Abstract

The utilization of coherent optical correlation techniques in the performance of matrix multiplication by optical analog methods has been investigated mathematically. Although the basic concepts have been known for some time, we have not been able to find explicit analyses in the existing literature. Since many correlations other than those corresponding to the desired matrix multiplication terms exist, methods of isolating the desired from the undesired terms are presented. Various spatial configurations for both the input and output arrays are discussed. For the special cases of 2 × 2 matrices the analyses are presented in greater detail. Using simple circ function distributions, the effects of finite-sized array elements and detector apertures are investigated.

© 1970 Optical Society of America

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  1. The following texts serve as general references for this material: J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968); A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968); A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962). Matrix multiplication is mentioned briefly on pp. 97 and 98 of the article by L. J. Cutrona, in Optical and Electrooptical Information Processing, J. T. Tippett et al., Eds. (Mass. Inst. of Technol. Press, Cambridge, 1965).
  2. This condition on y1and ŷ1is the most general and may be over-restrictive in any particular example. For any case where the order of the matrices is not infinity much less severe restrictions are imposed. For example, for 2 × 2 and 3 × 3 matrices we require only y1≠ ŷ1.
  3. Analyses equivalent to some of this material can be found, for example, on pp. 116–120 of Goodman (Ref. 1) and on pp. 32–35 of Papoulis (Systems and Transforms with Applications in Optics).
  4. C. B. Burckhardt, J. Opt. Soc. Amer. 59, 1544A (1969).

1969

C. B. Burckhardt, J. Opt. Soc. Amer. 59, 1544A (1969).

Burckhardt, C. B.

C. B. Burckhardt, J. Opt. Soc. Amer. 59, 1544A (1969).

Goodman,

Analyses equivalent to some of this material can be found, for example, on pp. 116–120 of Goodman (Ref. 1) and on pp. 32–35 of Papoulis (Systems and Transforms with Applications in Optics).

Goodman, J. W.

The following texts serve as general references for this material: J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968); A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968); A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962). Matrix multiplication is mentioned briefly on pp. 97 and 98 of the article by L. J. Cutrona, in Optical and Electrooptical Information Processing, J. T. Tippett et al., Eds. (Mass. Inst. of Technol. Press, Cambridge, 1965).

Papoulis,

Analyses equivalent to some of this material can be found, for example, on pp. 116–120 of Goodman (Ref. 1) and on pp. 32–35 of Papoulis (Systems and Transforms with Applications in Optics).

J. Opt. Soc. Amer.

C. B. Burckhardt, J. Opt. Soc. Amer. 59, 1544A (1969).

Other

The following texts serve as general references for this material: J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968); A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968); A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962). Matrix multiplication is mentioned briefly on pp. 97 and 98 of the article by L. J. Cutrona, in Optical and Electrooptical Information Processing, J. T. Tippett et al., Eds. (Mass. Inst. of Technol. Press, Cambridge, 1965).

This condition on y1and ŷ1is the most general and may be over-restrictive in any particular example. For any case where the order of the matrices is not infinity much less severe restrictions are imposed. For example, for 2 × 2 and 3 × 3 matrices we require only y1≠ ŷ1.

Analyses equivalent to some of this material can be found, for example, on pp. 116–120 of Goodman (Ref. 1) and on pp. 32–35 of Papoulis (Systems and Transforms with Applications in Optics).

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Figures (14)

Fig. 1
Fig. 1

Coherent optical detection system. The mathematical derivations in the text for simplicity assume all the focal lengths to be the same: f = f1 = f2.

Fig. 2
Fig. 2

Optical configuration of a matrix.

Fig. 3
Fig. 3

Optical configuration of b matrix.

Fig. 4
Fig. 4

Positions of the elements of the product matrix c = b × a.

Fig. 5
Fig. 5

Optical configurations for obtaining the product matrix c = a × b.

Fig. 6
Fig. 6

Separation of the rows and columns of the product matrix.

Fig. 7
Fig. 7

Optical configurations which yield the product matrix c = b × a in conventional mathematical form.

Fig. 8
Fig. 8

Optical configurations which yield the product matrix c = a × b in conventional mathematical form.

Fig. 9
Fig. 9

Optical configurations for operation of a matrix upon a vector.

Fig. 10
Fig. 10

Optical configurations for multiplication of 3 × 3 matrices.

Fig. 11
Fig. 11

Geometrical construction equivalent to the cross correlation of two circ functions.

Fig. 12
Fig. 12

Intensity distributions corresponding to the double and triple circ cross correlations.

Fig. 13
Fig. 13

Average intensity of the cross correlation outputs as a function of detector radius.

Fig. 14
Fig. 14

Total detector response as a function of detector radius.

Equations (45)

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f 2 ( x , y ) f 1 * ( x , y ) d x d y .
g ( x 0 , y 0 ) = f 2 ( x , y ) f 1 * ( x x 0 , y y 0 ) d x d y .
F { g ( x 0 , y 0 ) } = F 2 ( ξ 0 , η 0 ) F 1 * ( ξ 0 , η 0 ) ,
F { f 2 ( x , y ) } = F 2 ( ξ , η ) ,
F { f 1 * ( x , y ) } = F 1 * ( ξ , η ) .
F [ F { g ( x 0 , y 0 ) } ] g ( x 0 , y 0 ) = F { F 2 ( ξ 0 , η 0 ) F 1 * ( ξ 0 , η 0 ) } .
g ( 0,0 ) = F { F 2 ( ξ 0 , η 0 ) F 1 * ( ξ 0 , η 0 ) } x 0 = 0 , y 0 = 0 .
c l n = m b l m a m n .
f ( x , y ) = F ( ξ , η ) exp [ i 2 π ( ξ x + η y ) ] d ξ d η , F ( ξ , η ) = f ( x , y ) exp [ i 2 π ( ξ x + η y ) ] d x d y .
f ( x x 0 ) F ( ξ ) exp ( i 2 π x 0 ξ ) ;
f ( a x , b y ) ( 1 / | a b | ) F [ ( ξ / a ) , ( η / b ) ] ;
δ ( x x 0 ) δ ( y y 0 ) exp [ i 2 π ( ξ x 0 + η y 0 ) ] ;
f ( x 1 , y 1 ) A F ( x 2 / λ f , y 2 / λ f ) with A = ( 1 / λ f ) e i α ,
U 1 = a 11 δ ( x + x 1 + Δ x ) δ ( y y 1 ) + a 21 δ ( x x 1 + Δ x ) δ ( y y 1 ) + a 12 δ ( x + x 1 + Δ x ) δ ( y + y 1 ) + a 22 δ ( x x 1 + Δ x ) δ ( y + y 1 ) + r δ ( x ) δ ( y ) .
V 1 ( x 2 , y 2 ) = ( a 11 / λ f ) exp { i ( 2 π / λ f ) [ x 2 ( x 1 + Δ x ) + y 2 y 1 ] } + ( a 21 / λ f ) exp { i ( 2 π / λ f ) [ x 2 ( x 1 Δ x ) + y 2 y 1 ] } + ( a 12 / λ f ) exp { i ( 2 π / λ f ) [ x 2 ( x 1 + Δ x ) y 2 y 1 ] } + ( a 22 / λ f ) exp { i ( 2 π / λ f ) [ x 2 ( x 1 Δ x ) y 2 y 1 ] } + r / λ f .
ϕ 11 = β ( x 1 + Δ x ) + γ y 1 , ϕ 21 = β ( x 1 Δ x ) + γ y 1 , ϕ 12 = β ( x 1 + Δ x ) γ y 1 , ϕ 22 = β ( x 1 Δ x ) γ y 1 ,
V 1 = a 11 λ f exp ( i ϕ 11 ) + a 21 λ f exp ( i ϕ 21 ) + a 12 λ f exp ( i ϕ 12 ) + a 22 λ f exp ( i ϕ 22 ) + r λ f .
t = r a 11 ( λ f ) 2 exp ( i ϕ 11 ) + r a 21 ( λ f ) 2 exp ( i ϕ 21 ) + r a 12 ( λ f ) 2 exp ( i ϕ 12 ) + r a 22 ( λ f ) 2 exp ( i ϕ 22 ) + r a 11 * ( λ f ) 2 exp ( i ϕ 11 ) + r a 21 * ( λ f ) 2 exp ( i ϕ 21 ) + r a 12 * ( λ f ) 2 exp ( i ϕ 12 ) + r a 22 * ( λ f ) 2 exp ( i ϕ 22 ) + r 2 ( λ f ) 2 ,
U 2 = b 11 δ ( x + x ˆ 1 + Δ x ˆ ) δ ( y y ˆ 1 ) + b 12 δ ( x x ˆ 1 + Δ x ˆ ) δ ( y y ˆ 1 ) + b 21 δ ( x + x ˆ 1 + Δ x ˆ ) × δ ( y + y ˆ 1 ) + b 22 δ ( x x ˆ 1 + Δ x ˆ ) δ ( y + y ˆ 1 ) .
V 2 = b 11 λ f exp ( i ϕ ˆ 11 ) + b 12 λ f exp ( i ϕ ˆ 12 ) + b 21 λ f exp ( i ϕ ˆ 21 ) + b 22 λ f exp ( ϕ ˆ 22 ) ,
ϕ ˆ 11 = β ( x ˆ 1 + Δ x ˆ ) + γ y ˆ 1 , ϕ ˆ 12 = β ( x ˆ 1 Δ x ˆ ) + γ y ˆ 1 , ϕ ˆ 21 = β ( x ˆ 1 + Δ x ˆ ) γ y ˆ 1 , ϕ ˆ 22 = β ( x ˆ 1 Δ x ˆ ) γ y ˆ 1 .
V 3 = t V 2 = [ r b 11 a 11 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 11 ϕ 11 ) ] + [ r b 11 a 12 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 11 ϕ 12 ) ] + [ r b 11 a 21 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 11 ϕ 21 ) ] + [ r b 11 a 22 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 11 ϕ 22 ) ] + [ r b 12 a 11 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 12 ϕ 11 ) ] + [ r b 12 a 21 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 12 ϕ 21 ) ] + [ r b 12 a 12 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 12 ϕ 12 ) ] + [ r b 12 a 22 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 12 ϕ 22 ) ] + [ r b 21 a 11 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 21 ϕ 11 ) ] + [ r b 21 a 21 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 21 ϕ 21 ) ] + [ r b 21 a 12 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 21 ϕ 12 ) ] + [ r b 21 a 22 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 21 ϕ 22 ) ] + [ r b 22 a 11 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 22 ϕ 11 ) ] + [ r b 22 a 21 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 22 ϕ 21 ) ] + [ r b 22 a 12 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 22 ϕ 12 ) ] + [ r b 22 a 22 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 22 ϕ 22 ) ]
[ r b 11 a 11 * / ( λ f ) 3 ] exp [ i ( ϕ ˆ 11 ϕ 11 ) ] = A exp [ i ( β x ˆ 1 β Δ x ˆ + β x 1 + β Δ x + γ y ˆ 1 γ y 1 ) ] = A exp [ i β ( x ˆ 1 + x 1 Δ x ˆ + Δ x ) ] exp [ i γ ( y ˆ 1 y 1 ) ] ,
= A λ f δ ( x 3 + x ˆ 1 x 1 + Δ x ˆ Δ x ) δ ( y 3 y ˆ 1 + y 1 ) = [ r b 11 a 11 * / ( λ f ) 2 ] δ ( x 3 + x ˆ 1 x 1 + Δ x ˆ Δ x ) δ ( y 3 y ˆ 1 + y 1 ) .
c 11 = b 11 a 11 + b 12 a 21 ( 0 , y ˆ 1 + y 1 ) c 12 = b 11 a 12 + b 12 a 22 ( 0 , y ˆ 1 y 1 ) c 21 = b 21 a 11 + b 22 a 21 ( 0 , y ˆ 1 + y 1 ) c 22 = b 21 a 12 + b 22 a 22 ( 0 , y ˆ 1 y 1 ) .
c l n = m b l m a m n .
R p = 0 , A ( except R ) B } p = ± 1.
q = 0, 1, 2 ( reference to left of matrix center ) , q = 0 , 1 , 2 ( reference to right of matrix center ) .
s q .
x 1 = x ˆ 1 ( y 1 / y ˆ 1 ) ± 1 integer , Δ x = Δ x ˆ 0 < < 1 , 1 2 .
ϕ 11 = β ( x 1 Δ x ) ϕ 12 = β ( x 1 ) ϕ 21 = β ( x 1 Δ x ) ϕ 22 = β ( x 1 ) ,
ϕ ˆ 11 = β ( x ˆ 1 Δ x ˆ ) + γ ( y ˆ 1 ) ϕ ˆ 12 = β ( x ˆ 1 Δ x ˆ ) + γ ( y ˆ 1 ) ϕ ˆ 21 = β ( x ˆ 1 Δ x ˆ ) + γ ( y ˆ 1 ) ϕ ˆ 22 = β ( x ˆ 1 Δ x ˆ ) + γ ( y ˆ 1 ) .
c 11 at ( 0 , y ˆ 1 ) c 12 at ( Δ x ˆ , y ˆ 1 ) c 21 at ( 0 , y ˆ 1 ) c 22 at ( Δ x ˆ , y ˆ 1 ) .
c = b × a ( c = m b l m a m n ) ,
c 1 = b 11 a 1 + b 12 a 2 ( 0 , y ˆ 1 + y 1 ) c 2 = b 21 a 1 + b 22 a 2 ( 0 , y ˆ 1 + y 1 ) .
c = m b m a m
c = b 1 a 1 + b 2 a 2 ( 0 , y ˆ 1 + y 1 ) .
circ ( r ) = { b 0 r a 0 r > a r = ( x 2 + y 2 ) 1 2 F { circ ( r ) } = a J 1 ( 2 π a ρ ) ρ ,
C 1 ( s ) = { 2 a b c [ a cos 1 ( s / 2 a ) 1 2 s ( 1 s 2 / 4 a 2 ) 1 2 ] 0 s 2 a 0 s > 2 a ,
C 2 ( s ) = 0 p δ ( s 1 s ) C 1 ( s 1 ) d s 1 = p C 1 ( s ) ,
C 2 ( s ) = { 2 a b c p [ a cos 1 ( s / 2 a ) 1 2 s ( 1 s 2 / 4 a 2 ) 1 2 ] 0 s 2 a 0 s > 2 a .
C 3 ( s ) = 0 p ( s 1 s ) C 1 ( s 1 ) d s 1 .
T D R = 0 d I ( s ) 2 π s d s .
h ( x 0 , y 0 ) = f 2 ( x , y ) f 1 ( x 0 x , y 0 y ) d x d y . F { h ( x 0 , y 0 ) } = F 2 ( ξ 0 , η 0 ) F 1 ( ξ 0 , η 0 ) , h ( x 0 , y 0 ) = F { F 2 ( ξ 0 , η 0 ) F 1 ( ξ 0 , η 0 ) } , h ( 0,0 ) = F { F 2 ( ξ 0 , η 0 ) F 1 ( ξ 0 , η 0 ) } x 0 = 0 , y 0 = 0 .
h ( 0,0 ) = f 2 ( x , y ) f 1 ( x , y ) d x d y ,

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