Abstract

The analogy between electronic filters or networks and optical filters placed in the focal plane of an imaging arrangement has drawn attention to the potentialities of optical arrangements for information processing. This paper develops the theory of the filter in the focal plane whose transmission at each point is proportional to the intensity of the light from the object. This intensity filter—in imaging arrangements using coherent light sources—has the property of forming a “ghost image” of the whole object, when only a small fragment of the original object is shown. This ghost image can be utilized in principle for recognizing and locating a small fragment in the original object.

© 1963 Optical Society of America

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References

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  1. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, I.R.E. Trans. IT-6, 386 (1960).
  2. A. Maréchal, P. Croce, K. Dietsch, Opt. Acta 5, 25 (1958).
    [CrossRef]
  3. D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. A197, 454 (1949); Proc. Phys. Soc. B64, 449 (1951).
    [CrossRef] [PubMed]
  4. F. Seitz, Revs. Modern Phys. 26, 7 (1954).
    [CrossRef]
  5. R. Newman, W. W. Tyler, Solid State Phys. 8, 49 (1959).
    [CrossRef]

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, I.R.E. Trans. IT-6, 386 (1960).

1959 (1)

R. Newman, W. W. Tyler, Solid State Phys. 8, 49 (1959).
[CrossRef]

1958 (1)

A. Maréchal, P. Croce, K. Dietsch, Opt. Acta 5, 25 (1958).
[CrossRef]

1954 (1)

F. Seitz, Revs. Modern Phys. 26, 7 (1954).
[CrossRef]

1948 (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. A197, 454 (1949); Proc. Phys. Soc. B64, 449 (1951).
[CrossRef] [PubMed]

Croce, P.

A. Maréchal, P. Croce, K. Dietsch, Opt. Acta 5, 25 (1958).
[CrossRef]

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, I.R.E. Trans. IT-6, 386 (1960).

Dietsch, K.

A. Maréchal, P. Croce, K. Dietsch, Opt. Acta 5, 25 (1958).
[CrossRef]

Gabor, D.

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. A197, 454 (1949); Proc. Phys. Soc. B64, 449 (1951).
[CrossRef] [PubMed]

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, I.R.E. Trans. IT-6, 386 (1960).

Maréchal, A.

A. Maréchal, P. Croce, K. Dietsch, Opt. Acta 5, 25 (1958).
[CrossRef]

Newman, R.

R. Newman, W. W. Tyler, Solid State Phys. 8, 49 (1959).
[CrossRef]

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, I.R.E. Trans. IT-6, 386 (1960).

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, I.R.E. Trans. IT-6, 386 (1960).

Seitz, F.

F. Seitz, Revs. Modern Phys. 26, 7 (1954).
[CrossRef]

Tyler, W. W.

R. Newman, W. W. Tyler, Solid State Phys. 8, 49 (1959).
[CrossRef]

I.R.E. Trans. (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, I.R.E. Trans. IT-6, 386 (1960).

Nature (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. A197, 454 (1949); Proc. Phys. Soc. B64, 449 (1951).
[CrossRef] [PubMed]

Opt. Acta (1)

A. Maréchal, P. Croce, K. Dietsch, Opt. Acta 5, 25 (1958).
[CrossRef]

Revs. Modern Phys. (1)

F. Seitz, Revs. Modern Phys. 26, 7 (1954).
[CrossRef]

Solid State Phys. (1)

R. Newman, W. W. Tyler, Solid State Phys. 8, 49 (1959).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Optical filter analogous to electronic filter. The filter in F modifies each spatial Fourier component A(kl) independently.

Fig. 2
Fig. 2

Optical filter equivalent to Fig. 1, but symmetrical. Notice the reversal of the coordinate axes in the (x2y2) plane to simplify the computation.

Fig. 3
Fig. 3

Redundant information (filter A) for recognizing and partly locating the test object. The difference in photocurrents in P′ and P″ determines the phase of the square wave generated by the ghost image of A.

Equations (27)

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C ( x y ) = f ( x y ) g ( x + x , y + y ) d x d y
f ( x y ) = 1 2 π - + A ( k l ) exp [ i ( k x + l y ) ] d k d l .
A ( k l ) = 1 2 π - + f ( x y ) exp [ - i ( k x + l y ) ] d x d y ,
f ( x 0 y 0 ) = ( 1 2 π ) 2 - + f ( x y ) exp [ i k ( x 0 - x ) + i l ( y 0 - y ) ] d k d l d x d y .
g ( x y ) = 1 2 π - + B ( k l ) exp [ i ( k x + l y ) ] d k d l ,
C ( x y ) = ( 1 2 π ) 2 - + A ( k l ) B ( k l ) exp [ i ( k x + l y ) + i k ( x + x ) + i l ( y + y ) ] d x d y d k d l d k d l
C ( x y ) = 1 2 π - + A ( - k , - l ) B ( k l ) exp [ i ( k x + l y ) ] d x d l .
f ( x y ) exp ( i m 0 z - i ω 0 t ) + ½ U 0 exp ( i m 0 z - i ω 0 t ) .
f ( x y ) = 1 2 π - + A ( k , l ) exp i ( k x + l y ) d k d l ,
D ( x 2 y 2 ) = 1 2 π - + A ( k l ) A * ( k l ) B ( k l ) exp [ i ( k x 2 + l y 2 ) ] d k d l .
D ( x 2 y 2 ) = ( 1 2 π ) 2 - + f ( x y ) f * ( x y ) g ( x , y ) exp [ i k ( x 2 - x + x - x ) + i l ( y 2 - y + y - y ) ] d k d l d x d y d x d y d x d y .
D ( x 2 y 2 ) = ( 1 2 π ) 2 - + f ( x y ) f * ( x y ) g ( x 2 - x + x , y 2 - y + y ) d x d y d x d y .
D ( x 2 y 2 ) = ( 1 2 π ) 2 - + f ( x y ) f ( x y ) g ( x 2 - x + x , y 2 - y + y ) Δ x Δ y Δ x Δ y .
D 1 ( x 2 y 2 ) = ( 1 2 π ) 2 - + f 2 ( x y ) g ( x 2 ) ( Δ x ) 2 ( Δ y ) 2 ,
D 1 ( x 2 y 2 ) = C 1 g ( x 2 y 2 ) ,
C 1 = ( 1 2 π ) 2 - + f 2 ( x y ) Δ x Δ y .
D 2 ( x 2 y 2 ) = ( 1 2 π ) 2 - + f ( x 2 - x 0 , y 2 - y 0 ) f ( x y ) g ( x + x 0 , y + y 0 ) Δ x Δ y .
D 2 ( x 2 y 2 ) = C 2 f ( x 2 - x 0 , y 2 - y 0 ) ,
C 2 = ( 1 2 π ) 2 - + g 2 ( x + x 0 , y + y 0 ) ( Δ x ) 2 ( Δ y ) 2 .
f ( x - δ , y - ) f ( x + δ , y + ) = f 0 2 exp [ - 1 α 0 2 ( δ 2 + 2 ) ] ,
K 2 = [ D ( x 2 y 2 ) - d 2 f ( x 2 - x 0 , y 2 - y 0 ) ] 2 ,
K 2 = ( 1 2 π ) 4 - + f ( x y ) f ( x y ) g ( x 2 - x + x , y 2 - y + y ) × f ( x , y ) f ( x , y ) g ( x 2 - x + x , y 2 - y + y ) × d x d y d x d y d x d y d x d y - 2 d 2 ( 1 2 π ) 2 - + × f ( x ' y ) f ( x y ) g ( x 2 - x + x , y 2 - y + y ) f ( x 2 - x 0 , y 2 - y 0 ) d x d y d x d y + d 2 2 f 2 ( x 2 - x 0 , y 2 - y 0 ) .
K 2 = ( 1 2 π ) 4 f 0 6 π 2 3 α 0 4 d 2 l 2 - ( 1 2 π ) 2 2 d 2 f 0 4 2 π α 0 2 d 2 + d 2 2 f 0 2 .
( d 2 ) min = ( 1 2 π ) 2 2 π f 0 2 α 0 2 d 2 .
S = ( 1 2 π ) 2 2 π f 0 3 α 0 2 d 2 .
N = ( 1 2 π ) f 0 3 π 3 α 0 2 d l ,
S N = 2 3 d l .

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