Abstract

Deuterated cesium dihydrogen arsenate (CD*A) is used as a square-law device for optical processing. Placed in the Fourier transform plane, the device squares the incident optical radiation, thus producing the convolutions and correlations of the input functions at the output plane. The convolutions can be detected at the temporal angular frequency ωa + ωb and the correlations at ωaωb, where ωa and ωb are the frequencies of the input radiation. This paper describes the experiment setup, presents the convolution data obtained, and develops an associated mathematical model of the crystal interaction in an optical data processing system.

© 1978 Optical Society of America

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References

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  1. S. G. Lipson, P. Nisenson, Appl. Opt. 13, 2052 (1974).
    [CrossRef] [PubMed]
  2. G. H. Heilmeier, IEEE Trans. Electron Devices ED-23, 780 (1976).
    [CrossRef]
  3. G. G. Currie, I. Cindrich, C. D. Leonard, Proc. SPIE 83, 8 (1976).
    [CrossRef]
  4. N. K. Sheridon, IEEE Trans. Electron Devices ED-19, 1003 (1972).
    [CrossRef]
  5. D. Casasent, Proc. IEEE 65, 143 (1977).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. S. H. Lee, Appl. Phys. 10, 203 (1976).
    [CrossRef]
  8. R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
    [CrossRef]
  9. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), p. 186–198.

1977 (2)

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
[CrossRef]

1976 (3)

S. H. Lee, Appl. Phys. 10, 203 (1976).
[CrossRef]

G. H. Heilmeier, IEEE Trans. Electron Devices ED-23, 780 (1976).
[CrossRef]

G. G. Currie, I. Cindrich, C. D. Leonard, Proc. SPIE 83, 8 (1976).
[CrossRef]

1974 (1)

1972 (1)

N. K. Sheridon, IEEE Trans. Electron Devices ED-19, 1003 (1972).
[CrossRef]

Casasent, D.

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

Chekmenev, A. I.

R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
[CrossRef]

Cindrich, I.

G. G. Currie, I. Cindrich, C. D. Leonard, Proc. SPIE 83, 8 (1976).
[CrossRef]

Currie, G. G.

G. G. Currie, I. Cindrich, C. D. Leonard, Proc. SPIE 83, 8 (1976).
[CrossRef]

Eremeeva, R. A.

R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
[CrossRef]

Goodman, J. W.

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

Heilmeier, G. H.

G. H. Heilmeier, IEEE Trans. Electron Devices ED-23, 780 (1976).
[CrossRef]

Kudryashov, V. A.

R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
[CrossRef]

Lee, S. H.

S. H. Lee, Appl. Phys. 10, 203 (1976).
[CrossRef]

Leonard, C. D.

G. G. Currie, I. Cindrich, C. D. Leonard, Proc. SPIE 83, 8 (1976).
[CrossRef]

Lipson, S. G.

Mateev, J. N.

R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
[CrossRef]

Nisenson, P.

Sheridon, N. K.

N. K. Sheridon, IEEE Trans. Electron Devices ED-19, 1003 (1972).
[CrossRef]

Usacheva, T. G.

R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), p. 186–198.

Appl. Opt. (1)

Appl. Phys. (1)

S. H. Lee, Appl. Phys. 10, 203 (1976).
[CrossRef]

IEEE Trans. Electron Devices (2)

G. H. Heilmeier, IEEE Trans. Electron Devices ED-23, 780 (1976).
[CrossRef]

N. K. Sheridon, IEEE Trans. Electron Devices ED-19, 1003 (1972).
[CrossRef]

Proc. IEEE (1)

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

Proc. SPIE (1)

G. G. Currie, I. Cindrich, C. D. Leonard, Proc. SPIE 83, 8 (1976).
[CrossRef]

Sov. J. Quantum Electron. (1)

R. A. Eremeeva, V. A. Kudryashov, J. N. Mateev, T. G. Usacheva, A. I. Chekmenev, Sov. J. Quantum Electron. 7, 90 (1977).
[CrossRef]

Other (2)

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), p. 186–198.

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

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Figures (12)

Fig. 1
Fig. 1

Superposition multiplier response for λ1 = 2λ2.

Fig. 2
Fig. 2

Superposition multiplier response for λ1 = λ2 (experimental configuration).

Fig. 3
Fig. 3

Superposition multiplier experimental configuration.

Fig. 4
Fig. 4

Two-square input (λ = 1.06 μm, d = 1.3 cm).

Fig. 5
Fig. 5

Convolution outputs (λ = 0.53 μm).

Fig. 6
Fig. 6

Three-square collinear input (λ = 1.06 μm, d = 1.8 cm).

Fig. 7
Fig. 7

Joint transform output of Fig. 6 using crystal (λ = 0.54 μm).

Fig. 8
Fig. 8

Functional representation of the output plane distribution of Fig. 7.

Fig. 9
Fig. 9

Four-square input plane illumination in a square format (λ = 1.06 μm).

Fig. 10
Fig. 10

Output plane convolution distribution due to inputs shown in Fig. 9 (λ = 0.53 μm).

Fig. 11
Fig. 11

Four-square input illumination in a parallelogram format (λ = 1.06 μm).

Fig. 12
Fig. 12

Output plane distribution caused by inputs shown in Fig. 11.

Equations (13)

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P ¯ = 0 χ ¯ ¯ ( 1 ) E ¯ ,
P ¯ = 0 { χ ¯ ¯ ( 1 ) E ¯ + χ ¯ ¯ ¯ ( 2 ) [ E ¯ ] 2 + χ ¯ ¯ ¯ ¯ ( 3 ) [ E ¯ ] 3 + . } ,
P ¯ N . L . = 0 χ ¯ ¯ ¯ ( 2 ) [ E ¯ ] 2 .
2 E ¯ = μ 0 σ E ¯ t + μ 0 0 2 E ¯ t 2 + 0 μ 0 χ ¯ ¯ ¯ ( 2 ) [ E ¯ ] 2 t 2 . nonlinear term
E 0 2 ω ( l ) = j ω ( μ 0 ) 1 / 2 d [ E i ( ω ) ] 2 exp ( j Δ k l ) 1 j Δ k .
E out ( x , y ; t ) = α [ E in ( x , y ; t ) ] 2 ,
E i ( x , y ; t ) = 1 2 [ g 1 ( x , y ) * δ ( x a ) ] exp ( j 2 π c λ 1 t ) + c . c . + 1 2 [ g 2 ( x , y ) * δ ( x + a ) ] exp ( j 2 π c λ 2 t ) + c . c .
E i ( ω x , ω y ; t ) = 1 2 [ G 1 ( ω x , ω y ) exp ( j ω x a ) ] exp ( j 2 π c λ 1 t ) + c . c . + 1 2 G 2 ( ω x , ω y ) exp ( + j ω x a ) exp ( j 2 π c λ 2 t ) + c . c .
E 0 ( ω x , ω y ; t ) = α [ E i ( ω x , ω y ; t ) ] 2 = 1 2 [ G 1 ( ω x , ω y ) G 1 * ( ω x , ω y ) + G 2 ( ω x , ω y ) G 2 * ( ω x , ω y ) ] + 1 2 exp [ j 2 π ( c λ 1 c λ 2 ) t ] { G 1 ( ω x , ω y ) G 2 * ( ω x , ω y ) × exp [ j 2 π ( λ 2 + λ 1 λ 1 λ 2 ) x 0 f l a ] } + 1 2 exp [ j 2 π ( c λ 1 c λ 2 ) t ] { G 1 * ( ω x , ω y ) G 2 ( ω x , ω y ) × exp [ + j 2 π ( λ 2 + λ 1 λ 1 λ 2 ) x 0 f l a ] } + 1 4 exp [ j 2 π ( 2 c λ 1 ) t ] { G 1 ( ω x , ω y ) G 1 ( ω x , ω y ) × exp [ j 2 π ( 2 λ 1 ) x 0 f l a ] } + 1 4 exp [ j 2 π ( 2 c λ 1 ) t ] { G 1 * ( ω x , ω y ) G 1 * ( ω x , ω y ) × exp [ + j 2 π ( 2 λ 1 ) x 0 f l a ] } + 1 4 exp [ j 2 π ( 2 c λ 2 ) t ] { G 2 ( ω x , ω y ) G 2 ( ω x , ω y ) × exp [ + j 2 π ( 2 λ 2 ) x 0 f l a ] } + 1 4 exp [ j 2 π ( 2 c λ 2 ) t ] { G 2 * ( ω x , ω y ) G 2 * ( ω x , ω y ) × exp [ j 2 π ( 2 π λ 2 ) x 0 f l a ] } + 1 2 exp [ j 2 π ( c λ 1 + c λ 2 ) t ] [ G 1 ( ω x , ω y ) G 2 ( ω x , ω y ) ] + 1 2 exp [ j 2 π ( c λ 1 + c λ 2 ) t ] [ G 1 * ( ω x , ω y ) G 2 * ( ω x , ω y ) ] .
E 0 ( x , y ; f ) = F 1 [ α E 0 ( ω x , ω y ; t ) ] = 1 2 [ g 1 ( x , y ) * g 1 * ( x , y ) + g 2 ( x , y ) * g 2 * ( x , y ) ] + 1 2 δ ( f + f 1 f 2 ) [ g 1 ( x , y ) * g 2 * ( x , y ) * δ ( x a ) ] + 1 2 δ [ f ( f 1 f 2 ) ] [ g 1 * ( x , y ) * g 2 ( x , y ) * δ ( x + a ) ] + 1 4 δ ( f + 2 f 1 ) [ g 1 ( x , y ) * g 1 ( x , y ) * δ ( x a ) ] + 1 4 δ ( f 2 f 1 ) [ g 1 * ( x , y ) * g 1 * ( x , y ) * δ ( x + a ) ] + 1 4 δ ( f + 2 f 2 ) [ g 2 ( x , y ) * g 2 ( x , y ) * δ ( x + a ) ] + 1 4 δ ( f 2 f 2 ) [ g 2 * ( x , y ) * g 2 * ( x , y ) * δ ( x a ) ] + 1 2 δ ( f + f 1 + f 2 ) [ g 1 ( x , y ) * g 2 ( x , y ) ] + 1 2 δ [ f ( f 1 + f 2 ) ] [ g 1 * ( x , y ) * g 2 * ( x , y ) ] .
E 0 ( x , y ; f ) = 1 2 [ R 11 + R 22 ] + 1 2 [ R 12 * δ ( x a ) ] δ ( f + f 1 f 2 ) + 1 2 [ R 21 * δ ( x + a ) ] δ [ f ( f 1 f 2 ) ] + 1 4 [ C 11 * δ ( x a ) ] δ ( f + 2 f 1 ) + c . c . + 1 4 [ C 22 * δ ( x + a ) ] δ ( f + 2 f 2 ) + c . c . + 1 2 ( C 12 ) δ [ f + ( f 1 + f 2 ) ] + c . c .
P peak / cm 2 = E 0 π τ p ( 2 a 2 f 1 1.22 λ d 1 f 2 f 3 ) 2 < 50 MW / cm 2 ,
P avg / cm 2 ( τ p τ d ) P peak / cm 2 < 10 W / cm 2 .

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