Abstract

Coherent optical systems are well known for performing certain operations such as Fourier and Fresnel transformation and pattern recognition. The basis for pattern recognition in many coherent optical systems is the correlator. This paper discusses the effect of undersampling in an optical joint transform correlator which has a sampled-data input, detectors of size 1 × i resolution elements in the Fourier plane, and detectors of size 1 × 1 resolution elements in the correlation plane, where i is an integer. After the undersampling, it is shown that the correlation plane still contains the needed data, the shape of the correlation surfaces are unchanged, and the different correlation terms just move closer together. It is shown that the penalty for undersampling is that the amount of input data must be reduced by a factor of i. It is also shown that a digital simulation using a discrete Fourier transform gives an accurate prediction of the optical correlator performance. Results from the simulation are given to verify the theory.

© 1981 Optical Society of America

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References

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  1. A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

1979 (1)

D. Casasent, Proc. IEEE 67, 813 (1979).
[CrossRef]

1978 (1)

1977 (4)

J. W. Goodman, Proc. IEEE 65, 29 (1977).
[CrossRef]

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

L. Flores, D. Hecht, Proc. Soc. Photo-Opt. Instrum. Eng. 118, 182 (1977).

D. Casasent, A. Furman, Appl. Opt. 16, 285 (1977).
[CrossRef]

1970 (1)

1966 (1)

1964 (1)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

1962 (1)

Casasent, D.

Elliott, D. F.

D. F. Elliott, “Relationships of DFT Filter Shapes,” in Proceedings, Twenty-First Midwest Symposiums, Circuits and Systems, 14–15 Aug. 1978, (Iowa State U.), pp. 364–368.

Flores, L.

L. Flores, D. Hecht, Proc. Soc. Photo-Opt. Instrum. Eng. 118, 182 (1977).

Furman, A.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 65, 29 (1977).
[CrossRef]

Hecht, D.

L. Flores, D. Hecht, Proc. Soc. Photo-Opt. Instrum. Eng. 118, 182 (1977).

Leith, E. N.

Rau, J.

Upatnieks, J.

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Weaver, C. S.

Appl. Opt. (3)

IEEE Trans. Inf. Theory (1)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. IEEE (3)

J. W. Goodman, Proc. IEEE 65, 29 (1977).
[CrossRef]

D. Casasent, Proc. IEEE 67, 813 (1979).
[CrossRef]

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

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

L. Flores, D. Hecht, Proc. Soc. Photo-Opt. Instrum. Eng. 118, 182 (1977).

Other (1)

D. F. Elliott, “Relationships of DFT Filter Shapes,” in Proceedings, Twenty-First Midwest Symposiums, Circuits and Systems, 14–15 Aug. 1978, (Iowa State U.), pp. 364–368.

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Figures (8)

Fig. 1
Fig. 1

Pictorial representation of the JTC.

Fig. 2
Fig. 2

Block diagram of equivalent optical processing.

Fig. 3
Fig. 3

Plot of modulation function.

Fig. 4
Fig. 4

Integrating over several resolution elements in the Fourier plane collapses gaps between the correlation functions but does not collapse their size or shape.

Fig. 5
Fig. 5

Focal plane response of aperture limited sinusoidal input.

Fig. 6
Fig. 6

Response of DFT bins l − 1, l, and l + 1 to a spectral line.

Fig. 7
Fig. 7

Comparison of optical correlator and simulation outputs vs angle, Δω/2, for detectors of size (1 × i) resolution elements: (a) i = 1; (b) i = 3.

Fig. 8
Fig. 8

(a) Input functions starting at (0,0) and (0,342); (b) spectrum in Fourier plane integrated by detectors of size (1×3) resolution elements; (c) autocorrelation and cross-correlation surfaces demonstrating how proper positioning of input functions separates the correlation surfaces.

Equations (27)

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U ( x f , y f ) = t ( x 0 , y 0 ) exp [ j 2 π ( x 0 x f + y 0 y f ) λ f ] d x 0 d y 0 , U ( x f , y f ) 2 D [ t ( x 0 , y 0 ) ] ,
t ( x 0 , y 0 ) = n 1 = 0 M 1 n 2 = 0 N 1 exp [ jc ( n 1 , n 2 ) ] × WIND ( x 0 n 1 d x W x , y 0 n 2 d y W y ) ,
U ( x f , y f ) = Φ 1 ( d x , d y ) Φ 2 ( x f , y f , d x , d y ) 2 D WIND ( x 0 W x , y 0 W y ) , Φ 1 ( d x , d y ) = exp [ j π ( d x + d y ) λ f ] , Φ 2 ( x f , y f , d x , d y ) = n 1 = 0 M 1 n 2 = 0 N 1 exp [ jc ( n 1 , n 2 ) j 2 π × ( n 1 d x x f + n 2 d y y f λ f ) ] ,
WIND ( x 0 n 1 d x W x , y 0 n 2 d y W y ) ,
C ( k 1 , k 2 ) = n 1 = 0 M 1 n 2 = 0 N 1 c ( n 1 , n 2 ) W M n 1 k 1 W N n 2 k 2 , C ( k 1 , k 2 ) DFT 2 D [ c ( n 1 , n 2 ) ] ,
k 1 = 0 , 1 , , M 1 , k 2 = 0 , 1 , , N 1 , W M = exp ( j 2 π / M ) , W N = exp ( j 2 π / N ) , d x Δ x f λ f = 1 M , d y Δ y f λ f = 1 N .
U ( k 1 , k 2 ) U ( k 1 Δ x f , k 2 Δ y f ) = [ DFT 2 D c ( n 1 , n 2 ) ] × { phase term } ( 2 D { WIND [ ( x 0 n 1 d x W x ) , ( y 0 n 2 d y W y ) ] } ) .
C ( k ) = n = 0 N 1 c ( n ) W N k n .
C ̂ ( l ) = m = i 1 2 i 1 2 C ( il + m ) ,
( n ) = l = 0 N i 1 W N / i l n C ̂ ( l ) 2 = l = 0 N i 1 W N / i l n | m = i 1 2 i 1 2 C ( il + m ) | 2 , ( 9 )
c ( il + m ) = n = 0 N 1 c ( n ) W N n ( il + m ) ,
( n ) = i = 0 N i 1 W N / i l n | m = i 1 2 i 1 2 n = 0 N 1 c ( n ) W N n ( il + m ) | 2 C ̂ ( l ) .
C ̂ ( l ) = n 0 N 1 c ( n ) W N iln ( m = i 1 2 i 1 2 W N mn ) .
M i ( n ) = m = i 1 2 i 1 2 W N mn = 1 + 2 [ cos ( 2 π n N ) + cos ( 4 π n N ) + + cos ( 2 π i 1 2 ) n ] ,
m = 1 1 W N mn = 1 + 2 cos ( 2 π n N ) .
( n ) = i = 0 N i 1 W N / i l n | n = 0 N 1 c ( n ) W N iln M i ( n ) | 2 C ̂ ( l ) = DFT [ | C ̂ ( l ) 2 | ] , n = 0 , 1 , , ( N / i ) 1 .
n = 0 N 1 W N n ( k m ) = N δ km ,
( n ) = l = 0 N i 1 n = 0 N 1 a = 0 N 1 M i ( n ) M i * ( a ) c ( n ) c * ( a ) W N il n W N il ( n a ) = N i n = 0 N 1 c ( n ) c * ( n + n ) M i ( n ) M i * ( n + n ) . ( 16 )
gh ( n ) = n = 0 N i 1 g ( n ) h * ( n n ) .
{ rect [ y 0 ( N 1 ) d y / 2 N d y ] comb d y cos ( 2 π f y y 0 ) } = exp [ j π y f ( 1 1 / N ) / Δ y f ] sinc ( y f / Δ y f ) * rep N Δ y f [ ½ δ ( y f f y Δ y f ) + ½ δ ( y f + f y Δ y f ) ] ,
rect [ t P ] = { 1 | t | P / 2 0 otherwise ,
comb P = n = δ ( t nP ) ,
rep P [ c ( t ) ] = comb P * c ( t ) ,
{ exp [ j π y f ( 1 1 / N ) / Δ y f ] sin ( π y f / Δ y f ) / sin [ π y f / ( Δ y f N ) ] } * 1 / 2 [ δ ( y f f y Δ y f ) + δ ( y f + f y Δ y f ) ] .
D ( y f ) = k d i π 2 i π 2 A ( x ) dx ,
A ( x ) = exp [ j ( x + Δ ω / 2 ) ( 1 1 / N ) ] sin ( x + Δ ω / 2 ) sin [ ( x + Δ ω / 2 ) / N ] + exp [ j ( x Δ ω / 2 ) ( 1 1 / N ) ] sin ( x Δ ω / 2 ) sin [ ( x + 2 π l Δ ω / 2 ) / N ] ,
D ( k ) = m = 1 i 2 i + 1 2 A ( π m ) ,

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