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Tilted transducer arrays for wide-band accousto-optic deflectors
Volume 6, number 3
OPTICS COMMUNICATIONS
November 1972
TILTED TRANSDUCER ARRAYS FOR WIDE-BAND ACOUSTO-OPTIC DEFLECTORS* H. ESCHLER Siemens A G Forschungslaboratorien, 8 Miinchen 70, Germany Received 3 August 1972
Tilted transducers for wide-band acousto-optic deflectors have been studied in theory and experiment. These devices provide beam steering of the acoustic wave and broad-band efficient sound generation.
1. Introduction It is well known that the resolution of acoustooptic deflectors is equal to the product of acoustic transit time r and bandwidth A f N = Tar=
(a/v) a f ,
(l)
where a is the optical aperture, v the sound velocity. In order to increase the number if resolvable spots one can therefore enlarge both the optical aperture and bandwidth. However, for applications of deflectors as read-out devices in optical memories [ 1] and for optical data switching [2] where exact positioning of the deflected beam is required, one cannot raise a above a certain value due to thermal effects and inhomogeneities of the interaction medium which induce optical beam distortions and alteration of the deflection angle [3]. The striking temperature effects causing the shift of the light beam are thermal gradients between the transducer and the sound absorber and change of the elastic constants. Whereas the first effect decreases with rising optical aperture, the second is only a function of the temperature. For applications mentioned above the beam position generally has to be held constant to 10% o f the resolution limit X/a: A~o< X/lOa -~ A T . * Part of this paper was presented at the IEEE Ultrasonics Symposium, Miami Beach (1971). 230
(2)
If only the thermally induced change of the elastic constants is concerned, one obtains a trade-off between temperature and optical aperture (2). Moreover, acoustic absorption and restricted stability of the driving frequencies cause limitations of the acoustic transit time r. Consequently the only means of further enlargement of the deflector's resolution is magnification of the bandwidth Af. The bandwidth of acoustic deflectors is limited by the observance of the Bragg law and the transducer response. For octave bandwidth devices without acoustic beam steering the limitation of A f d u e to imperfect meeting of the Bragg angle has been calculated [4]: A f ~ ( 1, lno2/XoL )l/2 ,
(3)
where L is the transducer length. This equation holds only for weak l i g h t - s o u n d interaction. For bandwidths of more than 100 MHz fundamental transducer resonances of several hundred MHz are required. To provide the large acoustic diffraction angle in this case, transducer lengths have to be drastically reduced (3). This raises problems based on high sound powers and thermal losses. These effects can be partly overcome by acoustic wavefront rotation using phased-array transducers [5] yet stepped arrays provide technological difficulties. Moreover, such deflectors can only be operated within less than one frequency octave according to the limited transducer response.
Volume 6, number 3
OPTICS COMMUNICATIONS
November 1972
2. Advantages of tilted transducer arrangements These difficulties can be reduced by tilted transducer arrangements (fig. 1). On the sound medium several transducers are juxtaposed with adherent frequency ranges of efficient sound generation. To meet the Bragg condition within the whole bandwidth the transducers are tilted against each other by a small angle which is typically in the range of 1 0 - 4 0 minutes of arc. Similar to a phased-array arrangement the acoustic wavefronts are rotated. This can easily be understood by looking at the resulting sound waves for different frequencies. For simplicity we assume a two-transducer deflector but the subsequent considerations hold as well for multi-transducer arrangements. Looking at fig. 2a one realizes that around the lower "Bragg frequency" f01 only the sound wave generated by the first transducer contributes to light diffraction, because the second transducer's weak sound wave does not meet the Bragg angle. Similar effects occur around the second transducer's center frequency f02 (fig. 2c), where the first sound wave hardly causes any additional diffraction because of its weakness and because the Bragg condition is not fulfilled. While the vicinity off01 andfo 2 there are no unusual diffraction effects, the case f ~ ½(fol+f02) appears to be more complicated (fig. 2b). The sound power of both transducers is nearly equal and the combined wavefronts which form a sound field of double length compared with that of a single transducer, meet the Bragg law as well. Besides the ability of this transducer arrangement to rotate the acoustic wavefront one has a further ad-
undeftected beam deflected bean" sound absorber ~ ~I piezoet.ectric transducers for adjacent frequency ranges of efficient sound generation
Z / und medium
/
-
~ =
tunable f ~ ( ~ r f - generator
incident light beam
Fig. 1. Acoustic light deflector with tilted transducer arrangement (only one frequency is simultaneously present).
deflected beam
tower frequencies f~f01 second transducer shares Little becauseof uncorrect Bragg angle and Low response
Jeftected beam
~
J
-~
,efronts fulfil Bragg condition
sound wavefrontsof single transducers medium frequencies f ~fOl+'f02 2
~
t••
deflected
beam
higher frequencies f'~'fo2 first transducer shares little because of uncorrect Bragg angle and low response
Fig. 2. Rotation of acoustic wavefronts. vantage concerning the transducer response. In contrast to phased arrays of transducers the frequency range of efficient sound generation is not restricted to the resonance bandwidth of a single transducer but covers the sum of the bandwidths of all transducers with different operation ranges.
3. Calculation of the diffracted light intensity In order to get more information of the deflector's performance than could be provided by the former more intuitive considerations the diffraction efficiency of a tilted arrangement with two transducers was calculated for weak interaction of optical wave and sound wave. We assume a configuration where two 231
Volume 6, number 3
OPTICS COMMUNICATIONS
transducers of equal length L are juxtaposed on the medium side by side producing equal index variations An with a flat transducer response. This assumption which simplifies the calculation is not fulfilled in practice, but does not change the principal effects. The basic equations governing diffraction in the Bragg regime are [6] d¢o/dz -
with [-sin {K/(/'-¢Ol,2)} -] 2
J , and sin {Kf(~fo1)} r~l 2 = r?max Kf(/'_]01)
~kAn tpi= 0 ,
d~Pl/dZ + 12kAn ~0 = (2iK/L ) / q - f o ) ~1 ,
November 1972
(4}
sin {Kf(f-)'o2)}
Kf(f_.[02 )
X cos {Kf(2)':-J01-]02)} ,
(7)
where ~0 and tpl are the amplitudes of the zeroth and first order, L the interaction length, An the amplitude of the ultrasonically induced index variation, fo the Bragg frequency, K a material constant (K = ~ 7rLXo/ no 2) and k the light wave vector modulus. For weak interaction ~P0 is approximately constant and equal to unity. Eq. (4) changes into the form
"qmax = ( } k A n L ) 2 •
&Pl/dz - iatPl = - b ,
f = l(f01 +f02 )
which has the solution [~Ol(Z=0) = ¢1o, ~°l(Z=L) =
this mixing term is equal to the sum of the equal efficiency shares r/1 = 7/2. Fig. 3 shows the calculated efficiency versus frequency for a tilted transducer array with two elements for strong interaction of light and sound. This curve is the result of numerical evaluation of an analytically obtained formula. Around the medium frequency Ym, which is the average value of the Bragg frequencies )cOl and f02, a typical third hump can be seen due to this mixing term. By comparing the curves for r?1 + r/2 and for r/one realizes a considerable improvement of
=~IL] elL = ~10exp(iaL) + (b/ia) [1 - exp(iaL)] .
(5)
For two sound waves without mutual phase shift the solutions have the form p(1) =, (1) A 1L "Vl0"~l +B1 '
, (2) ,~(2)a + o elL = ' v 1 0 " 2 ~2"
Introducing tp]~ = ~'10~(2)and~]10)= 0, we obtain for the amplitude of the first order light wave ~01 = A 2B1 + B 2 .
Whereas r/1 and r/2 are simply the efficiency shares of each single transducer, 2r/12 is a mixing term which results from the mentioned diffraction by a combined wavefront formed by both sound fields (fig. 2b). For the particularly interesting frequency
(6)
771/lax-
--~----~/
Inserting the expressions for A and B in (6) yields
\
tp1 = exp(ia2L) (bl/ial) [1 - exp(ialL)] !~ I I
+ (b2/ia2) [l - exp (La2L)] . ..... From straightforward calculation using the cosine law and inserting the expressions for al, a2, b 1 and b2, we obtain for the light intensity diffracted into the first order r/= kO112= 7/1 + r/2 + 2r/12 , 232
t- . . . .
rto,,
I I I
+ ....
tm
Af-~ I I
I. . . .
to1+ ~ 2
-4--t---
~2 ~p
f ----,.,-
Fig. 3. Calculated efficiency for two-transducer arrangement (r/min = 50%). Solid line: total efficiency, dashed line: sum of efficiency shares.
Volume 6, number 3
OPTICS COMMUNICATIONS
bandwidth of the deflector due to the additional term, which appears as well in the formula derived for strong interaction (~o0vel) between light and sound. Deflectors of this kind have some additional properties concerning their electronic steering and relative bandwidth. Principally the transducers in a tilted transducer arrangement can be connected in parallel, because they behave like bandpass filters. In practice, it is advantageous to provide impedance transformation networks to transform the fairly low reactance of the parallel connected transducers. Transformation requirements are facilitated by the use of transducer materials with low dielectric constants (e.g., lithium sulphate). Moreover, acousto-optic deflectors using tilted transducers are suitable for operation in a frequency range of more than one octave. This is possible, because the normally disturbing second order spots remain weak due to operating the deflector in a purer Bragg mode than with a single transducer deflector.
4. Experimental deflector We have constructed an experimental deflector using water as a sound medium to demonstrate the specific properties of tilted transducer configurations (fig. 4). Two transducers consisting of lithium sulphate monohydrate were bonded to pieces of fused
tilted transducers 5ram 19mml thickness 75pro and 150p.m
i
incident laser beam
/
symmetrical, fused quartz pieces
Fig. 4o Experimental deflection cell.
rder
November 1972
2.0- ----
7/
1.5-
theoretical experimental
10-
r/2
-~ 0.5
lg
2'0
2'~
~b
3'~
~,o
~'sM,z~o
f ~
Fig. 5. Efficiency of experimental deflector.
quartz. These pieces were specifically shaped both to avoid disturbing sound reflections from the fused quartz-water interface and to provide the small tilting between the transducers. Both symmetrical pieces were pasted together. The optimum transducer lengths and thicknesses as well as the appropriate tilting angle were calculated by a computer program for the Bragg frequencies f01 = 20 MHz and f02 = 40 MHz. For measurements the two transducers were not connected in parallel but separately driven by two broadband power amplifiers which were supplied from the same rf signal generator. The actual Bragg frequencies differed slightly from the intended values (f02-f01 = 18 MHz instead of 20 MHz). Firstly, the efficiency shares r/1 and r/2 caused by each single sound field were measured while maintaining the sound power constant. The sound power was controlled by optical probing for the exact Bragg angle. The calculated and measured values show excellent agreement. In the medium frequency range from 24 MHz to 36 MHz the efficiency ~ caused by combined action of both transducers was measured and compared with theoretical results from numerical evaluation of eq. (7). Both curves show good agreement and the typical maximum in the vicinity of the medium frequency fin is clearly visible. Before the measurements the mutual phase shift between the two sound waves had to be carefully compensated for by inserting coaxial cables of proper length between driving amplifiers and transducers.
5. Conclusions By theoretical and experimental investigations the 233
Volume 6, number 3
OPTICS COMMUNICATIONS
proper operation of wide-band acousto-optic deflectors with tilted transducer arrays has been shown. The main advantage of these configurations is the possibility of increasing both the Bragg bandwidth by acoustic beam steering and the transducer response by using separate transducers operating in different frequency ranges. For fixed bandwidths the center frequencies may be considerably lowered, thereby reducing the problems which result from very thin transducers and acoustic absorption. High efficiency deflectors with bandwidths of several hundred MHz may be constructed using these techniques.
Acknowledgement The author wishes to thank Mr. R. Oberbacher,
234
November 1972
who constructed the deflection cell and Dr. P. Graf for helpful discussions.
References [l] H. Eschler, Frequenz 26 (1972) 124. [2] B. Hill, Nachr. Tech. Z. 23 (1970) 550. [3] G.A. Coquin, D.A. Pinnow and A.W. Warner, J. Appl. Phys. 42 (1971) 2162. [4] E.1. Gordon, Appl. Opt. 5 (1966) 1629. [5] G.A. Coquin, J.P. Griffin and L.K. Anderson, IEEE Trans. SU-17 (1970) 34. [6] W°RoKlein, B.D. Cook and W.G. Mayer, Acustica 15 (1965) 67.
OPTICS COMMUNICATIONS
November 1972
TILTED TRANSDUCER ARRAYS FOR WIDE-BAND ACOUSTO-OPTIC DEFLECTORS* H. ESCHLER Siemens A G Forschungslaboratorien, 8 Miinchen 70, Germany Received 3 August 1972
Tilted transducers for wide-band acousto-optic deflectors have been studied in theory and experiment. These devices provide beam steering of the acoustic wave and broad-band efficient sound generation.
1. Introduction It is well known that the resolution of acoustooptic deflectors is equal to the product of acoustic transit time r and bandwidth A f N = Tar=
(a/v) a f ,
(l)
where a is the optical aperture, v the sound velocity. In order to increase the number if resolvable spots one can therefore enlarge both the optical aperture and bandwidth. However, for applications of deflectors as read-out devices in optical memories [ 1] and for optical data switching [2] where exact positioning of the deflected beam is required, one cannot raise a above a certain value due to thermal effects and inhomogeneities of the interaction medium which induce optical beam distortions and alteration of the deflection angle [3]. The striking temperature effects causing the shift of the light beam are thermal gradients between the transducer and the sound absorber and change of the elastic constants. Whereas the first effect decreases with rising optical aperture, the second is only a function of the temperature. For applications mentioned above the beam position generally has to be held constant to 10% o f the resolution limit X/a: A~o< X/lOa -~ A T . * Part of this paper was presented at the IEEE Ultrasonics Symposium, Miami Beach (1971). 230
(2)
If only the thermally induced change of the elastic constants is concerned, one obtains a trade-off between temperature and optical aperture (2). Moreover, acoustic absorption and restricted stability of the driving frequencies cause limitations of the acoustic transit time r. Consequently the only means of further enlargement of the deflector's resolution is magnification of the bandwidth Af. The bandwidth of acoustic deflectors is limited by the observance of the Bragg law and the transducer response. For octave bandwidth devices without acoustic beam steering the limitation of A f d u e to imperfect meeting of the Bragg angle has been calculated [4]: A f ~ ( 1, lno2/XoL )l/2 ,
(3)
where L is the transducer length. This equation holds only for weak l i g h t - s o u n d interaction. For bandwidths of more than 100 MHz fundamental transducer resonances of several hundred MHz are required. To provide the large acoustic diffraction angle in this case, transducer lengths have to be drastically reduced (3). This raises problems based on high sound powers and thermal losses. These effects can be partly overcome by acoustic wavefront rotation using phased-array transducers [5] yet stepped arrays provide technological difficulties. Moreover, such deflectors can only be operated within less than one frequency octave according to the limited transducer response.
Volume 6, number 3
OPTICS COMMUNICATIONS
November 1972
2. Advantages of tilted transducer arrangements These difficulties can be reduced by tilted transducer arrangements (fig. 1). On the sound medium several transducers are juxtaposed with adherent frequency ranges of efficient sound generation. To meet the Bragg condition within the whole bandwidth the transducers are tilted against each other by a small angle which is typically in the range of 1 0 - 4 0 minutes of arc. Similar to a phased-array arrangement the acoustic wavefronts are rotated. This can easily be understood by looking at the resulting sound waves for different frequencies. For simplicity we assume a two-transducer deflector but the subsequent considerations hold as well for multi-transducer arrangements. Looking at fig. 2a one realizes that around the lower "Bragg frequency" f01 only the sound wave generated by the first transducer contributes to light diffraction, because the second transducer's weak sound wave does not meet the Bragg angle. Similar effects occur around the second transducer's center frequency f02 (fig. 2c), where the first sound wave hardly causes any additional diffraction because of its weakness and because the Bragg condition is not fulfilled. While the vicinity off01 andfo 2 there are no unusual diffraction effects, the case f ~ ½(fol+f02) appears to be more complicated (fig. 2b). The sound power of both transducers is nearly equal and the combined wavefronts which form a sound field of double length compared with that of a single transducer, meet the Bragg law as well. Besides the ability of this transducer arrangement to rotate the acoustic wavefront one has a further ad-
undeftected beam deflected bean" sound absorber ~ ~I piezoet.ectric transducers for adjacent frequency ranges of efficient sound generation
Z / und medium
/
-
~ =
tunable f ~ ( ~ r f - generator
incident light beam
Fig. 1. Acoustic light deflector with tilted transducer arrangement (only one frequency is simultaneously present).
deflected beam
tower frequencies f~f01 second transducer shares Little becauseof uncorrect Bragg angle and Low response
Jeftected beam
~
J
-~
,efronts fulfil Bragg condition
sound wavefrontsof single transducers medium frequencies f ~fOl+'f02 2
~
t••
deflected
beam
higher frequencies f'~'fo2 first transducer shares little because of uncorrect Bragg angle and low response
Fig. 2. Rotation of acoustic wavefronts. vantage concerning the transducer response. In contrast to phased arrays of transducers the frequency range of efficient sound generation is not restricted to the resonance bandwidth of a single transducer but covers the sum of the bandwidths of all transducers with different operation ranges.
3. Calculation of the diffracted light intensity In order to get more information of the deflector's performance than could be provided by the former more intuitive considerations the diffraction efficiency of a tilted arrangement with two transducers was calculated for weak interaction of optical wave and sound wave. We assume a configuration where two 231
Volume 6, number 3
OPTICS COMMUNICATIONS
transducers of equal length L are juxtaposed on the medium side by side producing equal index variations An with a flat transducer response. This assumption which simplifies the calculation is not fulfilled in practice, but does not change the principal effects. The basic equations governing diffraction in the Bragg regime are [6] d¢o/dz -
with [-sin {K/(/'-¢Ol,2)} -] 2
J , and sin {Kf(~fo1)} r~l 2 = r?max Kf(/'_]01)
~kAn tpi= 0 ,
d~Pl/dZ + 12kAn ~0 = (2iK/L ) / q - f o ) ~1 ,
November 1972
(4}
sin {Kf(f-)'o2)}
Kf(f_.[02 )
X cos {Kf(2)':-J01-]02)} ,
(7)
where ~0 and tpl are the amplitudes of the zeroth and first order, L the interaction length, An the amplitude of the ultrasonically induced index variation, fo the Bragg frequency, K a material constant (K = ~ 7rLXo/ no 2) and k the light wave vector modulus. For weak interaction ~P0 is approximately constant and equal to unity. Eq. (4) changes into the form
"qmax = ( } k A n L ) 2 •
&Pl/dz - iatPl = - b ,
f = l(f01 +f02 )
which has the solution [~Ol(Z=0) = ¢1o, ~°l(Z=L) =
this mixing term is equal to the sum of the equal efficiency shares r/1 = 7/2. Fig. 3 shows the calculated efficiency versus frequency for a tilted transducer array with two elements for strong interaction of light and sound. This curve is the result of numerical evaluation of an analytically obtained formula. Around the medium frequency Ym, which is the average value of the Bragg frequencies )cOl and f02, a typical third hump can be seen due to this mixing term. By comparing the curves for r?1 + r/2 and for r/one realizes a considerable improvement of
=~IL] elL = ~10exp(iaL) + (b/ia) [1 - exp(iaL)] .
(5)
For two sound waves without mutual phase shift the solutions have the form p(1) =, (1) A 1L "Vl0"~l +B1 '
, (2) ,~(2)a + o elL = ' v 1 0 " 2 ~2"
Introducing tp]~ = ~'10~(2)and~]10)= 0, we obtain for the amplitude of the first order light wave ~01 = A 2B1 + B 2 .
Whereas r/1 and r/2 are simply the efficiency shares of each single transducer, 2r/12 is a mixing term which results from the mentioned diffraction by a combined wavefront formed by both sound fields (fig. 2b). For the particularly interesting frequency
(6)
771/lax-
--~----~/
Inserting the expressions for A and B in (6) yields
\
tp1 = exp(ia2L) (bl/ial) [1 - exp(ialL)] !~ I I
+ (b2/ia2) [l - exp (La2L)] . ..... From straightforward calculation using the cosine law and inserting the expressions for al, a2, b 1 and b2, we obtain for the light intensity diffracted into the first order r/= kO112= 7/1 + r/2 + 2r/12 , 232
t- . . . .
rto,,
I I I
+ ....
tm
Af-~ I I
I. . . .
to1+ ~ 2
-4--t---
~2 ~p
f ----,.,-
Fig. 3. Calculated efficiency for two-transducer arrangement (r/min = 50%). Solid line: total efficiency, dashed line: sum of efficiency shares.
Volume 6, number 3
OPTICS COMMUNICATIONS
bandwidth of the deflector due to the additional term, which appears as well in the formula derived for strong interaction (~o0vel) between light and sound. Deflectors of this kind have some additional properties concerning their electronic steering and relative bandwidth. Principally the transducers in a tilted transducer arrangement can be connected in parallel, because they behave like bandpass filters. In practice, it is advantageous to provide impedance transformation networks to transform the fairly low reactance of the parallel connected transducers. Transformation requirements are facilitated by the use of transducer materials with low dielectric constants (e.g., lithium sulphate). Moreover, acousto-optic deflectors using tilted transducers are suitable for operation in a frequency range of more than one octave. This is possible, because the normally disturbing second order spots remain weak due to operating the deflector in a purer Bragg mode than with a single transducer deflector.
4. Experimental deflector We have constructed an experimental deflector using water as a sound medium to demonstrate the specific properties of tilted transducer configurations (fig. 4). Two transducers consisting of lithium sulphate monohydrate were bonded to pieces of fused
tilted transducers 5ram 19mml thickness 75pro and 150p.m
i
incident laser beam
/
symmetrical, fused quartz pieces
Fig. 4o Experimental deflection cell.
rder
November 1972
2.0- ----
7/
1.5-
theoretical experimental
10-
r/2
-~ 0.5
lg
2'0
2'~
~b
3'~
~,o
~'sM,z~o
f ~
Fig. 5. Efficiency of experimental deflector.
quartz. These pieces were specifically shaped both to avoid disturbing sound reflections from the fused quartz-water interface and to provide the small tilting between the transducers. Both symmetrical pieces were pasted together. The optimum transducer lengths and thicknesses as well as the appropriate tilting angle were calculated by a computer program for the Bragg frequencies f01 = 20 MHz and f02 = 40 MHz. For measurements the two transducers were not connected in parallel but separately driven by two broadband power amplifiers which were supplied from the same rf signal generator. The actual Bragg frequencies differed slightly from the intended values (f02-f01 = 18 MHz instead of 20 MHz). Firstly, the efficiency shares r/1 and r/2 caused by each single sound field were measured while maintaining the sound power constant. The sound power was controlled by optical probing for the exact Bragg angle. The calculated and measured values show excellent agreement. In the medium frequency range from 24 MHz to 36 MHz the efficiency ~ caused by combined action of both transducers was measured and compared with theoretical results from numerical evaluation of eq. (7). Both curves show good agreement and the typical maximum in the vicinity of the medium frequency fin is clearly visible. Before the measurements the mutual phase shift between the two sound waves had to be carefully compensated for by inserting coaxial cables of proper length between driving amplifiers and transducers.
5. Conclusions By theoretical and experimental investigations the 233
Volume 6, number 3
OPTICS COMMUNICATIONS
proper operation of wide-band acousto-optic deflectors with tilted transducer arrays has been shown. The main advantage of these configurations is the possibility of increasing both the Bragg bandwidth by acoustic beam steering and the transducer response by using separate transducers operating in different frequency ranges. For fixed bandwidths the center frequencies may be considerably lowered, thereby reducing the problems which result from very thin transducers and acoustic absorption. High efficiency deflectors with bandwidths of several hundred MHz may be constructed using these techniques.
Acknowledgement The author wishes to thank Mr. R. Oberbacher,
234
November 1972
who constructed the deflection cell and Dr. P. Graf for helpful discussions.
References [l] H. Eschler, Frequenz 26 (1972) 124. [2] B. Hill, Nachr. Tech. Z. 23 (1970) 550. [3] G.A. Coquin, D.A. Pinnow and A.W. Warner, J. Appl. Phys. 42 (1971) 2162. [4] E.1. Gordon, Appl. Opt. 5 (1966) 1629. [5] G.A. Coquin, J.P. Griffin and L.K. Anderson, IEEE Trans. SU-17 (1970) 34. [6] W°RoKlein, B.D. Cook and W.G. Mayer, Acustica 15 (1965) 67.