholarship, and a U.B.C. Graduate Scholarship. Thanks are due to Mr. C. Carter for designing the trigger unit used i n the discbarge apparatus, and to the s t a f f of the E l e c t r i c a l Engineering workshop for building many of the special components required i n this project. I am indebted to Mr. R. Parkinson and Mr. B. Wilbee for the use of their "full-wave" phase calculations which were used to plot the curves i n F i g . 4 . 1 9 . Also, I wish to thank Mr. R. Olsen and Mr. B. Wilbee for proofreading, and Miss A. Hopkins for typing t h i s thesis. F i n a l l y , to my wife Marian, I am grateful for help i n typing the preliminary draft, and for her patience during the writing of this thesis. x i i i 1. INTRODUCTION Plasma research has attracted much attention i n recent years due to various technological p o s s i b i l i t i e s of the plasma. One of the main factors responsible for the., rapid growth i n plasma research has been the quest for a method of co n t r o l l i n g the release of energy from thermonuclear fusion of l i g h t elements. Success i n this direction can be achieved only as a r esult of •detailed and. precise investigations of laboratory plasmas. There-fore, the role of plasma diagnostics i s an important one, since progress i n plasma research i s determined, to a large extent, by the stage of development of i t s measurement techniques and the adequacy of the accompanying theoretical interpretation. A comprehensive summary of some more common techniques for studying plasmas i s given i n a book edited by Huddlestone and Leonard Plasma diagnostics i s somewhat i n t e r - d i s c i p l i n a r y i n char-acter, borrowing i t s methods from many branches of science, including optics, spectroscopy, microwave technology, and f l u i d mechanics. Often, the methods are based on phenomena which are not d i r e c t l y related to i n t r i n s i c plasma properties, but depend on some well understood connection between such phenomena and the condition of the plasma. This thesis i s concerned with non-perturbing plasma mea-surements based on the interaction of high-frequency electromagnetic waves with the free charges of the plasma. The s i g n i f i c a n t parameter i n t h i s case i s the plasma frequency, co , which i n the simplest case P (2) of an i s o t r o p i c , cold, c o l l i s i o n l e s s plasma, i s given by (ne^ N e o m e / z where n i s the electron-number density e i s the electronic charge E q i s the permit t i v i t y of free space m i s the electron mass e When an electromagnetic wave propagates i n a plasma medium, the plasma appears as a frequency-dependent d i e l e c t r i c with a r e l a t i v e p e r mittivity, e , given by to 2 E (CO) = 1 - "I-to where co i s the angular frequency of the wave. Thus, i f co exceeds co , the wave w i l l propagate through the plasma, whereas i f co i s less than co , i t w i l l be attenuated. Since the plasma frequency of many laboratory plasmas l i e s i n the microwave region, many int e r e s t i n g measurements have been made i n thi s f r e -quency range. A detailed account of various microwave techniques (3) can be found i n a book by Heald and Wharton . Most of the tech-niques described can be c l a s s i f i e d into two categories: (i) reso-nant-cavity techniques and ( i i ) "free-space" beam techniques. Resonant-cavity techniques are useful at densities up to 11 -3 about 10 cm , above which the plasma frequency becomes so high that the necessary cavity dimensions become r e s t r i c t i v e l y small. The average density over the volume occupied by the plasma i s deduced from measurements of the s h i f t i n resonant frequency of the^ cavity due to the plasma. Free-space beam techniques, useful at densities up to about lO^cm -^, are preferred when the dimensions of the plasma are large i n comparison to a wavelength corresponding to the plasma frequency. Most of these techniques depend on measuring the phase change of a signal transmitted through the plasma, and give the integrated value of electron density along the path of propagation. However, they do not usually provide information about the s p a t i a l d i s t r i b u t i o n of electrons i n the plasma. Although a few data reduc-t i o n procedures^'^^ have been developed which allow some informa-tion about the density p r o f i l e s to be obtained, these depend on using very simple p r o f i l e models, and hence cannot provide a com-plete description of the electron-density p r o f i l e . Some of-these r e s t r i c t i o n s were avoided by applying an ( 6 ) oblique-incidence method to laboratory plasmas . It consisted of probing the plasma with a "narrow" beam of microwave radiation incident obliquely on the boundary of the plasma and i t s surrounding medium, and measuring the distance between entrant and emergent beams. This method was shown to be fe a s i b l e , and electron-density p r o f i l e s were obtained by using a theoretical model. Subsequent f 7) work has avoided the need for an assumed model, and the range of v a l i d i t y of the method has been more c l e a r l y defined. However, the improved method of reconstructing p r o f i l e s requires more data, and the o r i g i n a l experimental technique becomes tedious and inadequate for extensive plasma studies. The object of the present work i s to improve the experi-mental technique for implementing the oblique-incidence method, and to develop new microwave methods for determining accurate electron-density p r o f i l e s i n transient^non-uniform plasma columns. Chapter 2 of t h i s thesis i s concerned with developing an improved microwave ref r a c t i o n technique which w i l l allow the required • . 4 data to be obtained more e f f i c i e n t l y than by the previous technique. In Chapter 3, various methods for reconstructing p r o f i l e s from re f r a c t i o n data are described. Some of the exi s t i n g methods are modified i n order to make them more general, and a new method, based on a different interpretation of the same data, i s proposed. The methods are evaluated by applying them to hypothetical data .calculated for certain assumed p r o f i l e s . Also, electron-density p r o f i l e s are determined from experimental data obtained during the afterglow period of actual discharges i n argon. Chapter 4 describes a normal-incidence method for. deter-mining p r o f i l e s from measurements of Doppler s h i f t i n frequency of the signal reflected from a t o t a l l y - r e f l e c t i n g plasma. Several data-reduction procedures for determining electron-density p r o f i l e s from normal-incidence data are described and evaluated. Chapter 5 describes and discusses some results obtained from extensive r e f r a c t i o n measurements i n an afterglow' plasma for various discharge conditions. Also, several related measurements are carried out i n order to examine the v a l i d i t y of some assumptions made i n connection with the microwave methods. 2 . MICROWAVE REFRACTION TECHNIQUES FOR MEASURING ELECTRON-DENSITY PROFILES I N A LINEAR DISCHARGE 2.1 I n t r o d u c t i o n A l t h o u g h d i a g n o s t i c t e c h n i q u e s b a s e d on r e f r a c t i o n o f (8) r a d i o waves a r e w e l l known i n the f i e l d o f i o n o s p h e r i c p r o p a g a t i o n , d i r e c t a p p l i c a t i o n o f t h e s e t e c h n i q u e s t o l a b o r a t o r y p l a s m a s i s r e s -t r i c t e d by t h e s m a l l d i m e n s i o n s w h i c h a r e i n v o l v e d . However, by u s i n g a " n a r r o w " m i l l i m e t e r - w a v e beam i n c i d e n t o b l i q u e l y on t h e p l a s m a , K h a r a d l y ^ ^ showed t h a t r e f r a c t i o n measurements c o u l d be employed t o d e t e r m i n e t h e e l e c t r o n - d e n s i t y d i s t r i b u t i o n o f an i n h o -mogeneous p l a s m a . The method was a p p l i e d t o a c y l i n d r i c a l p l a s m a , w i t h t h e i n c i d e n t beam i n a p l a n e c o n t a i n i n g t h e a x i s o f the c y l i n -d e r , a l t h o u g h t h e method c o u l d a l s o be a p p l i e d i n t h e case where t h e p l a n e o f i n c i d e n c e i s t r a n s v e r s e t o t h e a x i s ^ ' ^ ^ . The method was o r i g i n a l l y . Implemented by u s i n g a s i m p l e e x p e r i m e n t a l t e c h n i q u e and the d a t a were r e l a t e d t o t h e e l e c t r o n - d e n s i t y p r o f i l e t h r o u g h a t h e o r e t i c a l model based on g e o m e t r i c a l o p t i c s ^ \ I n subsequent (7)" work , a d e t a i l e d d i s c u s s i o n on the l i m i t a t i o n s o f t h e method has been g i v e n , and a t e c h n i q u e of p r o f i l e a n a l y s i s has been d e s c r i b e d w h i c h a v o i d s the need f o r an assumed model o f t h e e l e c t r o n - d e n s i t y p r o f i l e . A l t h o u g h t h i s t e c h n i q u e a l l o w s more d e t a i l e d p r o f i l e s t o be d e t e r m i n e d , i t r e q u i r e s a d d i t i o n a l d a t a and hence t h e t e d i o u s and t i m e - c o n s u m i n g p r o c e d u r e of o b t a i n i n g d a t a by u s i n g t h e o r i g i n a l e x p e r i m e n t a l t e c h n i q u e becomes i n a d e q u a t e . I n the p r e s e n t w o r k , an a t t e m p t was made t o i m p r o v e t h e e x p e r i m e n t a l t e c h n i q u e s o ' t h a t t h e r e q u i r e d d a t a c o u l d be o b t a i n e d more e f f i c i e n t l y . 2 . 2 Description of the Method Referring to Fig. 2 . 1 , i t i s ea s i l y shown that the path of a ray incident obliquely on the boundary of an inhomogeneous plasma with a monotonically increasing density can be expressed • 4.V, -r ( 6 ) xn the form x P tan 0. J f ^ ^ C 2 . 1 ) 1 _ a i l i 0 n L c where 9 ^ i s the angle between the ray and the normal to the free-space/plasma interface n(x) describes the electron-density d i s t r i b u t i o n n i s the electron density at which t o t a l r e f l e c t i o n occurs c J The value of n depends on the angle of incidence and the applied c frequency, and i s given by 2 n = n cos 9 . c co x where n i s the c r i t i c a l electron density at which the ref r a c -co J t i v e index of the plasma i s zero ( i . e . , the plasma f r e -quency i s equal to the applied frequency). Thus, the ray w i l l penetrate the plasma to a depth "h", where n(h) = n . c The distance D between the entering and emerging rays i s then given by h D = 2 tan Q± j * ^ - y r ( 2 . 2 ) x 0 1 _ n(x) n c Pig. 2.1 ..Ray Path i n Inhomogeneous Plasma This equation relates the electron-density d i s t r i b u t i o n to the d i s -tance D which may be determined d i r e c t l y from experimental measure-(7) ments and forms the basis for various methods of p r o f i l e analysis . Since D also depends on the angle of incidence and on the probing frequency, the oblique-incidence method can be implemented, experi-mentally by measuring D as a function of the angle of incidence at a fixed f r e q u e n c y ^ \ or as a function of frequency at a fixed angle of i n c i d e n c e . In the case of a transient plasma*, the angle or the frequency i s used as a parameter and D i s measured as a func-* The term "transient plasma" i s used i n t h i s context to denote a plasma i n which the electron density changes with time but where the plasma as a whole i s stationary (no mass motion). t i o n of the time at which the received signal i s at a maximum. The experimental technique used o r i g i n a l l y involved using several angles of incidence and consisted of recording the signal received by a moveable antenna at various distances from the transmitting antenna. Both antennas were highly d i r e c t i o n a l horn-lens combinations located on the same side of the dishcarge tube, with t h e i r axes l y i n g i n a plane which contained the axis of the tube. For each position.of the receiving antenna, the discharge tube was f i r e d at least once, and the procedure was repeated for each angle of incidence. A drawback of t h i s method was that many discharges were required at each angle of incidence. Thus, reprodu-c i b i l i t y of the discharge was essential i n order for the experi-mental data to be r e l i a b l e . The method was also time-consuming and, therefore, not convenient for carrying out detailed investiga-tions of plasmas for a wide range of different conditions. Although interpretation of the data was generally easy, some ambiguities existed in the data taken at small separations between transmitting and receiving antennas^^, because direct r e f l e c t i o n s from the glass wall of the discharge tube tended to interfere with the reflected signal. The oblique-incidence method has also been implemented by using a fixed angle of incidence and a microwave beam with several frequency components'"^. In order to determine curves of distance against time, t h i s technique also required the discharge tube to be f i r e d for each position of the receiving device as i t was placed at various distances from the transmitting antenna. Although experi-mental data could be obtained more quickly than i n the previous case since only one angle of incidence was used, the experimental setup 9 was more complex, requiring microwave signals at several frequencies and a more elaborate receiving device for receiving the d i f f e r e n t signals. In this chapter, we are concerned with developing a simple experimental technique which w i l l allow s u f f i c i e n t data to define a complete curve of distance against time at a given angle of i n c i -dence to be obtained during a single discharge, while avoiding some of the d i f f i c u l t i e s associated with the previous methods. Two d i f -ferent techniques have been considered; the f i r s t involves using a large lens antenna for continuously monitoring the refracted micro-wave signal, while the second depends on multiple r e f l e c t i o n s within the discharge tube. 2.3 Modified Methods of Measurements 2 . 3 , 1 Lens Method This method was based on using a receiving antenna with an aperture large enough to detect the emerging beam at a l l r e l e -vant points along the discharge tube. Although a curved r e f l e c t o r or a lens antenna could be used for t h i s purpose, the lens was chosen because of i t s more lenient tolerances on surface shape, a d e s i r -able feature i n the millimeter-wavelength region where dimensional tolerances are quite severe. Absorbing s t r i p s were placed i n front of the lens to provide markers i n the waveform of the received s i g -nal i n order to i d e n t i f y the point at which the microwave signal emerged from the plasma. Proposed operation of the system i s i l l u s -trated i n F i g . 2.2. A lens was constructed from a polystyrene-based p l a s t i c foam with a r e l a t i v e d i e l e c t r i c constant of 1 . 6 . The front surface Fig. 2.2 Proposed Operation of Lens Method in.a Decaying Plasma of the lens was plane, 75 cm long and 10 cm wide, and the maximum thickness was about 25 cm at the lens axis. An H-plane sectoral horn with a phase-correcting lens i n i t s aperture was used at the foca l point of the lens. Pieces of microwave absorbing material were placed at several positions across the plane surface of the lens and the entire receiving system was supported i n a structure made from a very l i g h t polystyrene foam which was reasonably r i g i d , but not l i k e l y to.introduce spurious effects into the signal received by the lens. The antenna was set up adjacent to the discharge tube at some desired angle of incidence, with i t s axis l y i n g i n a plane that passed through the axis of the tube. The pattern of the s i g -nal received by the lens was measured by illuminating part of the lens aperture with a narrow microwave beam (obtained by using the same horn-lens combination normally used for the transmitting antenna) and recording the received signal strength as the horn was moved along a track p a r a l l e l to the discharge tube. Although the minima i n the received signal pattern caused by the absorbing s t r i p s were well defined and could be used to iden-t i f y the position of the microwave beam i n the above case, certain d i f f i c u l t i e s were encountered i n the oblique-incidence application. These were mainly caused by strong multiple r e f l e c t i o n s between the glass walls and the plasma c r i t i c a l l a y e r a t certain angles of incidence, and, to a minor extent, by large variations i n the amp-li t u d e of the signal emerging from different points along the tube... The l a t t e r made i t d i f f i c u l t to display the entire received signal on the oscilloscope screen with good resolution i n that portion of the trace where the signal was weak, while avoiding an "off-scale" deflection of the oscilloscope beam i n the region where the received 1 . signal was much stronger. Interference between multiple r e f l e c t i o n s and the main reflected signal produced extraneous dips i n the observed waveform which caused ambiguities i n the interpretation of results. Owing to the curvature of the glass walls, i t was found d i f f i c u l t , i n practice, to eliminate these r e f l e c t i o n s completely by using matching structures at the tube walls. However, since their amplitude was usually much smaller than that of the main s i g -nal i t was often possible to i d e n t i f y the minima which were due to absorption of the main signal by noting that t h e i r d-c l e v e l was usually much lower than that of the minima caused by interference. An oscillogram of the received waveform obtained during the afterglow period of an actual discharge i s shown i n Pig. 2.3a.* The two traces shown were obtained during separate discharges under the same conditions. A smoothed waveform was superimposed upon the actual trace to obtain an estimate of the waveform that might be obtained i n the absence of multiple r e f l e c t i o n s . This i s shown as a broken l i n e i n F i g . 2 . 3 b , which shows a small portion of the received waveform enlarged from the top trace • i n F i g . 2 . 3 a . By r e l a t i n g the minima i n t h i s hypothetical waveform to the positions of the absorbers i n front of the lens, several values of D as a function of time could be obtained. Points obtained on t h i s basis were not sharply defined, but were generally found to be i n reasonably good agreement with those obtained by applying the method used i n reference (6). However, the method was considered unsuitable for obtaining accurate measurements i n situations where multiple r e f l e c -tions could occur. * In this p a r t i c u l a r case, f i v e absorbing s t r i p s were placed i n front of the lens at points corresponding to D=2.7, 8.7, 16.1, 22.1, and 27.2 inches. The angle of incidence was 40° and the probing f r e -quency was 35.00GHz. The discharge was i n argon at a pressure of 30um Hg and at a discharge voltage of 3kV. A detailed description of the discharge apparatus and measuring procedure i s given i n Section 2.4. Fig. 2.3a Waveform of Signal Received by Lens Antenna Time scale, 500|is/cm Frequency, 35-OOOHz Q± - 40° Pressure, 30(im Hg Voltage, 3 k V 1 1 1 1 1 1 1 1 1 \ \\ / / / I / / / / / \ \ I \ I \ i i i i i \ 1 1 1 1 1 1 \\ \\ \ \ \ \ < // \ j V J i i i » J V j I ^ \ A \ \\ 1 1 1 1 1 1 • i i i i i i i 1 1 1 2.5 aO 3.5 U.O t. ms Fig. 2.3b Estimated Waveform in Absence of Multiple Reflections - - - Estimated Actual (enlarged from top trace i n Fig. 2 . 3 a ) 1.' 2 . 3 - 2 M u l t i p l e - R e f l e c t i o n Method T h i s t e c h n i q u e makes use o f t h e o c c u r r e n c e o f m u l t i p l e r e f l e c t i o n s . Owing t o d e n s i t y - g r a d i e n t s w i t h i n t h e p l a s m a , a n o b l i q u e l y - i n c i d e n t beam i s c o n t i n u o u s l y r e f r a c t e d by t h e p l a s m a u n t i l i t emerges a t some p o i n t on t h e b o u n d a r y , a d i s t a n c e D away f r o m t h e p o i n t o f e n t r y . A t t h i s p o i n t , p a r t o f t h e s i g n a l i s r e f l e c t e d b a c k i n t o t h e p l a s m a by t h e g l a s s w a l l o f t h e . d i s c h a r g e t u b e and t h i s wave p r o p a g a t e s a l o n g a p a t h s i m i l a r t o t h e p r e v i o u s one u n t i l i t emerges a t a d i s t a n c e 2D f r o m t h e p o i n t o f e n t r y , as i l l u s t r a t e d i n F i g . 2 . 4 . T h u s , a m i c r o w a v e beam may be r e f l e c t e d s e v e r a l t i m e s b e f o r e i t i s f i n a l l y d e t e c t e d by t h e r e c e i v i n g a n t e n n a . F o r a t r a n s i e n t p l a s m a , s e v e r a l s i g n a l s may be r e c e i v e d a t t h e same p o i n t , a t v a r i o u s t i m e s , c o r r e s p o n d i n g t o t h e m a i n and t h e m u l t i p l y -r e f l e c t e d s i g n a l s . I n t h e c a s e o f a d e c a y i n g p l a s m a , D i n c r e a s e s w i t h t i m e , and i d e n t i f i c a t i o n o f peaks i n t h e r e c e i v e d waveform w i t h t h e c o r -r e s p o n d i n g m u l t i p l e r e f l e c t i o n s i s s t r a i g h t f o r w a r d . The most d e l a y e d peak i n t h e waveform i s due t o t h e m a i n s i g n a l w h i c h i s r e c e i v e d d i r e c t l y , w h i l e t h e peaks o c c u r r i n g e a r l i e r a r e due t o m u l t i p l e r e f -l e c t i o n s . The v a l u e o f D f o r any g i v e n peak i s o b t a i n e d by d i v i d i n g t h e d i s t a n c e between t h e e n t r a n t and f i n a l emergent p o i n t s by t h e number o f t i m e s t h a t t h e beam i s r e f l e c t e d f r o m t h e p l a s m a c r i t i c a l l a y e r b e f o r e b e i n g d e t e c t e d . E x p e r i m e n t a l l y , i t was f o u n d t h a t , due t o v a r i o u s l o s s e s , . , s i g n a l s w h i c h were r e f l e c t e d more t h a n once a t t h e w a l l s o f t h e g l a s s tube were u s u a l l y t o o weak t o be d e t e c t e d . I n t h i s e x p e r i m e n t , m u l t i p l e r e f l e c t i o n s were enhanced by c o v e r i n g t h e g l a s s w a l l s o f t h e d i s c h a r g e tube w i t h m e t a l f o i l i n t h e r e g i o n between t h e t r a n s -l"5 F . i g . 2 . 4 M u l t i p l e R e f l e c t i o n s and R e c e i v e d Waveform i n a' D e c a y i n g P lasma mitting and receiving antennas. Depending on the separation between the antennas and on the length of the r e f l e c t i n g sheet, up to seven d i s t i n c t peaks could be observed i n some cases, although f i v e were more general. However, i t was occasionally necessary to reduce the number of multiple r e f l e c t i o n s by using a smaller r e f l e c t i n g sheet i n order to avoid interference between adjacent peaks. Additional information was obtained by using a second receiving antenna, A l , as shown i n Fig . 2.4. By a suitable choice of separations between the two receiving antennas and the transmitting antenna, a check on the accuracy of the data could also be obtained. The separations were usually chosen so that at least one multiply-reflected signal was received simultaneously by both antennas. As an example, for a r a t i o of separations of 3'4-, the t h i r d multiply-reflected signal was received by the f i r s t antenna at the same instant of time as the fourth was received by the second antenna. 2.4 Experimental Setup 2.4.1 Discharge Apparatus The plasma was produced by discharging a bank of capaci-tors between plane electrodes located i n a straight glass tube, as shown i n Fig . 2.5. Except for minor differences, the discharge apparatus was s i m i l a r to that described i n reference (12). The capacitor bank (l80uf, lOkV max.) was connected to the discharge tube by a copper s t r i p transmission l i n e , v i a an ig n i t r o n (type BK178) which was used as the main discharge switch. A trigger u n i t , which-produced pulses at intervals adjustable from a few seconds to over a minute, was used to trigger the ig n i t r o n and to i n i t i a t e the sweep of oscilloscopes used to display waveforms connected with the d i s -charge . G a s I n l e t C h a r g i n g R e s i s t o r - A A A -C a p a c i t o r B a n k I g n i t r o n /ft] ; T r i g g e r i n g C i r c u i t I n s u l a t i n g R i n g . G l a s s Tu be F i g . 2 . 5 D i s c h a r g e A p p a r a t u s 18 The discharge tube was of flameproof glass 30cm i n dia-meter and 150cm long, and was mounted with i t s axis perfectly ver-t i c a l . The e l e c t r i c a l connections to the discharge tube were made through two brass plates located at each end. The high-voltage conductor was connected to the upper end-plate which also served as an electrode. Gas was admitted into the system through a very small hole at the center of this plate. The lower end-plate con-tained a large port through which pumping took place, and i t also carried the bottom electrode, a f l a t brass plate 28cm i n diameter. Six brass rods, uniformly spaced around the tube, connected from the base plate to the return conductor of the transmission l i n e at the top of the tube. The return conductor was grounded; a l l ground connections i n the system were made to a common point i n order to, avoid ground loops i n the c i r c u i t . A stainless s t e e l resistance s t r i p , inserted i n the return conductor adjacent to the ground con-nection, allowed the waveform of the discharge current to be moni-tored by observing the voltage drop across the s t r i p . A capacitor voltage divider, connected across the transmission l i n e at a point adjacent to the tube, was used to measure the discharge voltage. The pumping system consisted of an oil-vapour d i f f u s i o n pump and a mechanical backing pump. During the experiments, gas (usually argon) was pumped continuously through the discharge tube. This served to remove impurities dislodged from the glass walls and the electrodes during the discharge. The flow rate and pressure i n the.system were controlled by adjusting a needle valve at the gas i n l e t and a baffle valve at the pumping port. Pressure i n the tube was measured with a P i r a n i gauge located just below the lower elec-trode. A second gauge measured the pressure i n the backing l i n e and provided information about the flow rate, since the capacity of the mechanical pump at various backing pressures was known. Also, t h i s reading provided a convenient reference when i t was desired to reproduce conditions i n the discharge tube i n order, to repeat certain experiments. Although various precautions were taken to keep the system "clean", i t i s recognized that the pres-ence of some impurities was unavoidable i n these experiments. How-ever, the main requirement of the system .for our par t i c u l a r measure-ments was re p r o d u c i b i l i t y of the discharge. In th i s connection, i t was found that by adhering to a careful and consistent operating procedure, extrememly good r e p r o d u c i b i l i t y could be obtained. 2.4.2 Microwave Setup Most of the measurements were carried out by using standard millimeter microwave components. In order to avoid stray pickup caused by large pulsed currents i n the discharge c i r c u i t , most of the measuring equipment was located inside a screened room. A typ-i c a l setup used for r e f r a c t i o n measurements i s shown i n Fig. 2.6. In some'experiments, i t was advantageous to use two antenna systems located on diametrically opposite sides of the discharge tube. The second system could be used for simultaneous measurements at a d i f -ferent p o l a r i z a t i o n , or the receiving aitenna only could be used to measure the signal transmitted through the plasma at oblique-inci-dence . The antennas used were optimum-gain pyramidal horns with" phase-correcting polystyrene lenses at thei r apertures. The aper-ture dimensions of the horns used i n the 8mm band were 7 . 5 x 6.0cm while those for the 4mm band were 3 . 5 x 2.8cm. The larger horns frequency meter ^ w attenuator sk? isolator klystron \\J directional coupler. power supply matching sections oscilloscope -w-detectors Pig. 2.6 Microwave Apparatus discharge tube reened room o were fabricated from sheet brass while the smaller ones were made by electroplating copper onto a stainless s t e e l form. The lenses were of the single-refracting-surface type, designed on the basis of geometrical optics so as to produce a constant-phase wavefront on the collimated side of the lens. To minimize r e f l e c t i o n s from the plane surface of the lens, and hence to reduce the p o s s i b i l i t y of standing waves i n front of the antenna-due to interaction with the plasma container or another antenna, this surface was matched with a simulated quarter-wavelength layer obtained by perturbing (13) the surface with l i n e a l grooves . Lenses were constructed with the corrugations oriented p a r a l l e l to the E - f i e l d vector. Several lenses with diff e r e n t groove dimensions were used i n order to ob-t a i n matching at various frequencies. The transmitting antenna was generally fixed near the top end 'of the discharge tube, with i t s aperture a few inches from the tube walls. The antenna axis lay i n a plane which passed through the axis of the discharge tube and midway between two adjacent return conductors of the discharge circuit,. A short section of f l e x -i b l e waveguide between the antenna and the waveguide feed allowed the angle of the horn to be adjusted to any desired value. Angles were measured by using a protractor which incorporated a s p i r i t l e v e l , thereby allowing each angle to be set independently of any other measurements. Angles of incidence set i n this manner were estimated to be accurate to within half a degree of the desired value. The receiving antenna was set below the transmitting an-tenna at the corresponding angle of incidence and with i t s axis l y i n g i n the same plane. For the purpose of comparison, some mea-surements were obtained by using the method of reference ( 6 ) , with a few minor differences i n the experimental apparatus. The r e c e i -ving antenna (a horn-lens combination i d e n t i c a l to that used for transmitting) was mounted on a carriage which was free to move along a track p a r a l l e l to the axis of the tube. .The position of the car-riage could be changed by means of an e l e c t r i c motor controlled remotely from inside the screen room. The control included a device which allowed the carriage to be moved i n precise increments of one inch. The receiving horn was connected to the return waveguide by a f l e x i b l e section which allowed the antenna to be moved over a r e l a -t i v e l y large range without changing any waveguide connections. Using th i s arrangement, the receiving antenna could be repositioned within the time i n t e r v a l between successive discharges, thereby speeding up the process of taking data. 2 . 4 . 3 Measuring Procedure Measurements involved detecting the received signal and display-ing i t on an oscilloscope. The trace was photographed with a 35imn oscilloscope camera. In most measurements the control c i r c u i t was adjusted to trigger the ignitron at 30 second int e r v a l s . The high-voltage power supply was set to charge the capacitor bank to the desired voltage i n t h i s i n t e r v a l . The oscilloscope was set up for single sweep operation and i t s time base was triggered a few micro-seconds before the discharge by a separate pulse from the control c i r c u i t . This started the sweep an instant before the discharge, thereby allowing the waveform of the microwave signal to be observed over the complete discharge. Times of various events connected with the discharge were measured from the instant at which breakdown 23 occurred. Although r e p r o d u c i b i l i t y of oscillograms was generally very good, about three oscillograms were usually recorded under the same-conditions i n order to avoid errors caused by the occasional e r r a t i c discharge. 2.5 Experimental Results 2.5.1 Preliminary Measurements Measurements were f i r s t carried out by applying the tech-nique used i n reference (6). Although this method i s tedious, i t can give good results over a large range of D, and therefore pro-vides a convenient reference with which results obtained by other methods can be compared. Waveforms of the received signal obtained during the afterglow period were sim i l a r to those shown i n reference (6). However, more detailed results were obtained i n t h i s case by measuring the refracted signal at one-inch increments along the discharge tube and by using a larger range for the angles of incidence (20° - 70°). The range of D over which measurements were made was usually from the- point of intersection of the antenna axes at the plasma-glass boundary ( i . e . , D=0) up to the point where a refracted signal could no longer be detected, except at large angles of i n c i -dence, where the l a s t refra.cted signal was sometimes obstructed by the bottom electrode of the discharge tube. In determining D, ref-raction of the beam by the glass walls of the discharge tube was accounted for on the basis of ray theory. Points obtained from the waveforms were used to plot "t/D"-curves, r e l a t i n g the time, t , at which the emerging signal was received, to the distance, D, between the incident and emergent beams. A t y p i c a l "t/D" curve, obtained at 9. = 40° and at a f r e -quency of 35.00GHz, i s shown i n F i g . 2.7. Although the data were generally well defined, some ambiguities occurred at small values of D, where direct r e f l e c t i o n s from the glass wall of the discharge tube interfered with the refracted signal. Several, points, corres-ponding to the various peaks i n the interference pattern, were plotted for each value of D i n t h i s region. 2.5.2 Results Obtained Using Multiple-Reflection Method The following results were obtained by using the multiple-r e f l e c t i o n method. Fig . 2.8 shows the received signa.l waveform obtained under the. same conditions as i n the previous measurements. The two traces in.the oscillogram were obtained simultaneously by using two separate receiving antennas, and.A^. The separation between the points where the axes of the transmitting and receiving antennas intersected the plasma-glass boundary was 18 inches for A^ (upper trace) and 24 inches for The most delayed peak i n each, tra.ce i s due to the main signal which i s received . d i r e c t l y . The other peaks are due to multiple r e f l e c t i o n s , and the value of D for any given'peak i s equal to the above distance divided by the number of times the beam i s reflected from the plasma c r i t i c a l layer before i t i s f i n a l l y detected. For example, the value of D corresponding to the t h i r d peak (counting from the right) i n the top trace i s 6 inches I t may be noted that the value of D for the fourth peak in the lower trace i s also 6 inches, and that the two peaks were received at the same instant of time. Points obtained from these waveforms were used to plot a "t/D" curve, shown i n Fig. 2.9. For the purpose of comparison, point obtained i n the preliminary measurements by using the f i r s t method 4.5 2.51 J 1 1 : 1 1 —I 0 . 5 10 15 20 25 30 D , i n F i g . 2.7 "t/D" Diagram ©i= 40° Frequency, 35.00GHz Pressure, 30um Hg Voltage, 3kV 26 Pig. 2.8 Signals Received by A 2 (upper trace) and A i Time scale, 500(is/cm Frequency, 35.00GHz g± _ 40° Pressure, 30um Hg Voltage,.3 k V 27 w e r e a l s o s h o w n i n t h i s d i a g r a m . T h e t w o s e t s o f r e s u l t s w e r e v e r y s i m i l a r , e x c e p t i n t h e r e g i o n c o r r e s p o n d i n g t o s m a l l v a l u e s o f D , w h e r e a r e l a t i v e l y l a r g e s c a t t e r w a s o b t a i n e d i n t h e p o i n t s d e t e r m i n e d b y t h e f i r s t m e t h o d . I n t h i s c a s e , t h e r e c e i v e d s i g n a l w a s n o t w e l l (7) d e f i n e d b e c a u s e i n t e r f e r e n c e c a u s e d b y b e a m o v e r l a p o b s c u r e d t h e s i g n a l ' s p e a k . T h i s d i f f i c u l t y , h o w e v e r , w a s n o t e n c o u n t e r e d b y t h e p r e s e n t m e t h o d , s i n c e t h e a c t u a l s e p a r a t i o n b e t w e e n t r a n s m i t t i n g a n d r e c e i v i n g a n t e n n a s w a s r e l a t i v e l y l a r g e . 2.6 S u m m a r y E x p e r i m e n t a l d a t a o b t a i n e d b y u s i n g t h e o r i g i n a l m e t h o d g a v e g o o d r e s u l t s a n d p r o v i d e d a c o n v e n i e n t r e f e r e n c e w i t h w h i c h t o c o m p a r e r e s u l t s o b t a i n e d b y o t h e r m e t h o d s . H o w e v e r , t h e m e t h o d h a d s e v e r a l p r a c t i c a l d i s a d v a n t a g e s . I n o r d e r t h a t e n o u g h d a t a c o u l d b e o b t a i n e d f o r p l o t t i n g a s i n g l e " t / D " c u r v e , t h e p o s i t i o n o f t h e r e c e i v i n g a n -t e n n a h a d t o b e c h a n g e d t e n s o f t i m e s a n d t h e d i s c h a r g e t u b e f i r e d a t e a c h p o s i t i o n . M e a s u r e m e n t s w e r e t h e r e f o r e t e d i o u s a n d t h e l a r g e n u m b e r o f d i s c h a r g e s c o u l d c a u s e a g r a d u a l d e t e r i o r a t i o n o f t h e d i s -c h a r g e a p p a r a t u s w h i c h , i n t u r n , c o u l d a f f e c t t h e a c c u r a c y o f t h e r e s u l t s . D a t a o b t a i n e d a t s m a l l s e p a r a t i o n s b e t w e e n t r a n s m i t t i n g a n d r e c e i v i n g a n t e n n a s w e r e u n r e l i a b l e b e c a u s e r e f l e c t i o n s f r o m t h e t u b e w a l l s i n t e r f e r e d w i t h t h e r e f r a c t e d s i g n a l . B o t h t h e l e n s m e t h o d a n d t h e m u l t i p l e - r e f l e c t i o n m e t h o d w e r e m o r e e f f i c i e n t a s t h e y e n a b l e d s e v e r a l p o i n t s o n t h e " t / D " c u r v e t o b e o b t a i n e d d u r i n g a s i n g l e d i s c h a r g e . H o w e v e r , p r a c t i c a l d i f f i -c u l t i e s w e r e e n c o u n t e r e d i n a p p l y i n g t h e l e n s m e t h o d d u e t o i n t e r -f e r e n c e c a u s e d b y m u l t i p l e - r e f l e c t i o n s . I t w a s f o u n d d i f f i c u l t i n p r a c t i c e t o e l i m i n a t e t h e s e r e f l e c t i o n s b y m a t c h i n g t h e t u b e w a l l s . T h i s w a s a l s o i n c o n v e n i e n t b e c a u s e t h e m a t c h i n g l a y e r s w e r e s e n s i t i v e to frequency and angle of incidence. Thus, the approach of using a receiving antenna such that i t s aperture covers the entire range of D.was not considered p r a c t i c a l i n situations where strong multiple re f l e c t i o n s can occur. Of the experimental techniques considered here, the multiple r e f l e c t i o n method was found to he the simplest and most accurate. The data were generally well defined and easily interpreted. More points were obtained than by the lens method, and these points were concen-trated i n the part of the "t/D" curve where the largest changes occurred. Unlike the f i r s t method, direct r e f l e c t i o n s from the wall of the discbarge tube did not affect the refracted signal and, there-fore, small values of D could be determined .more accurately. The high gain of the direc t i v e horn antennas allowed the measurements to be carried out at lower power levels than those required by the lens method. Also, the m u l t i p l e - r e f l e c t i o n method did not require access to the discharge tube over i t s entire length, a point which could be a s i g n i f i c a n t factor i n other experimental setups, although i t was not important i n t h i s particular experiment. 29-3. RECONSTRUCTION OF ELECTRON-DENSITY PROFILES FROM MICROWAVE REFRACTION DATA Electron-density p r o f i l e s can "be deduced from microwave (7) refraction data by several different methods- . Since a n a l y t i c a l solution of the equation which relates the electron-density d i s t r i -bution to experimentally measurable quantities i s usually very d i f -f i c u l t , most methods depend on c e r t a i n simplifying assumptions. One method involves using a two-parameter theoret i c a l model based on ray t r a c i n g ^ \ Although p r o f i l e s can be determined easi l y from data obtained at two angles of incidence, they do not, i n general, des-(7) cribe a plasma p r o f i l e accurately at a l l points. Another method , which avoids the need for an assumed model, depends on approximating the plasma by a series of laminae within which the electron density varies l i n e a r l y with position. In t h i s chapter, consideration i s given to: ( i ) generalizing the two parameter model to take account of various boundary conditions, ( i i ) applying c u r v e - f i t t i n g tech-niques to experimental data and determining p r o f i l e s by numerical integration, ( i i i ) using a different interpretation of microwave refraction data to reconstruct p r o f i l e s , (iv) evaluating the various reconstruction techniques by applying them to hypothetical data c a l -culated for certain assumed p r o f i l e s , and (v) applying the tech-niques to experimental data obtained i n a l i n e a r discharge. 3.1 Ray-Theory Analysis In t h i s analysis, the plasma w i l l be treated as a plane slab illuminated by plane waves. I t w i l l be assumed that ray theory can be used to describe .microwave propagation i n the plasma. A detailed discussion on the v a l i d i t y of these assumptions and t h e i r implications i n the oblique-incidence application i s given i n refer-30 ence ( 7 ) and w i l l not be considered here. It w i l l also be assumed that the plasma i s cold, i s o t r o p i c , and collision-free. In the sense that the distance travelled by an electron during one cycle of the microwave f i e l d i s n e g l i g i b l y small, the f i r s t assumption i s v a l i d . Since measurements are carried out at a time when no current flows through the plasma, and since the Earth's magnetic f i e l d i s extremely weak, the assumption that the plasma i s isotropic i s j u s t i f i e d . An approximate calculation of the c o l l i s i o n frequency of the plasma, using parameters t y p i c a l l y encoun-tered In these experiments, shows that t h i s frequency i s considerably less'than the very high microwave frequency used. Thus, the f i n a l assumption i s j u s t i f i e d i n t h i s case. The basic equation which describes the path of a ray i n c i -dent obliquely on the boundary of an inhomogeneous plasma with a monotonically Increasing density i s given by Eq. (2.1) and i s repeated here for convenience. x. p . tanO. y = dx (3.1) J r v ^ v ' o n c Providing that the maximum density of the plasma i s greater than n , c an incident ray w i l l penetrate.the plasma to a depth, h, such that n(h) = n c The distance between the entering and emerging rays Is then given by Eq. (2.2) which i s again repeated for convenience. h D = 2 tanO ± J d x , r (3.2) l 0 n c _ 231 Since the distance D may be determined d i r e c t l y from experimental measurements, t h i s equation forms the basis for various methods . of p r o f i l e reconstruction. The value of the electron density n Q i n Eq. (3-1) depends •on the angle of incidence and on the probing frequency according to the relationship n c = nco c o s 2 Q i ( 5 - 3 ) and, of course, i t i s equal to the c r i t i c a l density n c Q for zero angle of incidence. (Eor a cold c o l l i s i o n l e s s plasma, n c Q = 2 2 co e iDp/e , where co i s the applied angular frequency, m0 i s the electron mass, and e i s the electronic charge.) Because of this dependence on frequency and angle of incidence, p r o f i l e s may be deduced from measurements of D as a function of probing frequency (fixed angle) or as a function of the angle of incidence (fixed frequency). How-ever, the analysis i s s i m i l a r i n both situations,, and only the case where D i s known as a function of the angle w i l l be considered here. 3.2 Reconstruction Procedures 3.2.1 Using; a P r o f i l e Model The method of deducing p r o f i l e s used i n reference (6) involved representing the electron density i n terms of a p r o f i l e function, f, as follows n(x/R) = n m f(x/R) where n i s the maximum electron density at x = R m J f i s a monotonic function of x which i s zero at x/R = 0 and unity at x/R = 1. 32 Tv70 models were considered, namely f(x/R) = (x/R) a ( 3 . 4 ) and TCX \a 2Rj. f(x/R) = (sin ( 3 . 5 Substituting these functions f o r the electron d i s t r i b u t i o n into Eq. ( 3 . 2 ) and integrating gave expressions which related the d i s -tance, D, to the parameters, n^ and a. Taking the r a t i o of distances-corresponding to two angles of incidence allowed "a", and hence the p r o f i l e function, to be determined. (This result was approximate for p r o f i l e function ( 3 . 5 ) as the i n t e g r a l could not be evaluated i n closed form i n this case). was zero at the boundary, and i n applying the method to actual r e f -raction data, i t was assumed that this boundary was adjacent to the walls of the discharge tube. The p r o f i l e model w i l l now be modified to include the situations where: ( i ) ' the electron density has a f i n i t e value, n Q, at the boundary, as shown i n Pig. 3 . 1 a , ( i i ) the boundary i s a f i n i t e distance, x , from the walls of the discharge tube, as shown i n Pig. 3 . 1 b . ( i ) Modified Procedure, n Q / 0 In t h i s case, the electron-density d i s t r i b u t i o n w i l l be assumed to be of the form n = n Q + ( n m - n Q)f(x/R) as shov/n i n Fig. 3 . 1 a . Application of Snell's lav; of r e f r a c t i o n gives In that analysis, i t was assumed that the electron-density sin ©^ Pi s i n © v o r r\ s i n © ( 3 . 6 ) 33 F i g . 3 . 1 Assumed P r o f i l e Models (a) n o / -0 ( b ) x J O 34 w h e r e n = (1 - n / n ) 2 i s t h e i n d e x o f r e f r a c t i o n a t t h e p l a s m a O O CO r b o u n d a r y © r i s t h e a n g l e t h e r a y m a k e s w i t h t h e n o r m a l j u s t i n s i d e t h e b o u n d a r y . T h e e q u a t i o n o f t h e r a y p a t h i n t h e p l a s m a i s g i v e n b y x r t a n 0 d x y = I R E T T ( 3 . 7 ) J B . - A f (X/R )1 2 o • • n - n w h e r e A = — 2  n „ c o s © n. 1 o r c o . U s i n g t h e p r o f i l e f u n c t i o n g i v e n b y E q . ( 3 . 4 ) a n d m a k i n g t h e s u b s t i -t u t i o n ( x/R) = X g i v e s t h e f o l l o w i n g r e s u l t f o r t h e d i s t a n c e D , b e -tween t h e e n t e r i n g a n d e m e r g i n g r a y s . h/R D = 2Rtan© r J0 ( 1 - A X " } f d X B l ( 3 . 8 )  f V _ A Y ^ 2 T h i s e q u a t i o n i s o f t h e same f o r m a s t h a t o b t a i n e d i n r e f e r e n c e (6) f o r t h e same p r o f i l e m o d e l , b u t w i t h n Q = 0. T h e s o l u t i o n i s g i v e n b y D = 2Rtan© A - 1 ^ Z ( a ) Eta) = r < V a ) r ( 1 / 2 ) Ra + 2 ) a a n d w h e r e P d e n o t e s t h e gamma f u n c t i o n . S u b s t i t u t i n g f o r A a n d e x p r e s s i n g © r i n t e r m s o f ©^ g i v e s n 1/a D = 2 R s i n 0 . co 1 \ n - n V m o 1 1 5 _ 5 Z ( a ) ( 3 . 9 ) T h i s e q u a t i o n c o n t a i n s t h r e e unknown p a r a m e t e r s ; a , r \ o , and n f f l . These can be d e t e r m i n e d f r o m measurements o f D a t t h r e e d i f f e r e n t a n g l e s o f i n c i d e n c e , a c c o r d i n g t o t h e e q u a t i o n s and where D fb 1 D, K sin©-sin©. sin©. s i n 9 3 K - — - — a 2 1 s m 0 1 1 1 s m 9 , K • 2 n s m 0 2 • 2„ - s m 0 3 K (3.10) The above e q u a t i o n s can be s o l v e d g r a p h i c a l l y by p l o t t i n g s o l u t i o n c u r v e s (K as a f u n c t i o n o f D-^/T)^ and D^/D^) f o r a wide range o f p a r a m e t e r s . As an e x a m p l e , s u c h c u r v e s a r e g i v e n i n F i g . 3.2 f o r ©-^,©2, and ©^  e q u a l to 30°, 40°, and 50° r e s p e c t i v e l y and 2 f o r r^ o i n the range 0.75 t o 1.0. F o r g i v e n r a t i o s , D-^/D^ and ^ / D ( f r o m e x p e r i m e n t a l m e a s u r e m e n t s ) , two c u r v e s o f K as a f u n c t i o n o f 2 P| c a n be p l o t t e d by t a k i n g p o i n t s f r o m F i g . 3-2. The p o i n t a t 2 w h i c h t h e s e c u r v e s i n t e r s e c t d e t e r m i n e s K a n d q Q . The v a l u e o f n f f i c a n t h e n be c a l c u l a t e d f r o m E q . (3.9), and hence the p r o f i l e c a n be d e t e r m i n e d . ( i i ) M o d i f i e d P r o c e d u r e , x^ £ 0 x ' o The p r o f i l e may be r e p r e s e n t e d as f o l l o w s n = 0 f o r 0 < x <x Pig. 3.2 Solution Curves for P r o f i l e Model, n ^0 37 n n f1 m /x-x \ for x =s x =s R o I t i s convenient to s h i f t the coordinate system so that the point at which the ray enters the plasma coincides with the o r i g i n , as shown i n Fig. 3•lb, i . e . , X 1 = X - X o y' = y - x tan 9 . o 1 where (') denotes variables i n the new coordinate system-The problem then beccmes i d e n t i c a l to that considered i n reference ( 6 ) , where X q was zero. Thus, the equation .of the ray path i s given by X * tan©. dx' j r n f 0 1 - - a -n For the p r o f i l e function f ( § r ) = (§T) & = (X') a, D'is given by /n \ l / a D' = 2R' tan ©,( — ) £(a) iV n m ( 3 . 1 1 ) Expressed i n terms of the o r i g i n a l coordinates, Eq. ( 3 . 1 1 ) becomes 1/a D = 2 x tanO. + (R-x )( ^ ) £(a) ( 3 . 1 2 ) There are three unknown parameters i n the above equation, namely: a, x , and n . These can be determined from measurements of D at . o' m three d i f f e r e n t angles of incidence, according to the equations D , -2x tan©-, 1 o 1 D0-2x tan©0 2 o 2 tan©-, n n 1 c l tan9 0 n 0 2 c2 1/a 38 and D_-2x tan9 0 2 o _ D„-2x tan©, 3 o 3 tan© tan©-2 n n l / a Letting A = 2xo/I>2 and substituting n = n cos ©-, gives c co 1 to ;3 .13) and D-^tan92 D^tanQ-j^  D„tan9v 2 3 D„tan@0 3 2 ' D2 1 - —- A tan©-^  1 A tan©, 1 - A tan©. 1 A tan©.-, ^3 3 2„ cos Q 1 COS ©, 2^ COS ©, 1/ a cos 9 3 l / a (3 .14) For any three specified angles, solutions of Eq, (3 .14) may be plotted as a function of the r a t i o of distances, for various values of the parameter "A". A few such curves are shown i n Fig. 3 .3 for the angles 30°, 40°, and 50°, and for A i n the range from 0 . 0 to 0 . 2 5 . From these graphs, curves of A as a function of " l / a " may be plotted for any given r a t i o of distances. The solution for "A" and " l / a " i s determined from the intersection of two curves. The value of n m can then be calculated from Eq. ( 3 - 1 2 ) , hence determining the p r o f i l e . 3 . 2 . 2 A n a l y t i c a l Solution Equation (3 .2) can be expressed i n the following form D = 2 h I 0 sin©^ dx r \ 2 ( x ) - s i n 2 9 . (3 .15) I f D i s known as a function of 9^, r\(x) can be determined by inver-t i n g the above expression. By a suitable change of variables, the l / a 0 ! i 1 1 i ] 1 i i i H-H4-I i i 1 .' i 1 n 1 i ! ! ! ! ! ' i | ! i i -l - - 2 1 - A tanQ, .„ 1 - 2 A tan0 o d L w -f o r : 0^=30°, Q2=40°, 03=5O° cos Q, 1 :n a i f cos^0 o COS 0g lit: l j-f a 4—1 cos^9o — -1- \ 1 ! fr H I " XX. l I I ! I _LL T T T i l t T j j i i l l -UXLLiJ-illLUJU TfrjTr1"i j i ;J1I U 111 ! i i: i TTi — 1 1 2 3 4 F i g . 3.3 S o l u t i o n C u r v e s f o r P r o f i l e M o d e l , x /O 40 above equation can be written i n a form s i m i l a r to the Abel in t e g r a l (7) equation-, and inversion of the in t e g r a l yields the result 1 = I f ° ( S > ( 3 . 1 6 ) where S = sinO. I To determine the position of a layer with a given density, i t i s necessary to determine D(S) experimentally over the whole range of integration and evaluate Eq. (3.16) numerically. Repeating this pro-cedure for various densities allows a point-by-point reconstruction of the p r o f i l e . However, there are certain p r a c t i c a l d i f f i c u l t i e s which occur i n applying this method. One d i f f i c u l t y i s that the range of integration includes a region where experimental results are d i f f i c u l t to obtain. Con-sequently, some estimate about the shape of the D(O^) curve i s re-quired i n t h i s region. Experimentally i t was found that D could be determined with reasonably good accuracy at angles of incidence i n the range from about 20° to 7 0 ° . Outside this range, measurements were re s t r i c t e d by the f i n i t e size of the antennas and by poor d e f i -n i t i o n of the maxima i n the waveforms of the received signal. At large angles of incidence, i t was found i n the experiments that D tended toward zero as 9^ increased.** It w i l l be assumed i n t h i s analysis that D(90°) = 0 and that D(9^) in the unknown region may * The solution of this equation, i n an analogous problem concerned with determining ionospheric density d i s t r i b u t i o n s , i s given by Manning ( 1 4 ) . ** Using equation ( 3 . 2 ) , D was calculated as a function of 9j_ for several hypothetical electron-density'profiles. In the cases considered, i t was always found that, at 9j_ = 9 0 ° , D was zero i n those cases where the slope of the p r o f i l e function was f i n i t e at the boundary, and D was f i n i t e when the slope was zero at the boundary. be approximated by extending the experimental curve to zero at 0 i = 90°. Another d i f f i c u l t y i s that numerical integration of Eq. (3.16) requires that the experimental data be represented by a con-tinuous function D(S), while i n practice, D i s known only at discrete values of 0^. A method for handling this s i t u a t i o n , i n an analogous problem^^, involved approximating the experimental curve with two or three s t r a i g h t - l i n e segments. The int e g r a l could then be evaluated a n a l y t i c a l l y i n each l i n e a r region. In the present work, the p o s s i b i l i t y of applying some c u r v e - f i t t i n g technique to the experimental data i n order to obtain more accurate results i s con-sidered. (15) In one case, Newton's divided-difference formula was used to obtain a unique polynomial such that the r e s u l t i n g curve passed through every experimental point. A disadvantage of t h i s method was that the a n a l y t i c a l curve was sometimes o s c i l l a t o r y i n the region between experimental points, whereas i t was found experimentally that D usually varied monotonically with 0^. Since the degree of the polynomial was one less than the number of experimental points, using a larger number of points resulted i n polynomials with coef-f i c i e n t s of extremely large magnitude. Consequently, round-off errors became s i g n i f i c a n t , even when performing the calculations by computer. This technique was therefore-considered unsuitable i n the present application. The second c u r v e - f i t t i n g technique considered involved using the method of l e a s t - s q u a r e s ^ ^ - t o obtain a polynomial repre-sentation of the experimental data. The degree of the polynomial could be specified and there was no r e s t r i c t i o n on using a large number A of experimental points. The data t y p i c a l l y consisted of about th seven or eight points and i t was found that polynomials of 5 b or th 6 ' degree generally gave smooth curves which came close to the experimental points. These curves were very similar to those obtained by manually drawing smooth curves through the points. In view of the satisfactory results, obtained by applying t h i s method, the additional complexity of using more sophisticated c u r v e - f i t t i n g techniques was not considered worth-while, i n t h i s application. Furthermore, the points used i n the D ( 9 ^ ) curve were taken from t/D diagrams, which had already been smoothed by drawing smooth curves through a large number of experimental points. Having described the experimental data a n a l y t i c a l l y , the i n t e g r a l i n Eq. (3.16) can be evaluated numerically. However, the choice of numerical integration techniques which can be used i s r e s t r i c t e d because the integrand i s singular at the lower l i m i t . Although there are several ways of handling s i n g u l a r i t i e s within the (17) range of integration , i t was found convenient i n t h i s case to use a technique which did not "blow-up" at the singularity. Integra-t i o n was performed numerically by using Gaussian integration formu-(18) las . The numerical integration technique was tested by substi-tuting the exact function, D(S), for certain assumed p r o f i l e s into Eq. ( 3 .16), and evaluating the i n t e g r a l numerically.* I t was found that no s i g n i f i c a n t error was introduced by using t h i s integration technique. 3 . 2 . 3 • Step-by-Step^Method Using Piecewise-Linear Approximation This method i s described i n reference (7), and i s included here mainly for comparison purposes. B a s i c a l l y , the method involves * The i n t e g r a l was evaluated numerically by using a computer program, based on the Gauss 16-point integration formula, which was a v a i l -abl  from the U.B.C. Compu ing Centre. representing the plasma by a series of laminae within which the electron density varies l i n e a r l y with position. I f a ray enters the plasma from free space at some angle of incidence which results i n t o t a l r e f l e c t i o n at the boundary between the f i r s t and second lay-ers, the distance, D^ , between entrant and emergent rays i s given by = 4h-^  tan 9-^  where h-^  i s the thickness of the f i r s t layer. The density at the boundary i s related to the c r i t i c a l density and the angle of incidence according to Eq. (3.3), i . e . , 2 -n-, = n cos 9-, 1 co 1 I f the angle of incidence is. adjusted so that the ray-now penetrates two layers, the corresponding distance, E^, can be expressed as a function of h^, h^, and Using h.-^  from above, together with exper-imental values of and Q^, the thickness of the second layer, , can be found. S i m i l a r l y , using h-^  and h^, h^ can be found from experimental values of and 0^, and the procedure continues step-by-step u n t i l a l l the layers are determined. This method has the advantages that the need for an assumed model is avoided and that the calculations involved are s t r a i g h t f o r -ward and may be done without the aid of a d i g i t a l computer. The method i s s l i g h t l y r e s t r i c t e d i n that reconstruction of p r o f i l e s i s based on the assumption that the electron density i s zero at the plasma boundary and that t h i s boundary i s adjacent to the walls of the d i s -charge tube. 3.2.4 Step-by-Step Method Using Relative Measurements of "D" The previously described methods for reconstructing p r o f i l e s 44 f r o m m i c r o w a v e r e f r a c t i o n d a t a h a v e b e e n b a s e d o n m e a s u r i n g t h e d i s t a n c e , D , b e t w e e n t h e p o i n t s where a m i c r o w a v e beam e n t e r s a n d e m e r g e s f r o m t h e p l a s m a . The p r e s e n t s e c t i o n d e s c r i b e s a m e t h o d f o r d e t e r m i n i n g p r o f i l e s i n a t r a n s i e n t p l a s m a w h i c h d e p e n d s o n m e a s u r i n g t h e r a t e a t w h i c h D c h a n g e s , d D / d t , d u e t o m o t i o n o f t h e p l a n e a t w h i c h t o t a l r e f l e c t i o n o c c u r s . R e c o n s t r u c t i o n o f p r o f i l e s i n v o l v e s u s i n g a s t e p - b y - s t e p p r o c e d u r e b a s e d o n a p i e c e w l s e - l i n e a r r e p r e s e n -t a t i o n o f t h e p l a s m a , s i m i l a r t o t h a t u s e d i n t h e a b o v e m e t h o d . C o n s i d e r a p l a s m a w i t h a p i e c e w i s e - l i n e a r d e n s i t y v a r i a -t i o n i n w h i c h t h e d e n s i t y i n c r e a s e s m o n o t o n i c a l l y w i t h d i s t a n c e a n d i s e q u a l t o z e r o a t t h e b o u n d a r y . U s i n g t h e s u b s c r i p t " i " t o d e n o t e t h q u a n t i t i e s r e f e r r i n g t o t h e i l i n e a r l a y e r , t h e d e n s i t y v a r i a t i o n i n t h i s r e g i o n may be d e s c r i b e d a s f o l l o w s : n . ( x ) = n . ^ + m . U - x . ^ ) (3.1?) w h e r e x i - l ^ s ^ e s P a " t i a l c o o r d i n a t e o f t h e b o u n d a r y b e t w e e n t h e ( i - l ) a n d i ^ * 1 l a y e r s n^_2_ i s t h e e l e c t r o n d e n s i t y a t t h i s b o u n d a r y t h i s t h e d e n s i t y g r a d i e n t i n t h e i l a y e r . A r a y i n c i d e n t o b l i q u e l y o n t h i s p l a s m a a t a n a n g l e o f i n c i d e n c e 0-^, s o t h a t i t p e n e t r a t e s t o a p o i n t h-^ l o c a t e d w i t h i n t h e f i r s t l a y e r , w i l l e m e r g e a t a . d i s t a n c e f r o m t h e p o i n t o f e n t r y , a s i l l u s t r a t e d i n F i g . 3 . 4 . T h e c o r r e s p o n d i n g c r i t i c a l d e n s i t y a t t h e r e f l e c t i n g p l a n e , n T_I 1 8 g i v e n b y E q . ( 3 - 3 ) . The v a l u e o f o b t a i n e d b y s u b -s t i t u t i n g f o r n ( x ) f r o m E q . (3.17) i n t o E q . (3 .2) a n d i n t e g r a t i n g , i s g i v e n b y D1 = 4tan G± n^/n^ ' (3.18) y A 0 h-j x-j h2 *2 x 3.4 Piecewise-Linear Representation In a transient plasma, the position of the plane at which the incident ray turns around changes with time, and hence, D also changes. I f the time dependence of electron density i s known, the rate at which D changes can be determined by substituting t h i s r e l a -tion into the expression for D and d i f f e r e n t i a t i n g with respect to time. The v a r i a t i o n of density with time can be determined experi-mentally from microwave measurements at normal incidence , as w i l l be discussed i n Chapter 4- In the present analysis, i t w i l l be assumed that the electron density decays exponentially with time and uni-formly across the plasma cross section. (This type of dependence i s t y p i c a l of the afterglow period i n many plasmas.) The electron density w i l l be expressed i n the form n(t) = n ( t o ) e " a ( t _ t o ) ( 3 . 1 9 ) where t i s some arbit r a r y reference time a i s determined experimentally from a plot of density against time. Substituting for the above time dependence into Eq. (3.17) and d i f -f e r entiating with respect to time gives a general expression for the rate at which the density gradient changes, i . e . , dm. The rate at which changes can be obtained by d i f f e r e n t i a t i n g Eq. (3-18), and i s given by dD d(l/ro 1) dt = ^tanG-L = 4 tan Q± n ^ / i ^ (3.20) 47 Since dD^/dt can be determined e x p e r i m e n t a l l y (by measuring the s lope of a " t / D " c u r v e ) , the d e n s i t y gradient i n the f i r s t l a y e r can be determined from E q . ( 3 . 2 0 ) . A s i m i l a r r e l a t i o n s h i p e x i s t s between dD^/dt and f o r a ray which penetrates i n t o the second l a y e r before being r e f l e c t e d . In general, f o r a region c o n s i s t i n g of n " l i n e a r " l a y e r s , the va lue of dD /dt i s r e l a t e d to the d e n s i t y g r a d i e n t i n the n^1 l a y e r , m , a c c o r d i n g to dD = 4 n t a n 9 dt c n A ' + n 2m / n -, . (2--e) n / n -i \ ( 3 . 2 1 ) where A' = . n n-1 2 : . -, m. i = l l i n . , / n 1 - 1 ' c n . /n i ' c The r e s u l t i s d e r i v e d i n Appendix I . The s t e p - b y - s t e p procedure f o r r e c o n s t r u c t i n g p r o f i l e s i s as f o l l o w s : R e f r a c t i o n data obtained at an angle of i n c i d e n c e 9-^, corresponding to a dens i ty n are used to determine m-^ . A l i n e a r segment w i t h slope m-^  i s used to approximate the p r o f i l e i n the f i r s t l a y e r . U s i n g data obtained at a s l i g h t l y s m a l l e r angle of i n c i d e n c e ©2, corresponding to a d e n s i t y ^-C2> ^ n e s lope i s c a l c u l a t e d by s u b s t i t u t i n g i n t o E q . ( 3 . 2 1 ) , i . e . , 2cm 0 t a n 9 0 c2 2 m. V nc2J [} ~ ^ ) C^r ~ 4 n c 2 t a n G 2 A A ) and where n-^  i s the density at the boundary between the f i r s t and second layers. The value of n-^  l i e s i n the range n c T _ < : n ] _ n c 2 ' a n (^ a reasonable choice for n^ i s n l = ^ ( n c l + n c 2 } The second step i n the p r o f i l e i s approximated by a li n e a r segment with slope EL-, which extends from n^ to n 2, where n 2 = i ( n c 2 + n c 3) and n - corresponds to the c r i t i c a l density at the next smaller angle 0.2 of incidence, 9 , . 3 By repeating t h i s procedure for data obtained at decreasing angles of incidence, the remaining steps i n the p r o f i l e are recon-.structed. 3•3 Evaluation of Reconstruction Methods The above methods for determining p r o f i l e s w i l l be evalu-ated by applying them to hypothetical data calculated for certain assumed electron-density p r o f i l e s . These "data" are calculated by substituting the assumed p r o f i l e function into Eq. (3.2). For con-venience, i t i s assumed that n = n at x = R. ' co m The method based on using an assumed model w i l l be applied to a "sinusoidal" p r o f i l e . In one case, i t i s assumed that the elec-tron density at the boundary i s n = 0.2n , and the p r o f i l e i s des-j J o m e cribed by 49 'n(x) = n + (n -n ) sin (jtx/2R) v o ra o The value of the integral i n Eq. (3.2) for t h i s p r o f i l e can he ex-pressed i n series form sim i l a r to that given, i n reference (6), except that 9^  i s replaced by 9 , where sin9. 9 = s i n \ C O / In this case, "data" are calculated by using the f i r s t seven terms of the series, at points corresponding to 9^  = 20°, 40°, and 60°. Applying "modified procedure ( i ) " gives the p r o f i l e shown as a bro-ken l i n e i n Fig. 3.5. Although the value of n Q i n the reconstructed p r o f i l e i s not very accurate, the overall shape of the p r o f i l e com-pares reasonably well with the assumed p r o f i l e over a large region. For comparison purposes, p r o f i l e s were also determined from the same "data" by using the two-parameter models described i n reference (6). One model was of the form n=n m(x/R) a, while the second was of the form n=n msin a(itx/2R). Since these models require data at two f r e -quencies only, three p r o f i l e s can be obtained by using different com-binations of the above "data". Only the two extremes are plotted to show the region within which these p r o f i l e s l i e . A second case i s considered i n which the electron density i s assumed to be zero for a f i n i t e distance, x , from the boundary. The p r o f i l e function considered i s n = 0 for 0 <x <x M x - x )\ n = n sin -TTTS v for x $x s;R m \2(R-x )/ . o o In calculating "data" i n t h i s case, i t is'convenient to s h i f t the n/n O . C o 0.2 0.4 0.6 x/R 0.8 1.0 F i g . 3.5 Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using an Assumed P r o f i l e Model Actual p r o f i l e , n=n 0+(n m-n 0)sinUx/2R), Reconstructed p r o f i l e s obtained by using: ( i ) Modified model, W-0, using data at e i =20°, 40°, 60° Two-parameter model, n = n m ( x / R ) a , usin? data at: a) 9,=20°, 4 0 o b) 01=40°, 60° Two-parameter model, n=n msin a(jtx/2R) , data ( i i ) ( i i i a) 9i=20°, 40c b) 0,=40°, 60° coordinate system so that x' = x-x and y ! = y-x QtanG i. The cor-responding values of D' are then calculated i n the same manner as i n the case X q = 0. The required values of D for this problem are obtained by simply adding the term "2x —tan©^" to the values of D'. It i s assumed i n this problem that x ~ 0.2R, and the values of D ^ o are calculated at 9^ = 30°, 45°, and 60°. Using "modified procedure ( i i ) " gives the p r o f i l e shown as a broken l i n e i n Fig. 3.6. It i s seen that the value o f ' x determined by t h i s method i s larger than the actual value, and that the overall p r o f i l e does not compare very well with the assumed one. P r o f i l e s obtained by using the two-parameter model of reference (6) are also shown i n Fig. 3.6. One of these p r o f i l e s i s concave i n shape, while the actual p r o f i l e i s convex. It may be noted that the value of n or x , as determined J o o ' . by the modified procedure, cannot-be used as a r e l i a b l e measure of the actual condition at the plasma boundary. Rather, this parameter gives the assumed model an extra degree of freedom which allows i t to provide a better description of the p r o f i l e i n the region where t o t a l r e f l e c t i o n of the incident beam occurs. The calculated value of this parameter is. largely a consequence of the form of the assumed model. For example, i n t h i s case, the basic model i s of the form (x/R) a. In order that t h i s model may provide a reasonable approxi-mation to a p r o f i l e which i s convex i n shape, the parameter "a" must be less than unity. Therefore, for small values of "x/R" v(i.e., near the boundary), the reconstructed p r o f i l e has a steep gradient which decreases rapidly with distance. I f , on the other hand, the density gradient of the actual p r o f i l e i s approximately uniform near the boundary, then the calculated value of n Q w i l l tend to be less than the actual value (in the case of f i n i t e n ), or the calculated value ( i i ) a / / / / / s —"* / ^^^^ ^ ( _ i i i j a - -• ^ ' ( i i i ) b * § .' * / / / • / . / / / # / f • / //' / / 1 / / / '•'/ / / / / / / /// / / / / ' 1 1 1 1 1 1 0 0.2 0M ^ 0.6 x/R 0.8 1.0 . 3.6 Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using an Assumed P r o f i l e Model Actual p r o f i l e , n = s i n ( ^ ^ ° 2 \ X-0=0.1R •x, Reconstructed p r o f i l e s obtained by using: ( i ) Modified model. x 0^0, data at ©i=30°,45°,60 ( i i ) Two-parameter model, n=n m(x/R) a, data at: a) 0-^30°, 45° b) Oi= 4 5 ° , 60° ( i i i ) Two-parameter model, n=n msin a(jtx/2R), data . . a) 01=30°, 45° b).9 i=45°, 60° of x w i l l be greater than the actual value ( i n the case of f i n i t e o u x ). This was found to be true when the method was applied to hypo-t h e t i c a l data calculated for a sinusoidal p r o f i l e . The method based on calculating the p r o f i l e by.evaluating the a n a l y t i c a l expression i n Eq. (3.16) numerically i s applied to "data" calculated for the following p r o f i l e s : n = n sin(:itx/2R) m ' n = nffi (x/R) 2 "Data" are calculated at angles of incidence corresponding to nine values of n , spaced by equal intervals of 0.1 n.CQ. These data are expressed a n a l y t i c a l l y by using a least-square polynomial approxima-th tion of 7 degree. Substituting this function into Eq. (3.16), and evaluating the p r o f i l e point-by-point, by using numerical integration, gives the results shown i n Fig . 3.7 (denoted by crosses). I t i s seen that the computed points agree very well with the assumed p r o f i l e s i n these cases. (7) Applying the piecewise-linear method to the same "data" gives the results shown i n Fig . 3.7 (denoted by c i r c l e s ) . I t i s seen that this method also provides a very good approximation to the actual p r o f i l e . Results obtained by this method are s l i g h t l y more accurate than those obtained by.numerical integration i n the case of the sinusoidal p r o f i l e , while the opposite i s true for the parabolic p r o f i l e . Although the accuracy obtained by both methods i s comparable, i t may be noted that the piecewise-linear results are simpler to c a l -culate. The calculations are straightforward and can be done manually, while the numerical integration technique requires a d i g i t a l computer to be p r a c t i c a l . 54 1.0 0.8 0.6 n/n m OM 0.2 7 y ( i ) / 7 o U o / o 0 0.2 0.8 1.0 0.4 0.6 x/R F i g . 3.7 Comparison of Actual and Reconstructed P r o f i l e s ( i ) . Assumed p r o f i l e , n=n sin(:tx/2R) . ( i i ) Assumed p r o f i l e , n=nffl(x/R)2 x Points obtained by numerical integration o Points obtained by piecewise-linear method The method based on measurments of "dD/dt" i n a transient plasma i s applied to a p r o f i l e of the form, n = n f f l(x/R) 2. An expo-nential time dependence i s assumed, and the "data" are calculated by d i f f e r e n t i a t i n g the expression for D(9^,t) with respect to time. A nine-layer model i s used, and the resulting p r o f i l e i s shown in Fig. 3 -8 (curve i i ) . The reconstructed p r o f i l e provides a reasonably good approximation to the actual p r o f i l e i n t h i s case. A p r o f i l e reconstructed by applying the piecewise-linear method i s also shown in Fig. 3 . 8 (curve i i i ) . Although i t was usually found that the pre-sent method was not quite as accurate as the piecewise-linear method, thi s method i s useful for providing an additional check on experi-mental p r o f i l e s , since i t i s based on a different interpretation of refractio n data. 3 . 4 Experimental P r o f i l e s Using the improved oblique-incidence method described i n Chapter 2 , r e f r a c t i o n data were obtained i n an argon afterglow plas-ma, at several angles of incidence. The corresponding "t/D" curves are shown i n Fig. 3 . 9 - The same data are replotted i n Fig. 3 . 1 0 to show the r e l a t i o n between D and 0^ at certain instants of time. The data for the method based on the rate at which D changes were obtained by measuring the slope of the "t/D" curves, while the time dependence of electron density was determined from normal-incidence microwave measurements which w i l l be described i n Chapter 4 . Using data at t = 3 . 5 0 and 3 .75ms, electron-density pro-f i l e s were determined by applying the various methods described i n this chapter, and the result i n g p r o f i l e s are plotted i n Figs. 3-11 and 3 - 1 2 . Data used for obtaining p r o f i l e s by numerical integration 56 1.0 0.8 0.6 n/nm OA 0.2 S* // A • • 0.0 0.2 OU 0.6 x/R 0.8 1.0 F i g . 3.8 Comparison of Actual and Reconstructed P r o f i l e s '.—'. ( i ) Actual p r o f i l e , n=n m(x/R) 2 .—- ( i i ) Reconstructed from "dD/dt" data ' ( i i i ) Reconstructed by piecewise-linear method 5 59 1.50 1.25 1.00 n 1 n 1 3 -3 x 10 cm 0.75 0.50 025 0.0 0 ( i i ) a •/ ( i i ) b 7 i v ) / / * / y /// / y/ / t / / / / r I / 1 I IV / ••' // / / '/' 1 ill 1 i r 1 /'1 i ! ! > \ HI; In) if I f j 0 6 9 x , cm .12 15 F i g . 3.11 Electron-Density P r o f i l e s at t=3.50ms Obtained by using: ( i ) Modified p r o f i l e model, x /0, data at 9_= 30°,45°,60° ( i i ) Two-parameter model, n=r (x/R) , data at: a) 9 i = 3 0 ° , 4 5 ° b) 9^ 4 5 ° ,60° ( i i i ) Numerical integration of a n a l y t i c a l expression • (iv) Piecewise-linear method 1.50 ( i i )a..-'"' 0 3 6 9 12 . 1 5 x , cm Pig. 3.12 Electron-Density P r o f i l e s at t=3.75ms . ;• Obtained by using: ( i ) Modified p r o f i l e model, x ^ 0 , data at 9i=30°,45°,60° ( i i ) Two-parameter model, n=n vx/R) a, data at: a) 9^30°,45° m b) 9i=45°,60° ( i i i ) Numerical integration of a n a l y t i c a l expression . (iv) Piecewise-lihear method ' (v) Step-by-step method, using "dD/dt" were represented a n a l y t i c a l l y by using a 6 degree least-square polynomial approximation. The a n a l y t i c a l curves are compared with the actual data i n Fig. 3.13. The piecewise-linear. p r o f i l e was ob-tained by drawing a smooth curve through a nine-layer model. I t i s interesting to note that p r o f i l e s obtained by using a four-layer model i n t h i s case agreed very well with the more accurate nine-layer model, as i l l u s t r a t e d i n Fig. 3-14. P r o f i l e s based on an a assumed p r o f i l e model were determined from data at 9^  = 30°, 4 5 ° , and 60°. It i s seen that p r o f i l e s reconstructed by using numerical integration, by using the piecewise-linear method, and by the method based on measuring "dD/dt", are a l l very similar to each other. Since i t was shown i n the previous section that these methods can provide an accurate description of a p r o f i l e without introducing any spurious effects, i t i s reasonable to expect that the p r o f i l e s shown here provide a very good approximation of the actual electron-density d i s -t r i b u t i o n i n the plasma. The method based on using the modified model gave results which were i n reasonable agreement with the above results, over a f a i r l y large region of the p r o f i l e , although the results differed near the boundary. This method appeared to indicate that the plasma boundary was a f i n i t e distance from the walls of the discharge tube. However, as was explained i n the previous section, this value cannot be considered r e l i a b l e , as the method tends to give a false indication of X q for p r o f i l e s of the shape encountered i n these experiments. P r o f i l e s obtained by applying the two-pararneter model were also found to provide a reasonable approximation i n certain regions of the pro-f i l e , but could not provide an accurate overall description. It may V 3.50 m s " \ t = 3 . 7 5 m s 20 30 40 50 60 9 j , deg rees 70 80 90 Pig. 3 . 1 3 A n a l y t i c a l Representation of Experimental Data Using Least-Square Polynomial Approximation _ A n a l y t i c a l curve X x Experimental points 1.25 x , c m F i g . 3 . 1 4 P r o f i l e s Reconstructed by Using 'Piecewise-Linear Method . : Nine-layer model : Four-layer model, using data at 9.=30°, 40°, 50 50° 1 be noted that the region i n which these p r o f i l e s are most accurate appears to be i n the v i c i n i t y of the point where the incident beam i s t o t a l l y reflected.' 3.5 Summary 1. Modified P r o f i l e Model, n , x / 0 Although t h i s method i s based on using a th i r d parameter to include the effect of f i n i t e boundary conditions, the calculated values of n and x do not provide a true in d i c a t i o n of the actual o o ^ conditions at the boundary. Rather, t h i s parameter provides the p r o f i l e function with an extra degree of freedom which appears to result i n a s l i g h t l y better approximation of the p r o f i l e i n the regi.on where t o t a l r e f l e c t i o n of the probing beam occurs. 2• A n a l y t i c a l Solution This method can give reasonably good results. However, even i n the simple form used here, the calculations are cumbersom.e and require a d i g i t a l computer. 3• Step-by-Step Method, Using Piecewise-Linear Approximation This method was found to be accurate and simple to implement 4. Step-by-Step Method, Using Relative Measurements of "D" This method gave results quite similar to the above method, although s l i g h t l y less accurate i n the cases considered. I t i s not as desirable as the above method because i t requires additional data about the time dependence of the electron density. However, because i t i s based on a different interpretation of re f r a c t i o n data, i t can be useful to check results obtained by other methods. c 65 4 . DETERMINATION OF ELECTRON-DENSITY PROFILES FROM MEASUREMENTS AT NORMAL INCIDENCE 4.1 Introduction The s p a t i a l d i s t r i b u t i o n of electrons i n a plasma can be determined quite accurately from microwave re f r a c t i o n measurements, as discussed i n Chapters 2 and 3 - However, the oblique-incidence technique has certain shortcomings i n that i t i s rest r i c t e d to a line a r plasma which i s uniform along i t s length, and a large access to the plasma region i s required, i n order to obtain the required data. Although the second requirement can be reduced by using the improved technique described i n Chapter 2 , i t would obviously be of great advantage to determine plasma p r o f i l e s without this r e s t r i c t i o n . The size of the probing "window", can be greatly reduced by probing the plasma with a microwave beam, incident normally on the free-space plasma boundary, which i s considered to be perpendicular to the d i r -ection i n which the plasma density varies. Such normal-incidence techniques are well known, and a detailed account of the more common (3) ones Is given, i n a book by Heald and Wharton . Most of'the tech-niques are interferometric and depend on measuring the sh i f t i n phase cf the electromagnetic wave due to the plasma. They do not usually provide information about the l o c a l density of electrons at points within the plasma, but give only the integrated value along the path of the microwave beam. However, several methods have beer developed • for extending these techniques so as to obtain some information about the s p a t i a l density d i s t r i b u t i o n , and a b r i e f summary i s given below. One m e t h o d r e q u i r e s data about the phase s h i f t of the signal transmitted through the plasma at several frequencies and a simultaneous measurement of the peak electron density at the center 66 of the plasma column. The experimental values are compared with calculated curves of phase s h i f t plotted as a function of peak density for various hypothetical density p r o f i l e s , and the p r o f i l e model which f i t s the experimental, points best i s used to estimate the true d i s t r i b u t i o n . Another method , r e s t r i c t e d to situations where the plasma i s immersed i n a strong uniform magnetic f i e l d , involves propagating two waves of different polarizations over the same path and measuring the phase shi f t of each wave. This technique has the advantage that i t i s somewhat more sensitive than the pre-vious one and that more information i s obtained at each frequency. However, i t also requires that a very simple model of the p r o f i l e be adopted, and hence, can only provide a qu a l i t a t i v e estimate of the (19) p r o f i l e shape. A t h i r d method ; uses a multibeam microwave i n t e r - . ferometer focussed so t h a t p a r a l l e l beams pass through different regions of the plasma cylinder. The mean density i n each zone through which the focussed beam passes i s calculated on the basis of the measured phase changes i n the in d i v i d u a l interferometer beams, and the s p a t i a l d i s t r i b u t i o n i s deduced from these results. However, refraction of the beams which do not pass through the cylinder axis i s not accounted for. In certain situations (e.g., when the maximum density at the center of the plasma column i s . near the c r i t i c a l den-s i t y ) , refraction, of the "offset" beams can be considerable, and forms the basis of another diagnostic t e c h n i q u e ^ ' ^ . Therefore, i t i s doubtful that the results obtained by this method are meaningful i n such situations. Except for the special technique exploiting a strong mag-netic f i e l d , the above methods are r e s t r i c t e d to plasmas where the plasma frequency corresponding to the maximum density i s less than the probing frequency ( i . e . , the plasma i s transparent to the i n c i -dent radiation). When the applied frequency i s lower than the maximum plasma frequency, the incident wave cannot•penetrate through the plasma and i s t o t a l l y reflected at some point within the plasma. Determination of density p r o f i l e s i n th i s case has been considered on the basis of measuring the absolute phase difference between i n c i -dent and reflected "waves at several different f r e q u e n c i e s ^ 2 ^ . In practice, however, absolute phase differences cannot be measured as accurately as relative variations i n phase difference. This chapter describes a method for determining p r o f i l e s i n a transient plasma from r e l a t i v e phase measurements.. The method i s based on measuring the rate of change i n phase of the reflected signal due to the Doppler s h i f t i n frequency produced by the motion of the plasma c r i -t i c a l layer. Subject to certain assumptions about the plasma, i n f o r -mation about the electron-density d i s t r i b u t i o n i s deduced from phase data obtained at several frequencies. 4 . 2 Description of the Method The analysis i s carried out for the planar case i n which the plasma i s treated as a slab illuminated by plane waves. Although most experimental situations involve c y l i n d r i c a l geometry, th i s approxima-tio n i s reasonable when the diameter of the plasma Is large compared to the free-space wavelength of the probing signal. This requirement was generally satisfied, i n our particular experiment. It i s also assumed that the electron density does not change appreciably within a distance of one wavelength. However, even for a slowly varying p r o f i l e , the propagation characteristics of the plas-ma change abruptly i n the region where the density reaches the c r i t i -c a l value, i . e . , where the plasma frequency i s equal to the applied f r e q u e n c y . When a n i n c i d e n t w a v e i m p i n g e s u p o n t h i s a n o m a l o u s r e g i o n , i t i s s t r o n g l y r e f l e c t e d a n d i t s e x t e r n a l b e h a v i o u r i s v e r y s i m i l a r t o t h a t o f r e f l e c t i o n f r o m a s h a r p l y b o u n d e d h i g h - d e n s i t y p l a s m a , p r o v i d e d t h a t t h e maximum d e n s i t y i s s u f f i c i e n t l y g r e a t e r (21) t h a n t h e c r i t i c a l d e n s i t y . S u b j e c t t o t h e s e c o n d i t i o n s , r a y t h e o r y c a n g e n e r a l l y be u s e d t o d e s c r i b e p r o p a g a t i o n o f t h e wave i n t h i s s i t u a t i o n , a n d w i l l be a s s u m e d v a l i d i n t h i s a n a l y s i s . C o n s i d e r a p l a s m a s l a b i n w h i c h t h e e l e c t r o n d e n s i t y i n c r e a s e s m o n o t o n l c a l l y f r o m z e r o a t t h e b o u n d a r y , . x = 0, t o a m a x i -mum:, n ^ , a t x. = R , a s s h o w n i n R i g . 4.1. T h e e l e c t r o n - d e n s i t y d i s -t r i b u t i o n - i n t h i s p l a s m a i s c o n v e n i e n t l y r e p r e s e n t e d i n t e r m s o f a p r o f i l e f u n c t i o n , f , a s f o l l o w s 1 : ; n ( x / R ) = n m f ( x / R ) . (4.1) w h e r e f i s a m o n o t o n i c f u n c t i o n o f x w h i c h i s z e r o a t x / R = 0 a n d u n i t y a t x / R = 1. A n e l e c t r o m a g n e t i c wave i n c i d e n t n o r m a l l y o n t h e p l a s m a f r o m f r e e s p a c e (x<0) w i l l b e t o t a l l y r e f l e c t e d i f t h e maximum d e n s i t y , n , i s g r e a t e r t h a n t h e c r i t i c a l v a l u e , n c Q . The p h a s e c h a n g e o f t h e wave a s i t t r a v e l s f r o m t h e b o u n d a r y , x = 0, t o t h e c r i t i c a l l a y e r , x = x c a n d - b a c k t o . x = 0, i s g i v e n b y x c 0 = 2 J p p d x + 0 Q (4.2) 0 w h e r e (3 i s t h e p h a s e c o e f f i c i e n t o f t h e p l a s m a 0 = Jt/2 i s t h e p h a s e c h a n g e a s s o c i a t e d w i t h r e f l e c t i o n a t t h e c r i t i c a l l a y e r S i n c e we a r e i n t e r e s t e d i n r e l a t i v e p h a s e d i f f e r e n c e s , t h e t e r m 0 Q i s ' u n i m p o r t a n t i n t h i s a n a l y s i s a n d w i l l be o m i t t e d . 69 n .1 F i g . 4.1 Hypothetical Electron-Density P r o f i l e In a cold c o l l i s i o n l e s s plasma, 6^  i s given by Pp = ?o 1 - (w /(d)' p •(4.3) where P Q i s the f ree-space phase c o e f f i c i e n t 0) i s the angulai" plasm.a frequency w i s the angular, frequency of the probing signal. Rewriting Eq. (4.3) i n terms of the electron-density distribution gives (4.4) 6 (1 - n(x)/n ) Ko co Substituting Eq. (4.4) into (4.2) and omitting 0 gives the result 1 n{xl n co dx (4.5) Let us now define a factor, K, which depends only on the electron-density d i s t r i b u t i o n and the c r i t i c a l density as follows: K - A . 0 1 n(x) n co 2 dx (4.6) 0 can then be written as (4.7) In a- transient plasma, motion of the c r i t i c a l layer w i l l 0 = 2B K x ^ Ko c p r o d u c e a D o p p l e r s h i f t i n t h e f r e q u e n c y o f t h e r e f l e c t e d w a v e . I n m o s t c a s e s t h e v e l o c i t y o f t h e c r i t i c a l l a y e r i s much s m a l l e r t h a n t h e f r e e - s p a c e v e l o c i t y o f t h e w a v e . T h u s , t h e f r e q u e n c y o f t h e r e f -l e c t e d s i g n a l i s v e r y n e a r l y e q u a l t o t h e i n c i d e n t f r e q u e n c y , a n d h e n c e PQ - P 0 and K'-K, w h e r e ( ' ) d e n o t e s q u a n t i t i e s w h i c h r e f e r t o t h e r e f l e c t e d w a v e . The r e l a t i o n s h i p b e t w e e n , t h e r a t e a t w h i c h t h e p h a s e o f t h e r e f l e c t e d s i g n a l c h a n g e s a n d t h e v e l o c i t y o f t h e c r i t i c a l l a y e r i s t h e n o b t a i n e d b y d i f f e r e n t i a t i n g E q . (4.7), and i s g i v e n b y f | = 2B x | f + 2p K ^  (4.8) d t M o c d t " o d t I t i s now a s s u m e d t h a t t h e s h a p e o f t h e p r o f i l e c h a n g e s s l o w l y w i t h t i m e ( a r e a s o n a b l e a s s u m p t i o n i n many p h y s i c a l s i t u a t i o n s ) . O v e r a s m a l l i n t e r v a l o f t i m e , K i s t h e n n e a r l y c o n s t a n t , a n d E q . (4.8) may b e a p p r o x i m a t e d b y §| = 2 6 = 2 B K u (4.9) w h e r e u = d x / d t i s t h e v e l o c i t y o f t h e c r i t i c a l l a y e r . S u b -c s t i t u t i n g 6 Q = 2 i t f / c , w h e r e f i s t h e f r e q u e n c y o f t h e i n c i d e n t wave a n d c I s t h e p h a s e v e l o c i t y o f t h e wave i n f r e e s p a c e , g i v e s || = 4* f K u / c (4.10) S i n c e b o t h u a n d K d e p e n d i m p l i c i t l y on. t h e p r o f i l e s h a p e , t h e r a t e a t w h i c h t h e p h a s e o f t h e r e f l e c t e d s i g n a l c h a n g e s i s r e l a t e d t o e l e c t r o n - d e n s i t y d i s t r i b u t i o n . I f t h e r e f l e c t e d s i g n a l a n d a r e f e r -e n c e s i g n a l f r o m t h e s o u r c e a r e d e t e c t e d s i m u l t a n e o u s l y i n a s q u a r e -l a w d e t e c t o r , t h e o u t p u t w i l l p r o v i d e a m e a s u r e o f t h i s p h a s e c h a n g e . T h e f r e q u e n c y , E, o f t h e a m p l i t u d e v a r i a t i o n i n t h e o u t p u t i s r e l a t e d to the rate of change of 0 by F = 1 d0 2it dt Substituting from Eq. ( 4 . 1 0 ) into ( 4-U) gives F = 2 f K u/c * (4.H) ( 4 . 1 2 ) I t i s easy to determine F i n an in t e r v a l of time from experimental measurements. It i s now desired to relate u to the p r o f i l e shape. In this case i t i s convenient to represent the electron d i s t r i b u t i o n i n terms of the inverse function, g, as follows : x/R = g(n/n ) m' ( 4 . 1 3 ) where g i s a monotonic function of n which i s zero at n/n = 0 , o m and unity at n/n = 1. 0 ' m From Eq. ( 4 . 1 3 ) , by differentiation, we get 1 cbc R dt u R _d dt ' nco N n=n co ( 4 . 1 4 ) Substituting for u and K i n Eq. (4. '10) gives the relationship d$ _ 4*Rf d c . r 1 - n(x) dt cx c dt J n CO ox ( 4 . 1 5 ) * The s i m i l a r i t y between Eq. ( 4 . 1 2 ) and that obtained by considering the Doppler s h i f t i n frequency of a signal reflected from an object moving i n free space i s evident. For example, i n the case of an object moving away from a stationary source at a veloc i t y u, the frequency of the reflected signal i s given by f 1 =f (c-u)/(c-i-u). The difference i n frequency between t h i s signal and the incident s i g -nal i s F=f-f' =2uf / C H -U ) , and when u «c, F^2uf/c. This result i s iden t i c a l to Eq. ( 4 . 1 2 ) except for the factor K which i s due to the plasma. Thus, the rate at which the phase cf the reflected signal changes depends on the shape of the p r o f i l e and on the time dependence of the electron density. The v a r i a t i o n of electron density with time may he determined experimentally i n a straightforward manner from ( 2 2 ) microwave transmission-attenuation measurements by observing the time at which the maximum density reaches cutoff ( i . e . , n m = n c Q) at several frequencies. The v a r i a t i o n of density as a function of time and the absolute density at the center of the plasma column are ob-tained from a plot of n m against time. In th i s particular experiment i t was found that during a large portion of the afterglow period, the electron density decayed approximately exponentially with time. Since this i s t y p i c a l of the l a t e r part of the afterglow period for many plasmas, an exponential time dependence w i l l be assumed i n this anal-y s i s . Thus, the electron density can be expressed i n the form n(t) = n ( t 0 ) e " a ( t - t o ^ (4.16) where t i s some arbit r a r y i n i t i a l time used as a reference a i s determined from a plot of n against time. r m 0 . Other situations with a dif f e r e n t time dependence may be handled s i m i l a r l y by substituting the corresponding relationship i n place of the exponential one. 4 . 3 Reconstruction of P r o f i l e s For any particular d i s t r i b u t i o n , the rate at which the phase of the reflected signal changes may be obtained from Eq. ( 4 . 1 5 ) by substituting the appropriate relationship for the time dependence of the electron density and the value of n /n corresponding to the u co m . applied frequency. However, our present sit u a t i o n i s one i n which' 73 the p r o f i l e function i s unknown, and the only measurable quantities are the phase change and the time dependence of n . Three methods for obtaining p r o f i l e s from the measured data, based on different assumptions about the p r o f i l e function,, are described below. 4 . 3 . 1 Using a P r o f i l e Model One p o s s i b i l i t y i s to approximate the plasma p r o f i l e by a simple function with unknown parameters which may be determined from phase data. In this analysis, the form of the p r o f i l e function used i s the same as one previously used i n deducing p r o f i l e s from refraction d a t a ^ \ namely f(x/R) = (x/R) a ( 4 . 1 7 ) Although t h i s function i s incapable of representing a physical plasma adequately since i t s slope i s discontinuous at x/R = 1, i t has the advantage that the parameter "a" can be determined easily from experimental data at two frequencies. This particular p r o f i l e may bo described i n terms of.the inverse function as follows: g(n/n m) = ( n / n m ) l / a (4.18) The v e l o c i t y of the c r i t i c a l layer, u, may be obtained by substitu-ting the inverse function into Eq. ( 4 . 1 4 ) . Assuming an exponential time dependence of the form described by Eq. (4.16) gives the result u = ^ (n /n ) l / a • ( 4 . 1 9 ) a co m v J Now, u i s also related to phase data according'to Eq. ( 4 . 1 2 . ) , i . e . , u = cF/2fK Equating these two expressions gives E = 2aRfK ca n co l/ a n ( 4 . 2 0 ) m Since F i s d i r e c t l y measurable, and n^ i s obtained from a plot of density against time, "a" and K are the only unknown quantities i n Eq. ( 4 . 2 0 ) . Although K i s not e x p l i c i t l y known, i t i s a function of the p r o f i l e shape and the position of the c r i t i c a l layer. From Eq. ( 4 . 6 ) , K i n this p a r t i c u l a r case i s given by x K = — x c l K [' - f t ) ' dx Substituting X = x/x gives 1 K • J 0 1. - X a 2 dX ( 4 . 2 1 ) Thus, for p r o f i l e s of this form, K i s a function of the parameter "a" only and i s independent of x (and hence, independent of the ra t i o n /n • co' ir The value of "a" i s then readily evaluated from ex-perimental data. I f F^ and F^ are the experimental values obtained from measurements at two frequencies, f-^ and f ^, by using Eq. ( 4 . 2 0 ) we obtain F, -F v ^ + 1 ( 4 . 2 2 ) "a" i s therefore given by a 2 log ( f 1 / f 2 ) log (F 1f 27F^f " 7 ( 4 . 2 3 ) Thus, the pi-ofile function i s determined In the above case, the value of n was determined from transmission-attenuation measurements at higher frequencies than those used to determine the p r o f i l e function. In situations where sources of high enough frequency are not available., an approximate value of n may be calculated from Eq. ( 4 . 2 0 ) by using the experimental value of "a" obtained at lower frequencies and a calculated value of K. This approach ess e n t i a l l y assumes that the electron density obeys the same time relationship over a long period of time. The general expression for K i s determined below. Substi-tuting s i n 2 y = X a i n Eq. ( 4 . 2 1 ) leads"to j t / 2 K = | I cos 2y s i n z y dy ( 4 - 2 4 ) a 0 where z = — - 1. a. Equation. ( 4 . 2 4 ) can be reduced to i c / 2 K = J S i n Z y dy (4'25) 0 For z > - 1 ( i . e . , a > 0 ) , the i n t e g r a l may be expressed i n terms of gamma functions. Thus, Eq. ( 4 . 2 5 ) becomes KffLi r(f • * or, i n terms of "a", ^ ( 4 . 2 6 ) K = a+2 r(i +i \a 2 K i s plotted as a function of the parameter "a" i n Fig. 4 . 2 . The procedure for determining the electron-density d i s t r i -bution i n this case i s as follows: From phase measurements at two frequencies, "a" i s evaluated according to Eq. (4.23) • The corres-ponding value of K i s either calculated from Eq. (4.26) or determined d i r e c t l y from Fig. 4.2. Substituting the values of K and a into Eq. (4.20) for either frequency yields n and hence the electron-density d i s t r i b u t i o n . 4.3.2 Step-by-Step Method Using Piecewise-Linear Approximation Although the above method for determining p r o f i l e s i s easy to apply, the simple model used to represent the plasma cannot, i n general, provide a complete description of the p r o f i l e . In order-to avoid the a r t i f i c i a l r e s t r i c t i o n imposed by an assumed model, the present method involves using a step-by-step procedure for recon-structing, p r o f i l e s from data obtained at several frequencies. The method depends on using a piecewise-linear approximation of the pro-(7) f i l e i n which the plasma i s represented by a series of laminae, within which the electron density varies l i n e a r l y with distance and i s continuous at the boundary between adjacent layers. Reconstruction of the p r o f i l e s i s based on r e l a t i n g the vel o c i t y of the c r i t i c a l layer to the density gradient at the c r i t i c a l layer. The accuracy of the reconstructed p r o f i l e s may be improved by increasing the num-ber of laminae i n the piecewise-linear representation. The v e l o c i t y of the c r i t i c a l layer may be expressed i n terms of the density gradient, dn/dx, at the c r i t i c a l layer as follows , d(n/n ) _ dx ' m u " d(n/n ) ' dt (4.27) n=n co Using m to denote the gradient at the c r i t i c a l layer gives dn m = T~ dx _ nco nm !~d / 1 u d t i n T t T n=n L \ ra' co • (4.28) For an exponential time dependence, Eq. (4.28) gives the result m = n o:/u ( 4 . 2 9 ) co Substituting for u from Eq. ( 4 . 1 2 ) gives m = 2n af K/cF . . ' ( 4 . 3 0 ) co We s h a l l now define an apparent density gradient, m', which may be determined completely from experimental data, as follows: m' = 2 n c Q a f / c F ( 4 - 3 1 ) The actual density gradient i s then related to the apparent gradient according to ' m = Km' ( 4 . 3 2 ) Although K i s s t i l l unknown, since i t depends on the electron d i s t r i -bution i n the plasma, i t s value can be estimated on the basis of the piecewise-linear approximation. It w i l l also be assumed, that the density varies monotonically with distance and i s equal to zero at the plasma boundary, as shown i n F i g . 4 . 3 . Numerical subscripts are used to i d e n t i f y various quantities with the layer to which they refer. According to ray theory, an electromagnetic wave incident nor mally on the free-space/plasma boundary, at a frequency which corres-ponds to the c r i t i c a l density w i l l penetrate into the f i r s t laye u n t i l i t i s reflected, from a plane located at the point where the density i s equal to the c r i t i c a l value. Since the density v a r i a t i o n i s l i near throughout the entire region i n which this wave propagates, the corresponding factor K, evaluated from Eq. ( 4 . 6 ) , i s equal to 2/3 79 The d e n s i t y g r a d i e n t , ra^, a t t h e c r i t i c a l l a y e r may t h e n be d e t e r -m i n e d f r o m e x p e r i m e n t a l d a t a a c c o r d i n g t o E q . ( 4 . 3 0 ) . 1^ ~2 E i g . . 4 - 3 P i e c e w i s e - L i n e a r A p p r o x i m a t i o n . x S i m i l a r l y , p r o b i n g w a v e s o f h i g h e r f r e q u e n c i e s c a n p e n e t r a t e s e v e r a l l a y e r s b e f o r e b e i n g r e f l e c t e d . The c o r r e s p o n d i n g f a c t o r , K ^ , f o r a r e g i o n c o n s i s t i n g o f n " l i n e a r " l a y e r s , i s e v a l u a t e d i n A p p e n -d i x I I a n d t h e r e s u l t i s g i v e n b e l o w . K • = -n n - n , + -^n m' C c o n - 1 3 c o n 2 m ' x -, n n - 1 + n - n -, +~n m ' C c o n - 1 3 c o n 2m'x . n n - 1 + 2n D c o 3m' x -j 1 • n n - J (4.33) w h e r e n - 1 C = H i = l m. 1 n . 3/2 1 -n c o '1 -n i - l N n c o / 3/2' D '1 -n ) c o / t h The d e n s i t y g r a d i e n t a t t h e b o u n d a r y o f t h e n l a y e r , n i . , may now b e . d e t e r m i n e d f r o m e x p e r i m e n t a l d a t a b y s u b s t i t u t i n g t h e a b o v e v a l u e o f K n i n t o E q . ( 4 . 3 0 ) . T h e s t e p - h y - s t e p p r o c e d u r e f o r r e c o n s t r u c t i n g t h e p r o f i l e s i s a s f o l l o w s : P h a s e d a t a o b t a i n e d a t a f r e q u e n c y f , c o r r e s p o n d i n g t o a d e n s i t y n a r e u s e d t o d e t e r m i n e m-^. A l i n e a r s e g m e n t w i t h s l o p e m-^ i s u s e d t o a p p r o x i m a t e t h e p r o f i l e i n t h e f i r s t l a y e r . P r o m d a t a o b t a i n e d a t a h i g h e r f r e q u e n c y f ^ , c o r r e s p o n d i n g t o a d e n s i t y n c 2 , t h e a p p a r e n t s l o p e m^ i s c a l c u l a t e d f r o m E q . ( 4 . 3 1 ) . T h e f a c t o r i s t h e n o b t a i n e d b y u s i n g E q . ( 4 - 3 3 ) , i . e . , 2n „ [ 1 J-n c 2 " n l n c 2 3/2 n c 2 / a n d w h e r e n-^ i s t h e d e n s i t y a t t h e b o u n d a r y b e t w e e n t h e f i r s t a n d s e c o n d l a y e r s . T h e v a l u e o f n-^ l i e s i n t h e r a n g e n c 2 < n i < n C 2 a n d a r e a s o n a b l e c h o i c e f o r n ^ i s . n l = \ . ( n c l + n c 2 } T h e n e x t s t e p i n t h e p r o f i l e i s a p p r o x i m a t e d b y a l i n e a r s e g m e n t w i t h s l o p e w h i c h e x t e n d s f r o m n-^  t o w h e r e n 2 = \ ( * c 2 + n c 3 ) a n d n c o r r e s p o n d s t o t h e c r i t i c a l d e n s i t y o f t h e n e x t p r o b i n g f r e -q u e n c y , f ^ . B y r e p e a t i n g t h i s p r o c e d u r e f o r d a t a o b t a i n e d a t p r o g r e s -s i v e l y i n c r e a s i n g f r e q u e n c i e s , t h e r e m a i n i n g s t e p s i n t h e . p r o f i l e . a r e r e c o n s t r u c t e d . 4 . 3 . 3 U s i n g e s t M o d i f i e d P i e c e w i s e - L i n e a r A p p r o x i m a t i o n T h e a c c u r a t e d e t e r m i n a t i o n o f t h e f i r s t s t e p i n t h e r e c o n -s t r u c t i o n o f t h e p r o f i l e i s o f m a j o r i m p o r t a n c e b e c a u s e t h e r e m a i n i n g s t e p s e s s e n t i a l l y m a k e u s e o f t h i s ' i n f o r m a t i o n . T h e a c c u r a c y l a r g e l y depends on t h e r a t i o n , / n ( i . e . , on t h e c h o i c e o f p r o b i n g f r e q u e n c y ) c J- m and on t h e v a l i d i t y o f t h e l i n e a r a p p r o x i m a t i o n . Improved a c c u r a c y may be a c h i e v e d by u s i n g a l o w e r f r e q u e n c y so t h a t the p e n e t r a t i o n ' d e p t h i s r e d u c e d , o r by u s i n g a h i g h e r - o r d e r f u n c t i o n i n s t e a d o f t h e l i n e a r one t o o b t a i n a b e t t e r a p p r o x i m a t i o n t o the a c t u a l p r o f i l e . The f i r s t p o s s i b l i l i t y has the p r a c t i c a l d i s a d v a n t a g e t h a t t h e phase change o f t h e r e f l e c t e d s i g n a l a t l a r g e r w a v e l e n g t h s i s l e s s s e n s i t i v e t o m o t i o n o f the c r i t i c a l l a y e r . C o n s e q u e n t l y , m e a s u r i n g e r r o r s b e -come more s i g n i f i c a n t and tend t o o f f s e t t h e advantage g a i n e d by u s i n g the l o w e r f r e q u e n c y . The a p p r o a c h c o n s i d e r e d h e r e i s based on the second p o s s i b i l i t y , t h a t o f u s i n g a more a c c u r a t e a p p r o x i m a t i o n . A s e c o n d - d e g r e e f u n c t i o n i s used t o a p p r o x i m a t e t h e shape o f t h e p r o -f i l e i n the f i r s t s t e p o f t h e . r e c o n s t r u c t i o n . The d e t e r m i n a t i o n o f t h i s , f u n c t i o n , h o w e v e r , r e q u i r e s a d d i t i o n a l i n f o r m a t i o n about the p r o -f i l e . The p o s i t i o n o f t h e c r i t i c a l l a y e r can be deduced f r o m measurements o f t h e t o t a l change i n phase as the i n c i d e n t wave t r a v e l s f r o m t h e p lasma b o u n d a r y t o t h e c r i t i c a l l a y e r and back t o t h e b o u n -d a r y . I n t h e case o f a n a f t e r g l o w p l a s m a , t h i s measurement may be c o n v e n i e n t l y o b t a i n e d by o b s e r v i n g t h e phase change o f t h e r e f l e c t e d s i g n a l as t h e p l a s m a c r i t i c a l l a y e r moves f rom a p o i n t v e r y n e a r t h e w a l l o f t h e d i s c h a r g e t u b e ( e . g . , d u r i n g t h e v e r y e a r l y p a r t o f the a f t e r g l o w p e r i o d ) t o . a p o i n t f u r t h e r f r o m t h e w a l l , c o r r e s p o n d i n g t o the t i m e i n t h e a f t e r g l o w d u r i n g w h i c h the p r o f i l e i s t o be d e t e r -m i n e d . The t o t a l phase change c f t h e r e f l e c t e d s i g n a l i n t h i s r e g i o n i s g i v e n by E q . (4.7), i . e . , 0 :: ? 6 Kx ^ ' o c 82 The position of the c r i t i c a l layer, i n terms of the free-space wave-length of the probing signal, i s given by x =.- 0 X /4itK (4.35) c o Thus, the position of the c r i t i c a l layer can be determined from the measured phase change i n terms of the factor K. The slope of the p r o f i l e function at the c r i t i c a l density was determined previously. Assuming again that the density i s zero at the plasma boundary, the p r o f i l e function used to approximate the density d i s t r i b u t i o n must sa t i s f y the following conditions: n(x) =0 at x = 0 = n at x = x co . c and 4^ = m - at x = x dx c These conditions are s a t i s f i e d by a second-degree function of the form o n(x) - Ax" + Bx for 0 sx s x c (4.36) where A = m/x - n /x 2 c co c B = 2n /x - m co' c • In order to be physically r e a l i z a b l e , the p r o f i l e function must have a positive slope at the plasma boundary. This corresponds to the condition that B must be positive. The factor K i s given by K = - • f x c J . 2 - ,± 2 _ Ax + B3 n co dx Evaluating the in t e g r a l at the l i m i t s gives the result 8; x I 4A 8A /An CO jt/2 - s i n (B/m) > f o r A > 0 ( 4 - 3 7 ) and K = -B m x [4A 8A J^Kn •• In co m B - 2 -An co for A < 0 I t i s convenient to define a parameter, M , as follows: M = mx /n c co (4.38) Substituting for m and x c from Eq. . ( 4 . 3 2 ) and ( 4 . 3 5 ) gives M d i r e c t l y i n terms of experimentally measurable quantities, i . e . , K 0' co Expressed i n terms of M , Eq. ( 4 . 3 7 ) becomes ,2 -( 4 . 3 9 ) v- _ _ M^2 7372 and K M-2 4TM-.lT 8(M-1) M 8 ( 1 - M ) 3 / 2 jt/2. - s i n """(I - 1 for M>1 ( 4 . 4 0 ) In M 2-M-2yT=lT for M<1 The condition - B^O i s s a t i s f i e d when M ^ 2 . K i s plotted as a func-ti o n of M i n Eig. 4 . 4 . The reconstruction of the p r o f i l e then proceeds as follows: Using data obtained at the lowest frequency, the parameter M i s c a l -culated from Eq. ( 4 . 3 9 ) . The corresponding value of K i s then deter-mined from Eq. ( 4 . 4 0 ) and using t h i s value i n conjunction with the experimental data i n Eqs. ( 4 . 3 0 ) and ( 4 . 3 5 ) gives the density .gradient at the c r i t i c a l layer and i t s position. This information, i s then substituted into Eq. ( 4 . 3 6 ) to obtain a second-order approximation to the p r o f i l e shape i n the region between the plasma boundary and the »4 c r i t i c a l layer, and this i s used as the f i r s t step i n the reconstruc-t i o n . For the second step, a li n e a r segment whose slope corresponds to the density gradient determined above i s used to approximate the pr o f i l e i n the region between n and n^. As before, a reasonable choice for n^ i s n = ^(n -, + n ) 2 .2 c l c 2 wh ere K n i s the c r i t i c a l density.corresponding to the next higher probing frequency. 0.8 0.7 0.6 0.5 0.4 0 0.5 1.5 T.o M F i g . 4-4 K as a Function of the Parameter M for the P r o f i l e n(x) =.Ax2 + Bx The procedure used for reconstructing the remaining portion of the p r o f i l e i s es s e n t i a l l y the same as that used i n the previous section. However, since the f i r s t segment of the approximation i s not l i n e a r , the general expression for K i s modified from the previous ca.se and i s now given by K n 2 n -n -, -m' I+-=n m1E co n-1 n 3 co n 2m'x , n n-1 + 2 n -n -, -rn11+^n m' E co n-1 n 3 co n m1 x n n-1 2 2n D 1 co I 3m' x -, I n n-U (4.41) wh ere D = (1 - n n/n ) n-17 co 3/2 n-1 1 . „ m. 1=2 I n. 1 n co I = X Ax + Bx n co 3/2 dx n 1 - -n i-1 co 3/2 This result i s derived i n Appendix I I I . 4•4 Accuracy In order to evaluate the above methods for obtaining pro-f i l e information, we s h a l l use these methods to reconstruct p r o f i l e from hypothetical data which can be calculated for certain electron d i s t r i b u t i o n s . The following electron-density p r o f i l e s are con-sidered: (i ) n = n (x/R) 2 m ( i i ) n = n (x/R) 2 m ' ( i i i ) n = nfflsin(jrx/2R) 2 (iv) n = n sin (jrx/2R m 1 The electron density i s assumed to vary exponentially with time. 86 It i s seen from Eq. ( 4 . 1 2 ) that the phase data depend on the vel o c i t y of the c r i t i c a l layer and on the value of the factor K. Thus, by calculating these quantities and substituting back into Eq. ( 4 . 1 2 ) , the r e s u l t i n g values may be treated as experimental data. The v e l o c i t y of the c r i t i c a l layer is., obtained by d i f f e r e n t i a t i n g the appropriate function i n Eq. (4.14). , . while K i s obtained by inte-grating in Eq. ( 4 . 6 ) . Detailed calculations for the above p r o f i l e s appear i n Appendix IV and the r e s u l t i n g expressions are summarized i n Table IV.1. The f i r s t method, of deducing p r o f i l e s , based on using a p r o f i l e model, requires'experimental data at only two frequencies. These were chosen to correspond to n c 0 / n m = ' 0 . 3 and 0 . 5 i n one case and to n /n = 0 . 4 and 0 . 6 i n another case. The results obtained by co m applying t h i s method to calculated "data" for p r o f i l e s ( i i i ) and (iv) are shown i n Figs. 4 . 5 a and 4 . 5 b , respectively.' For p r o f i l e s ( i ) and ( i i ) , the p r o f i l e model i s exactly of the same form as the assumed p r o f i l e s and therefore the reconstructed curves are i n exact agreement with the true p r o f i l e s . Curves (a) show the p r o f i l e s obtained by using the actual value of n , as would be determined ex-• J 0 m . perimentally from a plot of n m against time. Curves (b)'were obtained by using a calculated value of n , corresponding to situations where n cannot be measured d i r e c t l y . I t i s seen, that one set of curves m J . . • may be more accurate than the other i n certain regions, depending on the shape of the actual p r o f i l e . But when the p r o f i l e i s unknown, as in an actual diagnostic problem, i t i s d i f f i c u l t to ascertain which set of curves i s l i k e l y to give a better o v e r a l l approximation. However, a large discrepancy between them, indicates that the model i s incapable of describing that p a r t i c u l a r p r o f i l e adequately. Examination, of the u 1.2 1.0 n_ n 0.8 m 0.6 0.4 0.2 V // / f / // /? '/ / /// / f /' i //// w V t 0 0.2 0.4 0.6 x/R 0.8 1.0 Pig. 4.5a Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using a P r o f i l e Model Actual p r o f i l e n=:nmsin(n:z/2R) Reconstructed from data at n^/n=0,3 and 0.5 C O / Reconstructed from data at nco/nm--0.4 and 0.6 Curves (a) obtained by using actual value of n m. Curves (b) obtained by vising calculated value of n^. 8 8 u 1.2 1.0 n 0.8 m 0.6 0.4 02 / / / • ( b ) / /f / / (a) //• / / / / / / / / / / / / / / / V //< ' / / i / / / / / / / / / // / / t /// f /// / 0.2 0.4 0.6 x/R 0.8 1.0 F i g . 4•5b C o m p a r i s o n o f A c t u a l a n d R e c o n s t r u c t e d P r o f i l e s O b t a i n e d b y U s i n g a P r o f i l e M o d e l A c t u a l p r o f i l e n = n m s i n 2 (itx/2R.) R e c o n s t r u c t e d , f r o m d a t a a t n c o / n m=0.3 a n d 0.5 _ R e c o n s t r u c t e d f r o m d a t a a t n c o / n m=0.4 a n d 0.6 C u r v e s ( a ) o b t a i n e d b y u s i n g a c t u a l - v a l u e o f n . C u r v e s ( b ) o b t a i n e d b y u s i n g c a l c u l a t e d , v a l u e o f n m . 89 curves also shows that the shape of the reconstructed p r o f i l e i s quite sensitive to the choice of probing frequencies. As i s charac-t e r i s t i c of most methods which are based on using a simple p r o f i l e model, t h i s method can give a reasonably good approximation i n spec-i a l cases, but i t i s not able to provide a complete description of the electron-density p r o f i l e i n general. Thus, the method may be adequate when only a rough estimate of the .profile shape i s required, but i t i s not suitable for carrying out a detailed i n v e s t i g a t i o n of the electron d i s t r i b u t i o n i n a plasma. The second method of reconstructing p r o f i l e s , based on the piecewise-linear approximation, requires phase data at several f r e -quencies. Hypothetical data were calculated' at the points n c o / n m = 0.2, 0.4, 0.6, and 0.8 for each of the assumed p r o f i l e s (corresponding to the experimental s i t u a t i o n i n which the r a t i o s of the probing f r e -quencies to the maximum plasma frequency are 0.45, 0.63, 0.77, and 0.89). Although the accuracy of the reconstructed p r o f i l e s depends on the number of layers i n the piecewise-linear approximation and on the choice of frequencies at which data are obtained, the above choice i s convenient from the view of experimental implementation. The reconstructed p r o f i l e s are shown i n Figs. 4.6a and 4.6b. The actual p r o f i l e s are shown as so l i d l i n e s , while the reconstructed ones are shown as broken l i n e s . I t i s seen that the shapes of the reconstructed curves are i n reasonably gocd agreement with the assumed p r o f i l e s , although the accuracy varies with the shape of the actual p r o f i l e . Since this method of reconstruction involves adding segments to pre-vious sections, the shape of the overall p r o f i l e i s sensitive to the accuracy of the f i r s t step i n the reconstructed p r o f i l e . This, i n turn, depends on the v a l i d i t y of the linear approximation. Clearly, 0 0.2 0M 0.6 0.8 1.0 x/R F i g . 4.6a Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using Piecewise-Linear Approximation i ) n = n f f i(x/R)J • i i ) n = n m(x/R)^ . Actual p r o f i l e -. — --Reconstructed p r o f i l e '91 1.0 0.8 0.6 n n m 0.4 0.2 . ( i i i ) / ' / / / / / / / / / / / / / / / ( i v ) / / / / / f t / f / / / / / / / / / / / / / y / / / // / 0 0.2 0.4 ,„ 0.6 0.8 1.0 F i g . 4.6b Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using Piecewise-Linear Approximation ( i i i ) n = n sin(jtx/2R) • ' (iv) •• n = n :sin2(TCX/2R) Actual p r o f i l e Reconstructed p r o f i l e • . the approximation becomes less accurate as the p r o f i l e shape deviates to either a convex or concave shape. However, the error for the con-cave case i s greater than for the convex case since the microwave beam penetrates deeper into the plasma and a much larger region.is approximated by a single l i n e a r segment, as i l l u s t r a t e d i n Fig . 4.7. Fig. 4.7 Effect of P r o f i l e Shape on Size of F i r s t Segment of Piece-wise-Linear Approximation The third method, using the modified piecewise-linear approx-imation, requires additional data about the t o t a l phase change of the reflected s i g n a l . i n the region between the plasma boundary and the c r i t i c a l layer. This information i s required only at the lowest probing frequency and i s calculated by substituting the corresponding values of x c and K into Eq. (4.7). The method was applied to hypo-t h e t i c a l data calculated at the same values of n /n as considered c o m . 93 i n the preceding case. The reconstructed p r o f i l e s are shown i n Figs. 4-8a and 4.8b and are seen to be i n extremely good agreement with the actual p r o f i l e s in a l l cases. This method i s more accurate than the previous one, but requires more data and the calculations are somewhat morelaborious. In reconstructing p r o f i l e s from exper-imental data, i t may be convenient to f i r s t use the simpler piece-wise-linear approximation and then examine the resu l t i n g p r o f i l e to establish whether using a second-order approximation i s l i k e l y to result i n a s i g n i f i c a n t improvement. In the case where the p r o f i l e shape d i f f e r s considerably from a l i n e a r shape or where the size of the f i r s t layer i s r e l a t i v e l y large, the second-order approximation can be used to give more accurate re s u l t s . 4.5 Experiments ' 4.5-1 Measurements Measurements were carried out during the afterglow period of a pulsed discharge i n argon. .The experimental arrangement i s shown i n F i g . 4 - 9 . The discharge apparatus and most of the microwave system were the same as those used for oblique-incidence measurements. The plasma column was -30cm i n diameter and 150cm long. Most of the microwave measurements for determining p r o f i l e s were carried out i n the frequency range 26.5 to 40.0GHz. In order to determine the time dependence of the maximum density, measurements were also carried out i n the 55 and 74GHz range. Most observations were made using p a r a l l e l polarization although perpendicular polarization was also used i n a few cases. As shown i n Fig. 4 . 9 , the reflected signal was picked up' by the transmitting antenna A-^  and was detected together with a reference signal which was obtained from the source v i a a d i r e c t i o n a l coupler. 0 0.2 0.4 0.S 0.8 .1.0 x/R F i g . 4.8a C o m p a r i s o n o f A c t u a l a n d R e c o n s t r u c t e d P r o f i l e s O b t a i n e d b y U s i n g M o d i f i e d P i e c e w i s e - L i n e a r A p p r o x -i m a t i o n i ) . n = ryU/R)* i i ) n = n ( x / R ) 2 . A c t u a l p r o f i l e . _ _ R e c o n s t r u c t e d p r o f i l e 0 0.2 0.4 ,^ 0.6 0.8 1.0 x/R Fig. 4.8b Comparison of Actual and Reconstructed P r o f i l e s Obtained by Using Modified Piecewise-Linear Approximation ( i i i ) n = n msin(itx/2R) (iv) -n = n msin2 (jtx/2R) Actual p r o f i l e Reconstructed p r o f i l e frequency meter attenuator, isolator klystron directional couplers matching section r • a — > reflection signal transmission signal "* d G t G C t o r J/L ' A horn-lens 1 /A a n * G n n a 'screened room F i g . 4 .9 Microwave Apparatus The amplitude of the r e s u l t i n g interference pattern provided a measure of the phase of the reflected signal. In some cases, the reference signal was obtained by introducing a mismatch i n the feeding wave-guide. The results i n t h i s case were very similar to those obtained ~by using part of the incident wave as a reference. The transmitted signal was received by antenna A^, and provided a measure of the time at which the maximum density was equal to the cutoff value. Both s i g -nals were displayed simultaneously on a double-beam oscilloscope and were recorded photographically. Measurements were carried out at a pressure of 45^ m Hg. The discharge voltage was 3kV and the i n t e r v a l between successive discharges was 30 seconds. The general forms of the displayed received signals during the afterglow period are shown i n Fig. 4-10. The upper trace shows the transmitted signal while the lower trace shows the interference pattern produced by the reflected signal. Each major d i v i s i o n rep-resents 1cm on the oscilloscope screen, and the time at which the discharge, starts coincides with the f i r s t v e r t i c a l l i n e on the l e f t side of the screen. This point was used as the time o r i g i n i n the measurements. 4.5.2 P r a c t i c a l Considerations It was shown i n Section 4.4 that the accuracy of the recon-structed p r o f i l e s i s affected by the choice of frequencies at which, data are obtained. However, this choice Is r e s t r i c t e d by certain p r a c t i c a l considerations. Microwave measurements of plasma properties (23) may be influenced by the experimental configuration , and various precautions.must be taken i n order that t h i s has minimum effects on the measurements. 98 Pig. 4 . 1 0 Reflected (lower trace) and Transmitted Microwave Signals Pressure, 45p.m Hg Voltage, 3kV P a r a l l e l P olarization (a) 30.14GHz l.Oms/cm (b) 30.14GHz 0.5ms/cm (c) 33.86GHz 0.5ms/cm In many experiments which involve propagation of micro-waves through bounded plasmas, the antenna system i s located outside the vacuum enclosure. Reflections from, the walls of the vacuum system (usually glass) can produce undesirable interference effects. When measurements are carried out at millimeter wavelengths and. when'the walls are r e l a t i v e l y thick, i t i s often possible to reduce these r e f -lections by selecting the operating frequencies so that the o p t i c a l thickness of the walls i s approximately equal to a multiple of h a l f -wavelengths at each frequency. Reflections may also be reduced by using matching structures, at the vacuum walls. However, t h i s arrange-ment i s frequency sensitive and becomes inconvenient when different frequencies are used. Also, matching becomes more d i f f i c u l t when the surface i s curved, as i n the case of a system with c y l i n d r i c a l geo-metry. However, for the par t i c u l a r measurements involved i n t h i s experiment, i t i s reasonably easy to distinguish, between effects due to r e f l e c t i o n s and those due to the plasma. Since we are interested i n measuring the rate of phase change of the main signal reflected from the plasma c r i t i c a l layer, these measurements v i l l i not be atfected by external r e f l e c t i o n s from the vacuum system walls as these are constant i n phase and amplitude. Thus, we are primarily concerned with internal r e f l e c t i o n s between the plasma and the vacuum walls. This effect w i l l be examined q u a l i t a t i v e l y by considering a doubly-reflected wave, that i s , a wave which i s received by the antenna after twc re f l e c t i o n s from the plasma c r i t i c a l layer, as shown i n Fig. 4.13. Since t h i s wave propagates along the same path, twice, i t s phase s h i f t w i l l be twice that of the main signal. Thus, when the two waves are detected simultaneously i n a square-law detector, the output waveform w i l l appear sim i l a r to that obtained when a sinusoidal wave cf funda-mental frequency and a second-harmonic wave are added. Because of 100 the harmonic phase relationship between the main and the multiply-reflected signals, the effect of r e f l e c t i o n s on the measurements can be avoided by measuring the phase change i n steps of 2JT radians. The power l e v e l of a multiply-reflected signal i s usually much lower than that of the main signal due to various effects such as scattering caused by the curvature of the plasma column, losses i n the plasma, and the r e l a t i v e l y low power r e f l e c t i o n c o e f f i c i e n t of the vacuum wall However, Fig. 4.10c shows an oscillogram of the received waveform obtained by using a frequency at which r e l a t i v e l y strong r e f l e c t i o n s from the walls were observed. It i s -seen that the perturbation on the waveform of the main signal i s similar to that produced by the addition of a second-harmonic.wave. Also, the r e l a t i v e amplitude of t h i s perturbation decreases as the position of the c r i t i c a l layer s h i f t s toward the center of the plasma column where curvature effects result i n increased scattering. A lower l i m i t on the frequency range which can be used i s imp os ed by the fact that phase measurements become less sensitive at longer wavelengths. However, the effect of t h i s r e s t r i c t i o n on the accuracy of the reconstructed p r o f i l e s i s reduced by using the modi-fied piecewise-linear approximation. --si-Antenna Vacuum Walls' Plasma C r i t i c a l Layer Fig. 4-11 Double-Reflected Wave i ' o i The maximum probing frequency i s limited by curvature effects which can become s i g n i f i c a n t when the c r i t i c a l layer i s near the center of the plasma column. Consequently, the phase data, are not r e l i a b l e f or obtaining information about, the density d i s t r i -bution i n this region. However, the maximum density at the center of the plasma column may be determined independently from transmis-sion-attenuation measurements. 4 . 6 Analysis of Experimental Results Typical oscillograms of the transmitted and reflected s i g -nals, obtained, at a probing frequency of 30 .14GHz, are shown i n Fig. 4.10a. These oscillograms are of the same form as those shown i n reference (6) and have a similar interpretation. The upper trace, showing the transmitted signal, provides information about the time i n the afterglow at which the plasma becomes transparent to the i n c i -dent radiation. In t h i s particular case, the plasma remains opaque and t o t a l l y r e f l e c t i n g u n t i l about. 4.75ms a f t e r the start of the d i s -charge. After t h i s time, the plasma becomes transparent and a trans-mitted signal can be detected. The undulations i n th i s signal are due to interference produced by re f l e c t i o n s from opposite walls of the glass tube. The precise instant at which transmission occurs i s not as well defined, as might be expected from the abrupt change i n the attenuation coefficient of the plasma when i t i s near the cutoff condition. This i s caused by various factors such as refraction and scattering due to curvature effects, internal r e f l e c t i o n s , and i n t e r -ference effects between r e f l e c t i o n s . The reflected signal (lower trace) contains information about the plasma during the cutoff period. Each cycle in-the ampli-102 tude variation' of the interference pattern corresponds to a phase change of 2ic radians and the frequency of t h i s v a r i a t i o n provides information about the v e l o c i t y of the c r i t i c a l layer. The amplitude of the envelope depends on the intensity of the reflected signal and provides information about the d i r e c t i o n " i n which the c r i t i c a l layer i s moving. In t h i s case, the decreasing amplitude indicates that- the c r i t i c a l layer i s moving toward the axis of the tube, where the cross section of the reflecting, surface becomes smaller. The point where the amplitude of the reflected s ignal reaches a minimum corresponds to the end of the cutoff period, as can be seen from the transmitted signal.. After t h i s time, the plasma i s transparent, and the interference pattern i s due to r e f l e c t i o n s from the glass walls on the opposite side of the transmitting antenna. These undulations can be used to measure the phase s h i f t introduced, by the plasma, and hence to study the v a r i a t i o n of electron density, as i n other i n t e r -ferometric methods^\ In t h i s case, however, the wave travels along the same path twice and therefore i t s phase charge w i l l be twice that measured by the usual microwave interferometer. The instantaneous frequency, F, of the interference pattern shown i n Fig. 4.10b, i s plotted against time i n F i g . 4.12. F i s obtained by calculating the rate at which the phase of the reflected signal changes i n steps of 2 i t radians and involves measuring the time i n t e r v a l between s i m i l a r l y situated points on the interference pattern. This procedure reduces the effect of multiple r e f l e c t i o n s from the glass walls of the discharge tube, as discussed i n Section 4.5.2. Two groups of steps are shown i n F i g . 4.12; one group was obtained by measuring between maxima i n the interference pattern, while the other was obtained by considering the minima. A s o l i d l i n e i s drawn 40 11 : 1 1 1 1 1 1 2.6 3.0 3.4 3.8 4.2 4.6 5.0 t , m s Fig. 4 . 1 2 F as a Function of Time Frequency, 30.14GHz Steps obtained by measuring between maxima Steps obtained by measuring between minima 104 through these steps to obtain a smooth f i t to the step diagram. Figure 4-13 shows a series of curves obtained s i m i l a r l y from measure-ments at five different frequencies. Although there i s some a r b i -trariness associated with drawing a curve through the steps, t h i s was generally done by f i t t i n g a smooth curve which passed near the middle of most steps. An alternative method for obtaining this-curve involves p l o t t i n g the phase of the reflected signal against time and measuring i t s slope at various points to determine the F/t curve. This was done i n a few cases and the res u l t i n g curves were very simi-l a r to those obtained by smoothing .the steps. The maximum electron density at the center of the discharge tube during the afterglow i s plotted as a function of time i n Fig . 4 . 1 4 . This was obtained by observing the time at which the cutoff period ended at several frequencies. The time"was measured from the trace of the reflected s i g n a l , as i t 'was usually better defined i n thi s case than i n that of the transmitted signal. Figure 4 . 14 also, shows a plot of the t o t a l s h i f t i n phase of the reflected signal i n the region between the plasma boundary and the c r i t i c a l layer as a, function of time during the afterglow. These values were estimated by observing the r e l a t i v e phase change of the reflected signal i n the time i n t e r v a l s t a r t i n g from a time i n the very early part of the afterglow. I t was ascertained that the position of the c r i t i c a l layer was very near the walls of the discharge tube and nearly sta-tionary during the early afterglow, as the amplitude of the reflected signal was large and there was no major change i n the waveform i n the i n t e r v a l of time immediately after the end of the active part (current flowing) of the discharge. The phase information shown i n Fig. 4 .14 was obtained at 26.5GHz, the lowest frequency used i n th i s set of measurements, and i s required when applying the modified 106 107 piecewise-linear approximation only. Using the above data, electron-density p r o f i l e s at t = 4.00ms were reconstructed and are plotted i n Fig. 4 .15. P r o f i l e s ' ( i i ) and ( i i i ) were determined by applying the step-by-step method using the piecewise-linear and the modified piecewise-linear approxi-mations respectively. The maximum density, n , at the center of the discharge tube was obtained independently from the plot of -n against time. P r o f i l e s ( i ) a and (i)b were obtained by using the p r o f i l e model n/n m = (x/R) a. Since this method requires data at two frequencies only, several values of "a" were calculated by using various combina-tions of the available data. P r o f i l e s ( i ) a and (i)b represent the extremes, and were obtained by using the independently determined value of n . P r o f i l e ( i ) c i s e s s e n t i a l l y the same as ( i ) a , but uses a calculated value of nffi obtained by applying Eq. (4.20). Of the pro f i l e s shown, p r o f i l e ( i i i ) , based on the modified piecewise-linear representation, i s the most accurate because i t makes f u l l use of the available data, and, as shown previously, t h i s representation i s able to provide a detailed description of the p r o f i l e shape. Using the modified piecewise-linear representation, p r o f i l e s at several d i f f e r -ent times during the afterglow were determined and are shown, i n F i g . 4.16. It i s seen that the actual shape of the p r o f i l e does not change appreciably with time, s a t i s f y i n g the assumption made i n Sec-tion 4.2. A few p r o f i l e s were also determined from data obtained at perpendicular polarization, and these were found to be very similar i n shape to the above ones. Figure 4-17 shows p r o f i l e s obtained under approximately the same conditions, by using the above method and the oblique-incidence ..method described i n Chapter 2. Although the num-ber density of electrons i s lower i n the second case, the shape of the 108 3.0 2.5 2.0 n M O 1 c m 3 1.5 1.0 0.5 ( i ) a > ^ y s y s s s *>' -*** y > :'(i)a / / ( i i i L ^ * y «* y y y y y y y y y _ ^ — y y y* y ///'/ / / / / V • 1''/ / V'// it// w 0 6 9 x, cm 12 15 Fig. 4 . 1 5 Electron-Density-Profiles at t = 4 . 0 0 r n s ( i ) P r o f i l e model, (a.b) using independent value of n. c5 using calculated value'of n m m - ' c_j — ( i i ) Piecewise-linear representation — ( i i i ) Modified piecewise-linear representation 109 Pressure, 45p-re. Hg Discharge voltage," 3kV 110 2 5 2 0 1 5 n .0.5 • / s / / / / / / / / ' s / / / / / / / / r / // // // 0 3 6 9 12 15 x, c m P i g . 4.17 C o m p a r i s o n o f P r o f i l e s f r o m N o r m a l - I n c i d e n c e and O b l i q u e - I n c i d e n c e M e a s u r e m e n t s P r e s s u r e , 45l~™ Ng D i s c h a r g e V o l t a g e , 3kV t = 4.00ms P r o f i l e s r e c o n s t r u c t e d by u s i n g : ; ( i ) M o d i f i e d p i e c e w i s e - l i n e a r method, n o r m a l — i n c i d e n c e d a t a (7) : ( i i . ) P i e c e w i s e - l i n e a r method , o b l i q u e -' i n c i d e n c e d a t a (obtained a t 35-00GHz) I l l electron-density d i s t r i b u t i o n i s very similar i n both cases. (The ratios of densities obtained by the two methods are almost constant.) As these measurements were carried out at different times, the d i s -crepancy i n actual electron-number density i s probably due to d i f f i -culty i n reproducing discharge conditions precisely. In. order to examine the s e n s i t i v i t y of the step-by-step method to experimental errors, a few p r o f i l e s were reconstructed by using deliberately erroneous data. One curve in. Fig. 4.18 shows the effect of using a value for a which i s 5% lower than that determined from a plot of n against time. A second curve shows the p r o f i l e obtained by using a value of the t o t a l phase change, 0, of the ref-lected signal which was %/2. radians greater than the measured value. In both cases, these errors have an almost negligible effect on the p r o f i l e shape.. A further error could be introduced by the fact that i n measuring 0, the plasma boundary was assumed to be at the position of the c r i t i c a l layer during the early afterglow. However, even during the very early part of the afterglow period, the c r i t i -cal layer must be a f i n i t e distance away from the walls of the d i s -charge tube i f the electron density i s zero at the walls. Consequently the measured phase does not include the phase s h i f t of the wave i n the region between the walls and the c r i t i c a l layer. In order to ascertain the effect of t h i s discrepancy, the distance between the walls and the c r i t i c a l layer during the early afterglow was estimated. In t h i s connection i t was assumed that the exponential time dependence measured during the l a t e r afterglow period was v a l i d throughout the entire afterglow and that the p r o f i l e shape did not change. Although rapid fluctuations i n the waveform of the reflected signal occurred during the discharge, only gradual changes were observed for.a r e l a -2.5 1.5 n x101W 1.0 -//••• - ^ ^ - v -A' / / // // // ft / 0 3 6 9 12 15 K , c m Fig. 4.18 Effect of Experimental Error on Reconstructed P r o f i l e t = 4.00ms Actual data ._ i t/2 error i n 0 57° error i n a ... t i v e l y long i n t e r v a l 'immediately af t e r this time. This was taken, as the st a r t of the afterglow, and the phase of the reflected signal was measured "by observing the phase change from this time, which for the parameters used i n these particular measurements, occurred about 1C0|JS a f t e r the start of the discharge. The position of the c r i t i c a l layer at t = 100|is was then found to be about 0.6mm from the walls of the discharge tube. Thus, the error introduced by neglecting the phase s h i f t i n the region between the tube walls and the c r i t i c a l layer during the early afterglow w i l l be less than %/2 radians, and th i s was shown above to have very l i t t l e effect on the shape of the reconstructed p r o f i l e . Although t h i s i s a crude estimate because of the simplifying assumptions which were made, i t does appear.to i n d i -cate that no s i g n i f i c a n t error i s introduced by measuring 0 i n t h i s manner. • 4.7 V a l i d i t y of Ray Theory and Effect of Curvature on Phase  Measurements The method described i n t h i s chapter was based on ray theory and the analysis was carried out for the planar case. The v a l i d i t y of these approximations w i l l now be examined by comparing "full-wave" and ray-theory results calculated for a plasma with an electron-density p r o f i l e s i m i l a r to that a c t u a l l y determined from, normal and oblique-incidence measurements. A c y l i n d r i c a l plasma with a parabolic density d i s t r i b u t i o n of the following-form i s considered: n = n m 2x x" eL R \R The radius, R, of the plasma column i s taken to be one and f i v e times 114 the free-space wavelength at the probing frequency. Using wave theory-', the phase s h i f t of the reflected signal i s computed for a plane wave polarized i n the d i r e c t i o n p a r a l l e l to the plasma axis,' at various values of n /n ( i . e . , at various penetrations into the co' m ' ^ plasma). Wave-theory results are also calculated for a plane-slab model with the same electron-density p r o f i l e . The ray-theory r e s u l t , obtained by substituting the above p r o f i l e function into the phase int e g r a l , Eq.. ( 4 . 5 ) , i s given by n \ 2 A phase term of it/2 i s included i n the ray-theory result to account for the phase change associated with r e f l e c t i o n at the c r i t i c a l layer, as discussed i n Section 4.2. Using the wave-theory results for .the c y l i n d r i c a l model as a reference, Eig. 4 .19 shows the phase error introduced by ( i ) applying ray theory and ( i i ) applying wave theory but approximating the plasma column with a plane slab. Examination of the curves for R = 5\ shows that no s i g n i f i c a n t phase error i s introduced by using these approximations up to the point where n /n i s about 0 . 9 8 . co m (This corresponds to a penetration x c/R—0 . 8 6 . ) In the experiments described here, the radius of the plasma cylinder was 15cm and. the probing frequency was usually i n the 35GHz range. This corresponds to R — 17A. , and therefore the effect of curvature i n the actual s i t u -ation w i l l be even less s i g n i f i c a n t than that indicated above. It i s interesting to note that, over most of the region * The method used to obtain the-wave-theory results i s described i n reference (24). ' 0 = co n m (n /n ) 2  v co m In W a v e t h e o r y , p l a n e m o d e l R a y t h e o r y / V s / / / o ol Or/ Qr/ y * / y y y y y y x y y y y r / / / / / / / / / / / / .^y * ^y y yy y / y / y / y y y y y~ y » / / / / / / / / / 1 / i J>" y y^ ^ — *y y y y y / i / I / / / / / / ( ! i I I I I • — / / S y ^~ ! 1 1 / • t T 1 • 1 1 - 1 i m - o. 1 1 < I 85 0. l l l i 9 0 n / n 0. n c o / n m I I I I ] 95 1.00 F i g . 4 . 1 9 Phase Error Due to Ray Theory and Effect of Curvature on Wave-Theory Results, for the P r o f i l e n(x ')=2x/R-(x/R) 2 considered, the ray-theory result i s actually.more accurate than the wave-theory result based on the planar approximation. Even in the consider-ably more severe case, R = A. , the ray-theory result i s s t i l l sur-p r i s i n g l y good, i n the region n /n m^0.90. However, since the t o t a l phase s h i f t i n t h i s case i s much less than i n the above case , the "percent-error" i n phase i s considerably greater. For example, at nc o / n m = 0.87, the absolute phase change of the reflected signal i n the region between the plasma boundary and the c r i t i c a l layer i s 172° and the phase error i s 7.8%. In the case R = 5^Q, the t o t a l phase change i s 1286° and the phase error due to using ray theory i s only 0.2%. 4.8 Summary It has been shown that measurements of the Doppler s h i f t i n frequency of the signal reflected from the plasma c r i t i c a l layer can be used to determine detailed electron-density p r o f i l e s i n an a f t e r -glow plasma. P r o f i l e s can be reconstructed from these data by either using a simple p r o f i l e model or by using a step-by-step procedure. P r o f i l e s obtained by using the model cannot provide a complete des-c r i p t i o n of the plasma'profile since t h e i r determination depends on two measurements only. However, the data-reduction procedure i s very simple and this representation may be adequate when an estimate only of the p r o f i l e shape i s required. The need for a p r o f i l e model, i s avoided by using the step-by-step method and the resulting p r o f i l e s are more accurate since data at several frequencies are used. Although good accuracy can be. obtained by using four or f i v e different f r e -quencies, more d e t a i l may be determined by using a larger number of .closely spaced frequencies. Although more calculations are involved than i n the case of the model, a l l calculations are straightforward and may be done without the aid of a computer. The required data can be obtained by using a simple micro-wave setup and interpretation of the data i s straightforward. The method i s suitable for studying plasmas with limited a c c e s s i b i l i t y , since a l l the required data can be obtained from'the waveform of the reflected signal. Because this method i s applicable to plasmas which are t o t a l l y r e f l e c t i n g , i t can be used to determine p r o f i l e s i n higher density plasmas than can those methods which depend on measuring the phase s h i f t of the transmitted wave. The method i s somewhat re s t r i c t e d by the assumptions which have been made. It i s based on the assumption that ray theory can be used to describe the propagation of the microwave beam i n the plasma. This requires mainly that the' electron density does not change appreciably within a wavelength, and, for the case of a c y l i n -d r i c a l plasma, that the diameter of the c r i t i c a l layer be large i n comparison with a wavelength. Based on the p r o f i l e s determined by this method and those determined by the oblique-incidence method, the f i r s t requirement appears to be s a t i s f i e d in t h i s experiment.. The second requirement was also generally s a t i s f i e d for the millimeter-wavelengths which were used. The method also assumes that the plasma can be treated as a plane slab, with the electron density varying monotonically i n one dimension only. Consequently, the method cannot be used to describe the p r o f i l e i n the region near the axis of the discharge tube. However, the maximum density, n , at the axis can be determined from attenuation-transmission measurements which are quite independent of the above assumptions. In this p a r t i c u l a r exper iment, the good agreement between the independent value of nffl and that.obtained by extending the reconstructed p r o f i l e to the axis appears to bear out the v a l i d i t y of the ray-theory approach. Also, "full-wave" calculations on a p r o f i l e similar to that found experi-. mentally showed that, except i n the immediate v i c i n i t y of the plasma axis, no s i g n i f i c a n t error was introduced by using ray theory. Sub-ject to the above l i m i t a t i o n s , the method can be used to provide a detailed description of the electron-density p r o f i l e i n a transient plasma. 5. OTHER ME AS UREME NT S 5.1 I n t r o d u c t i o n T h i s c h a p t e r d e s c r i b e s some s t u d i e s c a r r i e d out by i m p l e -m e n t i n g t h e o b l i q u e - i n c i d e n c e method of C h a p t e r 2 f o r d e t e r m i n i n g p r o f i l e s , and a l s o d e s c r i b e s an e x p e r i m e n t a l i n v e s t i g a t i o n o f cer-t a i n f a c t o r s w h i c h c o u l d a f f e c t t h e microwave r e s u l t s . E x t e n s i v e measurements were c a r r i e d out i n an a f t e r g l o w p l a s m a , u n d e r a l a r g e v a r i e t y of d i s c h a r g e c o n d i t i o n s . Some i n t e r -e s t i n g c h a r a c t e r i s t i c s were o b s e r v e d and p o s s i b l e e x p l a n a t i o n s were s u g g e s t e d . L a n g m u i r d o u b l e probe measurements were a l s o c a r r i e d out i n a few c a s e s i n o r d e r t o o b t a i n an i n d e p e n d e n t measurement o f t h e e l e c t r o n - d e n s i t y d i s t r i b u t i o n . S e v e r a l r e l a t e d measurements were c a r r i e d out i n o r d e r t o examine the v a l i d i t y o f some a s s u m p t i o n s made i n c o n n e c t i o n w i t h t h e m i c r o w a v e methods f o r d e t e r m i n i n g p r o f i l e s . The e f f e c t o f p o l a r i z a t i o n , e f f e c t o f a l t e r i n g t h e d i s c h a r g e - c u r r e n t waveform, and t h e p o s s i b i l i t y of a d e n s i t y g r a d i e n t i n the a x i a l d i r e c t i o n were i n v e s -t i g a t e d . Some microwave measurements were a l s o a t t e m p t e d d u r i n g t h e c u r r e n t - f l o w p e r i o d of t h e d i s c h a r g e . These a r e d e s c r i b e d and some o f the a s s o c i a t e d d i f f i c u l t i e s a r e d i s c u s s e d . 5 . 2 R e f r a c t i o n Measurements a t V a r i o u s P r e s s u r e s and D i s c h a r g e U s i n g t h e m u l t i p l e - r e f l e c t i o n method d e s c r i b e d i n C h a p t e r 2 , e x t e n s i v e measurements were c a r r i e d out i n an a f t e r g l o w p l a s m a f o r v a r i o u s d i s c h a r g e c o n d i t i o n s . The d i s c h a r g e v o l t a g e was u s u a l l y v a r i e d i n t h e range f rom a b o u t 2 . 9 t o 7 - 9 k V , w h i l e t h e p r e s s u r e was varied from 30 to about 300um Hg. The r e s u l t i n g experimental data were expressed i n the. form of "t/D" curves. Because of the large number of curves involved, the data were condensed by using D as a parameter, and by pl o t t i n g the time " t " as a function of pressure, P. Smooth curves were drawn through these points to obtain "t/P" diagrams. "t/P" curves obtained i n an argon plasma at 9^ = 30°, 40°, 50°, and 60°, and at D - 5 , 10, 1 5 , and 20 inches are given i n Appendix V. Several interesting characteristics can be observed from these curves. For example, the value of " t " at low pressures (~30um Hg) i s very insensitive to the discharge voltage. At i n -creasing pressures, the s e n s i t i v i t y increases, and, for most curves, " t " increases monotonically with pressure. However, curves, corres-ponding to 2.9kV were found to increase with pressure at low values, but decreased at higher, pressures. This point was investigated i n more d e t a i l at 9^ = 40° by using lower discharge voltages. The resulting "t/P" curves, shown i n F i g . V.2, exhibited the same beha-viour as i n the above case. Similar trends, although occurring at lower pressures, were also observed i n a nitrogen plasma, as shown, i n F i g . 7.5 A q u a l i t a t i v e explanation of these effe c t s , which appears consistent with the observed behaviour, i s given below. It may be argued that, at low pressures, the gas i n the discharge tube i s f u l l y ionized during the discharge, even at low discharge voltages. There-fore, increasing the voltage does not result i n an appreciable increase i n density, and hence has l i t t l e effect on the time " t " taken for the electron density to decay to some given value during the afterglow. However, at higher pressures, the energy of the the capacitor bank at low voltages may be i n s u f f i c i e n t for complete ionization. Therefore, increasing the discharge voltage would pro-duce an increase i n the electron density, with a related increase i n " t " . As the pressure i s increased further, the degree of. i o n i z a -tion at low voltages w i l l decrease due to the higher density of neutral atoms. I t was observed i n this case that the value of D, at some given time " t " , increased with pressure, indicating that the electron, density decreased. with pressure. Therefore, i t appears that the decay rate of the plasma i s higher i n th i s case because of the higher density of neutral p a r t i c l e s . Using data from, the "t/P". curves given i n Appendix'V, "t/D" diagrams were reconstructed f o r various pressures and voltages. Electron-density p r o f i l e s were reconstructed from these data by (7) applying the piecewise-linear method , using a four-layer approxi-mation. Some p r o f i l e s , obtained by drawing smooth curves through the piecewise-linear representation, are shown i n Pig. 5.1. Although small differences exist i n the shape of these p r o f i l e s , they a l l have i n common a characteristic "bulging-shape". If the electron-density d i s t r i b u t i o n during the afterglow period, i s determined by ambipolar d i f f u s i o n to the walls of the discharge tube, i t can be shown t h e o r e t i c a l l y * that, for a long c y l i n d r i c a l plasma, the shape of the electron d i s t r i b u t i o n should be that of a zero-order Bessel function, i . e . , n(x) = n mJ 0(2.4r/R) where r = x - R, and R i s the radius of the cylinder. Some pr o f i l e s from Pig. 5.1 were normalized to the maximum density, n , obtained by extending these curves to the axis, and the res u l t i n g * See, for example, BrownI 25) # 1.25 1.00 0.75 10 c m 0.50 0.25 0.0 ( i i ) ( i ) [iii) .. . (iv) y/ v/ • / 0 12 15 x , c m Fig. 5 . 1 Electron-Density P r o f i l e s i n an Argon Afterglow Plasma ( i ) • 2.9kV, 30um Hg, t=3.50ms ( i i ) 2 - 9 k V , 200[am Hg, t=6.25rns ( i i i ) - 7 . 0 k V , 50(im Hg, t=5.00ms (iv) 7.6kV, lOOurn Hg, t=7.75ms curves are compared with the J Q ( r ) curve i n F i g . 5.2. This diagram shows that the measured p r o f i l e s are consistently more "bulging" than the Bessel curve. Thus, i t appears that the electron-density decay cannot be explained by the usual type of ambipolar d i f f u s i o n . Although a detailed investigation of t h i s behaviour i s beyond the scope of this work, a possible contributing factor i s considered. In developing the theory of ambipolar d i f f u s i o n , one of the basic assumptions made i s that the ionic mobility i s constant. How-ever, measurements of the m o b i l i t i e s of atomic rare gas ions i n t h e i r (26 27} parent gas ' have shown that the mobility decreases with increa-sing e l e c t r i c f i e l d strength, becoming inversely proportional to the square root of the f i e l d at high f i e l d s . The solution to the ambi-polar d i f f u s i o n problem for c y l i n d r i c a l geometry, taking into account (28) variable ion mobility, has been given by Frost . The theory was worked out numerically by using a n a l y t i c a l approximations to the observed f i e l d dependence of ionic mobility. Results showed that ambipolar d i f f u s i o n i n t h i s case could be considerably reduced from that predicted.by using constant mobility, t h i s effect increasing with the value of a dimensionless discharge parameter "av"e/PR" where "a" i s a constant (0.0267mm-cm/volt for argon) V i s the energy of the electrons, i n v o l t s P i s the pressure R i s the radius of the plasma cylinder The above theory was applied to parameters encountered i n the present experiments to ascertain whether the effect of variable mobility could be s i g n i f i c a n t i n t h i s case. Probe measurements (described i n the next section) indicated that electron energies of the order of one electron-volt were encountered during the afterglow period, 1.0 0.8 0.6 n / n m 0.4 0.2 (i)y ft / / y / // / // / fa* / / ( i v ) -/ / •Z. 0 0.2 0.4 0.6 0.8 1.0 x/R F i g . 5 . 2 Normalized Electron-Density P r o f i l e s ( i ) 2.9kV, 30u.in Hg, t=3.50ms ( i i ) 2.9kV, 200iam Hg, t=6.25ms ( i i i ) 7 . 0 k V , lOOum Hg, t=7.75ms (iv) J Q ( 2 . 4 r/R) 125 and could be as high as 3eV during the early part of the afterglow. Using these values, a.nd other parameters t y p i c a l l y encountered i n . the experiments, the effect of variable ionic mobility on p r o f i l e shape was calculated numerically. The re s u l t i n g p r o f i l e s , plotted i n Fig. 5.3, show that the electron density i s everywhere greater than that indicated by the Bessel curve. Also, these p r o f i l e s bear a closer resemblance to those which were actually measured i n the experiments It may be noted that the parameter "aVe/PR" indicates that the effect of variable mobility should increase with decreasing pressure. Examination' of the normalized p r o f i l e s for 30 and 200p.m. Hg (curves ( i ) and ( i i ) i n Fig . 5.2) does indeed show that the pro-f i l e obtained at a higher pressure tends more toward the zero-order Bessel curve. No comparison i s made with curve ( i i i ) as the abso-lute value of density i s considerably lower i n this case (cf., F i g . 5.1), and hence the value of V g w i l l probably be lower also, thereby obscuring the effect of pressure. Since the p r o f i l e s were determined by using a four-layer model only, and since the observed differences i n p r o f i l e shape were •small, a detailed compa.rison was not attempted i n this case. How-ever, the shape of the electron-density d i s t r i b u t i o n and the trends observed under different discharge conditions appear to be consistent with those expected i f ambipolar d i f f u s i o n i s assumed and variable ionic mobility i s taken into account. This appears, at least par-t i a l l y , to account for the observed p r o f i l e s . 5.3 Langmuir Probe Studies Probe measurements were carried out i n a few cases i n order .1.0 0.8 0.6 n/n m 0.4 0.2 0 . • ' / / / ** 0 / / / /// / / / (i) / / / > / / / / d v ) / 0 t . / / - /--/-/ ^ y ~ ( i i ) / / , . . . , 'V— ( III) //// 7 0 0.2 0.4 0 .6 x / R 0.8 1.0 F i g . 5 . 3 Effect of Variable Ionic Mobility on P r o f i l e Shape Argon Pressure, 30(im Hg ( i ) V , = 3 v o l t s R=7.5cm (11). Vg=3volts ( i i i ) V e=lvolt (iv) J Q(2.4r/R) R=15cm R=15cm to provide an independent check on the electron-density p r o f i l e s and to examine the v a l i d i t y of certain assumptions made i n connection with some cf the microwave methods for determining p r o f i l e s . B asically, the probe method involves measuring the current collected by a small metallic electrode inserted into the plasma as a function of i t s voltage with respect to the plasma. A reasonably detailed account of the o r e t i c a l and experimental considerations associated with various probe techniques may be found i n reference ( 2 9 ) . . A l -though probes tend to perturb the plasma, they have the advantage that l o c a l measurements of electron density can be obtained, whereas most other diagnostic techniques give information averaged over a large volume of plasma. Absolute values of density determined from probe measurements are not usually ve.ry accurate due to such sources of error as uncertainty i n interpreting probe voltage-current char-a c t e r i s t i c s and uncertainty of the effective probe area. However, many of these effects are nearly constant and do not affect the use-fulness of probes for determining r e l a t i v e densities i n different parts of the plasma. In many gas discharges, there i s an electrode i n good e l e c t r i c a l contact "with the plasma which can be used as a reference point for potential when applying a bias voltage to the probe. How-ever, i n situations where the plasma potential changes with time, as i n a decaying plasma, i t i s d i f f i c u l t to maintain a constant probe-plasma potential difference. To avoid this d i f f i c u l t y i n the present experiment, the Langmuir double-probe m e t h o d w a s used. Two probes, biased with respect to each other but insulated from ground, "float"•with the plasma and therefore follow the change of plasma potential. By measuring the "probe current as a function of the d i f -f e r e n t i a l probe voltage, electron temperature and plasma densities 128 can be determined. Double f l o a t i n g probes also have the advantage that they draw less•electron current from the plasma and therefore represent less of a perturbation than a.single probe. The electron energy i s related to the probe characteristics according t o ^ ^ kT Tl /dV,\ e _ /—- p ( d 4 \ d i / Vd=0 where T g i s electron temperature . e i s electronic charge k i s Boltzmann's constant Z ] i p i s the peak-to-peak saturation current i - i s the net current flowing i n the double-probe c i r c u i t i s the potential applied across the probes /dv"A .is the effective r e s i s t i v i t y of the plasma at \di / .^-^g poir^ wh.ere both probes, are at the same potential Once kT g i s known, the electron density can be determined from ' 1 n Y± / m. \ 2 0.8A e \2k T p \ e where A^ i s the probe surface area, assumed equal for both probes m. i s the ion mass l 5 - 3 . 1 Experimental Setup The discharge system was similar to that described i n Sec-tion 2.4.1, except that the discharge tube was 15cm i n diameter and 75cm long, with a 2.5cm diameter side port located midway along the tube. Probes were introduced into the plasma through this port, v i a a moveable vacuum seal. Tiny spherical copper probes (0.18cm i n dia-' 129 meter), vacuum sealed through glass, were used. The c i r c u i t used to measure probe current and voltage was essentially the same as that described i n reference ( 3 2 ) . The probe voltage was swept at frequencies i n the audio range, through an i s o l a t i n g transformer which separated the probe c i r c u i t from the signal generator. To enable the probe-circuit potential to follow rapid changes i n plasma potential, the i s o l a t i o n transformer was s p e c i a l l y wound to- obtain a low capacitance between i t s secondary and the remainder of the transformer. Four cathode followers and two d i f f e r e n t i a l amplifiers were used to measure the probe current and voltage simultaneously. The saturation current was measured by applying a suf-f i c i e n t l y large voltage to the probes to saturate them alternately. Oscillograms of the current waveform obtained during the afterglow-period were similar to those shown i n reference (32) and w i l l not be given here. The value (dV^/di)y _Q was determined by adjusting "V^  to a small value compa.red with that necessary to produce current-saturation, and by taking the r a t i o of voltage and current recorded simultaneously. • '- -5 . 3 . 2 Results Fig . 5.4a shows some electron-density p r o f i l e s determined during the afterglow period i n an argon plasma. Electron densities were determined at quarter-inch in t e r v a l s , and the p r o f i l e s shown were obtained by drawing smooth curves through these•points. For comparison purposes, electron densities were normalized to the maxi-mum density at the tube axis and the t h e o r e t i c a l p r o f i l e n/n = * . m (2 4r\ J Q I — 1 i s also plotted. The shape of these p r o f i l e s suggests that some electrons diffuse part way into the side port of the discharge tube. To avoid t h i s effect, the glass shields supporting the probes •131 were bent so as to displace the current-collecting electrodes away from the axis of the side port. Repeating the measurements i n t h i s case gave the p r o f i l e s shown'in Fi g . 5.4b. These pr o f i l e s are quite dif f e r e n t from the previous case and bear a closer resemblance to those determined by using microwave methods. Similar trends were observed at other pressures. Since the p r o f i l e s determined from probe measurements were affected by the presence of a side port, the results obtained .in this case were not considered r e l i a b l e enough to be used for making a meaningful comparison with microwave results. I t was assumed i n Chapter 4 that the electron.density i n an afterglow plasma decayed exponentially with time and at a uniform rate across the plasma cross section. To investigate the v a l i d i t y of these assumptions, probes were used to measure the electron den-s i t y as a function of time, at various probe penetrations. I t was found that, plotted on a semi-logarithmic scale, the re s u l t i n g curves were very nearly l i n e a r and curves corresponding to various probe . positions were approximately p a r a l l e l . The above measurements were car-r i e d out during the middle portion of the afterglow period. When the measurements were extended to the early afterglow period, i t was found that the re s u l t i n g curves exhibited two d i s t i n c t " l inear regions", the one at e a r l i e r times having a steeper slope. This appeared to indicate that some additional d ecay mechanism was effec-tiv e during the early afterglow but became i n s i g n i f i c a n t during the la t e r part. Similar trends were observed i n measuring the maximum density by using microwaves at increasing frequencies. However, over a r e l a t i v e l y small time i n t e r v a l i n the afterglow period, the assumption of an exponential time dependence appeared to be j u s t i -f i e d . Since the decay curves for various probe positions were 132 n e a r l y . p a r a l l e l , the assumption of a uniform decay rate also appeared to be v a l i d . 5 . 4 Comparison of Results Obtained by Using P a r a l l e l and Perpendicu- l a r Polarizations The effect of polarization, on oblique-incidence data was examined by carrying out some measurements by using p a r a l l e l and per-pendicular polarizations simultaneously. Microwave power from a com-mon source was divided by means of a 3-dB d i r e c t i o n a l coupler and was used to feed two i d e n t i c a l horn antennas, appropriately oriented and located on diametrically opposite sides of the discharge tube. The received signals were detected separately and displayed simul-taneously on a double-beam oscilloscope. I t was established that no interaction occurred between the two systems. By inserting large attenuation i n the waveguide connecting to either transmitting anten-na, i t was found that no microwave signal was detected by the cor-responding receiving antenna, while the waveform of the signal re-ceived by the opposite antenna was unaffected. Typical waveforms of the received signals for several angles of incidence are shown i n . F i g . 5 . 5 . The upper trace was obtained by using perpendicular polarization while the lower was for the paral-l e l case. The measurements were carried out at a frequency of 35.00GHz, and the separation, D, between the transmitting and r e c e i -ving antennas was 9 inches.. The discharge was i n argon at a pressure of 30p.m. Hg and the capacitor bank voltage was 3kV. The main peak i n the waveform corresponds to the time at which the c r i t i c a l r e f r a c t i o n condition occurs within the plasma, and i s well defined i n most of the oscillograms. Smaller peaks, F i g . 5 . 5 Waveforms of Received Signal at Two Polarizations Upper trace, Perpendicular Polarization Lower trace, P a r a l l e l P o larization Pressure, 30\xm Hg Discharge Voltage, 3kV Frequency, 35.00GHz D = 9 inches (a) Qi=20°, 500us/cm (d) 9 i =50° 500us/cm (b) © i = 3 0 Q , 500iis/cm (e) 9 i =60°, 500us/cm (c) 9i=40°, 500us/cm (f) 9 i =70°, 1.0ms/cm which can be observed at e a r l i e r times i n many of the traces, are due to multiple r e f l e c t i o n s , as described i n Section 2.3-2. The difference i n r e l a t i v e amplitudes of the multiply-reflected s i g -nals for the two polarizations i s due to differ e n t reflection, coeffi-c i e n t s at the glass walls for each polarization. The signal re-o ceived a f t e r the main peak at (h = 20 i s due to reflec t i o n s from the back wall of the discharge tube after the plasma decays s u f f i c i e n t l y for the incident beam to penetrate through, i t . The time, t, at which the received signal i s maximum, i s measured from the instant at which the discharge starts.' I t i s seen that observations at the two polarizations give results which are i n good agreement with each other. Although t h i s was usually found to be the case, very small but consistent differences i n corresponding values of " t " could be observed under certain conditions. The s i m i l a r i t y between these two sets of results indicates (7) that no s i g n i f i c a n t error i s introduced by plasma waves , which tend to be set up when-the e l e c t r i c vector of the incident wave l i e s i n the plane of incidence ( p a r a l l e l p o l a r i z a t i o n ) . The small d i f -ferences which could be observed i n some cases probably can be accounted for on the basis of wave theory. However, as these d i f -ferences were s l i g h t , ray theory was considered adequate i n the present application. 5.5 Effect of Al t e r i n g Current Waveform on Afterglow Measurements A few measurements were carried out i n order to determine whether changing the current waveform i n the discharge c i r c u i t would produce any s i g n i f i c a n t change i n the electron-density d i s t r i b u t i o n during the afterglow period. I t was normally found i n these exper-iments that waveforms of current and voltage were o s c i l l a t o r y , simi-l a r to those obtained in reference (12). A sketch of t y p i c a l wave-forms obtained at a discharge voltage of about 3kV i s shown i n Fig. 5.6a, The voltage leads the current by a quarter cycle since the discharge c i r c u i t has a r e l a t i v e l y low resistance and i s pre-dominantly reactive at this time. The st a r t switch i n the discharge c i r c u i t consisted of an ignitron (cf., Section 2.4.1.) and i t was found that once the ignitron was made conducting, current continued to flow i n the c i r c u i t for about three half-cycles. The scheme adopted here- for al t e r i n g the current waveform involved closing a sh o r t - c i r c u i t i n g switch across the capacitor bank at the end of the f i r s t half cycle i n order to terminate current flow i n the discharge c i r c u i t . A three-electrode spark gap, consisting of two hemispherical electrodes and a trigger pin mounted 'in one of the electrodes, was used for th i s purpose. The spark gap was connected d i r e c t l y to the capacitor bank as shown i n F i g , 5.6b. Operation, of the system was as follows: During charging, the capacitor "C" was charged to approximately the same voltage as the capacitor bank. When the discharge tube was f i r e d , the capacitor bank voltage decreased rapidly while the voltage across "C" changed more gradually due to the r e s i s t o r "R". Consequently a potential difference developed between the trigger pin and the electrode, as i l l u s t r a t e d i n F i g . 5.6c. V/hen t h i s voltage difference reached the breakdown value, capacitor "C" discharged across the trigger pin and the resulting spark caused conduction between the main electrodes of the spark . gap. A small resistance was usually inserted i n the trigger pin connection i n order to reduce damage to the pin. By appropriately adjusting the value of "R", the trigger electrode could be made to f i r e approximately at the time'when the discharge voltage was at i t s most negative value and the current was approximately zero.. C u r r o nt (a) T y p i c a l Discharge Waveforms I g n i t r o n T o D i s c h a r g e _ . — , C a p a c i t o r T u b e B a n k -<5 o S p a r k | | G a p R ( h ) Spark-Gap Connection to Capacitor Bank T r i g g e r - P i n V o l t a g e . V o l t a g e D i f f e r e n c e (c) Voltage D i f f e r e n c e Between Trigger P i n and Spark-Gap Electrode (d) . Current Waveform when Spark Gap F i r e d at End of F i r s t H alf-Cycle Fig.-5.6 Operation of Spark-Gap Switch 137 This g e n e r a l l y caused the i g n i t r o n to e x t i n g u i s h , terminating cur-rent flow i n the discharge c i r c u i t , while the remaining energy i n the c a p a c i t o r bank was d i s s i p a t e d i n the spark-gap c i r c u i t . A t y p i -c a l current waveform obtained i n this case i s shown i n F i g . 5•6d. Probe and normal-incidence microwave measurements were c a r r i e d out during the afterglow period i n the cases where the spark gap was both connected and disconnected from the discharge c i r c u i t . I t was found that no d i f f e r e n c e between the two cases could be observed i n e i t h e r the waveforms of the r e f l e c t e d micro-wave s i g n a l or i n the probe voltage-current c h a r a c t e r i s t i c s . There-f o r e , i t was concluded that a l t e r i n g the current waveform i n t h i s manner produced no s i g n i f i c a n t change i n the el e c t r o n - d e n s i t y pro-f i l e . . . . 5.6 Test f o r Uniformity of E l e c t r o n Density i n A x i a l D i r e c t i o n of  Plasma Column. One of the ba s i c requirements f o r implementation of the oblique-incidence technique i n the l o n g i t u d i n a l plane i s that the e l e c t r o n density be uniform'in the d i r e c t i o n along the plasma a x i s , with a s p a t i a l dependence i n the r a d i a l d i r e c t i o n only. In the . present experiment, t h i s required mainly that measurements be c a r r i e d out i n a region f a r enough away from the electrodes to avoid end-e f f e c t s caused by c o o l i n g of the plasma due to the electrodes. How-ever, there i s a second . p o s s i b i l i t y which must be considered i n s i t u a t i o n s where a continuous flow of gas i s maintained during the discharge. I f the flow r a t e i s too high, a large pressure gradient may occur along the discharge tube, causing a v a r i a t i o n of e l e c t r o n density i n the a x i a l d i r e c t i o n . In order to e s t a b l i s h whether t h i s e f f e c t was s i g n i f i c a n t i n the present experiments, the f o l l o w i n g two 1 3 8 s e t s o f m i c r o w a v e measurements were c a r r i e d o u t . S i n c e t h e c u t o f f t i m e ( i . e . , t h e t i m e t a k e n f o r t h e m a x i -mum e l e c t r o n - d e n s i t y t o decay t o t h e c r i t i c a l v a l u e ) i s s e n s i t i v e t o p r e s s u r e , one t e s t f o r u n i f o r m i t y i n the a x i a l d i r e c t i o n was based on m e a s u r i n g t h i s t i m e at v a r i o u s p o s i t i o n s a l o n g t h e l e n g t h o f t h e d i s c h a r g e t u b e . The c u t o f f t i m e was d e t e r m i n e d f r o m n o r m a l -i n c i d e n c e measurements ( d e s c r i b e d i n C h a p t e r 4) by n o t i n g t h e t i m e a t w h i c h the l a s t r e f l e c t e d s i g n a l was r e c e i v e d . F i g u r e 5 .7 shows waveforms o f t h e r e f l e c t e d s i g n a l o b t a i n e d a t n i n e d i f f e r e n t p o s i t i o n s , s t a r t i n g f i v e i n c h e s f rom the bot tom e l e c t r o d e , and i n c r e a s i n g i n f i v e - i n c h i n c r e m e n t s . These t r a c e s do n o t show t h e c o m p l e t e waveform o f t h e r e c e i v e d s i g n a l , but o n l y an expanded p o r t i o n o f the waveform i n t h e r e g i o n where c u t o f f o c c u r s . The d i s c h a r g e was i n a r g o n a t a. p r e s s u r e of 30um Hg and t h e f l o w r a t e was a d j u s t e d t o be a p p r o x i m a t e l y t h e same as t h a t n o r m a l l y used i n t h e e x p e r i m e n t s . * I t i s seen t h a t t h e t i m e a t w h i c h t h e l a s t r e f l e c t e d s i g n a l was r e c e i v e d was n e a r l y c o n s t a n t i n a l l t h e t r a c e s e x c e p t i n t h e one c o r r e s p o n d i n g t o a p o s i t i o n 5 i n c h e s above t h e b o t t o m e l e c t r o d e . These measurements were r e p e a t e d a t t h e same • p r e s s u r e , b u t w i t h t h e f l o w r a t e i n c r e a s e d about f i v e t i m e s by com-p l e t e l y o p e n i n g t h e b a f f l e v a l v e a t t h e pumping p o r t . A l t h o u g h t h e d i s c r e p a n c y i n c u t o f f t i m e a t the p o i n t 5 i n c h e s above t h e l o w e r e l e c t r o d e was more p r o n o u n c e d i n t h i s c a s e , t h e t i m e s were a g a i n n e a r l y c o n s t a n t . I n t h e r e m a i n i n g t r a c e s . T h u s , i t a p p e a r s t h a t , . , o v e r a major p o r t i o n o f t h e d i s c h a r g e t u b e , no s i g n i f i c a n t change i n e l e c t r o n d e n s i t y i n t h e a x i a l d i r e c t i o n i s i n t r o d u c e d by p r e s s u r e g r a d i e n t . * The f l o w r a t e i n t h i s c a s e was e s t i m a t e d t o be 1 9 l i t r e s p e r s e c o n d . T h i s was d e t e r m i n e d f r o m i n f o r m a t i o n a b o u t t h e b a c k i n g p r e s s u r e and t h e pumping c a p a c i t y o f t h e b a c k i n g pump a t t h i s p r e s s u r e . ^ 1 3 9 F i g . 5 . 7 Waveforms o f R e f l e c t e d S i g n a l a t V a r i o u s P o s i t i o n s a l o n g t h e D i s c h a r g e Tube F r e q u e n c y , 29.8GHz D i s c h a r g e V o l t a g e , 5kV Time s c a l e , 250u.s/major d i v i s i o n 140 The e f f e c t of pressure gradient was al s o studied by applying the oblique-incidence technique i n the case where no gas flowed through the discharge tube (and hence the pressure gradient was zero). These" r e s u l t s were compared w i t h those obtained under the same co n d i t i o n s , but with gas flowing at the usual r a t e . I t was found that the r e s u l t i n g e l e c t r o n density p r o f i l e s were almost i d e n t i c a l , although the absolute value of e l e c t r o n d e n s i t y was s l i g h t l y higher i n the second case. The close s i m i l a r i t y i n the shape of the two p r o f i l e s i n d i c a t e d that the e f f e c t of density gradient i n the discharge tube was n e g l i g i b l e i n the second case. However, the s l i g h t l y higher absolute value appeared to i n d i c a t e that the pressure was higher as w e l l . Since i t was established by normal-incidence measurements that no s i g n i f i c a n t pressure gradient was apparent i n the main p o r t i o n of the tube, and since the pressure gauge was located below the lower e l e c t r o d e , . t h i s discrepancy i n pressure was probably caused by a small pressure drop i n the v i c i -n i t y of the electrode. Because of the r e l a t i v e l y small cross-sec-t i o n a l area between the electrode and the tube w a l l s , the higher flow v e l o c i t y i n t h i s region could r e s u l t i n a s i g n i f i c a n t pressure gradient. However, t h i s would not a f f e c t the u n i f o r m i t y of e l e c t r o n -density i n the a x i a l d i r e c t i o n over the main p o r t i o n of the discharge tube. 5.7 Measurements During the Current-Flow Period. A few oblique-incidence measurements were attempted during the current-flow period. Figure 5.8 shows some oscillograms obtained i n t h i s case at two d i f f e r e n t values of D, f o r 9^ = 40° and a probing frequency of 35-OOGHz. The discharge was i n nitrogen at a pressure 141 F i g . 5 . 8 Waveforms Obtained During Current-Flow Period Pressure, 45pm Hg Voltage, 3«7kV Frequency, 35.00GHz Q± = 40° Time scale, 20ps/major d i v i s i o n Upper trace, Current Waveform Lower trace, Received Microwave Signal 1 4 2 of 45jJ.ni Hg and the discharge voltage was 3-7kV. The lower trace i n each o s c i l l o g r a m shows the received microwave s i g n a l , while the upper trace shows the discharge-current waveform which was obtained simultaneously by" using a dual-beam o s c i l l o s c o p e . The trace corresponding to D=5 i n shows that a maximum i n the received waveforms occurs at about 20u.s a f t e r the s t a r t of cur-rent flow. At D=3 i n , the peak occurs at approximately t=28|is. For D;r2 i n , no peak could be observed, but rather the s i g n a l l e v e l increased g r a d u a l l y to a c e r t a i n value, and then remained nearly constant f o r the d u r a t i o n of the current-flow period. Wo r e f r a c t e d s i g n a l could be detected at values of D greater than about seven inches. The decrease i n D with time appears to i n d i c a t e that the plasma c r i t i c a l l a y e r moves outward, toward the w a l l s of the d i s -charge tube. However, poor d e f i n i t i o n of " t " made i t d i f f i c u l t t o obtain information about the p r o f i l e during t h i s time. Since a microwave s i g n a l could be received at small values of D only, i t -appears that the probing beam d i d not penetrate very f a r i n t o the plasma. Shallow penetration was a l s o i n d i c a t e d by phase measurements at normal incidence under the same co n d i t i o n s . I t was found that the phase change observed i n the i n t e r f e r e n c e pattern of the r e f -l e c t e d s i g n a l was only a few times re radians i n the time i n t e r v a l from the s t a r t of current flow u n t i l the e a r l y afterglow. I t was ascertained that the p o s i t i o n of the c r i t i c a l l a y e r during the e a r l y part of the afterglow was very near the tube w a l l s . Therefore the r e l a t i v e l y small phase s h i f t i n d i c a t e d that the maximum depth to which the wave penetrated was only a few wavelengths from the 143 w a l l s . At oblique incidence, the depth of penetration f o r the same case w i l l be even l e s s . In order t o obtain deeper penetration i n t o the plasma, higher frequencies (55 and 75GHz) were used and the discharge con-d i t i o n s were a l t e r e d . I t was found that, f o r discharge parameters i i n the v i c i n i t y of 7kV and 18u.m Hg, the probing wave penetrated f u r -ther i n t o the plasma than i n the previous case. Because of the l a r g e r discharge current i n t h i s case, some d i f f i c u l t y i n measuring the microwave s i g n a l was encountered due to s t r a y pickup. This d i f f i c u l t y , however, could be overcome by using a more e f f e c t i v e screened room f o r s h i e l d i n g the microwave apparatus. I t was found that f a i r l y deep penetration could a l s o be obtained by using a very low discharge voltage. Figure 5 .9 shows some waveforms of the received microwave s i g n a l obtained during a discharge i n argon at a voltage of 1.4kV and a pressure of 40jim Hg. The frequency of the probing wave was 53-5GHz, 0^ was 40° and D was 6.3 i n . Traces ( i ) and ( i i ) show the microwave s i g n a l received during the afterglow period; ( i ) was obtained by using a r e f l e c t i n g sheet to enhance m u l t i p l e r e f l e c t i o n s . Trace ( i i i ) shows the microwave s i g n a l r e -ceived during current flow, while ( i v ) shows the corresponding cur-rent waveform. Traces (v) and ( v i ) were obtained under the same co n d i t i o n s , except that a r e f l e c t i n g sheet was used to increase m u l t i p l e r e f l e c t i o n s . Two main peaks i n the microwave s i g n a l received during cur-rent flow can be observed i n waveform ( i i i ) . S i m i l a r waveforms could be obtained at other values of D. However, information about the e l e c t r o n - d e n s i t y p r o f i l e was d i f f i c u l t to obtain due to poor time r e s o l u t i o n . A s i m i l a r d i f f i c u l t y was encountered i n the oblique-144 incidence measurements described i n reference ( 6 ) . Placing a r e f -l e c t i n g sheet between the transmitting and receiving antennas to enhance multiple r e f l e c t i o n s usually resulted i n a waveform of larger amplitude and s l i g h t l y longer duration. However, i n the cases considered, these waveforms could not be resolved 'into i n d i -vidual peaks as i n the case of afterglow measurements. 145 6 . CONCLUSIONS Two new m e t h o d s f o r d e t e r m i n i n g e l e c t r o n - d e n s i t y p r o f i l e s i n a t r a n s i e n t p l a s m a c o l u m n h a v e b e e n d e v e l o p e d . The f i r s t , an i m p r o v e d m i c r o w a v e r e f r a c t i o n t e c h n i q u e , d e p e n d s o n m u l t i p l e r e f l e c t i o n s o f a n o b l i q u e l y - i n c i d e n t m i c r o -wave beam b e t w e e n t h e p l a s m a c r i t i c a l l a y e r a n d t h e w a l l s o f t h e d i s c h a r g e t u b e . I t o f f e r s s e v e r a l a d v a n t a g e s o v e r p r e v i o u s m e t h o d s ; i t i s s i m p l e r t o i m p l e m e n t e x p e r i m e n t a l l y , a l l o w s more d a t a t o be o b t a i n e d d u r i n g a s i n g l e d i s c h a r g e , a n d y i e l d s a c c u r -a t e a n d m o r e r e l i a b l e r e s u l t s . S e v e r a l m e t h o d s f o r r e c o n s t r u c t i n g p r o f i l e s f r o m r e f r a c t i o n d a t a a r e d e s c r i b e d a n d c o m p a r e d . The m e t h o d s a r e e v a l u a t e d b y a p p l y i n g t h e m t o h y p o t h e t i c a l d a t a c a l c u l a t e d f o r c e r t a i n a s s u m e d p r o f i l e s . The m e t h o d b a s e d o n u s i n g a p i e c e w i s e - l i n e a r r e p r e s e n t a t i o n o f t h e p l a s m a a p p e a r s t o be t h e m o s t e f f i c i e n t f o r r e c o n s t r u c t i n g p r o f i l e s , a n d g i v e s s u r p r i s i n g l y g o o d r e s u l t s , e v e n w h e n o n l y a f e w l a y e r s a r e u s e d i n t h e r e p r e s e n t a t i o n . B y a p p l y i n g t h e s e m e t h o d s t o e x p e r i m e n t a l d a t a , e l e c t r o n - d e n s i t y p r o f i l e s d u r i n g t h e a f t e r g l o w p e r i o d h a v e b e e n r e c o n s t r u c t e d . A l t h o u g h t h e r e a r e s m a l l d i f f e r e n c e s i n t h e p r o f i l e s o b t a i n e d b y v a r i o u s m e t h o d s , t h e g e n e r a l s h a p e o f t h e e l e c t r o n - d e n s i t y d i s t r i b u t i o n i s v e r y s i m i l a r , ) The s e c o n d m i c r o w a v e m e t h o d i s b a s e d o n . m e a s u r i n g t h e 146 Doppler s h i f t i n frequency of the r e f l e c t e d s i g n a l at normal incidence. This method has an advantage over methods based on measuring phase charge of the s i g n a l transmitted through a p l a s -ma i n that more accurate and d e t a i l e d p r o f i l e s can be obtained. Also, f o r the same probing frequencies, p r o f i l e s can. be deter-mined i n higher d e n s i t y plasmas. The method a l s o has the advan-tage that the required data can be obtained from r e l a t i v e phase, measurements. Unlike the oblique-incidence method, the present method i s not r e s t r i c t e d to s i t u a t i o n s where a r e l a t i v e l y large access to the plasma region i s a v a i l a b l e , as a l l the required data can be obtained through a small probing "window". Owing to deeper penetration at normal incidence, t h i s method allows pro-f i l e s to be determined, i n s l i g h t l y higher density plasmas than the oblique-incidence method. Several data-reduction procedures s u i t a b l e f o r r e c o n s t r u c t i n g p r o f i l e s from normal-incidence data have been developed and evaluated. Approximate p r o f i l e s could be determined from data at two frequencies by u s i n g a simple two-parameter model, while more d e t a i l e d p r o f i l e s could be reconstructed from data obtained at s e v e r a l frequencies by using a step-by-step procedure based on the p i e c e w i s e - l i n e a r approach. In cases where good accuracy could not b"e obtained by using the l i n e a r approximation, much better accuracy could be a t t a i n e d by using a second-degree f u n c t i o n to represent the p r o f i l e i n the f i r s t step of the r e c o n s t r u c t i o n . • The v a l i d i t y of ray theory i n t h i s a p p l i c a t i o n was examined by comparing ray-theory and "full-wave" r e s u l t s c a l c u l a t e d f o r , a p r o f i l e s i m i l a r to that determined from experimental measure-ments. I t was found t h a t , f o r the parameters used i n these exper-iments, no s i g n i f i c a n t e r r o r was introduced by using ray theory. 147 Using the improved refraction technique, extensive measure-ments were carried out i n an afterglow plasma for a large variety of discharge conditions. Although small differences i n p r o f i l e shape were found, a l l p r o f i l e s had. a ch a r a c t e r i s t i c "bulging" shape which could not be attributed to experimental error. I t was found that p r o f i l e s predicted t h e o r e t i c a l l y , by assuming the usual type of ambipolar d i f f u s i o n , could not provide an accurate description of the s p a t i a l d i s t r i b u t i o n of electrons i n this afterglow plasma. An explanation has been suggested which could account partly for the observed p r o f i l e s . An experimental investigation of several factors which could affect the measured p r o f i l e s was carried out. These were: the effect of pol a r i z a t i o n , the effect of a l t e r i n g the discharge-current waveform on the afterglow conditions, and the p o s s i b i l i t y of a density gradient i n the a x i a l d i r e c t i o n of the plasma cylinder. 148 A P P E N D I X I CALCULATION OF "dD/dt" FOR A PIECEWISE-LINEAR PROFILE Consider a. plasma with a p i e c e w i s e - l i n e a r density v a r i a t i o n i n the region between the plasma boundary and the point at which the e l e c t r o n density i s equal to the c r i t i c a l value at oblique incidence as i l l u s t r a t e d i n F i g . 1.1, The s p a t i a l d i s t r i b u t i o n of elec t r o n s i s described by n(x) = m-^ x f o r 0 ^  x ^x 1 = n-^  + nig (x-x-^) f o r x-^  $ x $ x^ = n n + m (x-x -, ) f o r x -, < x ^  h n-1 n n-1 n-1^ th where m. i s the slope of the i segment th n^ i s the e l e c t r o n d e n s i t y at the i boundary t h x^ i s the s p a t i a l coordinate of the i l a y e r h i s the s p a t i a l coordinate cf the c r i t i c a l d e nsity ai oblique incidence. n n n. n-1 n 1 m m 'n-1-0 X h '1 An-1 F i g . 1.1 Piecewise-Linear Representation x 149 The c r i t i c a l d e n s i t y at oblique incidence may be expressed i n the form n = n -, + m (h-x .. ) c n-1 n n-1 (1.2 The time dependence of the e l e c t r o n density i s assumed to be expon-e n t i a l , and i s expressed i n the form n ( t ) = n(t ) e - a ( t - t o ) (1.3) The distance between entrant and emergent beams i s given by substi-t u t i n g f o r n ( x ) from Eq. ( l . l ) i n t o Eq. (3.2), i . e . , D = 2tan9 n n dx 1 - m^x n c J X, J dx x .1 1 -rij+m^ (x-x-^) 2 n + h 1 dx Sn -1 n -, +m (x-x -i ) n-1 n n-1 n The s u b s c r i p t "n" i s used with D to denote the t o t a l number of laminae i n the p i e c e w i s e - l i n e a r representation. E v a l u a t i n g the above expression term-by-term gives the r e s u l t D = 4n tan9 n c n n m \ n ; n \ c / (1.4) wne re n-1 -, n ^—• m 1=1 l 1 - l i x l n 1 - -1 n In a plasma, decaying with time, e l e c t r o n d e n s i t i e s at the boundaries between adjacent l a y e r s w i l l decrease according to Eq. (1.3), .while the plane located at the c r i t i c a l density w i l l move inward. The 150 corresponding change i n D may be obtained by d i f f e r e n t i a t i n g Eq, (1.4) as f o l l o w s : dD -r~ = 4n tanO dt c n - n . 1 . d / _ n n - l dt m dt I n n \ c n + 1 n ; d(l/m dt S u b s t i t u t i n g f o r the exponential time dependence from Eq. (1.3) gives the f o l l o w i n g r e s u l t s f o r d e r i v a t i v e s of i n d i v i d u a l terms : dn. I dt d_ dt 1 -an. n. n an. /n i ' c dm. T T ~ = -am. dt l Combining terms gives the f i n a l r e s u l t dD r -rr- = 4n tanO /A' + TT~ dt c n A n 2m„ a n where A' n n-1 Tl 1=1 a m. l n. -i /n 1-1' c n. /n i / c n 2 1 i - 1 ^ n 2 1 n. I n APPENDIX I I CALCULATION OF "K" FOR A PIECEWISE-LINEAR PROFILE Consider a plasma with a p i e c e w i s e - l i n e a r density v a r i a -t i o n i n the region between the plasma boundary and the c r i t i c a l l a y e r , as i l l u s t r a t e d i n F i g . I I . 1 . The s p a t i a l d i s t r i b u t i o n of electrons i s described by n(x) = rn-^x n l + m 2 ^ x _ x i ^ f o r 0 <x ^x-^ f o r x-j^  ^  x <x 2 ( I I . l ) n , + m (x-x - i ) n-1 n n-1 f o r x -, < x <x n-1 ~ ^ c th where m^  i s the slope of the i segment t h n^ i s the e l e c t r o n density at the i boundary t h x^ i s the s p a t i a l coordinate of the i boundary x c i s the s p a t i a l coordinate of the c r i t i c a l l a y e r The c r i t i c a l d e n s i t y may be expressed i n the form n - n -, + m (x -x -,) .co n-1 n c n-1 (II.2) S u b s t i t u t i n g f o r n(x) i n Eq. (4.6) gives n x 1 0 m-^ x n co dx + 1 -n-j+m^  (x-x-^) n dx + 1^ x ... + / 1 -n -j +m (x-x n ) n-1 n n-1 n x co n-1 dx > 152 n n n co n-1] n 1 TH 2 / 0 x 1 n. m / x -i n-1 -Es-X F i g . I I . 1 Piecewise-Linear P r o f i l e The subscript "n" i s used w i t h K to denote:the t o t a l number of l a y e r s i n the re g i o n between the plasma boundary and the c r i t i c a l l a y e r . Evaluating- the above expression term-by-term gives the r e s u l t where 2n C 2n D g _ C O . . + CO 3 x c 3 x c m n n-1 , c = YZ ~ 1=1 m i D = (l - ^=1-n. 3/2 n 1 - n - 1 -i - 1 co, n CO 3/2-(II.3) 3/2 n *m ", which occurs i n the second term of Eq. ( I I . 3), i s not e x p l i c i t l y n known, but i s r e l a t e d to the measurable q u a n t i t y m^  according to Eq. (4.32), i . e . , m n K m» n n 153 S u b s t i t u t i n g x c from Eq. ( I I . 2 ) and using the above value of m i n Eq. ( I I . 3 ) gives the quadratic equation 3x -,m'K 2 + 3(n -n ..+§n ra'C)K - 2 n D=0 n-1 n n co n-1 3 co n n co Of the two s o l u t i o n s f o r K , one i s negative and i s discarded desired r e s u l t i s given by n K = n h -n ,+„n m'C co n-1 5 co n 2m' x , n n-1 + 2 -12 n -n n +=-n m' C co n-1 3 co n 2m'x _ n n-1 + 2n D ' 2 cp^  3m' x -, , n n-1 154 APPENDIX I I I CALCULATION OP "K" FOR MODIFIED PIECEWISE-LINEAR PROFILE Consider a plasma represented by n laminae, where the density v a r i a t i o n i n the f i r s t lamina i s described by a second-de-gree.function and i s l i n e a r i n the remaining n-1 laminae, as shown i n F i g , I I I . l . The s p a t i a l d i s t r i b u t i o n of e l e c t r o n s i n the f i r s t segment i s given by n(x) = Ax 2 + Bx . f o r 0 =sx ( I I I . l ) where A = m^/x^ - n-^/x^ B = 2n 1/x 1 - m± and m-^  i s the d e n s i t y gradient at the' boundary, x=x-^. The remaining p o r t i o n of the p r o f i l e i s i d e n t i c a l to that used i n Appendix I I , and i s described by Eq. ( I I . l ) . n . A o x- X , x X k l ^2 ^n-1 c F i g . I I I . l Modified Piecewise-Linear P r o f i l e 155 Substituting for n(x) i n Eq. (4.6) gives K = ^ V I + n x 1 - n j_+^2 (x _ x i ) n co dx + '(III.2) L l r-where 0 I X 1 -n -, 4-rn (x-x -, ) n-1 n n-1 -4-n co dx n-1 Ax2+Bx n co dx Evaluating the in t e g r a l i n the expression for I gives, for A>0 m l ( l - n l / n c o ) 2 - B 4A + 4An +B' co SA^An co " • -1 • ml • -1 B sin =i - sm co v^ An" +B2-co and for A <0 I = m, ( l - n , / n ) 2-B 1 v V co 4A + 4An +£> co 8 Aflxi co In m -,-2/-An ( l - n n / n ) 1 v CO 1 CO B - 2.£In V CO Evaluating Eq. (III.2) term-by-term gives n x c 2n D I - #n E + co co 3m. n (III.3) where E 'n-1 , t z m i i - n. l n ,3/2 co 3/2-1 n D = 1 - n-1 3/2 n co Substituting for m and x from Eq. (4.32) and (II.2) respectively i n Eq. (III.3) gives the quadratic equation. 156 3x ,m'K 2 + 3(n -n ,-m'I + |n m'E)K ~2n D = 0 J n-1 n n v co n-1 n 3 co n n co The negative s o l u t i o n i s discarded and the desired r e s u l t i s given by f o 2 K = n n -n ,-m11 + ^n m1E co n-1 n 3 co n 2 m n x n - l + n - n -,-m11 + ~n m' E co n-1 n 3 co n n-1.n-1 + 2n D co 3m' x -, n n-1 157 APPENDIX IV EVALUATION OF "u" and "K" FOR ASSUMED PROFILES ( i ) n = n m ( i i ) n = n m ( i i i ) n = n m ( i v ) n •= n m The f o l l o w i n g e l e c t r o n - d e n s i t y p r o f i l e s are considered (x/R)"2" The time dependence of the e l e c t r o n density i s . assumed to he expo-n e n t i a l , and i s expressed i n the form n ( t ) = n ( t o ) e - a ( t - t o ) The v e l o c i t y of the c r i t i c a l l a y e r , u, i s then obtained by d i f f e r e n t i a t i n g the appropriate f u n c t i o n i n Eq. ( 4 . 1 4 ) , i . e . , / n u = R dt co 1 S l ^ T t 7 ; m ( I V . l ) where g describes the e l e c t r o n - d e n s i t y p r o f i l e as a f u n c t i o n of d e n s i t y . - _ . . . For example, consider, p r o f i l e ( i i i ) which has a s i n u s o i d a l density v a r i a t i o n . Expressing the p r o f i l e in,terms of the inverse f u n c t i o n , g, gives . x R S u b s t i t u t i n g f o r g i n Eq. ( I V . l ) ? a n d d i f f e r e n t i a t i n g w i t h respect to time, gives u 2R d_ JC dt s i n ( - £ r W T -\ n r o .158 u = 2aR Tt n co m n co Results obtained for the other p r o f i l e s are given i n Table IV.1. The factor K was defined i n Eq. (4.6) as follows • K 1 o I - n co ax (IV.2) The detailed evaluation of K for each p r o f i l e i s given below, (1) n = n (x/R)* m Substituting for n and n i n Eq. (IV.2) gives co K = 1 0 ±1 2 dx 2 • — " Substituting y = l-(x/x ) 2 then gives the result K = *J / ( y 2 - l ) d y = 8 / 1 5 1 ( i i ) n = n m(x/R)' In this case K c Q L\ c/ dx Substituting sin(y) - x/x gives the result jt/2 K | / (1 + cos 2y) dy =. it/4 0 ( i i i ) n = n sin(itx/2R) m Using the equality sin(:tx/2R) = 2 sin' 1 and substituting 159 f o r n i n Eq. (IV.2) gives x K 0 n 1 + m 2n m n co n co 2 (Tt /-. , X\ s m (- [1 + dx L e t t i n g 0 = ^ ( l + ^) gives the r e s u l t R' 0. K = C J 1-m s i n 9 d 9 (IV.3) wh ere C = m = JCX c 2n m n +n co m n co 0 = f I 1 + sr, x The above expression f o r K i s an e l l i p t i c i n t e g r a l , and may he expressed in' standard form as f o l l o w s ; where K = C |^ E(0|m) - E(Tt/4|m)J 0 of the second kind (IV.4 ) (0Jm) = J |l-m s i n ^ 9J d 9 i s an e l l i p t i c i n t e g r a l Numerical t a b l e s f o r e l l i p t i c i n t e g r a l s are u s u a l l y given f o r the range 0^0 ^90° and 0 < m ^ l . However, we are con-cerned w i t h the s i t u a t i o n where n <n , and therefore the c o r r e s -co m' ponding value of the parameter m i n Eq. (IV.3 ) i s greater than u n j t y . In t h i s case, m can be replaced by a parameter l e s s than. u n i t y by us i n g the f o l l o w i n g p r o p e r t i e s of the e l l i p t i c i n t e g r a l (35) 160 E(0|m) = m2"E(0|m 1 ) - (m-l) F(0|m) ( I V . 5 ) and F(0|m) = m 2F(0|m 1 ) ( I V . 6 ) where s i n 0 = m2 s i n 0 F(0|m) 0 1-m s i n 0 d9 i s an e l l i p t i c i n t e g r a l of the f i r s t k i n d . S u b s t i t u t i n g Eqs. ( I V . 5 ) and ( I V . 6 ) i n t o Eq. ( I V . 4 ) , and l e t t i n g M = l/m gives K = C | M " " 2 _ [ E ( M ) - E ( 0 | M ) ] - M~~ 2~(l-M) [K(M)-F(9(M)]J ( I V . 7 ) 9 = s i n - 1 J n /(n_+n_)' m' • m co where • E ( M ) = E( J C/2|M) i s a complete e l l i p t i c i n t e g r a l of the second kind K ( M ) = F(JC/2|M) i s a complete e l l i p t i c i n t e g r a l of the f i r s t kind Rearranging Eq. ( I V . 7 ) gives K = 2 h n '/: . , n m' co s i n "'"(n In ) m co E(M)-E(9 M)-(n -n )/2n [ K ( M ) - F ( Q | M ) ] [ ( I V . 8 ) where M = (n +n )/2n m c ' m K can nov; be evaluated from a t a b l e of e l l i p t i c i n t e g r a l s by using Eq. ( I V . 8 ) . 2 ( i v ) n = n m s i n (itx./2R) S u b s t i t u t i n g f o r .n and l e t t i n g 9 = JCX/2R i n Eq. ( I V . 2 ) gives 0 K 2R rex • 0 1-m s i n 9 d9 ( I V . 9 ) 161 where 0 = JCX c/2R m = n /n n m' co The above expression i s an incomplete e l l i p t i c i n t e g r a l of the second k i n d , but' as i n the previous case, i t cannot be determined d i r e c t l y from t a b l e s because the parameter m i s greater than u n i t y . S u b s t i t u t i n g from Eqs. (IV.5) and (IV.6), Eq. (IV.9) may be re w r i t -ten i n terms- of tabulated complete e l l i p t i c i n t e g r a l of the f i r s t and s e c o n d kin d , i . e . , K 2R_ Jl'X m2E(m _ 1) - m 2(l--)K(m _ 1) . . m ' (IV.10) Expressed i n terms of p h y s i c a l q u a n t i t i e s , Eq. (IV.10) becomes n. A /n K = / n / - . n m' co 1 "*"/n /n •* co m s m _ \ m / - 1 - C O ' n m CO) m . .The above expressions are summarized i n Table IV.1 and the r e s u l t s are p l o t t e d as a f u n c t i o n of n c o / n m i n F i g . IV.1. 0 162 P r o f i l e Velocity 'K' n=n (x/R)" m 2aR(n /n V \ co' m 8/15 n=n (x/R)' mv ' |aR(n /n )" 1 v co irr it/4 . htx n=n sm b y F m V2R . 2/rex\ n = nm s i n [Wj 2aR re n co l_n 2-n m co n co aR re p. -n m co -i 2 2(2n /n ) T f — — — ^ — - — . E(M) s i n " ( n c Q / n m ) I E(©|M) n -n m co 2n m K(M)-F(9|M) (n /n ) 2 IT/ CO s m -1 '/n /n >/ co m E n 1 - co n. m n > co n m / K 'n co n m * M = (n + n )/2n v m. co ' m 9 = s i n 1 JiTJl n +n ) m co Table IV.1 Velocity, of C r i t i c a l Layer and the Eactor "K" for Various Electron-Density Distributions With an Exponential Time Dependence 0.8 0 . 7 0 . 6 0 . 5 0 . 4 ( i i i ) ( i i ) _ l i v ) •; ( i ) •' 9 • 0 F i g . IV.1. 0 . 2 0.4 0 . 6 . n r r A 0 . 8 1.0 c o ' m n n (x/R) m • - • ."K".as a Function of C r i t i c a l Density for Various Electron-Density P r o f i l e s . ( i ) (ii.) ( i i i ) n = n^(x/R)' (iv) n = n msin(jtx/2R) (v) n = n sin^(jtx/2R) n == n- (x/R), m /• • -m 164 APPENDIX V REFRACTION DATA OBTAINED UNDER VARIOUS DISCHARGE CONDITIONS The f o l l o w i n g data were obtained i n an afterglow plasma "by applying the m u l t i p l e - r e f l e c t i o n method described i n Chapter 2 . The frequency of the probing wave was 35.00GHz, and i t was p o l a r i z e d with the e l e c t r i c vector p a r a l l e l to the plane of incidence. The data obtained at low pressures were g e n e r a l l y w e l l defined, and r e p r o d u c i b i l i t y of these experiments was very good. However, at higher pressures, the data were not u s u a l l y as c l e a r , and r e p r o d u c i b i l i t y was not as good, p a r t i c u l a r l y at high discharge voltages. The l a t t e r e f f e c t was mainly due to d i f f i c u l t y i n s e t t i n g the pressure and keeping i t constant with time. At high pressures, i t was found that pressure u s u a l l y increased j u s t a f t e r a discharge, p o s s i b l y due to heating of the gas, and then decreased g r a d u a l l y to i t s o r i g i n a l value. In these measurements, the i n t e r v a l between d i s -charges was kept at 45 seconds, as t h i s u s u a l l y allowed s u f f i c i e n t time f o r the pressure to s t a b i l i z e . " ' The data f o r argon were obtained at approximately the f o l l o w i n g pressures: 30, 5 3 , 110, 160, 280-370, and, i n a few cases, 600u.m Hg. For ni t r o g e n , data were obtained at approximately 30, 5 3 , 110, and 150u.m Hg. Data could not be obtained at 30pm Hg and at low discharge voltages since the tube would not f i r e . A d d i t i o n a l measurements i n the range 70-lOOu.m Hg were c a r r i e d out i n t h i s case. 165 166 167 168 30 50 70 100 , , , 2 0 0 P , p m H g 400 600 Fig.. V.4d "t/P" Diagram, Q.=60( 181 D =15" 8j= 4 0 ° I—-—-  .' 1 i- - l . l — i i r. i .. , , , , i ,, , [ . „ , l l 30 50 7 0 100 ^ 200 400 600 r . j j m H g F i g . V.5 "t/P" Diagram f o r Nitrogen, 9. =40° 182 REFERENCES 1. Huddlestone, R.H., and Leonard, S.L., (eds.), "Plasma Diagnos- t i c Techniques", Academic Press, New York, 1965. 2. S p i t z e r , L., "Physics of F u l l y Ionized Gases", I n t e r s c i e n c e , New York, 1956. 3 . Heald, M.A-. , and Wharton, C.B., "Plasma Diagnostics with Microwaves", John Wiley and Sons, New York, 1965 . 4. Wharton, C.B., and Slager, D.M., "Microwave Determination of Plasma Density P r o f i l e s " , J . Appl. Phys., V o l . 3 1 , pp. 428-4 3 0 , February, I 9 6 0 . Motley, R., and Heald, M.A., "Use. of M u l t i p l e P o l a r i z a t i o n s f o r E l ectron-Density P r o f i l e Measurements i n High-Temperature Plasmas". Proc. Symp. on M i l l i m e t e r Waves (Polytechnic Press, New York), pp. 141-154, I960. 6. Kharadly, M.M.Z., "A New Millimetre-Wave Method f o r Determina-t i o n of Ele c t r o n - D e n s i t y P r o f i l e s i n a Linear Discharge", Proc. IEE, V o l . 110, pp. 1202-1210, J u l y , 1963. 7. Kharadly, M.M.Z., and C u l l e n , A.L., "Oblique-Incidence M i l l i -metre-Wave Plasma Di a g n o s t i c s " , Proc. IEE, V o l . 114, pp. 1035-1044, August, 1967. 8. Budden, K.G., "Radio Waves i n the Ionosphere", Cambridge Uni-v e r s i t y Press, 1961. 9 . Shmoys, J . , "Proposed Diagnostic Method f o r C y l i n d r i c a l Plasma", J. Appl. Phys., V o l . 32, pp. 689-695, A p r i l , 1961. 10. A n i c i n , B.A., "E l e c t r o n Density P r o f i l e s i n C y l i n d r i c a l Plasma", J . Research N a t l . Bur. Standards, V o l . 69D, pp. 721-727, May, 1965. 11. Dushin, L.A., Kononenko, V.I., and Sklbenko, A.I., " S p a t i a l D i s t r i b u t i o n of Plasma Density Studied by R e f r a c t i o n of Micro-wave Beam with Several Frequency Components", Zh. Tech. F i z . , Vo l . 3 6 , pp. 1842-1850, October, 1966. 12. Kharadly, M.M.Z., Howatson, A.M., and Brown, R., " M i l l i m e t r e -Wave Diagnostics of a Pinched Linear Discharge", Proc. IEE, Vo l . 110, pp. 1211-1220,. J u l y , 1963. 13- M o r i t a , T., and Conn, S.B., "Microwave Lens Matching by Simu-l a t e d Quarter-Wave Transformers", IRE Trans., V o l . AP-4, pp. 3 3 - 3 9 , January, 1956. 14. Manning, L.A., "The Determination of Ionospheric E l e c t r o n D i s -t r i b u t i o n " , Proc. IRE, V o l . 35, pp. 1203-1207, November, 1947. 15. Noble, B., "Numerical Methods I I " , I n t e r s c i e n c e , New York, p. 210, 1964. : 183 16. N i e l s e n , K.L., "Methods i n Numerical A n a l y s i s " , The Macmillan Company, New York, Chapter 8, 1956. 17. Noble, op^ c i t . , pp. 240 -244. 18. Wilkes, M.V. , "In t r o d u c t i o n to Numerical A n a l y s i s " , Cambridge U n i v e r s i t y Press, pp. 4 3 - 4 6 , 1966. 1 9 . 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Hornbeck, J.A., "The D r i f t V e l o c i t e s of Molecular and Atomic Ions i n Helium, Neon, and Argon", Phys. Rev., V o l . 84, pp. 615-620, November, 1951. 27. B i o n d i , M.A., and Chanin, L.M., " M o b i l i t i e s of Atomic and Molecular Ions i n the Noble Gases", Phys. Rev., V o l . 9 4 , pp. 910-916, May, 1954. 28. E r o s t , L.S., " E f f e c t of V a r i a b l e Ionic M o b i l i t y on Ambipolar D i f f u s i o n " , Phys. Rev.. V o l . 105, pp. 554-356, January, 1957. 29. Chen, F.F.,> "Plasma Diagnostic Techniques", (edited by R. Huddlestone and S. Leonard), Academic Press, New York, Chapter 4, 1965. 30. Johnson, E.O., and Malter, L., "A F l o a t i n g Double Probe Method f o r Measurements i n Gas Discharges", Phys. Rev., V o l . 80, pp. 56-68, October, 1950. 31. Bohm, D., Burhop, E.H.S., Massey, H.S.W., " C h a r a c t e r i s t i c s of E l e c t r i c a l Discharges i n Magnetic F i e l d " , (edited, by A. Guthrie and R.K. Wakerling), McGraw-Hill Book Company, New York, Chapter • • • 2, 1 9 4 9 . 184 3 2 . 01sen, O.D. , and Skarsgard, H.M.,. "Probe Studies of. Helium Afterglows Dominated by Coulomb I n t e r a c t i o n s " , Can. J . Phys., V o l . 4 1 , pp. 3 9 1 - 4 0 4 , 1 9 6 3 . 3 3 . Abramowitz, M., and Stegun, I.A., "Handbook of Mathematical  Functions", Dover P u b l i c a t i o n s , Inc., New York, p. 593, 1965. 

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