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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 Brit ish Columbia, I agree t h a t 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 Department o r by his representatives. It is understood that c o p y i n g o r p u b l i c a t i o n o f this thesis for financial gain shall not be allowed without my written permission. Department of P H Y S I C S The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date MOV- 2 ] |<f7k ABSTRACT The nonlinear theory of acoustic amplification i n piezoelectric 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 acousto-electric amplifying mechanism. The f i r s t direct evidence of mode-locking i n the phonon maser has been provided by applying optical processing techniques to signals obtained from laser d i f f r a c t i o n . A new and powerful technique for the analysis of optical signals, involving a combination of spatial and temporal f i l t e r i n g , has been developed. Further application of optical processing has allowed the f i r s t direct observation of acoustically induced space charge gratings i n CdS. Proper exploitation of the methods outlined should give insight into the underlying physical processes. In addition to the optical experiments, data has been presented I hat demonstrates the existence of two new photovoltaic effects i n CdS. The resulting photovoltages are several orders of magnitude larger than lhose produced by known photovoltaic effects. As yet, the physical processes involved are not properly understood. i i TABLE OF CONTENTS Page List of Tables v Li s t of Figures . v i List of Symbols x Acknowledgements xiv Foreword xv CHAPTER 1: Mode-Locking in CdS Phonon Masers 1 1.1 Introduction 1 1.2 The Phonon Maser as a Repetitive Pulse Generator 3 1.2.1 Nonlinear Gain Theory - The Saturable Absorber ... 7 1.3 Theory of the Optical Measurements 25 1.3.1 Crystal Optics - The Photoelastic Effect 25 1.3.2 Interpretation of the Optical Signal -Preliminary Considerations 35 1.3.3 Interpretation of the Optical Signal -The Mode-Locked Phonon Maser 48 1.4 Experimental Results 62 1.4.1 Experimental Apparatus and Techniques ; • • • ^ 6 1.4.2 Optical Verification of Mode-Locking 73 1.5 Optical Determination of the Normal Modes of Composite Cavities .... 89 1.6 Conclusions and Summary of Contributions 92 i i i Page CHAPTER 2: Application of Optical Processing to a Study of Acoustically Induced Space Charge Gratings i n CdS 93 2.1 Introduction 93 2.2 Application of Optical Processing 99 2.3 Experimental Results 107 2.4 Conclusions and Summary of Contributions 116 CHAPTER 3: New Photovoltaic Effects in CdS 117 3.1 Introduction 117 3.2 The Photoacousto Voltaic Effect 118 3.2.1 Discussion 121 3.3 The A.C. Electric Field Induced Photovoltaic Effect 127 3.3.1 Discussion 134 3.4 Conclusions and Summary of Contributions 138 APPENDIX A: Acoustic Standing Wave Pattern for Mode-Locked Operation 139 APPENDIX B: Physical Properties 144 APPENDIX C: Acoustic Bonds 148 REFERENCES 150 BIBLIOGRAPHY - Diffraction of Light by Ultrasonic Waves 153 iv LIST OF TABLES Table Page 1 Experimentally determined strain amplitudes and phases 82 for mode-locked operation of DC1 Bl Physical data for acoustic cavities 146 v LIST OF FIGURES Figure Page 1. (a)Block diagram for the repetitive pulse generator 4 (b)Qualitative i l l u s t r a t i o n of saturable absorber operation 5 2. Nonlinear gain as a function of |A| 20 3. Behavior of the phonon maser as a function of conductivity 21 and electric f i e l d 4. The indicatrix construction 27 5. Form of the index ellipse in an isotropic material 33 6. Schematic representation of optical system to be studied 37 7. Intensity of zero and f i r s t order diffraction for various 47 values of the parameter Q 8. Intensity pro f i l e of the dark f i e l d term 56 9. Theoretical image plane profiles of Q component for combined 60 spatial orders 4,5,6 ; 10. Display of' the acousto-electric current signal for b|_ crystal 63 24.01.02.04 11. (a)Current density vs. applied electric f i e l d for bj_ crystal 64 24.01.02.04 (b)Time display of acousto-electric current for points 65 indicated in FIG 11(a) 12. Experimental configuration for the optical measurements 67 v i Figure 13. (a)Mounted composite cavity (b) Typical focal plane diffraction pattern (c) Focal plane spatial f i l t e r 14. Photographs and traces of the image plane intensity distribution for DC1 15. D.C. dark f i e l d term for DC1 16. Experimental and corresponding theoretical image plane intensity profiles for DCl under different conditions of spatial and temporal f i l t e r i n g 17. Acoustic strain p r o f i l e reconstructed from experimental data 18. Frequency display of the acousto-electric current for DCl operated in a multimode manner 19. Mode structure of composite cavity DCl obtained by light diffraction 20. Optical data obtained from model system consisting of quartz transducers in a c e l l of d i s t i l l e d water 21. Intensity profiles obtained by imaging diffraction spots ±1 and 1,2 for cj_ crystal 24.06.06.01 22. Image plane signals for cj_ crystal 24.06.06.01 produced by imaging diffraction spots ±1 and 0,1 23. Image plane photographs for DC4 i l l u s t r a t i n g that various optical signals can be completely separated v i i Figure 24. Image plane signal for one part of DC4, produced by combining diffraction spots 0,1 both in the presence and absence of the external driving f i e l d 25. A typical glow curve obtained from a CdS crystal of the type used to construct phonon masers 26. Experimental configuration for measuring the photoacoustic voltage 27. (a)Photoacoustic voltage as a function of light intensity for bj_ crystal 29.04.01.02 (b)Dember voltage as a function of light intensity 28. (a)Spectral response of photoacoustic voltage for bj_ crystal 29.04.03.02 (b)Spectral response of the photocurrent for bj_ crystal 29.04.03.02 29. Behavior of photoacoustic voltage as a function of frequency, for DC5 30. Experimental configuration for measuring the A.C. electric f i e l d induced photovoltage 31. Spectral response of the D.C. photovoltage for bj_ crystal 29.04.01.02 32. (a)D.C. photovoltage as a function of A.C. driving amplitude (b)D.C. photovoltage as a function of driving frequency at constant amplitude v i i i 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, for strongly absorbed light 34. Some results of pulse measurements on the D.C. photovoltage 135 ix LIST OF SYMBOLS nonlinear parameter in acoustic gain theory optical amplitude total dark f i e l d amplitude focal plane optical amplitude th optical amplitude i n n diffraction spot near f i e l d (object plane) optical amplitude near f i e l d optical amplitude in absence of acoustic waves nonlinear acoustic gain; + parallel and - antiparallel to applied electric f i e l d components of the relative dielectric impermeability tensor and matrix, respectively change in the parameters B^ parameter indicating the degree of spatial synchronization of Fourier components IJJ elastic stiffness constant electric displacement thickness of acoustic cavity electron diffusion constant relative phases of mode-locked acoustic waves elect r i c f i e l d piezoelectric stress constant synchronous electric f i e l d = v_/u • s total electric f i e l d permittivity constant F acoustic trapping fraction T l a t t i c e loss coefficient Y acoustic wave velocity parameter = 1 - v n / v s » o r normalized angle of incident light (section 1.3.2) Y + acoustic wave velocity parameter: + pa r a l l e l and -antiparallel to applied electric f i e l d I total current density Ijyp total dark f i e l d intensity distribution """n. n+1 n+2 optical intensity distribution produced by combining ' ' diffraction orders n, n+1, n+2 •""n n+1 n+2 image plane signal obtained with spatial orders n, n+1, ' ' n+2 and temporal f i l t e r at 1^ Q image plane signal at frequency <% produced by imaging ' diffraction spots 0,1 I g synchronous current = n 0 c l v s J conduction current density J Bessel function of order n n K acoustic wave number AK acoustic phase shift produced by piezoelectric coupling k optical wave number k* = ksin6, where 6 is angle of diffraction principal dielectric constants L width of the acoustic cavity A acoustic wavelength X optical wavelength M amplitude modulation index x i e l e c t r o n m o b i l i t y conduction band e l e c t r o n density, or index of r e f r a c t i o n change i n e l e c t r o n density or index of r e f r a c t i o n 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 semiaxes of index 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 presence of s t r a i n s t r a i n induced changes i n index e l l i p s e semiaxes eq u i l i b r i u m e l e c t r o n density, or unperturbed r e f r a c t i v e index ordinary and extraordinary i n d i c e s of r e f r a c t i o n a c o u s t i c a l l y modified ordinary and extraordinary i n d i c e s of r e f r a c t i o n n o - no» n e ~ n e acoustic frequency o p t i c a l 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 frequency as modified by nonlinear theory d i f f u s i o n frequency = v|/D n modulation frequency components of the o p t o e l a s t i c matrix phase fu n c t i o n d e s c r i b i n g mode-locked ac o u s t i c waveform phase angle, or angle of incidence a c o u s t i c phase fu n c t i o n phase fu 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 of o p t i c a l amplitude x i i ij>n(z) Fourier components of ^(x,z,t) Q parameter characterizing diffraction q electron charge r r e f l e c t i v i t y of acoustic cavity walls p mass density S, strain amplitude a e l e c t r i c a l conductivity T acoustic stress Tn,n+l,n+2 component of I ^ - ^ at frequency a Tn,n+l,n+2 F o u r i e r transform of ^ fftnflri-2 8 phase angle = Kx - Qt, or angle of diffraction linear combination of acoustic phases = ^n-^n+l+^n+2 u mass displacement V = 2vcosQt V = (2vcosfit + v Q ) ( l + Mcosoijjjt) V n = 2vncos(nKd-nnt+6n) v, v n Raman - Nath parameter Vp electron d r i f t velocity v Q amplitude of optical phase variation produced by static space charge grating v g velocity of sound z^ _. components of the electro-optic matrix C phase parameter = (n QkL - wt) 0 n x i i i 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 in the i n i t i a l stages of the development of a theoretical model for the mode-locked phonon maser. I am also grateful to Lore Hoffmann for her capable assistance in the preparation of publications. The financial support of the National Research Council, the International Nickle Company, and the University of British Columbia is gratefully acknowledged. Finally, I thank my wife Marg for her continuing support and for her assistance in typing this thesis. xiv FOREWORD Chapters 1 and 2 document the results of optical diffraction and signal processing experiments, and contain the most important contributions of the thesis. Chapter 3 deals with new photovoltaic effects that are not yet understood. This chapter i s included as a permanent record of the experimental results, and to provide a base for future work. xv CHAPTER 1 MODE-LOCKING IN CdS PHONON MASERS 1.1 Introduction Phonon masers are solid state acoustic oscillators produced by polishing the faces of piezoelectric single crystals accurately f l a t and par a l l e l , to form high Q resonant structures which are strongly analogous to lasers. The most successful phonon masers have been fabricated from CdS. The application of a D.C. electric f i e l d of sufficient magnitude creates a situation in which net round-trip acoustic gain can be achieved by virtue of the piezoelectric 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 oscillation i s then spontaneously b u i l t up from the thermal background. The linear theory of acoustic amplification i n piezoelectric semiconductors (White 1962) has been well established, and reviews may be found in the literature (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. electric f i e l d , but there i s also 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 in piezoelectric potential wells produced by the acoustic wave. Maines and Paige (1970) reported that phonon masers operated under certain experimental conditions exhibited sharp spiking i n the time display of 2 the A.C. current. The frequency display of the current signal consisted of a harmonic series having amplitudes constant i n time, and a frequency spacing equal to the reciprocal of the round tr i p transit time of the crystal cavity. 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 optical lasers, predicted that the acoustic output should consist of narrow, high strain pulses. The v a l i d i t y of this prediction i s , however, far 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 piezoelectric potential associated with these f i e l d s i s much larger than the thermal energy, a purely Sinusoidal acoustic wave can result in an electron distribution with r i c h harmonic content (Gulayev 1970, Gurevich 1969). The nonlinear nature of the mode-locked regime i s clearly established by the presence of strong current saturation, indicating that the conduction electrons are trapped i n the potential wells associated with the acoustic wave, and are constrained to move with the velocity of sound. These considerations clearly indicate that the prediction of Maines and Paige (1970) should be verified by direct observation of the acoustic f i e l d s . This chapter deals with a unique set of optical experiments that have provided v e r i f i c a t i o n of mode-locked operation, by allowing direct observation of the resulting acoustic strain pulses. The general 3 method relies on a powerful combination of spatial and temporal f i l t e r i n g applied to optical signals obtained from laser diffraction. Before considering the optical measurements, i t is 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 class of devices known as Repetitive Pulse Generators. 1.2 The Phonon Maser. as.r.a Repetitive Pulse Generator It i s well known that a feedback loop containing the basic elements of an amplifier, f i l t e r , delay line, and a nonlinear element called a saturable absorber, behaves as a Repetitive Pulse Generator (RPG) (Cutler 1955). The mode-locked optical laser i s a well known example of such a device (Demaria et a l 1969). The basic block diagram for a RPG i s shown i n FIG. 1(a). The output consists of a train of identical pulses recurring at a rate determined by the loop delay, with shape determined principally by the characteristics of the f i l t e r and saturable absorber, and width controlled primarily by the bandwidth of the f i l t e r . The saturable absorber has a crucial role. It has the effect of emphasizing the highest amplitude parts i n the pulse circulating i n the feedback loop, while reducing the lower amplitudes. An element having the transmission characteristics shown in FIG 1 (a) w i l l serve the purpose. As indicated, intensities > I are passed almost unattenuated while lower ' o intensities are reduced. A highly qualitative i l l u s t r a t i o n of saturable absorber operation i s shown in FIG 1 (b). As a circulating pulse makes DELAY Intensity FIG 1(a).. Block diagram for the Repetitive Pulse Generator (RPG) . + r o c\i M n I I I + + + i i tN CM 1 1 + + I E A qualitative i l l u s t r a t i o n of the operation of a saturable absorber in both the time (upper) and frequency (lower) domains. Only the envelope of the pulses is shown in the time display. f c i s the center frequency or frequency of maximum gain, and f is the reciprocal of the round trip transit time T of the delay loop. 6 repetitive 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 limited by the bandwidth of the f i l t e r . The steady state pulse width i s approximately equal to the reciprocal of the system bandwidth. In the phonon maser, most of the required elements may be readily identified. The piezoelectric 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 profile of the phonon maser constitute the f i l t e r , and the round trip transit time of the resonant cavity serves as the loop delay. It remains to find something which w i l l provide the necessary saturable absorber effect. In the case of an ordinary optical laser, which also contains the above 3 basic elements, the saturable absorber must be physically added to the system for RPG operation. One thing that i s done, for example, is to place a c e l l of bleachable dye solution in the laser cavity. Such a dye becomes transparent at very high light intensities, thus providing the necessary characteristic indicated i n FIG 1 (a). For some lasers, 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 in a very incoherent manner, so that any pulses existing i n the system w i l l be broad and i l l - d e f i n e d . The pulse compression produced by addition of the saturable absorber forces the operating cavity modes into a fixed phase relationship. Hence the origin of the term mode-locking,In this manner picosecond optical pulses have been achieved.. For the phonon maser i t i s not possible to isolate a physical element which functions as a saturable absorber. We shall present a qualitative discussion which demonstrates that certain nonlinear properties of the acousto-electric amplifying mechanism provide the necessary behavior. 1.2.1 Nonlinear Gain Theory - The Saturable Absorber To establish that a mechanism which serves as the counterpart of the saturable absorber exists in the phonon maser, i t is necessary to formulate an expression for large amplitude gain. Since mode-locking is a strongly nonlinear process, closely related to such phenomena as D.C. current saturation (Smith 1963) and high electric f i e l d domain formation and propagation (Maines 1966, Haydl and Quate 1966), any attempt to extend the linear theory of acoustic amplification (White 1962) w i l l not produce results valid for this situation. The nonlinear theory of acoustic wave propagation in piezoelectric semiconductors has been discussed by many authors: Butcher (1971), Butcher and Ogg (1968, 1969, 1970), Gay and Hartnagel (1969, 1970), Gulayev (1970), Gurevich (1969), Tien (1968), Wanneberger (1970), as well as others. While the above treatments produce results that are largely similar, 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 results in a transparent and useful form. Since only qualitative expressions are required for 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 static approximation (ie. the curl of the elec t r i c f i e l d = 0) w i l l be employed. This has the effect of decoupling purely electromagnetic waves (propagating at the velocity of light) from acousto-electric waves (propagating at the velocity of sound). Since this corresponds to ignoring terms in the acoustic dispersion which are a factor of ~10 1 0 (the square of the ratio of the acoustic velocity to the velocity of light) smaller than other contributions (Hudson and White 1962), i t i s an . excellent approximation. We shall consider the case of uniform plane acoustic waves, propagating along the x axis.^ This reduces the problem to a one-dimensional situation. With these considerations the system of equations to be solved becomes (MKS units are employed) : Elastic wave equation: 3T = p 3^u (1.1) 8x 9t2 'This i s not as unrealistic as i t may appear. The physical situation of particular interest corresponds to pure shear waves whose wave vectors l i e along the crystallographic b-axis. Results of our optical experiments indicate that a one dimensional analysis i s quite reasonable i n this case. 9 Poisson's equation: 3D = q(n - n Q) 0-* 2 ) 8x P i e z o e l e c t r i c equations of st a t e : T = cS + eE (1.3) D = -eS + eE (1.4) Continuity equation: 3 J __ -q 9 (n - no) (15) 3x 3t Equation of current density: J = qnuE - q D n 3n ( 1 > 6 ) 3x Where T and S are the acoustic s t r e s s and s t r a i n , u i s the mass displacement, p i s the mass density, E and D are the e l e c t r i c f i e l d and e l e c t r i c displacement, q, u, D n are the magnitude of the ele c t r o n charge, the el e c t r o n m o b i l i t y and the 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, are the appropriate e l a s t i c s t i f f n e s s , 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 constants, n i s the conduction band e l e c t r o n density and n Q i s the eq u i l i b r i u m e l e c t r o n density. The sign convention of Butcher and Ogg (1968) i s employed i n 10 equations (1.1) - (1.6) : E, J and D are measured positive in the negative x direction. This implies that a positive D.C. el e c t r i c f i e l d (ie. pointing in the negative x direction) w i l l produce electron d r i f t , and ultimately acoustic amplification, in the positive x direction. In equations (1.2), (1.5) and (1.6) the effect of holes has been ignored. This i s a very good approximation for CdS which i s n-type and has a hole mobility and hole lifetime which are approximately 10 •L and 10 times smaller, respectively, than the corresponding electron parameters. The presence of the acoustic wave produces a variation An = (n-n Q) in the equilibrium electron density. It i s imp l i c i t l y assumed i n the preceeding equations that the space charge qAn is mobile. While this approximation i s satisfactory for our purposes, i t should be realized that qAn w i l l include a contribution due to modulation of the electron density in localized traps. Thus, in general, only a fraction 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: |_ ( J + | f ) = o (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 total current I, consisting of the conduction current J and the displacement current 3D/8t i s spatially invariant. ,This allows us to modify equation (1.6) in the form 11 I = qnyE + 9D - q Dn 3_n (1.8) 3 t 3x where, according to equation (1.7), I can at most be a function of time. In fact, as w i l l be indicated shortly, I may be assumed to have a constant value in the calculations to be performed. S, T, E, n and D are assumed to have plane wave dependence of the form D = D„ + D i e 1 6 + c c . (1.9) as ill u s t r a t e d for the electric displacement. Here 6 = Kx - fit, where K, £2 are the real wavenumber and angular frequency of the wave. D Q is time independent, and D^ is 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 nonlinearity in the previous equations i s contained solely in the term nquE i n the expression for current density (equations (1.6) and (1.8) ). Thus this 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) for E and making use of equation (1.2) to obtain E = 9t + D n 1—arT, C 1 - 1 0 ) a ( 1 + - ^ ) qn Q B.x ' where 0 = nQqp 12 The nonlinear behavior is now expressed by the denominator in equation (1.10), as may be seen by expanding this term in powers of (qn_) 1 8D/9x. With the use of equation (1.9) and the substitution ^ 1 (1.11) equation (1.10) may be written: g = 1 + S f ° ( ( v s + 1 K D n ) A e l 9 + c' c- ) (1.12) a ( 1 + ( - e 1 9 + c.c. ) ) where v„ = — = velocity of sound. In obtaining equation (1.12) the derivatives of Di have been neglected in comparison with the derivatives of the exponential term. From the previous discussion i t is apparent that a l l nonlinear behavior is now embodied in the parameter A. A has a simple physical interpretation. Solving equation (1.2) for n and using equations (1.9) and (1.11) we obtain (again neglecting derivatives of D x): z T . A i6 . A -ie . / - I no\ n = n c ( l + 2-e + 2 " e . ) (1.13) or, writing A in 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 and phase of the e l e c t r o n density wave. Since n i n equation (1.14) cannot be negative, the present a n a l y s i s imposes the f o l l o w i n g c o n s t r a i n t : 0 <_ |A| <_ 1 (1.15) In a broader sense, A provides an extremely convenient parameter f o r determining amplitude dependent expressions f o r the v a r i a b l e s involved, as f o r E i n equation (1.12). The l i n e a r theory of White (1962) may be recovered i n the l i m i t |A| -> 0. The D.C. and A.C. components of E may be extracted from equation (1.12) by performing the following phase averages: 1 2TT E Q = ^ jEd9 (1.16) 6 , 2TT E l = Jv /Ee 1 0d6 (1.17) 0 In the slowly varying envelope approximation already o u t l i n e d , D^ may be treated as a constant i n car r y i n g out the i n t e g r a t i o n s . A l s o , since D, E and n have the form shown i n equation (1.9), the RHS of equation (1.8) -may be treated as s o l e l y a f u n c t i o n of 8 f o r the i n t e g r a t i o n s i n (1.16) and (1.17). Thus I, which has been shown to be at most a f u n c t i o n of time, may be treated as a constant i n equations (1.16) and (1.17). 14 19 Using equation ( 1 . 1 2 ) and the standard s u b s t i t u t i o n z = e , the i n t e g r a l s i n ( 1 . 1 6 ) and ( 1 . 1 7 ) may be converted to contour i n t e g r a l s around the unit c i r c l e i n the complex plane. One then obtains: 1 - ( 1 - I) (1 - A Z ) 2\h . ( 1 . 1 8 ) E SA 1 + ( 1 - A O (1 (1 i f i l A l * ) 1 * D ( 1 . 1 9 ) The notation used i s the same as Butcher and Ogg (1968) : v s E s = -— i s the synchronous e l e c t r i c f i e l d , i . e . the f i e l d required to y produce an e l e c t r o n d r i f t v e l o c i t y equal to the v e l o c i t y of sound. I s = n Q q v s i s the synchronous current, u)n = — i s the d i f f u s i o n frequency. Equation ( 1 . 1 8 ) may be rearranged to y i e l d : 15 a - i ) cTTT^ = T ( 1- 2 0 ) where y = 1 = 1 • v n = electron d r i f t velocity (1.21) Substituting equation (1.20) into equation (1.19) yields a relationship between E-^  and D-^ : ifiDi ewJ , iQ y H (1.22) where <or [ 1 + (1 - |A| 2)^ ] (1.23) a a) = — c e is the conductivity relaxation frequency. To find the expression for acoustic gain, we obtain a wave equation by substituting equation (1.3) into (1.1) to yield (using v l = — ): P 3 2u _ 2 as 9t2 V s 9 X e_ 9E p 9x (1.24) Taking the derivative with respect to x on both sides of (1.24) and 9u using the relation S = 3x 16 9 2S 2 3 2S e 9 2E 3t2 v l 9x2 ~ p txT (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 in S-^  and the f i r s t derivatives of S-^  are therefore retained. We obtain: f f i . i e 3 s i -16 . 3 S 1 19 3 S 1 -16 e - e + v e - v c e at 9 t S 9x 9x K_ e . ie , _* -ie . t 2iv~ p C E l e + E l e > d - 2 6 ) Equating the coefficients of e on both sides of (1.26) 9 S 1 9 S 1 K e + V s = 2 i v " f E l C 1 - 2 7 ) 9 t 9x / l v s P With the aid 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: 17 3S 1 dS± + v = c ^ V g S ! (1.28) 3t 3x ( y + —- ) w h e r e „ - D ' , c " 2 7 s <7~ . (1-29) ton Si The real and imaginary parts of a are given by: Re a. 2v, YOJC (Op S2 (1.30) AK Im a. 2v„ ft •> Q Yz + '( + — c ) to D (oD (1.3D X = —^2 is the piezoelectric coupling coefficient. From the form of the general solution for equation (1.28), i t may be readily inferred that a in equation (1.30) i s the nonlinear acoustic gain coefficient, while AK in equation (1.31) specifies the phase s h i f t of the acoustic wave produced by piezoelectric coupling. 18 Equations (1.30) and (1.31) only d i f f e r from the corresponding expressions in linear theory by the substitution of for o)c [see equation (1.23)]. In the limit |A| -»• 0, to^ . -> wc and linear 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) in the expression for total round t r i p gain in the phonon maser, i t is necessary to account for acoustic waves travelling both par a l l e l and antiparallel to the applied D.C. el e c t r i c f i e l d E D. This i s achieved by setting y in equation (1.30) equal to: VD Y_ = 1 (1.32a) v for acoustic waves travelling antiparallel to E Q, and VD Y + = 1 + — (1.32b) v s for acoustic waves travelling p a r a l l e l to E o. The total round trip gain G in the phonon maser may be written (Gurevich 1969): G = a, + a_ + 2T - Q™L (1.33) d where a + (a_) i s given by substituting y+ (y_) into equation (1.30). F i s the l a t t i c e loss coefficient, r i s the acoustic r e f l e c t i v i t y of 19 the phonon maser cavity wall, and d is the cavity thickness. For the l a t t i c e loss T we use the empirical relation (Maines and Paige 1969): r = 6f2B (cm)"1 (1.34) where 6 = 1.11 x I O - 1 4 ; 6 = 1.51 In FIG 2(a) and (b) the value of G corresponding to the frequency of maximum gain is plotted as a function of |A|, under different conditions of conductivity and applied D.C. electic f i e l d . The parameters used in the calculation were = 4.38 x 109Hz, y = 280 cm 2/volt-sec., x = 0.0378 and v g = 1.759 x 10 5 cm/sec. (the value for u was calculated from the threshold electric f i e l d for bj_ crystal 24.01.02.04). A value of 1.5 was used for the end loss term -21nr/d. As is indicated in FIG 2, conditions can be realized such that G increases with increasing |A|, and hence with increasing acoustic amplitude. It i s this nonlinear property that provides the counterpart of the saturable absorber, and completes the analogy between the phonon maser and the repetitive 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 theoretical behavior of the phonon maser as a function of conductivity and applied electric f i e l d i s summarized in 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 electric f i e l d E 0 . E g is the synchronous e l e c t r i c f i e l d . The small area enclosed by a dotted line near the bottom of the figure corresponds to the regime where mode-locked operation was observed for b_[_ crystal 24.01.02.04. As is indicated, mode-locking was achieved at conductivities approximately an order of magnitude lower than predicted by theory. It i s worthwhile to discuss the limitations of the theory. F i r s t of a l l , consider the maximum allowable value for the amplitude E-^  of the self consistent A.C. electric f i e l d . From equation (1.19),with |A| = 1, we observe that l E l l M a x " E s I Y + ^ I '(1.35) For typical operating frequencies, i t is reasonable to neglect isl/u)D in comparison with y. Thus E l Max * Eo - E s (1-36) Since E Q/E S ~ 1.2-2.0, for the experiments performed, equation (1.36) predicts that l E ] J M a x < 0.5Eo. In fact, values of E^ of the order of 2E C nave been inferred 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 situation 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 in the following manner. If equation (1.18) is solved for I, the resulting expression may be written in the form: aE (1 - F) + I„F (1.37) o ° where 1 - (1 - \k\2)h (1.38) From equation (1.37) we see that the total current density I behaves as i f a fraction F of the electrons were constrained to move at the velocity of sound, while the rest exhibit normal ohmic response to the local D.C. e l e c t r i c f i e l d . Thus F may be considered to be a trapping fraction in the sense that i t specifies the portion of available electrons that are trapped i n the potential wells produced by the acoustic wave. Since F increases monotonically from 0 to 1 as |A| increases from 0 to 1, for |A| = 1 , I = I G independent of the applied f i e l d E Q. Thus the present theory takes us up to the point of complete current saturation. The region of mode-locked operation i s , in fact, well into the regime of current saturation. The above limitations may be traced to the fact that we have assumed a sinusoidal electron density distribution [equation (1 .14)] . As has been indicated in section 1.1, this w i l l not be a good assumption in the case of very large acoustic amplitudes. The theory may be generalized to incorporate a more physical charge density pro f i l e by writing a l l variables in the form of a general Fourier expansion, and 24 then proceeding in much the same manner as in this section (see Butcher and Ogg 1969). For r e a l i s t i c profiles, 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 pl a u s i b i l i t y argument for the existence of a "saturable absorber". 25 1.3 Theory of the Optical Measurements The necessary background for interpretation of the optical experiments is developed in this section. To demonstrate the usefulness of light diffraction in the study of acoustic f i e l d s , we begin by considering the photoelastic effect (Nye 1964). 1.3.1 Crystal Optics - The Fhotoelastic Effect The optical properties of an anisotropic crystal are conveniently embodied in a quadric surface known as the indicatrix. It may be defined in i t s general form by (Nye 1964): B i j X i X j = 1 ( 1 ' 3 9 ) where B^ _. are components of the relative dielectric impermeability tensor defined by 3E. B. - e 0 — • (1.40) J 3D. 3 i-E^ and are components of the electric f i e l d and electric displacement, and e Q i s the vacuum permittivity. If the coordinate axes x^, X £ , x^ are chosen to l i e along the crystallographic directions which correspond to the principal axes of the dielectric tensor, equation (1.39) reduces to 2 2 2 — + — + — = 1 (1.41) n l n2 n3 26 where = n|, = n| and = n^ are the principal dielectric constants, and n^, n^, n^ are referred to as the principal refractive indices. The quadric surface defined by equation (1.41) i s an ell i p s o i d with semiaxes n^, n^, n^. This surface has a valuable property that can be il l u s t r a t e d by the geometrical construction in FIG 4(a). Consider that ligh t propagating in direction CO (0 i s the ell i p s o i d origin) impinges on the crystal. A central section of the indicatrix i s made perpendicular to CO. This section w i l l be an ellipse and is referred to as the "index ell i p s e " . In FIG 4 the index ellipse i s emphasized by shading and in FIG 4(b) is viewed along the direction of incident light ( i . e . light i s propagating into the page). If the incident light has i t s ele c t r i c displacement vector D polarized in the direction indicated in FIG 4(b), i t w i l l propagate in 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 refractive indices equal to OA and OB, respectively. For OA =f= OB (the index ellipse i s not a circle) the medium exibits birefringence or double refraction, and when the light emerges from the crystal the two components and w i l l recombine to form, in general, e l l i p t i c a l l y polarized l i g h t . For light having i t s wave normal parallel to one of the indicatrix axes, the two independent waves in the material w i l l have refractive indices corresponding to the principal indices in equation (1.41). For example, light propagating along the x^ axis w i l l have refractive indices n., and n 0 [see FIG 4 (a)]. 27 'FIG 4. (a) Indicatrix construction for optical wave normal CO. The index ellipse i s shaded, (b) Index ellipse viewed along direction of light propagation. The polarization of the incident light i s specified by the electric displacement vector D. 28 The shape of the indicatrix depends upon the symmetry of the medium in question. For cubic or isotropic materials, which possess one principal refractive index, i t i s a sphere. Hence a l l central sections are circles and there i s no natural birefringence. For hexagonal, tetragonal and trigonal crystals there are two principal refractive indices, and the indicatrix is an ellipsoid of revolution having the form (1.42) where n Q and n £ are termed the ordinary and extraordinary refractive indices. In general, central sections of (1.42) w i l l be ellipses, implying the existence of natural birefringence. The central section perpendicular to the x^ axis is a c i r c l e , however, and for this unique direction there is no birefringence, x^ is called the principal or optic axis and such crystals are termed uniaxial. CdS has hexagonal crystal structure and hence i s an example of a uniaxial crystal. For CdS, x^, X£> x^ in equation (1.42) correspond to the a, b and c axes, respectively. For the remaining crystal classes there are 3 principal refractive indices and the indicatrix i s a t r i a x i a l e l l i p s o i d . The permittivity and dielectric constant, and hence the index of refraction are, in general, modified by the introduction of elastic strain in a crystal. Thus the presence of an acoustic wave w i l l be indicated by a modulation of the local refractive index which may, 29 in turn, be expressed in terms of small changes in the shape, size and orientation of the indicatrix. If we consider only effects that are linear in the applied strain, the change induced in the indicatrix may be written (Nye 1964): A B i 3 = P i j k l S k l ^ Pjyk^ a n d are components of the optoelastic and strain tensors. The strain tensor i s related to the mass displacements u^ by: 9 u . 3 u . S = jj ( _ i + — 1 ) (1.44) 3 3 x . 3 x . 3 i If the shear strains ( i / j) are redefined in equation (1.44) by omitting the factor of % , the indices in equation (1.43) may be contracted unambiguously according to the following scheme (Nye 1964) : 11 -* 1 32, 23 -> 4 22 -> 2 31, 13 -> 5 (1.45) 33 -> 3 . 21, 12 -> 6 This allows equation (1.43)to be represented i n matrix form. Under the influence of strain the indicatrix e l l i p s o i d represented by equation (1.41) w i l l be transformed, and the new e l l i p s o i d may be described by [refer to equation (1.39)]: 30 B1 X1 + B2 X2 + B3 X3 + 2 B4 X2 X3 + 2 B5 X1 X3 + 2 B6 X1 X2 = 1 (1-46) With reference to equations (1.41), (1.45) and (1.46), equation (1.43) may be written in matrix form: AB B i - l / n j p l l •• P16 AB2 B2 - l/n2 p21 •• p26 AB3 B3 -1/n* P31 P36 AB. 4 B4 P41 •• P46 AB5 B5 P51 P56 D B6 P61 •• P66 s„ 1 s„ 2 S o 3 X s. 4 Sr-5 S6 (1.47) When the form of the strain i s known, the new indicatrix may be found from equations (1.46) and (1,47). With this information one is able to determine the new polarizations and indices of refraction for a given optical wave normal, using the indicatrix property illustrated by FIG 4• The general problem provides a tedious exercise in algebra and analytic geometry. General solutions have been obtained for crystals with hexagonal 6mm symmetry by Vrba and Haering (1973), and provide a useful model for solving other crystal structures. A much simpler case, that of a shear wave propagating in an optically and acoustically isotropic medium, w i l l be treated to i l l u s t r a t e the use of the indicatrix. This situation i s of particular interest and ..the results w i l l be used in later sections. In the absence of strain the 31 indicatrix i s a sphere and equation (1.4.1) becomes: ~ 2 ( x 2 + x2, + x 2 ) = 1 (1.48) The coordinate system is chosen so that the acoustic wave normal l i e s along the x^ axis, with mass displacement along the Y.^ axis. Thus the only non-zero strain component is Sg. For an isotropic material the optpelastic matrix becomes (Nye 1964): p l l p12 P12 0 0 0 p12 p l l p12 0 0 0 p12 P12 P l l 0 0 0 0 0 0 p44 0 0 0 0 0 0 p44 0 0 0 0 0 0 P44 (1.49) where p., = ^(p,, - P 1 9 ) . Equation (1.47) then yields: ' * AB1 0 AB2 0 AB3 0 AB. 4 = 0 AB5 0 A B 6 P44 S6 (1.50) 32 From equations (1.48) and (1.50), the coefficients in equation (1.46) may be readily inferred: B 4 = B 5 = 0 (1.51) B 6 = p44 S6 The new indicatrix thus becomes: i 2 ( x 2 + x2. + x2, ) + 2p 4 4S 6 X ; Lx 2 = 1 (1.52) If X g is chosen as the optical wave normal, the appropriate index ellipse i s given by setting x^ = 0 in equation (1.52) to obtain: n§ ( X l + X2 } + 2P44 S6 X1 X2 = 1 ( 1 ' 5 3 ) The form of the index ellipse, both in the presence and absence of the perturbing strain, i s shown in FIG 5. As is indicated, the axes of the index ellipse are rotated by TT/4 with respect to the coordinate axes x^ and x 2- We therefore change to a rotated frame using the transfor-mation: x± = l/v/2 ( X J + x£ ) (1.54) x 2 = 1//2 ( x^ - x£ ) FIG 5. Form of the index ellipse in-an isotropic material for optical wave normal X g , both in the presence and absence of shear srain Sg. 34 so that the new coordinate axes x|, x^ are colinear with the axes of the ell i p s e . Equation (1.53) then reduces to the more standard form: v »2 t2 1 2 — + — = 1 (1.55) 2 2 n I n I I where n no <X - P44 S6 no^ n Q (1.56) <x + P44 S6 no ) 9 -5 For typical experimental parameters P^S^n^ s 10 , so to a very good approximation we may expand the square roots in (1.56) and retain only linear terms: n I = % ( 1 + P44 S6 no ) (1.57) n I I = % (1 - Js P 4 4S 6n2) It i s readily apparent from equation (1.57) that the changes i n refractive index produced by the strain (see FIG 5) are: A n I = - A n I I = h P44 S6 no ( 1 * 5 8 ) The time dependence of the acoustic wave has thus far been 35 neglected. FIG 5 Is essentially a picture of the index ellipse at one instant of time. For sinusoidal time dependence, we may imagine that at time h T later, where T i s the period of the acoustic wave, the index ellipse w i l l be a c i r c l e ; at time % T i t w i l l again be an ellipse, but with the major axis now lying along x^ ; and so on. In this manner, the local index ellipse may be visualized as pulsating with the frequency of the acoustic wave. It i s worth mentioning that, for a given strain, the wave normal of the incident light must be judiciously chosen. If i n the present case, for example, the optical wave normal i s taken as x^ or x 2, i t may be seen from equation (1.52) that the appropriate central sections ( x^ = 0 and x 2 = 0 ) are circular and completely unaltered by the strain. Thus a knowledge of the predeeding theory i s essential in determining the configuration for optical experiments. 1.3.2 Interpretation of the Optical Signal - Preliminary Considerations The intimate relationship between the refractive index ;and the local strain, outlined in the previous section, makes light diffraction a powerful tool in the study of the phonon maser. In order to proceed to an optical analysis of the strain profiles i n a mode-locked phonon maser, i t i s necessary to f i r s t establish the relationship between the optical signals derived by light diffraction, and the acoustic f i e l d s that provide the diffracting mechanism. This subject has an extensive literature and w i l l not be f u l l y developed here [the general theory of ultrasonic light diffraction has been discussed 36 by Bhatia and Noble (1953) and by Klein, Cook and Mayer (1965)]. A categorized bibliography is provided at the end of this thesis. The configuration of physical interest is depicted in FIG 6. An acoustic cavity of width L is illuminated by a unit amplitude plane light wave, propagating in the xz plane (plane optical wavefronts may be reasonably approximated in practise, since laser beams with wavefront distortions less than 0.2 optical wavelengths may be prepared by careful collimation). The optical waveform emerging from the far side (z = L) of the acoustic cavity i s transformed due to acoustically induced modulations in the local refractive index. The near f i e l d optical distribution (i.e. the optical signal at z = L), denoted A Q, is imaged by a converging lens. The direction of acoustic propagation is taken as the x-axis in FIG 6, and the acoustic waveform is assumed to have spatial variation only in the x-direction. Hence the near f i e l d distribution A q i s a function only of x and t. This assumption requires some comment. In practice, the f i e l d of view is determined by the cross section of the incident laser beam. Since the beam size used in experimental situations was smaller than the dimensions of the acoustic cavity, distortions in acoustic wavefronts introduced by edge effects were avoided and, as w i l l be verified by the experimental results, the assumption of a one dimensional system i s v a l i d . In the focal plane of a converging lens the criterion for Fraunhofer diffraction is satisfied (Goodman 1968). This i s i n t u i t i v e l y .clear since in front of the focal plane light waves are converging and 38 behind i t they are diverging. Hence at z = in FIG 6 we have plane optical wavefronts. In the limit of small diffraction angles 18°), the optical distribution in the focal plane, A^(x.',t),is related to the near f i e l d distribution by a one dimensional form of the familiar Fraunhofer diffraction integral (Goodman 1968): A f ( x \ t ) = £ W ' A 0 ( x , t ) e - i k x S i n 9 d x (1.59) W 2 where £ is a phase factor determined by the physical configuration and W is the:effective width of the optical beam. If we imagine two parallel rays, separated by distance x in the near f i e l d , and making an angle 6 with the z axis, the term exp(-ikx sin0) equals the difference in optical phase suffered by the two rays in propagating to point x' in the focal plane (see FIG 6). The beam width W is assumed to be large compared with the acoustic wavelengths so that a l l diffraction orders in the focal plane are well resolved and completely separated. For this situation, we impose no restrictions on the spatial f i l t e r i n g experiments to be carried out, by extending the limits of integration in (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 is equal to the focal length of the lens (Goodman 1968). With these considerations equation (1.59) becomes: 39 A 0(x,t)e i k' Xdx (1.60) 4 0 0 where k' = k sin 0. There i s a one to one relationship between x 1 and k' in equation (1.60) since specifying x' uniquely determines 9 and vice-versa. Hence equation (1.60) indicates that the optical distribution in the focal plane is the Fourier transform of the near f i e l d signal. This Fourier transforming capability i s a general property of converging lenses (Goodman 1968). In propogating from the focal plane to the image plane in FIG. 6, the optical signal suffers an inverse Fourier transform. For a d i s t o r t i o n f r e e optical system, there is then a one to one relationship (aside from a spatial magnification factor) between the optical distributions in the image plane and in the near f i e l d or object plane. With the preceeding considerations, i t remains to determine the form of A Q(x,t). We begin with the simple example of a sinusoidal, progressive acoustic wave. Assume that the cavity i s optically isotropic i n the absence of acoustic waves, and, to simplify notation, that the incident light i s polarized along one of the axes of the index elli p s e (See FIG. 5). Since the change in refractive index i s , to a good approximation, linear i n the local strain [see equation (1.58)], the refractive index w i l l have the form: n(x,t) = n D + An sin(Kx - fit) (o<z<L) (1.61) A f(x',t) = 40 where n 0 is the unperturbed refractive index, K and fi are the wavenumber and frequency of the acoustic wave. Since the refractive index has slow variation compared with optical frequencies, the wave equation governing propagation of the optical signal A in the acoustic cavity may be written (Raman and Nath 1936b) : V 2 A = n(x,t) 2 92A c 2 8t 2 (1.62) where c is the vacuum velocity of l i g h t . In the absence of acoustic f i e l d s , A w i l l propogate in the region o<z<L in the form a 0(x,z,t) = e i [ n ° k ( Z C O S * + x S i n •> (1.63) k,u are the optical wavenumber and frequency and (J> is the angle made by incident light with the z axis (see FIG. 6). The modulation of the optical wave produced by the presence of acoustic strains w i l l be slowly varying compared with the spatial and temporal dependence of (1.63). Thus in the presence of the acoustic wave the optical waveform may be described by: A(x,z,t) = a Q(x,z,t) Tjj(x,z,t) (1.64) 41 The envelope function ip(x,z,t) w i l l be periodic in space and time with the acoustic wave, and hence may be written as a Fourier series: CO *(x,z,t) = I * n(z) e i n ( K x - B t> (1.65) n ~ -oo subject to the boundary conditions v - 1 i|»n = 0 n ^ 0 at z=0 (1.66) The near f i e l d amplitude A Q(x,t) is given by setting z=L in equations (1.63) to (1.65). Substituting the resulting expressions into (1.60) we obtain for the focal plane optical signal: A f(x',t) = e i(n nkL cos<j> - cot) r , . -inftt ° L ' M L ' e n = -oo e - i (k' -ksintj) -hK) x (1.67) The integral in equation (1.67) i s non-zero only i f k' - ksin<J> - nK = 0 (1.68) Light w i l l appear in the focal plane only at those points which satisfy equation (1.68). Hence the index n labels the diffraction orders in the 42 focal plane. The optical intensity in the n*"*1 diffraction order i s given by equation (1.67) as i n = <j>na)|2 d.69) t i l Also, the angular frequency in the n order may be inferred from (1.67) : wn = to + nfi (1.70) Equations (1.63) to (1.65) may be substituted into (1.62). If terms in (An)^ and ij> (z) are neglected, coefficients of exp i[n(Kx-fit)J may be equated.to obtain: d»n(z) . v [T|, (Z)-I|) (Z)] = inQ(n-2 YH (z) (1.71) ^ — + — n-1 n+1 2L" n where: v = AnkL (1.72) Q = K 2L (1.73) n k o -n k sin* (1.74) Y = o T K as indicated by equation (1.69), the optical amplitude in the n t h 43 diffraction order in the focal plane i s proportional to il/ (L) . The n difference - di f f e r e n t i a l equation (1.71) was derived in the special case y = 0 (normal light incidence) by Raman and Nath (1936b), and in the more general case by Klein and Cook (1967) . In equation (1.71) adjacent optical 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 relationship, i.e. maintain spatial synchronization. The degree of synchronization i s determined by the coefficient, to be designated 8n, on the R.H.S. of equation (1.71) : a _ inQ (n-2Y) (1.75) This may be considered as a relative phase factor, and only those optical modes having the same, or nearly the same, value of this coefficient may be considered as being spatially coherent or synchronized, and hence able to exchange optical energy. Since 6 q = 0, an appreciable amount of optical energy can be coupled from the zeroth into the f i r s t order only i f one or both of the coefficients are very small. This condition can be satisfied in two regimes : (i) Q << 1, y - 0 ; ( i i ) Q » 1, y = ±%. (i) and ( i i ) are the two limiting situations for which analytic solutions to (1.71) are practical, and each case w i l l be discussed separately. 44 Raman - Nath Limit: Q<<1 For Q = 0 and with boundary conditions (1.66) the solution to (1.71) may be shown to be (Raman and Nath 1936 b) : il) (z) = J (Ankz) (1.76) where J i s a Bessel function of order n. For sufficiently small nonzero n values of Q, Bessel functions form good approximate solutions of (1.71). The regime of va l i d i t y of this approximation i s named for the two workers who f i r s t ' provided a theoretical basis for this type of diffraction (see Raman and Nath 1935 a,b and 1936 a,b). zero order into both f i r s t order modes (n=±l). In addition, for sufficiently 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 third, etc. From a comparison of equations (1.73) and (1.74) i t i s apparent that for Q small y can be very large, even for small values of the incident angle <J>. Thus we optimize the condition 8N ~ 0 (n^O) hy choosing cj> = 0 => y = 0 (normal light incidence) . The optical intensity th in the n diffraction order i s then given from equations (1.69), For this situation energy is symmetrically coupled from the (1.72) and (1.76) I = J 2(v) n n (1.77) and, setting <J> = 0 in equation (1.68), we obtain for the diffraction 45 angle G : sin 0 = y- (1.78) where A i s the optical wavelength in vacuo and A is the acoustic wavelength. In the Raman - Nath regime the acoustic wave only modulates the phase of the optical signal. The parameter v, referred to as the Raman - Nath parameter in this situation, is equal to the maximum acoustically induced phase sh i f t suffered by the light in traversing the acoustic cavity (see eqn. 1.72). A practical upper limit of the Raman - Nath regime i s given by (Klein and Cook 1967) : Q ~ 0.5 (1.79) For large values of v (ie. large values of acoustic strain) condition (1.79) may not be sufficient to ensure true R-N diffraction. For this reason the supplementary condition (Extermann and Wannier 1936) : Qv < 2 (1.80) is sometimes quoted. For our experimental situation, however, V<<1 and the nature of the diffraction i s adequately described by the magnitude of the parameter Q. 46 Bragg Limit : Q » l For this situation, the condition 6^  = 0 (n^O) can be satisfied only for n = 1 or n = -1. Thus for y = %(-%) light i s diffracted only into order +1(-1). Other combinations of n and y are not allowed for a simple sinusoidal acoustic wave. For n = ±2, y = ±1, for example, no light may be transferred from the zero order since only adjacent orders are directly coupled by equation (1.71). For y = ±Jg equation (1.74) may be x^ritten in the more transparent form: — = ±2A sin <j> (1.81) o Thus y = ±% implies that cf> satisfies the Bragg condition for specular optical reflection from the acoustic wave fronts. A practical lower limit for 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, analytic solutions to (1.71) are not practical. The gap between the two limiting cases has been spanned by Klein and Cook (1967), using numerical techniques. Some of their results are reproduced in FIG. 7. The curves illustrated give the percentage of optical intensity in the zero and two f i r s t order diffraction spots at normal incidence (y=0) , for different values of the parameter Q. By summing the intensities in the zeroth and f i r s t order in 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 optical intensity in the zero and two f i r s t order diffraction spots at normal incidence (y = 0), for various values of the parameter Q. (after Klein and Cook 1967). v is defined in equation (1.72). 48 light appears beyond the f i r s t order. In the range Q = 4 to Q = 7 the amount of light in the two f i r s t orders decreases considerably, and when Q = 10 there is almost no diffraction at normal incidence for small values of v (small acoustic strains). The values of Q encountered for mode-locked phonon maser operation can span the entire gamut from Raman - Nath to Bragg diffraction. However, the experimental results to be described in section 1.4 l i e in the range V<<1, for which second order diffraction i s completely negligible, and Q < 5. For these conditions, i t i s apparent from FIG. 7 (a) - (d) that f i r s t order diffraction does not change significantly from the Raman - Nath case. For this reason i t w i l l be assumed in a l l subsequent discussion that the effect of acoustic fields on a light wave may be described by an optical phase transformation, as in the Raman -Nath regime. 1.3.3 Interpretation of the Optical Signal - The Mode-Locked Phonon Maser In applying the results of the last section to a study of the strain p r o f i l e in a mode-locked phonon maser, the basic configuration illustrated in FIG. 6 w i l l be retained. The light i s noxtf assumed to propogate in the z direction so as to be normally incident on the acoustic waveform. As discussed in section 1.3.2, this i s the optimum configuration for Raman - Nath diffraction. Following the discussion at the end of section 1.3.2, we assume that the only effect of the acoustic strain profile is to produce variations in the optical phase, via acoustically induced modulations of 49 the refractive index. Since the relationship between the local strain and refractive index variation i s linear [eg. see equation (1.58)], the phase variation profile directly yields the acoustic strain p r o f i l e . If the phase variation of the acoustic cavity i s described by $(x,t), the near f i e l d light amplitude i s given by [ c f . equations (1.63) and (1.64)] A o(x,t) = ei(noWL " »t)el*(x,t) ( 1 8 3 ) For mode-locked operation, !the acoustic waveform is assumed to be a standing wave pattern consisting of a phase-locked harmonic series of sinusoidal waves. Using the results of Appendix A, the phase function in equation (1.83) may be written in the form: N $(x,t) '= I v n n=l sin[n(Kx-ftt) + 6 R] + sin[nK(x-2d) + nftt - « n] (1.84) where d is the thickness of the acoustic cavity, N i s the number of active or participating 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 strain amplitudes. The set of phases 6 n w i l l determine the exact shape of the phase variation, and hence the strain, p r o f i l e . Equation (1.84) may also be written in the form: 50 N >(x,t) = 2 I v nsin[nK(x-d)]cos(nKd - nOt + 6 n) (1.85) n=l Since the acoustic cavity alters only the phase of the incident li g h t , the strain f i e l d i s not directly v i s i b l e in the image plane of the focussing lens (refer to FIG 6). This problem may be overcome by means of focal plane spatial f i l t e r i n g techniques. If certain diffraction orders are physically removed (i.e. spatially filtered) in the focal plane, the effect 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 is given by the modified focal plane distribution passed by the spatial f i l t e r . By removing the zero order or undiffracted beam in the focal plane, for example, the dark f i e l d image is obtained (Born and Wolf 1959). In this situation, phase variations in the object plane produce intensity variations in the image plane. The dark f i e l d distribution w i l l be derived in two ways. F i r s t , we begin by substituting the phase function for a mode-locked waveform, equation (1.85), into equation (1.83): A Q(x,t) = e^exp.i f N I 2vnsin[nK(x-d)]cos(nKd-nftt+6n) n=l (1.86) where t, = (n QkL - wt) . Making use of the Bessel function relationship (Abramowitz and Segun 1968): iz sinO v T r \ imO m = _ o o equation (1.86) may be written: (1.87) N « A o(x,t) LC n I J p n[2v ncos(nKd-nfit+6 n)]e i P n n K ( x d ) (1.88) n=l p n=-= e Pl= J p (v x) o X • J P N ( V e x p 1 • N I npnK(x-d) n=l (1.89) where V n = 2vncos(nKd-nfit+6n). (1.90) A useful analogy may now be made with the case of a simple sinusoidal acoustic wave discussed in section 1.3.2. When equation (1.65) was substituted into equation (1.60) to obtain the focal plane optical distribution, i t was found that the coefficient of the spatial factor Kx in the exponential term of (1.65), n in this case, labelled the diffraction orders in the focal plane. By analogy, the diffraction order for equation (1.89) may be inferred : n diffraction order = £ np n=l n (1.91) This greatly simplifies the mathematics of spatial 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 find the focal plane distribution, perform spatial f i l t e r i n g , and then inverse Fourier 52 transform to obtain the image plane signal. Instead, we may use equation (1.90) to identify those components of A D(x,t) that correspond to the focal plane orders to be removed (passed) by the spatial f i l t e r , and hence obtain the modified image directly. It i s int 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 distribution i s not practical in 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. It i s therefore necessary to consider only effects that are f i r s t order in v . In addition, we may use the asymptotic form for Bessel functions of small argument (Abramowitz and Segun 1968): J (z) i — ( m > 0 ) (1.92) m z small m! — Equation (1.92) may be used for negative values of m by f i r s t using the relationship (Abramowitz and Segun 1968): J (z) = < - l ) m j (z) (1.93) -m m In f i r s t order, the parameters p n in 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 = +1 -> p. = 0 f o r i h (1.94) rn * i 53 The corresponding contribution from equation (1.89) is [using equation (1.92)]: A = e U |A einK(x-d) n 2 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 diffraction order +n. Similarly, the f i r s t order contribution to diffraction order -n is given by setting Pn=-1 in equation (1.94) and using (1.92) and (1.93) to obtain: A = - e i ? |* e-inK(x-d) 9 -n z The zero order amplitude must be treated separately and is given by setting p^ = 0 for a l l i . Using equation (1.92) we obtain: A = e i C (1.97) o Hence the intensity in the zero order, | A q | 2 , is equal to one. This indicates that, to f i r s t order i n the Raman - Nath parameters, the zero order beam is unaffected by the diffraction. Adding equations (1.95) and (1.96) we obtain: ie l CV nsin[nK(x-d)] (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), labels symmetric diffraction orders in the focal 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 : . N A ^ = i e 1 ^ I V nsin[nK(x-d)] (1.99) n=l or, using equations (1.85) and (1.90): ^ = i e l C *(x,t) (1.100) Thus the f i r s t order dark f i e l d amplitude i s proportional to the phase variation produced by the acoustic cavity. The corresponding optical intensity i s given by: I D F = * 2(x,t) (1.101) It i s , in fact, the intensity distribution, equation (1.101), that may be detected experimentally. The second method of deriving the dark f i e l d distribution i s much simpler, but gives l i t t l e insight into the mechanism of spatial f i l t e r i n g . Since we have assumed small phase variations, the exponential in equation (1.83) may be expanded to yield: A Q(x,t) = e 1 ? [ 1 + i$(x,t) ] (1.102) The dark f i e l d amplitude corresponds to removing the factor of 1 in 55 equation (1.102), to immediately yield equation (1.100). While this method of derivation i s much simpler, i t should be emphasized that i t i s useful only for effects that are f i r s t order in v . For higher order effects the factor of 1 in equation (1.102) can no longer be correctly identified as the amplitude of the zero order diffraction spot, and the image plane distribution must be derived using the f i r s t method. The dark f i e l d intensity distribution i s obtained by substi-tuting equation (1.84) or equation (1.85), into equation (1.101). The resulting expression i s very messy, having terms with a l l the possible combinations of spatial and frequency dependence produced by the nonlinear operation of squaring. If, however, we extract only the term which has no time dependence, we obtain the relatively simple expression (Smeaton, Hughes, Vrba and Haering 1976): N Io = I v n [ 1 " cos2nK(x-d)] (1.103) n=l This D.C. term is straightforward in the sense that i t contains none of the phases <5n [see equation (1.84) or (1.85)]. Thus i t s form depends only on the constituent amplitudes and not on the exact shape of the acoustic waveform. Equation (1.103) is plotted in FIG 8. The relative values of the amplitudes v n were inferred from experimental diffraction intensities, obtained for the f i r s t 13 active modes of a mode-locked phonon maser. Thus N = 13 in FIG 8. The width PW of the dark fringes i s inversely proportional to the number of participating modes N, and provides an FIG 8. Profile 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 is indicated, the basic periodicity i s A/2, where A is the wavelength of the acoustic fundamental. To obtain I Q in equation (1.103), a combination of spatial and temporal f i l t e r i n g was employed. That i s , we f i r s t modified the optical distibution in t h e focal plane, a n d then selected only one frequency component, the D.C. component in this case, to be examined in the image plane. By extending this general technique, information about the phases 6n, absent in equation (1.103), may be recovered. This is of considerable interest since these phases determine the exact shape and amplitude of the acoustic strain p r o f i l e . Consider, for example, that spatial orders n, n+1, and n+2 t i l (i.e. the n pair of diffraction spots symmetric to the zero order, t i l the (n+1) pair and so on) are combined, and the intensity variation of the temporal component corresponding to the acoustic fundamental frequency Si is scanned in the image plane. The total image plane intensity distribution 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) ' m=n + sin[mK(x-2d) + mSlt - 6 ] (1.104) Extracting only those terms from (1.104) which have frequency dependence Si, we obtain: 58 T = 2 n, n+1,n+2 a ncos(Kd+6 n + 1-6 n) + b ncos ( K d + 6 ^ - 6 ^ ) cosftt + 2 a nsin(Kd+6 -6 ) + bsin(Kd+S , _-6 n n+1 n n n+2 n + l sinftt (1.105) where a n = 2v nv n + 1sin[nK(x-d)]sin[(n+l)K(x-d)] b n = 2v n + 1v n + 2sln[(n+l)K(x-d)]sin[(n+2)K(x-d)] We write equation (1.105) in the form (1.106) n,n+1,n+2 = acos^t + bsinftt (1.107) where a,b are defined implicitly by comparison of equations (1.105) and (1.107). Introducing notation a = Csinn b = Ccosn C = (a 2 + b 2 ) ^ tann = a/b (1.108) equation (1.107) may be written: n,n+1,n+2 = Csin(Qt + n) (1.109) The Fourier transform of (1.109) has the following simple form in the frequency (ft') domain: 59 ft n,n+1,n+2 -iC 2 e i n6(ft'-ft) - e 1 TVft'+ft) (1.110) In practice, the magnitude of the component of (1.110) at ft'=ft i s monitored. Thus the detected signal becomes: cft n,n+1,n+2 C 2 (a 2+b 2)^ (1.111) Using the definitions of a and b [equations(1.105) and (1.107)] we obtain (Smeaton. and Haering 1976a): where a n, b n are defined by equation (1.106) and °n = 6n - 2 < Sn+l + 6n+2 The observed image plane profile should therefore depend on the magnitude of 0 n , a linear combination of the individual phases of the three combined orders. The behavior of equation (1.112) as a function of 0 i s illustrated in FIG 9, for n=A. The relative values of v„ were n ' n calculated from experimental diffraction intensities obtained for 61 mode-locked operation. The sensitivity of the theoretical profiles in FIG 9 to variations in 0^ indicates that i t i s feasible to recover phase information by optical means. The intensity profiles for other combinations of spatial and temporal f i l t e r i n g may be derived i n a manner similar to equation (1.112). 62 1.4 Experimental Results The CdS phonon maser crystals used in these experiments were oriented so that the b-axis was perpendicular to the polished cavity surfaces. For this orientation the active acoustic modes consist of shear waves whose K-vectors l i e along the b-axis. For the optical measurements, the active CdS crystals were coupled to passive fused quartz cavities by means of a high quality bond (described in Appendix C). Since the two materials have nearly the same acoustic impedance for the chosen CdS orientation, the double cavity 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 laser lig h t could be passed through i t . Such a high intensity probe could not be used in the active cavity since the large photo-currents produced would disrupt or prevent o s c i l l a t i o n . The A.C. acousto-electric current signal (refer to section 1.1) from the active crystal could be displayed either in the frequency regime, by means of a spectrum analyzer, or in the time regime by means of a sampling oscilloscope. An example of both displays i s shown in FIG 10, for mode-locked operation of phonon maser 24.01.02.04. The well defined waveform exhibited in the time domain in FIG 10(b), indicates that the harmonically related modes in FIG 10(a) are phase-locked. The D.C. I-V characteristics of the same phonon maser are displayed in FIG 11(a) for two different conductivities. The oblique arrows indicate the threshold for acoustic o s c i l l a t i o n . Above threshold, 63 TIME DOMAIN, 10 na no sec./cii v. Display of the acousto-electric current signal for b]_ phonon maser 24.01.02.04 (part of composite cavity DC1). (a) frequency domain, (b) time domain. Operating conditions: applied D.C. electric f i e l d =0.98 KV/cm, D.C. current density =11.1 mA/cm2 APPLIED D.C. ELECTRIC FIELD (KV/cm) FIG 11(a). D.C. current density vs. applied D.C. electric f i e l d for bj_ phonon maser 24.01.02.04 (part of composite cavity DCl) for two different conductivities a, b. [See text and FIG 11(b)]. 66 saturation of the D.C. current may be observed (refer to the discussion at the end of section 1.2). The small discontinuous steps in this region occur when the phonon maser makes sudden adjustments in i t s mode structure. To show the development of current spiking, the time display of the acousto-electric current was recorded at the points indicated in FIG 11(a), and may be seen in FIG 11(b). It was the observation of spiking and self-locking modes in the acousto-electric current, as evidenced in FIG 10 and 11(b), that f i r s t led to speculation that the acoustic output was also mode-locked, and hence should consist of narrow, high amplitude strain pulses (Maines and Paige 1970). 1.4.1 Experimental Apparatus and Techniques The configuration for the optical experiments i s shown schematically in FIG 12. The light source was an argon-ion laser o operated at 5145 A, with a typical output power of about 1 Watt. At this high power level, even stray scattered laser light was found to greatly disrupt o s c i l l a t i o n in the active cavity, and careful lig h t shielding was necessary. The incident light beams were carefully prepared to provide good collimation, and beam diameters were typically 4 - 5 mm. A mounted double cavity is shown in FIG 13(a). The goniometer mount allowed convenient orientation of the acoustic cavity for maximum diffraction efficiency. The focussing lens was required to subtend a s u f f i c i e n t l y large diffraction angle at the acoustic cavity that loss of optical information was negligible, and yet to provide a focal plane di f f r a c t i o n laser light onon maser / detector fused quartz focussi passive cavity lens focal plane ON image plane FIG 12. Experimental configuration for the optical measurements. 68 pattern of sufficient size to allow spatial f i l t e r i n g . An f/2.5, 15 cm focal length lens was found to satisfy these c r i t e r i a . A l l optical components were maintained clean and vibration free to minimize optical noise in the image plane. Fused quartz may be considered optically and acoustically isotropic, and the results of section 1.3.1 may be directly applied to i l l u s t r a t e a useful polarization property of shear acoustic waves. Consider, in FIG 5, that incident light i s polarized along the axis, i.e. perpendicular to the direction of acoustic propagation. If we l e t i , j be unit vectors along the x^, x^ axes in FIG 5, incident light of amplitude A may be represented by: A = ± (1 + j ) . (1.114) /I t i l Light diffracted into the m order has components along the axes of the index ellipse given by (refer to Vrba and Haering 1974): A T m = ~ ^ (VT m ) e x P ( i v T m) I,  ^ m -L," 1 J-j™ = ~ A ( V T T m ) e x p ( i v T X ) II,m ^2 m •LJ->Ttl II,m' (1.115) where V j m and v^^ m are the Raman - Nath parameters corresponding to the x|, x^ axes, respectively, in FIG 5. A^ i s the appropriate diffraction amplitude per unit incident amplitude, and i s given by equations (1.90) 69 and (1.95). The exponential factors in equation (1.115) represent the optical phase variation produced by the acoustic cavity. The two components in equation (1.115) add coherently in vector form (Vrba and Haering 1974). The total amplitude may then be written: A m /2 VvI,m ) e xP< i vI,m) 1 + V vII,m> e xP<iv I I j m) j (1.116) Equation (1.116) yields e l l i p t i c a l l y polarized l i g h t . The amount of e l l i p t i c i t y is extremely small, however, since the Raman - Nath parameters V I m » V I I m K < r^ n u s» t o a v e r y good approximation, the exponential factors in (1.116) may be replaced by 1. In addition, v-£ m = -V J J M from equations (1.58) and (1.72). Thus from equations (1.90) and (1.95) we see that Am ( vI, m> = ( | m | > ° } ( 1 ' 1 1 7 ) and equation (1.116) becomes: A m = ^ A m ( v T m ) ( i - j) (|m| > 0) (1.118) /2 mvvI,m^ Thus the polarization of the diffracted light i s rotated by 90° with respect to the incident l i g h t . This polarization f l i p also occurs i f the incident light i s polarized along x^ in FIG 5, i.e. p a r a l l e l to the direction of acoustic propagation. In either case, the polarization of 70 the zero order light [refer to equation (1.97)] is unaffected. Thus an exit polarizer may be used to greatly attenuate the strong background due to undiffracted light, while leaving diffracted light largely unaffected. The exit polarizer used was an air-spaced Glan-Thompson prism, designed for high intensity laser l i g h t . The useful apertxire of the polarizing prism was large, about 2.5 cm, to avoid truncation of the diffracted l i g h t , and dielectric coatings on the entrance and exit faces optimized transmission at 5145 A. The extinction ratio was about 3 x IO 4. The focal plane diffraction pattern consisted of a vert i c a l row of spots symmetrically arranged about the zero order or undiffracted beam. At higher operating levels,.60 - 70 diffraction spots could be distinguished by eye. In the f i r s t order approximation outlined in section 1.3.3, each acoustic mode only produced one symmetric pair 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, optically decoupled in the focal plane. The time-averaged intensity of the n t b order diffraction spot is given by equations (1.95) and (1.90) as v^/2 (in practice, we are not dealing with unit amplitude incident li g h t , and i t was necessary to divide by the zero order intensity to obtain v 2/2). The corresponding strain amplitude could then be obtained using equations (1.58) and (1.72). The form of the focal plane diffraction pattern i s il l u s t r a t e d in FIG 13(b). The weak spots la t e r a l l y displaced from the main ver t i c a l array were produced by diffraction of the acoustic waves, i.e. the active cavity effectively behaved as an aperature for acoustic waves radiating into the buffer, ( c f . Fraunhofer diffraction by a single s l i t ) . This secondary feature may be ignored since i t had negligible effect on the optical experiments performed. The apparatus used for focal plane spatial 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 for processing symmetric diffraction orders, and i t s width could be accurately adjusted by means of a micrometer. X-Y movement was controlled by adjusting screws, to provide accurate positioning in the focal plane, and the diffraction pattern was viewed through a low power microscope to ensure precise optical "surgery". A photomultiplier coupled to a horizontal s l i t was located in the image plane. The photomultiplier could be driven transverse to the optical axis, as indicated i n FIG 12, in order to investigate the spatial variation of light intensity in 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 optical 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 effective resolution of better than 1.5 microns. To examine A.C. components of the image plane distribution, the photomultiplier signal was processed by a Hewlett Packard model 8555A spectrum analyzer, operated i n zero-scan mode. The resulting FIG 13. (a) Composite cavity DC2 on i t s goniometer mount, (b) typical focal plane diffraction pattern, (c) focal plane spatial f i l t e r . 73 signal was essentislly the magnitude of the Fourier component for the temporal term to be examined [refer to equations (1.110) and (1.11)]. 1.4.2 Optical Verification 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 in fused quartz. The time and frequency displays of the acousto-electric current correspond to FIG 10. FIG 14(a) is simply a bright f i e l d photograph of DCl., FIG 14(b) shows the image plane intensity variation of the D.C. term [equation (1.103), FIG 8]. Its form agrees well with the theory of section 1.3.3, and this experimental evidence provided the f i r s t direct evidence of mode-locking of the acoustic signal in phonon masers (Smeaton, Hughes, Vrba and Haering 1976). The width of the resulting strain pulses, inferred from FIG 14(b), was about 23 microns, which corresponds to about 6 nanoseconds at the shear velocity of sound in fused quartz. By spatially f i l t e r i n g symmetric pairs of diffraction spots in the focal plane, each component of the sum in equation (1.103) could be examined individually. FIG 14(c) and (d) are the profiles of the components for n=l and n=2. If 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 intensity distribution for DCl, corresponding to (a) the bright f i e l d image of DC1, (b) the total 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 effective resolution, referred to this plane is about 1.5 microns. Operating conditions: applied D.C. electric f i e l d = 0.83 KV/cm, D.C. current density = 11.1 mA/cm2 Fundamental acoustic frequency = 11.71 MHz. (after Smeaton, Hughes, Vrba, and Haering 1976). 76 At higher operating conditions narrox^er strain pulses could be achieved. In FIG 15 the profile of the D.C. term is shown for 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 in FIG 15 is 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 inferred from the dark f i e l d measurements. The phonon maser was not very stable i n this regime, however, possibly due to heating effects, and could only be maintained at this l e v e l for short periods of time. The acoustic strain amplitudes used i n determining the theoretical profile in FIG 8 were obtained for the same operating conditions as in FIG 15. If A/2 is taken as 160.5 microns in FIG 8, the width PW is about 20.2 microns. This is sli g h t l y larger than the average dark fringe width in FIG 15, owing to the fact that only the f i r s t 13 acoustic amplitudes were used in the theoretical p r o f i l e . While the intensity pr o f i l e of the D.C. term (FIG 14(b), 15) provides an estimate of the strain pulse width, i t contains no information about the exact form of the strain p r o f i l e , since the phases <$n [equations (1.84), (1.85)] are not present in equation (1.103). By applying spatial and temporal f i l t e r i n g techniques of the type outlined in 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 their best f i t t i n g theoretical curves, are presented 78 i n FIG 16. This data was taken for mode-locked operation of DCl at the same fundamental frequency of 11.71 MHz as in FIG 10,14,15, and under the same operating conditions as in FIG 15. FIG 16(b) corresponds to the same conditions of spatial and temporal f i l t e r i n g as FIG 9. The success of the theoretical model i s readily apparent. It was usually possible to f i t the experimental profiles to within 10 degrees. By employing various combinations of spatial and temporal f i l t e r i n g , i t was determined that, to within experimental error, the phases 6 n formed an arithmetic series of the form ^,62,63, = 0,a,2a (1.119) where a = 238° for the situation depicted in FIG 16. The phases in (1.119) are measured relative to 6^  , i.e. a reference has been chosen corresponding to 6^  = 0. The strain amplitudes and phases obtained for the fundamental and the f i r s t 12 harmonics are given in TABLE 1. FIG 17 is the strain profile reconstructed from the data l i s t e d in TABLE 1. It should be mentioned that, due primarily to the fact that the measurements allowed only the magnitude of different linear combinations of the phases to be determined, the data obtained allowed for two possible strain profiles: that presented in FIG 17 and a second profi l e obtained by inversion. The physically correct pro f i l e was inferred from the form of the acousto-electric current signal for mode-locked operation. In the active cavity, the piezoelectric potential corresponding .to the strain profile in FIG 17 would be consistent with strong electron FIG 16. Experimental (upper) and corresponding theoretical (lower) image plane intensity profiles for DCl under different conditions of spatial and temporal f i l t e r i n g : (a) spatial orders 3,4,5 with temporal f i l t e r at fi; theoretical profile: 63 - 26^ + 6^  = 0 (b) " 4,5,6 " £2; " fi - 25. + 5, = 0 4 5 6 (c) " 1,3 " 2ft; " 3S1 - 63 = 115° (d) " 3,4,5,6 " • 2ft; " 6„ - 6. - 6. + 6, = 0 3 4 5 6 (e) " 4,5,6,7 11 2ft; " 6. - 6 - S, + 6_ = 0 (f) " 5,6,7,8 " 2ft; " 6C - 6, - 6., + 6_ = 0 5 6 7 8 In a l l cases the periodicity of the profiles is A/2 = 160.5 microns, but different scaling factors have been used. Operating conditions: applied D.C. el 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 strain amplitudes and phases for mode-locked operation of DCl under the conditions: fundamental acoustic frequency = 11.71 MHz, applied D.C. elec t r i c f i e l d =1.17 KV/cm, D.C. current density = 24 mA/cm2. (after Smeaton and Haering 1976a). ^ / / n STRAIN x 106 6 n (degrees) 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 Fig 17. The acoustic strain profile for DCl, reconstructed from the experimentally determined strain amplitudes and phases l i s t e d in TABLE 1. Operating conditions: applied D.C. electric 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). 03 CO \ \ 8^ 85 bunching, giving rise to dominant negative spikes in the acousto-electric current. This i s , in fact, what is generally observed [see FIG 11(b)], The width of the pulses in FIG 17 is about 23 microns, which corresponds to a pulse duration of about 6 nanoseconds at the shear velocity of sound in fused quartz. This i s somewhat larger than the width of the dark fringes for the corresponding dark f i e l d term, FIG 15, due primarily to the fact that only the f i r s t 13 modes were used in obtaining the profile in FIG 17. A more direct comparison may be made between FIG 17 and FIG 8, since the number and relative amplitudes of the acoustic modes are the same for both these prof i l e s . The width PW in FIG 8 is about 20.2 microns for A/2 = 160.5 microns, in reasonable agreement with the pulse width in FIG 17. Hence the dark f i e l d profile does, in fact, provide a reasonable estimate of the corresponding strain pulse width. As has been mentioned, the minimum pulse duration achieved, as inferred from dark f i e l d measurements, was about 3 nanoseconds. Also, i t was possible to achieve peak strains, as estimated from measurement of diffraction intensities, in excess of 5 x 10 ~*. Due to i n s t a b i l i t y of the phonon maser in this high operating range, detailed optical measurements were not possible. The acoustic pulses produced by a mode-locked phonon maser are unique for their narrow width. There is no other known source of nanosecond acoustic strain pulses. An additional 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 - 9 of about 10° Hz and a pulse width of 5 5 x 10 s e c . Hence i t produces "D.C." pulses which only contain about 1/2 cycle of the carrier. The composite cavity fundamental frequency, or reciprocal of the round-trip transit time, was 323.4 KHz for DCl. The frequency spacing of the active modes in strong mode-locked operation was usually close to a multiple of the reciprocal of the round-trip transit time of the active cavity, about 736 KHz. While i t was not possible to achieve true mode-locking at the composite cavity fundamental frequency, an interesting multimode form of operation could be achieved, and the corresponding acousto-electric signal i s shown in FIG 18. The display in FIG 18(a) consists of groups of lines separated by the composite cavity fundamental frequency. The modes within a single group [see FIG 18(b)] had amplitudes randomly varying i n time, indicating the absence of phase-locking. The acoustic modes participating in mode-locked operation were found to be harmonically related to better than 1 part i n IO 4. For single frequency osc 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 cavity (Hughes and Haering 1976). For fused quartz and the chosen orientation of CdS, the maximum deviation from harmonicity i s less than 3 KHz, and the amount of mode-pulling required to induce harmonicity for mode-locked operation i s minimal. On this basis, one would predict that mode-locking should be d i f f i c u l t to achieve in composite cavities 87 ( b ) FIG 18. Frequency display of the acousto-electric current for DCl operated under the conditions: applied D.C. electric f i e l d = 1.32 KV/cm, D.C. current density = 13.3 mA/cm? Horizontal scale is (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 for fused quartz as the material for the passive cavity, the maximum deviation from harmonicity i s greater than 100 KHz (Hughes and Haering 1976). A composite cavity formed with a bj_ phonon maser and a sapphire passive cavity.(DC3) was found, i n fact, not to display strong mode-locking characteristics. 89 1.5 Optical Determination of the Normal Modes of Composite Cavities The normal modes of composite cavities have previously been obtained by measurement of the active admittance as a function of frequency (Hughes and Haering 1976). In this situation, the composite cavity was located in one arm of a symmetrical admittance bridge. Since the bandwidth of the phonon maser is > 300 MHz, the bridge was required to be accurately balanced over a wide range in order to examine the cavity modes of interest. A much simpler, optical technique may be applied in many situations. Using a configuration similar to FIG 12, the normal modes of composite cavity DCl were measured by monitoring the intensity of diffracted light as a function of frequency. The active crystal was driven by a Hewlett Packard model 8601A sweep oscillator (maximum output voltage 3 V rms). No attempt was made to optimize signal to noise by use of phase sensitive detection or other techniques. A typical spectrum is shown in FIG 19. The spacing of the lines (~ 323 KHz) corresponds to the reciprocal of the round trip transit time of the composite cavity. The ratio of the line width to the center frequency for the modes in FIG 19 is a factor of 2 larger than for the e l e c t r i c a l impedance measurements of Hughes and Haering (1976). This is due to Q-spoiling of the active cavity produced by the presence of residual scattered laser l i g h t . With proper light shielding and optimization of the experimental arrangement, the optical technique should easily match 50.1 4 M H z I— 1MHz H KJ FIG 19. Mode structure of composite cavity DCl, obtained by optical diffrac 91 the resolution of the e l e c t r i c a l measurements. Aside from inherent simplicity, the optical technique has the advantage of wide frequency capability. Using a Hewlett Packard model 3200B VHF os c i l l a t o r to drive the active crystal, diffraction could be observed by eye for frequencies in excess of 400 MHz. In fact, i f the Bragg condition is optimized, there i s , in principle, no restriction on the frequencies to be examined. The major disadvantage of this technique is that i t restricts the form of the passive cavity. The passive cavity must be transparent for one of the laser lines available, and should have a thickness £ 200 microns. In addition, anisotropic materials must be oriented so that the appropriate optoelastic constants are non-zero. These constraints are satisfied, however, for many cases of practical interest. 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 direct evidence of mode-locking was provided by employing laser diffraction and spatial f i l t e r i n g techniques (Smeaton, Hughes, Vrba and Haering 1976), and by means of a unique combination of spatial and temporal f i l t e r i n g sufficient phase information was recovered to allow reconstruction of the mode-locked strain profile (Smeaton and Haering 1976a). The general technique of combining spatial and temporal f i l t e r i n g provides a powerful method of processing and analyzing optical signals. At high operating conditions, strain pulses of width - 3 nanoseconds and amplitude > 5 x 10 ~* were achieved. For future experiments, i t would be of interest 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 in this situation. Hence wider bandwidths and correspondingly narrower strain 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 piezoelectric semiconductors have received considerable experimental and theoretical 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 effects result from a nonlinear interaction of the electric f i e l d produced by an active acoustic mode of wave number K and angular frequency ft, with an externally applied, spatially invariant A.C. electric f i e l d of frequency equal, or harmonically related to ft. Of present interest are so-called 3 pulse echoes (Shiren et a l 1973) which are produced in the following manner. At t=0 an acoustic pulse of wave number K and angular- frequency ft. is introduced into a piezoelectric crystal. 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 via the piezoelectric interaction. At a later instant t=x, an R.F. pulse of frequency ft (or harmonically related to ft - Melcher and Shiren 1975) is applied to the crystal. Aside from piezoelectric effects, this second pulse w i l l produce a spatially invariant A.C. f i e l d in the crystal. If, at time t=x+T, another R.F. pulse is Applied, an echo pulse i s recorded at time t=2x+T. 94 In some recently proposed models (Melcher and Shiren 1975, Chaban 1975), localized electron traps play a fundamental role in the formation of echoes. It i s postulated that a nonlinearity in the system produces a redistribution of the trapped charge, which, in turn, produces a static electric f i e l d distribution whose spatial variation mirrors the periodicity of the propagating acoustic wave. There is some controversy over the nonlinear mechanism involved, and there are two basic models of the interaction: (i) Electron "bunching": In this model, set forth in theoretical papers by Chaban (1972, 1974, 1975), the periodic piezoelectric potential associated with the propagating acoustic wave gives rise to a modulation ("bunching") i n the conduction band electron density. The electron density then has the form [see equation (1.9)]: / .\ . i(Kx-ftt) , ' ,A N N n(x,t) = n Q + n-j^ e + c c . (2.1) where n Q i s the equilibrium electron density. At the instant t=x an A.C. electric f i e l d of the form „ islt , E l e + c' c- (2.2) is applied along the x-axis of the crystal. Since the current density, equation (1.6), contains a nonlinear 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 e i K 95 A periodic space charge w i l l be b u i l t up to compensate for this current. Hence trapping w i l l be more intense in some parts of the crystal than in others and, in the absence of the acoustic and e l e c t r i c a l signals, a bound space charge w i l l exist, having the spatial periodicity of the acoustic wave. ( i i ) Electric field-assisted detrapping (Shiren et al 1973, Shiren 1975, Melcher and Shiren 1975): In this model one considers the total electric f i e l d E i n the crystal at time t=x. It is given by the sum of the piezoelectric f i e l d (amplitude E 2) of the propagating acoustic wave, and the spatially uniform A.C. electric f i e l d : „ , . v „ islt , _ i(Kx-ftt) , .„ „. E t(x,t) = E^e + E 2e + c c . (2.3) It i s assumed that the total electric f i e l d i n equation (2.3) produces field-assisted detrapping of electrons i n i t i a l l y uniformly distributed in shallow traps. Since this process is highly nonlinear (Haering 1959), i t gives rise to many components in the conduction electron density harmonically related in time and space to the total electric f i e l d . In a manner similar to ( i ) , the nonlinear current density [equation (1.6)] will-then contain spatially nonuniform, time invariant terms and, in the absence of the perturbing f i e l d s , a stored periodic space charge pattern w i l l exist. For both (i) and ( i i ) , the charge grating w i l l decay with a time constant related to the lifetime of the traps involved. If the A.C. electric 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 electic f i e l d of the space charge grating to produce a backward propagating (time reversed) acoustic wave (as well as a forward wave), which is detected as a pulse echo at the surface of the crystal at time t=2r+T. Experimental data regarding acoustically induced space charge gratings has been obtained almost exclusively by pulse echo measurements, and i t i s of considerable interest to find an experimental technique that w i l l provide supplementary information. Since the electric f i e l d produced by a static space charge grating w i l l modulate the refractive index in a piezoelectric crystal, via the primary and secondary electro-optic effect (Nye 1964), optical measurements are indicated. The variation in the indicatrix produced by the electro-optic effect may be represented by (Nye 1964) [ c f . equation (1.43)]: AB.. = Z...E. (2.4) IJ x j k k where z^-^> E ^ are components of the electro-optic tensor and the electric f i e l d . Contracting the indices i j according to equation (1.45) allows equation (2.4) to be represented in matrix form, in a manner exactly analogous to equation (1.47). For hexagonal 6mm symmetry, appropriate for CdS, the electro-optic matrix has the form (Nye 1964): 97 0 0 z13 0 0 Z13 0 0 z33 0 Z42 0 z42 0 0 0 0 0 (2.5) For the case of an electric f i e l d p a r a l l e l to the c-axis, for example, the appropriate electric f i e l d component is E 3 and one obtains: A B , AB , A B , A B r A B , Z13 E3 Z13 E3 Z33 E3 0 0 0 (2.6) The coefficients in equation (1.39) become: B l = B2 = < n2 + Z13 E3 > B3 = ( n2 + Z33 E3 > e (2.7) = B 5 = B 6 - 0 98 no> n e are the ordinary and extraordinary r e f r a c t i v e indices [refer to equation (1.42)]. The new i n d i c a t r i x thus becomes: < rT2 + Z13 E3 > ( X ! + X2> + ( n-| + Z33 E3 } *l ( 2 ' 8 ) For l i g h t propagating along the b or x 2 a x i s , the appropriate index e l l i p s e i s given by: ( n2 + Z13 E3 > X ! + ( £ | + ^ 3 3 * 3 ) (2 . 9 ) and has semiaxes: n. = _ (1 + n 2 z 1 3 E 3 ) 2 (2.10) n e 1e z33 t3' n e (1 + n ^ Z o o E o ) 5 5 Since n 2 z 3 3 E 3 , n 2 z ^ 3 E 3 << 1, the square roots i n (2.10) may be expanded to y i e l d : A n o a ~ \ n o z 1 3 E 3 An e - - | n | z 3 3 E 3 (2.11) 99 Equation (2.11) may be directly compared with equation (1.58). This strong analogy with the photoelastic effect indicates that i t should be possible to apply the laser diffraction and spatial f i l t e r i n g techniques of CHAPTER 1, to directly observe acoustically induced space charge gratings. 2.2 Application of Optical Processing In principle, i t should be possible at low temperatures to observe diffraction from stored charge gratings which have been previously prepared by the application of suitable electric f i e l d s . In practice this has not proved possible with our crystals, presumably because the trapped charge is disturbed by the probing light used in the diffraction experiment. This i s consistent with the experimental findings of Shiren et,al (1973). For this reason, optical 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 this steady state situation, the acoustically induced charge gratings exist in the presence of l i g h t . However, i t is then necessary to devise a measurement technique which can distinguish between diffraction from a static charge grating and diffraction from the acousto-electric fields which are simultaneously present. The electric f i e l d in the crystal may be written: „ i f i t , _, r i(Kx - n t ) , i(Kx+ftt), , ,„ E(x,t) = E^e + ^[e + e 3 + c.c. ( 2 . 1 2 ) where x i s taken along the c-axis. It consists of a spatially invariant 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 right and l e f t going waves of -equal amplitude E 2- The latter fields are produced via the piezoelectric interaction of the crystal with the driving f i e l d . Equation (2.12) does not take into account the electric fields produced by modulations of the space charge. The static space charge grating i s produced from the fie l d s E-^  and E 2 in equation (2.12) by one of the nonlinear effects described in section 2.1, and, as was indicated, w i l l produce a periodic modulation of the lo c a l refractive index of the crystal via the electro-optic effect. It thus presents a fixed phase grating to incident laser light and results in d i f fraction. In the Raman - Nath regime (refer to section 1.3.2 of CHAPTER 1) the diffraction patterm w i l l consist of a row of spots symmetrically arranged about the zero order or undiffracted beam. The acousto-electric f i e l d E 2 w i l l also produce diffraction due to photoelastii (section 1.3 of CHAPTER 1) and electro-optic coupling. In the experiments performed, this diffraction was roughly two orders of magnitude more intense than that produced by the static space charge. Moreover, since the spatial periodicity was the same in both cases, both diffraction patterns were superimposed. It was possible to completely separate the two optical signals, however, by means of spatial and temporal f i l t e r i n g . This may be demonstrated by examining the optical phase transformation produced by the acoustic standing waveform. In analogy with equations (1.83) and (1.85), i t may be written: 101 . CO = i2vsinKx cosftt _ v w „ inKx e - l J n(2vcosfit)e (2.13) e n=_oo The Bessel function relationship, equation (1.87), has again been used. The optical amplitude in the order diffraction spot in the focal plane i s proportional to J n(2vcosftt). Consider that only diffraction spots n and m are combined and imaged. The image plane intensity distribution i s then proportional to ^.m = J n ( V ) + Jm ( V ) + 2 Jn ( V> J m(V) cos (n-m)Kx (2.14) where V = 2vcosSlt (2.15) If we extract those components of (2.14) that have no time variation, the f i r s t two terms w i l l contribute only a uniform background. From the well known relationship (Abramowitz and Segun 1968) J m(-z) = ( - l ) m J m ( z ) (2.16) and the form of V in equation (2.15), i t is apparent that the cross term in (2.14) w i l l only contain a D.C. component i f n+m i s an even number. Thus for n+m odd, the D.C. part of the image plane signal w i l l contain no terms with spatial variation. This i s rigorously true, even i f weak acoustically induced variations of the optical absorption coefficient are taken into account. 102 Consider a few s p e c i a l cases of p a r t i c u l a r i n t e r e s t : [1] Adjacent d i f f r a c t i o n orders, m = n+1 : the D.C. s i g n a l c o n s i s t s s o l e l y of a uniform background, produced by the f i r s t two terms i n equation (2.14). [2] Symmetric d i f f r a c t i o n orders, m = -n : using equation (1.93), i t i s apparent that the D.C. component of (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 of the dark f i e l d term, equation (1.103), see FIG 1 4 ( c ) , ( d ) ] . Thus f o r n=l, f o r example, the image plane 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 of a cosine pattern of bri g h t and dark f r i n g e s with p e r i o d i c i t y A/2, where A = 2TTK \ [3] m = n+2 : f o r t h i s s i t u a t i o n , the t h i r d term i n (2.14) provides a D.C. component proportional to cos2Kx. Thus cosine f r i n g e s appear having p e r i o d i c i t y A/2, s i m i l a r to [2] with symmetric orders ±1. The f r i n g e s w i l l be l e s s d i s t i n c t i n t h i s case, however, due to the presence of a uniform background D.C. component produced by the f i r s t tx^o terms i n (2.14). Image plane photographs x^hich i l l u s t r a t e the above r e s u l t s are shown i n FIG 20. This data was taken with a model system c o n s i s t i n g of two orthogonally arranged 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. The two transducers were driven simultaneously by a General Radio model 1211-C o s c i l l a t o r and a two dimensional d i f f r a c t i o n pattern, FIG 20(a), r e s u l t e d . In obtaining FIG 20(b)-(g) only the c e n t r a l v e r t i c a l row of d i f f r a c t i o n spots was used. I t should be noted that the d i f f e r e n t spots i n FIG 20(a) correspond 103 ( e ) ( f ) ' ( g ) FIG 20. Experimental data obtained from a model system consisting of two orthogonal quartz transducers (resonant frequency 10 MHz) mounted in a c e l l of d i s t i l l e d water, (a) focal plane diffraction pattern, (b) zero order light imaged with sound generator off. Image plane distributions produced by combining and imaging diffraction spots (c) 0,1,(d) 1,2,(e) ± l , ( f ) 0,2 and (g) 1,3. 104 to different orders of Raman - Nath diffraction 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 diffraction from a waveform consisting of many harmonically related components. In FIG 20(c) and (d) adjacent diffraction orders 0,1 and 1,2 , respectively, have been combined and imaged. By comparison with FIG 20(b), the zero order light with the sound generator off, i t i s apparent that no fringes are present, as was predicted i n [1]. In FIG 20(e) symmetric orders ±1 have been imaged. A pattern of periodicity A/2, consisting of alternating dark and bright fringes i s produced, in agreement with [2], In FIG 20(f) and (g) orders 0,2 and 1,3 , respectively, have been imaged. A fringe pattern of periodicity A/2 i s produced. As predicted in [3], the fringes are less distinct than in FIG 20(e). If, i n addition to (j>^  i n equation (2.13), a static phase grating i s present, such as would be produced by a fixed space charge pattern, the image plane signal w i l l be modified. A sinusoidal grating of period A = 2irK ^  w i l l produce a phase transformation i<f>s _ iv_sinKx e = e ° > (2.17) where V q is the amplitude of the optical phase variation (v Q << 1). <ps may be considered to be the fundamental component of the periodic space charge pattern. The phase (J>s (or the combination of <ps and <p )^ produces an image plane D.C. term of the form cosKx, i f diffraction 105 orders 0,1 (or any other pair of adjacent diffraction orders) are combined and imaged. In practice, i t was necessary to amplitude modulate the external oscillator and to phase-lock the image plane detector to the modulation frequency, in order to remove extraneous signals and optical noise in the image plane. With this in mind, the total phase transformation given by equations (2.13) and (2.17) may be written: exp I (2vcosfit + v Q)(1 + Mcoswmt)sinKx; = I J n[(2vcosftt + v Q ) ( l + Mcostomt)]e inKx n=_oo (2.18) where M is the modulation index (- 0.1 in practice) and u)m is the modulation frequency. The Bessel functions in (2.18) may be Taylor expanded to yield: Jn(2vcosf2t + v Q) + (2vcosf2t + v 0)Mcosw mt J^(2vcosftt + v Q) + (2.19) In practice, v D,v << 1 and v Q ~ v? It is therefore necessary to retain contributions to f i r s t order in v Q, and to second order in v. Only the f i r s t two terms in (2.19) need be considered since they contain a l l contributions to second order in both v and v Q . The derivative of the Bessel function in (2.19) may be expanded using the relation (Davis 1968): 106 zJ^(z) = nJ n(z) - z J n + 1 ( z ) (n = 0,1,2 ) (2.20) If diffraction orders 0,1 in equation (2.18) are combined and imaged, the resulting image plane intensity distribution i s proportional to [see equation (2.14)]: J 2 ( V ) + j J ( V ' ) + 2J 0(V')J 1(V ,)cosKx (2.21) where V = (2vcosftt + v Q)(1 + Mcosu^t) (2.22) Using equations (2.19), (2.20), the small argument expression for 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 in v): l^mQ = M(vQcosKx - v 2) (2.23) Thus the presence of the static space charge grating i s indicated by a cosine fringe pattern of periodicity A. The component -Mv2 in equation (2.23) i s a uniform background produced by acoustic diffraction effects. Its negative sign indicates that i t is ir out of phase with the amplitude modulating reference. 107 2.3 Experimental Results The experimental arrangement for optical measurements \<ras essentially the same as in 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 this configuration, the piezoelectically active, acoustic modes consist of longitudinal waves whose K-vectors l i e along the c-axis. The light source was a o 5 mW HeNe laser (X = 6328 A) and incident polarization was p a r a l l e l to the c-axis. Aside from the laser l i g h t , no ancillary illumination of the crystals was used. A s l i t width of 250 microns was used for the scanning photomultiplier in 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 profiles for cj_ crystal 24.06.06.01. The external oscillator in this case was an Arenburg model PG-650C. The photographs on the LHS of FIG 21(b) and (c) were obtained by imaging diffraction orders ±1. The spacing of the bright fringes is thus A/2 (refer to section 2.2). The photographs on the right 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 static phase grating. To obtain more reliable data, the technique outlined in section 2.2 was employed. The crystals were driven by a Hewlett Packard model 3200B VHF oscillator, 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 signal provided 108 FIG 21. (a) Bright f i e l d image of c]_ c r y s t a l 24.06.06.01, and i n t e n s i t y p r o f i l e s produced by imaging d i f f r a c t i o n spots ±1 (LHS) and 1,2 (RHS) f o r d r i v i n g frequencies (b) 15.619 MHz and (c) 19.062 MHz. 109 the reference for a P.A.R. model HR-8 lock-in amplifier, which was used to process the signal from the image plane detector. For typical operating -2 conditions, the Raman - Nath parameter v s 5 x 10 , as calculated from the diffraction intensities. An example of the image plane signal for c[_ crystal 24.06.06.01 is shown in FIG 22. The upper curve was obtained by imaging orders ±1. As has been indicated, the resulting pr o f i l e has periodicity A/2 and is presented as a ref erence. The lower curve i n FIG 22 was obtained by combining orders 0,1 (the variable amplitude of the optical signals is a reflection of the intensity p r o f i l e of the laser beam). The appearance of sinusoidal fringes of spacing A i s direct evidence for the presence of a fixed charge grating of the form sinKx [refer to equation (2.17)]. This sinusoidal term is superimposed on a larger negative (i.e. ir out of phase with respect to the amplitude modulating reference) background term, in 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 Q. Hence, with the aid of equations (1.2), (1.72) and (2.11), the peak electron density modulation associated with the static 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 in pulse echo measurements, performed at correspondingly higher frequencies and driving amplitudes (Shiren 1975) . To check these results, a double cavity (DC4) was constructed 'by bonding together two similarly oriented c_j_ CdS crystals. Either side FIG 22. Image plane signal for c[_ crystal 24.06.06.01, produced by imaging diffraction spots (a) + 1 and (b)^ 0,1. I l l of the resulting composite cavity could be driven by an external o s c i l l a t o r . This produced a situation in 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 driving f i e l d E-^  was present only in one part. Laser diffraction could be done in one or both parts of DC4 and, as i s demonstrated in FIG 23, the optical signals could be completely separated by focal plane spatial f i l t e r i n g . The image plane signal for one part of DC4, both in the presence and absence of the driving f i e l d E^, is presented in 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 art 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 establish whether the static charge grating was associated with trapped or mobile charge. However, the strong sample dependence of our measurements i s an indication of the importance of defects i n producing the observed results. To determine the presence of shallow traps of the type postulated by Melcher and Shiren (1975), thermoluminescence measurements were performed. CdS samples, freshly etched in concentrated HC1, were mounted in a LHe dewar at 4.2°K. After being illuminated with strong band gap ligh t , the crystals were maintained i n the dark. Once the afterglow had subsided (10 - 15 min. after illumination), the samples were warmed up and the luminescent intensity was recorded as a function of time. The resulting glow curves were analyzed using the techniques of Garlick 112 FIG 23. Image plane photographs of DC4, i l l u s t r a t i n g that different components of the optical signal may be examined individually, (a) a l l light has been imaged, (b) the background ligh t has been removed, (c) only the background light has been imaged, (d) the light from the lower member of DCA has been removed. FIG 2k, Image plane.signals obtained from the upper member of DC4 in FIG 23. The profiles correspond to (a) orders + 1 combined and imaged; and orders 0,1" imaged {b\ in the absence and (c) in I the presence of the external driving f i e l d . 114 and Gibson (1948), Grossweiner (1953) and Booth (1954). A typical spectrum is shown in FIG 25. As indicated, these measurements ve r i f i e d the existence of shallow traps with an ionization energy of about 20 meV, as required for the model proposed by Melcher and Shiren (1975) [refer to model ( i i ) in section 2.1]. In principle, the diffraction experiments could be extended to study the higher Fourier components of the static charge grating. This would be of interest since these components would y i e l d information about the nature of the underlying nonlinearity. In practice, this extention has not been possible with our crystals because the observed diffraction effects were too weak. 115 ~20meV ~150meV I FIG 25. A typical glow curve obtained from a CdS crystal of the type used to construct phonon masers. The points at which different temperatures occurred are indicated on the horizontal axis, and the calculated trap ionization energies corresponding to the major thermoluminescent peaks are given. 116 2.4 Conclusions and Summary of Contributions By means of the spatial and temporal f i l t e r i n g techniques outlined in CHAPTER 1, optical experiments were performed that allowed the f i r s t direct observation of acoustically induced space charge gratings in CdS (Smeaton and Haering 1976b). The information obtained by optical means should be of considerable supplimental value to data obtained by pulse echo measurements. Proper exploitation of the techniques described could provide insight into the underlying physical processes involved in the formation of the static charge gratings. 117 CHAPTER 3 NEW PHOTOVOLTAIC EFFECTS IN CdS 3.1 Introduction Photovoltaic effects constitute a class of phenomena in which light generates a voltage across a portion of a semiconductor. Such effects have been observed for several decades in CdS and other semi-conductors (for a review see Pankove 1971) . The samples used in the present experiments were polished phonon maser crystals, constructed from high r e s i s t i v i t y [10 1 0 - 10 1 2 (ficm) 1 in the dark] photoconductive CdS. For this situation, the dominant known photovoltage i s produced via the Dember effect (Williams 1962). The Dember voltage results when strongly absorbed radiation generates a high density of electron-hole pairs, which then diffuse away from the illuminated surface. The electrons have a higher mobility than the holes and hence extend further into the crystal, tending to make the surface positive with respect to the bulk. This effect, and the resulting voltage, are named for their discoverer (see Dember 1931) . In this chapter some i n i t i a l results from on-going experiments are presented to i l l u s t r a t e the existence of two new photovoltaic effects in CdS. As w i l l be shown, these effects provide D.C. voltages which are several orders of magnitude larger than the Dember voltage. As yet, the underlying physical mechanisms are not understood. 118 3.2 The Photoacousto Voltaic Effect A D.C. voltage, requiring the simultaneous presence of light and propagating acoustic fi e l d s , has been discovered i n CdS phonon masers. The basic experimental configuration for measurement of the photoacoustic voltage i s shown in FIG 26. Composite cavities, consisting of txro similarly oriented b_[_ or c_|_ phonon maser crystals, were used in the experiments. The active side of the double cavity 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. electric f i e l d to achieve phonon maser action. The free surface of the passive cavity was illuminated and the D.C. voltage appearing across the crystal was measured with a Keithley model 153 electrometer, or a Fluke model 8120A d i g i t a l voltmeter. The form of the photoacoustic voltage for DC5 as a function of light intensity i s shown in FIG 27(a). The corresponding short c i r c u i t photoacoustic current had a maximum value % 0.1 mA/cm2, and mirrored the light intensity dependence of the photoacoustic voltage. For comparison, passive cavity incident light r active cavity D.C. meter _ L _ oscillator or D.C. power supply . FIG 26. Experimental configuration for measuring the photoacoustic voltage. 119 FIG 27. (a) Photoacoustic voltage measured across b]_ crystal 29.04.03.02 (pass ive part of DC5) illuminated with 4880 A lig h t . The active part of DC5 (bj_ crystal 29.04.01.02) was operated in mode-locked fashion, under the conditions: applied D.C. elec t r i c f i e l d = 1.35 KV/cm, D.C. current density = 15 mA/cm2, fundamental acoustic frequency =18.2 MHz. (b) Dember voltage as a function of light intensity at 5145 A for bj_ crystal 33.09.01.01 (part of DC2). 120 o 1 — — — — — — — — o (a) - i 1 1 5 10 15 LIGHT INTENSITY (mW/cm2) io 1b2 i b 1 i b 1 LIGHT INTENSITY (mW/cm2) 121 the Dember voltage for bj_ crystal 33.09.01.01 is presented in FIG 27(b). The photoacoustic voltage was 3 to 4 orders of magnitude larger than the corresponding Dember voltage. For strongly absorbed lig h t , the sign of the photoacoustic voltage was generally the same as the Dember voltage, i.e. the illuminated side of the passive cavity was positive with respect to the dark side. However, anomalous sign reversals could be observed at low strain amplitudes and for certain acoustic frequencies. The spectral response of the photoacoustic voltage generally took the form illustrated in FIG 28(a), and displayed strong correllation with the response of the photoconductivity, FIG 28(b). This seems to indicate that the effect i s proportional to the number of mobile carriers. When the active crystal was driven by an external o s c i l l a t o r , the photoacoustic voltage only appeared at multiples of the reciprocal of the round trip transit time of the composite cavity (~ 1.28 MHz for DC5) . This is illustrated in FIG 29. At higher conductivities ( i . e . higher light intensities), the peak of the photoacoustic voltage shifted to lower frequencies [compare curves (1) and (2) in FIG 29], i n agreement with the known conductivity tuning characteristics of CdS phonon masers (Burbank 1971). When the passive crystal was illuminated at - 5200 A [the peak wavelength in FIG 28(a)] i t was possible to achieve photoacoustic voltages of up to 15 volts. 3.2.1 Discussion While the underlying physical mechanisms that produce the photoacoustic voltage are poorly understood, two qualitative models \ deserve mention: 122 FIG 28. (a) The spectral response of the photoacoustic voltage for b_[_ crystal 29.04.03.02 (part of composite cavity DC5) . The active part of DC5 (bj_ crystal 29.04.01.02) was driven at 7.55 MHz by an external o s c i l l a t o r . (b) The spectral response of the photocurrent for b]_ crystal 29.04.03.02 under the same conditions of illumination as (a). The light was provided by a 200 W tungsten halogen lamp in series with a Heath monochromator. 124 12.76 MHz FIG,29. Behavior of the photoacoustic voltage for DC5 as a function of the A.C. driving f i e l d applied to the active part of the composite cavity. Weakly absorbed (orange) light was used. The curves for two different conductivities (light intensities) are presented: a « (1) 2 x 10""4(ftcm)-1 and (2) 4 x 10~ 7 (ftcm) - 1. 125 (1) "Mode-pulling" The active modes in a phonon maser must satisfy the phase condition (Vrba and Haering 1973): ftd( - + — ) + 20 = 2mr (3.1) vs+ v s -where n is an integer, d is the cavity length, v g + ( v s_) 1 S t h e velocity of sound para l l e l (antiparallel) to the applied D.C. electric f i e l d . It is well known, from the linear theory of acoustic amplification in piezoelectric semiconductors (see, for example, McFee 1966), that the velocity of sound i s a function of both conductivity and electric f i e l d . This forms the qualitative basis for the tuning characteristics of phonon masers, since varying the conductivity or electric f i e l d modifies the velocity of sound and requires the phonon maser to adjust i t s frequency in order to satisfy (3.1). In the present case, illumination modifies the conductivity in the passive part of the composite cavity, thus changing the velocity of sound and spoiling the resonance condition, equation (3.1). It is conceivable that resonance can be restored, however, i f the crystal establishes a D.C. electric f i e l d of the proper magnitude, thus returning the velocity of sound to i t s value before illumination. This highly qualitative model does not establish the sign of the photoacoustic voltage. 126 (2) Electron transport In this model we assume that the piezoelectric potential wells associated with the acoustic wave act as "buckets" to transport optically created electrons from the illuminated surface to the dark surface. The resulting D.C. voltage should have the same sign as the corresponding Dember voltage. This model does not readily account for voltages of opposite sign to the Dember voltage, or for effects observed with weakly absorbed li g h t . It has not yet been possible to form a conclusive link between the experimental results and the preceeding models. 127 3.3 The A.C. Electric Field Induced Photovoltaic Effect A second D.C. voltage, requiring the simultaneous presence of light and an externally applied A.C. electric f i e l d , has been discovered in CdS crystals. The experimental configuration for measurement of this photovoltage i s shown in FIG 30. The capacitor C^ in FIG 30 served to D.C. decouple the external os 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. driving voltage. The samples used in these experiments were polished phonon maser crystals and hence had In diffused surfaces (see APPENDIX B) . The spectral response of the D.C. voltage i s shown in FIG 31. Similar to the photoacoustic voltage, i t contained one dominant peak, O shifted ~ 100 A to shorter wavelengths with respect to the peak in FIG 28(a). Also, the sign of the voltage generally reversed for wave-o lengths £ 5250 A. As a function of ligh t intensity, the D.C. voltage incident light r 1 1 i f CdS . L crystal ' , L Jvwsw-1 l / x . external I V i oscillator ~ 1 D.C. meter FIG 30. Experimental configuration for measuring the A.C. ele c t r i c f i e l d induced photovoltage. 128 5108A o o LO LO WAVELENGTH OF ILLUMINATION (A) FIG 31. Spectral response of the D.C. photovoltage for bj_ crystal 29.04.01.02 (part of DC5). The intensity of illumination was ~ 0.5 mW/cm2. The light was provided by a 200 W tungsten halogen lamp in series with a J a r r e l l Ash monochrometer. 129 saturated very quickly, in a manner similar to FIG 27. It displayed a quadratic dependence on the amplitude of the A.C. driving voltage, as illustrated in FIG 32(a), and as the driving frequency was increased the D.C. voltage f e l l off quickly, as i s shown in FIG 32(b). The D.C. photovoltage appeared at a l l frequencies, indicating that i t s presence was probably not related to the resonant acoustic structure of the crystal. At frequencies corresponding to acoustic resonances of the crystal, the photoacoustic voltage was also present. In FIG 33(a) the D.C. photovoltage is shown as a function of frequency for illumination with small amounts of weakly absorbed l i g h t . As the light level (and hence the conductivity) was reduced, the photoacoustic voltage became dominant [compare curve (3) in FIG 33(a) with curve (2) in FIG 29]. For higher light levels, and particularly in the case of strongly absorbed light, the nonresonant photovoltage dominated, as illustrated in FIG 33(b). To further investigate the behavior of this photovoltage, pulse measurements were performed at temperatures from 5°K to 300°K. The crystals used were mounted in a variable temperature LHe dewar, and were illuminated with white light provided by a 100 W Hg arc lamp. When the light was shut off, by means of a mechanical shutter, an R.F. pulse, whose width and delay time with respect to the end of the optical signal could be varied, was applied to the crystal. 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) for bj_ crystal 29.04.01.02 o (part of DC5). The illumination was at 5100 A, with an intensity of about 0.5 mW/cm2. The solid curve through the experimental points corresponds to V = (2.78 x 10 )V 2 . JJ • L» * .A. # • (b) The D.C. photovoltage as a function of driving frequency at constant amplitude for bj_ crystal 29.04.03.02 (part of DC5), illuminated with a small amount of weakly absorbed (orange) lig h t . 131 O 5 10 15 20 APPLIED A.C. VOLTAGE (V P P) 0 5 10 15 20 25 DRIVING FREQUENCY (MHz) 132 FIG 33. (a) The D.C. photovoltage for bj_ crystal 29.04.01.02 (part of DC5) as a function of frequency, for low levels of weakly absorbed (orange) light. Curves for three different conductivities (light levels) are presented: a - (1) 1.4 x 10~^(ftcm)~1, (2) 5 x 10~ 5(ftcm) _ 1 and (3) 4 x 10~ 7(flcm)" 1. (b) The D.C. photovoltage for b|_ crystal 29.04.01.02 as a function of frequency (the driving amplitude was not constant). o The crystal was illuminated with about 0.5 mW/cm2 at 5100 A. The separation of the small negative peaks i s equal to the reciprocal of the round t r i p transit time of DC5 (- 1.28 MHz). 134 The t r a i l i n g edge of the optical signal and the form of the R.F. pulse are shown in FIG 34(a). For a given set of conditions, i t was found that both the lifetime and amplitude of the pulse generated voltages increased by a factor of 2-3 in cooling the samples from room temperature to 100°K. At temperatures much below 70°K, however, the signals rapidly became weaker, and were almost nonexistent at 5°K. The voltage signals generated at - 80°K by R.F. pulses of different delay times are shown in FIG 34(b). If the time for generation tg was measured from the end of the optical signal [see lower traces in FIG 34(a), (c), (id)] the - t fx amplitude of the generated voltage decayed roughly as e 8 , where x ~ 10ms. The decay time of the voltage dependended strongly on the level of illumination. As i s shown in FIG 34(c) and (d), the amplitude of the generated voltage increased, while the decay time of the voltage decreased, with increasing light intensity. 3.3.1 Discussion Even less i s understood about the physical processes involved in this photovoltage than in the case of the photoacoustic voltage. Its dominant characteristics, however, do not appear to be strongly correlated with the acousto-electric properties of the CdS samples. The characteristic decay time illustrated in FIG 34(b) is approximately the same size as the electron lifetime, as determined using A.C. photoconductivity measurements. This suggests that the photovoltage is associated with mobile rather than trapped charge. The behavior ill u s t r a t e d i n FIG 34(c) and (d) is consistent with the crystal behaving as a battery with a 135 FIG 34. Some results of the pulse measurements: (a) lower trace shows t r a i l i n g edge of (negative) signal produced by sample illumination; upper trace shows the R.F. pulse (11 MHz), in this case having a length of 200 usee and a delay of 10 msec with respect to the end of the optical signal, (b) voltages produced by R.F. pulses of 1 msec width, applied at various delay times to c]_ crystal 33.05.01.01 (T = 80°K), and voltages produced in cj_ crystal 33.04.02.00 (upper traces) by 10 msec delayed R.F. pulses (width 200 usee) at T - 100°K for (c) weak illumination and (d) intensity of illumination increased by a factor of 10 with respect to (c). 136 Horizontal scale: 2ms/div. Upper trace - lOV/div. V e r t i c a l scale: _ n c,,/j.„ Lower trace - 0.5V/div. Horizontal scale: 5ms/div, V e r t i c a l scale: 0.5V/div. (a) (b) Horizontal scale: 2ms/div. Upper trace V e r t i c a l scale: Lower trace 20mV/div. 0.5V/div, Horizontal scale: 2ms/div. Upper trace Vertxcal scale: L o w e r t r a c e 0.2V/di 0.5V/di (c) (d) 137 variable internal impedance. At high light levels, the conductivity of the crystal i s larger. Hence i t s internal impedance is smaller and i t "discharges" more quickly than for low light levels. 138 3.4 Conclusions and Summary of Contributions Two new photovoltaic-type effects have been discovered in CdS. The voltages incurred are remarkable for their magnitude: photoacoustic voltages of about 15 V and A.C. electric f i e l d induced photovoltages of about 25 V were achieved with very small lig h t levels ( £ 1 mW/cm2). These voltages are several orders of magnitude larger than the normal photovoltages encountered at similar l i g h t levels i n CdS. At this time, the experimental results are not understood and more work is warranted in this area. 139 APPENDIX A ACOUSTIC STANDING WAVE PATTERN FOR MODE-LOCKED OPERATION In practice, the strain profiles of the phonon maser were observed by acoustic coupling to a passive cavity of fused quartz. This allowed the strain p r o f i l e to be optically probed in the passive cavity, without disturbing the phonon maser (refer to section 1.4 of CHAPTER 1). The acoustic strain at a point x in the passive cavity, represented in FIG Al , may be written as a superposition of an incident waveform F^, coupled from the phonon maser, and a waveform F^ _, which r \.J. •x = 0 - - X = x "V d-x ,-ir x = d FIG A l . has been spatially reflected from the surface x=d of the passive cavi We make the assumption that in steady state a l l waves moving in the , same direction maintain the same phase at any plane x = constant, for a l l times (i.e. no anomalous effect due to multiple reflections). 140 Then, to obtain the cor r e c t r e l a t i v e phase for F^ and F , i t may be i n f e r r e d from FIG A l that i t i s necessary to set x 2d-x f o r F . Assuming p e r i o d i c form f o r the acoustic s t r a i n , F may be x^ritten i n Fourier s e r i e s form: F ± ( x , t ) = I n=l^ a ncos[n(Kx-Slt)] + b n s i n [ n ( K x - f t t ) ] (Al) where a n , b n are the usual F o u r i e r c o e f f i c i e n t s and K, ft are the wave number and angular frequency of the acoustic fundamental. (A constant term has been omitted i n (Al) from the general form of the Fourier s e r i e s , since no D.C. s t r a i n s are associated with the acoustic waves.) Assuming no losses upon r e f l e c t i o n , i t i s apparent from the preceeding d i s c u s s i o n that F r ( x , t ) = F ±(2d-x,t) = I a ncos[nK(2d-x)-nftt] + b nsin[nK(2d-x)-nftt] n=l <• I n=l a ncos(2nKd)cos[n(Kx+ftt)] + a nsin(2nKd) sin[n(Kx+ftt)] + b nsin(2nKd)cos[n(Kx+fit)] - b ncos(2nKd)sin[n(Kx+ftt)] CO r «. = I c ncos[n(Kx+Slt)] + d nsin[n(Kx+f2t)] n=l J (A2) where c n = a ncos(2nKd) + b nsin(2nKd) d n = a nsin(2nKd) - b ncos(2nKd) (A3) 141 We now introduce phases 6 n in equation (Al) such that a n = S nsin6 n b_ = S cos6_ n n n (A4) where tan6 n = a n/b n (A5) a n d Sn = a n + bn <A6) Equation (Al) now takes the form: F i(x,t) = I S nsin[n(Kx-ftt) + 6 n] (A7) n=l In a similar manner, phases B n may be defined in equation (A2) such that: c n = C nsinB n d n = C ncosB n (A8) where tan3 n = c n / d n (A9) Making use of equation (A3) one obtains: 142 [ancos(2nKd) + b nsin(2nKd)] 2+ [a nsin(2nKd) - b ncos(2nKd)] 2 a 2 + b 2 n n S n from equation (A6) Hence n — n (A10) Using equation (A3) in equation (A9): tan8r ancos(2nKd) + b nsin(2nKd) a nsin(2nKd) - bncos(2nKd) tan6 n + tan(2nKd) tan6ntan(2nKd) - 1 using equation (A5) - tan(6 n + 2nKd) Hence gn = - <6n + 2 n K d> (All) With (A10) and ( A l l ) , equation (A2) now takes the form: F r(x,t) = + I S nsin[nK(x-2d) + nftt - 6 R] n=l (A12) From equations (A7) and (A12), the total acoustic strain may be written: F(x,t) = I S n n=l sin[n(Kx-ftt)+6_] + sin[nK(x-2d)+nftt-S ] (A13) 143 The boundary at x=d may be considered a free surface.. Thus, for the one-dimensional system considered, both the stress and strain must vanish at x=d. This can be satisfied by F(x,t) only i f the +sign is chosen in equation (A13). Equation (A13) may then be written in the alternate form: CO F(x,t) = 2£ Snsin[nK(x-d)]cos(nKd-nftt+6n) (A14) n=l The boundary at x=0 i s not free since the passive cavity i s bonded to the phonon maser at this point. The value of the strain at x=0 is determined by the parameters K,ft,5n and d in equation (A14). The acoustic modes participating in mode-locked operation may be considered as being harmonically related. Thus, for this case, the upper limit of the summation in equations (A13) and (A14) w i l l equal the number of active acoustic modes N. 144 APPENDIX B PHYSICAL PROPERTIES Material Parameters For CdS: Density 1 (10 3 Kg/cm3) p = 4.820 Elastic stiffness constants 1 (10 1 0 newton/m2) C l l = 9 , 0 7 » c i 2 = 5 - 8 1 » c i 3 = 5.10, c 3 3 = 9.38, c 4 4 = 1.504 Piezoelectric stress constants"'" (coulomb/m2) e 1 5 = -0.21, e 3 1 = -0.24, e 3 3 = 0.44 Dielectric constants (constant s t r a i n ) 1 = 9.02, K2 = 9.02, < 3 = 9.53 2 Refractive indices n Q = 2.506, n Photoelastic constants e 3,4 2.491 p n = 0.11, p 1 2 = 0.051, p 1 3 = 0.072, P 3 1 = 0.050, p 3 3 = 0.13, p 4 4 = 0.054 Electro-optic constants 5' 6 (10~ 1 2 m/V) zr\<-» C^ r^ ) ^ i ^ 5• 5 j z, e 2zf 42 "c "33 V "13 "33 Auld (1973) 2Neuberger (1969) Dixon (1967) Gainon (1964) 4 6 Maloney and Carleton (1967) Kaminow (1968) 145 Material Parameters For Fused Quartz: o Density (10 3 Kg/m3) p = 2.2 3 Refractive index n Q = 1.46 3 Photoelastic constant p ^ = 0.075 3 3 Shear acoustic velocity (10^ m/sec) v g = 3.76 Longitudinal acoustic velocity 3(10 3 m/sec) v^ = 5.95 The techniques used for fabrication of phonon masers have been detailed by Burbank (1971). The CdS crystals were oriented by means of X-ray diffraction to an accuracy of ±0.5°. For typical sample dimensions, a flatness of from 1/5 to 1/10 of an optical 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 diffusion. Techniques similar to those used for phonon masers were applied to fabricate acoustic cavities from fused quartz, sapphire, and Barium Fluoride. Composite acoustic cavities were bonded together using the technique outlined in APPENDIX C. The physical data for the samples mentioned in the main text of the thesis are l i s t e d in TABLE Bl . TABLE B l 146 - Physical Data For Acoustic Cavities t Acoustic Cavity DCl DC2 DC3 DC4 DC5 24.01.02.04 FQ1 (fused quartz) 33.09.01.01 FQ2 (fused quartz) 24.07.10.01 SI (sapphire) 27.02.04.02 27.01.01.02 29.04.01.02 29.04.03.02 Dimensions (mm) d = 1.193, b-axis X £ = 2.166 w = 1.999 d = 3.306 X £ = 10.781 w 10.760 X d = 1.090, b-axis X £ = 3.541 w = 2.775 d = 4.414 X £ = 7.844 X w 7.778 X d = 0.830, b-axis X £ = 2.228 w = 2.022 d 3.351, x,-axis X £ = 10.949 J. w = 7.087 X d 0.744, c-axis X £ : 3.252 X w = 2.288 X d = 0.819, c-axis X £ = 3.579 X w 3.207 d = 0.318, b-axis X £ = 3.210 w = 1.976 d 0.379, b-axis X £ = 4.731 w = 4.132 Surface Finish polished wire saw X X X X X X X X X X X X In diffused X X X X Unless otherwise indicated, the acoustic cavities are CdS crystals. k I bj_ crystal 33.08.06.00 was later bonded to DC2 to form a t r i p l e cavity. 147 Acoustic Cavity t 24.06.06.01 33.04.02.00 33.05.01.01 Dimensions (mm) d 1.265, c-axis X £ 7.225 X w — 2.170 X d = 0.376, c-axis X £ = 3.111 w = 1.838 d = 0.854, c-axis X £ 4.037 w 2.289 Surface Finish polished wire saw I In diffused X X X X X X 148 APPENDIX C -ACOUSTIC BONDS To provide high quality acoustic bonds between a CdS phonon maser and a second acoustic cavity (buffer), a cold welding procedure similar 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 crucial since the evaporated layers that formed the bond were only o ~ 3000 A thick. Following thorough ultrasonic washes i n acetone and subsequent rinsing 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 optical f l a t (also carefully cleaned) was used to check for the presence of dust particles using interference techniques. Systematic wiping of the surfaces was continued u n t i l no interference fringes could be observed, indicating that a l l dust particles :> 1000 A had been removed. • Following cleaning, the samples were mounted on a press in an —6 evaporator. When a vacuum of about 3 x 10 torr was achieved, a thin o o layer (~ 100 A) of Cr followed by a 1000 A layer of Au were evaporated o on the buffer, 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 diffused 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/cm2. After allowing several hours for the bond to form, the composite cavity could be removed from the evaporator. In the region of the bond the Au layer took on a silver color, indicating that alloying had taken place with the In layer. Using this technique, CdS phonon masers were successfully bonded to buffers of AI2O3, fused Si0 2, BaF 2 and CdS. 150 REFERENCES Abramowitz, M. and Segun, I,A., 1968. Handbook of Mathematical Functions. Dover, N.Y. Auld, B.A., 1973. Acoustic Fields and Waves in Solids. Wiley. Bhatia, A.B. and Noble, W.J., 1953. Proc. Roy. Soc. 220, ser. A, 356. Booth, A.H., 1954. Can. J. Chem. 32, 214. Born, M. and Wolf, E., 1959. Principles of Optics. Pergamon Press, N.Y. Burbank, M.B., 1971. Phonon Maser Action in CdS. Ph.D. thesis, S.F.U. Butcher, P.N., 1971. J. Phys. C: Solid St. Phys. 4_, 36. Butcher, P.N. and Ogg, N.R., 1968. J. Phys. D: Appl. Phys. 1_, 1271. 1969. J. Phys. D: Appl. Phys. 2, 333. • • '• 1970. J. Phys. C: Solid St. Phys. _3, 706. Chaban, A.A., 1972. JETP Lett. 15, 74. 1974. Sov. Phys. Solid State 15_, 2405. 1975. Sov. Phys. Solid State 17, 650. Cutler, CC., 1955. Proc. I.R.E. 43, 140. Davis, H.F., 1968. Fourier Series and Orthogonal Functions. Allyn and Bacon, Inc. Demaria, A.J., Glenn, W.H., Brienza, M.J. and Mack, M.E., 1969. Proc. IEEE 57, 2. Dember, H., 1931. Physik. Zeits. 32_, 554. Dixon, R.W., 1967. J. Appl. Phys. 38, 5149. Extermann, R. and Wannier, G., 1936. Helv. Phys. Acta. 9^, 520. Gainon, J.A.', 1964. J. Opt. Soc. Am. 54, 270. 151 G a r l i c k , G.F.J, and Gibson, A.F., 1948. Proc. Roy. Soc. A60, 574. Gay, R.K.L. and Hartnagel, H.L. , 1969. J. Phys. D: Appl. Phys. 2., 1589. 1970. J. Phys. D: Appl. Phys. 3_, 736. Goodman, J.W., 1968. Introduction to Fou r i e r Optics. McGraw-Hill. Grossweiner, L.J., 1953, J. Appl. Phys. 2k, 1306. Gulayev, Y.V., 1970. IEEE Trans. Sonics Ultrason. SU17, 111. 1970. Sov. Phys. S o l i d St. 12, 328. Gurevich, V.L., 1969. Sov, Phys. Semicond. 2, 1299. Haering, R.R., 1959. Can. J. Phys. 37, 1374. Haydl, W.H. and Quate, C.F., 1966. Phys. L e t t . 20, 463. Hudson, A.R. and White, D.L., 1962. J. Appl. Phys..33, 40. Hughes, R.C., 1974. Mode Structure i n Acoustic C a v i t i e s . Ph.D. t h e s i s , S.F. Hughes, R.C. and Haering, R.R., 1976. J. Acoust. Soc. Am. 59, 452. Kaminow, I.P., 1968. IEEE J. Quan. E l e c . QE4, 23. K l e i n , W.R., Cook, W.D. and Mayer, W.G., 1965. Acustica 15, 57. K l e i n , W.R. and Cook, W.D., 1967. IEEE Trans. Sonics U l t r a s o n . 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Sharma, A.K. and Wilson, M.G.F., 1970. Proc. IEEE 117, 2216. Shiren, N.S., 1975. IEEE Trans. Sonics Ultrason. SU22, 229. Shiren, N.S., Melcher, R.L., Garrod, D.K. and Kazyaka, T.G., 1973. Phys. Rev. Lett. 31, 819. S i t t i g , E.K. and Cook, H.D., 1968. Proc. IEEE L56, 1375. Smeaton, M.D., Hughes, R.C, Vrba, J. and Haering, R.R., 1976. Solid State Comrnun, 18, 465. Smeaton, M.D. and Haering, R.R., 1976a. Solid State Comrnun. (in press). 1976b. Solid State Comrnun. (in press). Smith, R.W., 1963. Phys. Rev. Lett. 9_, 87. Tien, P.K., 1968. Phys. Rev. 171, 970. Vrba, J. and Haering, R.R., 1973. J. Appl. Phys. 44_, 5213. 1973. Can. J. Phys. 51_, 1341. 1974. Can. J . Phys. 52, 1233. White, D.L., 1962. J. Appl. Phys. 33_, 2547. Willard, G.W., 1949. J. Acoust. Soc. Am. 21, 101. 'Williams, R., 1962. J. Phys. Chem. Solids 23_, 1057. Wonneberger, W., 1970. Phys. Stat. S o l i d i 40, 129. Yushin, N.K., Lemanov, V.V., and Agishev, B.A., 1975. Sov. Phys. Solid State 16, 1812. 153 BIBLIOGRAPHY - DIFFRACTION OF -LIGHT BY ULTRASONIC WAVES General Theory: Bhatia, A.B. and Noble, W.J., 1953. Proc. Roy. Soc. 220, ser. A, 356. Klein, W.R., Cook, W.D. and Mayer, W.G., 1965. Acustica 15, 57. Quate, C.F., Wilkinson, C.D.W. and Winslow, D.K., 1965. Proc. IEEE 53, 1604. Raman - Nath Diffraction: Nath, N.S.N, and Mueller, H., 1938. Nature 141, 37. Raman, CV. and Nath, N.S.N., 1935. Proc. Ind. Acad. Sci. A2, 406 1935. Proc. Ind. Acad. Sci. A2, 413. 1936. Proc. Ind. Acad. Sci. A3, 75. 1936. Proc. Ind. Acad. Sci. A3, 119. Vrba, J. and Haering, R.R., 1973. Can. J. Phys. 51_, 1341. : : 1973. Can. J. Phys. 51, 1359. 1974. Can. J. Phys. 52_, 1233. — 1974. Can. J. Phys. 5_2, 1239. Modifications to Raman - Nath Theory: Hargrove, L.E., 1967. IEEE trans. Sonics Ultrason. SU14, 33. Klein, W.R. and Hiedemann, E.A., 1963. Physica 29_, 981. Mertens, R., 1958. Ind. Acad. Sci. A48, 88. Mille r , R.B. and Hiedemann, E.A., 1958. J. Acoust. Soc. Am. 30_, 1042. Plancke-Schuyten, G., Mertens, R. and Leroy, 0., 1972. Physica 61, 299. 154 Comparison of Raman - Nath and Bragg Diffraction: Klein, W.R. and Cook, B.D., 1967. Trans. Sonics Ultrason. SU14, 123. Mayer, W.G. , 1964. J. Acoust. Soc. Am. 3_6, 779. Imaging Experiments: Goruk, W.S., Vella, P.J. and Stegeman, G.I.A., 1973. Phys. Lett. 45A, 357. Loeber, A.D. and Hiedemann, E.A., 1955. J. Acoust. Soc. Am. 28_, 27. Ris t i c , W.M., Zuliani, M.J.J., Stegeman, G.I.A. and Vella, P.J., 1972. Appl. Phys. Lett. 2JL, 85. Vella, P.J. and Stegeman, G.I.A., 1973. Appl. Phys. Lett. 23, 296. Zuliani, M.J.J., Ri s t i c , V.M., Vella, P.J. and Stegeman, G.I.A., 1973. J. Appl. Phys. 44, 2964. 

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