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Angular and frequency dependence of light diffracted by surface acoustic waves
Volume 10, number 4
OPTICS COMIVlUNICATIONS
April 1974
ANGULAR AND FREQUENCY DEPENDENCE DIFFRACTED BY SURFACE ACOUSTIC
OF LIGHT WAVES
M. SINOU Centre National d'Etudes des T~ldcommunieations, 92130 Issy les Moulineaux, France Received 14 November 1973 Revised manuscript received 17 December 1973
The intensity of light diffracted by surface acoustic waves is described by its spectrum, which contains all the harmonics of the acoustic frequency. A formulation is given, mainly for the first harmonic angular dependence, taking into account most of the experimental parameters. Calculated results are compared with experimental data. Optimum conditions for detection are given for a wide range of frequencies.
This paper describes calculations and experiments of diffraction in reflection of an optical beam by surface acoustic waves (SAW). The calculations concern a theoretical approach to experiments such as those performed by Korpel et al., using what they call "the knife-edge technique" [1 4]. Ill this technique, an optical beam is focused on a part of the SAW device under study. A description of the interaction can be given in a simple way: the small area of the surface that is probed acts as a mirror oscillating at the SAW frequency, causing the optical beam to be tilted around its mean position. By "cutting" the optical beam by an edge placed in front of an optical detector, an electric signal related to the deflected optical intensity can be obtained. The frequency, phase and amplitude o f the signal are related to the acoustic signal. With his model [2], Korpel et al. found that the relation between the acoustic amplitude and the optical intensity is linear. Calculations for the optical intensity can be formulated in terms of the experimental parameters. These calculations could be useful in the optimization of experiments such as those developed on the model of the "knife-edge technique". The parameters of the experiments are as follows: ~,, A = optical, acoustic wavelength, co, fZ = optical, acoustic angular frequency, a = acoustic amplitude, q~ = incidence angle of the optical beam, L = diameter of the gaussian optical beam whose 348
intensity near the surface is defined by: F(x) = exp(- 4x2/L2), 0 = angle in which dilTraction is measured, defined from the reflected beam direction. By using the scalar theory of diffraction applied to the far field case, Mayer et al. [7] have calculated the instantaneous intensity in the 0 direction:
;(o, t)= 717=
~
n=._~
x W,m Wn exp [(m - n)f2t],
(1)
where 4ha 7 = - ~ - cos
cos½0
(2)
and Wm = exp~-~--~cos[-[4nO 2~b2. 0 +..2Art)L2 cos2~] .
(3)
if, as occurs generally, the amplitude a is much smaller than the optical wavelength k, 7"~1.
(4)
Formula (1) shows that the optical intensity spectrum contains: a d.c. part 10 obtained for all terms in which
Volume 10, number 4
OPTICS COMMUNICATIONS
m = n, and a.c. t e r m s / r , with angular frequency rg2. In their papers Mayer et al. [ 5 - 8 ] considered only the d.c. part which leads to the well-known law of proportionality between the intensity of the first order and the acoustic power. Only a.c. terms are studied here, mainly the fundamental one, I 1 . They are obtained in (1) from all terms where n
-
rn
= +
(5)
r,
April 1974
~
0.2
055
2
,1)=20°
0.1
Z tu 0.1 ua >
003
which gives +oo
t)=
0
n=_oo
x W n W n+r e x p ( - + i r a t )
(6)
4X in 0 + 0 20 '~-~7-5 ~Cos 2 q ~ ~-
10
Fig. I. Spatial dependence of the first harmonic of the intensity for several values ofA = rr2L2/8A2cos20.
F o r r = 1, I 1 ( 0 , t) = 2"),exp[ A(~ 2 + 1)] s h ( A a ) cosg2t,
(7)
where c~ (4A/X)sin½0 cos =
24+0 2
(8)
and A = rr2L2/8A2 cos2~b.
(9)
Formula (7) was obtained by expansion of the Bessel functions to the first term, according to (4) and keeping only terms of the first order in 3' o f e q . (1). It can be seen that i f 0 is small, a is proportional to 0. A is a parameter that takes into account the squared ratio of the beam diameter to the acoustic wavelength. The amplitude of the intensity I 1 is plotted in fig. 1 in terms ofc~ for several values of A. Fig. 1 shows that, as A goes to zero, i.e. for small values of the spot size L, the intensity I 1 appears to become uniformly distributed in angular space. In that limiting case, a geometrical approach is possible, given by the moving mirror model: the total intensity I(0, t) in the 0 direction is zero at every time except when the ray is reflected exactly in the 0 direction, which occurs twice a period. The Fourier spectrum of such a signal is zero and especially the first component I1, which explains the result of fig. 1. Also, as A goes to infinity, i.e. for large values of the spot size L, the intensity I 1 goes to zero as shown in fig. 1. In that
Power
- edge)1 m~t°~er ~
Square
~
wave generator
Ref. Lock - in Amplifier
I
Acos~
Fig. 2. Experimental set up. 349
V o l u m e 10, n u m b e r 4
OPTICS COMMUNICATIONS
case, a grating model call be used which gives the d.c. c o m p o n e n t I 0 of the spectrum. Our calculations showed that in that case, the other colnponellts are zero. T h e experimental set up is shown in fig. 2. An optical beam, is focused by a lens on the SAW device under study. The size of the optical spot obtained in tire foca! plane is calculated from the focal length of the lens and the characteristics of the laser used, using the equations for gaussian beam propagation. A part of tlre energy from a r.f. generator is mixed with a low-frequency square wave to give an amplitcide-modulated signal. This signal is amplified in a power amplifier, and then sent to the input of the SAW devicu. The p h o t o m u l t i p l i e r (p.m.) signal is mixed with the ,>ther par! of the r.f. wave ill order togive al>w-treou~lk'} sigi!a!whose ulnplitudeis proportional to V c~!s ¢1~.where l is the aulp.litllde ~1 the r.f. C O l l l p o l l e i i l o f the p.m. o t l l p l l t a l l d tip the phclse of this c o n l p o n e l l t relative to tile leterence sigllti!. Ill order to avoid measurements of the reflected sigiitl[ from the power amplifier, an attenuator preceded tile amplifier in the line supplying energy to tile SAW device.
April 1974
Several experiments have been performed with this setup, all with an acoustic line of an Y c u t , X propagating quartz cr),stal equiped with an interdigital transducer with 20 fingers, operating near 20 MHz. Tile optical wavelength was 6328 ~,, giving a ratio A/X = 250. We first verified that, in our power range, the approximation of flu/damental amplitude linearity with acoustic amplitude was true. [11 other words, assuming tile losses ill the tiansducer and matching n e t w o r k to be constant, we verified that the detected signal was proportional to the square root o f the electrical power. Then, in order to verit).' tile ctuves plotted ill fig. 1, we used the same experimental setup, wifll a narrow slit in place of tile "knife-edge". "['he spot was moved ill front of this slit by turnit~,g the line at a known angle. Tile results are shown in fig. 3, where 0 has been ~epresented on the horizontal axis instead oi a, :lccording Io the value 250 of the ratio ,k,,'X, 9 was ,',~ke:~ equal to 20 °. Experimental points agree quite wei! wi!h file theoretical curve of fig. i corresptmding t<>L ~ 30 ;~, which is tile value calculated for the lells ctsed J_,'l ~',lis experimea~l. I h i s curve is drawn a',, a solid lille in fig. ~.
.~'/A = 2 5 0 .
L:30.
~,r~: 2 0 o
points
Theory
Slit >-
Experimental
dimension
Z
u.J 1. pZ
o
Ia.I >
< _..1 tlJ ee'
O.
0.5 OBSERVATION
1.
1.5 ANGLE
(DEGREES)
Fig. 3. Spatial dependence of the first harmonic of the intensity for several values of optical beam size L. Experiment and theory.
350
Volume 10, number 4
OPTICS COMMUNICATIONS
April 1974
i .t = 250. w
~
o~1.5
20 ° Experimental
.7
points
Theory
~,.
Ill
5 t.=7 0_= 12 4 D
I-n3~ U.J
I--
_z 2
0.=0 ~' q~
t
175 0.6t
. . . . . .
RATIO
OF
SPOT
o'.~
.
.
.
.
.
.
.
.
t
Sl ZE L TO A
A
Fig. 4. E f f i c i e n c y o f the " k n i f e - e d g e t e c h n i q u e " in t e r m s o f optical b e a m size L.
Thirdly, we verified the optimization conditions of the "knife-edge technique". The acoustic wavelength was kept constant, and only the optical beam diameter L and the p.m. aperture angle 00 were varied. In the "knife-edge technique", the measured intensity is the integral
with the size of tile probing spot indicated for each one. The best fitting curve is obtained for 00 = 0.5 °, which was the angle measured approximately in tile previous experiment. At the frequency used in the experiment, the optimum should be obtained for A = 0.2, which corresponds to L = 67 ~, which is approximately 0.4 A. t:l Finally, fig. 5 presents some results about the change . t o [ 1(0) dO. of the optimum conditions with frequency. The opti0 mum value of the parameter A was computed for each This integral was computed for several values of A acoustic wavelength A, for several values of the p.m. (i.e. of L/A) and 00 (fig. 4). Special attention was given angle 00. Some conclusions can be drawn from these to the interpretation of the experimental data, because, curves. First. the optimal value of A is always smaller by changing the lens, the incident optical intensity and than l, and corresponds to a beam size always smaller the point probed on the device are changed. In order to than about 0.8 A for optimum conditions. Second, the minimize errors, the spot was placed near the input optimum value of A is near 1 for high frequencies or transducer, assuming that, in this place, the amplitude small values of 00. This can be understood by studying is quite constant. The incident optical intensity was also fig. 1. For high frequencies or small p.m. angle, only the measured at each point and the result obtained was part of curves I(c~) near 0 need to be considered. They modified by a factor that took into account this change, can be approximated by straight lines of which a greater In fact, this change was small compared with the disperslope is obtained forA = 1, which gives the greatest insion of the results. In fig. 4, experimental points are shown tensity. 351
Volume 10, number 4
OPTICS COMMUNICATIONS
A=
~L~ 8 t ~ cos2~
April 1974
be i m p r o v e d by using a slit placed near the o p t i m u m value o f the angle 0 (fig. 3). T h a t m a y allow one to increase the signal to noise ratio in o r d e r to o b t a i n a wider d y n a m i c range o f acoustic a m p l i t u d e m e a s u r e m e n t s or to m a k e signal m e a s u r e m e n t s or processing easier.
1
O,8
References
-O.6
:0,4 -0,2
2o 1GHz~
5o
lOO "-Quartz
26o Y/X - i
5bo
16oo D 20MHz
4~A FREQUENCY
Fig. 5. Optimal conditions for the "knife-edge technique" in terms of the ratio A/K and the optical detector aperture 00. T h e above results (figs. 4 a n d 5) have b e e n o b t a i n e d b y using an edge. We t h o u g h t t h a t the e x p e r i m e n t could
352
[ 1 ] R. Adler, A. Korpel and P. Desmares, 1EEE J. Sonics and Ultrasonics SU-15 (1968) 157. [2] A. Korpel and P. Desmares, J. Acoust. Soc. Amer. 45 (1969) 881. [3] R.L. Whitman and A. Korpel, Appl. Opt. 8 (1969) 1567. [4] A.J. de Vries and R.L Miller, Appl. Phys. Lett. 20 (1972) 210. 15] W.R. Klein, B.D. Cook and W.G. Mayer, Acustica 15 (1965) 67. [6] W.G. Mayer, G.B. Lamers and D.C. Auth, J. Acoust. Soc. Am., 42 (1967) 1255. [7] R.J. Hallermeier and W.G. Mayer, J. Acoust. Soc. Am. 47 (1970) 1236. [8] R.J. Hallermeier and W.G. Mayer, IEEE J. Sonics and Ultrasonics SU-18, (1971) 176.
OPTICS COMIVlUNICATIONS
April 1974
ANGULAR AND FREQUENCY DEPENDENCE DIFFRACTED BY SURFACE ACOUSTIC
OF LIGHT WAVES
M. SINOU Centre National d'Etudes des T~ldcommunieations, 92130 Issy les Moulineaux, France Received 14 November 1973 Revised manuscript received 17 December 1973
The intensity of light diffracted by surface acoustic waves is described by its spectrum, which contains all the harmonics of the acoustic frequency. A formulation is given, mainly for the first harmonic angular dependence, taking into account most of the experimental parameters. Calculated results are compared with experimental data. Optimum conditions for detection are given for a wide range of frequencies.
This paper describes calculations and experiments of diffraction in reflection of an optical beam by surface acoustic waves (SAW). The calculations concern a theoretical approach to experiments such as those performed by Korpel et al., using what they call "the knife-edge technique" [1 4]. Ill this technique, an optical beam is focused on a part of the SAW device under study. A description of the interaction can be given in a simple way: the small area of the surface that is probed acts as a mirror oscillating at the SAW frequency, causing the optical beam to be tilted around its mean position. By "cutting" the optical beam by an edge placed in front of an optical detector, an electric signal related to the deflected optical intensity can be obtained. The frequency, phase and amplitude o f the signal are related to the acoustic signal. With his model [2], Korpel et al. found that the relation between the acoustic amplitude and the optical intensity is linear. Calculations for the optical intensity can be formulated in terms of the experimental parameters. These calculations could be useful in the optimization of experiments such as those developed on the model of the "knife-edge technique". The parameters of the experiments are as follows: ~,, A = optical, acoustic wavelength, co, fZ = optical, acoustic angular frequency, a = acoustic amplitude, q~ = incidence angle of the optical beam, L = diameter of the gaussian optical beam whose 348
intensity near the surface is defined by: F(x) = exp(- 4x2/L2), 0 = angle in which dilTraction is measured, defined from the reflected beam direction. By using the scalar theory of diffraction applied to the far field case, Mayer et al. [7] have calculated the instantaneous intensity in the 0 direction:
;(o, t)= 717=
~
n=._~
x W,m Wn exp [(m - n)f2t],
(1)
where 4ha 7 = - ~ - cos
cos½0
(2)
and Wm = exp~-~--~cos[-[4nO 2~b2. 0 +..2Art)L2 cos2~] .
(3)
if, as occurs generally, the amplitude a is much smaller than the optical wavelength k, 7"~1.
(4)
Formula (1) shows that the optical intensity spectrum contains: a d.c. part 10 obtained for all terms in which
Volume 10, number 4
OPTICS COMMUNICATIONS
m = n, and a.c. t e r m s / r , with angular frequency rg2. In their papers Mayer et al. [ 5 - 8 ] considered only the d.c. part which leads to the well-known law of proportionality between the intensity of the first order and the acoustic power. Only a.c. terms are studied here, mainly the fundamental one, I 1 . They are obtained in (1) from all terms where n
-
rn
= +
(5)
r,
April 1974
~
0.2
055
2
,1)=20°
0.1
Z tu 0.1 ua >
003
which gives +oo
t)=
0
n=_oo
x W n W n+r e x p ( - + i r a t )
(6)
4X in 0 + 0 20 '~-~7-5 ~Cos 2 q ~ ~-
10
Fig. I. Spatial dependence of the first harmonic of the intensity for several values ofA = rr2L2/8A2cos20.
F o r r = 1, I 1 ( 0 , t) = 2"),exp[ A(~ 2 + 1)] s h ( A a ) cosg2t,
(7)
where c~ (4A/X)sin½0 cos =
24+0 2
(8)
and A = rr2L2/8A2 cos2~b.
(9)
Formula (7) was obtained by expansion of the Bessel functions to the first term, according to (4) and keeping only terms of the first order in 3' o f e q . (1). It can be seen that i f 0 is small, a is proportional to 0. A is a parameter that takes into account the squared ratio of the beam diameter to the acoustic wavelength. The amplitude of the intensity I 1 is plotted in fig. 1 in terms ofc~ for several values of A. Fig. 1 shows that, as A goes to zero, i.e. for small values of the spot size L, the intensity I 1 appears to become uniformly distributed in angular space. In that limiting case, a geometrical approach is possible, given by the moving mirror model: the total intensity I(0, t) in the 0 direction is zero at every time except when the ray is reflected exactly in the 0 direction, which occurs twice a period. The Fourier spectrum of such a signal is zero and especially the first component I1, which explains the result of fig. 1. Also, as A goes to infinity, i.e. for large values of the spot size L, the intensity I 1 goes to zero as shown in fig. 1. In that
Power
- edge)1 m~t°~er ~
Square
~
wave generator
Ref. Lock - in Amplifier
I
Acos~
Fig. 2. Experimental set up. 349
V o l u m e 10, n u m b e r 4
OPTICS COMMUNICATIONS
case, a grating model call be used which gives the d.c. c o m p o n e n t I 0 of the spectrum. Our calculations showed that in that case, the other colnponellts are zero. T h e experimental set up is shown in fig. 2. An optical beam, is focused by a lens on the SAW device under study. The size of the optical spot obtained in tire foca! plane is calculated from the focal length of the lens and the characteristics of the laser used, using the equations for gaussian beam propagation. A part of tlre energy from a r.f. generator is mixed with a low-frequency square wave to give an amplitcide-modulated signal. This signal is amplified in a power amplifier, and then sent to the input of the SAW devicu. The p h o t o m u l t i p l i e r (p.m.) signal is mixed with the ,>ther par! of the r.f. wave ill order togive al>w-treou~lk'} sigi!a!whose ulnplitudeis proportional to V c~!s ¢1~.where l is the aulp.litllde ~1 the r.f. C O l l l p o l l e i i l o f the p.m. o t l l p l l t a l l d tip the phclse of this c o n l p o n e l l t relative to tile leterence sigllti!. Ill order to avoid measurements of the reflected sigiitl[ from the power amplifier, an attenuator preceded tile amplifier in the line supplying energy to tile SAW device.
April 1974
Several experiments have been performed with this setup, all with an acoustic line of an Y c u t , X propagating quartz cr),stal equiped with an interdigital transducer with 20 fingers, operating near 20 MHz. Tile optical wavelength was 6328 ~,, giving a ratio A/X = 250. We first verified that, in our power range, the approximation of flu/damental amplitude linearity with acoustic amplitude was true. [11 other words, assuming tile losses ill the tiansducer and matching n e t w o r k to be constant, we verified that the detected signal was proportional to the square root o f the electrical power. Then, in order to verit).' tile ctuves plotted ill fig. 1, we used the same experimental setup, wifll a narrow slit in place of tile "knife-edge". "['he spot was moved ill front of this slit by turnit~,g the line at a known angle. Tile results are shown in fig. 3, where 0 has been ~epresented on the horizontal axis instead oi a, :lccording Io the value 250 of the ratio ,k,,'X, 9 was ,',~ke:~ equal to 20 °. Experimental points agree quite wei! wi!h file theoretical curve of fig. i corresptmding t<>L ~ 30 ;~, which is tile value calculated for the lells ctsed J_,'l ~',lis experimea~l. I h i s curve is drawn a',, a solid lille in fig. ~.
.~'/A = 2 5 0 .
L:30.
~,r~: 2 0 o
points
Theory
Slit >-
Experimental
dimension
Z
u.J 1. pZ
o
Ia.I >
< _..1 tlJ ee'
O.
0.5 OBSERVATION
1.
1.5 ANGLE
(DEGREES)
Fig. 3. Spatial dependence of the first harmonic of the intensity for several values of optical beam size L. Experiment and theory.
350
Volume 10, number 4
OPTICS COMMUNICATIONS
April 1974
i .t = 250. w
~
o~1.5
20 ° Experimental
.7
points
Theory
~,.
Ill
5 t.=7 0_= 12 4 D
I-n3~ U.J
I--
_z 2
0.=0 ~' q~
t
175 0.6t
. . . . . .
RATIO
OF
SPOT
o'.~
.
.
.
.
.
.
.
.
t
Sl ZE L TO A
A
Fig. 4. E f f i c i e n c y o f the " k n i f e - e d g e t e c h n i q u e " in t e r m s o f optical b e a m size L.
Thirdly, we verified the optimization conditions of the "knife-edge technique". The acoustic wavelength was kept constant, and only the optical beam diameter L and the p.m. aperture angle 00 were varied. In the "knife-edge technique", the measured intensity is the integral
with the size of tile probing spot indicated for each one. The best fitting curve is obtained for 00 = 0.5 °, which was the angle measured approximately in tile previous experiment. At the frequency used in the experiment, the optimum should be obtained for A = 0.2, which corresponds to L = 67 ~, which is approximately 0.4 A. t:l Finally, fig. 5 presents some results about the change . t o [ 1(0) dO. of the optimum conditions with frequency. The opti0 mum value of the parameter A was computed for each This integral was computed for several values of A acoustic wavelength A, for several values of the p.m. (i.e. of L/A) and 00 (fig. 4). Special attention was given angle 00. Some conclusions can be drawn from these to the interpretation of the experimental data, because, curves. First. the optimal value of A is always smaller by changing the lens, the incident optical intensity and than l, and corresponds to a beam size always smaller the point probed on the device are changed. In order to than about 0.8 A for optimum conditions. Second, the minimize errors, the spot was placed near the input optimum value of A is near 1 for high frequencies or transducer, assuming that, in this place, the amplitude small values of 00. This can be understood by studying is quite constant. The incident optical intensity was also fig. 1. For high frequencies or small p.m. angle, only the measured at each point and the result obtained was part of curves I(c~) near 0 need to be considered. They modified by a factor that took into account this change, can be approximated by straight lines of which a greater In fact, this change was small compared with the disperslope is obtained forA = 1, which gives the greatest insion of the results. In fig. 4, experimental points are shown tensity. 351
Volume 10, number 4
OPTICS COMMUNICATIONS
A=
~L~ 8 t ~ cos2~
April 1974
be i m p r o v e d by using a slit placed near the o p t i m u m value o f the angle 0 (fig. 3). T h a t m a y allow one to increase the signal to noise ratio in o r d e r to o b t a i n a wider d y n a m i c range o f acoustic a m p l i t u d e m e a s u r e m e n t s or to m a k e signal m e a s u r e m e n t s or processing easier.
1
O,8
References
-O.6
:0,4 -0,2
2o 1GHz~
5o
lOO "-Quartz
26o Y/X - i
5bo
16oo D 20MHz
4~A FREQUENCY
Fig. 5. Optimal conditions for the "knife-edge technique" in terms of the ratio A/K and the optical detector aperture 00. T h e above results (figs. 4 a n d 5) have b e e n o b t a i n e d b y using an edge. We t h o u g h t t h a t the e x p e r i m e n t could
352
[ 1 ] R. Adler, A. Korpel and P. Desmares, 1EEE J. Sonics and Ultrasonics SU-15 (1968) 157. [2] A. Korpel and P. Desmares, J. Acoust. Soc. Amer. 45 (1969) 881. [3] R.L. Whitman and A. Korpel, Appl. Opt. 8 (1969) 1567. [4] A.J. de Vries and R.L Miller, Appl. Phys. Lett. 20 (1972) 210. 15] W.R. Klein, B.D. Cook and W.G. Mayer, Acustica 15 (1965) 67. [6] W.G. Mayer, G.B. Lamers and D.C. Auth, J. Acoust. Soc. Am., 42 (1967) 1255. [7] R.J. Hallermeier and W.G. Mayer, J. Acoust. Soc. Am. 47 (1970) 1236. [8] R.J. Hallermeier and W.G. Mayer, IEEE J. Sonics and Ultrasonics SU-18, (1971) 176.