Abstract

We present a general analytical treatment of optical correlation in correlators that use pixellated spatial light modulators with transmissive (or reflective) dead zones in both the input and filter planes. The active areas of the pixels modulate the light intensity while the dead zones transmit (or reflect) all of the light. Our model can predict the changes in the correlation peak and the signal-to-noise ratio with changes in dead zones, calculated in Part I [Appl. Opt. 32, 6527 (1993)] from computer simulations. This model is also a general one: It applies to correlators in which one spatial light modulator contaions only opaque dead zones while the other contains only transmissive dead zones; it also applies to the case in which any one spatial light modulator contains both opaque and transmissive dead zones.

© 1993 Optical Society of America

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References

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  1. P. D. Gianino, C. L. Woods, “Effects of spatial light modulator opaque dead zones on optical correlation,” Appl. Opt. 31, 4025–4033 (1992).
    [CrossRef] [PubMed]
  2. P. D. Gianino, C. L. Woods, this issue, “General treatment of spatial light modulator dead-zone effects on optical correlation. Part I. Computer simulations,” Appl. Opt. 32, 6527–6535 (1993).
    [CrossRef] [PubMed]
  3. J. A. David, E. A. Merrill, D. M. Cottrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094–1100 (1990).
    [CrossRef]

1993

1992

1990

J. A. David, E. A. Merrill, D. M. Cottrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Bunch, R. M.

J. A. David, E. A. Merrill, D. M. Cottrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Cottrell, D. M.

J. A. David, E. A. Merrill, D. M. Cottrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

David, J. A.

J. A. David, E. A. Merrill, D. M. Cottrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Gianino, P. D.

Merrill, E. A.

J. A. David, E. A. Merrill, D. M. Cottrell, R. M. Bunch, “Effects of sampling and binarization in the output of the joint Fourier transform correlator,” Opt. Eng. 29, 1094–1100 (1990).
[CrossRef]

Woods, C. L.

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Figures (1)

Fig. 1
Fig. 1

Block diagram of the correlator showing the different components and signals involved in the correlation process. The various signals are explained in the text.

Equations (42)

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k T = ( w + t s ) ( g + t F ) t dc ,
v x rect ( x / b x ) / b x ,
q x sinc ( x / c x ) / c x ,
m x sinc ( e x x / h x l x ) ,
r x rect ( x / L x ) ,
sinc ( α ) ( sin π α ) / π α .
V ξ FT ( v x ) sinc ( b x ξ 2 ) ,
Q ξ FT ( q x ) rect ( c x ξ 2 ) ,
M ξ ( e x / h x L x ) FT ( m x ) rect ( h x L x ξ 2 / e x ) ,
R ξ F T ( r x ) L x sinc ( L x ξ 2 ) .
w ( x 1 , y 1 ) = { [ n x = - n y = - δ ( x 1 - n x c x ) δ ( y 1 - n y c y ) s ( x 1 , y 1 ) ] ( b x v x ) ( b y v y ) } r x r y .
W ( ξ 2 , η 2 ) = ( 1 - Z 1 ) { [ S ( ξ 2 , η 2 ) ( V ξ V η ) ] R ξ R η } Q ξ Q η ,
w ( x 3 , y 3 ) = ( 1 - Z 1 ) { [ s ( x 3 , y 3 ) v x v y ] r x r y } q x q y .
D 1 ( x 1 , y 1 ) = rect ( x 1 c x ) rect ( y 1 c y ) - ( b x v x ) ( b y v y ) .
t s ( x 1 , y 1 ) = { [ n x = - n y = - δ ( x 1 - n x c x ) δ ( y 1 - n y c y ) ] D 1 ( x 1 , y 1 ) } r x r y
T s ( ξ 2 , η 2 ) = ( Z 1 R ξ R η ) ( Q ξ Q η ) ,
t s ( x 3 , y 3 ) = Z 1 ( r x r y q x q y ) .
G ( ξ 2 , η 2 ) = { [ n x = - n y = - δ ( ξ 2 - n x / L x ) δ ( η 2 - n y / L y ) × F * ( ξ 2 , η 2 ) ] M ξ M η } Q ξ Q η .
g ( x 3 , y 3 ) = ( 1 - Z 2 ) [ n x = - n y = - f * ( - x 3 + n x L x , - y 3 + n y L y ) m x m y ] q x q y ,
D 2 ( x 2 , y 2 ) = rect ( x 2 h x ) rect ( y 2 h y ) - rect ( x 2 e x ) rect ( y 2 e y )
D 2 ( ξ 2 , η 2 ) = rect ( L x ξ 2 ) rect ( L y η 2 ) - M ξ M η .
T F ( ξ 2 , η 2 ) = { [ n x = - n y = - δ ( ξ 2 - n x / L x ) δ ( η 2 - n y / L y ) ] D 2 ( ξ 2 , η 2 ) } Q ξ Q η .
t F ( x 3 , y 3 ) = Z 2 q x q y - Σ Σ ( 1 - Z 2 ) sinc ( n x e x h x ) sinc ( n y e y h y ) × 1 c x sinc ( x 3 - n x L x c x ) 1 c y sinc ( y 3 - n y L y c y ) ,
T dc ( x 2 , y 2 ) 1 - rect ( x 2 p x h x ) rect ( y 2 p y h y ) .
T dc ( ξ 2 , η 2 ) = 1 - rect ( L x ξ 2 p x ) rect ( L y η 2 p y )
t dc ( x 3 , y 3 ) = δ ( x 3 , y 3 ) - p x p y L x L y sinc ( p x x 3 L x ) sinc ( p y y 3 L y ) .
k T = k c + k s + k R + k FR ,
k c ( x 3 , y 3 ) F T - 1 ( W G T dc ) = [ w ( x 3 , y 3 ) g ( x 3 , y 3 ) ] t dc ( x 3 , y 3 ) ,
k s ( x 3 , y 3 ) FT - 1 ( W T F T dc ) = [ w ( x 3 , y 3 ) t F ( x 3 , y 3 ) ] t dc ( x 3 , y 3 ) ,
k R ( x 3 , y 3 ) FT - 1 ( T s T F T dc ) = [ t s ( x 3 , y 3 ) t F ( x 3 , y 3 ) ] t dc ( x 3 , y 3 ) ,
k FR ( x 3 , y 3 ) FT - 1 ( T s G T dc ) = [ t s ( x 3 , y 3 ) g ( x 3 , y 3 ) ] t dc ( x 3 , y 3 ) .
k c ( x 3 , y 3 ) = ( 1 - Z 1 ) ( 1 - Z 2 ) ( { [ s ( x 3 , y 3 ) v x v y ] r x r y } [ f * ( - x 3 , - y 3 ) m x m y ] q x q y ) t dc ,
k s ( x 3 , y 3 ) = ( 1 - Z 1 ) Z 2 ( { [ s ( x 3 , y 3 ) v x v y ] r x r x } q x q y ) t dc Z 2 w ( x 3 , y 3 ) t dc ,
k R ( x 3 , y 3 ) = Z 1 Z 2 ( r x r y q x q y ) t dc Z 2 t s ( x 3 , y 3 ) t dc ,
k FR ( x 3 , y 3 ) = Z 1 ( 1 - Z 2 ) ( { r x r y [ f * ( - x 3 , - y 3 ) m x m y ] } q x q y ) t dc .
k T 2 = ( 1 - Z 1 ) ( 1 - Z 2 ) A c + ( 1 - Z 1 ) Z 2 A s + Z 1 Z 2 A R + Z 1 ( 1 - Z 2 ) A FR 2 ,
k T 2 = ( A Z ) 1 ( A Z ) 2 A c + ( A Z ) 1 Z 2 A s + Z 1 Z 2 A R + Z 1 ( A Z ) 2 A FR 2 .
I p ( Z 2 ) I p ( 0 ) = 1 - ( 1 - A s / A c ) Z 2 2 ,
SNR ( Case C ) = ( A c ) p ( j { A c + [ Z 2 / ( 1 - Z 2 ) ] A s } j 2 ) 1 / 2 / n T ,
SNR ( Case C ) SNR ( Case B ) = [ j { A c + [ Z 1 / ( 1 - Z 1 ) ] A FR } j 2 ] 1 / 2 [ j { A c + [ Z 2 / ( 1 - Z 2 ) ] A s } j 2 ] 1 / 2 .
k T 2 ( ODZ , TDZ ) = ( AZ ) 1 2 ( 1 - Z 2 ) A c + Z 2 A s 2 = ( AZ ) 1 2 [ k T 2 ( Case C ) ] .
k T 2 ( TDZ , ODZ ) = ( AZ ) 2 2 ( 1 - Z 1 ) A c + Z 1 A FR 2 = ( AZ ) 2 2 [ k T 2 ( Case B ) ] .

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