AN INVESTIGANON OF UGHT DIFFRACTED BY WIDE, HlGH‘FREQUENCY ULTRASONIC BEAMS Thesis far the Gemee of M. S. MICHJGAN HATE UNIVERSITY W. Richard Main 1962 IHIIWIHHHI“IWUIHHIU“WHHIIIHHIHUIM 1 31293017164 L I BR A R Y Michigan St Univcrsity " vuvsics 1mm . h. AN INVESTIGATION OF LIGHT DIFFRACTED BY WIDE, HIGH-FREQUENCY ULTRASONIC BEAMS by W. Richard Klein AN ABSTRACT OF A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1962- Approved: 2 A. W. W. Richard Klein ABSTRACT The theory of light diffraction by a sinusoidal, progressive, ultrasonic wave presented by Raman and Nath (1) becomes invalid for high frequency, intense and/or wide sound beams. Mertens (2) has developed a more exact theory for predicting the light diffraction pattern under such conditions. An experimental investigation to determine the range of validity of Mertens' theory is described. REFERENCES 1. C. V. Raman and N. S. Na endra Nath Part I: Proc. Indian Acad. Sci. _, 406-412 (1955). 2. R. Mertens, Meded. Koninklijke Vlaaune Academia Wetenschappen lg, 1-57 (1950). AN INVESTIGATION OF LIGHT DIFFRACTED BY WIDE, HIGH-FREQUENCY ULTRASONIC BEAMS ‘ . r by 'W} Richard Klein ‘, t(. A THESIS Submitted to . Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1962 ’7 3; ,~ .3 1 5",... y, .1. ,3 ‘ ///,;-; /g.‘.—.‘- '5.) o e .0 ACKNOWLEDGMENTS The author wishes to extend thanks to Dr. E. A. Hiedemann for his assistance in planning this in- vestigation and invaluable guidance during its execution. Also, Dr. L. E. Hargrove, Dr. M. A. Breazeale, Dr. W. G. Mayer, and Mr. B. D. Cook have given freely of time and effort. The assistance given by the entire group is greatly appreciated. W. R. K. 11 TABLE OF CONTENTS Page EJTRODUCTIONOOOOOOOO00.0.0.0...00.0.0000000001 THE THEORY OF MERTENS........................4 EXPERIMENTAL APPARATUS.......................8 EXPERIMENTAL PROCEDURE......................12 PRESENTATION OF DATA........................l4 Zeroth order 14 First orders 21 DISCUSSION OF RESULTS.......................25 REFERENCESOOOOOO0.00.00.00.0000.0.000000000029 iii Table I. II. III. LIST OF TABLES Page Maximum values of the parameter'v and the quantity HIV for a deviation of less than 3‘% (absolute) between the values of light intensity predicted by the theory of Mertens and experimental values for the zeroth order............................28 Maximum values of the parameter v’and the quantity rrr for a deviation of less than *2 % (absolute) between the values of light intensity predicted by the theory of Raman and Nath and experimental values for the zeroth order........................28 Tabulation of an) and 3:00 for the zeroth and first orders.....................50 iv LIST OF FIGURES Figure Page 1. Diagram of the optical arrangement...........9 2. Diagram of the sound aperture...............ll 3. Zeroth order light intensity !§_ir for ”800624 (L82OO cm).000.0.00000000000000015 4. Zeroth order light intensity lg 1r for H . 00956 (L: 5.0 cm)ooooooo00.00000000000016 5. Zeroth order light intensity vs 1r for "81.26 (L=4OO cm)OOOOOOOOOOOOOOOOO 00000 17 6. Zeroth order light intensity gs if for “31.48 (L'407 cm)....00.00.000.0000000018 7. Intensity of light in the first and second minima of the zeroth order gs ultrasonic beam WidthOOOOOOOOOOOOO0.00.0000.19 8. Intensity of light in the first and second maxima of the zeroth order gs ultrasonic beam Width. 0 O O O l O O O O O O O O O O O O O O O O 0 2O 9. First order light intensity vs if for H = 00936 (LS 300 cm)oooooooooooooooooo0.0.22 10. First order light intensity vs 1r for ”.1026 (L‘4OO cm)......00.0.0000000000025 11. First order light intensity gs 1’ for H31048 (L‘407 cn)....00.00.00.00.00000024 INTRODUCTION In 1922, Brillouin (I) predicted that the Debye waves in a liquid should produce effects on light similar to the Bragg reflection. Looking for this effect Debye and Sears (2) found that a liquid striated by an ultrasonic beam acts as an optical grating. Lucas and Biquard (5) discovered the same effect independently at about the same time. Several authors have treated this subject on a theoretical basis with varying degrees of success. Notable among the early papers are two presenting two different views. Lucas and Biquard (3) attempted to explain the effect on the basis of refraction of the light due to the periodically varying refractive index, but they could not make any predictions about the intensities of the different diffraction orders. Bar's (4) experiments revealed the parameters on which the intensity of a diffraction order depends. B§r's results insPired Raman and Nath (5) to a different theoretical approach. The inter- action of the ultrasound with the light was considered to have the effect of a pure phase grating whose spacing was the same as the wavelength of the ultrasound. The results of the theory of Raman and Nath show that for a progressive ultrasonic l 2 wave with normal incidence the angles at which the diffraction orders should be observed are given by ”9‘3293}; roofing...” x and A. being the wavelengths of the light and the ultrasound respectively, and the intensity of the nth.diffraction order should be given by In .-. 'J’Z’Lv) where J;(v9 represents the nth order Bessel function whose argument'v'is given by .— . he. ; P represents the maximum variation in the refractive index due to the ultrasound, and L the width of the sound beam. In a later work, Raman and Nath (6) used Maxwell's equations to describe the propagation of the light through the medium containing ultrasound. An approximate solution of the resulting differential equation shows the predicted intensities to be in agreement with the results of their simplified theory described above. The validity of these results of Raman and Nath was verified experimentally by Sanders (7). Intensity predictions obtained using the theory of Raman and Nath appear to be limited in validity to those experimental arrangements where the refraction described by Lucas and Biquard may be 3 neglected. Experimentally, this requires a small gradient of refractive index and/or a narrow sound beam. The first requirement implies a moderate sound intensity and a moderate sound frequency. . Qualitatively, one might think of the observed light diffraction pattern as the resultant of the combined effects of a phase modulation whose magnitude is indicated by the parameter'v'and an intensity modulation whose magnitude is indicated by a quantity Hv’. Conditions to be placed on vand H to describe the range of validity of a theory, then, follow as a natural consequence. These conditions for the Raman and Nath theory have been expressed by Extermann and Wannier (8) as H e #211'1} 4< 2o . and Flat 4 2. In the above inequalities, it should be noted that the left hand members depend upon the square of the ultrasonic frequency.' As a consequence of this dependence, one must use a more complicated model than that of Raman and Nath in describing diffraction by high-frequency ultrasound even for relatively narrow sound beams. Several authors (8-10) have treated the diffraction problem in the range beyond that where the Raman and Nath theory is valid. Extermann and Wannier, 4 for example, have obtained solutions with what appears to be only a small degree of approximation, but numerical predictions of intensities for a given experimental arrangement are extremely difficult to obtain. Mertens (10) has made several simplifying assump— tions which lead to results in a form from which it is easier to obtain numerical results, but the solutions are less general and the theory appears to be limited in applicability. The purpose of this work is to examine the results of Mertens experiment- ally and to determine over what range of experimental conditions the theory is useful. THE THEORY OF MERTENS The Raman and Nath theory for the diffraction of light by ultrasonic waves is valid for the limited range of experimental conditions described above. Martens has extended this range by obtaining a more exact solution of the differential equation of Raman and Nath. The method used is to apply Maxwell's equations to the region of the medium containing the sound beam, along with the initial condition that the light has a plane wave-front at the plane of entrance to the sound beam. 5 The directions of travel of the light beam and the ultrasound are assumed to be normal. The refractive index is taken to be a linear function of the density of the medium. For a sinusoidally varying ultra- sonic beam,then, Mt.g,t,t) = p. + p ”(L‘W‘D‘t - 73-7 where p. is the refractive index of the medium, p the maximum variation of the refractive index, 0' the ultrasonic frequency, i? the propagation vector of the ultrasound, andff'the position vector within the medium. Maxwell's equations as applied to the medium are written: oi? _. Vlg=°tfil ‘V-MSOJ g.- v.fi.Jé.§__t'_§.), V.(r‘s)=o . at E represents the electric intensity and 3: the magnetic intensity. Upon elimination of 13 l t" —b v‘E:—L$_(" 5). V(V-E) (I) 6‘ at“ v-(H‘E) = 0 (2) Since 9'0: 1? , 1) being the frequency of the light, the refractive index is assumed to be time inde- pendent, and the time variation is reestablished in the final result. This gives the problem a stationary character. 6 Equation (1) may be written van-E- ' _y_:-(x,s,1,t) 3.31%. 4. V(V«F) (3) . C If the plane ultrasonic waves are traveling in the Xédirection 9(3’3’i‘t) a: Kat) , 1 put» I. .E -E EL ’ Substituting this into Eq. (3) gives ‘A I V‘E : 3:15:94 E _V( I. Q ”(XJE) E The second term on the right hand side is neglected and since it is small compared with the other terms (11). Also, if the light is traveling in the Z-direction these further simplifications result in the form 3'? Q"; F‘u,” 3‘? —— -—____ —— 9 a x" ‘ J i g c‘ at ‘- ( ) If E is written in the form .9? 111'! E 3 e +(fiJi3t) Eq. (4) becomes ‘ ‘ .1 4> + J 4:, = .. V..___1: H‘Ufi) 4’ )‘x‘ a 2 ,1 The solutions of this equation give the values of the amplitudes of the light in the various diffrac- tion orders. Since the purpose of this work is to check the results of Mertens' theory experimentally, the mathematical details will not be discussed here. It will suffice to say that Mertens has found solutions of Eq. (5) and Miller and Hiedemann (12) 7 have shown these solutions to be expressible in the form *3. = 3“?) ‘ SHA'(V) , Hl-BJV’) The two parameters ATV) and Bi?) may be expressed as the power series - A (V) = Z G. v.1”?! on ,, n’m and '2:; 6,,(11’) = 2: 5,5", Uzmm" where ”no (a) , (~')"‘(zm*"U-"“D a 6(zlm+n+lxm!)[(m+n).l] 3 b : ('0)"‘L 6 m «a (n+0XlOn-1)I[ m g ”0‘ Zh‘vOSfl-Gfl w flammaum-oqum .. -->!J The prime over the summation sign in Eq. (6) O. and indicates that the summation starts with m = 2 when. ha 0. The light intensity of the nth diffraction order is given by In 3 +0. 4,: III-r) «o H‘[ Afw’) o 2.1;”) B..“’"] . H" B:(1r) Tftv') . n‘ anhr) . H' 3:!” (v) where a Gnhr) 8 Adv) * 2'3“?!) BJV) Table III (appendix) gives values of 64v) and 8311’) for the zeroth and first orders. All three terms in Eq. (7) were used in the calculations made in 8 this investigation, but in the theory presented by Martens the last term is neglected. The experimental results show that the contributions of this term are negligible within the range where the theory is applicable. EXPERIMENTAL APPARATUS The standard optical arrangement for the observation of light diffraction by a progressive ultrasonic wave in a liquid medium was used. (See Fig. 1.) The light source, a 100 watt General Electric AH-4 mercury vapor lamp, was housed in a light tight enclosure. Lens L1, a condenser lens with an adjustable aperture, was used to focus the light on the slit. The aperture was used to control the intensity of the light passing through the system. A filter (not shown) designed to pass only the desired mercury green line was placed between L: and the slit. This filter was unnecessary for intensity readings since a second filter was contained in the photomultiplier, but it served to give a monochromatic source for visual observations. The lens, L2, was positioned so that the light passing through the tank, T, was collimated. A 4 mm square aperture was placed over the exit window of the tank. This mask covered the entire window, eliminating .psoaomcmnpm adoapmo esp Mo smnmnaa .a madmflm Eomm venom _ _ N._ :6 r. 330m xv.” 3:22:52ocn n 4 a _ _ _ _ _ _ _ " .l/mr\\. 10 stray light. A horizontal slit was placed between the tank and the lens, L5, to further limit the vertical dimension of the light field. Lens L3 was used to project the diffraction pattern onto the plane of the photomultiplier slit. The photo- multiplier-microphotometer was type 10-210 manufactured by the American Instrument Company. The tank used had a castor oil absorption well at one end to prevent reflections. The transducer was a 2 inch diameter, x-cut, air backed quartz crystal of fundamental frequency 1.76 Me which was driven in its third harmonic. The voltage across the transducer was measured with a General Radio vacuum tube voltmeter, type 1800B. The rf power to drive the transducer was supplied by a crystal controlled Johnson-Viking CDC transmitter. The width of the sound beam was controlled using a pair of hollow reflectors. (See Fig. 2.) The reflectors, R, were designed to present only a plane surface when seen frOm the transducer side. In order to prevent reflection of sound back onto the face of the transducer, the reflectors were inclined at an angle of 45° so that the sound was reflected down onto the faces of the two Plexiglas prisms, P. These prisms were oriented so that the sound reflected from their faces would be directed .ondpnnqn endow map Mo smnmmaq .N madem Ikdd oznom ‘ \\. / «‘ I 12 into the absorption well at the end of the tank. The dimensions of the sound aperture assembly made it necessary to work with the transducer at a minimum distance of 4 cm from the light beam. EXPERIMENTAL PROCEDURE. The results of Martens expressed by Eq. (7) show that it is possible to describe the experimental conditions using the two parameters 1; and H . It is necessary than to examine the theory over a range of these two parameters. Since the variation of refractive index is linear with quartz voltage, the parameter 1r can be varied by changing the quartz voltage. The parameter bi can be varied by changing either the sound beam width or the frequency. In this investigation the width was varied using the sound aperture described previously. One dis- advantage is that the range of V which can be attained experimentally decreases as the beam width is reduced. To insure that the directions of travel of the sound and the light were mutually perpendicular the quartz voltage was set so that about fifty per- cent of the light was in the zeroth order. The quartz was then rotated about a vertical axis and 15 simultaneously the light intensity was noted on the photomultiplier. The angular setting was taken to be correct when the light intensity was at a minimum. The sound aperture was positioned by projecting its shadow image onto a viewing screen. _ Using the schlieren technique, the optical aperture was located so that the light passed through the most uniform part of the sound field. Also, since the greater variation was in the vertical direction, a horizontal slit was used to limit the light field to about 0.4 mm in the vertical direction. When working at high sound intensities dissipated energy creates temperature differences and hence optical inhomogeneity within the medium resulting in undesirable optical refraction. Also, as the temperature of the system changes the acoustical and electrical impedances change giving unstable conditions. For this reason the system was allowed to come to thermal equilibrium between readings. In taking the readings the quartz voltage was varied in 10 volt intervals from zero to the maximum output voltage attainable. First the intensity of the zeroth diffraction order was observed. Data were taken for sound beam widths from 1.6 cm to 4.7 cm in 0.2 cm intervals 14 (the odd interval being that from.4.4 cm to 4.7 cm). Three sets of readings were then taken for the first order. PRESENTATION OF DATA ZEROTH ORDER Measured zero-order light intensities zg 1r are shown on the following pages. Four curves, represent- ative of the 16 taken are shown in Figs. 5-6. To normalize tha'V'scale it is necessary to assume that some characteristic point on the experi— mental curve corresponds to the same characteristic point on the theoretical curve since the values of If cannot be measured directly. When the second minimum on a curve could be reached experimentally this point was used. Otherwise the first minimum was used. Figure 7 shows the intensity of the light at the first and second minima of the zeroth order zg the width of the ultrasonic beam. Figure 8 shows the intensity of the zero-order maxima zg sound beam width. In determining Fl the following values were used: sound velocity 1500 m/sec refractive index of water 1.5} wavelength of light 5461 A frequency of ultrasound 5255 kc o o o o o Hmunesanmaxm nnnnn annesp .mnmpnmfi Lao o.m .2: $0.0 u 1 Homkp we zpflmcmpcfl ammfla guano npoaom .n ondmflm .o"" 0. ON Om 0v on 0m ON 00 cm 00. (‘70) AJJSNBLNI b m m w n E . ‘2'" u q a \o\“ o o 0 WI 664W \A\e oil a\ \‘6 0 VI \ 9 b «.9665... o o o o o amazesanmmxa : a n n .. .30on 35.322 Spoon» mama and swamm IVES o.n .1: onmd u I you... me hpflmnupwfl pnmfia.nmdno sponom .e mndmfih O. O O O O O «3 :0 v- m N ('70) AllSNBlNI (D p. 00 om 00. m> mpflmnmpnfl psmwa guano sponnm ®. .0 D .V m — d d H II 11!! \\& \fi 0 O [I \ 0 0 o // \\° // o c all \\e /‘ 0 9 \\O o aloud o .0 o o o o o amanEHHmQNm I I I I I mhovsp .QOpHuE bk 0 as u. SPSZ dens QnEJm IIIIoABO Oowc val-v OmoH ”I “0%: I .n madmflm O O O o h' ‘D I) q» (‘79) AllSNBlNI C) a) om 00. m w w m e m «I _ . . . \II \\o o/ [I \ / \O / \\ o It \\ III I \\ o e e I/ \e \ e o I, O /l|‘\“ O 0 o e e e o o o o o o anQmEHnmmxm IIIII known» .msmppmz zwomnp mama dun neanm '45 Rep}: 34.: noes. m> zpflmnmpsa pzmaa nodes sponmN .o madmwm o o o 0* 0 ID no <1- IO N (7.) AllSNBlNI C) p. Om om CO. (°/o) INTENSITY l4 l2. IO Figure-7. Intensity of light in the first and second minima of the zeroth order 1g ultrasonic beam width. Mertans' theory first minimum ----- second minimum Experimental first minimum + + + + + second minimum 0 o o o o DATA (continued) FIRST ORDERS Three sets of measurements of first order light intensities were made. These data are shown in Figs. 9-11. The average of the measured positive and negative first order light intensities is shown since an asymmetry was observed. However, in no case for points shown did these measured intensities differ by more than 5 percent (absolute). The 1’ scale was normalized as described for the zeroth order.. The rf voltage to u" ratios for the zeroth and first orders were compared and found to agree within experimental error. M u. .. 9» N R. l 100 .A M o I. ow ( 1 ON 0 o o o o anqmfinqua u n I n n apes ch 323.32 Page,» spam odd csEmm I 00 1:80 06. ".1866 u I no?» _ mp mpfimswpqfl psmgpvdyo pmnam .m madmflm I 00 LOO. 0.0 o o o Hansosflpqum nnnnn anomnp .onpme anomza npmz dam cdsmm .28 03¢ “1:34 at you...» MN mpamnmpza panH nodpo pmnfim 0H magmas on .um Auk Om hum Aun: kUSNBlNI (’lo) 00 r. (O U) ‘r H) o o o o o Hancoaahmmxm llllll mpoosp .QOppmu hhomflp Spsm dud Gwafifi .28 is .212: u... So: mH.mpflmsmch pszH amcpo papa. .HH madmflm 0. ON on Ow 00 cm ON 00 cm 00. (°/o) AJJSNBLNI DISCUSSION OF RESULTS Several discrepancies between the model upon which the theory of Martens is based and the actual experimental situation should be noted. First, in spite of the precautions taken, it is probable that some non~uniformity existed in the sound field. The observed intensity of a given diffraction order will then be some "average" of the intensities cor- responding to the local values of V over the light aperture. As a consequence, the extrema of the light intensity XE 1r curves will be less sharp, or less light will be observed in the maxima than expected and more light will be observed in the minima than expected. As the sound intensity increases the absolute spread in v’ would also increase. A second effect not considered in the theory of Mertens is the scattering of light within the medium and elsewhere in the optical system. Finally, Mertens' model is based upon diffraction by a sinusoidal ultrasonic wave. The presence of finite-amplitude effects will cause some deviation from intensities predicted for a sinusoidal waveform. Intensity measurements taken with the transducer at various distances from the optical aperture (giving various degrees of finite amplitude distortion) indicated that within experimental error the zeroth 25 26 order was unaffected by finite-amplitude effects. However, a finite-amplitude distorted sound wave does produce an asymmetric diffraction pattern. For this reason the agreement between theoretical predictions and experimental results is expected to be poorer for higher orders. A choice of *3 %'(absolute) as a criterion for satisfactory agreement between experiment and theory was made to allow for the above effects. As in the case of the Raman and Nath theory, the limits of validity of the theory of Martens can be expressed in terms of the parameters if and H . Table I shows the values of Wm“ , the maximum value of Ir for which the deviation between theory and experiment does not exceed $5 %. From Table I, it is seen that a reasonable upper limit for the validity of the theory of Mertens is :41r N .032707 .032603 .031410 .029465 .027269 .025404 .024443 .024846 .026880 .030572 .055253 .040874 .046559 .051561 .055202 .057001 .056756 .054604 .051008 .046697 .001624 .001564 .001347 .001000 .000592 .000222 .000013 .000087 .000545 .001440 .002650 .004187 .005826 .007332 .008458 .008985 .008772 .007798 .006184 .004193 31 .014898 .016573 .020137 .025267 ~031554 .037605 .043166 .047262 ~049359 .049161 .047068 .043270 .038682 .034261 -030973 .029614 .030706 .034356 .047687 .000415 .000083 .000014 .000316 .001038 .002142 .003511 .004947 .006205 .007040 .007326 .006879 .005775 .004198 .002459 .000952 .000086 .000205 .001513 .004015 lllllllllllHHIIHIIWHIHIH1|"IllllllllllllHlllllHlllll 17640412