Abstract

Many coherent optical processing techniques that are based on wave diffraction can be carried out using acoustic waves. An array of acoustic transducers serves as a coherent radiation source, spatial modulator, and lens. Complex input data are entered by amplitude and phase modulation of the rf transducer drives. The output is retrieved from a second transducer array. This paper discusses the surface acoustic wave (SAW) implementation of space and time Fourier transforms, the effect of material anisotropy, and the results of an experimental SAW rf spectrum analyzer fabricated on LiNbO3.

© 1983 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. A. A. Oliner, Ed., Acoustic Surface Waves (Springer, New York, 1978).
    [CrossRef]
  3. A. Korpel, Proc. IEEE 65, 48 (1981).
    [CrossRef]
  4. P. Hartemann, P. Cauvard, “Wavefront Synthesis and Reconstruction Using Acoustic Surface Waves,” in 1977 Ultrasonic Proceedings, IEEE Cat. 77CH1264-ISU (IEEE, New York, 1977), pp. 840–842.
    [CrossRef]
  5. C. S. Tsai, L. N. Nguyen, Proc. IEEE 62, 863 (1974).
    [CrossRef]
  6. A. Korpel et al.Proc. IEEE 54, 1429 (1966).
    [CrossRef]
  7. G. A. Coquin et al.IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
    [CrossRef]
  8. D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-17, 209 (1970).
  9. R. M. De La Rue et al., Electron. Lett. 9, 326 (1973).
    [CrossRef]
  10. M. S. Kharusi, G. W. Farnell, Proc. IEEE 60, 945 (1972).
    [CrossRef]
  11. C. P. Wen, IEEE Trans. Microwave Theory Tech. MIT-17, 1087 (1969).
    [CrossRef]
  12. B. J. Knorr, K. D. Kuchler, IEEE Trans. Microwave Theory Tech. MIT-23, 541 (1975).
    [CrossRef]

1981 (1)

A. Korpel, Proc. IEEE 65, 48 (1981).
[CrossRef]

1975 (1)

B. J. Knorr, K. D. Kuchler, IEEE Trans. Microwave Theory Tech. MIT-23, 541 (1975).
[CrossRef]

1974 (1)

C. S. Tsai, L. N. Nguyen, Proc. IEEE 62, 863 (1974).
[CrossRef]

1973 (1)

R. M. De La Rue et al., Electron. Lett. 9, 326 (1973).
[CrossRef]

1972 (1)

M. S. Kharusi, G. W. Farnell, Proc. IEEE 60, 945 (1972).
[CrossRef]

1970 (2)

G. A. Coquin et al.IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-17, 209 (1970).

1969 (1)

C. P. Wen, IEEE Trans. Microwave Theory Tech. MIT-17, 1087 (1969).
[CrossRef]

1966 (1)

A. Korpel et al.Proc. IEEE 54, 1429 (1966).
[CrossRef]

Cauvard, P.

P. Hartemann, P. Cauvard, “Wavefront Synthesis and Reconstruction Using Acoustic Surface Waves,” in 1977 Ultrasonic Proceedings, IEEE Cat. 77CH1264-ISU (IEEE, New York, 1977), pp. 840–842.
[CrossRef]

Coquin, G. A.

G. A. Coquin et al.IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

De La Rue, R. M.

R. M. De La Rue et al., Electron. Lett. 9, 326 (1973).
[CrossRef]

Farnell, G. W.

M. S. Kharusi, G. W. Farnell, Proc. IEEE 60, 945 (1972).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hartemann, P.

P. Hartemann, P. Cauvard, “Wavefront Synthesis and Reconstruction Using Acoustic Surface Waves,” in 1977 Ultrasonic Proceedings, IEEE Cat. 77CH1264-ISU (IEEE, New York, 1977), pp. 840–842.
[CrossRef]

Kharusi, M. S.

M. S. Kharusi, G. W. Farnell, Proc. IEEE 60, 945 (1972).
[CrossRef]

Knorr, B. J.

B. J. Knorr, K. D. Kuchler, IEEE Trans. Microwave Theory Tech. MIT-23, 541 (1975).
[CrossRef]

Korpel, A.

A. Korpel, Proc. IEEE 65, 48 (1981).
[CrossRef]

A. Korpel et al.Proc. IEEE 54, 1429 (1966).
[CrossRef]

Kuchler, K. D.

B. J. Knorr, K. D. Kuchler, IEEE Trans. Microwave Theory Tech. MIT-23, 541 (1975).
[CrossRef]

Nguyen, L. N.

C. S. Tsai, L. N. Nguyen, Proc. IEEE 62, 863 (1974).
[CrossRef]

Pinnow, D. A.

D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-17, 209 (1970).

Tsai, C. S.

C. S. Tsai, L. N. Nguyen, Proc. IEEE 62, 863 (1974).
[CrossRef]

Wen, C. P.

C. P. Wen, IEEE Trans. Microwave Theory Tech. MIT-17, 1087 (1969).
[CrossRef]

Electron. Lett. (1)

R. M. De La Rue et al., Electron. Lett. 9, 326 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

C. P. Wen, IEEE Trans. Microwave Theory Tech. MIT-17, 1087 (1969).
[CrossRef]

B. J. Knorr, K. D. Kuchler, IEEE Trans. Microwave Theory Tech. MIT-23, 541 (1975).
[CrossRef]

IEEE Trans. Sonics Ultrason. (2)

G. A. Coquin et al.IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-17, 209 (1970).

Proc. IEEE (4)

C. S. Tsai, L. N. Nguyen, Proc. IEEE 62, 863 (1974).
[CrossRef]

A. Korpel et al.Proc. IEEE 54, 1429 (1966).
[CrossRef]

A. Korpel, Proc. IEEE 65, 48 (1981).
[CrossRef]

M. S. Kharusi, G. W. Farnell, Proc. IEEE 60, 945 (1972).
[CrossRef]

Other (3)

P. Hartemann, P. Cauvard, “Wavefront Synthesis and Reconstruction Using Acoustic Surface Waves,” in 1977 Ultrasonic Proceedings, IEEE Cat. 77CH1264-ISU (IEEE, New York, 1977), pp. 840–842.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. A. Oliner, Ed., Acoustic Surface Waves (Springer, New York, 1978).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Two-dimensional Fourier transform processor: (a) bulk acoustic wave implementation; (b) optical equivalent.

Fig. 2
Fig. 2

Space Fourier transform processors. (a) Generic optical configuration. Input data array is entered via spatial light modulator with complex transmittance t0(x1,y1). Square-law detection at the output provides the power spectrum. (b) Equivalent acoustic processor. Input data array is entered by amplitude and phase modulation of the rf carrier via the array of input modulators. Coherent output transduction provides the complex Fourier transform.

Fig. 3
Fig. 3

Time Fourier transform processors. (a) Bragg cell spectrum analyzer using an acoustooptic delay line as a spatial light modulator. The input signal, in the form of a modulated rf carrier, generates acoustic waves that propagate along the delay line to modulate the amplitude and phase of the optical carrier. Square-law detection at the output provides the power spectrum of the signal. (b) Acoustic processor using a tapped delay line samples the input signal. (c) Acoustic processor with signal time delays incorporated by stepping the input transducer elements. The transducer elements are oriented to direct the acoustic energy into the +1 diffraction order.

Fig. 4
Fig. 4

Frequency plane filtering by successive Fourier transforms. (a) Optical space Fourier transforms. (b) Acoustic time Fourier transforms. A lens may be needed in the frequency plane to correct the phase. (c) Acoustic time Fourier transforms using a reflective frequency plane filter and exploiting the bidirectionality of the spectrum analyzer.

Fig. 5
Fig. 5

Application of the space Fourier transform to determine the direction of arrival of rf signals.

Fig. 6
Fig. 6

A 31-element dispersive and focusing SAW transducer. The sixteen contact pads are connected to a common drive line to excite all the IDT elements in parallel.

Fig. 7
Fig. 7

Construction of the dispersive and focusing SAW transducer.

Fig. 8
Fig. 8

Experimental seven-channel spectrum analyzer.

Fig. 9
Fig. 9

Acoustic power density along the focal arc for five single-frequency input signals.

Fig. 10
Fig. 10

Calculated (solid line) and measured (open circles) dispersion.

Fig. 11
Fig. 11

Frequency response measured with a laser probe positioned at the 100-MHz position along the focal arc.

Equations (5)

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U ( x z , y z ) = exp ( jkz ) j λ z exp [ j k 2 z ( x z 2 + y z 2 ) ] { U ( x 1 , y 1 ) × exp [ j k 2 z ( x 1 2 + y 1 2 ) ] × exp [ j 2 π λ z ( x z x 1 + y z y 1 ] dx 1 dy 1 ,
U ( x 1 , y 1 ) = A t 0 ( x 1 , y 1 ) exp [ j k 2 f ( x 1 2 + y 1 2 ) ] .
U ( x f , y f ) = A j λ f t 0 ( x 1 , y 1 ) exp [ j 2 π λ f ( x 1 x f + y 1 y f ) ] dx 1 dy 1 .
r n = ( f + n λ 0 2 ) ( 1 + γ 1 + γ α 2 2 ) for n = 0 , ± N ,
Δ x = f Δ W ( 1 + γ ) ( λ λ 0 ) ,

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