ABSTRACT A GENERAL TREATMENT OF FRAUNHOFER LIGHT DIFFRACTION BY ULTRASONIC GRATINGS by William Richard Klein The Raman and Nath theory of light diffraction by a sinusoidal, progressive, ultrasonic wave has been found to give valid results for the case of low ultrasonic frequency and narrow beam width. Since the time of its publication the treatment has been extended to include 1) arbitrary ultrasonic waveform for normal incidence of the light upon the sound field and low ultrasonic frequency and 2) sinusoidal ultrasonic waveform at oblique incidence with high ultrasonic frequency. General solutions of the diffraction problem for moderate ultrasonic frequencies from which numerical values could be easily obtained have not been found. In this work, the diffraction problem is treated for arbitrary waveform, ultrasonic frequency, ultrasonic beam width and angle of incidence. The resulting difference equations are then evaluated numer- ically using the Control Data 3600 computer. Experimental verification of these solutions as well as the high frequency solutions of the diff- raction problem is obtained using instantaneous pulse-optical techniques. A GENERAL TREATMENT OF FRAUNHOFER LIGHT DIFFRACTION BY ULTRASONIC GRATINGS by William Richard Klein A THESIS Submitted to Michigan State University in partial fulfillment of the requirements of the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 196A ACKNOWLEDGMENTS The author greatly appreciates the assistance received in the formulation and solution of the problem from Dr. E. A. Hiedemann. The "7' many discussions with Dr. B. D. Cook, Dr. W. G. Mayer and Dr. C. B. Tipnis as well as other members of the ultrasonics group at M. S. U. have also been extremely helpful. TABLE OF CONTENTS INTRODU CTION SUMMARY OF THE RESULTS OF RAMAN AND NATH TEDEOFUY Resolution Into Plane Waves The Wave Equation Analytic Solutions of the Difference-Differential Equation Numerical Solution of the Difference-Differential Equation Fresnel Interference THE EXPERIMENT RESULTS DISCUSSION APPENDIX The Computer Program BIBLIOGRAPHY ii 17 19 21 27 39 1L6 Figure 1h 15 16 18 19 20 21 Diagram of the LIST OF FIGURES Ultrasonic Diffraction Grating Orientation of the Partial Wave Vectors Diagram of the Diagram of the Diagram of the Light Light Light Light Light Light Light Light Light Light Light Light Light Light Light Light 0 Light 0 Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs Intensity vs = 1.50, = 1'50) V = Intensity vs = 1.50, Electronic a for Q a for a for a for a for Q: a for (D = a for a for 0= 0 for (Q = a for Q a for a for V for V for a for Optical Arrangement Photomultiplier Circuit Circuit Fixed 0.57, Fixed 0-57; Fixed 1.15, Fixed 1.15, Fixed 2.25, Fixed 2.25, Fixed 3.74, Fixed 3-715 Fixed 6.28, Fixed 6.28, Fixed 9-3, Fixed 9.3; Fixed 9-3, Fixed 9-3; Fixed (3, V and V; . V = 1.00, V2: 0.0h Intensity vs a for Fixed<2 , V and V! . (3 and V . V = 2.0 (2 and V . V = 3-0 (D and V . V = 2.0 (D and V . V = 3.0 (D and V . V = 2.0 (D and V . V = 3-0 C) and V . V = 2.0 (D and V . V = 3.0 (D and V . V = 2.0 (D and V . V=3.0 (D and V . V = 2.0 (D and V . v=3co Q and a. a = 0-5 (3 and a . a = 0.0 2.00, Vt: 0.15 a for Fixed 0, v and v. . v = 3.00, V, = 0.30 iii Page 22 22 2h 30 3o 31 31 32 32 33 33 31+ 31+ 35 35 36 36 37 37 38 INTRODUCTION The first generally successful theoretical approach to the problem of light diffraction by plane progressive ultrasonic waves was presented in a series of papers by Raman and Nath (1-5). In their generalized theory they resolved the diffracted light into plane, partial waves and by considering the partial differential equations governing the prop- agation of light in a quasi-homogeneous medium they arrived at a set of difference-differential equations which described the amplitudes of the various partial waves. The theory was successful in describing the diffraction pattern produced by sinusoidal, ultrasonic waves of relatively long wavelength, low intensity and narrow beam width. The shortcomings of the theory were that l) in general ultrasonic waves are not sinusoidal except under special conditions and 2) for the case of arbitrary ultrasonic wavelength, intensity and beam width the difference- differential equations could not be solved to obtain analytic expressions for the amplitudes of the diffraction orders. Zankel and Hiedemann (6) solved the problem of light diffraction by plane, progressive ultrasonic waves of arbitrary waveform subject to the conditions of relatively long sound wavelength, low intensity and narrow beam width. Their results show that the intensity of the light in a particular diffraction order can be written as an infinite series in which only a very few terms are of significant magnitude to require consideration. Attempts to overcome the second shortcoming of the theory have been made by several authors (7-10). However, none of these treatments is satisfactory for the solution of the diffraction problem under (1) completely arbitrary experimental conditions. The principal difficulty encountered is that in the case of arbitrary experimental conditions the physical descriptions at best lead to solutions which are extremely difficult to evaluate numerically. Several authors have successfully treated the problem using digital computer methods to obtain numerical solutions. Hargrove (11) recently has treated the case of normal incidence of the light upon the sound beam but otherwise general experimental conditions. His approach was to consider the sound as producing a series of successive diffractions as the light is propagated through it. The resulting diffraction inte- grals are evaluated using a digital computer. Mertens (12) has also treated the diffraction problem for the case of normal incidence using computer methods but unfortunately a choice of parameters was made which does not coincide with those parameters which can easily be varied experimentally. The purpose of the present work is to obtain a description of the Fraunhofer diffraction pattern under completely general experimental conditions. The resulting equations are then solved using a digital computer. Experimental verification of these numerical solutions along with several analytic solutions for limiting cases is given. (2) SUMMARY OF THE RESULTS OF RAMAN AND NATH The theory of Raman and Nath predicts that after being diffracted by an ultrasonic beam the light will propagate in discrete directions * described by n). Sin Bn=T (I) where A is the wavelength of the light and n is an integer. Equation (1) is seen to be the usual grating equation except that the grating spacing is replaced by the sound wavelength, A. . The refractive index of the medium in the region containing the sound field is described by . I 1 Fix,” 3 [1.0 - ,1. sm(kx-wt) (2) where f%)is the undisturbed refractive index in the absence of any sound field and [L the maximum variation in the refractive index produced by the ultrasound. The symbols kl and a; represent the magnitude of the propagation vector and the circular frequency of the ultrasound, respectively. The optical properties of the grating can be described by two parameters V= ka (3) and 12 Q= % (4) where k is the magnitude of the propagation vector for the light in vacuo and L. the width of the ultrasonic beam. The amplitudes of the light in the various diffraction orders, the * The convention chosen here for numbering diffraction orders differs from that of Raman and Nath. (3) (#TI’ are described by the difference-differential equation d¢n v _inQ 2&x'sin6 37004-16...) ‘n- . m ‘5’ "2'1.— subject to the boundary conditions at Z = O ¢o=v ¢n=0 (naéO) (6) The angle 9 is the angle of incidence of the light upon the sound beam. There are two special cases where Eq.(5) can be solved in terms of well known functions: Case I: Q << 1, 9 = 0. In this case, the r.h.s. of Eq. (5) can be neglected and the solutions at Z = L are ¢n= J_n(v) (7) where Jn(V) is the Nth order Bessel function of argument V. Under these conditions the only effect the sound beam has upon the light beam is to produce a modulation in phase. In those regions of the sound field where the refractive index is greater than its undisturbed value the phase of the light lags and where the refractive index is less than its undisturbed value the phase of the light leads that of the regions of average refractive index. This ultrasonic grating is equivalent to an idealized diffraction grating which is localized to a plane in space and produces only a phase modulation of the light which passes through it. For normal incidence this phase distribution is an exact reproduction of the refractive index distribution of the medium within the sound wave. (1+) Case II: C >> 1, Sin 6 : A/ZpoA' [See reference (13).] In this case the r.h.s. of Eq.(5) becomes extremely large for,!1 i 0 or 1 and the ¢n1 correSponding to these values of r1 are rapidly varying functions of Z which are nearly periodic in Z . As such, their absolute values are extremely small so that the only ¢n which must be considered are those for which fl : 0 or 1. Equation (5) thus reduceszto the set of two coupled equations d f-ng: 0 (e) and (I¢S I V - whose solutions at Z = l. are :30 cos-‘5’— (l0) and <1». -sin-2‘!- (II) The requirement on the angle of incidence given above is immediately recognizable as the Bragg condition and evidently the dominant effect of the sound beam upon the light beam is to produce a Bragg reflection from the plane wavefronts of the sound wave. Under experimental conditions described roughly by 0.5 > 1 it is possible to obtain analytic solutions of Eq. (32). For the case 0 << 1 with a sinusoidal sound wave and 8' = 0 Eq. (32) reduces to Cl¢> v in()a' ___r| __. _. : _. dz + L(¢ M ¢n+,) L 4:“ (38) where the term in n Q¢n has been neglected. Letting 2L 4:" = Xn exp(-in0az/2L) (38) becomes an+ 002 dz +2L cos 2L —(Xn-l —xn+l) Qaz , inQa +'2"E5"‘— Lz‘Xn-l "XnI-l)’ 2L _—x" ' (39) Using x Jn(X) = 2—n[an_,(X) + Jn+,(X)] (40) it is a simple matter to show that 5m (00/ 2) = v X" -n 00/2 This agrees with the result of Raman and Nath (2) which was obtained from (4)) geometric arguments. For normal incidence (a = 0) Xn' ¢n‘ in“” (42) or at the exit plane of the sound field (Z = L ) E = exp[i(wt—Iuokl_)] E) J_n(v) eprin(w't-k'x)] . (43) )1 n=-cn The light intensity in this plane is as I=EE I I 3MB 2 _&l)_n(v) eprIMw't - k'xlj I cosIv sithI-k'xfl — i sinIv sinb‘t-k‘x)” 2 = I (44) or the light intensity is independent of the x-coordinate showing that there is no intensity modulation of the light beam in this plane. The relative phase of the light as a function of the x-coordinate can be found by examining the quantity under the summation in Eq. (44). Let the relative phase be 52 . Then m . 1 | 0 g: -:O,n(v) sm [0(a) t-k )0] f0 J-n(v)cos[n(w't-k'x)] n= -ao _ _ sin[v sinUt-k'x)! (45) cosIv sink!) 4%)] so that the phase is given by (11+) n=-vsm(w'I-k'x) , (46) Thus, the phase of the light beam at the exit plane of the sound field has the functional form of the variation in the refractive index due to the ultrasound, or the phase distribution of the light is an image of the density distribution within the medium. For C) << 1, a = 0 and with a sound wave which can be described in terms of a Fourier sine series ( 8. = O) Zankel and Hiedemann (6) have shown that m. ¢n= Z J-n+2kfl+3k I..(V) J_k22(V ) J- k3(V3) (47) k21k31“ =- 00 For the case (3 >>l diffraction effects are not found to occur at normal incidence ( a = 0). In fact, when diffraction effects do occur at oblique incidence light is found in only a few diffraction orders. For (lav 1/2 with a harmonic sound wave all orders except the zeroth and first can be neglected. Eq. (32) reduces to the set of two difference- differential equations $30 - gi¢, = O (48) 9.2. -19. __ dz +2voL¢ - 2L (1 20)¢' (49) which have the solutions at Z = L ¢ = exp[IB/2] [cos(D/2) -- L’ZBSIIMD/Zfl (50) 0 and 4., =--[v3-exp[IB/2] sin(D/2) (5)) where (15) as-ZQII—za) (52) and Des x62 + v2 (53) For a = 1/2, 8 = 0 so that Eqs. (50) and (51) reduce to ¢O= co s(v/2) (54) and ¢.=- sin/Z) . (55) Equations (50) and (51) were obtained by Phariseau (13). In the general case where Q is neither very small nor very large no general solutions have been obtained so that the most practical approach to the problem appears to be numerical solution of Eq. (32). (16) Eggsgiggl Solution of the Difference-Differential Equation The Control Data 3600 computer was programed for numerical solution of Eq. (32). For purposes of computation the square of the r.h.s. of Eq. (32) was completed by substituting - - 2 {n - It)" exp(Ia Qz/ZL) (56) which gives d_€:n_ I (D . - dz - 2L E'vjffihj exphsj) - £n+j exp(-I8j)] i0 4.... 2L Let the sound field be equally divided into F’divisions in the z-direction (n—a)2 en .(57) where F’is a large integer. The z-coordinate of a point within the sound field can be described in terms of an integer p multiplied by the width of the divisions Z12 . The spatial derivative of {n at the point labeled by p is approximately p £p+l p-l d_‘5n z n ‘ 5n (58) dz 2A2 ' Substituting Eq. (58) into Eq. (57) and rearranging one obtains l -) CD {:5 {E 713 zv.[§£_jexp(Is.)—§fi+jepr—Is.)] +%(n-a)2 cf: .(59) The initial conditions used in Eq. (59) are 52:65:) (60) €3=Erg=0 n #0. (17) The 0Perations indicated in Eq. (59) are performed F3 times to obtain numerical values for the amplitudes of the various partial waves at the plane Z = L The intensity of the nth Fraunhofer diffraction order is given by In=I§fIZ .(6I) (18) (Ill-Ila ‘ILJII‘ EESSQEL Interference To determine the light intensity distribution in the Fresnel or near field interference pattern one can use the numerical values of the light amplitudes calculated from Eqs. (32) and (56) or Eq. (59) as the initial values of the (”(L) at the boundary 2: L second integration from Z: L and perform a to z = z} where z. is the value of the z-coordinate at which the Fresnel pattern is observed. In this region the Vj = C) so that Eq. (57) reduces to d5" i0 2 dz 2L“) 0:) En ( ) which has solutions ("(2') = (”(L)exp[-;£L(n-a)2(Z'-L)] ,(63) Replacing all of the phase factors which have been removed, one obtains E(x,z',t) = exp{-i[wI—,.Ok(z'cos 9 +xsin 0)” exp[£(n-a)2(Z'-L)] (54) 01' m I O I = g gnu.) exp[in(w I—kxI] n=-ao , 2 exP[-§'%(n2-2na)(z'- LI], . (65) TIN? first exponential in Eq. (65) shows that the Fresnel interference pattern moves with the sound field. The second describes the z dependence of the Fresnel pattern. For normal incidence, it is seen that there is a fundamental spatial periodicity in the z direction given by .9. 2L 2__._. 0 0:3314‘2': (66) k'2 which agrees with the usual result obtained using geometric means (14). (20) THE EXPERIMENT The experimental work in this investigation was confined to the deter- _ mination of the Fraunhofer diffraction pattern for various values of the eXperimental parameters. The regions of particular interest were those of moderately high sound frequency and wide ultrasonic beam characterized by (D > 0.5 for which little systematic experimental work has been done. In particular, an attempt was made to verify the numerical solutions obtained from Eq. (59) and the analytic solutions for Q >>l, namely, Eqs. (50), (51), (5%) and (55)- The standard optical arrangement for observing Fraunhofer diffraction was used (Fig. 2). The light source S was a General Electric AH 100w/2 mercury arc. A green filter F was used to select only the 5461 A wave- length. The light beam was collimated by lens L After the light 2. passed through the tank containing the medium through which the sound was propagated, the image of the source slit 81 was formed in the plane 1 of the photomultiplier slit so that the light in one diffraction order only was incident upon the photomultiplier. When making diffraction measurements at high sound intensities, the heating of the medium due to sound absorption causes the diffraction grating to be unstable. For this reason, pulsed ultrasound was used rather than a continuous wave as is the usual practice. By setting the pulse repetition rate at 60 sec-1 and the pulse duration at approximately 50 us the pulse duty cycle which is the fraction of the time the sound is on is 3 x 10-3. Even though the average power propagated through the medium is small the instantaneous sound pressures can be relatively large and the instantaneous diffraction grating is stable. The use of pulses, (21) (2 2) I 2.9 SI 1 IL)! ALB—Q é» pHorouumpusn . (Tm/ 5., Diagram a! the Opvical FIGURE 2. Arranaamant a. L VR-75 . R "'75 H‘s ovuooes ' moo: ., T ..s. II TO DC OSCILLO SCOPE Diagram of (ha PhoIomulIiplior FIGURE 3. Circuit (23) however, introduces other complications into the experiment. The problem is that the diffraction pattern only appears when the sound pulse crosses the light beam. Thus the average light intensity in any of the higher diffraction orders is reduced by a factor equal to the duty cycle compared to what it would be using the same sound pressure with continuous waves. The intensity of the stray light in the optical system becomes comparable in magnitude to the magnitude of the average intensity of the diffracted light in the higher orders. Also, the maximum change in the average intensity of the central order when the sound is turned on can only be equal to one minus a factor equal to the duty cycle. Thus the fluctuations in light intensity of the source and the noise associated with the electron- ics of the photomultiplier circuit make central order measurements impossible if one attempts to use time average measurements. Lester and Hiedemann (15), who studied finite amplitude effects using pulse-optical methods, were interested in higher diffraction order measurements and they found it possible to use time average light intensity measurements for these orders. However, in this work it was necessary to use central order measurements so that it was necessary to use a photomultiplier circuit capable of responding during the time interval in which the sound pulse crossed the light beam. The photomultiplier circuit shown in Fig. 3 proved to be satisfactory for this purpose. The photomultiplier tube was an RCA 931. The variable resistors Rd and RS allowed any dc bias across the terminals of the oscilloscope to be nullified. Also, their values determine in a large part the time constants of the circuit. Variable resistors of 20 K were found to be satisfactory. The light intensities were read directly from a Dumont 30h-A dc oscilloscope. (24) oscuLoscoPE CAI: ourpur OSCHLOSCOPE nr IMPEDANCE (::) MATCHING PULSER netwoax rnAnsoucen PNCmNULrIPuEII LIGHT (J) so CVCLE SVNC Diagram of IM Electronic FIGURE 4. Circuit (25) Figure A shows a block diagram of the complete electronic circuit used in making measurements. Since the intensity of the mercury arc varied with a frequency of 120 sec-1 it was necessary to trigger the sound pulse so that it crossed the light beam when the light intensity was a maximum. This was done by synchronizing the rf pulser with the line voltage by means of the gate output of a Hewlett Packard 150 oscilloscope. The repetition rate of 60 sec-1 ensured that the pulser was always triggered on the same half of the light cycle. A change in the triggering level of the oscilloscope allowed a variable phase shift between the light and the sound pulse. This same oscilloscope was used to measure the voltage across the trans- ducer. An Arenberg PG 650 pulsed oscillator was used to supply the rf pulses. The photomultiplier was mounted on a traversing screw allowing observation of a particular diffraction order. The variation in angle of incidence between the light and the sound field was accomplished by mounting the tank containing the sound field on a rotating platform to which was attached a long lever arm. The free end of the lever arm was coupled to a second traversing screw allowing measurment of the angle of incidence. This arrangement required a simple correction for refraction at the windows of the tank. An X-cut quartz p1ate,6 cm in diameter, suspended directly in the medium served as a transducer. The effective width of the transducer was varied by changing the width of the rear electrode. In order to compare the theoretical results with the measured values of light intensity it was necessary to obtain a range of values for the (26) parameters which describe the properties of the diffraction grating. The value of the Q parameter was fixed by choosing l) the medium, 2) the sound frequency and 3) the sound beam width. The value of the parameter V was varied by changing the rf voltage across the quartz crystal. The procedure followed in the first series of measurements was to set the quartz voltage so that the observed light intensity distribution agreed with the distribution predicted by the theory. For small values of the parameter (3 this was done at normal incidence. For larger values of Q it was done at a = 1/2. A comparison of the quartz voltages and the corresponding values of V showed them to be linear as was F expected. Once the desired value of the quartz voltage had been obtained in this manner, intensity measurements were obtained as a function of angle of incidence. In the second set of measurements the angle of incidence was set and the light intensities in the various orders were obtained as a function of quartz voltage. The results were fitted by assuming a linear relation- ship between the quartz voltage and the value of V . In the final set of measurements the transducer was moved away from the light beam a distance of 10 cm in order to obtain a finite amplitude distorted wound wave. The procedure used in the first set of measure- ments was then repeated. RESULTS Figures 5 - 21 give a comparison of the intensities of the zeroth and first Fraunhofer diffraction orders as predicted by the theory and as measured for different values of the experimental parameters. Figures 5 and 6 show the results for (D = 0.57. This corresponds to a sound frequency of 5.0 mc and a sound beam width of 2.0 cm. Figures 7 and 8 show the results for (D = 1.15. This corresponds to a sound frequency of 5.0 me and a sound beam width of h.0 cm. The experimental values shown in these four figures were obtained by Dr. C. B. Tipnis*. Water was used for a medium and standard continuous wave techniques were used. In the case of a sinusoidal sound wave (Figs. 5 - 18) the positive and negative first order light intensities as a function of angle of incidence are mirror images of each other about the a = 0 axis. For this reason, only one first order is shown. Figures 9 and 10 show the results for C) = 2.25. This corresponds to a frequency of 5.2 me and a sound beam width of 2.6 cm. The medium, Dow Corning 200 fluid (0.65 cs), was selected for its low sound velocity which allowed large values of (3 without resorting to extreme ultrasonic frequencies. From the first order curves it is seen that the intensity distributions are nearly symmetric about the Bragg angle. Figures 11 and 12 show the results for (D = 3.7h. This corresponds to a frequency of 5.2 mc and a sound beam width of 4.3 cm in a medium of Dow Corning 200 fluid. Here it is seen that the peak in the first order at a = 1/2 is becoming sharper indicating that Bragg reflection is a more important process. * Private communication (27) Figures 13 and 1h show the results for (D = 6.28. This corresponds to an ultrasonic frequency of 8.6 mc and a sound beam width of 2.6 cm. The medium used was Dow Corning 200 fluid. Figures 15 - 18 show results for Q = 9.3. This corresponds to an ultrasonic frequency of 8.6 mc and a sound beam width of h.7 cm. The medium used was carbon tetrachloride. In Figs. 15 and 16 the light intensities calculated from Eqs. (50) and (51) are shown as well as the numerical results obtained from Eq. (59). In Fig. 17 the light intensities calculated from Eqs. (5%) and (55) are shown. It is seen that the dominant effect of the sound field upon the light beam is that of a Bragg reflection. Figures 19 - 21 show the effects of a finite amplitude distorted sound grating on the diffraction pattern. The distortion was produced by allowing the sound to propagate through a distance of 10 cm before crossing the light beam. The amount of second harmonic content was calculated from the theory of Lester (16). The measurements were made in distilled water with (D = 1.50 corresponding to a frequency of 5.2M mc and a sound beam width of h.7 cm. In general, the experimental results are in very good agreement with the results obtained from the theory. It is seen that at large angles of incidence some deviation is observed. This may be due to the presence of sub-components within the radiation pattern of the transducer which travel at an angle with respect to the primary beam. In the theory the sound field is assumed to be uniform, to have plane wavefronts and to have distinct boundaries at its edges. In the actual radiation pattern the field is nonuniform, it may not have perfectly plane wavefronts and the sound pressure gradually diminishes to zero at the edges. 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IIIOTI Moll I o'IIIV OIDI’I I -'IIY 0.0!. I DISCUSSION In this work the general problem of Fraunhofer light diffraction by a progressive ultrasonic beam of arbitrary waveform has been treated. Solutions of the problem were obtained by numerical integration of the difference-differential equations which describe the growth of the light amplitudes in the various diffraction orders. In addition, analytic solutions have been reviewed for very low and very high ultrasonic frequency. The validity of the low frequency solutions has been shown elsewhere (6 , 17 and 18). The numerical solutions for general experimental con- ditions and the high frequency solutions have been shown to be valid in this work. The work undertaken here should make possible studies of light diff- raction by high frequency ultrasonic waves of finite amplitude using pulse optical techniques. In addition, with the current interest in modulation of laser beams the extension of the frequency range of the ultrasonic diffraction techniques should be of interest because the modulation frequencies of interest are generally rather high. (39) APPENDIX The Computer Program Following is the computer program in FORTRAN language used for numerical amplitude evaluation of the difference equations for the case of a finite distorted ultrasonic wave which can be described in terms of the fundamental and second harmonic components [Eq. (32)]. The harmonic components of order higher than second are neglected. The values of 8' and 83 are n and 0 respectively. Card number 1 2 201 3 202 u 203 5 20h 6 205 7 206 8 207 9 208 10 209 DIMENSIflN R1(25),R2(25),R3(25),H1(25),H2(25),H3(25) FflRMAT (I3) FgRMAT (F10.2) FgRMAT (F10.h) FgRMAT (1H ,ux,3(E12.5,3x) FgRMAT (1H0,9x,hHREAL,9x,9HIMAGINARY,6x,9HINTENSITY) FgRMAT (1HO,5HNMAX=,I3,3X,hHJIN=,I3) EDRMAT (1H0,uHFIN=,E9.3) FgRMAT (1H0,uHALF=,E9.3,3x,hHVEE=,E10.u,3x,uHQUE=,E10.h) FfiRMAT (F10.6) READ.201,MT0T D¢ 99 M=11MT¢T READ 201,NMAx READ 201,INT READ 201,JIN READ 202,ALF READ 203,QUE READ 209,FIN READ 203,VEE D¢ 101 I=1,INT PRINT 206,NMAX,INT,JIN PRINT 207,FIN PRINT 208,ALF,VEE, QUE PRINT 205 K=2*NMAX+5 D¢ 100 N=1,K R1(N)=0.0 R2(N)=0.0 R3(N)=0.0 H1(N)=0.0 H2(N)=0.0 H3(N)=0.0 (1+0) 100 CQ'NTINUE A R1(NMAx+3)=1.0 35 R2(NMAX+3)=1.0 36 NTEST=K-2 37 D0 102 J=1,JIN 38 D0 103 N=3,NTEST 39 CM=N-NMAx-3 A0 B=(CM+ALF)*(CM+ALF) A1 R3(N)=R1(N)+VEE*(R2(N-l)-R2(N+l))-FIN*(R2(N-2)-R2(N+2)) A2 l-QUE*B*H2(N) A3 H3(N)=H1(N)+VEE*(H2(N-l)-H2(N+l))-FIN*(H2(N-2)-H2(N+2)) AA 1+QUE*B*R2(N) A5 103 cgNTINUE A6 D0 10A N=3,NTEST A7 R1(N)=R2(N) A8 Hl(N)=H2(N) A9 10A C¢NTINUE 50 D¢ 105 N=3,NTEST 51 R2(N)=R3(N) 52 H2(N)=H3(N) 53 105 cgNTINUE 5A 102 cgNTINUE 55 D0 106 N=3,NTEST 56 A=R2(N)*R2(N)+H2(N)*H2(N) 57 PRINT 20A,R2(N),H2(N),A 58 106 C¢NTINUE 59 JIN=(JIN*(I+1))/I 60 DM=I 61 QUE=QUE*DM/(DM+1.0) 62 101 C¢NTINUE 63 99 CfiNTINUE 6A END 65 END ID 002 2D 010 3D 005 AD 100 5D 0.00 6D 0.0057 a 7D 0.000000 g 8D 0.0100 G A 2D' 010 3 3 3D' 005 2° AD' 100 3 5D' 0.00 60' 0.0115 7D' 0.000000 8D' 0.0100 (1+1) In this form the program will give the light amplitudes and inten- Sities in the various diffraction orders for a fixed angle of incidence and fixed (2 and it will increment V automatically. The machine symbols for the parameters are as follows: Card Card Card Card Card Card Card Card with Data cards must be prepared as follows: 1D: 7D: For the Parameter Symbol v / P VEE V: / P FIN Q/ P QUE a ALF P JIN This card gives the number of different values of parameters (MTOT) to be input into the machine. It is input as a three place integer. This card gives the number of orders (up to 10) to be cal— culated on either side of the central diffraction order (NMAX). It is to be input as a three place integer. This card gives the number of times the variable parameter is to be incremented (INT). It is input as a three place integer. This card gives the value of the parameter P (JIN). It is input as a three place integer. This card gives the value of the parameter a (ALF). It is input as a floating point number with two decimal places. It should be right adjusted so that the last digit falls on the tenth column of the data card. This card gives the value of CD (QUE). It is input as a floating point number with four decimal places. It should be right adjusted so that the last digit falls on the tenth column of the data card. This card gives the value of V2/ P(FIN). It is input as a floating point number with six decimal places. It should right adjusted so that the last digit falls on the tenth column of the data card. This card gives the value of V/P (VEE). It is to be prepared like card 6D. computing more than one set of data, cards 2D—8D are repeated values of the new parameters. (L12) The value of NMAX, the number of orders to be calculated, must be determined from experience. All of the orders which contain any signifi- cant amount of light must be included. For a value of Q of 2 or less the Bessel functions of argument V give a good indication of this value. For larger values of (D fewer orders need be included. Table I gives the values of NMAX determined from the computations made in this work. The value of JIN must be chosen in such a way that JIN > 2 Q (NMAX + ALF)2 JIN > 100 V The first and second columns on the print-out sheet give the real and imaginary parts of the amplitude in the various diffraction orders respectively. The third column gives the corresponding intensities. Example I: Suppose it is desired to calculate the light intensities in the various diffraction orders for values of the parameters as follows: V 1,2,3,A 0 1,2 V5 0 a O The calculations will be made by fixing Q and incrementing V . From Table I it is seen that NMAX = 6 and 5 respectively for the two values of (D . The conditions on JIN are satisfied by JIN = 100 (The minimum value of V is used in the second condition). (1+3) v (D 57 1.15 2 25 3.7A 6.28 9.3 1 3 3 3 3 2 2 2 A A A 3 3 2 3 5 5 A A 3 3 A 6 6 A A 3 3 Table 1: Values of NMAX for given values of (3 and V The data cards in the above example then read as follows: 002 006 00A 100 0.00 0.0100 .000000 0.0100 0 005 00A 100 0.00 0.0200 .000000 0.0100 0 Alternate Program To hold V , V. and Q constant and vary 0 remove cards 59, 60 and 61. In their place insert a single card reading ALF = ALF — (increment) where (increment) stands for the numerical value of the intervals of a . Example II: Suppose it is desired to calculate the light intensities for values of the parameters as follows: ,2 a ‘3 <= , 0.5, 1.0, 1.5, 2.0 < OOUJH 2 The card which replaces cards 59, 60 and 61 should read ALF = ALF ~ 0.50 (AA) From Table I, NMAX = 3 and A for"V ==l and 2 respectively and the requirements on JIN are satisfied by 200 and 300 reapectively. The data cards should read 002 003 005 200 2.00 0.0150 0.000000 0.0050 00A 005 300 2.00 0.0100 0.000000 0.0067 (AS) C.V. Raman A06 (1935). C.V. Raman A13 (1935)- C.V. Raman 75 (1936)- C.V. Raman 119 (1936)- C.V. Raman and N.S. Nagendra Nath, Proc. A59 (1936)- and N.S. and N.S. and N.S. and N.S. BIBLIOGRAPHY Nagendra Nath, Proc. Nagendra Nath, Proc. Nagendra Nath, Proc. Nagendra Nath, Proc. Indian Indian Indian Indian Indian K.L. Zankel and E.A. Hiedemann, J. Acoust. Soc. R. Mertens, Meded. Koninklijke Vlaamse Academie 37 pp- (1950)- P.H. van Cittert, Physica 3,590 (1937). R. Extermann and G. Wannier, Helv. Phys. Acta. 2, 520 (1936). 2.11. Wagner, z. Physik fl, 60A (1955). .E. Hargrove, J. Acoust. Soc. Am. 36, 323 (196A). . Mertens, Proc. Indian Acad. Sci. 55, 63 (1962). . Phariseau, Proc. Indian Acad. Sci. A3, 165 (1956). .D. Cook, J. Opt. Soc. Am. 53, A29 (1963). Acad. Sci. Acad. Sci. Acad. Sci. Acad. Sci. Acad. Sci. Am. 31, AA (1959). Wetenschappen 12, .W. Lester, J. Acoust. Soc. Am. 33, 1196 (1961). Sanders, Can. J. Research IA, 158 (1936). .G. Mayer, J. Acoust. Soc. Am. 36, 779 (196A). L R P B W.W. Lester and E.A. Hiedemann, J. Acoust. Soc. Am. 3&, 265 (1962). W F W (”6) IN \- IN so 3, 3; .II. V