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A high performance laser-excited interferometer for measuring electron densities Funk, Lynn Warren 1971

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A HIGH PERFORMANCE LASER-EXCITED INTERFEROMETER FOR MEASURING ELECTRON DENSITIES i by •' < Lynn Warren Funk B.Sc, University of Manitoba, 1965 M = S c , University of Manitoba, 1968 A Thesis Submitted i n P a r t i a l F ulfilment of the Requirements for the Degree o f Doctor of Philosophy i n the Department of Physics We accept t h i s thesis as conforming to the required standard The University of B r i t i s h Columbia August, 1971 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date ( i i ) ABSTRACT S i g n i f i c a n t improvements i n s e v e r a l aspects of l a s e r - e x c i t e d i n t e r f e r o m e t e r design have been i n c l u d e d i n t h i s instrument. The device u t i l i z e s p l a s m a - l i m i t i n g quartz tubes, and a concen-t r i c resonator design, to g r e a t l y s i m p l i f y i n t e r f e r o m e t r y i n the presence o f trans v e r s e g r a d i e n t s o f r e f r a c t i v e index, and improve beam o v e r l a p . The techniques o f f r a c t i o n a l f r i n g e s h i f t i n t e r f e r o m e t r y i n the time domain are incorporated, and provide order-of-magnitude improvements i n s e n s i t i v i t y ( J N d£ £ 5 x 1 0 l E cm - 2) and temporal r e s o l u t i o n (Aj ^  ±2 5 nsec.). E l e c t r o n i c c i r c u i t r y provides a d i r e c t , continuous readout o f e l e c t r o n d e n s i t y . Measurements have been made on a z-pinch discharge which are impossible i n any other way, confirming the usef u l n e s s o f the discharge as a sp e c t r o s c o p i c source, and r e v e a l i n g d e t a i l s of the p i n c h mechanism. ( i i i ) TABLE OF CONTENTS Page Abstract i i Table of Contents i i i L i s t of Figures v L i s t of Tables v i i Acknowledgements v i i i Chapter I Introduction 1 II The O p t i c a l Resonator 5 1 1 0 1 Introduction 5 1 1 0 2 Conventional Fabry-Perot Design Limitations 6 1 1 . 3 Structural I n s t a b i l i t y 13 1 1 . 4 Transverse Gradients i n Electron Density 14 III F r a c t i o n a l Fringe S h i f t Interferometry 21 I I I . l General P r i n c i p l e s 21 I I I o 2 C h a r a c t e r i s t i c s of the Reference Object 26 1 1 1 . 3 Fringe S h i f t Analysis 32 1 1 1 . 4 The integrating Frequency Modulation Detector 3 5 1 1 1 . 5 Other Systems of F r a c t i o n a l Fringe S h i f t Interferometry 4 8 IV The Z-Pinch 53 IV.1 General Description 53 IV.2 Discharge C h a r a c t e r i s t i c s 6 1 IV. 3 Experimental Objectives 6 2 V Experimental Procedures 6 4 V. l introduction 64 V.2 Resonator Alignment 64 V.3 Triggering C i r c u i t r y 6 8 V.4 Temporal and Radial P r o f i l e s 7 8 V.5 Discussion of Error and Rep r o d u c i b i l i t y 83 (iv) TABLE OF CONTENTS (cont'd.) Chapter Page VI Results 102 VI.1 introduction 102 VI.2 Summary of Results 102 VI. 3 The Model 104 VII conclusions 114 VII. 1 introduction 114 VII.2 A Summary of Resonator Improvements 114 VII.3 Observation of the Z-Pinch 117 Bibliography 120 Appendices I The Integrating Frequency Modulation Detector 122 II The i n t e r v a l Recognition Detector 126 III Hel Transitions and the Refractive Index 129 (v) LIST OF FIGURES Figure T i t l e Page I I - l Idealized Fabry-Perot 7 II-2 Mechanical Vibrations 10 II-3 Conventional Fabry-Perot showing o r i e n t a t i o n of Quartz Tubes 15 I I - 4 Optical Resonators 17 I I I - l Concentric Resonator with Reference Object 24 III-2 Quartz Block Ray Diagram 28 III-3 Integrating Frequency Modulation Detector 36 III-4 Signal Conditioning 38 III-5 Phase Discrepancy 44 III-6 Typical Oscilloscope Trace 46 I I I - 7 Moving Mirror Reference Objects 49 IV- 1 The Z-Pinch Discharge Schematic 54 IV-2 Vacuum Feedthroughs i n End Windows 57 IV- 3 Quartz Tube Mounts 59 V- l Discharge Tube and Interferometer 65 V-2 Triggering C i r c u i t r y 71 V-3 Interval Recognition Detector 73 V-4 Triggering Sequence 76 V-5 Electron Density D i s t r i b u t i o n i n the Discharge 79 V-6 Electron Density Temporal P r o f i l e 81 V-7 Electron Density Temporal P r o f i l e 81 V-8 Radial Electron Density P r o f i l e 84 V-9 Radial Electron Density P r o f i l e 84 V-10 Radial Electron Density P r o f i l e 84 V - l l Radius vs. Time Diagram for Current and Density Features 88 V-12 Radial Current D i s t r i b u t i o n 90 (vi) LIST OF FIGURES (cont'd,,) Figure T i t l e Page V-13 Magnetic F i e l d D i s t r i b u t i o n s 92 V-14 Magnetic F i e l d D i s t r i b u t i o n s 92 V-15 Magnetic F i e l d D i s t r i b u t i o n s 92 V-16 Maximum j x B vs, Time 96 V-17 Force on Current Shell Element vs. Radial P o s i t i o n at t = 7.0 u.sec. 98 VI-1 Log N_(l - r g / r ) vs. Log 4 r 109 AI-1 Integrating Frequency Modulation Detector C i r c u i t Diagram 124 AII-1 Interval Recognition Detector C i r c u i t Diagram 127 (vii) LIST OF TABLES Table Page I Z-Pinch Components 56 II Interferometer Components 69 AI Selected Hel Transitions 131 ( v i i i ) ACKNOWLEDGEMENTS I would l i k e to express my sincere appreciation to Dr. F o L . Curzon, for h i s guidance and encouragement during the course of t h i s experiment. I would l i k e to acknowledge the technical assistance of D.G. Sieberg" and J . Aazam-Zanganeh. Special thanks are due to D.M. Camm, a fellow graduate student, who graciously allowed me to draw on h i s e l e c t r o n i c s and computing experience. The contribution of T. Matthews, through h i s work i n the data reduction, i s g r a t e f u l l y acknowledged. The measurements of current and magnetic f i e l d appear i n t h i s thesis through the generosity and diligence of J. Pachner. The f i n a n c i a l support of the Atomic Energy Control Board of Canada i s g r a t e f u l l y acknowledged. - 1 -I. INTRODUCTION The Z-pinch discharge i s formed between two plane electrodes in a discharge vessel with cylindrical geometry. In this con-figuration, current flows i n i t i a l l y i n a thin shell on the inner boundary of the discharge vessel. Once a current path i s es-tablished in this shell, the current, _, grows u n t i l the azimuthal magnetic f i e l d , B, associated with i t , i s sufficiently large that the _ x B forces acting on the current begin to drive i t radially inward. This is the beginning of the collapse phase. The collapse continues u n t i l the current sheet has been brought to rest, near, or on the discharge axis. This i s termed the "pinch", and i t is from this effect that the device gets i t s name. The Z-pinch discharge has been the object of considerable scrutiny since i t was f i r s t introduced as a potential fusion reactor. Some of that interest has been generated by the mech-anics of the formation and collapse of the current sheet which gives i t i t s name. But, more significant perhaps, some of the interest has been directed toward using the discharge as a spectroscopic source. The advantages of the z-pinch as a spectroscopic source are many. Most important, i t i s one of the few devices in which large plasmas of high density and temperature can be formed. This makes i t useful for laboratory studies of many plasmas of astrophysical interest which are d i f f i c u l t or impossible to study in any other way. Almost as important is the fact that the device is relatively inexpensive, rugged, and produces plasmas with a very high degree of reproducibility. Finally, i t s dynamics are generally well understood. - 2 -Sadly, however, up to t h i s time i t has been impossible to u t i l i z e the l i n e a r pinch discharge as a c a l i b r a t e d spectroscopic source, because no diagnostic techniques existed for the accurate determination of those plasma parameters ( i . e . the electron density and temperature) upon which spectroscopic measurements are based. It was the objective of t h i s research, therefore, to devise a measuring system which would allow determination of electron density to an accuracy s u f f i c i e n t to permit the c a l i b r a t i o n of the pinch discharge as a spectroscopic source. This objective has been achieved. The s t a r t i n g point from which the desired diagnostic probe was developed i s the l a s e r - e x c i t e d Fabry-Perot interferometer, which w i l l hereafter be r e f e r r e d to as the las e r interferometer or ju s t as the interferometer. The introduction of the plasma into the interferometer changes the o p t i c a l path length between the resonator mirrors, by reason of the dependence of plasma r e f r a c t i v e index on electron density. The r e s u l t i n g changes i n the interferometer fringe patterns are used to reveal the electron density. The basic interferometer can be made to have a good high frequency response and so appears suited to diagnostics of pulsed d i s -charges l i k e the Z-pinch i n which the electron density i s varying r a p i d l y . There are, however, many l i m i t a t i o n s to be overcome before the interferometer can be made capable of the p r e c i s i o n measurements we w i l l demand of i t . These problems and t h e i r solutions form the bulk of the th e s i s . The f i r s t problem to be overcome i s the inherent i n -s t a b i l i t y of conventional interferometers against gradients i n electron density normal to the resonator axis. The elimination - 3 -of t h i s unsatisfactory feature by reducing the plasma length and improving the resonator geometry forms the basis of Chapter I I . The most important l i m i t a t i o n on the s e n s i t i v i t y of the interferometer l i e s i n i t s i n a b i l i t y to resolve changes i n the phase of the output signal of less than T T . This l i m i t a t i o n makes i t very d i f f i c u l t to observe slow changes of electron density. In addition the basic interferometer gives no i n d i c a t i o n of the sign of the change i n electron density. The overcoming of these l i m i t a t i o n s by a p p l i c a t i o n of the techniques of f r a c t i o n a l fringe s h i f t interferometry i n the time domain are described i n the f i r s t part of Chapter I I I . F i n a l l y , the output of a conventional Fabry-Perot i n t e r -ferometer requires a good deal of analysis, before the electron density can be extracted from the fringe patterns. The u t i l i -z ation of the f r a c t i o n a l fringe s h i f t techniques, however, encourages the use of automatic data analysis. The output of an interferometer incorporating these techniques consists of a burst of high frequency o s c i l l a t i o n s , i n which the e f f e c t of the changing electron density appears as a frequency modulation. An e l e c t r o n i c c i r c u i t has been constructed to perform the necessary frequency demodulation i n real-time, providing an output voltage which i s at a l l times d i r e c t l y proportional to the electron density i n the resonator. This, then, i s the s i g n i f i c a n t contribution of these in v e s t i g a t i o n s . The author has developed a direct-recording high s e n s i t i v i t y , high r e s o l u t i o n diagnostic device for the measurement of electron d e n s i t i e s . The device r e t a i n s the high frequency response which i s the c h i e f advantage of l a s e r -excited interferometers. I As was out l i n e d e a r l i e r , the j u s t i f i c a t i o n for the work i s the c a l i b r a t i o n of the Z-pinch as a spectroscopic source, and measurements were made on a t y p i c a l Z-pinch discharge to demon-strate the c a p a b i l i t i e s of the diagnostic device, The description of the plasma source i s found i n Chapter IV, and the ou t l i n e of the experimental procedures, presentation of data, and a discussion of sources of error i s found i n Chapter V. An analysis of the r e s u l t s and a discussion of a model for a portion of the collapse phase can be found i n Chapter VI. A summary of the important r e s u l t s , and some suggestions for future research are found i n Chapter VII. Two Appendices, dealing with the d e t a i l s of the e l e c t r o n i c c i r c u i t r y , and a t h i r d Appendix, estimating the contribution of Hel t r a n s i t i o n s to the change i n r e f r a c t i v e index, conclude the t h e s i s . - 5 -II The Optical Resonator II. 1 Introduction The optical resonator used in this experiment is of the Fabry-Perot type. In principle, Mach-Zehndsr and Michelson interferometers have somewhat better time resolution. However, the Fabry-Perot interferometer has the great me.rits of simple construction and easy alignment, and since _ g resolution times better than 10 sec. are not required, the choice was made to continue the evolution of the Fabry-Perot interferometer developed 1 2 at this Plasma Physics Laboratory. ' This chapter w i l l deal with the basic principles governing the operation of such a device, and with some of the drawbacks and d i f f i c u l t i e s which are encountered when i t is used to measure electron densities i n a z-pinch. It w i l l be seen that of the four serious problems enumerated, two are i n t r i n s i c to the resonator, while two are associated with the particular experimental arrangement. The responses made to overcome the latt e r w i l l be discussed in some detail, and the work done to alleviate the former w i l l be presented in Chapter III. - 6 -II.2 Conventional Fabry-Perot Design Limitations The l i m i t a t i o n s of a conventional Fabry-Perot soon became apparent when i t was used to measure electron density i n the collapse phase of a Z-pinch. The schematic of F i g . II-l(a) shows the e s s e n t i a l features of an i d e a l i z e d Fabry-Perot interferometer, and i l l u s t r a t e s the method used to apply the device to the measurement of electron densities i n a time-dependent plasma. Introduction of the plasma into the resonant c a v i t y a l t e r s n, the index of r e f r a c t i o n . This, i n turn varies the phase difference between the emergent beams (1 and 2 i n F i g . I I - l ) , thereby modulating the beam i n t e n s i t y observed by the photo-m u l t i p l i e r . As i s well known, the electron density i s r e l a t e d to the index of r e f r a c t i o n by the r e l a t i o n : e 2N n 2 = 1 - V (2.1) where n i s the r e f r a c t i v e index, e i s the e l e c t r o n i c charge, m i s the e l e c t r o n i c mass, e i s the d i e l e c t r i c p e r m i t t i v i t y o f e o free space, UJ i s the frequency of the r a d i a t i o n which i s used to o probe the plasma (in t h i s case the frequency of the 6328A t r a n s i t i o n i n neon, 3 x 1 0 1 5 s e c - 1 ) , and N i s the electron e density. Equation (2.1) i s v a l i d for the values of UJ far away from any c h a r a c t e r i s t i c frequency of the plasma, including c o l l i s i o n frequencies, plasma frequencies or atomic t r a n s i t i o n s . Knowledge of the plasma geometry, and of the way i n which the electron density varies along the resonator axis, can be combined with the change i n output i n t e n s i t y of the resonator to y i e l d the electron density. - 7 -Pig. I I - l Idealized Fabry-Perot (a) conventional plane - plane resonator, showing the e f f e c t of transverse gradients i n electron density. (b) cross section of laser beams at point A, showing separation of beams having d i f f e r e n t numbers of t r a n s i t s (beams 1 and 2). (c) dependence of transmittance (T) on change of phase, 0, for Fabry-Perot interferometer. Mx, M3 - laser c a v i t y mirrors M 2 / M 3 ~ resonator c a v i t y mirrors (both plane i n the conventional interferometer.) I A PLASMA - 9 -In a Z-pinch, N g i s best measured by passing the resonator axis p a r a l l e l to, but not nece s s a r i l y coincident with, the pinch axis, because along such d i r e c t i o n s i s constant along the l i n e of sight. This g r e a t l y s i m p l i f i e s i n t e r p r e t a t i o n of the experimental r e s u l t s . For a further discussion of t h i s feature, see Seco V-2. Figure II-l(a) reveals some of the fundamental l i m i t a t i o n s of the device. These are discussed below: (a) Extraneous o p t i c a l path length v a r i a t i o n I t i s c l e a r l y e s s e n t i a l that changes i n Jnd-t be due to the changing electron density and not to mechanical movement of the two mirrors M2 and M3 with respect to one another. Fortunately, the equipment used i n t h i s experiment has a charac-t e r i s t i c o s c i l l a t i o n period and amplitude which r e s u l t s i n an output modulation at approximately 1 KHz (see F i g . II-2). Since the discharge duration i s 50 u.sec, of which only the f i r s t 20 Lisec. i s of in t e r e s t , phase changes due to mechanical v i b r a t i o n during the observation period may be ignored. (b) Beam d e f l e c t i o n I t i s assumed that the l i g h t beams r e f l e c t e d between M2 and M3 overlap. When overlap i s poor (see F i g . I I - l ( b ) ) , the c a l c u l a t e d electron density i s not r e a d i l y r e l a t e d to the e x i s t i n g density along e i t h e r beam path. Furthermore, since the two beams can only i n t e r f e r e where they overlap, poor overlap reduces the degree of modulation possible. Complete beam over-lap can be assured i f the beams are always normally incident on the mirrors. I t i s c l e a r that the laser beam w i l l be r e f r a c t e d - 10 -F i g . II-2 Mechanical Vibrations This diagram i s an enlargement of a photo taken of the output of PM2, showing the e f f e c t of mechanical v i b r a t i o n s . V e r t i c a l S e n s i t i v i t y = 250 m V./div. Sweep Speed = 10 msec./div. (a) ground l e v e l (b) the l a s e r output with the resonator detuned (c) the l a s e r output with the resonator tuned - 12 -by gradients i n electron density normal to i t s d i r e c t i o n of propagation. I f M2 and M3 are plane mirrors, the resonator stops working even for a very small beam d e f l e c t i o n (for a 5 m. long resonator and a 2 mm. wide beam, the cu t o f f angle i s 0 .10 mrad.) . (c) Dependence of transmission on path length The dependence of the transmission of a simple Fabry-Perot interferometer on o p t i c a l path length i s i l l u s t r a t e d i n F i g . I I - l (c) . This dependence i s far from l i n e a r . The r e s u l t i s that the phase i s only known well at the points of peak transmission. In addition, the transmission i s found to be dependent on ca v i t y losses and the rate of change of c a v i t y length 4 5 . (d) The sign of dN_/dt The device i l l u s t r a t e d i n F i g . II-l(a) provides no i n d i c a t i o n of the sign of the change i n [ndt and hence on the sign of dN^/dt. To recapitulate, the four c h i e f l i m i t a t i o n s of conventional Fabry-Perot design as applied to the problems of electron density measurement are: a) Structural i n s t a b i l i t y b) I s n t a b i l i t y against transverse dN e/dr c) Non-uniform dependence of the t r a n s m i s s i v i t y change on o p t i c a l path length change. d) Uncertainty i n the sign of dN^/dt. These factors are considered i n more d e t a i l below. - 13 -II.3 Structural I n s t a b i l i t y Intensity modulations i n the resonator output due to mechanical vi b r a t i o n s are at too low a frequency to i n t e r f e r e d i r e c t l y with the measurement of plasma electron density. This fact was established i n Sec. 11.2(a)„ However, they make the phase of the system at the s t a r t of a measurement uncertain. The e f f e c t of t h i s i s i l l u s t r a t e d i n F i g . I I - l ( c ) . Consider the s i t u a t i o n which a r i s e s i f the resonator i s located at point A on t h i s diagram. Assuming that the electron density w i l l go from zero to some p o s i t i v e value, &Jndl w i l l be negative, and the i n t e n s i t y w i l l respond immediately to the changing phase. I f the resonator were to s t a r t from B, however, the electron density and the phase of the system could change s u b s t a n t i a l l y before the tra n s m i s s i v i t y of the resonator would respond. Since one can use only the peaks i n the tran s m i s s i v i t y as benchmarks for the phase i n c a l c u l a t i n g the electron density, one i s l e f t with an uncertainty of ± T i n the phase change required to produce the f i r s t observed peak i n the transmitted i n t e n s i t y . Since the t o t a l phase change observed during a run i s IOTT or les s , t h i s contributes quite a sizeable error to the peak electron density measurement. The lack of mechanical r i g i d i t y has one b e n e f i c i a l side e f f e c t . The fluctuations i n output i n t e n s i t y due to mechanical vib r a t i o n s are used to simp l i f y the i n i t i a l alignment procedure (see Sec. V o l ) . - 14 -II„4 Transverse Gradients i n Electron Density As was o u t l i n e d e a r l i e r , a gradient i n electron density normal to the resonator axis i s also a gradient i n the index of r e f r a c t i o n which def l e c t s the beam passing through i t . The magnitude of the d e f l e c t i o n depends on the size of the gradient and the path length of the beam i n the plasma. Given the geometry of the Z-pinch discharge and the advantages . of a l i g n i n g the interferometer p a r a l l e l to the axis of symmetry of the discharge, there i s l i t t l e that can be done about the magnitude of the gradient. However, the length of plasma i n the resonator can be shortened by the quartz tubes shown i n F i g . I I - 3 . 2 The l a s e r beam passes through the tubes, and as the discharge column collapses, a length of plasma i s intercepted between t h e i r ends. The separation of the tubes can be adjusted to l i m i t the amount of r e f r a c t i o n so that i t can be handled by the mirror system. A plane-plane mirror configuration, however, can accept only a very small angular deviation of the beam (for our system, 0.1 mrad.). (see F i g . II-4(a)). A piano-spherical mirror system 2 can accommodate much la r g e r angles, but only i f the r e f r a c t i o n can be l o c a l i z e d . The use of the quartz tubes to reduce the plasma length f u l f i l l s t h i s condition. I f the gradient of electron density i s uniform along the ray, and i f the angle of r e f r a c t i o n i s s u f f i c i e n t l y small, r e f r a c t i o n can be considered to be occurring at a point i n the centre of the plasma. We s h a l l r e f e r to t h i s point as the r e f r a c t i o n point (denoted i n F i g . II-3, as R.P.). I f the interferometer e x i t mirror, M3, i s spherical, with i t s centre of curvature at t h i s point, any ray - 15 -F i g . I I - 3 Conventional Fabry-Perot showing or i e n t a t i o n of Quartz Tubes MT , M2 , M3 - plane mirrors - 17 -F i g . II-4 Optical Resonators (a) Conventional plane-plane mirror configuration, including plasma to show e f f e c t of r e f r a c t i o n by transverse gradients i n electron density. The cross-section of output beams i n plane A demonstrates beam overlap. (b) Piano-spherical mirror configuration. The cross-sectional view shows poor overlap of beams 1 and 2, with poor interference as a r e s u l t . (c) Concentric mirror configuration. C o r r e c t l y aligned, the overlap of beams 1 and 2 should be exact, as shown i n cross-section A. Interference occurs only where beams 1 and 2 overlap, M2, M3 - p a r t i a l l y transmitting mirrors. L - converging lens. - 19 -which can c l e a r the confining quartz tube w i l l be normally incident on the e x i t mirror. I t now becomes.important to insure that the centre of curvature of the e x i t mirror i s located at the r e f r a c t i o n point, for i f i t i s not, the plane interferometer entrance mirror (Ma i n F i g . II-3) w i l l accentuate any r e f r a c t i o n s that occur, since the beam w i l l not return along i t s ingoing t r a j e c t o r y . Although the use of a piano-spherical mirror system i n -sures overlap of the emergent beams for larger angles of re f r a c t i o n , the q u a l i t y of the interference produced i s much l e s s . F i g . II-4 (b) shows the beam dimensions to be expected with such a resonator. Beam 2 i s spread over a much larger area than beam 1, with the r e s u l t that much of beam 2 i s l o s t . The f luctuations i n output i n t e n s i t y are therefore much smaller since t h i s parameter i s determined by the i n t e n s i t y of the weaker beam (beam 2) . Further s t a b i l i t y can be achieved by making both M2 and MQ spherical mirrors with t h e i r centres of curvature at the centre of the plasma. Ma i s r e a d i l y converted to a spherical mirror by mounting a long f o c a l length converging lens i n front of i t . In t h i s way, any ray which i s moving o f f axis as i t approaches M2 w i l l be bent back towards the axis. This condition could r e s u l t , as discussed above, from a misalignment of the mirror M3. The lens i s placed with i t s f o c a l point at the r e f r a c t i o n point. The use of the concentric mirror design has not only reduced the e f f e c t of transverse gradients i n electron density, but i t has also improved the degree of i n t e n s i t y modulation which can be achieved i n the output beam. This i s i l l u s t r a t e d i n F i g . II-4 (c) . In t h i s configuration, good alignment w i l l - 20 -r e s u l t i n complete overlap of beams one and two, at maximum possible energy density for each.. The use of quartz tubes, and the conversion to a concen-t r i c mirror system has o f f s e t the problems associated with transverse gradients i n electron density. The problem of the determination of the sign of dN^/dt, and of the non-uniform dependence of the change i n t r a n s m i s s i v i t y of the resonator on changes of o p t i c a l path length w i l l be dealt with i n the next chapter. - 2 1 -III Fractional Fringe Shift Interferometry I I I . l General Principles As was outlined in Sec. I I . 3 , attempts to measure directly the phase change produced by variations in electron density by observations of variations in transmitted intensity can be frustrated. This is because the change in transmission for a given change of phase depends on the unknown i n i t i a l phase. Where the phase change produced by the plasma is many multiples of 2TT, the percentage error due to the unknown i n i t i a l phase is small. In this experiment this method of minimizing the problem i s of very limited usefulness. The reason for this i s clear from the discussion of Sec. I I . 4 . To increase the change in phase, the plasma length has to be increased, which increases the refraction of the laser beam. Even with a concentric resonator of the type discussed above, the permissible beam deflection i s limited by the dimen-sions of the quartz tubes through which i t must pass and by the dimensions of the mirror. It is important, therefore, to keep the plasma length small, and as a result, the number of 2TT changes in phase is also small. We are therefore faced with the problem of measuring small changes in phase with a device which cannot readily resolve changes of phase smaller than 2 T T . A similar problem has been overcome by those workers who construct interferograms or holograms of transparent objects 8. The technique used involves the introduction of a second phase object (which w i l l be defined as the reference object) into the system. - 22 -This reference object produces s u f f i c i e n t l y large changes i n phase to create many interference fringes i n a regular pattern over the region where the t e s t object (in t h i s case, the plasma) produces only a few fringes. I f an interferogram i s taken with both objects i n the interferometer the t o t a l phase change w i l l be the sum of the phase changes due to each object. As a r e s u l t , regular fringes produced by the reference object w i l l be di s t o r t e d . From the d i s t o r t i o n , the phase s h i f t s produced by-the t e s t object are e a s i l y deduced. Thus, changes i n phase of TT/10 can be measured with ease. The problem i n t h i s experiment i s sim i l a r , but the difference i s also of i n t e r e s t . In t h i s case, the objective i s not the measurement of small changes i n phase as a function of posit i o n , but rather as a function of time. The "reference object" i n t h i s case must be one which produces a regular pattern of interference fringes i n time. The interference fringes are, of course, the o s c i l l a t i o n s of the tr a n s m i s s i v i t y of the o p t i c a l resonator at the la s e r wavelength. Therefore, a reference object i s required which w i l l produce a regular v a r i a t i o n i n the o p t i c a l path length between the mirrors M2 and M3 (Fig. II-4 (c)) which constitute the resonator. The simplest way to create such a reference object i s to impose a r e l a t i v e motion on the two mirrors. The easiest way of doing t h i s i s to move the e x i t mirror of the resonator. Various workers 7' 1 3 have mounted t h e i r return mirrors (M3 i n t h i s experiment) on loudspeaker cones, on p i e z o - e l e c t r i c c r y s t a l s , or on ro t a t i n g tables. The method i n use i n t h i s experiment i s somewhat d i f f e r e n t , and i t i s believed, both neater and more e f f i c i e n t . - 23 -The procedure has been to introduce a quartz block i n t o the resonator, and to rotate i t about an axis normal to the resonator axis (see F i g . I I I - l ) . As the angle of incidence of the lase r beam on the face of the quartz block (QB) changes, so too does the o p t i c a l path length within the resonator. This produces the required r a p i d o s c i l l a t i o n s i n the resonator tra n s m i s s i v i t y . As the plasma (PL i n F i g . I I I - l ) enters the resonator, i t w i l l change the o p t i c a l path length i n the resonator, and the frequency of the o s c i l l a t i o n s of output i n t e n s i t y . Knowing the sign of the change i n o p t i c a l path length produced by the r o t a t i n g quartz block, i t i s possible to deduce, from the sign of the change i n output frequency, the sign of dN e/dt. So long as the o s c i l l a t i o n frequency of the output due to the r o t a t i o n of the quartz block exceeds that produced by the changing electron density, there i s no ambiguity i n assigning changes i n fringe spacing (or period) to the influence of the v a r i a t i o n s i n electron density. - 24 -F i g . I I I - l Concentric Resonator with Reference Object QB - quartz block L i ' 2 ~ converging lenses with f o c a l points at R.P„ R.P. - Refraction point QT - Quartz tubes PL - Plasma column I - Interference f i l t e r PM - Photomultiplier PD - Photodiode TO TRIGGERING ELECTRONICS PHOTODIODE REFRACTION POINT QUARTZ TUBES M - 26 -III.2 C h a r a c t e r i s t i c s of the Reference Object In the preceding section we have o u t l i n e d the techniques of f r a c t i o n a l fringe s h i f t interferometry i n the time domain. This section w i l l deal with the c a l c u l a t i o n of the rate of change of o p t i c a l path length to be expected from the reference object, the ro t a t i n g quartz block. The quartz block i s mounted i n the resonator between the las e r e x i t mirror (M2) and the converging lens (L x) which together constitute a spherical mirror (Fig. I I I - l ) . Since the resonator axis i s horizontal, the axis of r o t a t i o n of the quartz block has also been made horizontal, to f a c i l i t a t e alignment. Rotation of the quartz block w i l l r e s u l t i n a v e r t i c a l displacement of the laser beam ( i l l u s t r a t e d i n F i g . I I I - l ) . This does not a f f e c t the resonator, however, as the converging lens (L1) corrects for t h i s displacement. In order to be sure that no horizontal d e f l e c t i o n occurs (which would r e s u l t i n the laser beam moving from one r a d i a l p o s i t i o n to another as i t moved through the plasma), the portion of l a s e r l i g h t r e f l e c t e d from the front surface of the quartz block (seen s t r i k i n g the photodiode i n F i g . I I I - l ) i s made to pass through the l a s e r e x i t spot on M2 as the block i s rotated. The above discussion assumes that the faces of the quartz block through which the beam passes are accurately plane and p a r a l l e l . The c h a r a c t e r i s t i c s of the quartz block used are: Mate r i a l : Fused Quartz Dimensions: 1.25 cm. x 2.50 cm. x 3.00 cm. Plan a r i t y : large faces plane to X/20 Par a l l e l i s m : large faces p a r a l l e l to 1". - 27 -T h e c h a n g e i n o p t i c a l p a t h l e n g t h d u e t o r o t a t i o n o f t h e q u a r t z b l o c k c a n b e e a s i l y c a l c u l a t e d ( s e e F i g . I I I - 2 ) . T h e o p t i c a l p a t h l e n g t h (L) b e t w e e n p o i n t s Ax a n d A a i s g i v e n b y : L = n 1 A 1 A 2 - n 1 B 1 C 1 + n a B 1 B 3 ( 3 . 1 ) w h e r e n 1 = r e f r a c t i v e i n d e x o f a i r a n d n a = r e f r a c t i v e i n d e x o f f u s e d q u a r t z . A s i s s t a n d a r d , l e t n b e t h e i n d e x o f r e f r a c t i o n o f q u a r t z r e l a t i v e t o a i r ( n x = 1.000) w h i c h g i v e s : L = A1hs - BlC1 + n B 1 B a ( 3 . 1 a ) w h e r e B X B 2 = . a n d B l C l = D C O S ( r P ) ( s e e F i g . I I I - 2 ) 1 1 cos(3 S u b s t i t u t i n g f o r B X B 2 a n d B 1 C 1 i n t o ( 3 . 1 a ) , we g e t : T Ti ^ D c o s ( g - 3 ) nD L = A , A 2 - ^ Q r + — ( 3 . 1 b ) 1 c o s f i cos£3 E x p a n d i n g c o s (ct-f3) a n d d i f f e r e n t i a t i n g w i t h r e s p e c t t o a l e a d s t o t h e e x p r e s s i o n : d L n P s i n g dB _ . dB — = j + D s i n a - D s i n a ^ d a c o s B d a d a D s i n B c o s g _ D s i n a s i n 5 B dp cos[3 c o s s B d a F r o m S n e l l ' s Law, we h a v e : s i n a = n s i n B ( 3 . 3 ) w h i c h , a f t e r some m a n i p u l a t i o n , y i e l d s : 1 i c o s B = - ( n 3 - s i n 2 a ) s ( 3 . 3 a ) n • o 1 s r n p = - s i n a ( 3 . 3 b ) - 28 -F i g . I l l - 2 Q u a r t z B l o c k R a y D i a g r a m B e a m e n t r a n c e a n d e x i t p o i n t s o n B i > 2 q u a r t z b l o c k f a c e s a - b e a m a n g l e o f i n c i d e n c e D - q u a r t z b l o c k t h i c k n e s s - 30 -and a f t e r d i f f e r e n t i a t i o n w i t h respect to a g i v e s : dp _ cosa da ncosp . S u b s t i t u t i n g (3.3a) i n t o (3.3c) we get: dft _ cosa  (n - s m 3a) S u b s t i t u t i o n o f (3.3a), (3.3b) and (3„3d) i n t o (3.2) g i v e s : dL n 2Dsingcosa (3.3c) (3.3d) + Dsina (3.4) (n 2 - s i n 2 a ) Equation (3.4) reduces t o : dL ^ . f. cosa 1 _. — = DsinaHl - TY (3.5) ^ L (n 2 - s i n 2 a ) 2 j To determine the frequency at which the output o f the o p t i c a l resonator w i l l o s c i l l a t e , r e c a l l t h a t an in c r e a s e i n the l e n g t h o f a double t r a n s i t o f the resonator ( i . e . M s to M3 to M 2 i n F i g . I I I - l ) of X w i l l produce a complete c y c l e o f the output. A change o f X/2 i n L i s e q u i v a l e n t . The number o f c y c l e s dN through which the output o s c i l l a t e s f o r an angular r o t a t i o n da i s , t h e r e f o r e : dN _ 2. j3L _ 2Dsina f, cosa -> . . da ~ X da " X t , a • 3 M ( (n 3 - sm 2a) S J F i n a l l y , the o s c i l l a t i o n frequency w i l l be, from the c h a i n r u l e o f d i f f e r e n t i a t i o n : *° = x i i (3.7, Since the p e r i o d of r o t a t i o n (T) of the quartz b l o c k i s t h a t observable most e a s i l y measured, we s u b s t i t u t e : - 31 -da 2TT dt ~ T i n t o (3.7) to get: 4irDsina f, cosa (3.8) f = h cosa A (3.9) *- ... 2 • _ 2 v c-J X T " (n* - s i n a a p I t should be noted t h a t the di s t a n c e B 2C T i n F i g . I I I - 2 can be w r i t t e n as: B s C s R = Dsin(a-p) ' 3 1 cosp • Dsinficosa ._ l r i. R = Dsina - (3.10) cosf3 S u b s t i t u t i n g f o r sin{3 (eqn. (3.3b)) and cosp (eqn.(3,3a)) we get: R = Dsina { l - r} (3.11) L (n 2 - s i n 2 a ) ^ J Comparison w i t h equation (3.9) g i v e s : This way o f expressing the r e l a t i o n between the observed frequency (f ) and the quartz b l o c k angular p o s i t i o n (a) and v e l o c i t y (2TT/T) i s u s e f u l because i t g i v e s an easy way t o c a l c u l a t e the magnitude o f the frequency to be expected. With a b l o c k t h i c k n e s s (D) of 1.25 cm. i t i s p o s s i b l e to get a v e r t i c a l d e f l e c t i o n (R) on the order of 0.8 cm. f o r a == 70°. The quartz b l o c k i s d r i v e n by a motor which g i v e s T = 5 x 10~ 3 s e c , y i e l d i n g an o s c i l l a t i o n frequency i n excess of 30 MHz. A s i m i l a r modulation frequency c o u l d be produced by d i s p l a c i n g one o f the resonator m i r r o r s along the resonator a x i s a t a v e l o c i t y of 10 m. s e c - 1 . Some o f the disadvantages - 32 -of such apparently simpler methods of producing reference fringes are discussed i n Sec„ III.5. III.3 Fringe S h i f t Analysis In Sec. I I I . l the p r i n c i p l e s of f r a c t i o n a l fringe s h i f t interferometry i n the time domain have been discussed. In Sec. I l l . 2 i t was shown that the output of the interferometer which includes a r o t a t i n g quartz block consists of a t r a i n of high frequency o s c i l l a t i o n s . In view of the rate at which the resonant c a v i t y length i s being modulated, these o s c i l l a t i o n s are nearly s i n u s o i d a l 4 5„ Discharge of the Z-pinch during such a burst of o s c i l l a t i o n s produces a changing electron density i n the resonator volume which modulates the o s c i l l a t i o n frequency. The temporal development of the electron density can be determined by measuring the change i n o s c i l l a t i o n frequency. This i s done as follows: The rate of o s c i l l a t i o n of the output of the interferometer (f) i s given by: <3-"> where L i s tht t o t a l o p t i c a l length of the resonator, and X = 6328 A. The length L i s made up of two parts, the f i r s t of which i s the "geometric" length L x, which i s the geometric distance between the resonator mirrors, and the additional o p t i c a l path through the quartz block. I f the geometric inter-mirror distance i s considered to be constant (see Sec. III.2a), then the only time-dependent length i n L x i s the path through the quartz block. - 33 -The second part, L 2 , of L i s the departure from the "geometric" length due to fluctuations i n r e f r a c t i v e index of the plasma i n the resonator. Let the geometric length of the plasma be Z, which gives: L 2 = (n-1)Z (3.14) and f = ~ ~ (L, + (n-l)Z) (3.15) cLL But -—^ has been c a l c u l a t e d i n Sec. I I I . 2 and i s given by dt eqn. (3.9) whence: 2 dL, 4TrDsina f, cosa {l - ^ A (3.16) fo X dt XT v , 3 . 2 , ^ . (n - sm^a) -In addition, Z i s f i x e d by the spacing of the quartz tubes, and (3.15) reduces to: 07 ri f = f + I T £ (n-D (3.17) o X dt The dependence of r e f r a c t i v e index on electron density i s well-known 1 4 and i s given by: e sN n 3 = 1 V (3.18) m e ur e o where n i s the r e f r a c t i v e index, m i s the e l e c t r o n i c mass, e e i s the e l e c t r o n i c charge, E q i s the d i e l e c t r i c p e r m i t t i v i t y of free space, uu i s the frequency of the probing radiation, and N i s the electron density. For N = 10 1 8 cm"3, a value e e much higher than any achieved i n t h i s experiment, and for uu = 2.99 x 1 0 1 5 s e c . - 1 , the quantity K = — - — v = 3.5 x 10" 2 2 cm.3 m E to2 e o - 34 -which makes KN = 3.56 x 10" 4. This quantity i s s u f f i c i e n t l y e small that we may do a binomial expansion of i n = (1 - KN ) 2 e and ignore a l l but the f i r s t - o r d e r terms. Thus, n = 1 - |KN (3.19) e and dN £ i " - 1 ' - - * * - d t * ( 3 - 2 0 ) Substitution of (3.20) into (3.17) gives: dN f = f o + f ( f ) ^ f (3-21) dN e and, solving for ^ , we get: dN - d f " ( f o " f > i i < 3 - 2 1 a ) Let C = X/K = 17.6 2 x 10 1 6 cm."3. Then, dN - d f = f ( f o - f > ( 3 ° 2 1 b ) Integrating (3.21b) with respect to time we get: t (t) = ^ I e which can be stated as: the instantaneous value of the electron density i s proportional to the t o t a l phase discrepancy between the perceived o s c i l l a t i o n and that which could be expected i n the absence of a plasma. The constant C Q i s set equal to zero by se t t i n g the o r i g i n of time at the instant when the voltage i s applied to the discharge tube. The electron density at t = o i s zero. N (t) = ~ fQ(£Q - f)dr + C q (3.22) - 35 -III.4 The Integrating Frequency Modulation Detector Equation (3.22) can be rewritten as: r t c t N (t) = ~ r f (T)dr - -r r f ( T ) d T ( 3 . 2 3 ) e Z J 0 ° z o The electron density can be computed i f i t i s possible to make a continuous comparison of the phase of the observed s i g n a l with the phase of the s i g n a l due to the reference object alone. Since these two signals are not simultaneously a v a i l a b l e from the same resonator, i t was decided to construct a c i r c u i t to perform int e g r a t i o n of f ( T ) , and to then manufacture the i n t e g r a l of f Q ( T ) e l e c t r o n i c a l l y , assuming f Q ( T ) to be constant. The e l e c t r o n i c device constructed to perform t h i s function i s described i n d e t a i l i n Appendix I. A functional block diagram appears i n F i g . III-3. The input s i g n a l i s the high-frequency, quasi-sinusoidal o s c i l l a t i o n denoted by f ( T ) i n (3.23). This s i g n a l i s amplified and an i d e n t i c a l s i g n a l of opposite phase i s generated. These two signals are used to t r i g g e r separate b i s t a b l e multivibrators or " f l i p - f l o p s " . These f l i p - f l o p s are turned "ON" by a positive-going zero-crossing i n the input voltage, and are turned "OFF" by the next positive-going zero-crossing. Thus the two f l i p - f l o p s are o s c i l l a t i n g at h a l f the frequency of the input s i g n a l , and t h e i r outputs d i f f e r i n phase by TT/4 . The signal conditioning process i s g r a p h i c a l l y i l l u s t r a t e d i n F i g . III-4. Each f l i p - f l o p has two antiphase outputs. The four signals thus available are used to trigger four separate monostable multi-vibrations, or "one-shots". The reason for t h i s choice of l o g i c c i r c u i t r y l i e s i n the f a c t that only the amplifiers are required to function at the f u l l input frequency; a l l other components need work at only h a l f the input - 36 -F i g . III-3 Integrating Frequency Modulation Detector. INPUT SIGNAL FL IP -FLOP (I) OUTPUT Q F L I P - F L O P ( I ) OUTPUT Q FL IP - FLOP (2) OUTPUT Q F L I P - F L O P (2) OUTPUT Q ONE-SHOT( l ) OUTPUT ONE-SHOT (2) OUTPUT 0 N E - S H 0 T ( 3 ) OUTPUT ONE-SHOT (4) OUTPUT - 38 -F i g . III-4 Signal Conditioning GATE PULSE — • IN SIGNAL IN AMPLIFIER AND . PHASE INVERTER I FLIP-FLOP (I) r ONE-SHOT (I) ONE-SHOT (2) ONE-SHOT ONE-SHOT CONSTANT CURRENT SOURCE INTEGRATOR SIGNAL OUT - 4 0 -frequency. The system therefore can handle high frequency data using r e l a t i v e l y low-frequency components. The outputs of the four one-shots are found to be independent of t r i g g e r i n g rate up to 1.8 x 10 7 pulses per second. This corresponds to an input frequency of 3 6 MHz., much higher than any used i n pr a c t i c e . The one-shot outputs are inverted and fed into the integrator. I t should be noted that one of the four one-shots f i r e s for each zero-crossing of the input s i g n a l . The integrator design i s the creation of D.M. Camm, a fellow graduate student. The p r i n c i p l e behind the design i s that each one-shot pulse can be used to draw the same quantity of charge from a charged capacitor. Thus, i f the input frequency i s constant, the current being drawn from the capacitor w i l l be constant. I f , at the same time, the same constant current i s being fed onto the capacitor from another source, the voltage on the capacitor w i l l be constant. Should the input frequency (f) decrease, the capacitor w i l l begin to accumulate charge. I f f increases, the capacitor w i l l begin, to lose charge. The charge on the capacitor w i l l continue to change as long as f d i f f e r s from that value which i s required to balance the current input ,from the constant current source. I f t h i s frequency i s defined as the base frequency ( f ^ ) / then i t i s cl e a r that the current flowing onto the capacitor w i l l be proportional to the difference, f^ - f. Further, the change of voltage across the capacitor over a time •i n t e r v a l At w i l l be proportional to the t o t a l change i n phase between f^ and f during the time i n t e r v a l At. The e l e c t r o n i c device performs the integ r a t i o n : V(t 3) - V(t,) = Av = C, J* (f - f)dx ( 3 . 2 4 ) - 41 -Equation (3.24) i s i d e n t i c a l to equation (3.22), except for the subst i t u t i o n of f, for f . The value of f, , determined b o b by the magnitude of the constant current fed onto the integrating capacitor, i s time-independent. The value of f , determined by the angular p o s i t i o n and frequency of r o t a t i o n of the quartz block i s a time-dependent quantity. The value of N g(t) can be ca l c u l a t e d from equations (3.22) and (3,24) as follows: From equation (3.24).: t t - J f ( T ) d T = - J ( f b ( T ) d T (3,25) o 1 o which, on su b s t i t u t i o n into (3,22), y i e l d s : N (t) = ^^1° + f f ( f , ( T ) - f , ( T ) ) d r (3.26) e C. Z Z «• o b 1 o Thus, the electron density can be ca l c u l a t e d exactly from the change i n voltage i n the case where f^ has been adjusted to be equal to f , o Since the second term on the r i g h t hand side of (3.26) i s time dependent, i t i s necessary to show that i t can be made small, and that i t can be approximated. t Let 0(t) = f (f (T) - f ( T ) ) d r (3.27) J o b o The base frequency f^ i s constant. Let i t be assumed that f^ i s set, so that f (0) = f, o b Then, f o ( r ) = f b + (f o(0))]< Aa df (0) - f h + i ~ - ~ (3.28) b T da - 42 -Substituting (3.28) into (3.27), we get: 0( t ) = J. t df o (0) 2TTT da T dT = (3.29) o The quantity 0(t) i s the number of complete cycles of f , which occur between t = 0 and t = t, which are not compensated for by f ^ . The number of uncompensated current pulses drawn from the integrating capacitor i s therefore 20(t) i n the same time i n t e r v a l . These uncompensated pulses are d i s t r i b u t e d uniformly over the time i n t e r v a l , and i f N q ( T ) = 0, the signal w i l l be a smoothly varying voltage which st a r t s at (0) and which has: as i t s f i n a l value. The only thing which remains to be done i s to show that the signal described by (3.30) varies slowly, compared to the fluctuations associated with v a r i a t i o n s of the electron density i n the discharge. D i f f e r e n t i a t i n g equation (3.9) with respect to a, we get: The function 0(t) can now be evaluated, using t y p i c a l experimental values for T(T = 6 msec.) and a (40° < a < 7 5°). The value of t i s established by the time over which the electron density i s to be observed, and t h i s i s 20 (isec. for t h i s experiment. o V1 ( T ) = V1 (0) + C x 0 ( T ) (3.30) (3.31) - 43 -The values o f 0(t) and the change i n f as a f u n c t i o n of a have been p l o t t e d i n F i g . I I I - 5 . N otice t h a t the t o t a l number of uncompensated o s c i l l a t i o n s observed over the viewin g i n t e r v a l i s never more than nine f o r values o f a which are o f experimental i n t e r e s t . This number i s o f the same order as the number of f r i n g e s to be expected from the changing e l e c t r o n d e n s i t y , but the l a t t e r w i l l occur i n a time i n t e r v a l one to two orders o f magnitude s m a l l e r . The f l u c t u a t i o n s i n i n t e g r a t o r v o l t a g e due to changing e l e c t r o n d e n s i t y are t h e r e f o r e r e a d i l y d i s t i n g u i s h a b l e from the s l o w l y and c o n s t a n t l y v a r y i n g b a s e l i n e due to the time dependence o f f Q . The experimental r e s u l t s bear out the c o n c l u s i o n s o f these t h e o r e t i c a l c o n s i d e r a t i o n s . Figure I I I - 6 i s an enlarge-ment o f a t y p i c a l o s c i l l o s c o p e t r a c e . The upper waveform i s the discharge c u r r e n t waveform, and the lower i s the v o l t a g e output o f the I n t e g r a t i n g Frequency Modulation Detector. In t h i s case, the assumption has been made t h a t the e l e c t r o n d e n s i t y i s zero a t the p o i n t o f o b s e r v a t i o n d u r i n g the f i r s t few microseconds o f the discharge. The curvat u r e of the v o l t a g e waveform i s t h e r e f o r e e n t i r e l y due to the f l u c t u a t i o n s i n the second term o f the r i g h t - h a n d s i d e of equation (3.26) . This curve has been e x t r a p o l a t e d , and the e l e c t r o n d e n s i t y i s g i ven by the v e r t i c a l s e p a r a t i o n o f the. observed waveform .and the e x t r a p o l a t e d b a s e l i n e . I t i s c l e a r t h a t the i n t e g r a t i n g c i r c u i t cannot be l e f t on c o n s t a n t l y , as f ^ - f > : > 0, except when a , the angle o f i n c i d e n c e o f the l a s e r beam on the quartz b l o c k , i s c l o s e to the value r e q u i r e d f o r making measurements. For t h i s reason, i t i s convenient t o s w i t c h on the constant c u r r e n t source j u s t b efore the discharge i s s t a r t e d . The d e t a i l s o f how t h i s i s done are d i s c u s s e d i n Sec. V.3. - 44 -F i g . III-.5 Phase Discrepancy Plot of 0(t) as a function of the angle of incidence (a) of the laser beam on the quartz block, showing the r e l a t i o n to the change i n the o s c i l l a t i o n rate due to the quartz block (Af Q) during the course of one 20 p,sec. observation. - 46 -F i g . III-6 Typical Oscilloscope Trace This figure i s an enlargement of a t y p i c a l o s c i l l o s c o p e trace. The upper waveform i s the discharge current and the lower, the output voltage of the Integrating Frequency Modulation Detector. - 48 -III.5 Other Systems of Fr a c t i o n a l Fringe S h i f t Interferometry In comparing the ro t a t i n g quartz block as a reference object with the other techniques which have been reported, the following w i l l be used as c r i t e r i a : 1) The degree to which good beam overlap can be maintained during the course of the measurement. (For a discussion of beam overlap, see Sec. II.2)„ 2) The maximum frequency reported for the system. The f i r s t system to be considered i s that reported by Baker and co-workers 7. In t h i s system (see F i g . III-7(a)) the resonator end mirror i s plane, and mounted so that i t can be rotated. I t i s immediately obvious that beam overlap i n t h i s device w i l l be s u f f i c i e n t to produce good interference and good fringe amplitude over only a very small range of angular p o s i t i o n s . I f we use as a c r i t e r i o n of good beam overlap, that the area i n which the beams overlap i s 1/3 of the c r o s s - s e c t i o n a l area of one beam, then such a system has good beam overlap for only -that time i n t e r v a l when the beam centres are displaced by one radius or l e s s . For a i m . long resonator and a 2 mm. diameter beam, t h i s condition of good beam overlap w i l l e x i s t over only 1 mrad. of the c y c l e . The quartz block system as ou t l i n e d above w i l l achieve good beam overlap over nearly TT radians, although the measurement window i s usually only about 250 mrad. . (45° > a > 60°) . Furthermore, the system of Baker, et a l , i s far more susceptible to vibrations,, inasmuch as i t involves d i r e c t movement of the mirror. Once again, i f the normal to the mirror face diverges from the res-onator axis at an angle i n excess of 1 mrad., beam overlap i s poor. Since the mirror i t s e l f i s being moved, i t i s d i f f i c u l t to see how such v i b r a t i o n s can be suppressed. - 49 -Fi g . III-7 Moving Mirror Reference Objects (a) PM - photodetector MX,M2 - laser c a v i t y mirrors R.Mo - rot a t i n g plane mirror (b) ID - Infrared detector M 1 7M 3 - laser c a v i t y mirrors M3 - corner mirror M4 - plane mirror A - rotatable table M , M D M ' .. 1 IVI / / L A S E R M M 2 ID C0 2 L A S E R (a) R O T A T I N G M I R R O R (b) - 51 -The frequency of o s c i l l a t i o n achieved i n t h i s system i s only 1 MHz. The low o s c i l l a t i o n frequency i s due to the detection technique, but even with improvements i n t h i s area, the changes of o p t i c a l path length are taking place i n a i r rather than i n quartz. Another system i s that which has the interferometer end mirror (M3 i n F i g . II-3) mounted on a loudspeaker cone 1 0 1 1 . Since t h i s system moves the mirror i n a s t r a i g h t l i n e , i t ought to be f e a s i b l e to maintain beam overlap for an i n d e f i n i t e period. However, the f a c t that the mirror i s being r a p i d l y moved must cause vi b r a t i o n s and i f these vibra t i o n s cause r o t a t i o n o f the normal to the mirror by as l i t t l e as 1 mrad. (for the same conditions as i n the previous system), beam overlap w i l l be poor. Such problems were reported. The frequency reported for t h i s system when used as a plasma diagnostic t o o l 1 1 was 90 KHz. This was achieved at a speaker d r i v i n g frequency of 50 Hz. and a mirror displacement of approximately 1 mm. I t can be used therefore on very tenuous or very slowly varying plasmas. Frequencies of up to 5 MHz. have been reported 1 0 l S but i n these cases the t o t a l mirror displacement was much less than X, the wavelength of the laser r a d i a t i o n . Such systems no longer correspond to the r o t a t i n g quartz block system, and further comparison i s p o i n t l e s s . Similar considerations may be applied to the systems which mount the mirror on a p i e z o - e l e c t r i c c r y s t a l 9 1 3 1 3 . xThe maximum f r i n g i n g rate reported i s orders of magnitude less than that achieved with the ro t a t i n g quartz block. The system which comes c l o s e s t to equalling the per-formance of the r o t a t i n g quartz block i s the rotating corner mirror of Herold and Jahoda 9. I t i s i l l u s t r a t e d i n F i g . III-7(b). - 52 -This system has excellent beam overlap s t a b i l i t y , both to rotation' of the table (A) and to v ibrat ion. In addit ion, the frequencies which could be achieved with such a device are at least as good as those achieved with a rotating quartz block. The chief drawback of the system is that i t requires a large, care-fu l l y balanced table which can be spun at re lat ive ly high speeds. In this respect, the quartz block system is superior because of i t s mechanical s impl ic i ty . 4 26 A variant on the fract ional fringe technique has been devised ' , by dNe which both the sign and magnitude of /dt can be determined. The method involves comparison of two fringe patterns produced by beams with a known phase difference. A 1/8 - wave plate is inserted into the interferometer with the plane of polarization of the laser beam ( l inearly polarized) bisecting the angle between the fast and slow axes of the plate. In this way, two beams of equal intensity , orthogonal polarization and TT/2 phase difference are produced. The interference fringes produced for each p o l a r i -zation are measured by photomultipliers behind suitably oriented polar izers. The disadvantage of this configuration l i es in the duplication of detection equipment and the data reduction which must be done to obtain prof i les of N g from the fringe patterns. For these reasons, i t is fe l t that rotating block technique is to be preferred. - 53 -IV The Z-Pinch IV.1 General Description The Z-pinch discharge under study i n t h i s experiment i s of conventional design 1, with minor modifications to accommodate the interferometer (see F i g . IV-1). The s p e c i f i c a t i o n s of the components can be found i n Table I. The brass electrodes (A) have been provided with 3/8" wide s l o t s (I), cut along a diameter of the electrode to within 1/16" of the circumference, to permit i n s e r t i o n of the quartz tubes (H), and to allow these tubes to be moved from one r a d i a l p o s i t i o n to another 2. In addition, the quartz tube mountings have been modified to allow adjustment i n both r a d i a l and a x i a l d i r e c t i o n s without opening the vacuum system. The vacuum feedthroughs for adjusting the quartz tubes are shown i n F i g . IV-2. Notice the square s l o t s cut into the centres of the brass i n s e r t s to accommodate the mechanical conn-ection to the quartz tube mounts. This mechanical connection i s achieved using f l e x i b l e s t e e l cable, of the type used i n speedometers. The r a d i a l adjustment of the quartz tubes i s accomplished by means of the rack and pinion arrangement shown i n F i g . IV-3. The end of the drive cable i s inserted d i r e c t l y into the centre of the gear (A i n F i g . IV-3). Rotation of the exte r n a l l y acc-es s i b l e part of the mechanical feedthrough (Fig. IV-2, A) causes the gear to move along the rack, and the quartz tube to move along a radius. - 54 -F i g . IV-1 The Z-Pinch Discharge Schematic A. Electrodes B. Pyrex Discharge Tube Walls C. High Voltage Power Supply D. Main Capacitor Bank E. Main Spark Gap Switch F. Quartz Envelope of U l t r a v i o l e t Source G. Return Conductor (Brass Gauze) H. Quartz Tubes I. Diametral Slot DISCHARGE VESSEL 200k H W V 51-5 F ; 12 kV D 10 M 200k / T -^VVW^VWT] ^ W V j l O M - W IOM - V W ] IOM -^ WNHIOM •AVVI IOM r A / W M O M -I20M " V W 30 k END - 56 -Table I - Z-Pinch Components Discharge Tube M a t e r i a l : Pyrex Length: 76.2 cm. Inner Diameter: 15.0 cm. Outer Diameter: 17.0 cm. I n t e r e l e c t r o d e Length: 61.6 cm. Vacuum System Mech a n i c a l Pump: D i f f u s i o n Pump: Vacuum Gauges: Base P r e s s u r e : Leak Rate: Cenco - HYVAC 14 O i l - Type 17 B a l z e r 2 V a c u s t a t McLoed 1 a t 0-1 T o r r . 1 a t 0-10 T o r r . CVC - P i r a n i Type GP-110 1 u.Hg. 5 LLHg./min. E l e c t r i c a l Power Supply: C a p a c i t o r s : Leads: E l e c t r o d e s : Return Conductor: V o l t a g e Measurement: Sorenson Model 10 20-30 (0-20 kV; 0-30 mA.) 5 NRG Type 203 T o t a l c a p a c i t y 51.5 u.F. 1/16" x 10 cm. x 1.3 m. (copper) Brass Brass Gauze Simpson microammeter i n s e r i e s w i t h p r e c i s i o n 500 MQ r e s i s t o r . - 57 -Fig. IV-2 Vacuum Feedthroughs in End Windows GLASS ^ or ^ BRASS - 59 -F i g . I V - 3 Quartz Tube Mounts The mechanisms (A) are designed to hold the quartz tube mounts i n place i n s i d e the discharge electrodes. <Q-POSITIONING MECHANISM (MATERIAL: BRASS) - 61 -The quartz tube i s held i n place between four "0"-ring r o l l e r s (see F i g . IV-3). A x i a l movement can then be achieved by r o t a t i o n of these r o l l e r s . One r o l l e r on each assembly, designated as the "drive r o l l e r " i n F i g . IV-3, i s lengthened, to accept the square end of the f l e x i b l e drive cable. The drive cables are f i x e d f i r m l y into the drive r o l l e r and r a d i a l adjustment gear, but are only inserted into t h e i r respective s l o t s on the vacuum feedthroughs, i n order not to hamper disassembly and reassembly by the vacuum system. IV.2 Discharge C h a r a c t e r i s t i c s A l l measurements were done on the plasma created when a charging voltage of 12 kV. (±0.05 kV.) and a f i l l i n g pressure of 4 Torr. of helium were used. Under these conditions the current o s c i l l a t i o n had a period of 22.4 u.sec„, and a peak value of 17 5 kA. The discharge current was measured with a Rogowski c o i l c o n s i s t i n g of a 12 cm. length of RG63-A/U delay l i n e with the outer conductor removed. This c o i l i s inserted between the leads carrying the discharge current, with the c o i l axis normal to the d i r e c t i o n of current flow. The c o i l output was integrated using a passive RC integrator (time constant = 1.1 msec.) to give a signal proportional to the discharge current. This s i g n a l was then c a l i b r a t e d by manual integration of the current waveform, using the r e l a t i o n : - 62 -J l d t = Q (4.1) where I i s the current flowing i n the discharge, and Q i s the t o t a l charge stored on the capacitors at the i n i t i a t i o n of the discharge. I t was found to be h e l p f u l to move the photomultiplier and recording oscilloscopes away from the discharge tube, to reduce the e f f e c t of the radio-frequency noise generated by the discharge. In addition, f e r r i t e i s o l a t o r s were used on a l l c o axial cables connected to these o s c i l l o s c o p e s to break up R.F. ground loops. Additional f e r r i t e i s o l a t o r s were placed i n the mains connections to the oscilloscopes and photomultiplier power supply. The f e r r i t e i s o l a t o r s are constructed by wrapping the cables around a f e r r i t e r i n g . In the case of coaxial cable, the outer sheath i s i n d u c t i v e l y l i n k e d to the f e r r i t e , and has a correspondingly high inductance. The centre conductor i s shielded from the f e r r i t e by the outer sheath. In the case of the mains connection, of course, a l l the mains l i n e s have a high inductance. Thus, high frequency signals propagating along the ground connection or on the mains are attenuated. IV.3 Experimental Objectives As has been previously mentioned the breakdown of a Z-pinch i s characterized by a p r e f e r e n t i a l current flow at the walls of the discharge v e s s e l . Once breakdown has occurred the a x i a l current j_ grows r a p i d l y and i n t e r a c t s with i t s own - 63 -azimuthal magnetic f i e l d , B. The r e s u l t i n g j x B force drives the current s h e l l r a d i a l l y inward. Associated with the r a d i a l collapse of the current s h e l l i s the development of i o n i z a t i o n . The ionized gas c a r r i e s some of the a x i a l current and i s trapped by the magnetic f i e l d . In t h i s way, the gas i n the discharge vessel i s heated, ionized, trapped and' compressed by the c o l l a p s i n g current s h e l l . To understand the dynamics of the collapse, i t i s necessary to know both the d i s t r i b u t i o n of mass and the d i s t r i -bution of current i n the discharge, i n order to c a l c u l a t e the forces acting on the gas. The d i s t r i b u t i o n of current has been measured accurately by J . Pachner of t h i s laboratory, using a t h r e e - c o i l probe 3. This probe has a s p a t i a l r e s o l u t i o n which i s a factor of four better than the conventional s i n g l e c o i l probe. The current measurements were not made on the same Z-pinch as the la s e r interferometrie measurements of electron density. The two discharge systems were, however, e f f e c t i v e l y i d e n t i c a l , i n e l e c t r i c a l as well as geometric c h a r a c t e r i s t i c s . I t was with the inte n t i o n of determining the mass d i s t r i b u t i o n i n the discharge, that l a s e r interferometric measure-ments have been made. The la s e r interferometer i s most s e n s i t i v e when used as a detector of electron density. Since plasmas are quasi-neutral, however, the measurement of the electron d i s t r i b u t i o n gives that of the ions as well, and hence the d i s t r i b u t i o n of mass d i r e c t l y a f f e c t e d by Lorentz forces. - 64 -V Experimental Procedures V. 1 Introduction This chapter w i l l deal with the d e t a i l s of the experimental procedures. The f i r s t section describes the way i n which the interferometer i s aligned. This procedure has two facets: f i r s t , the alignment of the laser beam p a r a l l e l to the discharge tube axis, and second, the alignment of the resonator mirrors to produce beam overlap and good resonance. The next section describes the t r i g g e r i n g e l e c t r o n i c s used to ensure that the discharge breakdown occurs when the resonator configuration i s appropriate for measurement. The t h i r d section presents d e t a i l s of the measurements, with sample data and graphical reductions from t h i s data. The chapter closes with a discussion of errors, V.2 Resonator Alignment The Fabry-Perot interferometer measures the integrated electron density along the resonator axis. To deduce the electron density from Fabry-Perot fringe patterns, i t i s necessary that something be known about the d i s t r i b u t i o n along the resonator axis. In the Z-pinch, t h i s means ali g n i n g the resonator axis p a r a l l e l to the discharge tube axis, as, along any such l i n e , the electron density i s constant. The resonator axis i s made p a r a l l e l to the discharge tube axis as follows: The lens L x (see F i g . V-l) i s removed, and the graduated s t e e l rods (R i n F i g . V - l , ru l e d every 0.02 5") are inserted into the discharge tube through the ports (A) i n the discharge v e s s e l . The ports are 1/4" I.D, and are located 22 cm. on eith e r side of the discharge tube centre. As F i g . V - l shows, - 65 -Fi g . V - l Discharge Tube and Interferometer The view i s a horizontal, diametral cross-section, i l l u s t r a t i n g the alignment procedure. PM2 — alignment photomultiplier PMl - resonator monitoring photomultiplier MX,M2 - l a s e r c a v i t y mirrors M2,M3 - resonant c a v i t y mirrors QB - quartz block (seen from above) , L 2 - lenses E - discharge electrodes G - pyrex discharge vessel R,R - graduated rods A, A - tube ports R Mi ! 11 3-P M 2 LASER QUARTZ BLOCK M l =LT| M 3 •« 44 cm J 51 L 2 P M , - 67 -the rods are i n the same h o r i z o n t a l p l a n e , The rods are pushed i n t o the v e s s e l u n t i l t h e i r ends abut on the opposing w a l l . The p o i n t s a t which the l a s e r beam s t r i k e s the. rods are noted. The r e s o n a t o r a x i s i s then a d j u s t e d u n t i l these p o i n t s are the same d i s t a n c e from the d i s c h a r g e tube w a l l . The r e s o n -a t o r a x i s i s then p a r a l l e l to the d i s c h a r g e tube a x i s . From the known i n n e r diameter o f the d i s c h a r g e v e s s e l , and the d i s t a n c e o f the r e s o n a t o r a x i s from the tube w a l l , the l o c a t i o n o f the r e s o n a t o r a x i s , r e l a t i v e to the d i s c h a r g e tube a x i s , i s e a s i l y deduced. The o p t i c a l bench which supports the r e s o n a t o r and l a s e r may be moved l a t e r a l l y w i t h r e s p e c t t o the t a b l e on which i t stand s . The p o s i t i o n o f the o p t i c a l bench f o r which the r e s o n a t o r and d i s c h a r g e axes would be c o i n c i d e n t i s noted, and h e n c e f o r t h the r a d i a l p o s i t i o n o f the r e s o n a t o r a x i s i s determined by i t s d i s t a n c e from the d i s c h a r g e a x i s p o i n t . Once t h i s procedure i s completed, the rods are removed and the s i d e p o r t s s e a l e d up. M i r r o r M3 (see F i g . V - l ) i s now a d j u s t e d to send the l a s e r beam back on i t s e l f . The accurac y o f t h i s alignment i s e a s i l y monitored w i t h PM2. The b e t t e r the alignment, the b e t t e r the beam o v e r l a p , and the b e t t e r the c o u p l i n g from the e x t e r n a l r e s o n a t o r to the l a s e r . The mechanical v i b r a t i o n s o f the e x t e r n a l r e s o n a t o r m i r r o r s produce changes i n the phase o f the r a d i a t i o n b e i n g f e d back, and these v i b r a t i o n s are s u f f i c i e n t l y slow t h a t the feedback v a r i a t i o n s produce v a r i a t i o n s i n l a s e r power. Once the amplitude o f the o s c i l l a t i o n s observed by PM2 have been maximized by the adjustment o f M 3, l e n s L t i s r e i n s e r t e d i n t o the system. T h i s l e n s i s then a d j u s t e d to r e -maximize the l a s e r power f l u c t u a t i o n s . T h i s new maximum i s - 68 -larger than the previous one, as the presence of L x converts the resonator into a concentric one with correspondingly better beam overlap. This procedure ensures that L T does not produce a small divergence between the resonator axis and the discharge tube axis. The quartz tubes are now r o l l e d into p o s i t i o n so as to enclose the la s e r beam; a suitable spacing between t h e i r ends i s chosen, and the resonator alignment i s complete. F i n a l alignment consists of d i r e c t i n g the transmitted beam to the face of photo-m u l t i p l i e r PMl, using lens L 3 (Fig. V-l) to keep the beam from diverging. I t i s necessary that L 3 have a f o c a l length such that the centre of curvature of mirror M3, located at the r e f r a c t i o n point, be focussed on the photomultiplier surface. The s p e c i f i c a t i o n s of the interferometer components are found i n Table II, on the following page. In the course of the experiment, i t became apparent that better interference conditions, and hence greater amplitude of the resonator output o s c i l l a t i o n s , could be obtained when the resonator length (the distance between mirrors M2 and M3, F i g . V - l , t y p i c a l l y 5m.) was an i n t e g r a l multiple of the 35.5 cm. laser c a v i t y length. This e f f e c t has been observed by others 3- 6, and has been at t r i b u t e d to the e x c i t a t i o n of more than one a x i a l mode i n the laser c a v i t y . V.3 Triggering C i r c u i t r y The previous section has described the alignment of the resonator and the introduction of the reference object. This reference object, the quartz block, must, be i n such a p o s i t i o n that the resonator i s aligned when the discharge i s to be - 69 -TABLE II - Interferometer Components Laser: Spectra-Physics Model 130C o Continuous He-Ne (6328A) Beam Power: 1.0 mW. -2 Beam Diameter (at e points) Beam Divergence: 0.7 mrad. 1.4 mm. Lenses: L 2 f o c a l length = + 1.27 m« foc a l length = + 2.5m. Interference F i l t e r : Baird-Atomic; 43 A bandpass Photomultiplier: Oscilloscopes: External Mirrors (Ma): Type Substrate Surface Coating Radius of Curvature Diameter o R e f l e c t i v i t y at 6328 A about 6330 A; Maximum transmission 60% P h i l l i p s 150 CVP, Risetime: 8 nsec. Fluke Model 412B high voltage power supply Tektronix Type 551 Dual Beam Spherical Fused quartz M u l t i - l a y e r d i e l e c t r i c 1.0 or 2.0 meters 3.8 cm. 60% This configuration produces 30 MHz o s c i l l a t i o n s with an amplitude of 10% of the background beam i n t e n s i t y . - 70 -triggered. The angular p o s i t i o n and frequency of r o t a t i o n of the quartz block must also be such as to provide a resonator output o s c i l l a t i o n frequency, f , approximately equal to f^, as defined i n Sec. IV.4. This section w i l l describe the way i n which t h i s i s done. The basic t r i g g e r i n g c i r c u i t r y i s shown i n F i g . V-2. The t r i g g e r i n g source i s a photodiode, set up to intercept the l i g h t r e f l e c t e d from the front face of the quartz block. As the quartz block i s rotated, the beam i s swept over the photodiode, producing one pulse per cycle of block r o t a t i o n . (The r o t a t i n g assembly includes a geometrically s i m i l a r opaque block as a counterbalance.) These pulses are shaped, and then fed into an i n t e r v a l recognition detector (IRD) which produces an output pulse as soon as the pulse i n t e r v a l , T, i s equal to or less than some preset value. Figure V-3 shows a block diagram of the IRD, and a d e t a i l e d c i r c u i t diagram i s given i n Appendix I I . Coincid-ence between an input pulse and the one-shot pulse i s recognized by the NAND gate. The coincidence output t r i g g e r s the one-shot which provides the output pulse. To allow the quartz block to rotate through a small angle before the discharge i s triggered, the IRD output pulse can be delayed by the delay u n i t shown i n Figure V-2. This configuration allows the small adjustments necessary to ensure that f = f^ when the discharge i s triggered (see Section IV.4). The operating procedure requires that the quartz block d r i v i n g motor be turned on when a l l other preparations have been made to f i r e the discharge ( i . e . , the f i l l i n g pressure has been reached, capacitors f u l l y charged, and o s c i l l o s c o p e camera shutters opened). As the quartz block angular v e l o c i t y increases, the pulse r e p e t i t i o n rate from the photodiode increases, u n t i l i t i s s u f f i c i e n t to t r i g g e r the IRD. - 71 -Fig. V-2 Triggering Circuitry M1,M2 - laser cavity mirrors M 2 » M 3 - resonator mirrors Lj^Lg - lenses PD - photodiode (FPT-100) IF - interference f i l t e r PM - photomultiplier M-IF LI 1 -i PM AMPLIFIER AND PULSE SHAPER _ n _ VOLTAGE BREAK PULSE FROM DISCHARGE TUBE INTERVAL RECOGNITION DETECTOR _TL DELAY J~~L TRIG I THYRATRON AND VOLTAGE DOUBLER o. SCOPE A BEAM I INPUT (f-t)dr FREQUENCY MODULATION DETECTOR BEAM 2 INPUT DISCHARGE CURRENT CONSTANT CURRENT SOURCE GATE IN + GATE OUT BEAM 2 1 TRIG IN SCOPE B INPUT I-BEAM I INPUT RESONATOR I OSCILLATIONS * TO PHOTON TRIGGERED SPARK GAP - 73 -I n t e r v a l R e c o g n i t i o n D e t e c t o r F i g . V-3 (a) Block Diagram F i g . V-3 (b) Pulse Sequence R - e x t e r n a l v a r i a b l e r e s i s t o r which determines T, the i n t e r v a l r e c o g n i z e d . INPUT hKUM PHOTODIODE A i PULSE SHAPER B DELAY VARIABLE ONE - SHOT D NAND GATE t E O N E - S H O T OUTPUT (a) B A / | W V n h n TIME (b) - 75 -The delayed IRD output goes to the thyratron u n i t and voltage doubler, which de l i v e r s a voltage pulse to the spark l i g h t source of a photon-triggered pulse generator (see F i g . I I - l ) . U l t r a v i o l e t photons from the l i g h t source pass through an enclosing quartz bulb and illuminate the electrodes of the a i r spark i n the t r i g g e r pulse g e n e r a t o r . 2 2 . The output of t h i s generator f i r e s the main spark gap switch. Using an u l t r a v i o l e t l i g h t source to f i r e the t r i g g e r pulse generator ensures that low voltage t r i g g e r i n g and measuring e l e c t r o n i c s are completely decoupled e l e c t r i c a l l y from the high power discharge c i r c u i t , which reduces the noise picked up by the measuring c i r c u i t s . The u l t r a v i o l e t l i g h t source cannot be used to t r i g g e r the main gap d i r e c t l y for two reasons. F i r s t , the technique w i l l only t r i g g e r gaps which are operated at a few hundred v o l t s below the breakdown voltage. This requires accurate electrode placement which i s impossible i n the main gap because of electrode erosion. Second, the quartz bulb could not withstand the temperatures generated i n the main spark gap. The f i n a l t r i g g e r i n g sequence i s shown i n F i g . V-4. The delayed IRD pulse (Fig. V-4(a)) goes to the thyratron and voltage doubler, by which i t t r i g g e r s the discharge, and to the t r i g g e r input of scope "B" (for t r i g g e r i n g c i r c u i t r y block diagram, see F i g . V-2) . As. a r e s u l t , scope "B" begins to sweep immediately, and a f t e r a short delay through the t r i g g e r i n g spark gaps ( t ^ i n F i g . V-4), the discharge breaks down. The + GATE output of scope "B" i s used to turn on the constant current source i n the IFMD (see F i g . IV-3). The delay t ^ i s s u f f i c i e n t to ensure that the IFMD i s functioning by the time the discharge f i r e s . The IFMD stays on as long as scope "B" continues to sweep. This sweep time must be greater than t (approximately 4 u.sec.) plus the - 76 -Fig. V-4 Triggering Sequence The delayed IRD pulse triggers oscilloscope "B", and the voltage spike "A" triggers oscilloscope "A". The photograph is of the "A" trace. 50 V-oLn -12 kV DELAYED IRD PULSE VOLTAGE ACROSS DISCHARGE TUBE + 20 V| 0 + GATE OUT SCOPE B DISCHARGE CURRENT IFMD OUTPUT TIME SHADED AREA CONSTITUTES PHOTOGRAPHIC RECORD - 78 -duration of the observation (set by the duration of "A" scope (see Fig- V-4) sweep, usually 20 u.sec. The breakdown of the main t r i g g e r spark gap puts the f u l l capacitor voltage across the discharge electrodes. This voltage pulse (A i n F i g . V-4(b)) i s used to t r i g g e r scope "A", on which i s displayed the discharge current and the electron density at the observation point. This display i s photographed, and appears as shown i n F i g . V-4(d,e). V.4 Temporal and Radial P r o f i l e s The techniques by which the temporal p r o f i l e s of electron density are obtained from photographic records, produced as i n Sec. V.2, have been discussed i n Sec. III.4. Temporal p r o f i l e s have been obtained at twelve r a d i a l p o s i t i o n s : at x = 0.7 5 cm., and at 0.5 cm. i n t e r v a l s from r = 1.00 cm. to r = 6.00 cm. The r e s u l t s are shown i n F i g . V-5, which i s a perspective view of the temporal p r o f i l e s arranged i n proper sequence. Each temporal p r o f i l e represents an average over several shots done under i d e n t i c a l conditions. Figures V-6 and V-7 show the temporal p r o f i l e s generated at two r a d i a l p o s i t i o n s . These have been included to i l l u s t r a t e the plasma features, whose dynamics are discussed i n Chapter VI, and to i l l u s t r a t e the accuracy and r e p r o d u c i b i l i t y of the r e s u l t s . The s o l i d l i n e s represent i n d i v i d u a l shots, and the c i r c l e s show the average over these i n d i v i d u a l shots. The i n t e r e s t i n g features which have been l a b e l l e d are the precursor, the main peak and the r e f l e c t i o n peak. The other feature of i n t e r e s t i s the very large "a x i a l spike", which appears on the axis of the discharge at the time when the precursor a r r i v e s on axis. As can be seen i n F i g . V-5, t h i s feature i s expanding out from the a x i s . - 79 -Fig. V-5. Electron Density Distribution in the Discharge. - 81 -Electron Density Temporal P r o f i l e s F i g . V-6: at r = 3.00 cm. Fi g . V-7: at r = 4.00 cm. 150 H FIG. V-7 •3 q/ERRGE OF SHOTS: 12-5 5 10-Oi < < CL 14/06/71'3493 M/06'71/3494 14/06/71/3495 a z o cr r -O UJ 5 0 2-5 0 0 2O0 TIME ( ysecs) - 83 -Radial p r o f i l e s are now obtained by taking a cross-section of the surface of F i g . V-5 at selected times. This technique has been used to generate the r a d i a l p r o f i l e s presented i n Figs. V-8, 9, 10. Figure V - l l shows the t r a j e c t o r i e s of the four features pointed out on Figs, V-5, 6, 7. In addition, the t r a j e c t o r i e s of the current peak (obtained from the measurements made by J. Pachner) and of the leading edge of the precursor are plot t e d . The current and magnetic f i e l d measurements made by Mr. Pachner which are used i n the course of t h i s thesis are found i n the following f i g u r e s . Figure V-12 gives the r a d i a l current d i s t r i b u t i o n , Figs. V-13, 14, 15 give the r a d i a l magnetic f i e l d d i s t r i b u t i o n , F i g . V-16 shows the maximum value of j x B observed as a function of time, and F i g . V-17 shows a t y p i c a l force p r o f i l e at t = 7.0 u.sec. V. 5 Discussion of Error and Repro d u c i b i l i t y The greatest source of error i n the absolute values of electron density generated i n t h i s experiment i s the measure-ment of the photographs. The large number of photographs to be reduced and the large number of data points on each photo precluded measurement by hand. Instead, the curves i n the photos were traced out on an el e c t r o n i c d i g i t i z e r , and the co-ordinates of the curve were recorded on punched cards which were l a t e r analyzed on the IBM 360/67 at the U.B.C. Computing Centre. The l i m i t i n g factor i n the p r e c i s i o n of t h i s process i s the accuracy with which the d i g i t i z e r operator can accurately follow the curves on the photos. - 84 -R a d i a l E l e c t r o n D e n s i t y P r o f i l e s F i g . V - 8 t = 1 . 0 L x s e c . t o t = 5 . 0 i x s e c . F i g . V - 9 t = 6 . 0 u.sec. t o t = 1 0 . 0 u.sec. F i g . V - 1 0 t = 1 1 . 0 L i s e c . t o t = 1 5 . 0 p . s e c . 9 0 I E !-° CD O 2-0 1-0 0 0 o t = 1 0 /isec 8 0 • = 2 - 0 jLlsec • = 3 - 0 jusec 7 0 V = 4 - 0 j isec • = 5 - 0 Usee 6 0 -5 0 -4 0 3 0 -7 R, R(cm) - 88 -F i g . V - l l Radius vs. Time Diagram for Current and Density Features E o - 90 -F i g . V-12 Radial Current D i s t r i b u t i o n 3 0 - 0 £ 22-5 < 2 15-0 7-5 004-0 o t = 1-0 j isec • = 2 0 fJLsec • = 3 - 0 /Usee V = 4 - 0 jLfsec • = 5 - 0 )Usec 4 5 - 0 ~ o t = 6 0 usee • = 7 - 0 fjisec • = 8 - 0 jUsec V = 9 - 0 /isec • = 1 0 0 jLisec - 92 -Magnetic Field Distributions Fig. V-13 Fig. V-14 Fig. V-15 t = t = t = 1.0 p.sec. to t = 6.0 LLsec. to t = 8.0 Lisec. to t = 5.0 LLsec. 7.0 [xsec. 10.0 LLsec. 0-6 He 4 torr r (cm) r (cm) - 96 -Fig. V-16 Maximum j x B vs time MAXIMUM J X B ( x 10 NT ) J> Ol 0) - 98 -Force on Current Shell Element vs. Radial Position: t = 7.0 |j.sec. FORCE ON ELEMENT OF CURRENT SHELL ( x l 0 3 N t ) - 100 -In addition, the baseline for each curve must be drawn i n by hand, as discussed i n Section III.4. This introduces an additional uncertainty, but only for the value of at l a t e r times. The value of the main peak density should be quite accurate. I t i s quite d i f f i c u l t to assign, a p r i o r i , a value to the error to be expected from these sources, but i t i s ce r t a i n to be less than 10%. It i s also d i f f i c u l t to. estimate the measuring accuracy by comparing runs done under i d e n t i c a l conditions. Some properties of the plasma are sensi t i v e to the i n i t i a l discharge conditions, and the accuracy with which the f i l l i n g pressure could be set was only about ±10%. This may a l t e r the plasma temperature s l i g h t l y , which w i l l produce much larger fluctuations i n the electron density (N q has an exponential dependence on temper-ature) . For t h i s reason, fluctuations i n c a l c u l a t e d temporal p r o f i l e s from shot to shot (see F i g . V-6, 7) are a t t r i b u t e d to a lack of shot-to-shot plasma r e p r o d u c i b i l i t y rather than to measuring inaccuracies. The measurement of time from the photographs i s much more accurate. This measurement i s l i m i t e d i n accuracy by the l i n e a r i t y of the oscil l o s c o p e time-base (less than ±100 nsec. over the ce n t r a l 80% of the screen as measured with a c a l i b r a t e d , c r y s t a l - c o n t r o l l e d o s c i l l a t o r ) . As w i l l be seen i n the ensuing chapter, the data analysis w i l l be concerned almost e n t i r e l y with the v e l o c i t i e s of the s i g n i f i c a n t features indicated i n Figs. V-6, 7, and very l i t t l e with t h e i r amplitudes. Furthermore, the errors associated with the absolute electron density values do not figure so prominently i n the measurement of r e l a t i v e changes i n N q during a run. Thus, i f the electron density i s sa i d to be slowly varying for several - 101 -microseconds, the absolute density i s accurate to ±10%, but the slope of dN /dt is much more accurately known. - 102 -VI Results VI. 1 Introduction This chapter begins with a presentation of the s i g n i f i c a n t features of the data presented i n Sec. V.4. I t contains a model which can be used to account for the observed features. I t concludes with an example of how the dynamics of the discharge can be used to reveal the c h a r a c t e r i s t i c s of the plasma i n which they move. VI.2 Summary of Results Examination of the electron density d i s t r i b u t i o n as a function of time and r a d i a l p o s i t i o n , i l l u s t r a t e d i n F i g . V-5, brings to the attention four features of importance. F i r s t i s the r e s u l t that the r a d i a l electron density d i s t r i b u t i o n i s divided into two d i s t i n c t regions. The d i v i s i o n occurs at r = 2.50 cm. In the region of r greater than t h i s value, an ionized layer implodes toward the discharge axis, as one would expect i n a conventional Z-pinch. For r less than 2.50 cm., there i s no evidence of t h i s c o l l a p s i n g s h e l l . I t has been brought completely to rest, and only a small f r a c t i o n penetrates to the inner zone. The second feature i s that the degree of i o n i z a t i o n , with the exception of the "a x i a l spike" region, i s low. At room temperature, the number density at 4 Torr. i s : 1.33 x 101 7 cm.'3. Nowhere i n the outer region does the density exceed 8 x 10 1 8 cm.-' ( i . e . 60% of the f i l l i n g density). - 103 -Third, i t can be seen that, i n the outer regions of the plasma, following the passage of the i o n i z a t i o n s h e l l , the electron density at a given radius remains constant for periods of several microseconds. (see Figs. V-6, 7 i n the i n t e r v a l between main and r e f l e c t i o n peaks). F i n a l l y , there i s the "precursor". I t can be seen f u l l y separated from the "main peak" at t = 5.1 u.sec. and r = 4.50 cm., and thereafter propagates i n to the axis at a constant v e l o c i t y . The tremendous burst of i o n i z a t i o n i n the " a x i a l spike" i s associated with the a r r i v a l on axis of t h i s precursor. Further d e t a i l s of the c o l l a p s i n g i o n i z a t i o n s h e l l s are evident i n F i g . V - l l . F i r s t , i t i s seen that the "main peak" v e l o c i t y i s constant, at 7.3 Km/sec, from the f i r s t observation at r = 6.0 cm to r = 4.0 .cm., at which point i t begins to decelerate, and the current peak decelerates simultaneously. However, i t i s easy to see from F i g . V - l l , that the "main peak" i s outside the current peak, and that the spacing between them increases from 5 mm. to 9 mm. i n the time i n t e r v a l 4.5 u.sec. to 8.0 u.sec. Furthermore, l e s s than 10% of the ionized gas i s a c t u a l l y coincident with 7 5% of the current. Although both current peak and "main peak" are brought to rest, the i o n i z a t i o n feature stops at r = 3.0 cm. while the current peak stops at r = 2.50 cm. In addition, although deceleration begins simultaneously for the two peaks, the i o n i z a t i o n peak stops more gradually than the current peak, reducing the separation of the two from 9 mm. at 8.0 u.sec to 5 mm. at 9.5 p.sec. - 104 -The i o n i z a t i o n t r a j e c t o r i e s indicate that the "precursor" leading edge arri v e s on axis at approximately 8.5 Lisec, and that the " r e f l e c t i o n peak" originates on axis at t h i s time. F i n a l l y / the r a d i a l current p r o f i l e s indicate the presence of large a x i a l current densities at t = 7.0 u.sec, at which time the r a d i a l electron density p r o f i l e s show no s i g n i f i c a n t a x i a l i o n i z a t i o n . Ionization on the discharge axis does not become s i g n i f i c a n t u n t i l the a r r i v a l of the precursor, a microsecond l a t e r . To recapitulate, the important facts which the model must explain are: 1) the development of the precursor 2) the constant v e l o c i t i e s of the current and i o n i z a t i o n s h e l l s 3) the abrupt arrest of these two features, and 4) the low percentage i o n i z a t i o n . VI.3 The Model This model i s put forward to explain the observed dynamics of the Z-pinch discharge i n 4 Torr. helium. I t consists of a shock wave driven by the co l l a p s i n g current sheet. The chara c t e r i s t i c s , of the shock front are di c t a t e d by the require-ment that the k i n e t i c pressure of the shock-heated gas ahead of the current sheet equal the magnetic pressure exerted on the current sheet by Lorentz forces. The model ou t l i n e d i n the preceding paragraph i s capable of explaining the state of dynamic equilibrium which appears i n the discharge during the collapse phase, and which i s revealed i n the constancy of the current and i o n i z a t i o n sheet v e l o c i t i e s - 105 -over much of the c o l l a p s e . I t explains the generation and behaviour of the precursor. The reason for the abrupt h a l t of the current sheet can be deduced from the model, and i t provides an explanation of the " r e f l e c t i o n peak" mentioned above. The shock front which i s a leading feature of the model i s i d e n t i f i e d with the leading edge of the "precursor". The "precursor" feature i s f i r s t observed at r = 4.5 cm. The t r a j e c t o r y of the peak electron density i n the "precursor", and of the leading edge of that feature have been p l o t t e d i n F i g . V - l l . The v e l o c i t y of the front i n the lab frame i s constant and i s measured to be 12 Km./sec. The sound speed i n room temperature helium i s 1.0 Km./sec, which gives, for the shock wave, a Mach number, M, of 12. We (18) apply the theory of steady, strong shocks i n an i d e a l gas to c a l c u l a t e the thermo-dynamic properties of the gas behind the shock. £2. = 2g_ Pi g+i vP _ g-1 v x " g+1 (M 3-l) (6.1) (6.2) where p, v and T r e f e r to the gas pressure, mean flow v e l o c i t y r e l a t i v e to the shock, and temperature, res p e c t i v e l y . The subscript 1 r e f e r s to conditions ahead of the shock, and the subscript 2, to conditions behind i t . The parameter g i s the r a t i o of "specific heats, which i s 1.67 for an i d e a l gas. The parameter M, as already defined, i s the Mach number. - 106 -In our case: p x = 4 Torr. = 567 nt./m2 v x = 12 Km./sec. T1 = 290°K. Application of 6.1, 2 and 3 y i e l d s P 2 = 1.0 x 10 s nt./m2 = 710 Torr. V 2 = 3 Km./sec. T 2 = 13,300°K Cal c u l a t i o n of the magnetic pressure, P, on the current sheet during the time i n t e r v a l when that sheet has a constant v e l o c i t y (from 5.0 Lxsec. to 7.0 uLsec), gives, for a magnetic f i e l d of 0.51 W./m2: R 2 P = ^ — = 1.04 x 10 5 nt./m 2. 2|la (The magnetic forces per u n i t volume can be written as grad (B 2/2 LL-O) provided that j_ x JB i s large only i n a t h i n s h e l l . Figure V-17 shows t h i s assumption to be v a l i d ) . I t i s evident that P = p 3, so that we impose on the model the requirement that the gas k i n e t i c pressure ahead of the current sheet be balanced by the magnetic pressure acting on i t from the rear. As a r e s u l t of the c y l i n d r i c a l geometry, we expect that the balance between gas k i n e t i c pressure and magnetic pressure must soon be destroyed, since the gas flow i s everywhere r a d i a l l y inward. The flow compresses the gas and increases i t s pressure. The maximum i n t e r n a l pressure which the current sheet can provide i s B 2/2 u.0, which i s e s s e n t i a l l y c o n t r o l l e d by the discharge c i r c u i t . The measurements c l e a r l y indicate that the v e l o c i t y - 107 -of the current s h e l l i s constant. Hence, gas must be flowing out through the piston, to keep the gas pressure equal to the magnetic pressure. I t i s i n s t r u c t i v e at t h i s point to make an analogy be-tween a shock wave being driven by a leaky piston, and a water wave supported by the movement of a sieve. The viscous drag of the water flowing through the sieve determines the sieve v e l o c i t y for a given applied force. I t also determines the height to which water w i l l p i l e up ahead of the sieve. In the Z-pinch, the force i s applied on the ions, through the electrons. A constant v e l o c i t y of the current sheet w i l l be achieved when the "viscous drag" on the ions, from c o l l i s i o n s with neutrals, i s equal to j x B, the applied force. S i m i l a r l y , the viscous drag on the neutrals determines the pressure gradient that w i l l be established. Mathematically, j_ x B_ = grad p. I t can be seen, however, that the piston i s not "leaky" for the electrons whose presence we detect i n the precursor. This can be shown, i f we assume that the number of ions and electrons between the piston and the shock front i s not changed (e.g. by recombination, Joule heating, shock i o n i z a t i o n , e t c . ) . Then N (precursor) i s proportional to 1/V, where V i s the e volume occupied by the electrons and ions. The volume, V, i s given by: V = 7r(r 3 - r s s ) l (6.5) where r i s the radius of the c y l i n d r i c a l piston, r i s the p s radius of the shock front, and £ i s the discharge length. I f N i s the constant number of electrons trapped, then the density, N i s : - 108 -B » _ — a (6. 6 ) e V r ( r j - r ; ) t r + r Let r = P — , where we i d e n t i f y r with the measurement point, then: N 1 1 ., e = ~t 4^ (1 - r /r ) ( 6 ' 7 ) s p Figure VI-1 shows a log-log p l o t of N @. (1 - r s / r ^ ) versus r . The slope of the s t r a i g h t l i n e drawn i n F i g . VI-1 i s -1.9, which supports the assertion that there i s l i t t l e leakage of electrons and ions through the current piston, so that the pressure equilibrium i n the shock-heated gas i s maintained by the leakage of neutrals. I t should be pointed out that the r e l a t i v e v e l o c i t y of the shock front with respect to the shock heated gas means that the compressing gas, which i s observed by monitoring the increase i n peak electron density of the precursor, a c t u a l l y l i e s i n a r i n g whose inner radius i s greater than the shock front radius. Thus the value chosen for r i n equation 6.7 should not be the s shock front radius, but some s l i g h t l y larger number. This w i l l tend to decrease the slope of the graph of F i g . VI-1 towards -2. The jump i n observed at the shock front i s a t t r i b u t e d to the compression of the small number of electrons already present i n t h i s volume. This does not m a t e r i a l l y a f f e c t the value of N (see equation 6.6) as the t o t a l number of electrons i n a t h i n s h e l l at r = 1.0 cm. i s a factor of 50 l e s s than the t o t a l number i n a s h e l l of the same density and thickness at r = 7.0 cm. To review the features of the model thus far elucidated: an equilibrium i s established i n which the shock-heated gas pressure ahead of a leaky piston exactly balances the magnetic pressure behind i t . This model i s constructed on the basis of - 109 -Fig. VI-I Log N .(1 - r /r ) vs. Log r e s p 3 0 r (cm) - I l l -the constant current sheet and shock front v e l o c i t i e s , and i t i s capable of explaining the generation of the precursor. The state of dynamic equilibrium evidenced during the i n t e r v a l between f i v e and seven microseconds i s badly upset, however, during the next microsecond. The maximum j_ x B p r o f i l e as a function of time (Fig. V-16) shows a substantial drop i n the magnetic force i n t h i s time i n t e r v a l . The pressure gradient i n the shock-heated gas i s now greater than j_ x B_ and the neutral flow out through the piston i s increased. This flow increases the viscous drag, which brings the current sheet to an abrupt h a l t . The precursor shock c a r r i e s on to the axis, where i t r e f l e c t s back into the shock-heated gas. This r e f l e c t e d shock cannot be detected between r — 0 and r = 2.5 cm. because i t i s moving through hot un-ionized gas. I t i s a weak shock and cannot produce i o n i z a t i o n . (The shock v e l o c i t y i s 13 Km./sec, and the l o c a l sound speed 6.7 Km./sec.). For r greater than 2.5 cm. (see F i g . V - l l ) the r e f l e c t e d shock i s moving i n p a r t i a l l y ionized gas, and the r e s u l t i n g compression can be detected as a jump i n electron density. This i s the " r e f l e c t i o n peak". The " r e f l e c t i o n peak" whose t r a j e c t o r y i s p l o t t e d i n Figure V - l l , i s a-promising diagnostic t o o l . I t i s possible to measure the density r a t i o across the shock, and i t s v e l o c i t y i n the lab frame. From t h i s data, we may c a l c u l a t e some of the thermodynamic properties of the gas into which the shock i s passing. Since the shock i s so weak, i t cannot be producing s i g n i f i -cant i o n i z a t i o n . Instead, the jump i n N q across the shock must r e f l e c t a compression of the gas as a whole, including the - 112 -e l e c t r o n i c component, which can be observed. Further, the "main peak" into which the shock i s passing has just come to rest, which allows us to equate the front v e l o c i t y i n the lab frame to that i n the frame of the gas. From the conservation equations for a weak shock (g, the (17) r a t i o of s p e c i f i c heats, constant through the f r o n t ) , we have Pa g+i where px and p 2 are the densities of the gas ahead of and behind the shock, respectively, and M i s the Mach number, defined as: M = - . (6.9) c The sound speed i n the gas into which the shock i s moving i s c, and v i s the front v e l o c i t y . The value of c at an a r b i -t r a r y temperature, T, can be r e l a t e d to the known speed of sound, c , at some other temperature T by: o o O T ° (6.10) Solving 6.10 for c and su b s t i t u t i n g into 6.9 we get: = 2 -c 1 T M = ir w m (6.ii) 'o Substituting 6.11'into 6.8 and solving for T, we get: - • . f e ) " M £ - H ' ] T = T„ I — / | 1 + I — - 1 J (6.12) We choose T = 290°K. and c =1.0 Km./sec. Measurement of o o Fi g . V - l l , gives v = 13 Km./sec, and F i g . V-5 gives p^Ps =0.7. The gas i s d e f i n i t e l y ionized at t h i s point and g i s not 5/3. - 113 -However, su b s t i t u t i o n of g = 5/3 into 6.12 gives T = 28,000°K, while setting g = 1.1 only raises the value of T to 33,000°K. (17) Comparison with ca l c u l a t e d values of g for a hot helium plasma o shows that for any temperature i n excess of 16,000 K, g has a value between 1.1 and 1.2. In t h i s way we estimate the temperature of the gas into which the shock i s passing i s 32,000°K. This value i s s u b s t a n t i a l l y higher than the gas temperature behind the imploding shock. However, the gas o whose temperature has just been measured at 32,000 K has been brought to r e s t from a r a d i a l v e l o c i t y of 6-7 Km./sec. and the thermalization of t h i s k i n e t i c energy i s thought to be responsible for the elevated temperature. - 114 -VII Conclusions VII.1 Introduction This chapter begins with a b r i e f resume of the important improvements made i n the design of l a s e r - e x c i t e d Fabry-Perot interferometers for plasma diagnostics which have been made i n the course of t h i s experiment. Some suggestions are included for ways i n which the device may be further improved. The chapter closes with a b r i e f review of the important observations which have been made on the collapse of a Z-pinch i n 4 Torr. helium with the interferometer, and includes some remarks on the d i r e c t i o n which future research could take. VII.2 A Summary of Resonator Improvements When the work which t h i s thesis describes was begun, i t was apparent, that i n order to be able to elucidate the plasma c h a r a c t e r i s t i c s i n the Z-pinch collapse, a means of measuring electron density had to be developed which had a high frequency response, high temporal resolution, high s e n s i t i v i t y , and which was not too d i f f i c u l t to operate. Such a device i s now a v a i l a b l e as a r e s u l t of t h i s work. The s t a r t i n g point for the device was a conventional l a s e r - e x c i t e d Fabry-Perot interferometer. An analysis of the shortcomings of such an instrument was made, and work was begun to overcome them. The f i r s t problem overcome was the problem of i n t e r -ferometry i n the presence of transverse gradients i n electron density. The use of quartz tubes to reduce the plasma length, (2) pioneered by S. Medley , was adopted. The resonator geometry was modified from the unstable planoconcave to the stable - 115 -c o n c e n t r i c form . T h i s improvement not o n l y reduced the e f f e c t o f the t r a n s v e r s e g r a d i e n t s , b u t made alignment e a s i e r and improved beam o v e r l a p . Next, the techniques o f f r a c t i o n a l f r i n g e s h i f t i n t e r -ferometry i n the time domain were a p p l i e d to the problem o f measuring e l e c t r o n d e n s i t i e s . The i n t e r f e r o m e t e r which i n c l u d e d t h i s improvement had a s e n s i t i v i t y and temporal r e s o l u t i o n an or d e r o f magnitude b e t t e r than the c o n v e n t i o n a l i n t e r f e r o m e t e r a t times when the e l e c t r o n d e n s i t y was r a p i d l y v a r y i n g , and s e v e r a l o r d e r s o f magnitude b e t t e r where the e l e c t r o n d e n s i t y was s l o w l y v a r y i n g o r v a r y i n g r a p i d l y by s m a l l amounts. T h i s improvement a l s o p e r m i t t e d a unique d e t e r m i n a t i o n o f the s i g n o f the change i n o p t i c a l path l e n g t h to be made. F i n a l l y , e l e c t r o n i c c i r c u i t r y was developed t o enable the user o f the i n t e r f e r o m e t e r t o d i s p l a y d i r e c t l y on an o s c i l l o s c o p e the i n s t a n t a n e o u s v a l u e o f the e l e c t r o n d e n s i t y . Comparison o f the m o d i f i e d , l a s e r - e x c i t e d F a b r y - P e r o t i n t e r f e r o m e t e r , i n c l u d i n g the improvements d i s c u s s e d above, w i t h c o m p e t i t i v e techniques f o r measuring e l e c t r o n d e n s i t i e s i n plasmas i s i n s t r u c t i v e . The b e s t c o m p e t i t i v e method i s the technique o f measuring l i n e - b r o a d e n i n g . In o r d e r to make l i n e -b r oadening measurements, a very expensive, h i g h d i s p e r s i o n spectrometer i s r e q u i r e d . For a p p l i c a t i o n to a p u l s e d , non-r e p r o d u c i b l e d i s c h a r g e , a m u l t i - c h a n n e l d e t e c t i o n system would be e s s e n t i a l . In s p i t e o f a l l t h i s expensive equipment, the data a n a l y s i s would be d i f f i c u l t and s u b j e c t to many u n c e r t a i n t i e s ( i . e . s p a t i a l inhomogeneities, nature o f broadening mechanism). In the end, the system depends on measuring v e r y s m a l l changes i n the widt h o f a l i n e o f p r o b a b l y u n c e r t a i n shape. - 116 -The interferometric system, on the other hand, i s simple, d i r e c t , and most important, provides a continuous and d i r e c t recording of the electron density<, No integration over a number of shots i s required, and only a minimum of analysis i s necessary to get out the r e s u l t s . As a f i n a l comparison of the two techniques, l e t i t be noted that the r e f l e c t e d shock, which i s measured with s u f f i c i e n t accuracy to allow i t s use as a diagnostic probe of the plasma, would never have been detected i n a line-broadening measurement. The lase r interferometer i s now a very accurate, high frequency, and straight-forward way of measuring the r a p i d changes i n electron density observed i n the collapse phase of a z-pinch. A further improvement of the system would be achieved i f the output frequency of the resonator due to the rota t i n g quartz block could be measured at a l l times, instead of being approximated e l e c t r o n i c a l l y as at present. A second resonator, including the quartz block but not the plasma could achieve t h i s . The second resonator would feed a second IFMD. Direct subtraction of the outputs of the two IFMD's would then provide exact c o r r e c t i o n for the instantaneous value of f , thus r e a l i z i n g the f u l l accuracy of f r a c t i o n a l fringe s h i f t interferometry i n the time domain. The same e f f e c t could be achieved by using the output of the plasma-free resonator to determine the magnitude of the current fed onto the IFMD integra t i n g capacitor. - 117 -VII.3 Observation of the Z-Pinch The claims for the improvement of the resonator have been substantiated i n the course of measurements made for that purpose on the Z-pinch i n 4 Torr. helium. This p a r t i c u l a r d i s -charge was chosen because of i t s p o t e n t i a l as a spectroscopic source, and because several other workers had observed p e c u l i a r i t i e s i n the dynamics of the discharge which had not been noted i n other discharges. The abrupt arrest of the current sheet had been reported by T a m ^ 2 ° \ and the separation of the current sheet and the i o n i z a t i o n region was also reported, but no r e a l l y good measurements on the c o l l a p s i n g i o n i z a t i o n s h e l l had been made. We have observed the separation of current sheet and i o n i z a t i o n sheet, and the h a l t i n g of both some distance o f f axis. We have also observed a precursor shock front and the r e f l e c t i o n of t h i s shock o f f the axis and back out into the plasma. From the observed dynamics of the discharge the following model of the collapse process has been developed: 1) The current sheet acts as a leaky piston to drive a shock wave. 2) The r e s u l t i n g configuration i s an equilibrium one, with neutral gas flow through the piston to maintain equilibrium pressure and density ahead of the piston i n the c y l i n d r i c a l c o l l a p s e . 3) The equilibrium configuration i s established through the mechanism of ion-neutral "viscous drag". The j_ x B_ force on the ions balances the ion-neutral viscous drag. The neutral-ion viscous drag sustains the pressure gradient which drives the shock. - 118 -4) The current sheet comes to r e s t when the magnetic pressure declines and i s no longer s u f f i c i e n t to balance the pressure gradient i n the shock-heated gas. 5) The driven shock wave r e f l e c t s from the axis. Passing o through the ionized gas, i t shows a temperature of 32,000 K. Observations of the i o n i z a t i o n s h e l l show less than 60% i o n i z a t i o n i f the t o t a l density i n the s h e l l i s only equal to the f i l l i n g density. I t i s possible that t h i s i o n i z a t i o n represents only a remnant of the o r i g i n a l i o n i z a t i o n produced by Joule heating at the walls, which has been swept up by the t r a i l i n g edge of the magnetic f i e l d . A l t e r n a t i v e l y t h i s i o n i z a t i o n i s due to an i o n i z a t i o n mechanism coupled to the neutrals leaking out of the current piston. This cannot be decided from the data at hand and should be the object of fur the r s tudy. Another object of further study should be the d i s t r i b u t i o n of neutral atoms through the plasma. As has been discussed, the flow of neutral atoms determines the equilibrium attained between the current sheet and the shock-heated gas. One way i n which the neutral d i s t r i b u t i o n could be determined would be to measure the change i n index of r e f r a c t i o n of the plasma at two d i f f e r e n t wavelengths. The difference between the two measurements w i l l give an i n d i c a t i o n of the e f f e c t of neutrals. The method discussed above i s not very s e n s i t i v e , and perhaps other superior methods could be found. F i n a l l y , these measurements confirm the s u i t a b i l i t y of a Z-pinch as a spectroscopic source. The presence of a r e l a t i v e l y quiescent plasma of 32,000°K. provides a source for the - 119 -c a l i b r a t i o n of line-broadening measurements, p a r t i c u l a r l y as the electron density and temperature are independently known. This provides an i n t e r e s t i n g prospect for future work. I t would be of i n t e r e s t to see i f v a r i a t i o n of the i n i t i a l f i l l i n g pressure i n the discharge vessel could be used to vary the temperature and density of t h i s stable plasma. (23) Work has been done by Roberts with a view to sub-s t a n t i a t i n g the Z-pinch as a spectroscopic source. He measured electron densities i n a Z-pinch using laser interferometry, but i n h i s case the resonator was aligned normally to the discharge axis. This necessitated Abel unfolding of the r e s u l t i n g fringe patterns, to f i n d the electron density d i s t r i b u t i o n . This i s c l e a r l y l e s s s a t i s f a c t o r y than the present technique. Further, he d i d not u t i l i z e the techniques of f r a c t i o n a l fringe s h i f t interferometry i n the time domain, and the accuracy of h i s measurements i s correspondingly l e s s . - 120 -BIBLIOGRAPHY Medley, S.S., Curzon, F.L. and Daughney, C.C, Rev. S c i . i n s t r . 36, 713 (1965) Medley, S.S., J . Appl. Physics 41, 142 (1970) Curzon, F.L., Pachner, J . J r . , and Tarn, Y.K.S., Can. J . Phys, 48, 1370 (1970) Williamson, J.H. and Medley, S.S., Can. J. Phys. 47, 515 (1969) Dangor, A.E. and F i e l d i n g , S.J., J. Phys. D. Appl. Phys. 3, 413 (1970) Jahoda, F.C., et a l . Appl. Opt. 6, 8 (1967) Baker, D.A., Hammel, J.E. and Jahoda, F.C., Rev. S c i . Instr. 36, 395 (1965) Sawyer, G.A., Finlayson, V.A., Jahoda, F.C. and Thomas, K.S., Phys. Fluids l£, 1564 (1967) Herold, H. and Jahoda, F.C., Rev. S c i . Instr. 40, 145 (1969) Thomas, K.S., Phys. Fluids 11, 1125 (1968) Kricker, W.A. and Smith, W.I.B., Phys. L e t t . 14, 102 (1965) Witteman, W.J., Appl. Phys. L e t t . 10, 347 (1967) Gibson, A. and Reid, G.W., Appl. Phys. L e t t . _5, 19 5 (1964) Holt, E.H. and Haskell, R.E., 1965. Foundations of Plasma Dynamics (The Macmillan Company, New York). Gerardo, J.B., Verdeyen, J.T. and Gusinow, M.A. J. Appl. Phys. 36' 2 1 4 6 (1965) - 121 -16. Bolwejn, P.T. , Peek, Th. H. and Alkemade, C. Th. J . , Phys. Lett . 23, 88 (1966) . 17. Ahlborn, B. and Salvat, M., Z. Fur Naturf. _22, 260 (1967). 18. Oswatitsch, K. 1956, Gas Dynamics (Academic Press, Inc. , New York). 19. Dimoff, K. and Tarn, Y . K . S . , Can. J . Physics 48, 884 (1970). 20. Tam, Y . K . S . , (1967) Doctoral Thesis, University of B r i t i sh Columbia. 21. Ashby, D.E.T.F. and Jephcott, D.F . , J . Appl. Phys. 36_ 29 (1965). 22. Kogelnik, H, and L i , T. , Appl. Opt. ,5. 1 5 5 0 (1966).. 23. Roberts, T .E . (private communication). Submitted to Phys. Fluids for publication. 24. Griem, H.R. 1964. Plasma Spectroscopy (McGraw-Hill Book Company, New York). 25. Fork, R. L. and Bradley, L . C . , Appl. Opt. 3_, 137 (1964). 26. Rowley, P.D. , Rev. Sc i . Inst. 41, 313 (1970). - 122 -Appendix I - The Integrating Frequency Modulation Detector The operating p r i n c i p l e s of the I.F.M.D. are found i n Chapter IV.4, so they w i l l not be elaborated upon here. Figure AI-1 shows the complete c i r c u i t diagram of the device. C a l i b r a t i o n C a l i b r a t i o n of the I.F.M.D. output i n terms of Ne/V, or the number of electrons per cubic centimetre per v o l t of de f l e c t i o n , involves the determining of two constants i n equation 4.26. These constants are C-j_ and Z, where Z i s the length of plasma which i s being observed, and where C^ depends on the amount of charge being removed from the integrator for each zero-crossing of the resonator output. The constant Z i s equal to the quartz tube separation and i s e a s i l y measured. The value of C-, i s determined using equation 4.24, which where f b , you w i l l r e c a l l , i s the constant base frequency at which I.F.M.D. i s balanced. Consider the e f f e c t i f f (T) i s i s : V ( t 2 ) - V ( t J = AV = C x f ( f b ( T ) - f ( T ) ) d r (4.24) replaced by a known constant frequency, f ^ . AVx = C t J* f B ( T ) d r C j f 1 ( r ) d r = C l ( ^ - f,) ( t 3 (AI.l) I f a second, also known frequency, f s , i s now provided: AV3 = C1 ( f b - f 3 ) ( t 3 - t t ) (AI.2) - 123 -In both cases, the v o l t a g e outputs are s t r a i g h t l i n e s o f constant s l o p e . I f S x = AVl/(t3 - tj)- and S a = A V 2 ( t 3 - t j , then; SL - S 2 = C x ( f b - f\) - C, ( f b - f 2 ) = C, ( f 2 - f j (AI.3) I f Sj and S 2 are measured, C\ i s e a s i l y c a l c u l a t e d from the known d i f f e r e n c e f 2 - f x . - 124 -F i g u r e AI-1 I n t e g r a t i n g Frequency-Mo d u l a t i o n D e t e c t o r C i r c u i t Diagram * 5 v + I 5 v * l 5 v MONOSTABLE MULTIVIBRATORS 0 - 126 -Appendix I I - The I n t e r v a l R e c o g n i t i o n D e t e c t o r The b a s i c f u n c t i o n s o f the IRD are g i v e n i n Chapter V.3, The complete c i r c u i t diagram o f the d e v i c e i s g i v e n i n F i g u r e A I I - 1 . - 127 -F i g u r e A I I - 1 I n t e r v a l R e c o g n i t i o n D e t e c t o r C i r c u i t D i a g r a m •5v .20-25v _ T L i _ r 3 12 I 4 MC846P SHAPER a SCHMITT TRIGGER 14 4 . It _TL d 2 2k -AAA-~!_r 1 2 3 12 MC 846P K> NANQ GATE ( | FLIP-FLOP 14 7 RI PS VARIABLE CONTROL FOR THE UNIT SI SELECTS HIGH OR LOW RANGE FOR Q 3 IN CONJUNCTION WITH RI. A - LOW, B - HIGH PBI RESETS FLIP-FLOP 2 2k —I 1 1 ^PBI • 5 » _n_ 9 14 6 ONE-SHOT lOOpF -vAAA-2 2k •22JJF MC85IP ONE-SHOT ICj ^ H 9 14 II 330pF - B -10 k 05JJF • 5v 2N3704 I I i 2N3704 READY \ 6 v LAMP - 129 -Appendix III - He'I Transitions and the Refractive Index (24) The plasma r e f r a c t i v e index xs given by : a N. f. n - l = " e M 2 A I I I . l o e i (LU -ttfp where n i s the r e f r a c t i v e index, e i s the e l e c t r o n i c charge, i s the e l e c t r o n i c mass, UJ^ i s the frequency of a r a d i a t i v e t r a n s i t i o n , i s the density o f atoms i n the ground state of the t r a n s i t i o n , f i s the t r a n s i t i o n o s c i l l a t o r strength, and UJ i s the frequency of the laser r a d i a t i o n . The plasma frequency (10^) i s : e 2N cu2 = AIII.2 p e M o e where i s the electron density. Substituting AIII.2 into A I I I . l , we get: 1 w 3 N i n - l = -- 2 2 f. -rr AIII.3 1 (UJ 2 -4 ) Since cu » UJ. , x UJ S - UJ | « 2UJ(UJ - UK) AIII.4 which, on s u b s t i t u t i o n into AIII.3, gives: 1 UJ 2 UJf. N. n - 1 = -- Z . - 1 AIII.5 2 i ~ 2 UJ-UJ. N 2UJ x e UJ 2 But — ^ — i s the contribution to the change i n r e f r a c t i v e index 2UJ3 due to the electrons, which allows us to write: - 130 -u>f. N. (n-1). . . = - (n-1) 2 - r 1 AIII.6 t o t a l 2 e i UJ-UJ. N 1 l e L e t t i n g a . be the r a t i o of the r e f r a c t i v e index change due to th *^* the i t r a n s i t i o n to that due to the electrons, we have , ojf. N. 1 1 1 7VXTX n a. = — — AIII.7 l 2 UJ-U). N l e I t i s now necessary to evaluate a. for a l l Hel t r a n s i t i o n s l (Hell t r a n s i t i o n s have been ignored because of the very high temperature required to excite them). A l i s t of Hel t r a n s i t i o n s which have both large values of f. and small values of UHUU. i s 1 uuf_ 1 found i n Table A l , which includes the value of 1 for each (24) 1 of these (from Griem , p. 363), Notice that the two l i n e s with la r g e s t are also those which are c l o s e s t to 6328A", the l a s i n g t r a n s i t i o n . They dominate the r e f r a c t i v e index contribution of the t r a n s i t i o n s of Hel. The energies associated with the 2*P l e v e l , the 2 3P l e v e l and the f i r s t i o n i z a t i o n p o t e n t i a l are 21.13 eV., 20.87 eV. and 24.46 eV., r e s p e c t i v e l y . These three excited states are s u f f i c i e n t l y close together that P. L . E jnay be considered to e x i s t among them. Applying Saha's equation to the 2 3p and i o n i z e d states, we f i n d : N N 2g 27rm kT 3/2 /T, „ . „ m  e + 1+ r e e\ - (E -E ) /kT " gT n ? ) e + e A I I I° 8 where N + i s the ion density (equal to the electron density) , N3 i s the density of atoms i n the 23 P state, g + (g + = 2) i s the - 131 -TABLE AI 2 LS - 3 XP 5,015.7 0.17 -0.82 2!p - 3 1 D 6,678.1 0.73 13.2 2 1 P - 4 1D 4,921.9 0.12 -0.54 2 3P - 33D 5,87 5.6 0.62 -8.6 2 3P - 4 3D 4,471.5 0.12 -0.41 - 132 -s t a t i s t i c a l weight of the ionized state and g 3 (g 3 = 9) that of the 2 3P state, h i s Planck's constant, k i s Boltzmann's constant and (E - E 3) i s the energy difference between the two states, g i v i n g : 3 o 59 A T ^ 2 — = 6o97 x l O " 2 1 r-Tx AIII.9 N e 2 ( k T e ) J / 2 where kT i s now expressed i n eV., and the densities are e measured i n c m - 3 „ The temperature of the plasma i s not l i k e l y to be s u f f i c i e n t to produce L.T.E. between the 23 P and ground states, which w i l l tend to produce an overpopulation of the ground state r e l a t i v e to the 2 3P state. In t h i s event: N3 < N — e " E 3 / ^ T e AIII.10 ° g o where N and g are the density and s t a t i s t i c a l weight, o 3 o . J 3 respectively, of the ground state. Combining AIII.10 with AIII . 9 , we get: The function g 3N 17 .28AT e < 6.97 X 10~ 2 1 r - r r - A I I I . l l g o (kT e) 3 / 2 _17.28/kT e e (kT ) 3 / 2 e has a maximum value o f 5.71 x 10 . for Thus, kT = 11.5 eV. e g 3N — 1 < 3.98 x 10" 2 3 AIII.12 g o - 1 3 3 -For a f i l l i n g p r e s s u r e o f 4 T o r r . , and a n e u t r a l compression r a t i o o f four , we get, as the maximum v a l u e o f N Q , N < 4 . 8 x 1 0 1 7 era" 3 o S i nce g 3 = 9 , — < 1 0 , which g i v e s g o N 3 < 4 . 4 x 1 0 " 3 A I I I . 1 3 N e Thus, f o r the 2 3 P - 3 3 D t r a n s i t i o n , a 3 < 1 . 8 9 x 1 0 ~ 3 A I I I . 1 4 A f a c t o r o f thr e e r e d u c t i o n i n the s t a t i s t i c a l weight, UJf i p a r t i a l l y o f f s e t by an i n c r e a s e i n — o f l e s s than a f a c t o r o f two, g i v e s an a f o r the 2 X P - 3lD t r a n s i t i o n which i s about one h a l f o f t h a t g i v e n i n A I I I . 1 4 . Thus the maximum p o s s i b l e c o n t r i b u t i o n o f Hel t r a n s i t i o n s to the change i n plasma r e f r a c t i v e index i s l e s s than 2%. T h i s "worst c a s e " presumes an e l e c t r o n temperature a t l e a s t a f a c t o r o f f i v e h i g h e r than i s l i k e l y to e x i s t i n the Z-pinch, and the a c t u a l c o n t r i b u t i o n i s t h e r e f o r e so sm a l l as to be e n t i r e l y n e g l i g i b l e . 

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