Relaxation Oscillations In Stimulated Raman-Scattering And Brillouin-Scattering

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Content RELAXATION OSCILLATIONS IN STIMULATED RAMAN AND BRILLOUIN SCATTERING by Richard Vincent Johnson A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Physics) January 1973 INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original docum ent. While the most advanced technological means to photograph and reproduce this docum ent have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the docum ent photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you com plete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication th at the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will fjnd a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being p h o to g rap h e d the photographer followed a definite m ethod in "sectioning" the material. It is custom ary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Departm ent, giving the catalog number, title, author and specific pages you wish reproduced. University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 46106 A Xerox Education Company I t 73-14,414 JOHNSON, Richard Vincent, 1945- RELAXATION OSCILLATIONS IN STIMULATED RAMAN AND BRILLOUIN SCATTERING. University of Southern California, Ph.D., 1973 Physics, optics University Microfilms, A X E R O X Com pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. U N IV E R S IT Y O F S O U T H E R N C A L IF O R N IA THE GRADUATE SCHOOL UNIVERSITY PARK LOS A NGELES. CALIFORNIA 9 0 0 0 7 This dissertation, written by _ . JRi char d Vine en t. John a on.......... under the direction of h .la .. Dissertation C om mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of requirements of the degree of D O C T O R OF P H IL O S O P H Y o4e c ?^^'? 7 i a %o D/am Date. 1973 DISSERTATION COMMITTEE PLEASE NOTE: Some pages may have i nd i st i net print. Filmed as received. University Microfilms, A Xerox Education Company ACKNOWLEDGMENTS The author wishes to express his deepest appreciation to Drs. John Nodvik, Harriet Forster, and John Marburger for their kind ness, patience, and support; to the Jansons for the same; and to the National Aeronautics and Space Administration, the National Science Foundation, and the United States Army for making his graduate years and research finan cially possible. And a final bow to the U.S. taxpayer, who ultimately footed the bill. ii TABLE OP CONTENTS ACKNOWLEDGMENTS .................................... LIST OP ILLUSTRATIONS .............................. CHAPTER I. INTRODUCTION ................................ A. Definition of the Problem B. History of Other Models II. DERIVATION OP THE SCATTERING EQUATIONS . . . . A. Brillouin Scattering B. Raman Scattering C. The Combined Equations III. THE ANALYTICAL SOLUTIONS.................... A. The Step Pump Pulse B. The Steady State Solutions C. Two Waves-Pump and Forward Stokes with No Dispersion D. Two Waves-Pump and Forward Stokes with Dispersion E. Two Waves-Pump and Backward Stokes in a Cell of Infinite Length F. Two Waves-Pump and Backward Stokes in a Cell of Finite Length IV. THE NUMERICAL SOLUTIONS .................... A. Three Wave Scattering B. Four Wave Scattering V. EXPERIMENTAL VERIFICATION OF THE THEORY . . . A. Pump Pulses of Finite Length B. Nonuniform Pump Cross Section C. Designing the Experiment ii v 1 1 6 14 14 26 27 29 29 20 26 41 48 55 67 67 72 77 77 82 90 iii APPENDIX A. THE ELECTROSTRICTIVE FORCE DENSITY........... 99 B. THE FLUID DENSITY AUTOCORRELATION FUNCTION . . 102 1. The Classical Linear Response Theory 102 2. The Fluetuation-Dissipation Theorem 102 2. The Philosophy of the Approach 104 k. The Perturbation Forces for Hydrodynamic Fluids 106 5. The Hydrodynamic Equations 108 6. The Density Autocorrelation Function 111 7. The Brillouin Noise Intensity 116 C. THE NUMERICAL INTEGRATION TECHNIQUE ........... 117 BIBLIOGRAPHY ...................................... 126 LIST OP ILLUSTRATIONS Figure 1. A hypothetical experiment to measure the time dependence of stimulated Raman and/or Brillouin scattering .......................... 2. The stimulated sound wave vector kg .......... J> . The two wave backward stokes (pump and backward Stokes) and three wave (pump, forward Stokes, and backward Stokes) steady state dependence of the backward Stokes intensity at the cell entrance upon the scattering cell length . . . . 4. The two wave forward Stokes (pump and forward Stokes) scattering with no dispersion spatial dependence for successive times .............. 5. The two wave forward Stokes (pump and forward Stokes) scattering with no dispersion spatial dependence in the steady state ................ 6. A Minkowski diagram of the two wave forward Stokes (pump and forward Stokes) scattering with no dispersion of a pump pulse of width T . 7. The two wave backward Stokes (pump and backward Stokes) scattering in a cell of infinite length spatial dependence for successive times . . . . 8. The two wave backward Stokes (pump and backward Stokes) scattering in a cell of infinite length time dependence of the backward Stokes intensity at the scattering cell entrance .............. 9. The two wave backward Stokes (pump and backward Stokes) scattering in a cell of finite length spatial dependence for successive times .... 10. The two wave backward Stokes (pump and backward Stokes) scattering in a cell of finite length temporal dependence of the backward Stokes light at the cell entrance .......................... Page 12 18 32 38 40 45 51 54 59 64 v Figure Page 11. The temporal structure of the three wave (pump, forward Stokes, and backward Stokes) backward Stokes intensity at the cell entrance for equal gains and various pump to noise intensity ratios ............................... 68 12. The temporal structure of the three wave (pump, forward Stokes, backward Stokes) backward Stokes intensity at the cell entrance for various gain ratios, scaled to L p .......... 69 13. This is the same as the previous figure, only scaled to Lg instead of L p ................ 70 14. The three wave (pump, forward Stokes, and backward Stokes) spatial dependence for successive times................................ 72 15. The four wave (pump, forward Raman, backward Raman, and backward Brillouin) temporal structure of the backward Brillouin intensity at the cell entrance............................ 75 16. The temporal structure of the two wave backward Stokes (pump and backward Stokes) scattering in a cell of finite length backward Stokes intensity for a skewed Gaussian pump pulse ... 79 17. The hypothesized dependence of the modulation depth upon the pump width to pulsation period ratio.......................................80 18. The temporal structure of the three wave (pump, forward Stokes, and backward Stokes) backward Stokes intensity for a skewed Gaussian pump pulse................................... 81 19. The approximate temporal structure of the backward Stokes power for various models .... 86 20. The steady state Brillouin gain gf i in N2 . . . . 95 21. The acoustical phonon lifetime in N2 ......... 96 22. The steady state gain gg for the Raman vibrational transition ^( 1) in H2 .............. 97 23. The optical phonon lifetime for the Raman vibrational transition Q^(l) in H2 ........... 98 v i Figure Page 24. The frequency spectrum of the density fluctuation autocorrelation function ........ 112 25. A Minkowski space-time diagram showing the characteristics of the light scattering equations.................................. 118 26. The square grid of points assumed in the difference equations of the numerical inte gration, and how these points are labeled . . . 121 27. The integration of the (1 = 2,J= 2) point . . . 12} vii CHAPTER I INTRODUCTION A. Definition of the Problem The laser has provided, for the first time in history, intense and coherent light at one well defined frequency. With the laser, many facets of the intricate interactions of electromagnetic waves and matter could be studied with a subtlety and clarity unknown in prelaser technology. Among the most often observed and interesting of these phenomena is the inelastic scattering of the laser "pump" energy. The frequency shift of the scattered photons occurs in a well defined spectrum which is charac teristic of the scattering medium. The frequency shift corresponds to the frequency of an excitation of the scattering medium, such as sound waves. CJ = LJ ±CJ laser Stokes medium (1A.1) pump light excitation where CJ represents the angular frequency. The plus sign in the above equation defines Stokes scattering, while the minus sign defines anti-Stokes scattering. In a quantum mechanical description of Stokes scattering, a laser pump photon is annihilated and a scattered Stokes photon and a quantum of scattering medium excitation are created. In anti-Stokes scattering, a quantum of medium excitation and a laser photon are annihilated and the Stokes photon is created. Stokes scattering can be further classified by the nature of the medium excitation. The two most common forms are Brillouin scattering, with sound waves (acous tical phonons) as the medium excitation, and Raman scattering, with either short range molecular vibrations (optical phonons), molecular rotational states, or excited electronic states as the medium excitation. The frequencies of the Brillouin shift, typically 10^ herz, are generally considerably smaller than the frequencies of the Raman shift, typically lO1^ herz. The Stokes light can exist in many "orders.” The pump, or "zeroth order," generates Stokes light downshifted once relative to the pump. This light, in turn, can grow to such intensities that it can generate "second order" Stokes light downshifted twice relative to the original pump frequency. The energy flow can be diagrammed as follows: Pump — ►First order — ► Second order — ►Third order — ► Furthermore, the medium excitations generated in Stokes scattering can couple with the pump (or any other mode) to drive anti-Stokes light at frequencies higher than the pump. As if this weren't complex enough, the first order Raman light can generate both second order Raman light and Brillouin light, and vice versa, so that cross coupling between Raman and Brillouin orders is possible. Since the second and higher Stokes and the anti-Stokes orders cannot grow until the first order Raman and Brillouin modes have developed, and since the anti-Stokes index matching requirement for the scattering geometry considered in this paper cannot be satisfied for normally dispersive media, and for the sake of simplicity, only the pump and first order Raman and Brillouin modes will be considered. Stokes light, like all scattered light, is generated in all directions. However, only that light which remains within the volume occupied by the pump in the scattering medium can continue to grow to appreciable intensities. If the scattering medium were a long thin tube, or if the pump were a thin and well defined beam, then all scattered light traveling obliquely with respect to the pump would exit through the sides of the scattering cell or the pump beam before coaxing much energy from the pump. Hence, only modes traveling collnearly with the pump beam, either in the forward direction (the same direction as the propagation of pump light), or in the backward direction, will be considered. The four modes of interest in this thesis, along with the initials by which they will be identified, are: The forward Brillouin mode is ignored because the gain for scattering in the forward direction is considerably smaller than that for scattering in the backward direc tion, as will be shown in the next chapter. The problem is still horrendously complex because the fields are functions of four independent variables: the three spatial coordinates and time. However, if the beams are not focusing or defocuslng, i.e., if they are well collimated into parallel rays, and if diffraction is not critical, the dominant dependence will be on the spatial coordinate 'z,* measured along the length of the scattering cell from the entrance window, and on the time 't.' Dependence on the remaining two spatial coordinates will be Ignored during most of the develop ment. The scattering equations will contain partial derivatives with respect to distance and time. The convention here is that partials with respect to distance Mode Symbol Pump (laser) Forward Raman Backward Raman Backward Brillouin P F define 'spatial growth,1 while partials with respect to time define 'temporal growth.1 If the light fields were to be suddenly turned on, the stimulated medium excitation would not instantaneously jump to its steady state value. Rather, the excitation would grow in a time characteristic of the scattering medium, a time which will be named the 'medium response time.' This growth is described by a differential equation for the medium excitation, an equation with an absorptive term inversely proportional to the response time (e.g., equation (2A.15c*) below). If, however, the time scale characteristic of the transfer of energy between light modes is long compared with the medium response time, then the differential equation for the medium response has an especially simple solution because, to a very good approximation, the medium can always "follow'1 the slow fluctuations in the driving light fields. The medium response in such an adiabatic approximation is simply proportional to the instantaneous values of the light fields. This approximation is impor tant since only the scattering equations for the light modes remain, decoupled from a complicated medium response integral. Such an adiabatic approximation eliminates one dependent variable and allows the transformation of complex scattering equations in the field amplitudes 6 and phases into real equations In the Intensities. The resulting intensity equations for Raman and Brillouin scattering are especially interesting because, for some mode combinations, analytical solutions exist, solutions which are not tractable for the more general 'non-adiabatic' cases. Hence, we shall restrict our attention to transient effects with characteristic times long compared with the medium response times. Finally, dispersion will be Ignored in the computer solutions, although not in the analytical work. In a dispersive medium, the pump, Raman, and Brillouin modes will travel with different speeds. If the medium is strongly dispersive, and if the pump pulse is of short duration, then the forward Raman light which was initially generated inside of the pump pulse will quickly "walk away” from the pump, terminating further growth of the Stokes mode in this portion of the Stokes pulse. This effect should be negligible for all but the shortest of pump pulses. B. History of Other Models The Stokes light does not conjure up its energy out of thin air, but rather siphons it out of the pump reservoir. The larger the Stokes mode, the more it can draw from the pump, and the more noticeable the depletion of the pump's reservoir of energy will be. To describe accurately the "high conversion" limit when a significant fraction of the pump energy has been devoured by the Stokes light, a model must include a pump mode equation relating the growth of the Stokes modes to the depletion of the pump energy, rather than assume a fixed pump intensity unaffected by the growth of the Stokes light. A model with a fixed pump intensity is linear in the dependent variables, whereas a model incorporating pump depletion is necessarily nonlinear. Since a linear model is far more tractable than a nonlinear model, most of the early papers on Brillouin and Raman scattering^ reported only linear models. The solution behavior for coupled linear differential equa tions is exponential, and much of the early work was devoted to a study of the dispersion relations which apply in the low conversion limit. However, if saturation is important, then the pump depletion can only be described by a nonlinear equation. The steady state behavior of the scattering, after all transient effects have died away, is of special interest because the time dependence is absent, leaving ^See, for example, Y. R. Shen and N. Bloembergen, Phys. Rev. 137, A1786 (1965); N. Bloembergen and P. Lallemand, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. fi. fannenwald (McGraw-Hill Book Company, Inc., New York, 1966), pp. 137-1^2; and D. L. Bobroff, J. Appl. Phys. 36, 1?60 (1965). 8 only the one spatial coordinate. Since equations In one Independent variable are In general considerably easier to solve than those In two Independent variables, the next models to be considered In the literature were steady state high conversion systems. An excellent example of such a steady state model 2 for Brillouin scattering la that of C. L. Tang. Three aspects of this model are Important: high phonon absorption, pump pulse depletion, and spontaneous scattering 'seed' Intensity. Many Brillouin scattering media, Including many liquids at room temperature, have high sound absorption at the ultrasonic frequencies stimulated In the Brillouin scattering. Noting this, Tang invokes an approximation to eliminate the sound wave equation, leaving Just the differential equations for the light mode intensities. This approximation Is closely related to the adiabatic approximation mentioned earlier. The second aspect of Tang's model, a pump mode equation which makes the model nonlinear and valid in the high conversion limit, has been discussed previously. Stimulated Stokes scattering equations typically describe the growth, not the onset, of the Stokes light. The initial appearance of the Stokes energy must be 2C. L. Tang, J. Appl. Phys. 29^5 (1966). introduced by some additional physical mechanism. Rather than arbitrarily invoking an artificial "seed" of Stokes energy Injected into the scattering system by some unspecified agency, Tang attributes the initial appearance of Stokes energy to spontaneous scattering from thermal density fluctuations in the scattering medium. This leads to a "noise generator" Intensity in the scattering equations, in addition to the stimu lated term. See the next chapter for a more complete exposition. Tang's model, and similar ones for Raman scat tering, totally ignores temporal growth for mathematical simplicity. Obviously, such steady state models cannot possibly describe transient effects. For example, pump pulses with time durations comparable to or shorter than the response time of the medium will, at least initially, stimulate smaller Stokes fields than pulses of longer duration because of the "inertia" of the medium. Hagenlocker, Mlnck, and Rado^ have attempted to describe such transient effects by defining an effective "transient gain" which incorporates the response time of the medium and the width of the pump pulse, without resorting to the complete analysis of E. E. Hagenlocker. R. W. Minck, and W. G. Rado the necessary mathematical equations. Although their model leads to close agreement with Brillouin data, such a model is clearly at best a crude approximation to the complete transient behavior. 4 N. Kroll has derived an analytical solution to the transient equations for stimulated Brillouin scattering for a step pulse pump time profile at the scattering cell entrance, but in the low conversion limit when the pump energy is assumed to be a fixed parameter. Similar solutions have been reported for Raman scattering by Carman, Shimizu, Wang, and Bloem- bergen-' for variously shaped pump pulses, but again only in the low conversion limit. To my knowledge, the only model to date (January 1973) in the literature incorporating both temporal and spatial gain and pump depletion with the attendant nonlinear equations in the two independent variables, other than a report of the results of this dissertation 7 is that of Maier, Kaiser, and Giordmaine for backward M. Kroll, J. Appl. Phys. 36, J>k (1965) 5C. S. Wang, Phys. Rev. 182, 482 (1969 b R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen (unpublished). ^R. V. Johnson and J. H. Marburger, Phys. Rev. A 4 1175 (1971). ^M. Maier. W. Kaiser, J. A. Giordmaine, Phys. Rev. 177, 580 (1969). scattered Raman light. Indeed, the only difference between their model and that considered in Chapter III Is the choice of the mechanism required to Initiate scattering. They introduce seed pulses of various shapes into the rear of a scattering cell filled with Initially undepleted pump energy. This choice of an initial condition is a consequence of their interest in the growth of baclcward scattered Raman light in self focusing, but it implies certain growth behavior (e.g., backward Stokes intensities larger than the incident pump intensity) which is not found in the model considered herein. For scattering in an experiment such as that diagrammed in Figure 1, an essentially new class of transient phenomena involving the nonlinear interactions of the light modes and independent of any phonon lifetimes has been found by this author. For first order stimulated Brillouin scattering (SBS) only, at high pump intensities, pulsations with a period equal to the photon round trip time in the cell have been observed in the analytical solutions for the backward scattered Stokes light intensity at the scattering cell entrance. Similarly, for first order stimulated Raman scattering (SRS) only, at high pump intensities in a scattering cell of sufficient length, pulsations ////// / / / INPUT PULSE/ //JSHAPE/// LASER RAMAN-BRILLOUIN SCATTERING CELL BB BR PHOTODETECTORS Figure 1: A hypothetical experiment to measure the time dependence of stimulated Raman and/or Brillouin scattering. The letters P, F, BR, and BB represent the pump, forward Raman, baclcward Raman, and backward Brillouin modes respectively. 13 with a period equal to 2Lp/cL (where Lp Is the "gain length" of the forward scattering process, defined In equation (3C.6) below, and cL Is the speed of light In the scattering medium) have been observed In the numerical solutions for the backward Stokes Intensity at the cell entrance. This "gain length" is approxi mately Inversely proportional to the pump Intensity, and Independent of the cell length. CHAPTER II DERIVATION OP THE SCATTERING EQUATIONS A. Brillouin Scattering The derivation of the Brillouin scattering equations will be considered in detail in this section because some of the expressions derived herein are not found in the standard literature. The Amplitude Equations Light may be scattered off of sound wave density fluctuations through the density dependence of the dielectric constant: £ ( p ) — £ « U +■ T L (2A.1) where & is the dielectric constant,p is the fluid density, and p o the density in the absence of sound. The elasto-optic constant IT and sound condensation & are defined by T = Q , r 3p The density fluctuations of the sound generate a polarization ^ ound: 14 (2A.2) (2A.3) 15 ^Sound ” 4^ ^ ^ (2A.4) The wave equation for propagation of light in a medium simultaneously supporting a sound wave is where the scattering medium is assumed to be nonmagnetic, i.e., permeability yUL=l, and the approximation V^*E:=0 has been invoked. E is the electric field amplitude, c the speed of light in vacuo, c^r:c/n^ the speed of light in the medium, and n^ the refractive index. The sound waves may, in turn, be driven by light energy through the electrostrictive force density which is the divergence of the Maxwell stress tensor. This force density expression and the related stress tensor are derived in Appendix A. The sound wave equation, obtained from the linearized hydrodynamic equations (continuity, Navier-Stokes, and entropy) and the equation of state, is ( 2 A. 6) where OC is the viscosity damping term and Cg is the speed of sound in the medium. 16 The light field Is assumed to consist of two well-defined spectral components: the laser pump with angular frequency and wave vector ( c j p r kp), and the scattered Stokes light with angular frequency and wave vector (cjg,kB). f-*- 1 (Vp'r-cjpt) = R e j^ E p ( z ,t) e p P f C BW e - J (2A.8) Ed and E are complex functions of the amplitudes and * B phases, functions which vary slowly in time and space compared with the rapidly oscillating plane wave factor. The stimulated sound condensation, similarly, is l(-i) = Re [SCz,t) ei ( ^'r-a4t)j (2A.9) Brillouin scattering obeys energy and momentum conser vation : CJp ~ CJ q +■ CJS (2A.10a) kp = lfB 4- ks (2A.10b) Since the frequency shift COg for Brillouin scattering is small, the difference in the magnitude of the wave vectors kp and kg due to dispersion is quite small. The Brillouin scattering gain, as will be seen later, is proportional to that constant which couples the light fields into the sound wave equation, i.e., the constant gg in equation (2A.15c) below. This, in turn, 17 is proportional to the magnitude of kg. As can be seen from Figure 2, kg is negligibly small for scat tering into the forward directions, compared with the For backward scattering, kg=2kp, its maximum value. Hence, the gain for scattering in forward directions is negligible compared with that for backward directions, and only backward Brillouin scattering will be considered. The backward scattering geometry shown in Figure 2 will be assumed. Substituting equations (2A.8) and (2A.9) into (2A.5)> and ignoring second derivatives in the slowly varying envelopes, one finds the following scattering equations for the two light modes: where the asterisk denotes the complex conjugation k stimulated by scattering into the backward directions O (2A.lla) (2A.llb) operator and i= Only those terms in the driving force which are closely matched in frequency and wave vector to the driven field of interest have been retained. The scattering equation for the sound condensation wave, similarly, is (2A.llc) FORWARD SCATTERING (low gain geometry) ^ , ^ ► BACKWARD SCATTERING (high gain geometry) ¥ _______ Jfp_ Figure 2: The stimulated sound wave vector kg Is much smaller for forward scattering of Brillouin light than for backward scattering. Since the Brillouin gain gB is proportional to the magnitude of kg, the gain for backward Brillouin scattering is much larger than the gain for forward Brillouin scattering. Therefore, only backward scattering of Brillouin light will be considered. where is a phenomenological damping term. If damping is primarily due to the cC viscosity term of equation (2A.7), then g^ and ac’are related by $ 3 - CJs ks a' (2A.12a) (2A.12b) Define coupling constants gj^ and g2: _ CJD T 9 1 it T q* = ** (2A.12c) U * 3 M P ' C ? Scattering equations for phenomena traveling with speed c^ may usually be expressed most simply in terms of independent variables (x,y): X = i ( C u t - h Z ) (2A.13a) Y - j [ ( C i t ~ Z ) (2A.13b) Similarly, phenomena traveling with the speed cg may be expressed most simply in terms of (£ ,7| ): £:iCcst+z) ^ i( c iL -‘ )y (2A.l4a) n - - (2A-14,>) In terms of (x,y) and the constants (g^gg^g^)* the scattering equations are The last equation, (2A.15c) for the sound waves, may also be written Noise and Statistical Averaging These equations for stimulated Brillouin scattering describe the growth of the modes, not their onset. The initiating mechanism which provides the "seed" energy must be postulated upon additional physical considerations. In almost all experiments, this mechanism is the sponta neous scattering from noise fluctuations. Thermal density fluctuations, the leading noise source in Brillouin scattering, dominate over quantum fluctuations because the frequency shift O s is too small. For a typical Brillouin scattering experiment, t)CJs is some ten thousand times smaller than kT, the thermal energy at room temperature. Therefore, the sound condensation term in the two light wave equations (2A.15a&b) consists not only of the stimulated sound of equation (2A.15c), but also an additional "noise generator," i.e., the random noise condensation (2A.15c*) function N(z,t) corresponding to those Fourier components of the noise field which obey the energy and momentum conservation equations (2A.10). These equations (2A.15a&b), thus amended, are no longer soluble in their present form because the noise condensation N, a complex function of random phases and amplitudes, can never be specified completely for any given scattering experiment. However, the statistical characteristics of N are (assumed to be) known. They are discussed in Appendix B for fluids obeying the hydrodynamic equations. We are interested in field equations which are statistically averaged over all allowable N. If the implicit dependence of the light fields on N is ignored, however, such an averaging leaves only the stimulated scattering equations, devoid of all noise kernels. Clearly, a somewhat more cautious approach is imperative. The amended scattering equations are The middle equation, and indeed all three, can easily be integrated: ~ i^i EpCS+fiif (2A.l6b) (2A.16c) (2A.l6a) 22 f y*y E eU y) = i j , J j y ' £ ? (* ;/') [ $ K y ' ) + N f y / ' ) ] (2A.17) y *Y» where yQ is chosen such that Eg(x,yo):=0. Equations (2A.l6b) and (2A.17) may be combined In a suggestive way: £‘ f t = 3 *JV W ' ' H * t • [S(x,ftS‘*(r/ /) + • Sb,j/)N*6r,f+ M /p 6 ,y ) +U(t,f)k*(//f l Let us apply the statistical averaging process to this equation, assuming the following: <&,> - Ep <S> = S ' , ' r (2A.19) < Eb> - Eg <A/> = 0 where the brackets O signify the aforementioned statis tical averaging. < S N*> = <NS*> = o (2A.20) follows from assumptions (2A.19)* since N is the only term which we assume fluctuates measurably, and Its average Is zero. The last term in equation (2A.18) involves ^N(x,y*)N*(x,y)^ , which is nothing more than the autocorrelation function for the thermal condensation, a known and real function. The middle cross terms drop out, leaving the first term in the stimulated sound alone, and the last term in the noise autocorrelation function alone. The first term corresponds to stimulated scattering, and the last to spontaneous scattering. The scattering equation for Eg can thus be written: Similarly, the scattering equation for Ep can be written: These are the desired scattering equations. The Adiabatic Approximation The stimulated sound equation (2A.l6c) can readily be integrated: (2A'22) where has been chosen such that )-0. Under certain circumstances, listed in the next paragraph, the e' ^ ) term will be very sharply peaked compared with the light fields, and a standard adiabatic S Q j r f ~ l 3 L E p C ^ * ? ) E q (2A.23) 3* approximation may be invoked Sf J ,* ? ) = As explained in Chapter I, this approximation will significantly simplify the mathematics of the problem. Compared with the exponential, the light fields must vary slowly both in space and in time for the approximation to be valid. The first condition is satisfied when a typical sound absorption length g^ ^ is much smaller than a typical Brillouin interaction length (gB^ump)-1' where gB is the gain for the Brillouin scattering, defined in equation (2A.25a) below, and Ipump is the maximum pump intensity; The second condition is satisfied when the first condition is met and when the pump pulse varies slowly in time compared with a typical sound absorption time, i.e., pump intensity. Define a Brillouin scattering gain gB and noise intensity aB: where the autocorrelation function is also assumed to be sharply peaked compared with the field amplitudes (see 3e (2A.24a) (2A.24b) where ) is the fastest time rate of change of U V 25 Appendix B). The scattering equations may now be written. In terms of pump and backward Brillouin Intensities P and B, 0 = -Je P(fltas) (2A.26.) I® = H}«P(8 + a6) (2A.26b) Because the sound absorption for many liquids at room temperature Is large at the frequencies associated with Brillouin scattering, the first condition (2A.24a) is often satisfied for such liquids. The second condition (2A.24b) may also be satisfied by a judicious design of the pump parameters. The simplified equations (2A. 2 6) are of interest because an analytical solution exists, as will be demonstrated in the next chapter, and also because fewer system parameters need be specified. However, these conditions (2A.24) are satisfied only in a limited number of experiments. The assumptions implicit in the Brillouin adiabatic approximation are in general the weakest link in this theory, and their validity should be scrupulously examined before submitting this theory to a quantitative test. The developement of this adiabatic approximation was inspired by C. L. Tang, who invoked a similar approximation in his steady state theory. 26 B. Raman Scattering The Raman scattering equations are derived in an entirely analogous manner. The pump and Stokes-shifted modes couple to drive a medium excitation, which in turn modulates the dielectric constant and thereby drives the light modes. The light consists of two distinct spectral components, the pump and Stokes, which obey the energy and momentum conservation equations (2A.10). However, in Raman scattering, the wave vector kg for the medium excitation is not associated with a traveling wave, since such excitation as the Raman-induced optical phonons is localized on a molecule (or small cluster of molecules). Therefore, equation (2A.10b) can be satisfied equally well for forward and for backward scattered Stokes light. The gains for scattering in the forward and the backward directions are equal, and both modes must be considered. The frequency shift for Raman scattering is large compared with the Brillouin shift, so large, in fact, that the phonon quantum energy twdy is comparable to the thermal energy kT at room temperature. The quantum noise in the light modes is therefore an important part of the total Raman noise, and a quantum mechanical approach to the Raman scattering equations is mandatory. Since the Raman medium excitation is localized compared with Brillouin scattering, the Raman phonon lifetime is considerably shorter than that for Brillouin phonons, and the Raman adiabatic approximation is quite accurate for all but the shortest pump pulses or patho logical systems. The equations of Raman scattering considered hereafter are where P, F, and BR are the intensities of the pump, forward Raman, and backward Raman modes respectively. The scattering medium parameters gR and a^ are the Raman gain and noise Intensity. where A is the wavelength of the light,h is Planck's constant, is the frequency width of the Raman line, and A f l is the solid angle observed by the photo- detection equipment. C. The Combined Equations The combined equations describing both Raman and Brillouin scattering are j* " - j R P C F + S f f + Z a i d - j B P C B B i - c l B ) (2C.la) gy = PCf +" BK + ■ (2B.la) (2B.lb) (2B.lc) AiAs A A A3 (2B.2) 28 §x - + 3r p c a«> 0 = i - f r P C B R + a d | ^ = +Jb P ( 8 B f a B) (2C.ld) (2C.lb) (2C.lc) These equations will be considered in various combina tions in the next two chapters. The smallest number of modes which can be considered, consistent with the assumption of high pump energy conversion, is two: one pump mode and one Stokes mode. These 2W (two wave) equations may be further classified by the direction of the Stokes mode relative to the pump: 2WPS (two wave, forward Stokes), and 2WPS (two wave, backward Stokes). Furthermore, it will prove convenient to subclassify the 2WPS into 2WPSND (two wave, forward Stokes with no dispersion) and 2WFSD (two wave, forward Stokes with dispersion), and the 2WBS into 2WBSC0C (two wave, backward Stokes in a scattering cell of infinite length) and 2WBSPC (two wave, backward Stokes in a scattering cell of finite length). Analytical solutions exist for the 2W and the steady state problems, and will be considered in the next chapter. The numerical solutions for the 3W (three modes: pump, forward Stokes, and backward Stokes) and the 4w (the full set of equations (2C.1)) problems will be presented in Chapter IV. CHAPTER III THE ANALYTICAL SOLUTIONS Some combinations of the scattering equations (2C.1) admit analytical solutions. In this section, the properties of these solutions will be studied both for their own intrinsic interest and for the insight they provide into the more complex solutions generated by computer. A. The Step Pump Pulse The transient solutions are obtained for arbitrary pump time profiles at the scattering cell entrance window. To investigate the solution behavior, a specific example of a step pulse profile is considered, i.e., If the transient behavior for this simple time profile, hereafter referred to as the "step pump pulse," can be understood, then the solution behavior for more complex pump pulses can hopefully be inferred. However, a step pulse violates the adiabatic assumption that the pump varies slowly in time compared with a phonon lifetime. Therefore, for our purposes, "step pump for t < O for t> 0 29 30 pulse" will denote a pump intensity profile at the cell entrance window which grows from zero intensity to I in times much shorter than the transient times of interest in this work, but much longer than the medium response times. B. The Steady State Solutions Before investigating the analytical solutions of the transient models, let us first consider the steady state solutions of the various models considered. Two Wave-Forward Stokes The equations for the steady state 2WFS model are mathematically analogous to those for the transient 2WFSND model, and so discussion will be deferred until Section C of this chapter. Two Wave-Backward Stokes The equations are d £ = - ^ B P ( B + a ^ ) (5B. la) d B - - o g P ( B + 3 e) (3B.lb) d z J The integrals are easily evaluated: Jz ( P- b) = O * Pf»~) - B<*~) = P&) - B &) (3B.2a) f ' l r s P l z ) J-. C Z ~ f ’ I 'TT'£P£o) + aS- &cd-7rj = J J b (3B.2b) ™ = 5 Z r (3B.3.) ^ r - P f r > a » L l - « - g _J (3B>3b) where J = ~ (3B.3c) '3sXj? P =• ~ ^ (3B.3d) B(0) Is not known, but B(L) Is known when L is the length of the scattering cell since the stimulated light inten sity must be zero at the rear window, where it starts its growth from pure noise. In Figure 3 is plotted B(0) as a function of cell length L for a pump to noise intensity ratio I0/ag of 10^ and lO1^. Note that, when the distance is scaled to L 8 = 4 & - T J r~ & ( ® » J b ( } B A ) < * B ( ft*)* £ 3) 3? ?6> \ the Stokes intensity profile is essentially insensitive to changes in the pump to noise ratio of over twelve orders of magnitude. Three Waves The equations are =. - (3B.5a) BxkutdrJ Stokes = 10 Figure 3: The 2WBS (A) and jM equal gain and noise level (B) steady state dependence of the backward Stokes intensity at the cell entrance upon the scattering cell length. The abcissa is the cell length in units of Lg, defined in equation (3B.4), while the ordinate is the Stokes intensity B In v* units of the incident pump intensity IQ. w a f = -3 b P(b +<3b) (3B.5c) Two Integrals are easily evaluated: (P-t-F-8) = o => P C o ' ) + P ( s £ - B (o) - Ffe) - + R z ) - BCz) (3B.6a) S ® y F ^3® (F+2f)3 = P = ^ G r 38 lajc.(&+2f)J = 7 > ( t f £ + a F ) 3B(B (0 ) + a & ? = (FCi\+5r)3^6?V5g|^F(3B.6b) Thus, if any one of the three intensities is known at a point, the other two may be readily determined for that same point. All that remains is to determine the spatial dependence of one of the three modes by performing the remaining integration. Before considering the final integration, let us consider the first two integrals in an infinite scattering cell. Equations (2B.6) become, when z=o0, B grows monotonlcally from noise at the rear of the scattering cell to a maximum Intensity B(0) at the front of the cell. For any given cell length L, B(0) is the maximum backward Stokes intensity in the cell. B(0) is a maximum for all possible cell lengths when the length is infinite (i.e., when there is the most scat tering cell through which to grow). Similarly, F(L) is the maximum forward Stokes intensity for any point in the cell of length L, and F(L) is a maximum for all possible cell lengths when L is infinite. B(0) and P(oo) in equations (3B.7) therefore represent the upper bounds on the possible growth of the two Stokes modes for all possible cell lengths in this steady state model. The third and final integral must be evaluated to determine the spatial distributions of any of the modes. However, solution of this integral for arbi trary gains and noise intensities is nontrivial. If an analytic solution could be found, it would involve the unknown B(0) value, which is found by setting B(L) equal to zero when L is the cell length. Generally, the final integral must be evaluated by numerical tech niques, as was done for Figure 3. In Figure 3 is plotted B(0) (which is identically equal to F(L) for the equal gains and noise intensities model) as a function of the cell length L for pump to noise intensity ratios I0/a 4 16 of 10 and 10 for the 2W equal noise, equal gain case (i.e., that which describes pure Raman scattering). The cell length is scaled to Lq of equation (3B.4). 35 Note how insensitive the Stokes intensity profile is to changes in the pump to noise ratio over twelve orders of magnitude when the cell length L is scaled to Lg. Pour Waves The equations are: -<j„P(F+BR+2.S.O - (?B.8a) +q„P(P+3s) (5B.8b) 3z J j|(f_ _g„p(BK+a(i) cjBB q„P(8B4 (?B.8d) d z ' ^ The first three integrals are easily found: i (P+F-Bf?-Bb) = o => o P(?)+ % ) - (3B.9a) o ^ i?J£bi?6>) +-<9ij| =- +■ ^■5^ 9b^ <JZ - P = £ [ £ l t j * ( ? % ^ B ) j = Z > jBR(g) + 3yJ j * gS^-t^gJ 3^ (:5 *9c) If the intensity of one mode is known at“a point, the intensities of all modes for that point may be inferred from these equations (3B.9). The spatial dependence of all modes is determined when the spatial dependence 36 of one mode is determined, and this requires the solution of the remaining integral. Generally, no analytical solution can be found, but a numerical solution is easily generated. No such solutions are generated here because, as will be explained in Section B of Chapter IV, the 4w problem can usually be reduced to a 3W problem. C. Two Waves-Forward Stokes With No Dispersion We now consider models with temporal as well as spatial growth, starting with the simplest case, the 2WPSND model. The equations to be solved, in terms of coordinates x and y defined in equation (2A.13), and the boundary conditions are: f f ' ^ (3C.la) |£ s + q f P(F + <2P) (3C.lb) o X J P M = (Pdf) (30<10) = ° (3C.ld) where P(t) is the (known) time envelope of the laser pump pulse entering the front of the scattering cell. The first integral can be found by inspection: |,(P+F)=0^ - 9 ( % ) (3C>2a) Substitute this into (3C.la) and integrate: 37 The solutions are: ->£0r7; P(X,y) - — ■ ------------- (3C.3a) D - * r z(x~y) F(x y ) - 3e.tlr ----- 2 p where (3C.3b) r = + S r Cjc.jc) 0 - +■ e (3COd) • Cut In terms of coordinates z and t: -gFrz- l i ------ (3C.4a) r * P -3^*7 f (?.-$ = ir>1" ^ «L (^c.Ub) ' D Z - P O k - Z/cd -b 2p (3C.4c) D - ,- ^F- - +- e- 3F (3C.4d) Consider now the special case of a step pump pulse: P(z,-1)= <£+. fr) e 3.p — ^ when rcz,fc)= 3P £t- g- + c-3 P <»+*=)*• J ht>0 (30.5a) (3C.5b) o (}A.l) Figure 4 shows a series of "snapshots" of the spatial dependence of the pump and Stokes modes after 1 . 0 X \ t - -1-0 tt Cu 1.0 t - 2 .0 1£ o s Z.0 \.o 1 . 0 \ o s 3 0 1 . 0 Figure 4: The spatial dependence of the 2WFSND pump (-----) and forward Stokes ( -----) intensities for successive times, assuming a step pump pulse. The abscissa is scaled to Lp, while the ordinate is scaled to the incident pump int intensity ratio I0/ap is ensi^y IQ. The pump to noise successive times. In the top view, the pump pulse has just entered the cell, and the Stokes pulse (which Is building exponentially from the noise level at the front of the cell) has not yet developed any Intensity compar able to that of the pump. As a consequence, the pump has not been noticeably depleted. In the next view, the Stokes light at the leading edge of the pulse has just grown to an intensity comparable to the pump. This intensity is sufficient to siphon an appreciable amount of pump energy into the Stokes mode, causing a noticeable dip in the pump intensity at the leading edge. The bottom views show that the pump has transferred almost all of its energy into the Stokes mode shortly after reaching a distance Lp, into the scattering cell. If one defines L as that distance at which half of the pump intensity has been depleted (and, by the energy conservation equation (3C.2a), that distance at which the Stokes mode has grown to half of the original pump intensity), then the analytical solutions (3C.5) provide the following expression for L^: This is similar to the Lg scaling length of equation In Figure 5 is exhibited the steady state exchange of energy between the pump and Stokes modes for various (3B.4). I « e > n N . /.r 1-0 2-0 o.r Figure 5: The 2WFSND spatial dependence of the pump (-----) and forward Stokes (----- ) intensities in the steady state. The abscissa and ordinate are scaled as in Figure 4. Note the rapid shift of energy from the pump to the forward Stokes mode after distance Lp. pump to noise ratios I0/ap. Note that the solutions are, as usual, quite insensitive to the pump to noise ratio. Note also that the Interval in the scattering medium wherein most of the energy is transferred, region B in the figure, is rather small compared with the distance Lp. D. Two Waves-Pump and Forward Stokes with Dispersion Consider the same problem as the previous section, only now the pump and forward Stokes beams propagate with different speeds c0 and cj, where cD is the pump speed of light, and c^ is the Stokes speed of light in the scattering medium. This is a much more complex mathematical problem. The equations are: ^ l t f l l = + 3* ^ ^ ^ (2D. lb) Define coordinates x and y: C*Ci 2 (2D.2a) I Ca X" = \C ,-C o\ r / - S < > C J - ± - — C ' — 2 (2D. 2b) ( \C<-C4 L lc ' - c °I The equations and their boundary conditions may now be written: D s K F + Z r ) Equation (3D.3a) can also be written: I J_ £P _ I 2 L. F_ -Jf p ST' J f ft > L ; <*>.«o Take the partial derivative of equation (3D.4) with respect to y: i / o j £ p]= w = £ But, by equation (3D.3b)» this can be rewritten as: P f i- i- 3^1 _ F(F+3f) - ~ -£> P Qyj (2D.6) Q C J_ J_ 1 ^ ^ L F+ > p3>/ J # ( i ) M - ( r ) ' 3* where w(y) is the "constant of integration." Define a function h(y): f ^ 7' If ‘ JV') i / \ p 7, b ) i p - \ 4 e e (j d . 8 ) yz'f, J Equation (3D.7) can now be written as: Defining f(x) as the "constant of integration" of this last equation, we now have (2D.9) ft*) + Wf) Similarly, ^ (3D.10b) H)6 +■ Wjr) This reduction of the partial differential equations (3D.3) to the ordinary differential equations (3D.10), within the context of stimulated Stokes scattering, was first adopted Q by Maier, Kaiser, and Giordmaine. The functions f(x) and h(y) are to be determined by requiring equations (3D.10) to obey the boundary condi tions: / n * d|*(y) P ( ^ ” -j%$r The linear differential equations for f and h represented by (3D.11) can readily be solved: > £ P ^ ) W = > +• ( / A + T ) t y ) (2D.12) V ( f ) = eXpff/jy' ( 3D. 1 3) £Y*) = + W*') (3D.Wa) J jrW. I A ) = W x.) v <fp C R a <- D ' 1 4 b ) where ®Maier, Kaiser, and Giordmaine, op. cit. 44 and the problem has been reduced to quadratures. The constants f(xQ) and h(x0) have never been defined, but the intensities P and P are independent of their values so long as the sum f(xc)+h(x0) is nonzero. Having found the function f and h, the intensities P and P may be derived from equation (3D.10), but in practice these intensities are more easily derived by combining equations (3D.10) and (3D.11): ^ ‘ * W t f l 1 Consider now as a speciric example the rectangular pump pulse of width T: vTct) - < n . (3D. 16) {,0 orfcenu.se The solutions for normal and anomalous dispersion for such a pump pulse are readily derived from the above equations. J ' * ' * fir \ t ,-r ft* = e fir 0 < x i 7 T Z J =& f r , ' <3D-17> ^p <2p (V-T©) where x0cO and Q = —^ T — ^ £> (3D.18) I c,-c.| The functions f(x) and h(y) can similarly be found. 45 The resulting solutions are: Region I (O^x ^-9,0<y <9) (see Figure 6): Pfyf) = Jo / aF \ / To S (3D.20a) (t.+aF ) e ' + ( 5 ^ 1 p/vy)- r ' - 1 - - (;5D*2 0 b ) ^ 1 + ap Region II (x >9,0 ^y ^9): (tty)-I. W . M f e ) ( i- . . . W /y)=£» ^pCfc+ap^srt ^ (3D. 21b) | ^ Q j F0b+ap^-. Region III (0 <x <9,y ^.0): P6,/)= 3PCiii-apV e - 1 ffyy)- X© r« ' e f Region IV (x>9,y<.0): (3D.22a) (3D.22b) ? ( X , f ) - 0 (3D.23«) 1 - e ______________ Ffy/) = Z ^ a A ro/ - > ^ ) ^ D . 23b) V /<£ f a H ^ -1) z« c,(t-r) NORMAL DISPERSION (°l>c0) 46 z- c , ( t - T ) I: 0<X<6}0<y<0 It: *>*, o<y<0 TET: £><*<©, z = Cit IE: X>6,y<° x*o Z=C,(f-Tl y = c,c»T ANOMALOUS DISPERSION (00>°l) z--c.a-f) y = c.CoT TL r: IL: Ht: HT; Z=Cet /* ° 0<x<0 ,o<y<e y< o }o< y<& o<X<* , y> 0 K< O y>& Figure 6: A Minkowski space-time diagram of the 2WFSD scattering of a pump pulse of width T, showing the four solution regions both for normal dispersion and for anomalous dispersion. 47 As indicated, the solutions are divided into four regions, diagrammed in Figure 6. Regions III and IV contain Stokes light only, propagating without change from regions I and II respectively. Region I corresponds to the initial penetration of the pump beam into the scattering medium. Region II is the locus of space-time points occupied by the pump light after all the Stokes light which grew from the cell entrance window has passed through the pump pulse. The only Stokes light left inside the pump region is that which grew from one edge of the pump pulse as it traveled through the medium. A study of the solutions indicates that most of the pump energy is confined to the front of the scattering cell to a depth of Lp (equation (3C.6)) if but that most of the pump energy propagates through the cell beyond Lp with only a gradual degradation if In the first case, the Stokes light building from one edge of the pump pulse can grow to such intensities that the pump is depleted within depth Lp in the cell before the Stokes light reaches the other edge of the In the second case, the pump pulse is shorter than this pump. Only a thin 48 sliver, and the Stokes light runs the entire gamut of the pump pulse before it can significantly deplete the pump. A curious feature of the solutions is that, in region I of Figure 6, they are identical to the 2WFSND solutions (2C.5). In particular, they have no time dependence. E. Two Waves-Pump and Backward Stokes in a Cell of Infinite Length Solutions to the 2WBSOOC equations have been found by the author by another approach before publi cation of the Maier, Kaiser, and Giordmaine paper, ^ but their method of attack is employed herein because it is by far the more elegant. In terms of coordinates x and y, assuming dispersion, where * = + (JE.la) y “ (3E-lb) where cQ and c^^ are the pump and Stokes speed of light, the scattering equations and boundary conditions are: 8f (3E.2a) ^Maier, Kaiser, and Giordmaine, op. cit. 49 ( 3 E . 2 0 ) I 3 ^ y ^ ) = O (3E.2d) The solutions may be written in the form: * * ‘P = “ <3E-*> (3E.3b) B 6 0 O + <3b “ * < s ^ r~_ using arguments similar to those of last section. The functions f and h are determined by forcing the above expressions to satisfy the boundary conditions: p/C.K. L.) - ' f y u f c r (?E-aa) <»•*«>> (5E.5a) These are readily integrated: K G ) - /?&)- J e + ^ ) ] ■ ■ $ >»■»> yso Juggling equations (^E.4), one can write P f y ) . P C (*.« R^v/\- 2» h t y ) (3E.6b) b t ' B 50 and the problem has been reduced to quadratures. Consider now the special case of a step pump pulse: In Figure 7 Is shown the spatial profiles of the pump and Stokes modes at four successive times. In the top view, shortly after the pump has entered the scattering medium, the Stokes light Is growing exponentially from noise from the front edge of the pump pulse to the front window of the scattering cell. The Stokes light just reaching the front of the cell In this top view has finally grown to Intensities comparable to the Initial pump. This surge of Stokes light Is now large enough to carve a considerable chunk of energy from the pump light, as shown by the noticeable dip In the pump profile near the cell entrance. In the remaining views, the Stokes light at the cell entrance has essentially reached Its asymptotic limit. The pump, as It enters the cell, Is overwhelmed by this strong Stokes light. It Is quickly drained of its strength as it fights Its way deeper into the cell. The only where 3.-r.«'*0**a,)y ‘ftj,()f-y) 1 f B >,)) = a«[i- e‘3oCC” i,)yJ ( 5 ^ c3s*>«7> 1 + S r > f i . K o £>r t>o (3A.1) t= 2 7s 1 51 / Figure 7: The spatial dependence of the 2WBSooc pump ( -----) and backward Stokes (-----) intensities for successive times, assuming a step pump pulse. The abscissa is scaled to Lg, while the ordinate is scaled to the incident pump intensity I_. The pump to noise intensity ratio Io/ag is 1Q15. 52 significant interval of practically undepleted pump energy is a pulse at the front edge of the pump light, correspond ing to the initial growth region of the Stokes light when the Stokes intensity is still negligible compared with that of the pump. This pulse has a width Lp~2Lg, where Lg is the scaling distance defined by equation (3B.4). Within this pulse the Stokes light grows exponentially with a spatial growth rate ggIQ from the noise level aB at the front edge of the pulse to the back edge: ■ kjB ^oLp B - B . % e g* T o ( 3E.8) where the factor of £ enters the exponential because the pump and Stokes pulses are moving relative to each other. The Stokes light remains in the pump pulse only half as long as it would if the pump were not sweeping past it. Note from equation (3E.2b) that most of the growth of the Stokes light on a logarithmic scale occurs inside this front pump pulse. When the Stokes light passes from the front edge of this pulse to the rear, it is growing expon entially and at its fastest rate anywhere in the cell because the pump is essentially undepleted in this region. But once the Stokes light steps out of this leading pump pulse, just before it grows to intensities comparable to the original pump intensity, and just before it can start depleting the pump on a grand scale, this Stokes light ceases its rapid growth because there is no more pump 53 energy to nurture it (at least not comparable with the initial growth phase). The Stokes light then settles down to a much more sedate growth rate in the remaining interval between the back edge of the pump pulse and the front cell window. The growth rate slowly increases towards the cell entrance as the pump intensity increases. The pump and Stokes light both follow a hyperbolic spatial dependence in the region outside the leading pump pulse, i.e., the region where the e ®B^o^aB)y term is negligibly small. This can be seen from equations (3E.7) when the e®BaB^x_y^ term is replaced by ^l-gBaB(x-y)J , an eminently reasonable approximation for all but astronomically large cell depths. The Stokes light grows to intensities practically equal to the original pump intensity just before leaving the scat tering cell. The Stokes light in this model never grows to intensities larger than the original pump intensity. The pump, just after it enters the scattering cell, is initially depleted very heavily because of the large Stokes light at the front. As it travels deeper into the cell, however, its depletion rate is diminished because both it and the Stokes light have smaller intensities deeper inside the cell than at the front. In Figure 8 is shown the intensity of the backward Stokes mode at the cell entrance as a function of time for various pump to noise intensity ratios. As usual, the intensity profile is practically insensitive to changes of OS 1.0 2.0 O.S *ITme Figure 8: The 2WBSOOC time dependence of the backward Stokes light intensity at the scattering cell entrance window, assuming a step pump pulse. The abscissa is scaled to 2Lq/cl, while the ordinate is scaled to the incident pump intensity IQ. 55 pump to noise ratios over several orders of magnitude. P. Two Waves-Pump and Backward Stokes in a Cell of Finite Length This is the most important section of the whole thesis because the 2WBSFC case is the simplest physical system in which the relaxation pulsations appear, and the only one considered in this paper for which analytical solutions have been found. Hopefully, this will be the easiest model in which to understand the basic pulsation mechanism. The scattering solutions are therefore analyzed in some detail in an attempt to understand most clearly this pulsation mechanism. With the same definitions of coordinates x and y given in the last section, the scattering equations and boundary conditions for a scattering cell of length L are: f£ = -jgPCff-t-ae) OF.ia) |fi= ^ P C S t a * ) (3F.lb) pU - j ) = P C % t p <*•!«> B(x=r l)= 0 <3P-ld> where xsy -f-L corresponds to the rear of the scattering cell. Proceeding as before, ^ d h C y ) = t S v C t ( 3 P - 2 a ) ' T srn svj a8 = (3F-5b) jb Jiy If f(x) and h(x) were knovm over the interval 0^x-£L, then f(x) could be generated over the next interval, L^-x£2L, by the integral (3F.4a). Knowing f(x) over this interval, h(x) could be generated over this same interval by the integral (2F.4b), and so forth. Thus, if the solutions over the first interval, 0*. x ^.L, were known, all succeeding solutions could be found. However, the solutions for the first interval are known. For times O^t^L/c^ and for points z inside the scat tering cell, the solutions must be identical with the corresponding 2WBSO°C model, since information about the finiteness of the scattering cell cannot affect the mode interaction until the pump has reached the back of the cell (assuming that the pump intensity inside the cell is zero for time t<0). Hence, this problem is essentially solved. The solutions, with 57 the following definitions Vt = la r g e st in te g e r ^ ~ (3F.5a) L-y = la r g e s t' 'nte^e'T - (3F.5b) Ut>= X-VxL (^F.5c) W; = y - ^ L (5F.5d) f(U+-l^ , O^U^L (5F.6a) h*V) = hlu+PL) , o±U±L (3F.6b) can be written r u u n , N 0 “ -38>«' 6 ^ " ^ ~j8j du' ( m £ cS<'')t ( l / 1 ^ «"»«' (3P.7b) M=o with initial conditions •Hu') = D t o f k ^ J e 9B<38U -h(o) (}F.8a) h ° (.u ) = h£>) - qtc Cfe)-f4t^7» ""-1 ' £,/£«> „\ • I < V (P^ "') e (3F.8b) u'-o The constant8 f(O) and h(0) are undefined, but the intensities P and B are Independent of these so long as f(0)4-h(0) ^rO. To illustrate the above technique of solution, consider the step pump pulse, equation (35A.1). By trial and error, one finds the following general form for the functions f and h in an arbitrary interval: r° x » 6U f (U) = U ( M j a e + e ) (^P.9a) h (u)- L ur( M ^ e + a ) (3F.9b) ><=o where 4P ^ Uu - (*- 1>L) 4BCZ*t-2d , c _ r-/- _ -iEs- J (3P.9c) ^ - 1 ' r*f<3F and where the coefficients M and N obey the following recursion relations: M 0 U = f U ^ (j5F.10a) - ji M f/ (*.10b) (j5P.10c) M / - <5 £ (3P.i0d) N * = J CJF.lOe) /*=» = S A /u _ | CJF.lOf) y* y where L* gB(lo+-a0). The solutions presented in the following figures were generated by Just this routine. In Figure 9 is 59 t= 2 Lb c L t - 5~ Lb_ r 3 cL _ q l~B C L t - i / ^ CL Figure 9: The spatial dependence of the 2WBSFC pump (-----) and backward Stokes (-----) intensities for successive times, assuming a step pump pulse and a cell length L -5Lg. The abscissa and ordinate are defined as in Figure 7. The pump to noise ratio is also the same. shown the spatial distributions of the two modes at four successive times. Because the pump pulse has not yet left the rear window of the scattering cell, the finiteness of the cell length has not yet Influenced the mode interaction. The solutions in the top two views proceed exactly as in the comparable infinite cell case considered in the previous section. The pump enters the cell, generates baclcward Stokes light, which in turn builds to significant intensities at the front of the cell. This Stokes light depletes the pump, forming a pulse of width Lp — 2Lg at the front of the pump signal, just as in the previous example. In the second view, the backward Stokes mode has by now grown to an intensity at the cell entrance comparable to that of the incoming laser light. At the same time, the front edge of the laser pulse has just touched the back window of the scattering cell. In the time between views two and three, the front pump pulse has completely passed through the rear window. This drastically affects subsequent growth of the backward Stokes mode because most of the pump energy, and hence most of the Stokes growth, occurs in this pump pulse region. As this pump light leaves the cell, the pump energy contained in this exiting pulse becomes no longer available to the Stokes mode. This is especially critical because the Stokes 61 growth in this front pulse was exponential, the largest growth rate anywhere in the scattering cell. After the pulse has left the cell, the Stokes light can grow only from the much depleted pump values, as in the bottom two views. One expects (correctly) that the backward Stokes intensity will exhibit a dip at time t corresponding to the exit of this leading pump pulse. Such a dip is indeed found in the bottom view. The mode profile does not long stay depleted, however. Because the Stokes light is greatly reduced in intensity at the cell entrance, it can no longer deplete the pump as vigorously as before. The pump no longer drops in intensity so drastically upon entering the cell, and indeed remains almost undepleted, as shown in the bottom view. In other words, a new pump pulse is created at the cell entrance window after time L/cl has elapsed since the old pump pulse has left the cell exit window. As in the first pulse, the pump light will extend into the scattering medium essentially undepleted until the backward Stokes light can grow from its depleted level at the front of this new pump pulse to intensities comparable to the incoming pump light. The second pump pulse, therefore, can extend into the medium a distance Lp', defined by 6 = a a r0 > 2 (iv) 62 (3P.11) where a', the new "effective noise," is the intensity D of the depleted Stokes light as it enters the front edge of this second pump pulse. Since there is pump energy, however depleted, between the front of the new pump pulse and the rear window of the scattering cell, aB' will in general be larger than aB, and Lp' will be smaller than Lp. Since the noise can vary over several orders of magnitude without drastically changing Lp, we shall assume that Lp’ = Lp. After the rear edge of this second pump pulse has been chiseled out of the solid block of incoming laser energy by the once more powerful Stokes light, this second pulse sweeps through the scattering cell towards the exit window. As with the first pump pulse, almost all of the growth of the Stokes mode on a logarithmic scale occurs within this one interval of practically undepleted pump energy. When the second pump pulse leaves the scattering cell, the Stokes light again takes a dive, which allows a third pump pulse to enter into the cell, and so on. In this manner, a train of pump pulses, and correspond ing Stokes pulses, can be generated in a scattering cell of finite length. Such pulses are never observed in a cell of infinite length because the initial pump pulse never leaves the cell, I.e., the pump energy Is always available to the Stokes light. Hence, the Stokes light never diminishes in intensity at the cell entrance, and no new pump pulse can be formed. The oscillation mechanism cannot occur unless the transfer of energy from the undepleted pump pulse to the backward Stokes mode is suddenly disrupted after a distance L from the front of the cell. In this case, the disruptive mechanism is clearly the finite length of the scattering cell. The parameters of the scattering system are the laser Intensity IQ, the backward Stokes gain gg, noise intensity ag, and cell length L. The behavior of the pulsations is best described in terms of the lengths L and Lg, where Lg is defined in terms of IQ, gg, and ag by equation (3B.4). In Figure 10 is shown the dependence of the pulsations upon these two length parameters. The period of oscillation is determined solely by the cell length L and the speed of light in the scattering medium: T0scillation=:2I'/cL* Th® lonSer the cell, the longer the period of oscillation must be. Similarly, the width of the Stokes pulse is determined by the ratio L/Lg. The higher the Stokes gain, the shorter Lg is relative to L, and the faster the pulse will rise. As a consequence, the higher the ratio L/L0, the wider the Stokes pulse. Backward Stokes io 2- l-S~ Figure 10: The temporal structure of the 2WBSFC backward Stokes intensity at the cell entrance, assuming a step pump pulse. The abscissa is scaled to 2L/cl* where L is the length of the scattering medium, while the ordinate is scaled to the incident pump intensity I . The pump to noise ratio I0/*B is 10 • In Pigure 10 are shown pulsations corresponding to L/L_ ratios ranging from two to ten. What is the D nature of the pulsations when this ratio Is considerably larger than ten or less that two? As the ratio approaches one from values larger than unity, the distance Lg becomes larger compared with L, and the Stokes light takes longer times to build up to its asymptotic limit compared with the pulsation period. Hence, the pulse width becomes more and more narrow, until the ratio L/Lg reaches unity. When Lg“ L, the Stokes pulse waits the full pulsation period until it Just begins to grow to a fraction of its asymptotic intensity before it is knocked down again at the end of the pulse period. For L/Lg ratios near unity, therefore, the Stokes pulses look like little spikes with a maximum intensity only a fraction of the initial pump intensity. When Lg is larger than L, the Stokes light cannot build to the intensities needed to drain the pump thoroughly because the Stokes light leaves the front cell window too soon, i.e., the cell is simply too short. As a consequence, the pump never exhibits noticeable depletion, and in particular, the pump pulses of length Lp never form because the cell is shorter than Lp. Hence, the pulsation mechanism breaks down if the cell length is too short. In the other extreme, when L is considerably longer than Lg, the Stokes light burrows through more and more pump light while traveling from the cell exit window to the front of the second and subsequent pump pulses. Therefore, the effective seed energy ag1 of Stokes light entering the second pump pulse is larger for larger L/Lq ratios. Graphically, the dips between Stokes pulses "fill in" because the seed energy is much larger, and the pulsations quickly degrade into the steady state limit. This naturally raises the question of the stability of the Stokes pulses. Are there combinations of the L/Lg ratio for which the pulsations appear to be fairly stable? In this 2WBSFC model, on the basis of all the solutions generated so far, the answer appears to be no. The pulsations always tend to degrade to the steady state limit. The greatest stability appears for L/Lg ratios close to unity. The stability is upset because the Stokes light does not go to zero intensity at the end of a pulse period, but rather to the seed intensity ag1. The pulsation stability will be greatest for the smallest aB', and ag' will be the smallest when there is the least distance between the cell exit window and front of the pump pulse through which the Stokes light can grow. This distance is the least when Lb Is closest in length to L. CHAPTER IV THE NUMERICAL SOLUTIONS The transient 3W and 4w problems unfortunately appear to be analytically intractable. Therefore, representative problems have been solved by numerical integration techniques discussed in Appendix C. The results of these computer programs are presented below. A. Three Wave Scattering In Figure 11 is presented the backward Stokes inten sity at the cell entrance as a function of time for the 3W problem in a scattering cell of infinite length, with equal forward and backward gains and noise intensities, assuming a step pump pulse at the cell entrance. Note the presence of pulsations similar to those observed in the 2WBSFC solu tions of Figure 10. Apparently the forward Stokes mode terminates the flow of energy from the pump to the backward The equations under consideration are: P<B+<3b) 67 0.6 E l - Figure 11: The temporal structure of the 3W equal gain and noise level backward Stokes intensity at the cell entrance assuming a step pump pulse. The abscissa is scaled to 2Lp/c-> while the ordinate is scaled to the incident pump intensity IQ. This i® a numerical solution of the scattering equations. Backward Stokes Intensity 0.5 1. 0 Figure 12: The temporal structure of the JW backward Stokes intensity at the cell entrance for various backward to forward gain ratios, assuming a step pump pulse. The abscissa and ordinate-are scaled as in Figure 11. The pump to noise intensity ratio I /a is . This is a numerical solution. vo Backward Stokes (.o Figure 13: This is the same as the previous figure, only with the abscissa scaled to 21^/c-^ instead of 2Lp/c^. This is a numerical solution. —j o 71 mode after length Lp, the 2WFS interaction length defined in equation (^C.6). This length Lp replaces the length L in the comments of the last section of the previous chapter. This is demonstrated most clearly in Figure 12, where the backward Stokes intensity at the cell entrance for various gain ratios is plotted against the time scaled to the forward gain. Contrast this to Figure 1J5 wherein the same solutions are plotted against time scaled to the backward gain. Note the similarity between Figures 10 and 12. All pulses in Figure 12 are cut off at the same time, 2Lp/cB, compared with the time 2L/c^ in Figure 10. For systems with gain ratios less than unity, LB is larger than Lp, and the backward Stokes light leaves the pump before it can drastically deplete the laser light. Hence there will be no pulsations when gg<gp. To investigate in more detail this pulsation mecha nism, consider Figure 14, wherein is exhibited a coarse "movie" of a typical JM interaction. In the top frame, the forward Stokes light has not grown to intensities compara ble with the pump, and so the pump and backward Stokes light interact essentially as in the comparable 2WBS case, with the backward Stokes light at the cell entrance reach ing the asymptotic intensity IQ shortly after time 2LB/cL. The pump in this initial scattering period is, as usual, etched into a pulse of undepleted pump light with width Lp=i2Lb. In the second frame, this pump energy has just Intensity 72 t , w £ t= 2.0 ZCu t= 2.6 £5. 0 8.0 Figure 14: The spatial dependence of the 3W pump (----- ), forward Stokes (-----), and backward Stokes (-----) intensities for successive times, assuming a step pump pulse. The abscissa is scaled to Lp, while the ordinate is scaled to the incident pump intensity ID. The pump to noise intensity ratio IQ/a is io , while the backward to forward noise ratio and gain ratio are l/l and 5/1. This is a numerical solution. 73 reached depth Lp, and the forward Stokes light is just beginning to overwhelm the pump. After the forward Stokes mode has wrenched the greater portion of the pump energy away from it, the pump is no longer capable of nurturing the budding backward mode, and we find our disruptive mechanism responsible for the relaxation pulsations. The only difference between this 3W solution and the previous 2WBSFC solutions worth mentioning is the question of stability of the pulsations. For no L/Lg ratio in the 2W case were any of the pulsations apparently stable. However, in the 3W problems, the pulsations generated by computer for the Lp»Lg models appear to be quite stable, with no noticeable tendency to degrade into the steady state limits. Of course, for Lp/Lg ratios significantly greater or smaller than unity, the solutions readily tend towards the steady state with relatively few or no pulses. B. Four Wave Scattering The numerical solutions for the 4w scattering problems for various interesting combinations of the medium parameters are essentially identical with the numerical solutions of the related 3W problems (the two forward modes of the 4w model and that backward mode with the greater gain). The mathematical explanation is straightforward: q r i \"7 If the Intensity of one of the backward modes Is known at a given space-time point, the intensity of the other backward mode at the same point can Immediately be determined by equation (4B.1). Define g y to be the larger of the Raman and Brillouin gains, and g< to be the smaller. Let B^. and a^ be the intensity and noise of the mode corre sponding to g^, and similarly for B< and a< . For large signal to noise ratios, Typical values for observable B/a ratios range from 10^ to lO1^. For a large class of problems, the Stokes intensity corresponding to the mode with the smaller gain is considerably smaller than the intensity of the mode with the larger gain, even for gain ratios g </g>> which are fairly close to unity. For these problems, the mode with the smaller gain will not significantly affect the interaction of the remaining three modes, and so it can safely be ignored. The full problem, therefore, can often be reduced to a corresponding 2W problem with little loss in accuracy. This is demonstrated in Figure 15, wherein is shown the time dependence of the backward Stokes light at the cell Jjnfcens'X] Figure 15: The temporal structure of the 4W equal noise, gi of ___ __ equal noise, S w S ] 5/1> backward Brillouin intensity at the cell entrance assuming a step structure the noise pump pulse. The same curve is also the backward Stokes intensity at the cell entrance for the related 5W model, i.e., backward to forward gain ratio gg/gp of 5/1. The abscissa is scaled to 2L-/cL, and the ordinate is scaled to I . The pump to noise ratio I0/a is 10 • These are numerical solutions. entrance for a typical combination of the medium parameters. The 3W and 4w numerical solutions map precisely on top of each other, with no noticeable variation. Incidentally, the small scale time struc ture at the top of the pulses is probably due to the numerical integration errors from using a grid Interval somewhat larger than that necessary to generate smooth solutions. Also note the transition from oscillatory behavior to steady state behavior after four or five well-defined pulses. CHAPTER V EXPERIMENTAL VERIFICATION OF THE THEORY The models considered above have several unphys ical characteristics to keep the mathematics as simple as possible. Before relating these results to exper iment, we must first consider the expected character istics of more physical models. Therefore, in the first section, we will consider pump time profiles at the cell entrance window which are not step pulses, while in the next section we will investigate nonuniform intensity distributions over the beam cross section. Finally, in the last section we will list the design parameters for a typical experiment and relate these to specific examples of N^ and H^ at room temperature. A. Pump Pulses of Finite Length A good approximation to the time profile of many a Ci-switched laser is This is the product of a ramp function, with a Gaussian distribution, scaled to have peak intensity 0 - T o r t< O h r t > 0 (5A.1) 77 78 I0 at time Tpump. Do these pulsations exist for such a pump pulse, and If so, what are their character istics? Clearly, If the pump width Tpump Is shorter than 2Lg/cL, the baclcward Stokes light would leave the pump energy too soon to saturate, and no pulsations could occur. But If were several times longer pump than the pulsation period, the behavior of the backward Stokes light is not Immediately clear from the above theory. Consider the numerical solutions of a typical 2WBSFC model in Figure 16. Note that pulsations with a period 2L/cl are indeed superimposed upon the basic pump pulse structure. I suspect that the longer the pump pulse (i.e., the more slowly the pump envelope changes in time), the smaller the modulation depth In the resulting Stokes light because the closer the oscillation system can follow the variations in the pump intensity. Figure 17 illustrates this hypothesis. Unfortunately, the computer solutions for this phase of the problem are quite time consuming, hampering a more extensive investigation of non-step pump pulses of reasonably long duration. A 2W solution for the skewed Gaussian pump pulse is presented in Figure 18. This represents some three hours of IBM 360 model 65 computation time. Unfortunately, there is much fine structure which is probably mesh noise, i.e., fluctuations which could be smoothed if H 8 it to 2o 2*1 2^ 3X 36 HO 5Z 5* 5& Figure 16: The temporal structure of the 2WBSFC backward Stokes intensity t ) at the cell entrance for a skewed Gaussian pump pulse ( -----). The abscissa is scaled to 2 1 ^/c-^t while the ordinate is scaled to I., the peak pump intensity. The cell length L=4Ln, while the pump width T=5(2L/cl). Note that the backward Stokes envelope is delayed relative to the incident pump by approximately (2Lb/cl), the "reflection time" of the backward Stokes system. This is a numerical solution. T- 0 , 1 T = 5 ( f c ) r - x Time Figure 17: The hypothesized dependence of the modulation depth of the 2WBSFC backward Stokes intensity at the cell entrance upon the pump width to pulsation period ratio T/(2Lb/cIj). oo o Irfenji'tv 1.0 1 / backward Stokes Intensity at the cell entrance for a skewed Gaussian pump. The abscissa is scaled to 2Lb/cjj, while the ordinate is scaled to I0, the peak pump intensity. The backward to forward gain ratio g_Vg is 2/1, while the pump to noise intensity ratio ID/a is 10 . The F pump width T is UObu/c-^, where Lg is the backward Stokes gain length defined for the peak pump intensity I0. This is a (crude) numerical solution. 82 a smaller interval were chosen. However, a decrease in interval size by a factor of two increases the computation time by a factor of four. I therefore present Figure 18 without any claim as to its accuracy. What is expected in Figure 18 is a set of pulses superimposed upon the skewed Gaussian pump envelope, much like Figure 16, but pulses with varying periods of oscillation. This feature is hinted at in the left portion of Figure 18. Note that the pulses should be most densely packed towards the peak of the pump where the pump intensity is the highest, and therefore the forward Stokes interaction length L , a function of the F pump intensity, is the shortest. B. Nonuniform Pump Cross Section Another assumption made in the above theory is the uniformity of the pump intensity over any plane perpendicular to the direction of propagation. No laser has such uniformity. Instead, the most typical intensity profile is Gaussian, with cylin drical symmetry: where r is the distance from the axis of symmetry. Since the pulse width and, for 2W models, the oscil lation period are pump intensity dependent, the temporal (5B.1) 83 structure of the Stokes power BP(t) will be somewhat different from that predicted for uniform intensity distribution. r- B F & = f x T r d f * 8£r, ?=*-fc) (5B.2) r-o is the backward Stokes power at the cell entrance as a function of time. The r dependence of the backward Stokes intensity B enters because of the r dependence of the incident pump pulse. Assume the following pump profile: Co Coc ~ t< 0 rp „rf r , w, f l . ^ ^ t>0 m At any given r, the Intensity is uniform, with a step pulse time profile. Hence the various solutions generated in the previous two chapters are valid here, if the incident pump Intensity IQ is replaced _r2 /r2 with IQe ' . 2WBS<«C: Consider first the 2WBS60C solutions given by equations (3E.7). This may be written, within the present context, as cs ^ r -J8(X»e t • ^-Cub 8<y,?=°1 ts) = ----------- -------J (5B.4) 84 This can be substituted into equation (5B.2), the Integral performed, and the power found. Even for this most simple of cases, however, the integration is far from trivial. Keeping in mind Figure 8, let us therefore approximate equation (5B.4) by S i z . e X C t , (5B .5) where . r5 Cr /h) = ' ‘I* + J s and X&jTh) ~ f A jA 0 o ik e r o * i5e ? (5B .7) W t)-j'a w .k - I . e ~ f t i j T i l f ) ) f - o = :&> - P C ^ V T * - ^ ’ p T7 ~ ® 7 ^ y e " e J (5B.8) r-o where r_j_ is defined by , r N '($ ) 2 (r = « > t - 2 T e => e - — (5B.9) O and P0 —T^TTR is the power in the Incident pump beam. 85 d for £■ < 2 "5^3 PB&) ^ £r i? 2 7 l( o ) (5B.10) This is diagrammed In Figure 19. The physical interpretation of this solution is fairly straightforward. Speaking rather loosely, a pump of uniform intensity IQ is "reflected back" at the Stokes shifted frequency by the scattering cell after time 2Lq/ c^ . If the pump has a Gaussian intensity profile, the center will begin to "reflect" the pump energy after time 2Tg(r=-0). As the time progresses, more and more of the initial pump is "reflected," until, after an infinitely long time, all of the pump is returned as backward Stokes light. Unlike the intensity, the Stokes power does not "turn on" suddenly but rather builds hyperbolically over times on the order of What happens if we expect pulsations in the intensity? One system we have found for which pulsations are expected is the 2WBSFC model. Let us therefore consider the following approximation to the 2WBSFC solutions (3F.9): 2 'ZTgCr^O). 2WBSFC: (5 B .1 1 ) Power l.o 1.0 A 3W o S Figure 19: The approximate temporal structure of the backward Stokes power at the cell entrance for various models. oo o\ 87 where /7X,(r/R) is still defined by equation (5B.6), B and 'T - — 2L/c is the period of pulsations for a cell L i i of length L. Define: U - s n & l/e s f in te g e r ~ (5B.12) T ~ t - D T u (5B.1J) Then t b & ) , T l )= ; K > 37ifr) 6 o l i e r u > he. (5B.1U) a ^ ^ ^ « ■ W &«- T > 2 7 5 A ) jfl . - (5B .15) f=tV f=tf> £ tfcfirtai ze where r_^_ is defined by T = t - P T L = - /T+>2 e ^ = 2. Ty(r=o) (5B .16) t ~ » T u C> r. ££{2L 1 £ 2T8k) Tl .jU-ise (5E'17) The tim e p r o file o f the Stokes power i s shown in Figure 19. Note th a t the p u lse stru ctu re i s the same as th a t fo r th e 2W BS0OC s o lu tio n , repeated w ith a period *77^. The pulsations appear to be as stable as the corresponding 88 intensity pulsations. 3W: A much more complicated model Is the 3W because the pulsation frequency for each value of r is different, unlike the previous case where the frequency was a uniform *7^ over the total cross section. The 3W pulsation period is different, of course, because the period is defined by the forward scattering length, which in turn is a function of the initial pump intensity. This intensity, in turn, is dependent upon the r chosen. Consider the following approximation to the backward Stokes intensity: j Z V R (5B.18) where is still defined by equation (5B.6), and For a given r, 2 7T|j(r/R) is the time between the entry of the pump energy in the cell and the emergence of backward Stokes light at the cell entrance, while Z J0„ {r/R ) r is the amount of time before the Stokes light "turns off" again. Define a function % to have the value unity when the Stokes light is "on," and zero when the Stokes light is "off": 89 Define i l l A = s n d / d l d e a e r — t- 2X~b (5B.20a) , i « « ? * ) Wr)*Si»dfcsT7ttfe«r 5i i - 2 T p (W i) (5B.20b) J 2tp flft) ftf,T,,7»= f 1 * A (r)* * r) (<5 $ ji(r) = Xr) (5 ) At any time t> 2'Cg(r =0), there is always at least one region of the cell entrance window for which the Stokes light is "on." These "on" regions have the shape of rings or disks. There will be j j i ( r =0) such regions at any given time. The ^ th region is r> r^” , where (5B.22a) (5B.22b) — " f ~ define r^ and r^ ; i.e., - 2~? ^ r~^- &r 9(<"=.) (5B.23a) 6~ (.%) = ITbM ^ af<>-i) feftW) (5B.23b) ^ t: The innermost radius of the innermost ring, , is either zero or nonzero depending upon whether the r = Q point is "on" or "off." If it is "off," then yL4(r=0)=- r =0), and the radius is defined by However, if it is "on,” then r =-0, and £. _ 2ufrJ lU .C * o ) 90 - T (5B.24) - ( ¥ ) ! ; L PB(t) = Fo ^ U 1 £ ) ' _ c~ ( . f ) j (5».89) 7S» For yx(r =0) ^ ^(r =0), t>2Z^(r=0), reft* = a C i - (5B. 26a) For j j S . r - 0 ) =-U(r =0), t >2tB(r -0), PBftr) - Pe (5B.26b) For t<2 t^(r-0), BP(t)=0. (5B.26c) These solutions are shown in Figure 19. Note that the backward Stokes power exhibits pulsations with a period equal to the on-axls (i.e., r —0) period 2^p(r=0), and that these pulsations quickly decay into the steady state value, regardless of the stability of the inten sity pulsations. C. Designing the Experiment Consider a typical experiment consisting of a laser pump, scattering medium, and light sensing equipment, as shown in Figure 1. Assume that the pump has the time profile of equation (5A.1). The pump parameters of primary Interest are the peak Intensity 91 I0 and the width while the parameters for the ° pump' scattering cell are the gains and noise levels gR, gB, a , and a , and the cell length L. Prom these numbers R B can be defined four time parameters: Tp, Tg, TCell' and T The time T„ is the forward gain time, which is pump r always the Raman gain time, Tp=r 2LRaman/cL, where Iceman is the gain distance of equations (3B.4) and (3C.6). The time TB is the backward gain time, Tg=: 21^/0^, where Lg is the shorter of the (LRaman* I^rmouin^ gain lengths. The cell time Tce^^ is 2L/cl, the round trip time for a photon in the scattering cell. If either of (Tp^p* Tcell^ is smaH er than T0, the back- ward Stokes light cannot grow large enough to saturate the pump, and no pulsations will occur. Define Tpu^se to be the smaller of (Tp, Tcell). The pulsations will be noticeable only if T „ is sufficiently long pump compared with Tpulae, say, several periods. If Tpulse= Tcen> the period will not be intensity dependent, but rather cell length dependent. One must be careful to keep all of these times long compared with the medium response times (at least an order of magnitude). In an experiment designed to test the above theory, the only permissible nonlinear effect is, of course, the Stokes scattering. The presense of self focusing, for instance, will drastically alter the temporal structure, thereby rendering the above theory almost useless. Unfortunately, most liquids with large Stokes gains, such as carbon disulfide and benzene, are also self-focusing, and hence not suitable candidates for a quantitive test of this theory. A scattering medium which does meet the above requirement is N2 at room temperature. It has the additional advantage of an adjustable gain, i.e., a gain which can be varied at will over several orders of magnitude simply by varying the pressure. The dependence of the Brillouin gain for Ng upon the pressure is shown in Figure 20, from the Hagenlocker, Mlnck, and Rado paper. Consider a laser with peak intensity IQ megawatts/cm2. The time for such an intensity, assuming a noise level of 10"^ watts/cm2 and a pressure of 100 atmospheres, is 100/lo nanoseconds. The pulsations can only appear in a cell which is longer that CjTg, i.e., Consider, for example, a pump intensity of 500 megawatts/ a cell 15 centimeters long. The pulsation period, ZL/c-^, will be 1 nanosecond, and the pump must be several nanoseconds long to generate any pulsations. This would be an ideal experimental' test, except for one serious flaw: the sound absorption time for cm2. The minimum cell length is 6 centimeters. Choose 9? N2 at room temperature and 100 atmospheres Is 10 nano seconds, one order of magnitude larger than the expected period, and two orders of magnitude larger than tb- Long phonon lifetimes are, unfortunately, a characteristic of most gases. If the pressure were changed from 100 atmospheres to 10, gg would decrease two orders of magnitude and TB and I*crltical would Increase two orders of magnitude. The sound absorption time would decrease to 1 nanosecond, a sufficiently small period compared with Tg. The required cell length, however, now would jump to 6 meters for a 500 megawatt/cm2 pump, and the pump width must be a minimum of 80 nanoseconds. If the intensity could be increased, the required cell length and pump pulse duration would correspondingly decrease. Remember, though, that the models of this theory presuppose scattering from unfocused pump light. Just as N2, with its high Brillouln gain and low Raman gain, is an interesting candidate for 2WBS scat tering, H2 is an interesting test of the 3W theory. Unfortunately, H2 also suffers from long phonon life times, even for Raman scattering, but the possibility of controlling the pulsation period by merely adjusting the pressure is quite intriguing. Ignoring the normal transient effects due to long phonon lifetimes, the period of pulsation of H q at 100 atmospheres and room 9k temperature pumped by light of intensity IQ megawatts/cm2 is 2 ^iseconds/l0. Thus, a pump of intensity of 500 megawatt s/cm2^ should generate a pulse train with period of 4 nanoseconds in a cell not shorter than S ^ (5C.2) i.e., about one meter. A drop in pressure of one atmosphere would increase the critical length and the period of pulsation some two orders of magnitude. The pressure dependence of the N2 Brillouin gain and phonon lifetime and H2 Raman gain and corre sponding phonon lifetime is given in Figures 20-23, from the Hagenlocker, Minck, and Rado paper. Br-.IU* 3*;« g 8 (J£r) 95 ISO K. loo K . -0 -II IO Pressure 6t*n.) Figure 20: The steady state Brillouin gain gB for nitrogen. From the Hagenlocker, Minck, and RSdo paper. -JO /O' Figure 21: The acoustical phonon lifetime in nitrogen. From the Hagenlocker, Minck, and Rado paper. vo ov 97 <n ID Figure 22: The steady state gain gR for the Reuman vibrational transition QjU) in hydrogen. From the Hagenlocker, Minck, and Raao paper. -i a . to -8 10 Raman Q,CO Transition Hydrogen 5-00 K SOK. <0 lo ** Pressure°(atm.) '°C to' to Figure 2 3: The optical phonon lifetime for the Raman vibrational transition Q,(l) in hydrogen. From the Hagenlocker, Minck, and Rado paper. vo 00 APPENDIX A THE ELECTROSTRICTIVE FORCE DENSITY The derivation In this appendix Is a slight rephrasing of the derivation given by Landau and Lifshltz Q in their Electrodynamics of Continuous Media. The force per unit volume F exerted on an isotropic fluid dielectric by an external electrostatic field is equal to the gradient of a stress tensor O'* the Maxwell stress tensor: P = 3 1 a T (A-l) The Maxwell stress tensor can be found by considering the work done in deforming the fluid an infinitesimal amount. Consider a plane-parallel slab of fluid uniform in composition, density, and temperature, with thickness h and normal vector n. Assume that the electric field within the fluid is due to conducting planes on the surfaces of this layer of fluid. Subject one of these conducting planes to a virtual displacement , keeping Q L. D. Landau and E. M. Lifshltz, Electrodynamics of Continuous Media (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, I9 6 0 ), pp.64-69; see also W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley Publishing Company, Inc., Reading Massachusetts, 1965), pp. 92-95* 99 100 the potential on the conducting planes unchanged, and assuming the process to be Isothermal. A force Is exerted by the conducting plane on a unit area of fluid surface, and this force does work -0"ucnj c^i* This work is equal to the increase in the free electrical energy J u dV, where U= fcE (A.2) 8 7 r p _ i.e., h€E /81T per unit surface area. <A-3> JTi = (A*4) 3 - T ^ r <A,5) In an homogenous deformation, with z the distance from the fixed conducting plane, the virtual displacement of each point in the fluid is (z/h)£ . Matter which was at r**(z/h)£ is now at r, and the change in electrical potential < j) is S(j>- (fir) - ~ g'£ ( A,6) The variation in the electric field is 101 ' T ) *'* *" (*■») The electrostrictive force is due to the (TE2/81T)^\. 1K term of the Maxwell stress tensor, i.e., ^eclmfrtJite = V ( w f ) . ( A ' 1 0 > APPENDIX B THE FLUID DENSITY AUTOCORRELATION FUNCTION The response of a system to an external distur bance can be described by a correlation function, or, if the disturbance leads to a sufficiently slow variation in space and time of all physical quantities, by the hydrodynamic equations. This equivalence is stated in the fluctuatlon-dissipation theorem. We shall use this theorem and the hydrodynamic equations for a one component fluid to derive the density autocorrelation function in the limit of slow variations. This autocorrelation function will then by used to evaluate the Brillouln noise intensity ag of Chapter II in the adiabatic approx imation of C. L. Tang. 1. The Classic il Linear Response Theory^ The response of an isolated system, initially in equilibrium, to a sufficiently small external force S f is linear: + 00 t) (b.i) - oO ^For a derivation of the equations in this section, see R. Kubo, J. Fhys. Soc. Japan 12, 570 (1957)* 102 102 where S b is the shift from the equilibrium of a dynamical variable B Induced by the perturbation. All dynamical variables are assumed to have no external time dependence, only that of the unperturbed Hamiltonian.10 When the external force defines a perturbation Hamiltonian of the form &K(t ) - - fjr FKr,t) SF(r,t) (b.2) then the susceptibility has the form 41 . (B>3) O oiharvtse where the brackets {* j denote the Poisson bracket operator, while the brackets denote the statistical equilibrium ensemble average. Note that this linear response is quite independent of the nature of the equilibrium ensemble distribution. 2. The Flue tuatlon-Dissipat ion Theorem11 If the equilibrium ensemble is canonical, then it can readily be shown that < ^ Wfi) (b.m 10For a more concise statement of this time depend ence assumption, see Kubo. 11For both a classical and quantum mechanical state ment and derivation of the fluctuation dissipation theorem, see Kubo. 104 where the dot represents differentiation with respect to time. In terms of Fourier components, work is that we can find the susceptibility, at least in the limit of slow spatial and temporal fluctuations, from our knowledge of the hydrodynamic equations and the equation of state. This is sufficient information to define the correlation function in the limit of the validity of the hydrodynamic equations. 3. The Philosophy of the Approach There are at least two philosophical approaches possible in a problem such as this. Both contain identical physical information. The first and most widely used approach is to calculate the response to a continuously applied sinusoidally varying perturbation force. The second approach is to apply adiabatically the perturbation force from the infinite past to time t=0, abruptly cut off the force at t— 0, and then calculate the decay of the system for positive times. We will eschew the first approach in favor of the second because the information contained in the equation of state, information necessary X J k«) <b-5) This is a classical statement of the fluctuation- dissipation theorem The significance of this theorem to our present 105 for a complete description of sound propagation, can most easily be incorporated if the second approach is selected. As usual, the transforms of the dynamical variables are more easily manipulated than the variables themselves. Since our interest is in the decay of the fluctuations for positive times, it will prove convenient to choose a (modified) Laplace transform over the positive time domain rather than a Fourier transform over the whole time domain: <§B(k,z)r j d r e k jcfte (B.6) 0 V where z is a complex variable, Im z/0. The fluctuations decay from their initial values at time t — 0, initial values which are defined by the perturbation force at time t: 0: cTFfO = f j i r e J F ( r t t ~ o ) (B 7) _ X m ( £~z) - X id J tiTo) < t f ( D ;Z- (B-8) where is the static wave-number dependent susceptibility and V(k,z) is the analytic continuation M of the susceptibility onto the complex z plane: X j K z ) -- f h i (b.9) The fluctuation-dissipation theorem can be written 3BCk*>+ i6 ) snV) (B *10) 106 1 9 4. The Perturbation Forces for Hydrodynamic Fluids The perturbation force must satisfy two criteria: l) it must maintain the system in local equilibrium at all times, and 2) it must define a perturbation Hamiltonian of the form of equation (B.2), where the dynamical variable A of equation (B.2) is one of the two variables in the correlation function of interest. Finding such a force is not always trivial, as the present problem will show. The electrostrlctive force of Appendix A falls the second criterion. Indeed, no obvious physically meaningful force which selectively stimulates sound waves and satisfies the above two criteria has suggested itself to this author. Therefore, let us invent one, using the technique 12 of Kadanoff and Martin. The key is "local equilibrium." Consider a true equilibrium represented by the grand canonical where f(p,q) is the ensemble distribution function in phase space, p and q the conjugate momenta and coordinates of the Hamiltonian ^f0 of the unperturbed system, J x ! the restatements of the quantum mechanical derivation of L. P. Kadanoff and P. C. Martin, Ann. Fhys. (N.Y.) 24, 419 (1963). ensemb? (B.ll) chemical potential per molecule, 7? the number of molecules, 12 Sections 4-6 of this appendix are classical 107 v'the velocity of the system (assumed to be small), g(r) =. V|0 (r) the momentum density, and p(?) the mass density. Assume v-0 in true equilibrium. In local equilibrium, the various thermodynamic parameters such as T and yu/ can be defined locally, but they no longer need be uniform throughout the system. They will differ slightly from the true equilibrium v a l u e s £ ? ] in local equilibrium as compared with ^L^#T0,oJ in true equilibrium. A reasonable candidate for an ensemble distribution function describing local equilibrium is £ , \ ~ XT ( ^ 'M' h ) ~ f <B.12) where (B.l^a) sm= -fepf ] where £ is the energy density, JA. the chemical potential per unit mass. Equation (B.13b) suggests the possibility of using the chemical potential as the perturbing force in the density autocorrelation derivation. However, for a one-component fluid, the chemical potential has no physical significance, and so incorporating it into the hydrodynamic equations as a driving force would prove difficult. However, the chemical potential (the Gibb's free energy per unit mass) is easily related to the 108 pressure, which Is physically meaningful. pdu - ~ dj - f JP =* (B.l4a) rr v v ' dp = (e>p-yY>)dI (B. l4b) in true equilibrium. In local equilibrium, define a pressure fluctuation (deviation from the equilibrium value) by Jp(r,t) = p . 4- (b.14o) where the subscript V indicates the true equilibrium value. Define also a density of heat energy transfer (see the Kadanoff and Martin paper for details): Cjirt) - - ^ £ = - p<0« (B.15) The perturbation Hamiltonian can now be written = — jdr + relating the dynamical variables [p,q,g] with their "conjugate forces" ; ; Jvj 5. The Hydrodynamic Equations The hydrodynamic equations are J§“ ~V 'J (B.17a) |f " “ + p 7T + ^ ^ (B.17b) ff - -fa>+f}v-7 + v*t (B.17C) 109 where Vf, $ , and K are the f i r s t and second v i s c o s i t i e s and th e therm al c o n d u c tiv ity . These eq u ation s rep resen t th e con servation o f m ass, momentum, and energy. The lin e a r iz e d eq u ation s may be w r itte n 1 ^ 9 - X V*3T = O (B .l8 b ) at 1 where q i s d efin ed in equation (B .1 5 ). Performing a F ourier-L ap lace transform o f equation (B .6) upon eq u ation s (B .1 8 ), iZH'Zttylr1) (Ba9a) -f? ^fjrl2)+KiflT(c:z)= -?<*) ( B . 1 9 b ) where D . = * s ( B .20) ‘ P- R e c a ll from S e c tio n 3 th a t our in t e r e s t i s in the tim e e v o lu tio n o f th e d e n s ity and h ea t flu c tu a tio n p and q from th e ir i n i t i a l v a lu es d efin ed a t tim e t r O by th e d istu r b in g " forces'1 p and T. Let us make th e fo llo w in g s u b s titu tio n s : = f f l s P (5-?) + r M p ^ (B .2ia ) T C W - | p 3 PM + ^ (, ^ (3 .2 1 b ) P( f ) = § f l ^ p ^ + M I FT(^ ( B - 2 i o ) 110 These substitutions serve the added purpose of Incorpo rating the equation of state Information into the hydro- These equations represent sound propagation and heat diffusion combined. Our interest is in sound propagation alone. The sound propagation can be isolated from the heat diffusion when the wave number k is sufficiently small, i.e.; when where D is either diffusion constant. In this limit; ignoring terms in the diffusion constants of second dynam* " * * ’ (B.22b) (B.23) power, the density equation becomes r j!lc * + P r U -% )k 2 - ’ * % ] (b.24) J P-r k3- 1 6. The Density Autocorrelation Function Combining equations (B.10), (B.l6), and (B.24), we find for the autocorrelation function of the density fluctuations from equilibrium The frequency dependence of this autocorrelation function is shown in Figure 24. By standard Brillouin scattering theory, the Fourier component of the scattered light intensity1-5 l(ks,6j) is simply proportional to We see in Figure 25 the central Rayleigh scattering line, the Brillouin Stokes line, and the Brillouin anti-Stokes line. Our interest is in the Stokes line, i.e., the 13 See, for example, Chaper 14 of L. D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, i9 6 0). Figure 24: The frequency dependence of the density fluctuation auto- m correlation function, showing the central Rayleigh scattering peak and £ the two Brillouin scattering (Stokes and anti-Stokes) peaks. IIP 113 Ul ____________ = 2p.K T C<o‘-cfl<y + Lu> k*r)1 ( b. 27) term. Only those components lying in the forward direction in a solid angle ( ^fl)SOund and close to frequency can contribute to the Brillouin scattering under consideration. <<fP fz,t) = I ' r°° \ (2i)^JdtJe < p M > (b.2 8) 0)sO$- ^ -{-Coylex. o jn ju ^ J ’ e. where is the frequency width of our measuring instru ments. The k integral is readily performed. Note that there are four poles in the complex k plane: ( ^ - c / k ’A cs(t-L) ( W K k D M (B29) _ i _ where jfo = £ ( | f ) 2£ g i §££ (B.JO) C-s ^ Cs1 -/ Cs where we have assumed that € « l , i.e., that the sound damping does not dominate the sound propagation. Note that the integrand is sharply peaked for k close to ^ and negligibly small otherwise. To a very good approxi mation, we can extend the lower limit of integration from k = O t o k = -o » : . . o ;lc(z-t') k‘fY ^ ~ J c* ( M u - & )& £ > {% j (B-31) Jcso -o a This last integral is readily evaluated by the standard contour integration techniques: . 114 tTa' 1 . V_ Art* 'g(z-*') I Z-Z'l ' « * (w <? e * <b.*> 4s-«o The only term in the CJ integrand which changes signifi- cantly over the narrow range is e 5 This will give a negligible contribution except when So, to a good approximation, _ / • v Br’ ti/ouin Arf$C S*uto£ <«rP(z,t) * tip) Trp.KTto.fiOt W c , 3 (&/e. e **> JO e. s + - C o m p lex' conjugate e'33'*’1 (».») 87ra C* i . + c&"tplex conjugate. This elegant expression indicates that two points in space-time can be correlated only if a forward-traveling sound wave can connect them, an obvious condition since we have considered only forward-traveling sound waves as the mechanism to propagate energy and information. This is embodied in the A £(z-z’)-c (t-t')7 term. The e~®3lz"z' l s term expresses sound absorption, where we identify the g of equation (2A.llc) with 3 = > = ft? 'S 7. The Brillouin Noise Intensity The Brillouin noise intensity a^ of equation 115 (2A.25b) is defined in terms of ^N(x,y)N*(x' ,y' « r We must now relate this correlation function with that of equation (B.32). v r I (ksZ-<Jst) - M r , t ) = KeLNU,t)e J i (ksi'Ust) * -!(ksZ-<Jst) - zN (Z jt)e + \ U ( £ ' 0 e . y t') ~ ^ Nfejt) N f e j i ) e. + o/exCc*\j<ju*atk 4 iC ksa -Z 'J-tia& 't'rf • h -q M fe fiN fr ltO e . f £«wpl«X C e n ^ o y T e - (B.3 6) t*)^ is, by Galilean invariance, a function only of [(z-z*), (t-t»)]. Therefore, the ensemble average of the first term on the right of equation (B.3 6) and its complex conjugate drops out, since both exponentials have a [(z+z* ),(t+t»)] dependence. < M r , t ) l a r ' j t ' ) } - , y 1, ,* v\ .{k.Cz-zV^^J 3-< e 1 , ■ - f , Comparing equations (B.3 3) and (B.37)> we identify <N< z , t ) U ir(z'J t')}.f zi * * 4 . - a f r j t . z 'l ZT^poC^ L -» (b>38) Substituting this into equation (2A.25b), we find the following expression for the noise intensity aB for fluids obeying the hydrodynamic equations: UW, a 116 i. ■2 - c K-rfra)3 Ji ds ZTT^P.C,1 (B.39) where, for the scattering geometry under consideration, sound. , f \ (A a) = (^fl) (B.40) APPENDIX C THE NUMERICAL INTEGRATION TECHNIQUE The scattering equations of this, as with any physical system, Involve the transport of physical quantities, such as energy, with well defined speeds of propagation. Such scattering equations belong to 14 a class of differential equations called hyperbolic. An Important concept In the theory of such equations is that of "characteristics," which, In this context, correspond to the possible trajectories traced by the light modes in a Minkowski space-time diagram. Figure 25 Illustrates the two groups of characteristics for the light mode equations considered in this thesis, ignoring dispersion. The speed of propagation cL is a constant Independent of the intensities of the modes, so the characteristics are straight lines. Each scattering equation can be Integrated along Its respective characteristic line by precisely the same techniques applicable to ordinary differential equations in only one independent variable. Indeed, the only ^See, for example, Chapter 8 of J. Mathews and R. L. Walker, Mathematical Methods of Physics (W. A. Benjamin, Inc., New York, 19b4). 117 118 Figure 25: A Minkowski space-time diagram showing the characteristics of the light scattering equations, i.e., the paths traced by the forward- scattered and the backward-scattered light. 119 Influence that the two-dimensionality of the model has on the solution behavior is through the presence in the "driving force" term of intensities of modes traveling in the opposite direction. Only through these driving terms is information transferred from one characteristic line to another of the same set. Otherwise, as in the 2WFSND case, the solution generated along one character istic line is quite independent of the solutions generated along neighboring characteristics of the same set, and the problem is effectively one-dimensional rather than two- dimensional. Fourth order Runga Kutta is the numerical integration technique chosen for these computer programs because of its reputation for excellent stability. I have strong confidence in the accuracy of the numerical solutions presented herein for three reasons: 1. The Runga Kutta is found to be one of the most stable techniques for a wide variety of problems. 2. All computer solutions quickly tend to a definite limit as the grid size is reduced. This is clearly a necessary condition for the accuracy of the numerical solutions. 3. All of the main features of the numerical solutions can be successfully explained by quite reasonable arguments based upon the analytical solutions of Chapter III. This last argument shall be considered the strongest one in support of the computer solutions. 120 By proper scaling of the "z" and "t" axes, the characteristic lines can be drawn as mutually perpendicular. Assuming the same grid interval H for both characteristics, a square array of points is generated. Each point will be identified by the ordered pair of integers (l,J) defined such that See Figure 26. The point at the origin is defined to be (1,1) rather than (0,0) for convenience in computer indexing (the "DO" loops in FORTRAN do not allow lower limits of zero). are defined on the J = 1 line of points, while the boundary conditions for the forward modes are defined on the I— 1 line of points. All mode intensities are defined on the (I = l,J=l) point. The forward mode intensities can be integrated along the J -1 line (since the backward mode intensities are known on all points of this line) as if the pump and forward Raman scattering equations were two coupled ordinary differential equations in only one inde- (C.la) (C.ld) (C.lc) (C.lb) The boundary conditions for the backward modes Figure 26: The square grid of points assumed in the difference equations of the numerical integration, and how these points are labeled. 122 pendent variable. This generates all the mode Intensities on the J = 1 line. The backward mode Intensities at the (l = l,J=2) point may be found by Integrating along the characteristic from the known values at the (I = 2,J— 1) point. The forward mode Intensities are known at both points, making the Integration especially easy compared with that for the next point to be considered. The evaluation of the Intensities at the (l:=2,J=2) point, on the other hand, Is far more complex. The Integration technique used for this point Is the same as that used for all "Interior" points, I.e., for all J> 1 and 1 < I < Inlay points, where XmftY Is the largest I considered In the computer program. Hence, some attention to the Integration of this point Is useful. All mode Intensities are known at the (I=OjJ = 1) and the (ir 1,J=2) points (see Figures 26 and 27). The forward mode intensities may be found at the (I = 2,J = 2) point by Integrating along the forward mode characteristic line from the known values at the (Irl,J=2) point. Similarly, the backward mode values at the (I=2,J=2) point are integrated from the known values at (I —3,J= 1). Unfortunately, the Integration of the forward modes by the Runga Kutta technique requires knowledge of the backward mode intensities not only at the known (1 = 1,J=2) point but also at the unknown (l = 2,J~2) point. ^ c r=2.J'2 ® IA'kafaT< p « t ^ J| - _ l ( baJcuidftl Mode, r=2.vut) X=J 3 _ _ ® . _ _ _ _ > j $ -------------- 5 . i r=.±,J-l Figure 27: The integration of the (X=2,Jr:2) point. TTie array is rotated 45° clockwise with respect to the previous figure. All mode intensities are known at 0 points (either by previous integration or by boundary conditions), some mode intensities are known at (£» points, and no mode intensities are known at • points. Similarly, the backward mode integration also requires knowledge of the forward mode intensities at the (I=2,J=2) point. The resolution of the problem is to integrate both the forward and the backward modes simultaneously as if they were all a coupled set of ordinary differential equations. That is, the first estimate of the forward modes at (I — 2,J = 2) is made solely from the (I^l,Jr-2) information, and the first estimate of the backward modes at the (l=2,J-=2) will be made strictly from the (I=3> j —1) information. The second estimate of the forward Stokes will depend upon both the (I=1,J=2) information and the first estimates of the (li-2,Jr2) intensities for all modes. Third estimates will depend upon second estimates, and fourth estimates upon third in the usual Runga Kutta philosophy. By such a technique can all mode intensities at the (I=2,J=2) point be approximated from known data, and by such a technique can all subsequent points on the Jrr2 line be evaluated. The last J point considered in the line requires some special attention. If the problem is that of a finite cell, the last J point corresponds to the far end of the cell, and the boundary conditions specify the backward Stokes intensities at this point. All that remains is the straightforward integration of the forward light modes. If the problem is that of a infinite length cell, the 125 integration grid is cut off after some distance for convenience. The above-mentioned integration technique requires information not contained in the computer memory, namely the mode intensities just beyond the last point of the J=rl line. However, these may readily be estimated by an appropriate extrapolation technique, and the integration completed as before. In an entirely analogous manner, each succeeding J line can be generated from the last known J line. Usually the computer need store only one line of points, or at most two, to perform the integration. Intensities on a J line are computed, the points of interest read out, and then a new line of points is generated, successively replacing the "old" intensities on subsequent points with the "new" intensities. Since at most two lines of points need be stored at any one time, rather than the whole array, considerably less memory space is needed. The disadvantage is that each block of numbers corresponding to each J line is associated with a forward characteristic line, not a constant "z" or a constant "t" line. To plot a constant "t" line, numbers from varying positions in many different blocks of J line points must be used, rather than a linear succession of numbers in one block. This is but a slight inconvenience compared with the drastic reduction in required computer memory space. BIBLIOGRAPHY Bloembergen, N., and Lallemand, P. L. "Light Waves with Exponential Gain." Physics of Quantum Electronics. Edited by P. L. Kelley, B. tax, and P. JE. Tannenwald. New York: McGraw-Hill Book Company, Inc., 1966. Bobroff, D. L. 1 9 6 5. J. Appl. Phys. 3 6: 1760-9. Carman, R. L.; Mack, M. E.; Shimizu, F.; and Bloembergen, N. 1969. Phys. Rev. Letters 23: 1327-9. Carman, R. L.; Shimizu, F.; Wang, C. S.; and Bloembergen, N. 1970. Technical Report No. 603. Cambridge, Massachusetts: Harvard University. Hagenlocker, E. E.; Minck, R. W.; and Rado, W. G. 1967. Phys. Rev. 154: 226-33. Johnson, R. V., and Marburger, J. H. 1971. Phys. Rev. A 4: 1175-82. Kadanoff, L. P., and Martin, P. C. 1963. Ann. Phys. (N.Y.) 24: 419-69. Kroll, N. M. 1965. J. Appl. Phys. 3 6: 3^-43. Kubo, R. 1957. J. Phys. Soc. Japan 12: 570-86. Maier, M.; Kaiser, W.; and Giordmaine, J. A. 1969. Phys. Rev. 177: 580-9 9. Maier, M.j Rother, W . ; and Kaiser, W. 1966. Phys. Rev. Letters 23: 83-5. Maier, M.j Rother, W.; and Kaiser, W. 1967. Appl. Phys. Letters 10: 80-2. Shen, Y. R., and Bloembergen, N. 1965. Phys. Rev. 137: A1787-1805. Tang, C. L. 1966. J. Appl. Phys. 34: 2945-5 5. Wang, C. S. 1969. Phys. Rev. 182: 482-94. Zubarev, D. N. 1962. Soviet Physics-Doklady 6: 776-8. 126

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Creator Johnson, Richard Vincent (author)

Core Title Relaxation Oscillations In Stimulated Raman-Scattering And Brillouin-Scattering

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Degree Doctor of Philosophy

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Advisor Marburger, John H. (committee chair), Porto, Sergio P.S. (committee member), Wagner, William G. (committee member)

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