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Mode-locking and other nonlinear effects in CdS Phonon Masers Smeaton, Melvin Douglas 1976

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MODE-LOCKING AND OTHER NONLINEAR EFFECTS IN CdS PHONON MASERS by MELVIN DOUGLAS SMEATON B.Sc.  (Hon.). University of Alberta, 1971  M.Sc, Simon Fraser University, 1973  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA November, 1976  (5) Melvin Douglas Smeaton, 1976  In presenting this thesis in partial  fulfilment of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I a g r e e  that  the Library shall make it freely available for r e f e r e n c e and study. I further agree that permission for extensive copying o f this  thesis  for scholarly purposes may be granted by the Head o f my D e p a r t m e n t o r by his representatives. of  It  is understood that c o p y i n g o r p u b l i c a t i o n  this thesis for financial gain shall not be allowed without my  written permission.  Department of  PHYSICS  The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  MOV- 2 |<f7k ]  ABSTRACT  The  nonlinear  theory of acoustic a m p l i f i c a t i o n i n p i e z o e l e c t r i c  semiconductors has been applied to the CdS phonon maser, to demonstrate that mode-locking In such a device can be predicted as a consequence of i t s structure and nonlinear  properties  of the a c o u s t o - e l e c t r i c  amplifying  mechanism. The f i r s t d i r e c t evidence of mode-locking i n the phonon maser has been provided by applying  o p t i c a l processing  techniques to s i g n a l s  obtained from l a s e r d i f f r a c t i o n . A new and powerful technique f o r the analysis of o p t i c a l s i g n a l s , i n v o l v i n g a combination of s p a t i a l and temporal f i l t e r i n g , has been developed. Further a p p l i c a t i o n of o p t i c a l processing d i r e c t observation  has allowed the f i r s t  of a c o u s t i c a l l y induced space charge gratings  i n CdS.  Proper e x p l o i t a t i o n of the methods outlined should give i n s i g h t i n t o the underlying  p h y s i c a l processes.  In addition to the o p t i c a l experiments, data has been presented I hat demonstrates the existence  of two new photovoltaic  e f f e c t s i n CdS.  The r e s u l t i n g photovoltages are several orders of magnitude l a r g e r than lhose produced by known photovoltaic involved are not properly  e f f e c t s . As yet, the p h y s i c a l processes  understood.  ii  TABLE OF CONTENTS  Page L i s t of Tables  v  L i s t of Figures  .  L i s t of Symbols  x  Acknowledgements  xiv  Foreword CHAPTER 1:  vi  xv  Mode-Locking i n CdS Phonon Masers  1  1.1  Introduction  1  1.2  The Phonon Maser as a Repetitive Pulse Generator  3  1.2.1 1.3  Nonlinear Gain Theory - The Saturable Absorber ... 7  Theory of the O p t i c a l Measurements  25  1.3.1  Crystal Optics - The Photoelastic E f f e c t  25  1.3.2  Interpretation of the O p t i c a l Signal Preliminary Considerations  1.3.3  Interpretation of the O p t i c a l Signal The Mode-Locked Phonon Maser  1.4  1.5  Experimental Results  48 62  1.4.1  Experimental Apparatus and Techniques  1.4.2  O p t i c a l V e r i f i c a t i o n of Mode-Locking  ; • • • ^6 73  O p t i c a l Determination of the Normal Modes of Composite C a v i t i e s  1.6  35  .... 89  Conclusions and Summary of Contributions  iii  92  Page CHAPTER 2:  A p p l i c a t i o n of O p t i c a l Processing to a Study of A c o u s t i c a l l y Induced Space Charge Gratings i n CdS  93  2.1  Introduction  93  2.2  A p p l i c a t i o n of O p t i c a l Processing  99  2.3  Experimental Results  107  2.4  Conclusions  116  CHAPTER 3:  and Summary of Contributions  New Photovoltaic E f f e c t s i n CdS  117  3.1  Introduction  117  3.2  The Photoacousto V o l t a i c E f f e c t  118  3.2.1  121  3.3  3.4 APPENDIX A:  Discussion  The A.C. E l e c t r i c F i e l d Induced Photovoltaic E f f e c t  127  3.3.1  134  Discussion  Conclusions  and Summary of Contributions  138  Acoustic Standing Wave Pattern f o r Mode-Locked Operation  139  APPENDIX B:  P h y s i c a l Properties  144  APPENDIX C:  Acoustic Bonds  148  REFERENCES  150  BIBLIOGRAPHY - D i f f r a c t i o n of Light by U l t r a s o n i c Waves  153  iv  LIST OF TABLES  Table 1  Page Experimentally determined s t r a i n amplitudes for mode-locked operation of  Bl  and phases  82  DC1  Physical data for acoustic c a v i t i e s  v  146  LIST OF FIGURES  Figure  Page  1. (a)Block diagram f o r the r e p e t i t i v e pulse generator  4  (b)Qualitative i l l u s t r a t i o n of saturable absorber operation 2. Nonlinear  5  gain as a function of |A|  20  3. Behavior of the phonon maser as a function of conductivity and e l e c t r i c  21  field  4. The i n d i c a t r i x construction  27  5. Form of the index e l l i p s e i n an i s o t r o p i c material  33  6. Schematic representation of o p t i c a l system to be studied  37  7. Intensity of zero and f i r s t order d i f f r a c t i o n f o r various  47  values of the parameter Q 8. Intensity p r o f i l e of the dark f i e l d term  56  9. Theoretical image plane p r o f i l e s of Q component f o r combined  60  s p a t i a l orders 4,5,6  ;  10. Display of' the acousto-electric current s i g n a l f o r b|_ c r y s t a l  63  24.01.02.04 11.  (a)Current density vs. applied e l e c t r i c f i e l d f o r bj_ c r y s t a l  64  24.01.02.04 (b)Time display of acousto-electric current f o r points  65  indicated i n FIG 11(a) 12. Experimental configuration f o r the o p t i c a l measurements  vi  67  Figure 13. (a)Mounted composite  cavity  (b) Typical f o c a l plane d i f f r a c t i o n pattern (c) Focal plane s p a t i a l  filter  14. Photographs and traces of the image plane i n t e n s i t y d i s t r i b u t i o n for DC1 15. D.C.  dark f i e l d term f o r DC1  16. Experimental and corresponding t h e o r e t i c a l image plane i n t e n s i t y p r o f i l e s f o r DCl under d i f f e r e n t conditions of s p a t i a l and temporal  filtering  17. Acoustic s t r a i n p r o f i l e reconstructed from experimental data 18. Frequency display of the acousto-electric current f o r DCl operated i n a multimode manner 19. Mode structure of composite  cavity DCl obtained by l i g h t  diffraction 20. O p t i c a l data obtained from model system c o n s i s t i n g of quartz transducers i n a c e l l of d i s t i l l e d water 21. Intensity p r o f i l e s obtained by imaging d i f f r a c t i o n spots ±1 and 1,2  f o r cj_ c r y s t a l 24.06.06.01  22. Image plane signals f o r cj_ c r y s t a l 24.06.06.01 produced by imaging d i f f r a c t i o n spots ±1 and 23. Image plane photographs f o r DC4  0,1  i l l u s t r a t i n g that various  o p t i c a l signals can be completely separated  vii  Figure 24. Image plane signal f o r one part of DC4, produced by combining d i f f r a c t i o n spots 0,1 both i n the presence and absence of the external d r i v i n g  field  25. A t y p i c a l glow curve obtained from a CdS c r y s t a l o f the type used to construct phonon masers 26. Experimental configuration f o r measuring  the photoacoustic  voltage 27. (a)Photoacoustic voltage as a function of l i g h t i n t e n s i t y for bj_ c r y s t a l 29.04.01.02 (b)Dember voltage as a function of l i g h t  intensity  28. (a)Spectral response of photoacoustic voltage f o r bj_ c r y s t a l 29.04.03.02 (b)Spectral response of the photocurrent f o r bj_ c r y s t a l 29.04.03.02 29. Behavior of photoacoustic voltage as a function of frequency, for  DC5  30. Experimental configuration f o r measuring  the A.C. e l e c t r i c  f i e l d induced photovoltage 31. Spectral response of the D.C. photovoltage f o r bj_ c r y s t a l 29.04.01.02 32. (a)D.C. photovoltage as a function of A.C. d r i v i n g amplitude (b)D.C. photovoltage as a function of d r i v i n g frequency at constant amplitude  viii  Figure  Page  33. (a)D.C. photovoltage as a function of frequency,for small  132  amounts of weakly absorbed  light  (b)D.C. photovoltage as a function of frequency, f o r strongly absorbed  light  34. Some r e s u l t s of pulse measurements on the D.C. photovoltage  ix  135  LIST OF SYMBOLS  nonlinear parameter optical  i n acoustic gain theory  amplitude  t o t a l dark f i e l d  amplitude  f o c a l plane o p t i c a l  amplitude th  o p t i c a l amplitude i n n near f i e l d  d i f f r a c t i o n spot  (object plane) o p t i c a l amplitude  near f i e l d o p t i c a l amplitude i n absence of acoustic waves nonlinear acoustic gain; + p a r a l l e l and - a n t i p a r a l l e l to applied e l e c t r i c f i e l d components of the r e l a t i v e d i e l e c t r i c impermeability tensor and matrix, r e s p e c t i v e l y change i n the parameters B^ parameter i n d i c a t i n g the degree of s p a t i a l synchronization of Fourier components IJJ e l a s t i c s t i f f n e s s constant e l e c t r i c displacement thickness of acoustic cavity electron d i f f u s i o n constant r e l a t i v e phases of mode-locked acoustic waves electric  field  piezoelectric  stress constant  synchronous e l e c t r i c f i e l d total electric field permittivity  constant  =  v_/u s  •  F  acoustic trapping  T  l a t t i c e loss  Y  acoustic wave v e l o c i t y parameter = 1 - n / » normalized angle of incident l i g h t (section 1.3.2)  Y  fraction  coefficient v  v  o  r  s  +  acoustic wave v e l o c i t y parameter: + p a r a l l e l and a n t i p a r a l l e l to applied e l e c t r i c  field  I  t o t a l current density  Ijyp  t o t a l dark f i e l d i n t e n s i t y  """n. n+1 n+2 ' '  o p t i c a l i n t e n s i t y d i s t r i b u t i o n produced by combining d i f f r a c t i o n orders n, n+1, n+2  •""n n+1 n+2 ' '  image plane signal obtained with s p a t i a l orders n, n+1, n+2 and temporal f i l t e r at  1^ Q '  image plane s i g n a l at frequency <% produced by imaging d i f f r a c t i o n spots 0,1  I  synchronous current  g  J J  =  n  distribution  c 0  l  v s  conduction current density n  Bessel function of order n  K  acoustic wave number  AK  acoustic phase s h i f t produced by p i e z o e l e c t r i c  k  o p t i c a l wave number  k*  =  ksin6, where 6 i s angle of d i f f r a c t i o n  p r i n c i p a l d i e l e c t r i c constants L  width of the acoustic  A  acoustic wavelength  X  o p t i c a l wavelength  M  amplitude modulation index  xi  cavity  coupling  electron mobility c o n d u c t i o n band e l e c t r o n d e n s i t y ,  o r i n d e x of r e f r a c t i o n  change i n e l e c t r o n d e n s i t y o r i n d e x o f r e f r a c t i o n principal refractive indices semiaxes of i n d e x e l l i p s e f o r i s o t r o p i c m a t e r i a l i n the p r e s e n c e o f s t r a i n s t r a i n i n d u c e d changes i n index e l l i p s e  semiaxes  e q u i l i b r i u m e l e c t r o n density, o r unperturbed r e f r a c t i v e index ordinary  and  extraordinary  indices of r e f r a c t i o n  a c o u s t i c a l l y m o d i f i e d o r d i n a r y and i n d i c e s of r e f r a c t i o n n  o  - o» n  acoustic optical  n  e  ~  n  extraordinary  e  frequency frequency  c o n d u c t i v i t y r e l a x a t i o n frequency  =  a/e  c o n d u c t i v i t y r e l a x a t i o n f r e q u e n c y as m o d i f i e d by nonlinear theory d i f f u s i o n frequency modulation  =  v|/D  n  frequency  components of the o p t o e l a s t i c m a t r i x phase f u n c t i o n d e s c r i b i n g mode-locked a c o u s t i c waveform phase a n g l e , or a n g l e o f  incidence  a c o u s t i c phase f u n c t i o n phase f u n c t i o n d e s c r i b i n g s t a t i c  space charge g r a t i n g  envelope f u n c t i o n d e s c r i b i n g a c o u s t i c m o d i f i c a t i o n o p t i c a l amplitude  xii  of  ij> (z)  Fourier components of ^(x,z,t)  Q  parameter characterizing d i f f r a c t i o n  q  electron charge  r  r e f l e c t i v i t y of acoustic c a v i t y walls  p  mass density  S,  s t r a i n amplitude  a  e l e c t r i c a l conductivity  T  acoustic stress  n,n+l,n+2  component of I  n  T  T  n,n+l,n+2  8 0  F  o  u  r  i  e  r  ^  -  transform of  phase angle  =  ^ at frequency a  ^ n fft  flri  -2  Kx - Qt, or angle of d i f f r a c t i o n  l i n e a r combination of acoustic phases  n  u  mass displacement  V  =  2vcosQt  V  =  (2vcosfit + v ) ( l + Mcosoijjjt)  =  2v cos(nKd-nnt+6 )  V  n  v, v  n  =  ^ ^n+l ^n+2 -  +  n  Q  n  n  Raman - Nath parameter  Vp  electron d r i f t v e l o c i t y  v  amplitude of o p t i c a l phase v a r i a t i o n produced by s t a t i c  Q  space charge grating v  g  v e l o c i t y of sound  z^_.  components of the e l e c t r o - o p t i c matrix  C  phase parameter  xiii  =  (n kL - wt) Q  ACKNOWLEDGEMENTS  I wish to thank my research supervisor, Dr. R.R. Haering, for a great many productive discussions, and for being a source of encouragement and motivation throughout the course of my research. Thanks are due also to Dr. J . Vrba, who participated i n the i n i t i a l stages of the development of a t h e o r e t i c a l model for the mode-locked phonon maser. I am also g r a t e f u l to Lore Hoffmann for her capable assistance i n the preparation of publications. The f i n a n c i a l support of the National Research Council, the International Nickle Company, and the University of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. F i n a l l y , I thank my wife Marg for her continuing support and for her assistance i n typing this t h e s i s .  xiv  FOREWORD  Chapters 1 and 2 document the r e s u l t s of o p t i c a l d i f f r a c t i o n and signal processing experiments, and contain the most important contributions of the t h e s i s . Chapter 3 deals with new photovoltaic e f f e c t s that are not yet understood. This chapter i s included as a permanent record of the experimental r e s u l t s , and to provide a base f o r future work.  xv  CHAPTER 1  MODE-LOCKING IN CdS PHONON MASERS  1.1  Introduction Phonon masers are s o l i d state acoustic o s c i l l a t o r s produced by  p o l i s h i n g the faces of p i e z o e l e c t r i c s i n g l e c r y s t a l s accurately f l a t and p a r a l l e l , to form high Q resonant structures which are strongly analogous to l a s e r s . The most successful phonon masers have been f a b r i c a t e d from CdS. The a p p l i c a t i o n of a D.C. e l e c t r i c f i e l d of s u f f i c i e n t magnitude creates a s i t u a t i o n i n which net round-trip acoustic gain can be achieved by v i r t u e of the p i e z o e l e c t r i c coupling between d r i f t i n g conduction electrons and l o c a l acoustic f i e l d s . Acoustic o s c i l l a t i o n i s then spontaneously b u i l t up from the thermal background. The l i n e a r theory of acoustic a m p l i f i c a t i o n i n p i e z o e l e c t r i c semiconductors (White 1962) has been well established, and reviews may be found i n the l i t e r a t u r e (Gurevich  1969, McFee 1966).  Operation of the phonon maser i s generally observed by monitoring the current passing through the device. There i s a D.C. component to this current, due to the applied D.C. e l e c t r i c f i e l d , but there i s a l s o an A.C. component present when the phonon maser i s operating. This acousto-electric current i s produced by the tendency of electrons to group i n p i e z o e l e c t r i c p o t e n t i a l wells produced by the acoustic wave. Maines and Paige (1970) reported that phonon masers operated under c e r t a i n experimental conditions exhibited sharp spiking i n the time d i s p l a y of  2  the A.C. current. The frequency display of the current s i g n a l consisted of a harmonic s e r i e s having amplitudes constant  i n time, and a frequency  spacing equal to the r e c i p r o c a l of the round t r i p t r a n s i t time of the c r y s t a l c a v i t y . From these observations Maines and Paige (1970) concluded that the phonon maser was operating i n a mode-locked regime, and, by analogy with mode-locked o p t i c a l l a s e r s , predicted that the acoustic output should consist of narrow, high s t r a i n pulses. The v a l i d i t y of t h i s p r e d i c t i o n i s , however, f a r from obvious, since the mode-locked regime i s strongly nonlinear,  and there i s no  one-to-one correspondence between the frequency spectrum of the acoustoelectric current and that of the acoustic f i e l d s . When the p i e z o e l e c t r i c p o t e n t i a l associated with these f i e l d s i s much l a r g e r than the thermal energy, a purely Sinusoidal acoustic wave can r e s u l t i n an electron d i s t r i b u t i o n with r i c h harmonic content (Gulayev 1970, Gurevich 1969). The nonlinear nature of the mode-locked regime i s c l e a r l y established by the presence of strong current saturation, i n d i c a t i n g that the conduction electrons are trapped i n the p o t e n t i a l wells associated with the acoustic wave, and are constrained  to move with the  v e l o c i t y of sound. These considerations c l e a r l y i n d i c a t e that the p r e d i c t i o n of Maines and Paige (1970) should be v e r i f i e d by d i r e c t observation of the acoustic f i e l d s . This chapter deals with a unique set of o p t i c a l experiments that have provided v e r i f i c a t i o n of mode-locked operation, by allowing d i r e c t observation of the r e s u l t i n g acoustic s t r a i n pulses. The general  3  method r e l i e s on a powerful combination of s p a t i a l and temporal  filtering  applied to o p t i c a l signals obtained from laser d i f f r a c t i o n . Before considering the o p t i c a l measurements, i t i s worthwhile to consider what mode-locking means and why the phonon maser may be mode-locked. This information may be provided by making a very f r u i t f u l analogy between a phonon maser and a general c l a s s of devices known as R e p e t i t i v e Pulse Generators. 1.2 The Phonon Maser. as.r.a Repetitive Pulse  Generator  I t i s w e l l known that a feedback loop containing the b a s i c elements of an a m p l i f i e r , f i l t e r , delay l i n e , and a nonlinear element c a l l e d a saturable absorber, behaves as a R e p e t i t i v e Pulse Generator  (RPG)  (Cutler 1955). The mode-locked o p t i c a l l a s e r i s a w e l l known example of  such a device (Demaria et a l 1969). The basic block diagram f o r a RPG  i s shown i n FIG. 1(a). The output c o n s i s t s of a t r a i n of i d e n t i c a l pulses recurring at a rate determined  by the loop delay, with shape  determined  p r i n c i p a l l y by the c h a r a c t e r i s t i c s of the f i l t e r and saturable absorber, and width c o n t r o l l e d p r i m a r i l y by the bandwidth of the f i l t e r . The saturable absorber has a c r u c i a l r o l e . I t has the e f f e c t of  emphasizing  the highest amplitude parts i n the pulse c i r c u l a t i n g i n  the feedback loop, while reducing the lower amplitudes. An element having the transmission c h a r a c t e r i s t i c s shown i n FIG 1 (a) w i l l serve the purpose. As indicated, i n t e n s i t i e s > I are passed almost unattenuated while lower ' o i n t e n s i t i e s are reduced. A highly q u a l i t a t i v e i l l u s t r a t i o n of saturable absorber operation i s shown i n FIG 1 (b). As a c i r c u l a t i n g pulse makes  DELAY  Intensity FIG 1(a)..  Block diagram for the Repetitive Pulse Generator  (RPG) .  +  ro I  c\i I  M I  +  +  n  tN  +  1  i  i  1  CM + +  A q u a l i t a t i v e i l l u s t r a t i o n of the operation of a saturable absorber i n both the time (upper) and frequency (lower) domains. Only the envelope of the pulses i s shown in the time display. f  c  i s the center frequency or frequency of maximum gain, and f i s the  reciprocal of the round t r i p t r a n s i t time T of the delay loop.  I  E  6  r e p e t i t i v e passes through the saturable absorber, i t becomes more and more compressed. Correspondingly, i t s frequency spectrum becomes broader and broader  u n t i l i t i s f i n a l l y l i m i t e d by the bandwidth of the f i l t e r .  The steady s t a t e pulse width i s approximately  equal to the r e c i p r o c a l of  the system bandwidth. In the phonon maser, most of the required elements may  be  r e a d i l y i d e n t i f i e d . The p i e z o e l e c t r i c coupling of the acoustic f i e l d to d r i f t i n g electrons provides an amplifying mechanism, a combination of the cavity  Fabry - Perot resonances  and the gain p r o f i l e of the phonon  maser c o n s t i t u t e the f i l t e r , and the round t r i p t r a n s i t time of the resonant c a v i t y serves as the loop delay. I t remains to f i n d something which w i l l provide the necessary saturable absorber  effect.  In the case of an ordinary o p t i c a l l a s e r , which also contains the above 3 b a s i c elements, the saturable absorber must be p h y s i c a l l y added to  the system f o r RPG operation. One  thing that i s done, f o r example,  i s to place a c e l l of bleachable dye s o l u t i o n i n the laser c a v i t y . Such a dye becomes transparent at very high l i g h t i n t e n s i t i e s , thus providing the necessary c h a r a c t e r i s t i c indicated i n FIG  1 ( a ) . For some l a s e r s ,  the gain p r o f i l e encompasses hundreds of cavity modes. In general a l l these modes w i l l be running but i n a very incoherent manner, so that any pulses e x i s t i n g i n the system w i l l be broad and The pulse compression  ill-defined.  produced by a d d i t i o n of the saturable absorber  forces the operating c a v i t y modes into a f i x e d phase r e l a t i o n s h i p . Hence the o r i g i n of the term mode-locking,In t h i s manner picosecond  o p t i c a l pulses have been achieved.. For the phonon maser i t i s not possible to i s o l a t e a p h y s i c a l element which functions as a saturable absorber. We s h a l l present a q u a l i t a t i v e discussion which demonstrates that c e r t a i n nonlinear properties of the a c o u s t o - e l e c t r i c amplifying mechanism provide the necessary 1.2.1  behavior.  Nonlinear Gain Theory - The Saturable  Absorber  To e s t a b l i s h that a mechanism which serves as the counterpart of the saturable absorber exists i n the phonon maser, i t i s necessary to formulate an expression f o r large amplitude gain. Since mode-locking i s a strongly nonlinear process, c l o s e l y related to such phenomena as D.C. current saturation (Smith 1963) and high e l e c t r i c f i e l d domain formation and propagation (Maines 1966, Haydl and Quate 1966), any attempt to extend the l i n e a r theory of acoustic a m p l i f i c a t i o n (White 1962) w i l l not produce r e s u l t s v a l i d f o r this s i t u a t i o n . The nonlinear theory of acoustic wave propagation i n p i e z o e l e c t r i c semiconductors discussed by many authors: Butcher  has been  (1971), Butcher and Ogg (1968, 1969,  1970), Gay and Hartnagel (1969, 1970), Gulayev (1970), Gurevich  (1969),  Tien (1968), Wanneberger (1970), as w e l l as others. While the above treatments produce r e s u l t s that are l a r g e l y s i m i l a r , the unique approach used by Butcher and Ogg (1968, 1969, 1970) w i l l be adopted since i t i s quite straight forward and provides r e s u l t s i n a transparent and u s e f u l form. Since only q u a l i t a t i v e expressions are required f o r our purposes,  the f i n i t e thickness of the phonon maser  8 and the ensuing e l e c t r i c a l boundary conditions w i l l be ignored. For a discussion of these boundary conditions see Sharma and Wilson  (1970).  The quasi s t a t i c approximation ( i e . the c u r l of the e l e c t r i c f i e l d = 0) w i l l be employed. This has the e f f e c t of decoupling purely electromagnetic waves (propagating at the v e l o c i t y of l i g h t ) from a c o u s t o - e l e c t r i c waves (propagating at the v e l o c i t y of sound). Since t h i s corresponds to ignoring terms i n the acoustic dispersion which are a f a c t o r of ~10  1 0  (the square of the r a t i o of the acoustic v e l o c i t y to the v e l o c i t y of l i g h t ) smaller than other contributions (Hudson and White 1962), i t i s an . excellent approximation. We s h a l l consider the case of uniform plane acoustic waves, propagating along the x a x i s . ^ This reduces the problem to a one-dimensional s i t u a t i o n . With these considerations the system of equations to be solved becomes (MKS units are employed) :  E l a s t i c wave equation:  3T 8x  =  p 3^u 9t2  (1.1)  'This i s not as u n r e a l i s t i c as i t may appear. The p h y s i c a l s i t u a t i o n of p a r t i c u l a r i n t e r e s t corresponds to pure shear waves whose wave vectors l i e along the c r y s t a l l o g r a p h i c b-axis. Results of our o p t i c a l  experiments  i n d i c a t e that a one dimensional analysis i s quite reasonable i n t h i s case.  9  Poisson's  equation: 3D 8x  0-* )  q(n - n )  =  2  Q  P i e z o e l e c t r i c equations of s t a t e :  Continuity  T  =  cS  +  D  =  -eS  (1.3)  eE  +  (1.4)  eE  equation: 3J 3x  __ -q 9 (n - no)  Equation of current J  (15)  3t  =  density: qnuE - q D  Where T and S a r e t h e a c o u s t i c  n  3n 3x  (  E and D a r e t h e e l e c t r i c  d i s p l a c e m e n t , q, u, D  a r e t h e magnitude o f the e l e c t r o n c h a r g e , t h e  electron  >  6  )  s t r e s s and s t r a i n , u i s the mass d i s p l a c e m e n t ,  p i s t h e mass d e n s i t y , n  1  f i e l d and e l e c t r i c  m o b i l i t y and t h e e l e c t r o n d i f f u s i o n c o e f f i c i e n t , J i s the  conduction current,  c, e, e, a r e t h e a p p r o p r i a t e  p i e z o e l e c t r i c and p e r m i t t i v i t y c o n s t a n t s , d e n s i t y and n  Q  i s the e q u i l i b r i u m e l e c t r o n  elastic  stiffness,  n i s t h e c o n d u c t i o n band  electron  density.  The s i g n c o n v e n t i o n o f B u t c h e r and Ogg (1968) i s employed i n  10 equations (1.1) - (1.6)  : E, J and D are measured p o s i t i v e i n the  negative x d i r e c t i o n . This implies that a p o s i t i v e D.C. e l e c t r i c  field  ( i e . pointing i n the negative x d i r e c t i o n ) w i l l produce e l e c t r o n d r i f t , and ultimately acoustic a m p l i f i c a t i o n , i n the p o s i t i v e x d i r e c t i o n . In equations (1.2), (1.5) and (1.6) the e f f e c t of holes has been ignored. This i s a very good approximation f o r CdS which i s n-type and has a hole m o b i l i t y and hole l i f e t i m e which are approximately 10 • and 10 L  times  smaller, r e s p e c t i v e l y , than the corresponding electron parameters. The presence of the acoustic wave produces a v a r i a t i o n An = (n-n ) Q  i n the equilibrium electron density. I t i s i m p l i c i t l y assumed i n the preceeding equations that the space charge qAn i s mobile. While t h i s approximation i s s a t i s f a c t o r y f o r our purposes, i t should be r e a l i z e d that qAn w i l l include a contribution due to modulation of the e l e c t r o n density i n l o c a l i z e d traps. Thus, i n general, only a f r a c t i o n of the space charge produced by the acoustic wave w i l l be mobile (see McFee 1966). Combining equations (1.2) and (1.5) to eliminate the term i n electron charge, we obtain:  |_  ( + |f) = o J  (1.7)  oX  Expression  (1.7) has general v a l i d i t y . For our one-dimensional system  i t implies that the t o t a l current I, c o n s i s t i n g of the conduction current J and the displacement current 3D/8t i s s p a t i a l l y i n v a r i a n t . ,This allows us to modify equation (1.6) i n the form  11  I = qnyE  +  9D  -  3t  q Dn 3_n 3x  (1.8)  where, according to equation (1.7), I can at most be a function of time. In f a c t , as w i l l be indicated shortly, I may be assumed to have a constant value i n the c a l c u l a t i o n s to be performed. S, T, E, n and D are assumed to have plane wave dependence of the form D  =  D„  +  Die  1 6  +  cc.  (1.9)  as i l l u s t r a t e d f o r the e l e c t r i c displacement. Here 6 = Kx - fit, where K, £2 are the r e a l wavenumber and angular frequency of the wave. D independent, and D^  Q  i s time  i s assumed to be a slowly varying function of x and  t compared with the exponential. The D.C. component E  Q  corresponds to  the applied d r i f t f i e l d required to operate the phonon maser. The n o n l i n e a r i t y i n the previous equations i s contained s o l e l y i n the term nquE i n the expression f o r current density (equations (1.6) and (1.8) ). Thus t h i s term must be e x p l i c i t l y retained to extend the theory into the large amplitude regime. Following Butcher and Ogg (1968), we begin by solving equation (1.8) f o r E and making use of equation (1.2) to obtain  E  =  9t  +  D  n  1—arT, ^ )  a ( 1 + qn where 0 = n qp Q  Q  B.x '  C - ) 1  10  12 The nonlinear behavior i s now  expressed by the denominator  i n equation (1.10), as may be seen by expanding of (qn_)  1  this term i n powers  8D/9x. With the use of equation (1.9) and the s u b s t i t u t i o n  ^ 1  (1.11)  equation (1.10) may be written:  g  =  1  +  S  f°  (  ( v  s  +  1 K D  n)  a ( 1 + ( - e  A  1 9  e  l  + ' -  9  c  c  )  (1.12)  + c.c. ) )  where v„ = — = v e l o c i t y of sound. In obtaining equation (1.12) the derivatives of Di have been neglected i n comparison with the d e r i v a t i v e s of the exponential term. From the previous discussion i t i s apparent that a l l nonlinear behavior i s now embodied i n the parameter A. A has a simple physical i n t e r p r e t a t i o n . Solving equation (1.2) f o r n and using equations (1.9) and  (1.11) we obtain (again neglecting d e r i v a t i v e s  of D ) : x  n  =  n  z T . A i6 . A ( l + 2-e 2" +  c  e  -ie . . )  /-I  o\  (1.13) n  or, writing A i n polar form |AJ e ^: 1  n  =  n  Q  [ 1 + | A| cos(9+<j>) ]  (1.14)  13 Thus A s p e c i f i e s  the amplitude  Since n i n equation  and phase of the e l e c t r o n d e n s i t y wave.  (1.14) cannot  be n e g a t i v e , the p r e s e n t  analysis  imposes the f o l l o w i n g c o n s t r a i n t :  0 <_ |A| <_ 1  In a broader  (1.15)  sense, A p r o v i d e s an e x t r e m e l y  determining amplitude as f o r E i n e q u a t i o n  parameter f o r  dependent e x p r e s s i o n s f o r the v a r i a b l e s i n v o l v e d , (1.12). The l i n e a r  recovered i n the l i m i t  t h e o r y o f White  (1962) may  be  |A| -> 0.  The D.C. and A . C . components equation  convenient  (1.12) by p e r f o r m i n g  E  Q  =  o f E may be e x t r a c t e d f r o m  the f o l l o w i n g phase  1  2TT  ^  jEd9  averages:  (1.16)  6  E  l  =  In the s l o w l y v a r y i n g envelope  ,  2TT  J  /Ee d6 0  (1.17)  1 0  v  approximation  a l r e a d y o u t l i n e d , D^  may  be t r e a t e d as a c o n s t a n t i n c a r r y i n g out the i n t e g r a t i o n s . A l s o , s i n c e D, E and n have t h e form shown i n e q u a t i o n -may and  ( 1 . 9 ) , t h e RHS o f e q u a t i o n  (1.8)  be t r e a t e d as s o l e l y a f u n c t i o n o f 8 f o r t h e i n t e g r a t i o n s i n (1.16) (1.17).  Thus I , which has been shown t o be a t most a f u n c t i o n o f time,  may be t r e a t e d as a c o n s t a n t i n e q u a t i o n s  (1.16) and  (1.17).  14  (1.12)  Using equation the i n t e g r a l s i n ( 1 . 1 6 ) around  the u n i t c i r c l e  and  (1.17)  and  the s t a n d a r d s u b s t i t u t i o n z = e may  be c o n v e r t e d to c o n t o u r  i n the complex p l a n e . One  1  -  (1  -  (1 -  .(1.18) Z  (1  ifi  The n o t a t i o n used s = -— y  -  integrals  A 2\h )  S  (1  ,  then o b t a i n s :  I)  E A 1 +  19  A O  (1  lAl*) * 1  i s the same as B u t c h e r and Ogg  (1.19)  D  (1968) :  v  E  s  i s the synchronous produce  I  = n qv  s  u)  Q  n  = —  s  an e l e c t r o n  electric drift  i s the synchronous  field,  may  r e q u i r e d to  v e l o c i t y e q u a l to the v e l o c i t y o f sound.  current,  i s the d i f f u s i o n f r e q u e n c y .  Equation ( 1 . 1 8 )  i . e . the f i e l d  be r e a r r a n g e d t o y i e l d :  15  a-i)  cTTT^ where y = 1  = 1  =  T  •  (1  v  n  = electron d r i f t v e l o c i t y  -  20)  (1.21)  Substituting equation (1.20) into equation (1.19) y i e l d s a r e l a t i o n s h i p between E-^ and D-^:  ifiDi  , iQ y H  (1.22)  ewJ  where  <o [ 1 + (1 - |A| )^ ] 2  r  a  a) = — c e  (1.23)  i s the conductivity r e l a x a t i o n frequency.  To f i n d the expression f o r acoustic gain, we obtain a wave equation by s u b s t i t u t i n g equation (1.3) into (1.1) to y i e l d (using v l = — ) : P  3 u 9t2 2  _  2 V  as  s 9  X  Taking the d e r i v a t i v e with respect using the r e l a t i o n  S =  9u 3x  e_ 9E p 9x  to x on both sides of (1.24) and  (1.24)  16 9S 32 2  2 3 S l 9x2  e 9 E txT  2  v  t  2  ~  p  (1.25)  The plane wave forms of S and E [see equation (1.9)] may  now  be substituted into equation (1.25). In keeping with the slowly varying envelope approximation, only the dominant terms on both sides of the equation w i l l be retained. On the LHS'there are no terms i n S-^ and the f i r s t derivatives of S-^ are therefore retained. We obtain:  ffi.ie  i  3 s  e  -  at  -16  e  C  +  v  E  e  S  1  9 S  +  V  +  =  9x  e  3 S  -  9x  v  1 -16 e  c  9x  E  t  d- )  e  2 6  on both sides of (1.26)  1  s  1 19  i e , _* - i e . l >  Equating the c o e f f i c i e n t s of e  9t  3 S  9t  K_ e . 2iv~ p l  9 S  .  K e 2 i v " f l s P E  C - )  / l v  With the a i d of equation (1.22) and using equation (1.4) i n the form = -eS-L + eE-^ , E-^ may be related to S-^ and one f i n a l l y obtains:  1  27  17 3S  dS  ±  1  + 3t  v  (1.28)  c ^ V g S !  =  3x  ( y + —- ) w  h  e  r  „  e  -  " 27  c  '  D  <7~ .  s  ton  The r e a l and imaginary parts of a  , (1  -  29)  Si  are given by: YOJ  C  Re a.  (1.30)  2v, (Op  z Y  AK  X  Im a.  =  —^2  S2  •> Q + + ft '( to  D  (o  —  c  )  D  (1.3D  2v„  i s the p i e z o e l e c t r i c coupling c o e f f i c i e n t .  From the form of the general solution f o r equation r e a d i l y i n f e r r e d that a i n equation  (1.28), i t may be  (1.30) i s the nonlinear acoustic  gain c o e f f i c i e n t , while AK i n equation  (1.31) s p e c i f i e s the phase s h i f t  of the acoustic wave produced by p i e z o e l e c t r i c coupling.  18 Equations (1.30) and (1.31) only d i f f e r from the corresponding expressions i n l i n e a r theory by the s u b s t i t u t i o n of (1.23)]. In the l i m i t |A| -»• 0, to^. -> w  c  f o r o) [see equation c  and l i n e a r theory i s recovered.  The frequency of maximum gain i s given by (w^ton)  [White 1962]. From  equation (1.23) i t i s apparent that the frequency of maximum gain i s reduced by about 30% as |A| increases from 0 to 1. To use equation (1.30) i n the expression f o r t o t a l round t r i p gain i n the phonon maser, i t i s necessary to account f o r a c o u s t i c waves t r a v e l l i n g both p a r a l l e l and a n t i p a r a l l e l to the applied D.C. e l e c t r i c f i e l d E . This i s achieved by setting y i n equation (1.30) equal to: D  V  Y_  =  1  D  (1.32a)  v  for acoustic waves t r a v e l l i n g a n t i p a r a l l e l to E , and Q  D 1 + — s V  Y  =  +  (1.32b)  v  for acoustic waves t r a v e l l i n g p a r a l l e l to E . o  The t o t a l round t r i p gain G i n the phonon maser may be w r i t t e n (Gurevich 1969): G  where a  +  =  a, + a_ + 2T - Q™L d  (a_) i s given by s u b s t i t u t i n g y  +  (1.33)  (y_) into equation (1.30).  F i s the l a t t i c e loss c o e f f i c i e n t , r i s the acoustic r e f l e c t i v i t y of  19 the phonon maser cavity w a l l , and d i s the c a v i t y thickness. For  the  l a t t i c e l o s s T we use the empirical r e l a t i o n (Maines and Paige 1969):  r  where  6 = 1.11  x IO  =  6f2 (cm)"  - 1 4  In FIG 2(a) and  B  ;  (1.34)  1  6 =  1.51  (b) the value of G corresponding to the  frequency of maximum gain i s p l o t t e d as a function of |A|, under d i f f e r e n t conditions of conductivity and applied D.C.  electic  The parameters used i n the c a l c u l a t i o n were  x 10 Hz,  y = 280 cm /volt-sec., x = 0.0378 and v 2  value f o r u was  g  = 4.38  = 1.759  x 10  field.  9  5  cm/sec. (the  calculated from the threshold e l e c t r i c f i e l d f o r bj_  c r y s t a l 24.01.02.04). A value of 1.5 was  used for the end loss term  -21nr/d. As i s indicated i n FIG 2, conditions can be r e a l i z e d such that G increases with increasing |A|, and hence with increasing acoustic amplitude. It i s t h i s nonlinear property  that provides  the  counterpart  of the saturable absorber, and completes the analogy between the phonon maser and  the r e p e t i t i v e pulse generator. The phenomenon of  mode-locking can thus be predicted as a natural consequence of the physical properties and structure of the phonon maser. The t h e o r e t i c a l behavior of the phonon maser as a function of conductivity and  applied  e l e c t r i c f i e l d i s summarized i n FIG 3. The region of o s c i l l a t i o n i s indicated by cross-hatching. The regime of the "saturable absorber" corresponds to the shaded area near the top of the region of o s c i l l a t i o n .  Oc  21  FIG 3.  Behavior of the phonon maser as a function of conductivity o and applied e l e c t r i c f i e l d E . E 0  g  i s the synchronous e l e c t r i c  field.  The small area enclosed by a dotted l i n e near the bottom of the f i g u r e corresponds to the regime where mode-locked operation was  observed  for  was  at  b_[_ c r y s t a l 24.01.02.04. As i s indicated, mode-locking  achieved  c o n d u c t i v i t i e s approximately an order of magnitude lower than  predicted by theory.  It i s worthwhile  to discuss the l i m i t a t i o n s of the theory. F i r s t  of a l l , consider the maximum allowable value f o r the amplitude E-^ of the s e l f consistent A.C. |A| =  e l e c t r i c f i e l d . From equation  (1.19),with  1, we observe that  l ll ax E  M  "  E  s  IY + ^  I  '(1.35)  For t y p i c a l operating frequencies, i t i s reasonable to neglect isl/u)  D  i n comparison with y. Thus  E  Since E / E Q  S  E  C  Max  *  E  o  - s  M a x  <  (1-36)  E  ~ 1.2-2.0, f o r the experiments performed,  predicts that l ] J 2E  l  equation (1.36)  0.5E . In f a c t , values of E^ of the order of o  nave been i n f e r r e d from experimental measurements. As a further consideration, the.behavior of the t o t a l current  density as a function of |A| should be examined. We observe f i r s t of a l l from equation (1.14) that |A| = 1  corresponds to a s i t u a t i o n of  •complete electron depletion at the minima of the electron density wave.  23 This corresponds to D.C. current saturation as may be seen more concretely i n the following manner. I f equation (1.18) i s solved for I, the r e s u l t i n g expression may be written i n the form:  aE  o  (1 - F) + I„F  1 - (1 -  where  (1.37)  °  (1.38)  \k\ ) 2  h  From equation (1.37) we see that the t o t a l current density I behaves as i f a f r a c t i o n F of the electrons were constrained  to move a t the  v e l o c i t y of sound, while the r e s t e x h i b i t normal ohmic response to the l o c a l D . C . e l e c t r i c f i e l d . Thus F may be considered  to be a trapping  f r a c t i o n i n the sense that i t s p e c i f i e s the portion of a v a i l a b l e electrons that are trapped i n the p o t e n t i a l wells produced by the acoustic wave. Since F increases monotonically increases from 0 to 1, f o r |A| = 1 , f i e l d E . Thus the present Q  from 0 to 1 as |A|  I = I G independent of the applied  theory takes us up to the point of complete  current saturation. The region of mode-locked operation i s , i n f a c t , w e l l into the regime of current saturation. The above l i m i t a t i o n s may be traced to the f a c t that we have assumed a s i n u s o i d a l electron density d i s t r i b u t i o n [equation  (1.14)].  As has been indicated i n section 1.1, t h i s w i l l not be a good assumption i n the case of very large acoustic amplitudes. The theory may be generalized to incorporate a more physical charge density p r o f i l e by w r i t i n g a l l v a r i a b l e s i n the form of a general Fourier expansion, and  24 then proceeding  i n much the same manner as i n t h i s section (see Butcher  and Ogg 1969). For r e a l i s t i c p r o f i l e s , however, numerical methods must be applied (see Butcher and Ogg 1970 and Tien 1968). Our purpose has not been to describe mode-locked operation per se, but to present a p l a u s i b i l i t y argument f o r the existence of a "saturable  absorber".  25 1.3  Theory of the Optical Measurements The necessary background f o r i n t e r p r e t a t i o n of the o p t i c a l  experiments i s developed  i n this s e c t i o n . To demonstrate the usefulness  of l i g h t d i f f r a c t i o n i n the study of acoustic f i e l d s , we begin by considering the photoelastic e f f e c t (Nye 1964). 1.3.1  C r y s t a l Optics - The F h o t o e l a s t i c E f f e c t The o p t i c a l properties of an a n i s o t r o p i c c r y s t a l are conveniently  embodied i n a quadric surface known as the i n d i c a t r i x . It may  be defined  i n i t s general form by (Nye 1964):  B  ij i j X  X  =  1  ( 1  '  3 9 )  where B^_. are components of the r e l a t i v e d i e l e c t r i c impermeability tensor defined by 3E.  B.  -  e  0  J  — 3D.  •  3  E^ and and e  Q  (1.40) i-  are components of the e l e c t r i c f i e l d and e l e c t r i c  displacement,  i s the vacuum p e r m i t t i v i t y . I f the coordinate axes x^, X £ , x^  are chosen to l i e along the c r y s t a l l o g r a p h i c d i r e c t i o n s which correspond to the p r i n c i p a l axes of the d i e l e c t r i c tensor, equation (1.39) reduces to  2  2  —+  —+  n  l  n  2  2 — n  3  =  1  (1.41)  26 where  = n|,  = n| and  = n^ are the p r i n c i p a l d i e l e c t r i c constants,  and n^, n^, n^ are referred to as the p r i n c i p a l r e f r a c t i v e i n d i c e s . The quadric surface defined by equation (1.41) i s an e l l i p s o i d with semiaxes n^, n^,  n^.  This surface has a valuable property that can be i l l u s t r a t e d by the geometrical construction i n FIG 4(a). Consider that l i g h t propagating i n d i r e c t i o n CO (0 i s the e l l i p s o i d o r i g i n ) impinges  on  the c r y s t a l . A c e n t r a l section of the i n d i c a t r i x i s made perpendicular to CO. This section w i l l be an e l l i p s e and i s referred to as the "index e l l i p s e " . In FIG 4 the index e l l i p s e i s emphasized by shading and i n FIG 4(b) i s viewed along the d i r e c t i o n of incident l i g h t ( i . e . l i g h t i s propagating into the page). I f the incident l i g h t has i t s e l e c t r i c displacement vector D polarized i n the d i r e c t i o n indicated i n FIG 4 ( b ) , i t w i l l propagate  i n the medium as the two independent  components  and D^, which are polarized along the axes of the index e l l i p s e . D^, D£ w i l l have r e f r a c t i v e indices equal to OA and OB, For OA =f= OB  respectively.  (the index e l l i p s e i s not a c i r c l e ) the medium e x i b i t s  birefringence or double r e f r a c t i o n , and when the l i g h t emerges from the c r y s t a l the two components  and  w i l l recombine to form, i n  general, e l l i p t i c a l l y polarized l i g h t . For l i g h t having i t s wave normal p a r a l l e l to one of the i n d i c a t r i x axes, the two independent waves i n the material w i l l have r e f r a c t i v e indices corresponding to the p r i n c i p a l indices i n equation (1.41). For example, l i g h t propagating along the x^ axis w i l l have r e f r a c t i v e indices n., and n  0  [see FIG 4 ( a ) ] .  27  'FIG 4.  (a) I n d i c a t r i x construction f o r o p t i c a l wave normal CO. The index e l l i p s e i s shaded, (b) Index e l l i p s e viewed along d i r e c t i o n of l i g h t propagation. The p o l a r i z a t i o n of the incident l i g h t i s s p e c i f i e d by the e l e c t r i c displacement vector D.  28 The  shape of the i n d i c a t r i x depends upon the symmetry of the  medium i n question. For cubic or i s o t r o p i c materials, which possess one p r i n c i p a l r e f r a c t i v e index, i t i s a sphere. Hence a l l central  sections  are c i r c l e s and there i s no natural birefringence. For hexagonal, tetragonal and t r i g o n a l c r y s t a l s there are two p r i n c i p a l r e f r a c t i v e indices, and the i n d i c a t r i x i s an e l l i p s o i d of revolution  having the form  (1.42)  where n  Q  and n  £  are termed the ordinary and extraordinary r e f r a c t i v e  indices. In general, central sections of (1.42) w i l l be e l l i p s e s , implying the existence of natural birefringence. The central  section  perpendicular to the x^ axis i s a c i r c l e , however, and f o r t h i s unique d i r e c t i o n there i s no birefringence, x^ i s c a l l e d the p r i n c i p a l or optic axis and such c r y s t a l s are termed u n i a x i a l . CdS has hexagonal c r y s t a l structure and hence i s an example of a u n i a x i a l c r y s t a l . For CdS, x^,  X£> x^ i n equation (1.42) correspond to the a, b and c axes,  respectively.  For the remaining c r y s t a l classes there are 3 p r i n c i p a l  r e f r a c t i v e indices and the i n d i c a t r i x i s a t r i a x i a l e l l i p s o i d . The  p e r m i t t i v i t y and d i e l e c t r i c constant, and hence the index  of r e f r a c t i o n are,  i n general, modified by the introduction  of e l a s t i c  s t r a i n i n a c r y s t a l . Thus the presence of an acoustic wave w i l l be indicated  by a modulation of the l o c a l r e f r a c t i v e index which may,  29 i n turn, be expressed  i n terms of small changes i n the shape, s i z e and  o r i e n t a t i o n of the i n d i c a t r i x . I f we consider only e f f e c t s that are l i n e a r i n the applied s t r a i n , the change induced i n the i n d i c a t r i x may be written (Nye 1964):  A B  Pjyk^  a n  d  i3  =  Pi  l kl  ^  S  j k  are components of the o p t o e l a s t i c and s t r a i n tensors.  The s t r a i n tensor i s related to the mass displacements u^ by:  9u.  S  = 3  jj ( _ i  3x. 3  3u. +  —1  )  (1.44)  3x. i  If the shear s t r a i n s ( i / j ) are redefined i n equation (1.44) by omitting the f a c t o r of % , the indices i n equation (1.43) may be contracted unambiguously according to the following scheme (Nye 1964) :  11 -* 1  32, 23 -> 4  22 -> 2  31, 13 -> 5  33 -> 3  . 21, 12 -> 6  (1.45)  This allows equation (1.43)to be represented i n matrix form. Under the influence of s t r a i n the i n d i c a t r i x e l l i p s o i d represented by equation (1.41) w i l l be transformed, and the new may be described by [refer to equation (1.39)]:  ellipsoid  30  B  1 1 X  +  B  2 2 X  +  B  3 3 X  +  2 B  4 2 3 X  X  +  2 B  5 1 3 X  X  +  2 B  6 1 2 X  X  =  (1-46)  1  With reference to equations (1.41), (1.45) and (1.46), equation (1.43) may be written i n matrix form:  AB  B  AB  2  B  AB  3  B  AB. 4 AB  5  D  i 2  - l/nj  p  - l/n2  3 -1/n* B  4  B  5  B  6  ll  ••  P  16  p  21 ••  p  26  P  31  P  36  P  41  P  51  P  61  •• 4 6 P  P  56  •• 6 6 P  s„  1  s„  2  So  X  3  (1.47)  s.  4  Sr-  5  S  6  When the form of the s t r a i n i s known, the new i n d i c a t r i x may be found from equations (1.46) and (1,47). With this information one i s able to determine the new p o l a r i z a t i o n s and indices of r e f r a c t i o n f o r a given o p t i c a l wave normal, using the i n d i c a t r i x property i l l u s t r a t e d by FIG 4• The general problem provides a tedious exercise i n algebra and a n a l y t i c geometry. General solutions have been obtained f o r c r y s t a l s with hexagonal 6mm symmetry by Vrba and Haering (1973), and provide a useful model f o r solving other c r y s t a l structures. A much simpler case, that of a shear wave propagating i n an o p t i c a l l y and a c o u s t i c a l l y i s o t r o p i c medium, w i l l be treated to i l l u s t r a t e the use of the i n d i c a t r i x . This s i t u a t i o n i s of p a r t i c u l a r i n t e r e s t and ..the r e s u l t s w i l l be used i n l a t e r sections. In the absence of s t r a i n the  31 i n d i c a t r i x i s a sphere and equation (1.4.1) becomes:  ~  2  ( x  + x, + x  2  2  2  )  =  1  (1.48)  The coordinate system i s chosen so that the acoustic wave normal l i e s along the x^ axis, with mass displacement along the Y.^ a x i s . Thus the only non-zero  s t r a i n component i s Sg. For an i s o t r o p i c material the  o p t p e l a s t i c matrix becomes (Nye 1964):  where p., = ^(p,, - P  1 9  p  ll  p  12  P  12  p  12  p  ll  p  12  p  12  P  12  P  ll  0  0  0  0  0  0  0  0  0  0  0  0  0  0  p  0  0  0  0  p  0  0  0  0  0  44  44  (1.49)  0 P  44  ) . Equation (1.47) then y i e l d s :  '  *  AB  1  0  AB  2  0  AB  3  0  AB. 4 AB A B  0  5  6  (1.50)  0  =  P  44 6 S  32 From equations (1.48) and (1.50), the c o e f f i c i e n t s i n equation (1.46) may be r e a d i l y i n f e r r e d :  B  B  = B  4  6  =  p  =  5  0  (1.51)  44 6 S  The new i n d i c a t r i x thus becomes:  i  ( x  2  2  + x . + x, ) + 2 p S 2  2  4 4  6 X ; L  x  2  =  1  (1.52)  I f X g i s chosen as the o p t i c a l wave normal, the appropriate index e l l i p s e i s given by s e t t i n g x^ = 0 i n equation (1.52) to obtain:  n§  (  X  l  +  X  2  }  +  2  P44 6 1 2 S  X  X  =  1  ( 1  '  5 3 )  The form of the index e l l i p s e , both i n the presence and absence of the perturbing s t r a i n , i s shown i n FIG 5. As i s indicated, the axes of the index e l l i p s e are rotated by TT/4 with respect to the coordinate axes x^ and x - We therefore change to a rotated frame using the t r a n s f o r 2  mation: x  ±  =  l/v/2  ( J + x£ ) X  (1.54) x  2  =  1//2  ( x^ -  x£ )  FIG 5.  Form of the index e l l i p s e in-an i s o t r o p i c material f o r o p t i c a l wave normal X g , both i n the presence and absence of shear s r a i n Sg.  34 so that the new coordinate axes x|, x^ are colinear with the axes of the e l l i p s e . Equation (1.53) then reduces to the more standard form:  »2  v  — n  1  t2  +  2  I  — n  2  = 1  2  n  where  (1.55)  II  n <  X  o  - P44 6 o^ S  n  (1.56) n <  x +  Q  P44 6 o S  n  )  -5  9  For t y p i c a l experimental parameters P^S^n^ s 10  , so to a very good  approximation we may expand the square roots i n (1.56) and r e t a i n only l i n e a r terms: n  I  %  =  ( 1  P44 6 o  +  S  n  )  (1.57) n  II =  %  (1 - Js P S n 2 ) 4 4  6  I t i s r e a d i l y apparent from equation (1.57) that the changes i n r e f r a c t i v e index produced by the s t r a i n (see FIG 5) are:  A n  I  =  -  A n  II  =  h  P44 6 o S  n  ( 1  *  The time dependence of the acoustic wave has thus f a r been  5 8 )  35 neglected. FIG 5 Is e s s e n t i a l l y a picture of the index e l l i p s e at one instant of time. For s i n u s o i d a l time dependence, we may imagine that at time h T l a t e r , where T i s the period of the acoustic wave, the index e l l i p s e w i l l be a c i r c l e ; at time % T i t w i l l again be an e l l i p s e , but with the major axis now l y i n g along x^ ; and so on. In t h i s manner, the l o c a l index e l l i p s e may be v i s u a l i z e d as pulsating with the frequency of the acoustic wave. I t i s worth mentioning  that, f o r a given s t r a i n , the wave  normal of the incident l i g h t must be j u d i c i o u s l y chosen. I f i n the present case, f o r example, the o p t i c a l wave normal i s taken as x^ or x , i t may be seen from equation (1.52) that the appropriate c e n t r a l 2  sections ( x^ = 0 and x  2  = 0 ) are c i r c u l a r and completely unaltered  by the s t r a i n . Thus a knowledge of the predeeding theory i s e s s e n t i a l i n determining the c o n f i g u r a t i o n f o r o p t i c a l  experiments.  1.3.2 Interpretation of the Optical Signal - Preliminary Considerations The intimate r e l a t i o n s h i p between the r e f r a c t i v e index and ;  the l o c a l s t r a i n , outlined i n the previous section, makes l i g h t d i f f r a c t i o n a powerful tool i n the study of the phonon maser. In order to proceed to an o p t i c a l analysis of the s t r a i n p r o f i l e s i n a modelocked phonon maser, i t i s necessary to f i r s t e s t a b l i s h the r e l a t i o n s h i p between the o p t i c a l signals derived by l i g h t d i f f r a c t i o n , and the acoustic f i e l d s that provide the d i f f r a c t i n g mechanism. This subject has an extensive l i t e r a t u r e and w i l l not be f u l l y developed  here  [the general theory of u l t r a s o n i c l i g h t d i f f r a c t i o n has been discussed  36 by Bhatia and Noble (1953) and by K l e i n , Cook and Mayer (1965)]. A categorized bibliography The An acoustic  i s provided at the end of t h i s t h e s i s .  configuration  of physical i n t e r e s t i s depicted i n FIG  6.  cavity of width L i s illuminated by a unit amplitude plane  l i g h t wave, propagating i n the xz plane (plane o p t i c a l wavefronts  may  be reasonably approximated i n p r a c t i s e , since laser beams with wavefront d i s t o r t i o n s less than 0.2 c o l l i m a t i o n ) . The of the acoustic  o p t i c a l wavelengths may  be prepared by  careful  o p t i c a l waveform emerging from the f a r side (z = L)  cavity i s transformed due  to a c o u s t i c a l l y induced  modulations i n the l o c a l r e f r a c t i v e index. The near f i e l d o p t i c a l d i s t r i b u t i o n ( i . e . the o p t i c a l signal at z = L ) , denoted A ,  i s imaged  Q  by a converging lens. The i n FIG 6, and  d i r e c t i o n of acoustic propagation i s taken as the x-axis the acoustic waveform i s assumed to have s p a t i a l v a r i a t i o n  only i n the x - d i r e c t i o n . Hence the near f i e l d d i s t r i b u t i o n A function only of x and  t. This assumption requires  q  is a  some comment. In  p r a c t i c e , the f i e l d of view i s determined by the cross section of  the  incident l a s e r beam. Since the beam s i z e used i n experimental s i t u a t i o n s was  smaller than the dimensions of the acoustic  cavity, d i s t o r t i o n s i n  acoustic wavefronts introduced by edge e f f e c t s were avoided and, be v e r i f i e d by the experimental r e s u l t s , the assumption of a  as w i l l  one  dimensional system i s v a l i d . In the f o c a l plane of a converging lens the c r i t e r i o n for Fraunhofer d i f f r a c t i o n i s s a t i s f i e d (Goodman 1968). This i s i n t u i t i v e l y .clear since i n front of the f o c a l plane l i g h t waves are converging and  38 behind i t they are diverging. Hence at z =  i n FIG 6 we have plane  o p t i c a l wavefronts. In the l i m i t of small d i f f r a c t i o n angles  18°),  the o p t i c a l d i s t r i b u t i o n i n the f o c a l plane, A^(x.',t),is related to the near f i e l d d i s t r i b u t i o n by a one dimensional form of the f a m i l i a r Fraunhofer d i f f r a c t i o n i n t e g r a l (Goodman 1968): W A (x\t) f  =  £ ' A (x,t)e0  i k x S i n 9  dx  (1.59)  W 2  where £ i s a phase factor determined by the physical configuration and W i s the:effective width of the o p t i c a l beam. I f we imagine two p a r a l l e l rays, separated by distance x i n the near f i e l d , and making an angle 6 with the z axis, the term exp(-ikx sin0) equals the difference i n o p t i c a l phase suffered by the two rays i n propagating to point x' i n the f o c a l plane (see FIG 6). The beam width W i s assumed to be large compared with the acoustic wavelengths so that a l l d i f f r a c t i o n orders i n the f o c a l plane are w e l l resolved and completely separated. For t h i s s i t u a t i o n , we impose no r e s t r i c t i o n s on the s p a t i a l f i l t e r i n g  experiments to be carried  out, by extending the l i m i t s of integration i n (1.59) to ± «>. In addition, i t may be shown that the phase factor E, reduces to unity when the separation between the near f i e l d region (object plane) and the lens i s equal to the f o c a l length of the lens (Goodman 1968). With these considerations equation (1.59) becomes:  39  A (x',t) f  =  A (x,t)e ' dx i k  (1.60)  X  0  4 00  where k' x  1  =  k s i n 0. There i s a  one to one r e l a t i o n s h i p between  and k' i n equation (1.60) since specifying x' uniquely determines  9 and vice-versa. Hence equation (1.60) indicates that the o p t i c a l d i s t r i b u t i o n i n the f o c a l plane i s the Fourier transform of the near f i e l d s i g n a l . This Fourier transforming c a p a b i l i t y i s a general property of converging lenses (Goodman 1968). In propogating from the f o c a l plane to the image plane i n FIG. 6, the o p t i c a l s i g n a l s u f f e r s an inverse Fourier transform. For a d i s t o r t i o n f r e e o p t i c a l system, there i s then a one to one r e l a t i o n s h i p (aside from a s p a t i a l magnification factor) between the o p t i c a l d i s t r i b u t i o n s i n the image plane and i n the near f i e l d or object plane. With the preceeding considerations, i t remains to determine  the  form of A ( x , t ) . We begin with the simple example of a s i n u s o i d a l , Q  progressive acoustic wave. Assume that the cavity i s o p t i c a l l y i s o t r o p i c i n the absence of acoustic waves, and, to s i m p l i f y notation, that the incident l i g h t i s polarized along one of the axes of the index e l l i p s e (See FIG. 5). Since the change i n r e f r a c t i v e index i s , to a good approximation, l i n e a r i n the l o c a l s t r a i n [see equation (1.58)], the r e f r a c t i v e index w i l l have the form:  n(x,t) = n  D  + An sin(Kx -  fit)  (o<z<L)  (1.61)  40  where n  0  i s the unperturbed r e f r a c t i v e index, K and fi are the wavenumber  and frequency of the acoustic wave. Since the r e f r a c t i v e index has slow v a r i a t i o n compared with o p t i c a l frequencies, the wave equation governing propagation of the o p t i c a l s i g n a l A i n the acoustic cavity may be w r i t t e n (Raman and Nath 1936b) :  V  2  A  n(x,t)  =  c  9A  2  2  8t  2  (1.62)  2  where c i s the vacuum v e l o c i t y of l i g h t . In the absence of acoustic f i e l d s , A w i l l propogate i n the region o<z<L i n the form  a (x,z,t) = e  i [ n  0  °  k  ( Z  C  O  S  *  +  x  S  i  n  •>  (1.63)  k,u are the o p t i c a l wavenumber and frequency and (J> i s the angle made by incident l i g h t with the z axis (see FIG. 6). The modulation of the o p t i c a l wave produced by the presence of acoustic s t r a i n s w i l l be slowly varying compared with the s p a t i a l and temporal dependence of (1.63). Thus i n the presence of the acoustic wave the o p t i c a l waveform may be described by:  A(x,z,t)  =  a ( x , z , t ) Tjj(x,z,t) Q  (1.64)  41 ip(x,z,t)  The envelope function  w i l l be periodic i n space and time with  the acoustic wave, and hence may  be written as a Fourier s e r i e s :  CO  *(x,z,t)  =  I * (z) e n ~ -oo  -  i n ( K x  n  >  (1.65)  z=0  (1.66)  Bt  subject to the boundary conditions  vi|»  n  1  at =  0  n ^ 0  The near f i e l d amplitude A ( x , t ) i s given by s e t t i n g z=L i n equations Q  (1.63) to (1.65). Substituting the r e s u l t i n g expressions into (1.60) we obtain for the f o c a l plane o p t i c a l s i g n a l :  A (x',t) f  =  e  i ( n k L cos<j> - cot) r , n  °  L  n  =  ' M  -oo  . -inftt L  '  e  e  - i (k' -ksintj) -hK) x (1.67)  The i n t e g r a l i n equation (1.67) i s non-zero only i f  k' - ksin<J> - nK  =  0  (1.68)  Light w i l l appear i n the f o c a l plane only at those points which s a t i s f y equation (1.68). Hence the index n labels the d i f f r a c t i o n orders i n the  42 f o c a l plane. The o p t i c a l i n t e n s i t y i n the n*"* d i f f r a c t i o n order i s 1  given by equation (1.67) as  i  <j> a)|  =  n  d.69)  2  n  Also, the angular frequency i n the n  til  order may be inferred  from  (1.67) : w  =  n  to + nfi  (1.70)  Equations (1.63) to (1.65) may be substituted into  (1.62).  If terms i n (An)^ and ij> (z) are neglected, c o e f f i c i e n t s of exp i[n(Kx-fit)J may be equated.to  d»n(z) ^ —  obtain:  . v + —  [T|, (Z)-I|) (Z)] n-1 n+1  =  i n Q ( n - 2 H (z) 2L" n  (1.71)  Y  where: v  =  AnkL  (1.72)  Q  =  K L n k o  (1.73)  =  -n k s i n * o K  2  Y  as indicated  (1.74)  T  by equation (1.69), the o p t i c a l amplitude i n the n  th  43  d i f f r a c t i o n order i n the f o c a l plane i s proportional to il/ (L) . The n difference  - d i f f e r e n t i a l equation (1.71) was derived i n the special  case y = 0 (normal l i g h t incidence) by Raman and Nath (1936b), and i n the more general case by K l e i n and Cook (1967) . In equation (1.71) adjacent o p t i c a l modes are coupled by the parameter v . Interaction  and transfer of energy between these modes  w i l l only take place i f they maintain a constant phase r e l a t i o n s h i p , i . e . maintain s p a t i a l synchronization. The degree of synchronization i s determined  by the c o e f f i c i e n t , to be designated 8 , on the R.H.S. n  of equation (1.71) :  a  _  inQ (n-2 )  (1.75)  Y  This may be considered as a r e l a t i v e phase factor, and only those o p t i c a l modes having the same, or nearly the same, value of t h i s c o e f f i c i e n t may be considered as being s p a t i a l l y coherent or synchronized, and hence able to exchange o p t i c a l energy. Since 6  q  = 0, an appreciable amount of o p t i c a l energy can  be coupled from the zeroth into the f i r s t order only i f one or both of the c o e f f i c i e n t s  are very small. This condition can be s a t i s f i e d i n  two regimes : ( i ) Q << 1, y - 0 ; ( i i ) Q » the two l i m i t i n g s i t u a t i o n s  1, y = ±%. ( i ) and ( i i ) are  f o r which a n a l y t i c solutions  p r a c t i c a l , and each case w i l l be discussed separately.  to (1.71) are  44  Raman - Nath L i m i t :  Q<<1  For Q = 0 and with boundary conditions (1.66) the s o l u t i o n to  (1.71) may be shown to be (Raman and Nath 1936 b) :  il) (z) =  (1.76)  J (Ankz)  where J i s a Bessel function of order n. For s u f f i c i e n t l y small nonzero n values of Q, Bessel functions form good approximate solutions of (1.71). The regime of v a l i d i t y of this approximation  i s named f o r the two  workers who f i r s t ' provided a t h e o r e t i c a l basis f o r t h i s type of d i f f r a c t i o n (see Raman and Nath 1935 a,b and 1936 a,b). For this s i t u a t i o n energy i s symmetrically coupled from the zero order into both f i r s t order modes (n=±l). In a d d i t i o n , f o r s u f f i c i e n t l y large values of the coupling parameter v , energy i s coupled from the f i r s t order to the second order, the second order to the t h i r d , e t c . From a comparison of equations for  Q small  (1.73) and (1.74) i t i s apparent that  y can be very large, even f o r small values of the incident  angle <J>. Thus we optimize the condition 8 N ~ 0 (n^O) cj>  = 0  => y  hy choosing  = 0 (normal l i g h t incidence) . The o p t i c a l i n t e n s i t y  th i n the n  d i f f r a c t i o n order i s then given from equations  (1.69),  (1.72) and (1.76) I n  =  J (v) n 2  (1.77)  and, s e t t i n g < J > = 0 i n equation (1.68), we obtain f o r the d i f f r a c t i o n  45  angle G : sin 0  =  y-  (1.78)  where A i s the o p t i c a l wavelength i n vacuo and A i s the acoustic wavelength. In the Raman - Nath regime the acoustic wave only modulates the phase of the o p t i c a l s i g n a l . The parameter v, referred to as the Raman - Nath parameter i n t h i s s i t u a t i o n , i s equal to the maximum acoustically  induced phase s h i f t suffered by the l i g h t i n traversing  the acoustic cavity (see eqn. 1.72). A p r a c t i c a l upper l i m i t of the Raman - Nath regime i s given by (Klein and Cook 1967) : Q ~ 0.5  (1.79)  For large values of v ( i e . large values of acoustic s t r a i n ) condition (1.79) may not be s u f f i c i e n t to ensure true R-N d i f f r a c t i o n . For t h i s reason the supplementary condition (Extermann and Wannier 1936) : Qv < 2 i s sometimes quoted. For our experimental s i t u a t i o n , however, V<<1 and the nature of the d i f f r a c t i o n i s adequately described by the magnitude of the parameter Q.  (1.80)  46  Bragg Limit : Q » l For this s i t u a t i o n , the condition 6^ = 0 (n^O) can be s a t i s f i e d only f o r n = 1 or n = -1. Thus f o r y = %(-%) l i g h t i s d i f f r a c t e d only into order +1(-1). Other combinations  of n and y are  not allowed f o r a simple sinusoidal acoustic wave. For n = ±2, y = ±1, for  example, no l i g h t may be transferred from the zero order since only  adjacent orders are d i r e c t l y coupled by equation (1.71). For y = ±Jg equation (1.74) may be x^ritten i n the more transparent form:  —  o  =  ±2A s i n < j >  (1.81)  Thus y = ±% implies that cf> s a t i s f i e s the Bragg condition f o r specular o p t i c a l r e f l e c t i o n from the acoustic wave f r o n t s . A p r a c t i c a l lower l i m i t f o r the Bragg regime i s given by (Willard, 1949) :  Q  I  4?r  (1.82)  For values of Q intermediate to the Raman - Nath and Bragg regimes, a n a l y t i c solutions to (1.71) are not p r a c t i c a l . The gap between the two l i m i t i n g cases has been spanned by K l e i n and Cook (1967), using numerical techniques. Some of t h e i r r e s u l t s are reproduced  i n FIG. 7.  The curves i l l u s t r a t e d give the percentage of o p t i c a l i n t e n s i t y i n the zero and two f i r s t order d i f f r a c t i o n spots at normal incidence (y=0) , for  d i f f e r e n t values of the parameter Q. By summing the i n t e n s i t i e s i n  the zeroth and f i r s t order i n FIG. 7, i t may be seen that second and higher orders are depleted as Q increases, u n t i l at Q = 4 almost no  47  FIG 7.  Percentage of o p t i c a l i n t e n s i t y i n the zero and two f i r s t order d i f f r a c t i o n spots at normal incidence (y = 0), f o r various values of the parameter Q. (after K l e i n and Cook 1967). v i s defined i n equation  (1.72).  48  l i g h t appears beyond the f i r s t order. In the range Q = 4 to Q = 7 the amount of l i g h t i n the two  f i r s t orders decreases considerably, and when  Q = 10 there i s almost no d i f f r a c t i o n at normal incidence for small values of v (small acoustic s t r a i n s ) . The values of Q encountered for mode-locked phonon maser operation can span the e n t i r e gamut from Raman - Nath to Bragg d i f f r a c t i o n . However, the experimental r e s u l t s to be described i n section 1.4 the range V<<1,  l i e in  for which second order d i f f r a c t i o n i s completely  n e g l i g i b l e , and Q < 5. For these conditions, i t i s apparent from FIG. 7 (a) - (d) that f i r s t order d i f f r a c t i o n does not change s i g n i f i c a n t l y from the Raman - Nath case. For t h i s reason i t w i l l be assumed i n a l l subsequent discussion that the e f f e c t of acoustic f i e l d s on a l i g h t wave may  be described by an o p t i c a l phase transformation,  as i n the Raman -  Nath regime. 1.3.3  Interpretation of the Optical Signal - The Mode-Locked Phonon Maser In applying the r e s u l t s of the l a s t section to a study of  the s t r a i n p r o f i l e i n a mode-locked phonon maser, the basic configuration i l l u s t r a t e d i n FIG. 6 w i l l be retained. The l i g h t i s noxtf assumed to propogate i n the z d i r e c t i o n so as to be normally incident on the acoustic waveform. As discussed for  i n section 1.3.2, this i s the optimum configuration  Raman - Nath d i f f r a c t i o n . Following  the discussion at the end of section 1.3.2, we  assume that the only e f f e c t of the acoustic s t r a i n p r o f i l e i s to produce v a r i a t i o n s i n the o p t i c a l phase, v i a a c o u s t i c a l l y induced modulations of  49  the r e f r a c t i v e index. Since the r e l a t i o n s h i p between the l o c a l s t r a i n and r e f r a c t i v e index v a r i a t i o n i s l i n e a r [eg. see equation (1.58)], the phase v a r i a t i o n p r o f i l e d i r e c t l y y i e l d s the acoustic s t r a i n p r o f i l e . I f the phase v a r i a t i o n of the acoustic cavity i s described by $ ( x , t ) , the near f i e l d l i g h t amplitude i s given by [ c f . equations (1.63) and (1.64)]  A (x,t) o  =  e  i(noWL " »t) l*(x,t) e  (  1  8  3  )  For mode-locked operation, the acoustic waveform i s assumed !  to be a standing wave pattern c o n s i s t i n g of a phase-locked harmonic series of s i n u s o i d a l waves. Using the r e s u l t s of Appendix A, the phase function i n equation (1.83) may be written i n the form:  $(x,t)  '=  N I v n=l  n  sin[n(Kx-ftt) + 6 ] R  + sin[nK(x-2d) + nftt - « ] n  (1.84)  where d i s the thickness of the acoustic c a v i t y , N i s the number of a c t i v e or p a r t i c i p a t i n g acoustic modes and ft, K are the angular frequency and wave number of the fundamental component. The phase amplitudes v  n  are the Raman - Nath parameters discussed i n section 1.3.2, and are  proportional to the corresponding s t r a i n amplitudes. The set of phases 6  n  w i l l determine the exact shape of the phase v a r i a t i o n , and hence the  s t r a i n , p r o f i l e . Equation (1.84) may also be written i n the form:  50 N >(x,t)  2 I v sin[nK(x-d)]cos(nKd n=l  =  n  - nOt + 6 ) n  (1.85)  Since the acoustic cavity a l t e r s only the phase of the incident l i g h t , the s t r a i n f i e l d i s not d i r e c t l y v i s i b l e i n the image plane of the focussing lens (refer to FIG 6). This problem may  be overcome by  means of f o c a l plane s p a t i a l f i l t e r i n g techniques. I f c e r t a i n d i f f r a c t i o n orders are p h y s i c a l l y removed ( i . e . s p a t i a l l y f i l t e r e d ) i n the f o c a l plane, the e f f e c t i s to present a modified Fourier transform of the near f i e l d signal to the image plane [refer to discussion following equation (1.60)]. Thus the image viewed w i l l correspond to a pseudo-object whose Fourier transform i s given by the modified f o c a l plane d i s t r i b u t i o n passed by the s p a t i a l  filter.  By removing the zero order or undiffracted beam i n the f o c a l plane, f o r example, the dark f i e l d image i s obtained (Born and Wolf 1959). In t h i s s i t u a t i o n , phase v a r i a t i o n s i n the object plane produce i n t e n s i t y v a r i a t i o n s i n the image plane. The dark f i e l d d i s t r i b u t i o n w i l l be derived i n two ways. F i r s t , we begin by s u b s t i t u t i n g the phase function for a mode-locked waveform, equation (1.85), into equation  f  A (x,t) Q  =  e^exp.i  (1.83):  N I 2v sin[nK(x-d)]cos(nKd-nftt+6 ) n=l n  n  (1.86)  where t, = (n kL - wt) . Making use of the Bessel function r e l a t i o n s h i p Q  (Abramowitz and Segun 1968):  v T r \ imO  i z sinO m  (1.87)  =_oo  equation (1.86) may be w r i t t e n : N  «  I J [2v cos(nKd-nfit+6 )]e n=l p = -  LCn  A (x,t) o  pn  n  i P n n K ( xd  n  )  (1.88)  n  =  J  e  where V  Pl=  n  o  p  (v ) x  X  • P J  N  (  V  e  x  p  1  • N I np K(x-d) n=l  (1.89)  n  (1.90)  = 2v cos(nKd-nfit+6 ). n  n  A useful analogy may now be made with the case of a simple sinusoidal acoustic wave discussed i n section 1.3.2. When equation (1.65) was substituted into equation (1.60) to obtain the f o c a l plane o p t i c a l d i s t r i b u t i o n , i t was found that the c o e f f i c i e n t of the s p a t i a l factor Kx i n the exponential term of (1.65), n i n t h i s case, l a b e l l e d the d i f f r a c t i o n orders i n the f o c a l plane. By analogy, the d i f f r a c t i o n order for equation (1.89) may be inferred : d i f f r a c t i o n order  n = £ np n n=l  (1.91)  This greatly s i m p l i f i e s the mathematics of s p a t i a l f i l t e r i n g . It i s not necessary to Fourier transform the near f i e l d to f i n d the f o c a l plane d i s t r i b u t i o n , perform s p a t i a l f i l t e r i n g , and then inverse Fourier  52 transform to obtain the image plane s i g n a l . Instead, we may  use  equation  (1.90) to i d e n t i f y those components of A ( x , t ) that correspond to the D  f o c a l plane orders to be removed (passed) by the s p a t i a l f i l t e r ,  and  hence obtain the modified image d i r e c t l y . It i s i n t u i t i v e l y clear, from the cumbersome form of equation (1.89), that an analytic expression for the dark f i e l d d i s t r i b u t i o n i s not p r a c t i c a l i n the most general case. However, we are concerned with small values of the Raman - Nath parameters v . -2 For the experiments to be discussed, v  n  <, 10  radians. I t i s therefore  necessary to consider only effects that are f i r s t order i n v addition, we may  . In  use the asymptotic form for Bessel functions of small  argument (Abramowitz and Segun 1968): J (z) m  Equation (1.92) may relationship  i  —  z small  (  m > 0 )  m!  (1.92)  —  be used f o r negative values of m by f i r s t using the  (Abramowitz and Segun 1968):  J  -m  (z)  =  < - l ) j (z) m  In f i r s t order, the parameters p  (1.93)  m  n  i n equation (1.89) may  only take the  values 0, + 1. Furthermore, i f one of these parameters takes the value +1 or -1, a l l others must be zero. Let p n r  =  +1  ->  p. * i  =  0  f o r i h  (1.94)  53 The corresponding contribution from equation (1.89) i s [using equation (1.92)]: A  =  n  e  |A 2  U  e  inK(x-d)  From equations (1.91) and (1.94) i t may be seen that (1.95) constitutes the f i r s t order contribution to d i f f r a c t i o n order +n. S i m i l a r l y , the f i r s t order contribution to d i f f r a c t i o n order -n i s given by s e t t i n g P =-1 n  i n equation (1.94) and using (1.92) and (1.93) to obtain:  A  =  -n  -e  i ?  |* -inK(x-d) z e  9  The zero order amplitude must be treated separately and i s given by s e t t i n g p^ = 0 f o r a l l i . Using equation (1.92) we obtain:  A  =  o  e  (1.97)  i C  Hence the i n t e n s i t y i n the zero order, | A | , i s equal to one. This 2  q  indicates that, to f i r s t order i n the Raman - Nath parameters, the zero order beam i s unaffected by the d i f f r a c t i o n . Adding equations (1.95) and (1.96) we obtain:  ie V sin[nK(x-d)] lC  n  (1.98)  >It i s apparent from equations (1.95) and (1.96) that the index n, as i t i s used i n (1.98), l a b e l s symmetric d i f f r a c t i o n orders i n the f o c a l  54 plane. Hence the dark f i e l d amplitude i s found by summing (1.98) over a l l values of n except n = 0 :  .  A^  or,  =  N  i e ^ I V sin[nK(x-d)] n=l  (1.99)  1  n  using equations (1.85) and (1.90):  ^  =  ie  *(x,t)  l C  (1.100)  Thus the f i r s t order dark f i e l d amplitude i s proportional to the phase v a r i a t i o n produced by the acoustic c a v i t y . The corresponding o p t i c a l i n t e n s i t y i s given by:  I  D F  =  * (x,t)  (1.101)  2  It i s , i n f a c t , the i n t e n s i t y d i s t r i b u t i o n , equation (1.101), that may be detected experimentally. The second method of deriving the dark f i e l d d i s t r i b u t i o n i s much simpler, but gives l i t t l e insight into the mechanism of s p a t i a l f i l t e r i n g . Since we have assumed small phase v a r i a t i o n s , the exponential i n equation (1.83) may be expanded to y i e l d :  A (x,t) Q  =  e  1 ?  [ 1 + i$(x,t) ]  The dark f i e l d amplitude corresponds to removing the f a c t o r of 1 i n  (1.102)  55 equation (1.102), to immediately y i e l d equation (1.100). While t h i s method of  derivation i s much simpler, i t should be emphasized that i t i s useful  only for e f f e c t s that are f i r s t order i n v . For higher order e f f e c t s the factor of 1 i n equation (1.102) can no longer be c o r r e c t l y i d e n t i f i e d as the amplitude of the zero order d i f f r a c t i o n spot, and the image plane d i s t r i b u t i o n must be derived using the f i r s t method. The dark f i e l d  i n t e n s i t y d i s t r i b u t i o n i s obtained by s u b s t i -  tuting equation (1.84) or equation (1.85), into equation (1.101).  The  r e s u l t i n g expression i s very messy, having terms with a l l the possible combinations  of s p a t i a l and frequency dependence produced by the  nonlinear operation of squaring. I f , however, we extract only the term which has no time dependence, we obtain the r e l a t i v e l y simple expression (Smeaton, Hughes, Vrba and Haering 1976): N I  This D.C. of  o  =  I n n=l v  [ 1  " cos2nK(x-d)]  (1.103)  term i s straightforward i n the sense that i t contains none  the phases <5 [see equation (1.84) or (1.85)]. Thus i t s form depends n  only on the constituent amplitudes and not on the exact shape of the acoustic waveform. Equation (1.103) i s plotted i n FIG 8. The r e l a t i v e values of the amplitudes v  n  were i n f e r r e d from experimental d i f f r a c t i o n i n t e n s i t i e s ,  obtained f o r the f i r s t 13 a c t i v e modes of a mode-locked phonon maser. Thus N = 13 i n FIG 8. The width PW of the dark fringes i s i n v e r s e l y proportional to the number of p a r t i c i p a t i n g modes N, and provides an  FIG 8.  P r o f i l e of the dark f i e l d term, equation (1.103), plotted using the amplitudes of the f i r s t 13 active acoustic modes for mode-locked operation of DCl (refer to section 1.4.2).  57 estimate of the mode-locked pulse  Xsridth  (refer to section 1.2). As i s  indicated, the basic p e r i o d i c i t y i s A/2, where A i s the wavelength of the acoustic fundamental. To obtain I  Q  i n equation (1.103), a combination of s p a t i a l  and temporal f i l t e r i n g was  employed. That i s , we f i r s t modified the  o p t i c a l d i s t i b u t i o n i n t h e f o c a l plane, a n d then selected only one frequency component, the D.C.  component i n t h i s case, to be examined  i n the image plane. By extending t h i s general technique, information about the phases 6 , n  absent i n equation (1.103), may  be recovered. This  i s of considerable i n t e r e s t since these phases determine  the exact  shape and amplitude of the acoustic s t r a i n p r o f i l e . Consider, f o r example, that s p a t i a l orders n, n+1,  and  n+2  til  ( i . e . the n  pair of d i f f r a c t i o n spots symmetric to the zero order,  til  the (n+1)  pair and so on) are combined, and the i n t e n s i t y  variation  of the temporal component corresponding to the acoustic fundamental frequency Si i s scanned i n the image plane. The t o t a l image plane  intensity  d i s t r i b u t i o n i s given by [refer to equation (1.99)]:  •""n, n+1, n+2  ( < '  n+2  r  I v l sin[m(Kx - Sit) + m=n +  & ) m  sin[mK(x-2d) + mSlt - 6 ]  (1.104)  Extracting only those terms from (1.104) which have frequency dependence Si, we obtain:  58 T n, n+1,n+2  + 2  = 2  a cos(Kd+6 n  n+1  - 6 ) + b cos ( K d + 6 ^ - 6 ^ ) n  n  a sin(Kd+6 -6 ) + bsin(Kd+S , _-6 n n+1 n n +2 + l n  where  sinftt  n  a  =  n  2v v n  n+1  cosftt  (1.105)  n  sin[nK(x-d)]sin[(n+l)K(x-d)] (1.106)  b  n  =  2v  n+1  v  n+2  sln[(n+l)K(x-d)]sin[(n+2)K(x-d)]  We write equation (1.105) i n the form  n,n+1,n+2  =  acos^t + bsinftt  (1.107)  where a,b are defined i m p l i c i t l y by comparison of equations  (1.105) and  (1.107). Introducing notation  a  =  Csinn  C  =  (a + b ) ^ 2  2  (1.108) b  =  Ccosn  tann  = a/b  equation (1.107) may be w r i t t e n :  n,n+1,n+2  =  Csin(Qt + n)  (1.109)  The Fourier transform of (1.109) has the following simple form i n the  59 frequency (ft') domain:  ft n,n+1,n+2  In p r a c t i c e ,  -iC 2  e 6(ft'-ft) - e Vft'+ft) in  1T  (1.110)  the magnitude of the component of (1.110) at ft'=ft i s monitored.  Thus the detected signal becomes:  c  ft n,n+1,n+2  (a +b )^  C 2  2  2  (1.111)  Using the d e f i n i t i o n s of a and b [equations(1.105) and (1.107)] we obtain (Smeaton. and Haering 1976a):  where a , b n  n  are defined by equation (1.106) and  °n  =  6  n -  2<S  n+l + n+2 6  The observed image plane p r o f i l e should therefore depend on the magnitude of 0  n  , a l i n e a r combination of the i n d i v i d u a l phases of the  three combined orders. The behavior of equation (1.112) as a function of 0n i s i l l u s t r a t e d i n FIG 9, ' for n=A. The r e l a t i v e values of v„ n were calculated  from experimental d i f f r a c t i o n i n t e n s i t i e s obtained f o r  61 mode-locked operation. The s e n s i t i v i t y of the t h e o r e t i c a l p r o f i l e s i n FIG 9 to variations  i n 0^ indicates that i t i s f e a s i b l e to recover  phase information by o p t i c a l means. The i n t e n s i t y p r o f i l e s f o r other combinations of s p a t i a l and temporal f i l t e r i n g may be derived i n a manner s i m i l a r to equation  (1.112).  62 1.4  Experimental Results The CdS phonon maser c r y s t a l s used i n these experiments were  oriented so that the b-axis was perpendicular  to the polished c a v i t y  surfaces. For t h i s o r i e n t a t i o n the active acoustic modes consist of shear waves whose K-vectors l i e along the b-axis. For the o p t i c a l measurements, the a c t i v e CdS c r y s t a l s  were  coupled to passive  fused  quartz c a v i t i e s by means of a high q u a l i t y bond (described i n Appendix C ) . Since the two materials have nearly the same acoustic impedance f o r the chosen CdS o r i e n t a t i o n , the double c a v i t y modes were nearly harmonic (Hughes and Haering 1976). The passive cavity provided a convenient means of examining the acoustic f i e l d of the phonon maser, since high power l a s e r l i g h t could be passed through i t . Such a high i n t e n s i t y probe could not be used i n the a c t i v e cavity since the large photocurrents produced would disrupt or prevent o s c i l l a t i o n . The A.C. acousto-electric current s i g n a l (refer to section 1.1) from the a c t i v e c r y s t a l could be displayed e i t h e r i n the frequency regime, by means of a spectrum analyzer, or i n the time regime by means of a sampling o s c i l l o s c o p e . An example of both displays i s shown i n FIG 10, f o r mode-locked operation of phonon maser 24.01.02.04. The w e l l defined waveform exhibited i n the time domain i n FIG 10(b), indicates that the harmonically  related modes i n FIG 10(a)  are phase-  locked. The D.C. I-V c h a r a c t e r i s t i c s of the same phonon maser are displayed i n FIG 11(a)  f o r two d i f f e r e n t c o n d u c t i v i t i e s . The oblique  arrows i n d i c a t e the threshold f o r acoustic o s c i l l a t i o n . Above threshold,  63  TIME DOMAIN, 10 na no sec./ciiv. Display of the a c o u s t o - e l e c t r i c current s i g n a l f o r b]_ phonon maser 24.01.02.04 (part of composite c a v i t y DC1). (a) frequency domain, (b) time domain. Operating conditions: applied D.C. e l e c t r i c f i e l d =0.98 KV/cm, D.C. current density =11.1 mA/cm  2  APPLIED D.C. ELECTRIC FIELD (KV/cm) FIG 11(a).  D.C. current density vs. applied D.C.  e l e c t r i c f i e l d f o r bj_ phonon maser 24.01.02.04  (part  of composite cavity DCl) f o r two d i f f e r e n t conductivities a, b. [See text and FIG 11(b)].  66 saturation of the D.C.  current may  at the end of section 1.2).  The  be observed (refer to the  discussion  small discontinuous steps i n t h i s region  occur when the phonon maser makes sudden adjustments i n i t s mode structure. To show the development of current of the acousto-electric current was i n FIG 11(a), and may  spiking, the time display  recorded at the points  be seen i n FIG 11(b). I t was  indicated  the observation  spiking and s e l f - l o c k i n g modes i n the acousto-electric current, evidenced i n FIG 10 and acoustic output was  of  as  11(b), that f i r s t led to speculation that  the  also mode-locked, and hence should consist of  narrow, high amplitude s t r a i n pulses (Maines and Paige 1970). 1.4.1  Experimental Apparatus and Techniques The  configuration f o r the o p t i c a l experiments i s shown  schematically  i n FIG 12. The  l i g h t source was  an argon-ion l a s e r  o  operated at 5145  A, with a t y p i c a l output power of about 1 Watt. At  t h i s high power l e v e l , even stray scattered l a s e r l i g h t was greatly disrupt o s c i l l a t i o n i n the a c t i v e c a v i t y , and s h i e l d i n g was  necessary. The  found to  careful light  incident l i g h t beams were c a r e f u l l y  prepared to provide good c o l l i m a t i o n , and beam diameters were t y p i c a l l y 4 - 5 mm.  A mounted double c a v i t y i s shown i n FIG 13(a). The  goniometer  mount allowed convenient o r i e n t a t i o n of the acoustic c a v i t y f o r maximum d i f f r a c t i o n e f f i c i e n c y . The  focussing lens was  required  to subtend a s u f f i c i e n t l y  large d i f f r a c t i o n angle at the acoustic cavity that l o s s of o p t i c a l information  was  n e g l i g i b l e , and yet to provide a f o c a l plane d i f f r a c t i o n  detector  laser light  ON  onon maser  /  fused quartz passive cavity  FIG 12.  focussi lens  focal plane  Experimental configuration f o r the o p t i c a l measurements.  image plane  68  pattern of s u f f i c i e n t s i z e to allow s p a t i a l f i l t e r i n g . An f/2.5, 15 cm f o c a l length lens was  found to s a t i s f y these c r i t e r i a . A l l o p t i c a l  components were maintained clean and v i b r a t i o n free to minimize  optical  noise i n the image plane. Fused quartz may  be considered o p t i c a l l y and a c o u s t i c a l l y  i s o t r o p i c , and the r e s u l t s of section 1.3.1  may be d i r e c t l y applied  to i l l u s t r a t e a useful p o l a r i z a t i o n property of shear acoustic waves. Consider, i n FIG 5, that incident l i g h t i s polarized along the  axis,  i . e . perpendicular to the d i r e c t i o n of acoustic propagation. I f we l e t i , j be unit vectors along the x^, x^ axes i n FIG 5, incident l i g h t of amplitude A may  be represented by:  A  =  ± (1 + j ) .  (1.114)  /I  t i l  Light d i f f r a c t e d into the m  order has components along the axes of the  index e l l i p s e given by (refer to Vrba and Haering 1974):  A  TI,m  ~^ ^ m( T -L," ) P(  =  V  m  e x 1  m  i v  T  m) J-j™ (1.115)  =  II,m  where V j  m  and v^^  m  ~ A (VTT )exp(iv ) ^2 • -> II,m' m  LJ  m Ttl  T X  are the Raman - Nath parameters corresponding to  the x|, x^ axes, respectively, i n FIG 5. A^ i s the appropriate d i f f r a c t i o n amplitude per u n i t incident amplitude, and i s given by equations (1.90)  69 and ( 1 . 9 5 ) . The exponential factors i n equation (1.115) represent the o p t i c a l phase v a r i a t i o n produced by the acoustic cavity. The two components i n equation (1.115) add coherently i n vector form (Vrba and Haering 1974). The t o t a l amplitude may then be written:  A  V I,m v  m  /2  )ex  P< I,m) iv  Equation (1.116) y i e l d s e l l i p t i c a l l y  1  + V II,m> v  e x  P<iv  I I j m  (1.116)  ) j  polarized l i g h t . The amount of  e l l i p t i c i t y i s extremely small, however, since the Raman - Nath parameters V  I m » II m V  K  <  r  ^  nus  »  t  o  a  v e r  y good approximation, the exponential  factors i n (1.116) may be replaced by 1. In addition, v-£  m  = -VJJ  M  from equations (1.58) and ( 1 . 7 2 ) . Thus from equations (1.90) and (1.95) we see that  A  m I, >  =  (v  m  (  |  m  | >  °  }  ( 1  '  1 1 7 )  and equation (1.116) becomes:  A  m  = ^A (v ) ( i - j) m I,m^  /2  m  vv  T  m  (|m| > 0)  (1.118)  Thus the p o l a r i z a t i o n of the d i f f r a c t e d l i g h t i s rotated by 90° with respect to the incident l i g h t . This p o l a r i z a t i o n f l i p also occurs i f the incident l i g h t i s polarized along x^ i n FIG 5, i . e . p a r a l l e l to the d i r e c t i o n of acoustic propagation. In either case, the p o l a r i z a t i o n of  70 the zero order l i g h t  [refer to equation (1.97)] i s unaffected. Thus an  e x i t p o l a r i z e r may be used to greatly attenuate the strong background due to undiffracted l i g h t , while leaving d i f f r a c t e d l i g h t  largely  unaffected. The e x i t p o l a r i z e r used was an air-spaced Glan-Thompson prism, designed f o r high i n t e n s i t y l a s e r l i g h t . The u s e f u l apertxire of the p o l a r i z i n g prism was large, about 2.5 cm, to avoid truncation of the d i f f r a c t e d l i g h t , and d i e l e c t r i c coatings on the entrance and exit faces optimized transmission at 5145 A. The e x t i n c t i o n r a t i o was about 3 x IO . 4  The f o c a l plane d i f f r a c t i o n pattern consisted of a v e r t i c a l row of spots symmetrically arranged about the zero order or undiffracted beam. At higher operating levels,.60 - 70 d i f f r a c t i o n spots could be distinguished by eye. In the f i r s t order approximation  outlined i n  section 1.3.3, each acoustic mode only produced one symmetric p a i r of  spots. Thus the f i r s t two spots symmetric to the zero order were  produced by the acoustic fundamental, the next pair by the f i r s t harmonic, and so on. Hence the various acoustic modes were, to second order i n the Raman - Nath parameters, o p t i c a l l y decoupled plane. The time-averaged i s given by equations  i n t e n s i t y of the n  t b  i n the f o c a l  order d i f f r a c t i o n spot  (1.95) and (1.90) as v^/2 ( i n p r a c t i c e , we are  not dealing with unit amplitude  incident l i g h t , and i t was necessary  to d i v i d e by the zero order i n t e n s i t y to obtain v / 2 ) . The corresponding 2  s t r a i n amplitude could then be obtained using equations  (1.58) and (1.72).  The form of the f o c a l plane d i f f r a c t i o n pattern i s i l l u s t r a t e d in FIG 13(b). The weak spots l a t e r a l l y displaced from the main v e r t i c a l array were produced by d i f f r a c t i o n of the acoustic waves, i . e . the a c t i v e cavity e f f e c t i v e l y behaved as an aperature f o r acoustic waves r a d i a t i n g into the buffer, ( c f . Fraunhofer d i f f r a c t i o n by a s i n g l e s l i t ) .  This  secondary feature may be ignored since i t had n e g l i g i b l e e f f e c t on the o p t i c a l experiments performed. The apparatus used f o r f o c a l plane s p a t i a l f i l t e r i n g i s shown i n FIG 13(c). The V - s l i t was designed f o r processing  symmetric  d i f f r a c t i o n orders, and i t s width could be accurately adjusted by means of a micrometer. X-Y movement was c o n t r o l l e d by adjusting screws, to provide accurate p o s i t i o n i n g i n the f o c a l plane, and the d i f f r a c t i o n pattern was viewed through a low power microscope to ensure precise optical  "surgery". A photomultiplier coupled to a h o r i z o n t a l s l i t was located  i n the image plane. The photomultiplier could be driven  transverse  to the o p t i c a l axis, as indicated i n FIG 12, i n order to investigate the s p a t i a l v a r i a t i o n of l i g h t i n t e n s i t y i n the image plane. The s l i t widths used were <_ 25 microns, and the length of the s l i t was about 5 mm.  The o p t i c a l gain (magnification) provided by the focussing lens  was about 17, so that the s l i t width, referred to the near f i e l d or object plane, provided an e f f e c t i v e r e s o l u t i o n of better than 1.5 microns. To examine A.C. components of the image plane d i s t r i b u t i o n , the photomultiplier s i g n a l was processed by a Hewlett Packard model 8555A spectrum analyzer, operated i n zero-scan mode. The r e s u l t i n g  FIG 13. (a) Composite cavity DC2 on i t s goniometer mount, (b) t y p i c a l focal plane d i f f r a c t i o n pattern, (c) focal plane s p a t i a l  filter.  73 signal was e s s e n t i s l l y the magnitude of the Fourier component f o r the temporal term to be examined 1.4.2  [refer to equations (1.110) and (1.11)].  O p t i c a l V e r i f i c a t i o n of Mode-Locking In FIG 14 photographs and traces of the image plane signal  are presented for mode-locked operation of composite cavity DCl (refer to Appendix B), with a fundamental acoustic frequency of 11.71 MHz.  The  corresponding wavelength A of the acoustic fundamental was 321 microns i n fused quartz. The time and frequency displays of the acousto-electric current correspond to FIG 10. FIG 14(a) i s simply a bright  field  photograph of DCl., FIG 14(b) shows the image plane i n t e n s i t y of the D.C.  variation  term [equation (1.103), FIG 8]. I t s form agrees w e l l with  the theory of section 1.3.3, and t h i s experimental evidence provided the f i r s t d i r e c t evidence of mode-locking of the acoustic signal i n phonon masers (Smeaton, Hughes, Vrba and Haering 1976). The width of the r e s u l t i n g s t r a i n pulses, inferred from FIG 14(b), was about 23 microns, which corresponds to about 6 nanoseconds at the shear v e l o c i t y of sound i n fused quartz. By s p a t i a l l y f i l t e r i n g  symmetric pairs of d i f f r a c t i o n spots  i n the f o c a l plane, each component of the sum i n equation (1.103) could be examined i n d i v i d u a l l y . FIG 14(c) and (d) are the p r o f i l e s of the components f o r n=l and n=2. I f a reference was chosen at a maximum of the n=l component, the cosine components for n odd were found to be (nearly) u out of phase with those for n even [compare FIG 14(c) and ( d ) ] , as i s predicted by equation (1.103).  74  FIG 14.  Photographs and traces of the image plane i n t e n s i t y d i s t r i b u t i o n for DCl, corresponding to (a) the bright f i e l d image of DC1, (b) the t o t a l D.C. dark f i e l d term, (c) the n = 1 and (d) n = 2 components of the dark f i e l d term. The lengths indicated refer to the near f i e l d or object plane. The detector s l i t width, or e f f e c t i v e r e s o l u t i o n , referred to this plane i s about 1.5 microns. Operating conditions: applied D.C.  e l e c t r i c f i e l d = 0.83 KV/cm,  D.C. current density = 11.1 mA/cm Fundamental acoustic frequency 2  = 11.71 MHz.  (after Smeaton, Hughes, Vrba, and Haering 1976).  76 At higher operating conditions narrox^er s t r a i n pulses could be achieved. In FIG 15 the p r o f i l e of the D.C.  term i s shown f o r mode-  locked operation of DCl with the same acoustic fundamental as i n FIG 14, but at a higher operating voltage. The average width of the dark fringes i n FIG 15 i s about 17.5 microns, corresponding to a pulse length of about 4.6 nanoseconds. At' yet higher applied f i e l d s , pulse lengths of about 3 nanoseconds were i n f e r r e d from the dark f i e l d measurements. The phonon maser was not very stable i n t h i s regime, however, p o s s i b l y due to heating e f f e c t s , and could only be maintained at t h i s l e v e l for short periods of time. The acoustic s t r a i n amplitudes used i n determining the t h e o r e t i c a l p r o f i l e i n FIG 8 were obtained f o r the same operating conditions as i n FIG 15. I f A/2  i s taken as 160.5 microns i n FIG 8,  the width PW i s about 20.2 microns. This i s s l i g h t l y larger than the average dark fringe width i n FIG 15, owing to the f a c t that only the f i r s t 13 acoustic amplitudes were used i n the t h e o r e t i c a l p r o f i l e . While the i n t e n s i t y p r o f i l e of the D.C.  term (FIG 14(b), 15)  provides an estimate of the s t r a i n pulse width, i t contains no information about the exact form of the s t r a i n p r o f i l e , since the phases <$  n  [equations (1.84), (1.85)] are not present i n equation (1.103). By applying s p a t i a l and temporal f i l t e r i n g techniques of the type outlined i n section 1.3.3, i t has been shown (Smeaton and Haering 1976a) that the necessary phase information can be recovered. Examples of experimental data, together with t h e i r best f i t t i n g t h e o r e t i c a l curves, are presented  78 i n FIG 16. This data was taken f o r mode-locked operation of DCl at the same fundamental frequency of 11.71 M H z as i n FIG 10,14,15, and under the  same operating conditions as i n FIG 15. FIG 16(b) corresponds to the  same conditions of s p a t i a l and temporal f i l t e r i n g as FIG 9. The success of  the t h e o r e t i c a l model i s r e a d i l y apparent. I t was usually possible to  f i t the experimental p r o f i l e s to within 1 0 degrees. By employing various combinations of s p a t i a l and temporal filtering, phases 6  n  i t was determined that, to within experimental error, the formed an arithmetic series of the form  ^,62,63,  =  0,a,2a  (1.119)  where a = 238° f o r the s i t u a t i o n depicted i n FIG 16. The phases i n (1.119) are measured r e l a t i v e to 6^ , i . e . a reference has been chosen corresponding to 6^ = 0. The s t r a i n amplitudes and phases obtained f o r the  fundamental and the f i r s t 12 harmonics are given i n TABLE 1. FIG 17  i s the s t r a i n p r o f i l e reconstructed from the data l i s t e d i n TABLE 1. It should be mentioned the  that, due primarily to the f a c t that  measurements allowed only the magnitude of d i f f e r e n t l i n e a r  combinations of the phases to be determined, the data obtained allowed for  two possible s t r a i n p r o f i l e s : that presented i n FIG 17 and a second  p r o f i l e obtained by inversion. The p h y s i c a l l y correct p r o f i l e was i n f e r r e d from the form of the acousto-electric current signal for mode-locked operation. In the a c t i v e cavity, the p i e z o e l e c t r i c p o t e n t i a l corresponding .to the s t r a i n p r o f i l e i n FIG 17 would be consistent with strong electron  FIG 16. Experimental (upper) and corresponding theoretical  (lower) image plane i n t e n s i t y p r o f i l e s  for DCl under d i f f e r e n t conditions of s p a t i a l and temporal  filtering:  (a) s p a t i a l orders 3,4,5 with temporal f i l t e r atfi;t h e o r e t i c a l p r o f i l e : 63 - 26^ + 6^ = 0  (b)  "  4,5,6  "  £2;  "  fi 4  (c)  "  1,3  "  (d)  "  3,4,5,6  "  (e)  "  4,5,6,7  (f)  "  5,6,7,8  11  "  •  2ft;  "  2ft;  "  2ft;  2ft;  "  "  3S  - 25. + 5, = 0 5 6 - 63 = 115°  1  6„ - 6. - 6. + 6, = 0 3 4 5 6 6. - 6  6  C  5  - S, + 6_ = 0  - 6, - 6., + 6_ = 0 6  7  In a l l cases the p e r i o d i c i t y of the p r o f i l e s i s A/2 = 160.5 microns, b u t d i f f e r e n t  8  scaling  factors have been used. Operating conditions: applied D.C. e l e c t r i c f i e l d = 1.17 KV/cm, D.C current density = 24 mA/cm? The fundamental acoustic frequency = 11.71 MHz.  82 TABLE 1 Experimentally determined s t r a i n amplitudes and phases f o r mode-locked operation of DCl under the conditions: fundamental acoustic frequency = 11.71 MHz, applied D.C. e l e c t r i c f i e l d =1.17 KV/cm, D.C. current density = 24 mA/cm . (after Smeaton and Haering 1976a).  ^  2  / n  STRAIN x 10  1  5.87  0  2  6.62  238  3  5.47  115  4  4.39  353  5  3.33  230  6  2.45  108  7  1.68  345  8  1.28  223  9  1.06  100  10  0.86  338  11  0.71  215  12  0.55  93  13  0.50  330  6  6  n  (degrees)  /  Fig 17.  The acoustic s t r a i n p r o f i l e f o r DCl, reconstructed  from  the experimentally  determined  s t r a i n amplitudes and phases l i s t e d i n TABLE 1. Operating conditions: applied D.C.  03 CO  e l e c t r i c f i e l d =1.17 KV/cm, D.C. current density = 24 mA/cm? Fundamental acoustic frequency = 11.71 MHz. (after Smeaton and Haering 1976a).  \  \  ^8  85 bunching, giving r i s e to dominant negative spikes i n the acousto-electric current. This i s , i n f a c t , what i s generally observed [see FIG 11(b)], The width of the pulses i n FIG 17 i s about 23 microns, which corresponds to a pulse duration of about 6 nanoseconds at the shear v e l o c i t y of sound i n fused quartz. This i s somewhat larger than the width of the dark fringes f o r the corresponding dark f i e l d term, FIG 15, due p r i m a r i l y to the f a c t that only the f i r s t 13 modes were used i n obtaining the p r o f i l e i n FIG 17. A more d i r e c t comparison may be made between FIG 17 and FIG 8, since the number and r e l a t i v e amplitudes of the acoustic modes are the same f o r both these p r o f i l e s . The width PW i n FIG 8 i s about 20.2 microns for A/2 = 160.5 microns, i n reasonable agreement with the pulse width i n FIG 17. Hence the dark f i e l d  profile  does, i n f a c t , provide a reasonable estimate of the corresponding s t r a i n pulse width. As has been mentioned, the minimum pulse duration achieved, as i n f e r r e d  from dark f i e l d measurements, was about 3 nanoseconds.  Also, i t was possible to achieve peak s t r a i n s , as estimated from measurement of d i f f r a c t i o n i n t e n s i t i e s , i n excess of 5 x 10 ~*. Due to i n s t a b i l i t y of the phonon maser i n t h i s high operating range, detailed o p t i c a l measurements were not p o s s i b l e . The acoustic pulses produced by a mode-locked phonon maser are unique f o r t h e i r narrow width. There i s no other known source of nanosecond acoustic s t r a i n pulses. An a d d i t i o n a l unique property may be seen i f one considers the product of center frequency, or frequency  86 of maximum gain, and pulse width. The phonon maser has a center frequency n of about 10° Hz and a pulse width of 5 5 x 10  -9  s e c . Hence i t produces  "D.C." pulses which only contain about 1/2 cycle of the c a r r i e r . The composite cavity fundamental frequency, or r e c i p r o c a l of the round-trip t r a n s i t time, was 323.4 KHz f o r DCl. The frequency spacing of the a c t i v e modes i n strong mode-locked  operation was usually close to  a multiple of the r e c i p r o c a l of the round-trip t r a n s i t time of the a c t i v e c a v i t y , about 736 KHz. While i t was not possible to achieve true mode-locking at the composite cavity fundamental frequency, an i n t e r e s t i n g multimode form of operation could be achieved, and the corresponding acousto-electric s i g n a l i s shown i n FIG 18. The display i n FIG 18(a) consists of groups of l i n e s separated by the composite c a v i t y fundamental frequency. The modes within a s i n g l e group [see FIG 18(b)] had amplitudes randomly varying i n time, i n d i c a t i n g the absence of phase-locking. The acoustic modes p a r t i c i p a t i n g i n mode-locked  operation  were found to be harmonically related to better than 1 part i n IO . 4  For single frequency o s c i l l a t i o n , composite cavity modes are not harmonic, but d i f f e r from harmonicity by an amount determined by the acoustic impedance mismatch between the two members of the composite c a v i t y (Hughes and Haering 1976). For fused quartz and the chosen o r i e n t a t i o n of CdS, the maximum deviation from harmonicity i s less than 3 KHz, and the amount of mode-pulling required to induce harmonicity f o r modelocked operation i s minimal. On t h i s basis, one would predict that mode-locking should be d i f f i c u l t to achieve i n composite c a v i t i e s  87  (b) FIG 18.  Frequency display of the acousto-electric current for DCl operated under the conditions: applied D.C. e l e c t r i c f i e l d = 1.32 KV/cm, D.C. current density = 13.3 mA/cm? Horizontal scale i s (a) 0.5 MHz/div., (b) 20 KHz/div.  whose members d i f f e r considerably i n acoustic impedance. For example, i f sapphire i s substituted f o r fused quartz as the material f o r the passive c a v i t y , the maximum deviation from harmonicity 100 KHz (Hughes and Haering  i s greater than  1976). A composite c a v i t y formed with a  bj_ phonon maser and a sapphire passive cavity.(DC3) was found, i n f a c t , not to display strong mode-locking c h a r a c t e r i s t i c s .  89 1.5  O p t i c a l Determination of the Normal Modes of Composite C a v i t i e s The normal modes of composite c a v i t i e s have previously been  obtained by measurement of the a c t i v e admittance as a function of frequency (Hughes and Haering 1976). In this s i t u a t i o n , the composite cavity was located i n one arm of a symmetrical admittance bridge. Since the bandwidth of the phonon maser i s > 300 MHz,  the bridge was  required  to be accurately balanced over a wide range i n order to examine the cavity modes of i n t e r e s t . A much simpler, o p t i c a l technique may be applied i n many s i t u a t i o n s . Using a configuration s i m i l a r to FIG 12, the normal modes of composite cavity DCl were measured by monitoring the i n t e n s i t y of d i f f r a c t e d l i g h t as a function of frequency. The active c r y s t a l  was  driven by a Hewlett Packard model 8601A sweep o s c i l l a t o r (maximum output voltage 3 V rms). No attempt was made to optimize s i g n a l to noise by use of phase s e n s i t i v e detection or other techniques. A t y p i c a l spectrum i s shown i n FIG 19. The spacing of the l i n e s (~ 323  KHz)  corresponds to the r e c i p r o c a l of the round t r i p t r a n s i t time of the composite  cavity. The r a t i o of the l i n e width to the center frequency f o r the  modes i n FIG 19 i s a factor of 2 larger than f o r the e l e c t r i c a l impedance measurements of Hughes and Haering (1976). This i s due to Q-spoiling of the a c t i v e c a v i t y produced by the presence of r e s i d u a l scattered l a s e r l i g h t . With proper l i g h t s h i e l d i n g and optimization of the experimental arrangement, the o p t i c a l technique should e a s i l y match  50.1 4  MHz  I— 1MHz H  KJ FIG  19.  Mode structure of composite cavity DCl, obtained by o p t i c a l d i f f r a c  91 the r e s o l u t i o n of the e l e c t r i c a l measurements. Aside from inherent s i m p l i c i t y , the o p t i c a l technique  has  the advantage of wide frequency c a p a b i l i t y . Using a Hewlett Packard model 3200B VHF o s c i l l a t o r to drive the a c t i v e c r y s t a l , d i f f r a c t i o n could be observed by eye for frequencies i n excess of 400 MHz.  In f a c t ,  i f the Bragg condition i s optimized, there i s , i n p r i n c i p l e , no r e s t r i c t i o n on the frequencies to be examined. The major disadvantage  of t h i s technique i s that i t r e s t r i c t s  the form of the passive cavity. The passive c a v i t y must be transparent for one of the l a s e r l i n e s a v a i l a b l e , and should have a thickness £ 200 microns. In a d d i t i o n , a n i s o t r o p i c materials must be oriented so that the appropriate o p t o e l a s t i c constants are non-zero. These constraints are s a t i s f i e d , however, f o r many cases of p r a c t i c a l i n t e r e s t .  92 1.6  Conclusions and Summary of Contributions By applying the nonlinear theory of Butcher and Ogg (1968,  1969, 1970), i t was shown that mode-locking can be predicted as a natural consequence of the structure of the phonon maser and nonlinear properties of the acousto-electric amplifying mechanism. The f i r s t d i r e c t evidence of mode-locking was provided by employing laser d i f f r a c t i o n and s p a t i a l f i l t e r i n g techniques (Smeaton, Hughes, Vrba and Haering 1976), and by means of a unique combination of s p a t i a l and temporal f i l t e r i n g  sufficient  phase information was recovered to allow reconstruction of the modelocked s t r a i n p r o f i l e (Smeaton and Haering 1976a). The general technique of combining s p a t i a l and temporal f i l t e r i n g provides a powerful method of processing and analyzing o p t i c a l  signals.  At high operating conditions, s t r a i n pulses of width - 3 nanoseconds and amplitude > 5 x 10 ~* were achieved. For future experiments, i t would be of i n t e r e s t to investigate mode-locking for phonon masers at low temperatures. The l a t t i c e  loss term, which reduces the acoustic band-  width (Burbank 1971), should be smaller i n this s i t u a t i o n . Hence wider bandwidths and correspondingly narrower s t r a i n pulse widths should be achieved.  93 CHAPTER 2  APPLICATION OF OPTICAL PROCESSING TO A STUDY OF ACOUSTICALLY INDUCED SPACE CHARGE GRATINGS IN CdS  2.1  Introduction Acoustic echo phenomena i n p i e z o e l e c t r i c semiconductors have  received considerable experimental and t h e o r e t i c a l attention (Yushin et a l 1975, Chaban 1972, 1974, 1975, Shiren et a l 1973, Shiren 1975, Melcher and Shiren 1975, Maerfeld and Tournois 1975). These e f f e c t s r e s u l t from a nonlinear i n t e r a c t i o n of the e l e c t r i c f i e l d produced by an a c t i v e acoustic mode of wave number K and angular frequency ft, with an e x t e r n a l l y applied, s p a t i a l l y invariant A.C. e l e c t r i c f i e l d of frequency equal, or harmonically related to ft. Of present i n t e r e s t are s o - c a l l e d 3 pulse echoes (Shiren et a l 1973) which are produced i n the following manner. At t=0 an acoustic pulse of wave number K and angular- frequencyft.i s introduced i n t o a p i e z o e l e c t r i c c r y s t a l . This may be achieved by means of an external transducer, or, more commonly, by applying an R.F. pulse to the sample, thus producing an acoustic pulse v i a the p i e z o e l e c t r i c i n t e r a c t i o n . At a l a t e r instant t=x, an R.F. pulse of frequency ft (or harmonically related to ft - Melcher and Shiren 1975) i s applied to the c r y s t a l . Aside from p i e z o e l e c t r i c e f f e c t s , t h i s second pulse w i l l produce a s p a t i a l l y invariant A.C. f i e l d i n the c r y s t a l . I f , at time t=x+T, another R.F. pulse i s Applied, an echo pulse i s recorded at time t=2x+T.  94 In some recently proposed models (Melcher and Shiren 1975, Chaban 1975), l o c a l i z e d electron traps play a fundamental r o l e i n the formation  of echoes. I t i s postulated that a n o n l i n e a r i t y i n the system  produces a r e d i s t r i b u t i o n of the trapped charge, which, i n turn, produces a s t a t i c e l e c t r i c f i e l d d i s t r i b u t i o n whose s p a t i a l v a r i a t i o n mirrors the p e r i o d i c i t y of the propagating acoustic wave. There i s some controversy over the nonlinear mechanism involved, and there are two basic models of the i n t e r a c t i o n : (i)  Electron "bunching": In this model, set f o r t h i n t h e o r e t i c a l  papers by Chaban (1972, 1974, 1975), the periodic p i e z o e l e c t r i c p o t e n t i a l associated with the propagating acoustic wave gives r i s e to a modulation ("bunching") i n the conduction band electron density. The electron density then has the form [see equation (1.9)]:  / .\ n(x,t)  where n  Q  =  n  Q  . i(Kx-ftt) , + n-j^e + cc.  ' ,A (2.1) N N  i s the equilibrium electron density. At the instant t=x an A.C.  e l e c t r i c f i e l d of the form  „ E  islt ,  l  e  + ' c  (2.2)  c  i s applied along the x-axis of the c r y s t a l . Since the current equation (1.6), contains a nonlinear  density,  term proportional to the product of  -(2.1) and (2.2), i t w i l l have a time invariant component of the form  i e  K  95 A p e r i o d i c space charge w i l l be b u i l t up to compensate f o r t h i s current. Hence trapping w i l l be more intense i n some parts of the c r y s t a l than i n others and,  i n the absence of the acoustic and e l e c t r i c a l s i g n a l s ,  a bound space charge w i l l e x i s t , having the s p a t i a l p e r i o d i c i t y of the acoustic wave. (ii)  E l e c t r i c f i e l d - a s s i s t e d detrapping  1975,  (Shiren et a l 1973,  Shiren  Melcher and Shiren 1975): In t h i s model one considers the t o t a l  electric field E  i n the c r y s t a l at time t=x. I t i s given by the sum  the p i e z o e l e c t r i c f i e l d  (amplitude  and the s p a t i a l l y uniform A.C.  „ , .v E (x,t) t  =  E) 2  of the propagating  electric  of  acoustic wave,  field:  „ islt , _ i(Kx-ftt) , E^e + Ee + cc.  .„ „. (2.3)  2  I t i s assumed that the t o t a l e l e c t r i c f i e l d i n equation  (2.3) produces  f i e l d - a s s i s t e d detrapping of electrons i n i t i a l l y uniformly d i s t r i b u t e d i n shallow traps. Since t h i s process i s highly nonlinear (Haering 1959), i t gives r i s e to many components i n the conduction electron density harmonically r e l a t e d i n time and space to the t o t a l e l e c t r i c f i e l d . In a manner s i m i l a r to ( i ) , the nonlinear current density [equation  (1.6)]  w i l l - t h e n contain s p a t i a l l y nonuniform, time invariant terms and, i n the absence of the perturbing f i e l d s , a stored p e r i o d i c space charge pattern w i l l e x i s t . For both ( i ) and  ( i i ) , the charge grating w i l l decay with  a time constant r e l a t e d to the l i f e t i m e of the traps involved. I f the A.C.  e l e c t r i c f i e l d i s again pulsed at time t=T+T, i t w i l l  interact  96 with the periodic e l e c t i c f i e l d of the space charge grating to produce a backward propagating (time reversed) acoustic wave (as w e l l as a forward wave), which i s detected as a pulse echo at the surface of the c r y s t a l at time t=2r+T. Experimental data regarding a c o u s t i c a l l y induced space  charge  gratings has been obtained almost exclusively by pulse echo measurements, and i t i s of considerable i n t e r e s t to f i n d an experimental technique that w i l l provide supplementary produced  information. Since the e l e c t r i c  field  by a s t a t i c space charge grating w i l l modulate the r e f r a c t i v e  index i n a p i e z o e l e c t r i c c r y s t a l , v i a the primary and secondary  electro-  o p t i c e f f e c t (Nye 1964), o p t i c a l measurements are indicated. The v a r i a t i o n i n the i n d i c a t r i x produced by the e l e c t r o - o p t i c e f f e c t be represented by (Nye 1964)  AB..  =  IJ  where ^-^> z  E  may  [ c f . equation (1.43)]:  Z...E. xjk  (2.4)  k  ^ are components of the e l e c t r o - o p t i c tensor and the e l e c t r i c  f i e l d . Contracting the indices i j according to equation (1.45) allows equation (2.4) to be represented i n matrix form, i n a manner exactly analogous  to equation (1.47). For hexagonal 6mm  symmetry, appropriate f o r CdS,  o p t i c matrix has the form (Nye 1964):  the e l e c t r o -  97 0  0  z  13  0  0  Z  13  0  0  z  33  0  Z  z  42  0  (2.5)  0  42  0  0  0  0  For the case of an e l e c t r i c f i e l d p a r a l l e l to the c-axis, f o r example, the appropriate e l e c t r i c f i e l d component i s E  AB, AB,  Z  13 3  Z  13 3  Z  33 3  and one obtains:  E  E  E  AB,  0  AB  0  r  AB,  The c o e f f i c i e n t s  3  (2.6)  0  i n equation (1.39) become:  B  l  =  B  3  =  =  B  2  =  n2 e  (  +  B  5  < n2  Z  =  +  Z  13 3 > E  (2.7)  33 3 > E  B  6  -  0  98 n  o>  n  e  a r e t h e o r d i n a r y and e x t r a o r d i n a r y r e f r a c t i v e  equation  [ r e f e r to  ( 1 . 4 2 ) ] . The new i n d i c a t r i x thus becomes:  < rT2 + 1 3 3 Z  E  For l i g h t propagating ellipse  indices  >  (  X  !  +  X  2>  +  (  n-|  +  along the b or x  Z  33 3 E  axis,  2  }  *l  the a p p r o p r i a t e  (  2  '  8  )  index  i s g i v e n by:  (  n2  +  Z  13 3 E  > ! X  +  £ | + ^33*3  (  )  (2.9)  and has semiaxes:  n  . =  _ (1 +  n2z  E )  1 3  2  3  (2.10) n  n  Since n z 2  3 3  E  e  n^ZooEo) e 33 3'  (1 +  , n z^ E  <<  2  3  3  3  e 1  z  5 5  t  1, t h e s q u a r e r o o t s i n (2.10) may be  expanded t o y i e l d :  A n  o  ~ \  a  n  o 13 3 z  E  (2.11) An  e  -  -|n|z  3  3  E  3  99 Equation (2.11) may  be d i r e c t l y compared with equation (1.58). This  strong analogy with the p h o t o e l a s t i c e f f e c t indicates that i t should be possible to apply the laser d i f f r a c t i o n and s p a t i a l f i l t e r i n g of CHAPTER 1, to d i r e c t l y observe a c o u s t i c a l l y induced space  techniques charge  gratings. 2.2  A p p l i c a t i o n of O p t i c a l Processing In p r i n c i p l e , i t should be p o s s i b l e at low temperatures  to  observe d i f f r a c t i o n from stored charge gratings which have been previously prepared by the a p p l i c a t i o n of s u i t a b l e e l e c t r i c f i e l d s . In p r a c t i c e t h i s has not proved p o s s i b l e with our c r y s t a l s , presumably because the trapped charge i s disturbed by the probing l i g h t used i n the d i f f r a c t i o n experiment. This i s consistent with the experimental findings of Shiren e t , a l (1973). For t h i s reason, o p t i c a l measurements were made with cj_ CdS phonon masers which were driven at a resonant frequency by an external o s c i l l a t o r . In t h i s steady state s i t u a t i o n , the a c o u s t i c a l l y induced charge gratings e x i s t i n the presence of l i g h t . However, i t i s then necessary to devise a measurement technique which can d i s t i n g u i s h between d i f f r a c t i o n from a s t a t i c charge grating and d i f f r a c t i o n from the a c o u s t o - e l e c t r i c f i e l d s which are simultaneously present. The e l e c t r i c f i e l d i n the c r y s t a l may  E(x,t)  =  „ i f i t , _, i ( K x - n t ) , E^e + ^[e r  +  be w r i t t e n :  e  i(Kx+ftt), , 3 + c.c.  ,„ (2.12)  where x i s taken along the c-axis. I t consists of a s p a t i a l l y i n v a r i a n t  100 term of frequency 0, and amplitude E^, related to the driving f i e l d of the external  o s c i l l a t o r ; and an acousto-electric  f i e l d of frequency 0.  and wave number K, consisting of r i g h t and l e f t going waves of equal amplitude E - The l a t t e r f i e l d s are produced v i a the p i e z o e l e c t r i c 2  i n t e r a c t i o n of the c r y s t a l with the d r i v i n g f i e l d . Equation (2.12) does not take into account the e l e c t r i c f i e l d s produced by modulations of the space charge. The s t a t i c space charge grating i s produced from the f i e l d s E-^ and E  2  i n equation (2.12) by one of the nonlinear e f f e c t s described  i n section 2.1, and, as was indicated, w i l l produce a periodic modulation of the l o c a l r e f r a c t i v e index of the c r y s t a l v i a the e l e c t r o - o p t i c e f f e c t . I t thus presents a f i x e d phase grating to incident l a s e r l i g h t and r e s u l t s i n d i f f r a c t i o n . In the Raman - Nath regime (refer to section 1.3.2 of CHAPTER 1) the d i f f r a c t i o n patterm w i l l consist of a row of spots symmetrically arranged about the zero order or undiffracted beam. The acousto-electric  field E  2  w i l l also produce d i f f r a c t i o n due to photoelastii  (section 1.3 of CHAPTER 1) and e l e c t r o - o p t i c coupling. In the experiments performed, t h i s d i f f r a c t i o n was roughly two orders of magnitude more intense than that produced by the s t a t i c space charge. Moreover, since the s p a t i a l p e r i o d i c i t y was the same i n both cases, both d i f f r a c t i o n patterns were superimposed.  I t was possible  to completely separate the  two o p t i c a l signals, however, by means of s p a t i a l and temporal This may be demonstrated  filtering.  by examining the o p t i c a l phase  transformation produced by the acoustic  standing waveform. In analogy  with equations (1.83) and (1.85), i t may be written:  101 CO  .  e  i2vsinKx cosftt  =  _  e  -  v n  w  „  inKx  l J n (2vcosfit)e =_oo  (2.13)  The Bessel function r e l a t i o n s h i p , equation (1.87), has again been used. The o p t i c a l amplitude i n the  order d i f f r a c t i o n spot i n the f o c a l  plane i s proportional to J (2vcosftt). Consider that only d i f f r a c t i o n n  spots n and m are combined and imaged. The image plane i n t e n s i t y d i s t r i b u t i o n i s then proportional to  ^.m  where  V  =  =  J  n  (  V  )+  J  m  (V)  +  2 J  n > J ( V ) cos (n-m)Kx ( V  m  (2.14)  2vcosSlt  (2.15)  If we extract those components of (2.14) that have no time v a r i a t i o n , the f i r s t two terms w i l l contribute only a uniform background. From the well known r e l a t i o n s h i p (Abramowitz and Segun 1968)  J (-z) m  =  (-l) J (z) m  m  (2.16)  and the form of V i n equation (2.15), i t i s apparent that the cross term i n (2.14) w i l l only contain a D.C. component i f n+m i s an even number. Thus f o r n+m odd, the D.C. part of the image plane s i g n a l w i l l contain no terms with s p a t i a l v a r i a t i o n . This i s r i g o r o u s l y true, even i f weak a c o u s t i c a l l y induced v a r i a t i o n s of the o p t i c a l absorption c o e f f i c i e n t are taken into account.  102 C o n s i d e r a few [1]  Adjacent  s p e c i a l cases o f p a r t i c u l a r  d i f f r a c t i o n o r d e r s , m = n+1  s o l e l y o f a u n i f o r m background, produced equation [2] is  : the D.C.  by  the f i r s t  interest: signal two  terms i n  (2.14).  Symmetric d i f f r a c t i o n o r d e r s , m = -n apparent  t h a t the D.C.  component o f  : using equation  (1.103),  see FIG  (1.93), i t  (2.14) i s p r o p o r t i o n a l to  (1 - cos2nKx) [ c f . i n d i v i d u a l components o f the dark f i e l d equation  consists  term,  1 4 ( c ) , ( d ) ] . Thus f o r n = l , f o r example,  the  image p l a n e i n t e n s i t y d i s t r i b u t i o n c o n s i s t s o f a c o s i n e p a t t e r n o f b r i g h t and [3]  dark f r i n g e s w i t h p e r i o d i c i t y A/2,  m = n+2  D.C.  : for this  situation,  similar  f r i n g e s w i l l be l e s s d i s t i n c t  in  the t h i r d  term  i n (2.14) p r o v i d e s a  a u n i f o r m background D.C.  t o [2] w i t h symmetric o r d e r s ± 1 .  i n t h i s c a s e , however, due component produced  by  The  t o the  the f i r s t  presence  tx^o terms  (2.14). Image p l a n e photographs x^hich i l l u s t r a t e  shown i n FIG 20. T h i s d a t a was two  \  component p r o p o r t i o n a l to cos2Kx. Thus c o s i n e f r i n g e s appear  h a v i n g p e r i o d i c i t y A/2,  of  where A = 2TTK  o r t h o g o n a l l y arranged  mounted i n a c e l l  t a k e n w i t h a model system c o n s i s t i n g  quartz transducers  of d i s t i l l e d  the above r e s u l t s  w a t e r . The  (resonant frequency  two  FIG 2 0 ( b ) - ( g )  10  MHz)  a  two  20(a), r e s u l t e d . In o b t a i n i n g  o n l y the c e n t r a l v e r t i c a l row  u s e d . I t s h o u l d be noted  of  t r a n s d u c e r s were d r i v e n  s i m u l t a n e o u s l y by a G e n e r a l Radio model 1211-C o s c i l l a t o r and d i m e n s i o n a l d i f f r a c t i o n p a t t e r n , FIG  are  t h a t the d i f f e r e n t  of d i f f r a c t i o n spots s p o t s i n FIG  20(a)  was  correspond  103  (e) FIG 20.  (f)  '(g)  Experimental data obtained from a model system consisting of two orthogonal quartz transducers (resonant frequency 10  MHz)  mounted i n a c e l l of d i s t i l l e d water, (a) f o c a l plane d i f f r a c t i o n pattern,  (b) zero order l i g h t imaged with sound generator o f f .  Image plane d i s t r i b u t i o n s produced by combining  and  imaging  d i f f r a c t i o n spots (c) 0,1,(d) 1,2,(e) ± l , ( f ) 0,2 and (g)  1,3.  104 to d i f f e r e n t orders of Raman - Nath d i f f r a c t i o n from a simple sinusoidal acoustic waveform. In contrast, a l l spots i n FIG 13(b) correspond to f i r s t order Raman - Nath d i f f r a c t i o n from a waveform consisting of many harmonically related components. In FIG 20(c) and (d) adjacent d i f f r a c t i o n orders 0,1 and 1,2 , respectively, have been combined and imaged. By comparison with FIG 20(b), the zero order l i g h t with the sound generator o f f , i t i s apparent that no f r i n g e s are present, as was predicted i n [1]. In FIG 20(e) symmetric orders ±1 have been imaged. A pattern of p e r i o d i c i t y A/2, c o n s i s t i n g of a l t e r n a t i n g dark and bright fringes i s produced, i n agreement with [2], In FIG 20(f) and (g) orders 0,2 and 1,3 , respectively, have been imaged. A f r i n g e pattern of p e r i o d i c i t y A/2 i s produced. As predicted i n [3], the fringes are less d i s t i n c t than i n FIG 20(e). I f , i n addition to (j>^ i n equation (2.13), a s t a t i c phase grating i s present, such as would be produced by a f i x e d space  charge  pattern, the image plane s i g n a l w i l l be modified. A sinusoidal grating of period A = 2irK ^ w i l l produce a phase transformation  e  where V  q  i<f>s  _ =  e  iv_sinKx °  >  (2.17)  i s the amplitude of the o p t i c a l phase v a r i a t i o n ( v << 1). Q  <p may be considered to be the fundamental component of the periodic s  space charge pattern. The phase (J> (or the combination of <p and <p^) s  s  produces an image plane D.C. term of the form cosKx, i f d i f f r a c t i o n  105 orders 0,1 (or any other pair of adjacent d i f f r a c t i o n orders) are combined and imaged. In p r a c t i c e , i t was necessary to amplitude modulate the external o s c i l l a t o r and to phase-lock the image plane detector to the modulation frequency, i n order to remove extraneous signals and o p t i c a l noise i n the image plane. With t h i s i n mind, the total phase transformation given by equations (2.13) and (2.17) may be written:  exp I (2vcosfit + v ) ( 1 + Mcosw t)sinKx; Q  =  I n  =_oo  m  J [(2vcosftt + v ) ( l + Mcosto t)]e n  Q  inKx  (2.18)  m  where M i s the modulation index (- 0.1 i n practice) and u) i s the modulation m  frequency. The Bessel functions i n (2.18) may be Taylor expanded to yield: J (2vcosf2t + v ) n  Q  +  (2vcosf2t + v )Mcosw t J^(2vcosftt + v ) 0  m  +  Q  (2.19)  In p r a c t i c e , v , v << 1 and v D  Q  ~ v? I t i s therefore necessary to r e t a i n  contributions to f i r s t order i n v , and to second order i n v. Only the Q  f i r s t two terms i n (2.19) need be considered since they contain a l l contributions to second order i n both v and v . The derivative of the Q  Bessel function i n (2.19) may be expanded using the r e l a t i o n (Davis 1968):  106 zJ^(z)  =  nJ (z) - z J n  n + 1  (z)  (n = 0,1,2  )  (2.20)  I f d i f f r a c t i o n orders 0,1 i n equation (2.18) are combined and imaged, the r e s u l t i n g image plane i n t e n s i t y d i s t r i b u t i o n i s proportional to [see equation (2.14)]:  J (V) + 2  where  V  =  jJ(V')  + 2J (V')J (V )cosKx  (2.21)  ,  0  1  (2vcosftt + v ) ( 1 + Mcosu^t)  (2.22)  Q  Using equations (2.19), (2.20), the small argument expression f o r the Bessel function, equation (1.92), and extracting the component of (2.21) at frequency u ^ , the appropriate image plane signal i s (to f i r s t order in v  Q  and second order i n v ) :  l^  m Q  =  M(v cosKx - v )  (2.23)  2  Q  Thus the presence of the s t a t i c space charge grating i s indicated by a cosine fringe pattern of p e r i o d i c i t y A. The component  -Mv  2  i n equation  (2.23) i s a uniform background produced by acoustic d i f f r a c t i o n e f f e c t s . I t s negative sign indicates that i t i s ir out of phase with the amplitude modulating reference.  107 2.3  Experimental Results The experimental arrangement  for o p t i c a l measurements \<ras  e s s e n t i a l l y the same as i n FIG 12. The samples used were c_[_ CdS phonon masers, driven by an external o s c i l l a t o r . For t h i s configuration, the p i e z o e l e c t i c a l l y active, acoustic modes consist of l o n g i t u d i n a l waves whose K-vectors l i e along the c-axis. The l i g h t source was a o  5 mW HeNe laser (X = 6328 A) and incident p o l a r i z a t i o n was p a r a l l e l to the c-axis. Aside from the laser l i g h t , no a n c i l l a r y i l l u m i n a t i o n of the c r y s t a l s was used. A s l i t width of 250 microns was used for the scanning photomultiplier i n the image plane (refer to FIG 12), and the magnification provided by the focussing lens was about 27. A l l measurements were made at room temperature. FIG 21 shows examples of image plane p r o f i l e s f o r cj_ c r y s t a l 24.06.06.01. The external o s c i l l a t o r i n this case was an Arenburg model PG-650C. The photographs on the LHS of FIG 21(b) and (c) were obtained by imaging d i f f r a c t i o n orders ±1. The spacing of the bright fringes i s thus A/2  (refer to section 2.2). The photographs on the r i g h t  side of the figure were obtained by imaging orders 1,2. Weak, poorly defined fringes of spacing A may be discerned, suggesting the presence of a s t a t i c phase grating. To obtain more r e l i a b l e data, the technique outlined i n section 2.2 was employed. The c r y s t a l s were driven by a Hewlett Packard model 3200B VHF o s c i l l a t o r , which was amplitude modulated at 3 KHz by a Wavetek model 114 o s c i l l a t o r . The amplitude modulating s i g n a l provided  108  FIG 21.  (a) B r i g h t f i e l d profiles 1,2  image of c]_ c r y s t a l  24.06.06.01, and  produced by imaging d i f f r a c t i o n  (RHS) f o r d r i v i n g  frequencies  spots ±1  (b) 15.619 MHz  intensity  (LHS) and and ( c ) 19.062  MHz.  109 the reference f o r a P.A.R. model HR-8 l o c k - i n amplifier, which was used to process the s i g n a l from the image plane detector. For t y p i c a l operating -2 conditions, the Raman - Nath parameter v s 5 x 10  , as calculated  from  the d i f f r a c t i o n i n t e n s i t i e s . An example of the image plane signal f o r c[_ c r y s t a l 24.06.06.01 i s shown i n FIG 22. The upper curve was obtained by imaging orders ±1. As has been indicated, the r e s u l t i n g p r o f i l e has p e r i o d i c i t y A/2 and i s presented as a r e f erence. The lower curve i n FIG 22 was obtained by combining  orders 0,1 (the v a r i a b l e amplitude of the o p t i c a l signals i s  a r e f l e c t i o n of the i n t e n s i t y p r o f i l e of the l a s e r beam). The appearance of s i n u s o i d a l fringes of spacing A i s d i r e c t evidence f o r the presence of a f i x e d charge grating of the form sinKx [refer to equation (2.17)]. This sinusoidal term i s superimposed on a larger negative ( i . e . ir out of phase with respect to the amplitude modulating reference) background term, i n agreement with equation (2.23). By comparing the amplitude of these two signals i t was possible, with the aid of equation (2.23), to estimate the value of v . Hence, with the a i d of equations (1.2), (1.72) Q  and (2.11), the peak electron density modulation associated with the s t a t i c space charge grating could be calculated, and was found to be ~ 10  1 2  electrons/cm  3  at ft/2ir equal to 33 MHz. Much higher modulations  have been reported i n pulse echo measurements, performed higher frequencies and d r i v i n g amplitudes  at correspondingly  (Shiren 1975) .  To check these r e s u l t s , a double c a v i t y (DC4) was constructed 'by bonding together two s i m i l a r l y oriented c_j_ CdS c r y s t a l s . Either side  FIG 22.  Image plane signal f o r c[_ c r y s t a l 24.06.06.01, produced by imaging d i f f r a c t i o n spots (a) + 1 and (b)^ 0,1.  Ill of  the r e s u l t i n g composite cavity could be driven by an external o s c i l l a t o r .  This produced a s i t u a t i o n i n which the acousto-electric f i e l d E£ [see equation (2.3)] was of nearly constant amplitude throughout the composite cavity, but the external d r i v i n g f i e l d E-^ was present only i n one part. Laser d i f f r a c t i o n could be done i n one or both parts of DC4 and, as i s demonstrated  i n FIG 23, the o p t i c a l signals could be completely separated  by f o c a l plane s p a t i a l f i l t e r i n g . The image plane s i g n a l f o r one part of DC4, both i n the presence and absence of the d r i v i n g f i e l d E^, i s presented i n FIG 24. The fact that the fringes require the simultaneous presence of E-^ and E  2  [equation (2.3)] indicates that they are not an  a r t i f a c t of the acousto-electric f i e l d alone. For  the measurement technique used, i t was not possible to  e s t a b l i s h whether the s t a t i c charge grating was associated with trapped or mobile charge. However, the strong sample dependence of our measurements i s an i n d i c a t i o n of the importance of defects i n producing the observed r e s u l t s . To determine the presence of shallow traps of the type postulated by Melcher and Shiren (1975), thermoluminescence  measurements were  performed. CdS samples, f r e s h l y etched i n concentrated HC1, were mounted i n a LHe dewar at 4.2°K. A f t e r being illuminated with strong band gap l i g h t , the c r y s t a l s were maintained i n the dark. Once the afterglow had subsided (10 - 15 min. a f t e r i l l u m i n a t i o n ) , the samples were warmed up and the luminescent i n t e n s i t y was recorded as a function of time. The r e s u l t i n g glow curves were analyzed using the techniques of G a r l i c k  112  FIG  23.  Image plane photographs of DC4, i l l u s t r a t i n g that d i f f e r e n t components of the o p t i c a l s i g n a l may be examined  individually,  (a) a l l l i g h t has been imaged, (b) the background l i g h t has been removed, (c) only the background l i g h t has been imaged, (d) the l i g h t from the lower member of DCA has been removed.  FIG  2k,  Image plane.signals obtained from the upper member of DC4 i n FIG 23. The p r o f i l e s correspond to (a) orders + 1 combined and imaged; and orders 0,1" imaged {b\ i n the absence and (c) i n  I  the presence of the external driving  field.  114 and Gibson (1948), Grossweiner (1953) and Booth (1954). A t y p i c a l spectrum i s shown i n FIG 25. As indicated, these measurements v e r i f i e d the existence of shallow traps with an i o n i z a t i o n energy of about 20 meV,  as  required for the model proposed by Melcher and Shiren (1975) [refer to model ( i i ) i n section 2.1]. In p r i n c i p l e , the d i f f r a c t i o n experiments could be extended to study the higher Fourier components of the s t a t i c charge grating. This would be of i n t e r e s t since these components would y i e l d information about the nature of the underlying n o n l i n e a r i t y . In p r a c t i c e , t h i s extention has not been possible with our c r y s t a l s because the observed d i f f r a c t i o n e f f e c t s were too weak.  115  ~20meV ~150meV  I  FIG 25.  A t y p i c a l glow curve obtained from a CdS c r y s t a l of the type used to construct phonon masers. The points at which d i f f e r e n t temperatures occurred are indicated on the horizontal axis, and the calculated trap i o n i z a t i o n energies corresponding to the major peaks are given.  thermoluminescent  116 2.4  Conclusions and Summary of Contributions By means of the s p a t i a l and temporal  filtering  techniques  outlined i n CHAPTER 1, o p t i c a l experiments were performed that allowed the f i r s t d i r e c t observation of a c o u s t i c a l l y induced space charge gratings i n CdS (Smeaton and Haering 1976b). The information obtained by o p t i c a l means should be of considerable supplimental value to data obtained by pulse echo measurements. Proper e x p l o i t a t i o n of the techniques described could provide insight into the underlying physical processes involved i n the formation of the s t a t i c charge gratings.  117 CHAPTER 3  NEW PHOTOVOLTAIC EFFECTS IN CdS  3.1 Introduction Photovoltaic e f f e c t s constitute a class of phenomena i n which l i g h t generates a voltage across a portion of a semiconductor. Such e f f e c t s have been observed f o r several decades i n CdS and other semiconductors (for a review see Pankove 1971) . The samples used i n the present experiments were polished phonon maser c r y s t a l s , constructed from high r e s i s t i v i t y  [10  1 0  - 10  1 2  (ficm)  1  i n the dark] photoconductive  CdS. For t h i s s i t u a t i o n , the dominant known photovoltage i s produced v i a the Dember e f f e c t (Williams 1962). The Dember voltage r e s u l t s when strongly absorbed r a d i a t i o n generates a high density of electron-hole p a i r s , which then d i f f u s e away from the illuminated surface. The electrons have a higher mobility than the holes and hence extend further into the c r y s t a l , tending to make the surface p o s i t i v e with respect to the bulk. This e f f e c t , and the r e s u l t i n g voltage, are named f o r t h e i r discoverer (see Dember 1931) . In t h i s chapter some i n i t i a l r e s u l t s from on-going are  experiments  presented to i l l u s t r a t e the existence of two new photovoltaic e f f e c t s  i n CdS. As w i l l be shown, these e f f e c t s provide D.C. voltages which are several orders of magnitude larger than the Dember voltage. As y e t , the underlying physical mechanisms are not understood.  118 3.2  The Photoacousto V o l t a i c E f f e c t A D.C.  voltage, requiring the simultaneous  presence of l i g h t  and propagating acoustic f i e l d s , has been discovered i n CdS phonon masers. The basic experimental configuration f o r measurement of the photoacoustic voltage i s shown i n FIG 26. Composite c a v i t i e s , consisting of txro s i m i l a r l y oriented b_[_ or c_|_ phonon maser c r y s t a l s , were used i n the experiments.  The a c t i v e side of the double c a v i t y was used as a  source of acoustic waves and was either driven by an external o s c i l l a t o r , or operated with a D.C.  e l e c t r i c f i e l d to achieve phonon maser a c t i o n .  The free surface of the passive cavity was  i l l u m i n a t e d and the  D.C.  voltage appearing across the c r y s t a l was measured with a K e i t h l e y model 153 electrometer, or a Fluke model 8120A d i g i t a l voltmeter. The form of the photoacoustic voltage f o r DC5  as a function  of l i g h t i n t e n s i t y i s shown i n FIG 27(a). The corresponding short c i r c u i t photoacoustic current had a maximum value % 0.1 mA/cm , and mirrored the 2  l i g h t i n t e n s i t y dependence of the photoacoustic voltage. For comparison,  p a s s i v e cavity active cavity incident light r D.C. meter  oscillator or D.C. power supply  _ L _ . FIG 26. Experimental configuration f o r measuring the photoacoustic voltage.  119  FIG 27. (a) Photoacoustic  voltage measured across b]_ c r y s t a l 29.04.03.02  (pass ive part of DC5) illuminated with 4880 A l i g h t . The a c t i v e part of DC5 (bj_ c r y s t a l 29.04.01.02) was operated i n mode-locked fashion, under the conditions: applied D.C. e l e c t r i c f i e l d = 1.35 KV/cm, D.C. current density = 15 mA/cm , fundamental 2  acoustic frequency =18.2  MHz.  (b) Dember voltage as a function of l i g h t i n t e n s i t y at 5145 A for bj_ c r y s t a l 33.09.01.01 (part of DC2).  120  o  1  ————————  o  (a)  - i  1  1  5 10 15 LIGHT INTENSITY (mW/cm ) 2  io  1b  2  i b  1  i b  LIGHT INTENSITY (mW/cm ) 2  1  121 the Dember voltage for bj_ c r y s t a l 33.09.01.01 i s presented  i n FIG 27(b).  The photoacoustic voltage was 3 to 4 orders of magnitude l a r g e r than the corresponding Dember voltage. For strongly absorbed l i g h t , the sign of the photoacoustic voltage was generally the same as the Dember voltage, i . e . the illuminated side of the passive c a v i t y was p o s i t i v e with respect to the dark side. However, anomalous sign r e v e r s a l s could be observed at low s t r a i n amplitudes and f o r c e r t a i n acoustic frequencies. The s p e c t r a l response of the photoacoustic voltage generally took the form i l l u s t r a t e d i n FIG 28(a), and displayed strong c o r r e l l a t i o n with the response of the photoconductivity, FIG 28(b). This seems to indicate that the e f f e c t i s proportional to the number of mobile c a r r i e r s . When the a c t i v e c r y s t a l was driven by an external o s c i l l a t o r , the photoacoustic voltage only appeared at multiples of the r e c i p r o c a l of the round t r i p t r a n s i t time of the composite c a v i t y (~ 1.28 MHz for DC5) . This i s i l l u s t r a t e d i n FIG 29. At higher c o n d u c t i v i t i e s ( i . e . higher  light  i n t e n s i t i e s ) , the peak of the photoacoustic voltage s h i f t e d to lower frequencies [compare curves (1) and (2) i n FIG 29], i n agreement with the known conductivity tuning c h a r a c t e r i s t i c s of CdS phonon masers (Burbank 1971). When the passive c r y s t a l was illuminated a t - 5200 A [the peak wavelength i n FIG 28(a)] i t was possible to achieve  photoacoustic  voltages of up to 15 v o l t s . 3.2.1  Discussion While the underlying p h y s i c a l mechanisms that produce the  photoacoustic voltage are poorly understood, two q u a l i t a t i v e models \ deserve mention:  122  FIG 28. (a) The spectral response of the photoacoustic voltage f o r b_[_ c r y s t a l 29.04.03.02 (part of composite cavity DC5) . The a c t i v e part of DC5 (bj_ c r y s t a l 29.04.01.02) was driven at 7.55 MHz by an external  oscillator.  (b) The s p e c t r a l response of the photocurrent f o r b]_ c r y s t a l 29.04.03.02 under the same conditions  of i l l u m i n a t i o n as ( a ) .  The l i g h t was provided by a 200 W tungsten halogen lamp i n series with a Heath monochromator.  124  12.76 MHz  FIG,29. Behavior of the photoacoustic voltage f o r DC5 as a function of the A.C. d r i v i n g f i e l d applied cavity. Weakly absorbed  (orange) l i g h t was used. The curves f o r  two d i f f e r e n t conductivities a « (1) 2 x 10"" (ftcm) 4  -1  to the a c t i v e part of the composite  ( l i g h t i n t e n s i t i e s ) are presented:  and (2) 4 x 10~ (ftcm) . 7  -1  125 (1) "Mode-pulling" The active modes i n a phonon maser must s a t i s f y the phase condition (Vrba and Haering 1973): ftd( - + — ) + 20 s+ sv  =  2mr  (3.1)  v  where n i s an integer, d i s the cavity length, v  ( _) v  g +  of sound p a r a l l e l ( a n t i p a r a l l e l ) to the applied D.C.  s  1 S  th  e  velocity  electric f i e l d . It  i s well known, from the l i n e a r theory of acoustic a m p l i f i c a t i o n i n p i e z o e l e c t r i c semiconductors (see, f o r example, McFee 1966), that the v e l o c i t y of sound i s a function of both conductivity and e l e c t r i c  field.  This forms the q u a l i t a t i v e basis f o r the tuning c h a r a c t e r i s t i c s of phonon masers, since varying the conductivity or e l e c t r i c f i e l d the v e l o c i t y of sound  modifies  and requires the phonon maser to adjust i t s  frequency i n order to s a t i s f y  (3.1).  In the present case, i l l u m i n a t i o n modifies  the conductivity  i n the passive part of the composite cavity, thus changing the v e l o c i t y of sound and s p o i l i n g the resonance condition, equation (3.1). I t i s conceivable  that resonance can be restored, however, i f the c r y s t a l  establishes a D.C.  e l e c t r i c f i e l d of the proper magnitude, thus returning  the v e l o c i t y of sound to i t s value before i l l u m i n a t i o n . This highly q u a l i t a t i v e model does not e s t a b l i s h the sign of the voltage.  photoacoustic  126 (2) Electron transport In t h i s model we assume that the p i e z o e l e c t r i c p o t e n t i a l wells associated with the acoustic wave act as "buckets" to transport o p t i c a l l y created electrons from the illuminated surface to the dark surface. The r e s u l t i n g D.C. voltage should have the same sign as the corresponding Dember voltage. This model does not r e a d i l y account for voltages of opposite sign to the Dember voltage, or f o r e f f e c t s observed with weakly absorbed light. I t has not yet been possible to form a conclusive l i n k between the experimental r e s u l t s and the preceeding models.  127 3.3  The A.C. E l e c t r i c F i e l d Induced Photovoltaic E f f e c t A second D.C. voltage, requiring the simultaneous presence of  l i g h t and an externally applied A.C. e l e c t r i c f i e l d , has been discovered in CdS c r y s t a l s . The experimental configuration f o r measurement of t h i s photovoltage i s shown i n FIG 30. The capacitor C^ i n FIG 30 served to D.C.  decouple the external o s c i l l a t o r from the rest of the c i r c u i t .  The combination of L and  formed a low pass f i l t e r to protect the  D.C. voltmeter from the A.C. d r i v i n g voltage. The samples used i n these experiments were polished phonon maser c r y s t a l s and hence had In d i f f u s e d surfaces (see APPENDIX B) . The s p e c t r a l response of the D.C. voltage i s shown i n FIG 31. Similar to the photoacoustic voltage, i t contained one dominant peak, O  s h i f t e d ~ 100 A to shorter wavelengths with respect to the peak i n FIG 28(a). Also, the sign of the voltage generally reversed f o r waveo  lengths £ 5250 A. As a function of l i g h t i n t e n s i t y , the D.C. voltage  incident  r  1  CdS crystal  . '  light L vwsw-  if  1  J  1  L  ,  . external I V i oscillator l / x  ~1 D.C. meter  FIG 30. Experimental configuration f o r measuring the A.C. e l e c t r i c induced photovoltage.  field  128  5108A  o o  LO LO  WAVELENGTH O F ILLUMINATION (A) FIG 31. Spectral response of the D.C. photovoltage f o r bj_ c r y s t a l 29.04.01.02 (part of DC5). The i n t e n s i t y of i l l u m i n a t i o n was ~ 0.5 mW/cm. The l i g h t was provided by a 200 W tungsten halogen 2  lamp i n series with a J a r r e l l Ash monochrometer.  129 saturated very quickly, i n a manner similar to FIG 27. I t displayed a quadratic dependence on the amplitude of the A.C. d r i v i n g voltage, as i l l u s t r a t e d i n FIG 32(a), and as the d r i v i n g frequency was increased the D.C.  voltage f e l l o f f quickly, as i s shown i n FIG 32(b). The D.C.  photovoltage appeared at a l l frequencies, i n d i c a t i n g  that i t s presence was probably not related to the resonant acoustic structure of the c r y s t a l . At frequencies corresponding to acoustic resonances of the c r y s t a l , the photoacoustic voltage was also present. In FIG 33(a) the D.C.  photovoltage i s shown as a function of frequency  for i l l u m i n a t i o n with small amounts of weakly absorbed l i g h t . As the l i g h t l e v e l (and hence the conductivity) was reduced, the photoacoustic voltage became dominant [compare curve (3) i n FIG 33(a) with curve (2) i n FIG 29]. For higher l i g h t l e v e l s , and p a r t i c u l a r l y i n the case of strongly absorbed l i g h t , the nonresonant  photovoltage dominated,  as i l l u s t r a t e d i n FIG 33(b). To further investigate the behavior of this photovoltage, pulse measurements were performed at temperatures  from 5°K to 300°K.  The c r y s t a l s used were mounted i n a v a r i a b l e temperature LHe dewar, and were illuminated with white l i g h t provided by a 100 W Hg arc lamp. When the l i g h t was shut o f f , by means of a mechanical shutter, an  R.F.  pulse, whose width and delay time with respect to the end of the o p t i c a l s i g n a l could be v a r i e d , was applied to the c r y s t a l . The R.F. pulses were obtained from a Hewlett Packard model 3200B VHF o s c i l l a t o r , which was pulse modulated by a Hewlett Packard model 214A  pulse generator.  130  FIG 32. (a) The D.C. photovoltage as a function of the  amplitude of  the applied A.C. voltage (12 MHz) f o r bj_ c r y s t a l 29.04.01.02 o  (part of DC5). The i l l u m i n a t i o n was at 5100 A, with an i n t e n s i t y of about 0.5 mW/cm. The s o l i d curve through the experimental 2  points corresponds to V JJ • L» *  =  (2.78 x 10  )V  .  2  .A. # •  (b) The D.C. photovoltage as a function of d r i v i n g frequency at constant amplitude f o r bj_ c r y s t a l 29.04.03.02 (part of DC5), illuminated with a small amount of weakly absorbed light.  (orange)  131  O  5 10 15 20 APPLIED A.C. VOLTAGE (V ) PP  0  5  10 15 20 25 DRIVING FREQUENCY (MHz)  132  FIG 33. (a) The D.C. photovoltage f o r bj_ c r y s t a l 29.04.01.02 (part of DC5) as a function of frequency, f o r low l e v e l s of weakly absorbed  (orange) l i g h t . Curves f o r three d i f f e r e n t c o n d u c t i v i t i e s  ( l i g h t levels) are presented: a - (1) 1.4 x 10~^(ftcm)~ , 1  (2) 5 x 10~ (ftcm) 5  _1  and (3) 4 x 10~ (flcm)" . 7  1  (b) The D.C. photovoltage f o r b|_ c r y s t a l 29.04.01.02 as a function of frequency (the d r i v i n g amplitude was not constant). o  The c r y s t a l was illuminated with about 0.5 mW/cm at 5100 A. 2  The separation of the small negative peaks i s equal to the r e c i p r o c a l of the round t r i p t r a n s i t time of DC5 (- 1.28 MHz).  134 The t r a i l i n g edge of the o p t i c a l s i g n a l and the form of the R.F. pulse are shown i n FIG 34(a). For a given set of conditions, i t was  found  that both the l i f e t i m e and amplitude of the pulse generated voltages increased by a factor of 2-3 i n cooling the samples from room temperature to 100°K. At temperatures much below 70°K, however, the signals r a p i d l y became weaker, and were almost nonexistent at 5°K. The voltage signals generated at - 80°K by R.F. pulses of d i f f e r e n t delay times are shown i n FIG 34(b). I f the time f o r generation tg was measured from the end of the o p t i c a l signal [see lower traces i n FIG 34(a), ( c ) , (id)] the - t fx amplitude of the generated voltage decayed roughly as e x ~ 10ms.  8  , where  The decay time of the voltage dependended strongly on the  l e v e l of i l l u m i n a t i o n . As i s shown i n FIG 34(c) and (d), the amplitude of the generated voltage increased, while the decay time of the voltage decreased, with increasing l i g h t 3.3.1  intensity.  Discussion Even less i s understood about the p h y s i c a l processes involved  i n t h i s photovoltage than i n the case of the photoacoustic voltage. Its dominant c h a r a c t e r i s t i c s , however, do not appear to be strongly correlated with the acousto-electric properties of the CdS samples.  The  c h a r a c t e r i s t i c decay time i l l u s t r a t e d i n FIG 34(b) i s approximately the same s i z e as the electron l i f e t i m e , as determined using A.C. photoconductivity measurements. This suggests that the photovoltage i s associated with mobile rather than trapped charge. The behavior i l l u s t r a t e d i n FIG 34(c) and (d) i s consistent with the c r y s t a l behaving as a battery with a  135  FIG 34. Some r e s u l t s of the pulse measurements: (a) lower trace shows t r a i l i n g edge of (negative) s i g n a l produced by sample i l l u m i n a t i o n ; upper trace shows the pulse (11 MHz),  R.F.  i n t h i s case having a length of 200 usee and  a delay of 10 msec with respect to the end o f the o p t i c a l s i g n a l , (b) voltages produced by R.F. pulses of 1 msec width, applied at various delay times to c]_ c r y s t a l 33.05.01.01 (T = 80°K), and voltages produced i n cj_ c r y s t a l 33.04.02.00 (upper traces) by 10 msec delayed R.F.  pulses (width 200 usee) at T - 100°K for  (c) weak i l l u m i n a t i o n and  (d) i n t e n s i t y of i l l u m i n a t i o n increased  by a factor of 10 with respect to ( c ) .  136  H o r i z o n t a l s c a l e : 5ms/div,  H o r i z o n t a l s c a l e : 2ms/div. Upper t r a c e - l O V / d i v . Vertical scale: _ c,,/j.„ Lower t r a c e - 0.5V/div.  Vertical scale:  0.5V/div.  n  (a)  (b)  Horizontal scale:  H o r i z o n t a l s c a l e : 2ms/div. Vertical  scale:  Upper Lower  (c)  trace trace  20mV/div. 0.5V/div,  Vertxcal  scale:  2ms/div.  Upper L  o  w  e  r  (d)  trace t  r  a  c  e  0.2V/di 0.5V/di  137 v a r i a b l e i n t e r n a l impedance. At high l i g h t l e v e l s , the conductivity of the c r y s t a l i s larger. Hence i t s i n t e r n a l impedance i s smaller and i t "discharges" more quickly than f o r low l i g h t l e v e l s .  138 3.4  Conclusions and Summary of Contributions Two new photovoltaic-type e f f e c t s have been discovered i n  CdS. The voltages incurred are remarkable for t h e i r magnitude: photoacoustic voltages of about 15 V and A.C.  electric field  induced  photovoltages of about 25 V were achieved with very small l i g h t l e v e l s ( £ 1 mW/cm ). These voltages are several orders of magnitude larger 2  than the normal photovoltages encountered  at s i m i l a r l i g h t l e v e l s i n CdS.  At t h i s time, the experimental r e s u l t s are not understood i s warranted  i n t h i s area.  and more work  139 APPENDIX A  ACOUSTIC STANDING WAVE PATTERN FOR MODE-LOCKED OPERATION  In p r a c t i c e , the s t r a i n p r o f i l e s of the phonon maser were observed by acoustic coupling to a passive c a v i t y of fused This allowed  quartz.  the s t r a i n p r o f i l e to be o p t i c a l l y probed i n the passive  c a v i t y , without disturbing the phonon maser (refer to section 1.4 of CHAPTER 1). The acoustic s t r a i n at a point x i n the passive c a v i t y , represented  i n FIG A l , may be written as a superposition of an incident  waveform F^, coupled  from the phonon maser, and a waveform F^_, which  •x = 0  "V  --X = x  d-x  r  \.J.  ,-ir  x = d  FIG A l .  has been s p a t i a l l y r e f l e c t e d from the surface x=d of the passive cavi We make the assumption that i n steady state a l l waves moving i n the , same d i r e c t i o n maintain the same phase at any plane x = constant, f o r a l l times ( i . e . no anomalous e f f e c t due to multiple r e f l e c t i o n s ) .  140 Then, to o b t a i n the c o r r e c t r e l a t i v e phase f o r F^ and F , i t may inferred  from FIG A l t h a t i t i s n e c e s s a r y to s e t x  2d-x  Assuming p e r i o d i c form f o r the a c o u s t i c s t r a i n , F x^ritten i n Fourier series  F (x,t) ±  where a , n  b  for F . may  be  form:  = I a cos[n(Kx-Slt)] + b sin[n(Kx-ftt) ] n=l^ n  (Al)  n  a r e the u s u a l F o u r i e r c o e f f i c i e n t s and K, ft a r e the wave  n  number and a n g u l a r f r e q u e n c y of the a c o u s t i c fundamental. term has been o m i t t e d i n ( A l ) from series,  be  s i n c e no D.C.  (A c o n s t a n t  the g e n e r a l form o f the F o u r i e r  s t r a i n s a r e a s s o c i a t e d w i t h the a c o u s t i c waves.)  Assuming no l o s s e s upon r e f l e c t i o n , i t i s apparent  from  the p r e c e e d i n g  d i s c u s s i o n that  F (x,t) r  =  F (2d-x,t) ±  = I a cos[nK(2d-x)-nftt] + b sin[nK(2d-x)-nftt] n=l <• n  n  a cos(2nKd)cos[n(Kx+ftt)]  I n=l  n  + a s i n ( 2 n K d ) sin[n(Kx+ftt)] n  + b sin(2nKd)cos[n(Kx+fit)] - b cos(2nKd)sin[n(Kx+ftt)] n  CO =  where  I n=l  c  n  n  «.  r  c cos[n(Kx+Slt)] + n  (A2)  d sin[n(Kx+f2t)] n  J  =  a cos(2nKd) + n  b sin(2nKd) n  (A3) d  n  =  a sin(2nKd) - b cos(2nKd) n  n  141 We now introduce phases 6  a  =  n  i n equation (Al) such that  n  S sin6 n  n  (A4) b_n  where  a n d  =  tan6  S  n  Sn cos6_n  =  n  =  a  a /b n  (A5)  n  <A6)  n + n b  Equation (Al) now takes the form:  F (x,t)  =  i  I S sin[n(Kx-ftt) + 6 ] n=l n  In a s i m i l a r manner, phases B  c  =  n  n  n  (A7)  may be defined i n equation (A2) such that:  C sinB n  n  (A8) d  where  n  =  tan3  n  C cosB n  =  n  c /d n  n  Making use of equation (A3) one obtains:  (A9)  142 [a cos(2nKd) + b sin(2nKd)] + 2  n  a  Hence  n  + b n  2  2  n  —  S  [a sin(2nKd) - b c o s ( 2 n K d ) ] n  n  from equation (A6)  n n  (A10)  n  Using equation (A3) i n equation (A9):  tan8  a cos(2nKd) + n  b sin(2nKd) n  r  a sin(2nKd) - b cos(2nKd) n  n  tan6  + tan(2nKd)  n  tan6 tan(2nKd) n  - 1  using equation (A5)  - t a n ( 6 + 2nKd) n  Hence  g  n  =  - <n 6  +  2 n K d  With (A10) and ( A l l ) , equation (A2) now  F (x,t)  =  r  (All)  >  takes the form:  + I S sin[nK(x-2d) + nftt n=l n  6] R  (A12)  From equations (A7) and (A12), the t o t a l acoustic s t r a i n may be written:  F(x,t)  =  I S n=l  n  sin[n(Kx-ftt)+6_] + sin[nK(x-2d)+nftt-S ]  (A13)  2  143 The boundary at x=d may be considered a free surface.. Thus, for the one-dimensional  system considered, both the s t r e s s and s t r a i n  must vanish a t x=d. This can be s a t i s f i e d by F(x,t) only i f the +sign i s chosen i n equation  (A13). Equation  (A13) may then be w r i t t e n i n the  alternate form: CO  F(x,t)  =  2£ S sin[nK(x-d)]cos(nKd-nftt+6 ) n=l n  n  (A14)  The boundary at x=0 i s not f r e e since the passive c a v i t y i s bonded to the phonon maser at t h i s point. The value of the s t r a i n at x=0 i s determined by the parameters K,ft,5 and d i n equation (A14). The acoustic n  modes p a r t i c i p a t i n g i n mode-locked operation may be considered as being harmonically r e l a t e d . Thus, f o r this case, the upper l i m i t of the summation i n equations acoustic modes N.  (A13) and (A14) w i l l equal the number of a c t i v e  144 APPENDIX B  PHYSICAL PROPERTIES  Material Parameters For CdS: Density  1  (10 Kg/cm ) 3  p = 4.820  3  E l a s t i c s t i f f n e s s constants C  ll  c  3 3  =  »  9 , 0 7  c  i2  = 9.38, c  =  -  5  1  8 1  (10 »  c  i3  =  1 5  = -0.21, e  newton/m ) 2  5.10,  = 1.504  4 4  P i e z o e l e c t r i c stress constants"'" e  1 0  3 1  (coulomb/m ) 2  = -0.24, e  3 3  D i e l e c t r i c constants (constant s t r a i n ) = 9.02, K Refractive indices n  = 9.02, <  2  p  n  P  3 1  = 0.11, p  1 2  = 0.050, p  3 3  Electro-optic constants ' 5  42  Auld (1973) 2  Neuberger (1969)  = 9.53  2.491  e 3,4 constants  Photoelastic  1  2  = 2.506, n  Q  3  = 0.44  "c  6  = 0.051, p  1 3  = 0.072,  = 0.13, p  4 4  = 0.054  (10~  12  m/V)  z r\<-» V C^r^) "13 ^i ^ "33 e  5• 5j "33 z,  Dixon (1967) 4 Maloney and Carleton (1967)  2z  f  Gainon (1964) 6 Kaminow (1968)  145 M a t e r i a l Parameters For Fused Quartz: o  Density (10  3  Kg/m ) 3  p =  2.2  =  1.46  3 Refractive index  n  Q  3 Photoelastic constant  p^ 3  Shear acoustic v e l o c i t y  = 0.075 3  (10^ m/sec)  Longitudinal acoustic v e l o c i t y ( 1 0 3  The  3  v  g  =  m/sec)  3.76 v^ =  5.95  techniques used for f a b r i c a t i o n of phonon masers have been  d e t a i l e d by Burbank (1971). The CdS  c r y s t a l s were oriented by means of  X-ray d i f f r a c t i o n to an accuracy of ±0.5°. For t y p i c a l sample dimensions, a f l a t n e s s of from 1/5  to 1/10  of an o p t i c a l wavelength could be achieved  for the polished surfaces. Transparent e l e c t r i c a l contacts were provided by means of In d i f f u s i o n . Techniques s i m i l a r to those used for phonon masers were applied to f a b r i c a t e acoustic c a v i t i e s from fused sapphire,  quartz,  and Barium F l u o r i d e . Composite acoustic c a v i t i e s were bonded  together using the technique outlined i n APPENDIX C. The p h y s i c a l data for the samples mentioned i n the main text of the thesis are l i s t e d i n TABLE B l .  146 TABLE B l  -  Physical Data For Acoustic Cavities t  Acoustic Cavity 24.01.02.04 DCl FQ1 (fused quartz)  33.09.01.01  Dimensions  (mm)  Surface F i n i s h polished wire saw In diffused  d = 1.193, b-axis £ = 2.166 w = 1.999  X  d = 3.306 £ = 10.781 w 10.760  X X  d = 1.090, b-axis £ = 3.541 w = 2.775  X  FQ2 (fused quartz)  d = 4.414 £ = 7.844 w 7.778  X X X  24.07.10.01  d = 0.830, b-axis £ = 2.228 w = 2.022  X  d  X  DC2  DC3 SI  (sapphire)  3.351, x,-axis  £ = 10.949 w = 7.087  27.02.04.02  27.01.01.02  29.04.01.02 DC5 29.04.03.02  J.  d  X X  X X X  X  X  0.744, c-axis 3.252 w = 2.288  X X X  X  d = 0.819, c-axis £ = 3.579 w 3.207  X X  X  d = 0.318, b-axis £ = 3.210 w = 1.976  X  d  X  £  DC4  X X X  :  0.379, b-axis  £ = 4.731 w = 4.132  X X X X X  Unless otherwise indicated, the acoustic c a v i t i e s are CdS c r y s t a l s . k  I  bj_ c r y s t a l 33.08.06.00 was l a t e r bonded to DC2 to form a t r i p l e c a v i t y .  147  Acoustic Cavity t 24.06.06.01  33.04.02.00  33.05.01.01  Dimensions  d £ w  (mm)  polished  1.265, c-axis 7.225 2.170  X X X  d = 0.376, c-axis £ = 3.111 w = 1.838  X  d = 0.854, c-axis £ 4.037 w 2.289  X  —  Surface F i n i s h wire saw I In diffused  X  X X  X X  X  148 APPENDIX C -  ACOUSTIC BONDS  To provide high q u a l i t y acoustic bonds between a CdS phonon maser and a second acoustic cavity (buffer), a cold welding  procedure  s i m i l a r to that of S i t t i g and Cook (1968) was employed (Hughes 1974). The two samples to be bonded were f i r s t thoroughly cleaned. This step was  c r u c i a l since the evaporated layers that formed the bond were only o  ~ 3000 A thick. Following thorough u l t r a s o n i c washes i n acetone and subsequent r i n s i n g i n d i s t i l l e d water, a f i n a l cleaning was performed i n a laminar flow box, using lens tissues moistened with ether. An o p t i c a l flat  (also c a r e f u l l y cleaned) was used to check f o r the presence of dust  p a r t i c l e s using interference techniques. Systematic wiping of the surfaces was continued u n t i l no interference fringes could be observed, i n d i c a t i n g that a l l dust p a r t i c l e s :> 1000 A had been removed. •  Following cleaning, the samples were mounted on a press i n an —6 evaporator. When a vacuum of about 3 x 10 t o r r was achieved, a thin o  o  layer (~ 100 A) of Cr followed by a 1000 A layer of Au were evaporated o  on the b u f f e r , to form a durable e l e c t r i c a l contact. 2000 A of In was then l a i d down on the phonon maser, thus making e l e c t r i c a l contact with i t s d i f f u s e d In surface. Following evaporation, and while s t i l l under vacuum, the evaporated surfaces of the two samples were pressed together  149 with a pressure - 200 Kg/cm . After allowing several hours f o r the bond 2  to form, the composite  cavity could be removed from the evaporator. In  the region of the bond the Au layer took on a s i l v e r color, i n d i c a t i n g that a l l o y i n g had taken place with the In layer. Using t h i s technique, CdS phonon masers were successfully bonded to buffers of AI2O3, fused S i 0 , BaF 2  2  and CdS.  150 REFERENCES  Abramowitz, M. and Segun, I,A., 1968. Handbook of Mathematical Dover, N.Y. 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