Abstract

In a recent work, an operator algebra was developed for the description of axially symmetrical coherent optical systems. The present work extends the operator representation to incorporate cylindircal lenses and is used to analyze the transforming properties of an arbitrarily oriented cylindrical lens. The results provide the basis for synthesizing various systems for signal processing, such as convolution and correlation operations.

© 1979 Optical Society of America

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References

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  1. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra.” J. Opt. Soc. Am. (in print).
  2. A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
    [CrossRef]
  3. F. P. Carlson, R. E. Francois, Proc. IEEE 65, 10 (1977).
    [CrossRef]
  4. D. Psaltis, D. Casasent, Appl. Opt. 18, 163 (1979).
    [CrossRef] [PubMed]
  5. B. E. A. Saleh, Appl. Opt. 17, 3408 (1978).
    [CrossRef] [PubMed]
  6. R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl. Opt. 16, 739 (1977).
    [CrossRef]
  7. J. W. Goodman, P. Kellman, E. W. Hansen, Appl. Opt. 16, 733733 (1977).
    [CrossRef] [PubMed]

1979 (1)

1978 (1)

1977 (3)

1966 (1)

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
[CrossRef]

Carlson, F. P.

F. P. Carlson, R. E. Francois, Proc. IEEE 65, 10 (1977).
[CrossRef]

Casasent, D.

Francois, R. E.

F. P. Carlson, R. E. Francois, Proc. IEEE 65, 10 (1977).
[CrossRef]

Goodman, J. W.

Hagler, M. O.

Hansen, E. W.

Kellman, P.

Krile, T. F.

Marks, R. J.

Nazarathy, M.

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra.” J. Opt. Soc. Am. (in print).

Psaltis, D.

Saleh, B. E. A.

Shamir, J.

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra.” J. Opt. Soc. Am. (in print).

Vander Lugt, A.

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
[CrossRef]

Walkup, J. F.

Appl. Opt. (4)

Proc. IEEE (2)

A. Vander Lugt, Proc. IEEE 54, 1055 (1966).
[CrossRef]

F. P. Carlson, R. E. Francois, Proc. IEEE 65, 10 (1977).
[CrossRef]

Other (1)

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra.” J. Opt. Soc. Am. (in print).

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Figures (5)

Fig. 1
Fig. 1

Basic configuration of an inclined cylindrical lens.

Fig. 2
Fig. 2

Convolution with 2-D pre-FT: L, π/4 oriented cylindrical lens of focal length f; L1, spherical lens of focal length 2f.

Fig. 3
Fig. 3

Output plane photoscan on line y = 0 for two 1-D input functions g(x,y) and h(x,y): (a) convolution of two identical double slits; (b) as (a), but with double slits of different spacing; (c) autocorrelation of a nonsymmetrical three-slit system; (d) autoconvolution of the same.

Fig. 4
Fig. 4

Convolution with 1-D pre-FT: L1, spherical, and L2, cylindrical, lenses of focal length 2f; L and L3 cylindrical lens of focal length f.

Fig. 5
Fig. 5

Singe input plane convolution: L1, spherical lens of focal length f′(= d2); L2, cylindrical lens of focal length f(= d1).

Equations (99)

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Q [ 1 ; a ] g ( x , y ) = exp [ ( j k / 2 ) a x 2 ] g ( x , y ) ,
Q [ 2 ; a ] g ( x , y ) = exp [ ( j k / 2 ) a y 2 ] g ( x , y ) ,
Q [ a ] = Q [ 1 ; a ] Q [ 2 ; a ] = Q [ 2 ; a ] Q [ 1 ; a ] .
F [ 1 ] g ( x , y ) = g ( x , y ) exp ( - 2 π j x x ) d x .
V [ 1 ; s ] g ( x , y ) = g ( s x , y ) ,
F = F [ 1 ] F [ 2 ] = F [ 2 ] F [ 1 ] ,
V [ s ] = V [ 1 ; s ] V [ 2 ; s ] = V [ 2 ; s ] V [ 1 ; s ] .
Q [ 1 ; a ] Q [ 1 ; b ] = Q [ 1 ; a + b ] ;
Q - 1 [ 1 ; a ] = Q [ 1 ; - a ] ;
V [ 1 ; s 1 ] V [ 1 ; s 2 ] = V [ 1 ; s 1 s 2 ] ;
V - 1 [ 1 ; s ] = V [ 1 ; 1 / s ] ;
V [ 1 ; s ] Q [ 1 ; a ] = Q [ 1 ; s 2 a ] V [ 1 ; s ] ;
F [ 1 ] V [ 1 ; s ] = 1 / s V [ 1 ; 1 / s ] F [ 1 ] ;
F - 1 [ 1 ] = V [ 1 ; - 1 ] F [ 1 ] = F [ 1 ] V [ 1 ; - 1 ] ;
F [ 1 ] F [ 1 ] = V [ 1 ; - 1 ] .
I ( x , y ) I ( x ) I ( y ) = 1.
Q [ 1 ; a ] * = Q [ 1 ; a ] I * = exp [ ( j k / 2 ) a x 2 ] * ,
F Q [ a ] g ( x , y ) F Q [ a ] * * F g ( x , y ) [ F Q [ a ] I ( x , y ) ] * * [ F g ( x , y ) ] F Q [ a ] I ( x , y ) * * F g ( x , y ) .
Q [ 1 ; a ] * g ( x ) = exp [ ( j k / 2 ) a ( x - x ) 2 ] g ( x ) d x = exp [ ( j k / 2 ) a x 2 ] exp [ ( j k / 2 ) a x 2 ] g ( x ) × exp [ ( - j k / 2 ) 2 x x ] d x .
exp [ ( j k / 2 ) a x 2 ] g ( x ) exp { - 2 π j [ ( a / λ ) x ] x } d x = V [ 1 ; ( a / λ ) ] F [ 1 ] Q [ 1 ; a ] g ( x ) .
Q [ 1 ; a ] * = Q [ 1 ; a ] V [ 1 ; a / λ ] F [ 1 ] Q [ 1 ; a ] .
F [ 1 ] Q [ 1 ; a ] I ( x ) = ( j λ / a ) 1 / 2 Q [ 1 ; - λ 2 / a ] I ( x ) .
F [ 1 ] Q [ 1 ; a ] = ( j λ / a ) 1 / 2 Q [ 1 ; λ 2 / a ] * F [ 1 ] ,
F [ 1 ] Q [ 1 ; a ] = ( j λ / a ) 1 / 2 Q [ 1 ; - λ 2 / a ] V [ 1 ; - λ / a ] × F [ 1 ] Q [ 1 ; - λ 2 / a ] F [ 1 ] .
Q [ 1 ; a ] * = ( i λ / a ) F - 1 [ 1 ] Q [ 1 ; - λ 2 / a ] F [ 1 ] .
R [ d ] = exp ( j k d ) j λ d Q [ 1 / d ] V [ 1 / λ d ] F Q [ 1 / d ] .
R [ d ] = R [ 1 ; d ] R [ 2 ; d ] = R [ 2 ; d ] R [ 1 ; d ] .
R [ 1 ; d ] = exp ( j k d / 2 ) ( j κ d ) 1 / 2 Q [ 1 ; 1 / d ] V [ 1 ; 1 / λ d ] F [ 1 ] Q [ 1 ; 1 / d ] ,
R [ 1 ; d ] = exp ( j k d / 2 ) F - 1 [ 1 ] Q [ 1 ; - λ 2 d ] F [ 1 ] ,
u o = L [ f ] u i .
L s [ f ] = Q [ - 1 / f ] ,
L [ α ; f ] g ( x , y ) = exp [ ( j k / 2 f ) x 2 ] g ( x , y ) ,
x = x cos α + y sin α .
L [ α ; f ] = C [ - sin 2 α / f ] Q [ 1 ; - cos 2 α / f ] Q [ 2 ; - sin 2 α / f ] ,
C [ a ] g ( x , y ) = exp [ j ( k a / 2 ) x y ] g ( x , y ) .
L [ 0 ; f ] = Q [ 1 : - l / f ] ;
L ( π / 2 ; f ) = Q [ 2 ; - 1 / f ] .
L [ α ; f ] L [ - α ; f ] = Q [ 1 ; - cos 2 α f ] Q [ 2 ; - sin 2 α f ] .
T = R [ d 2 ] L [ α ; f ] R [ d 1 ] .
T = R [ 1 ; d 2 ] R [ 2 ; d 2 ] Q [ 1 ; - 1 / f ] R [ 1 ; d 1 ] R [ 2 ; d 1 ] .
T = R [ 1 ; d 2 ] Q [ 1 ; - 1 / f ] R [ 1 ; d 1 ] R [ 2 ; d 1 + d 2 ] .
g ( x , y ) = g ( x ) I ( y ) .
T g ( x , y ) = R [ 1 ; d 2 ] Q [ 1 ; - 1 / f ] R [ 1 ; d 1 ] g ( x ) R [ 2 ; d 1 + d 2 ] I ( y ) .
F [ 2 ] I ( y ) = δ ( y ) .
R [ 2 ; d 1 + d 2 ] I ( y ) = F - 1 [ 2 ] Q [ 2 ; - λ 2 ( d 1 + d 2 ) ] F [ 2 ] I ( y ) ] = F - 1 [ 2 ] Q [ 2 ; - λ 2 ( d 1 + d 2 ) ] δ ( y ) ,
R [ 2 ; d ] I ( y ) = F - 1 [ 2 ] δ ( y ) Q [ 2 ; 0 ] = 1 = I ( y ) ,
T A I ( x , y ) = A R [ 1 ; f ] Q [ 1 ; - 1 / f ] I ( x ) I ( y ) .
T A ( x , y ) = A exp ( j k f ) ( j λ f ) 1 / 2 Q [ 1 ; 1 / f ] V [ 1 ; 1 / λ f ] F [ 1 ] I ( x ) = A exp ( j k f ) ( j λ f ) 1 / 2 Q [ 1 ; 1 / f ] δ ( x / λ f ) = A exp ( j k f ) ( j λ f ) 1 / 2 δ ( x , λ f ) ,
R [ f ] L [ α ; f ] A I ( x , y ) = A exp ( j k f ) ( j λ f ) 1 / 2 δ ( x cos α + y sin α λ f ) .
L [ π / 4 ; f ] = Q [ 1 ; - 1 / 2 f ] Q [ 2 ; - 1 / 2 f ] C [ - 1 / f ] = Q [ - 1 / 2 f ] C [ - 1 / f ] .
T = R [ 2 f ] Q [ - 1 / 2 f ] C [ - 1 / f ] .
T = exp ( 2 k j f ) 2 j λ f Q [ 1 / 2 f ] V [ 1 / 2 λ f ] F Q [ 1 / 2 f ] Q [ - 1 / 2 f ] C [ - 1 / f ] ,
T = exp ( 2 j k f ) 2 j λ f Q [ 1 / 2 f ] V [ 1 / 2 λ f ] F C [ - 1 / f ] .
u ( x , y ) = F C [ - 1 / f ] g ( x , y ) = F [ 1 ] F [ 2 ] C [ - 1 / f ] g ( x , y ) .
u ( x , y ) = F [ 1 ] - exp [ - j ( k / 2 f ) x y ] g ( x , y ) exp ( - j 2 π y y ) d y = F [ 1 ] - g ( x , y ) exp [ - j 2 π ( y + x / 2 λ f ) y ] d y .
g ( x , y ) = g 1 ( x ) g 2 ( y ) .
u ( x , y ) = F [ 1 ] g 1 ( x ) G 2 ( y + x / 2 λ f ) ,
u ( x , y ) = G 1 ( x ) * { F [ 1 ] G 2 ( x / 2 λ f + y ) } = G 1 ( x ) * { F [ 1 ] V [ 1 ; 1 / 2 λ f ] G 2 ( x + y ) } ,
u ( x , y ) = G 1 ( x ) * 2 λ f V [ 1 ; - 2 λ f ] F - 1 [ 1 ] G 2 ( x + y ) .
u ( x , y ) = 2 λ f G 1 ( x ) * V [ 1 ; - 2 λ f ] exp ( - 2 π j x y ) g 2 ( x ) .
T g 1 ( x ) g 2 ( y ) = exp ( 2 j k f ) j Q [ 1 / 2 f ] V [ 1 / 2 λ f ] { G 1 ( x ) * V [ 1 ; - 2 λ f ] exp ( - 2 π j x y ) g 2 ( x ) } .
T g 1 ( x ) g 2 ( y ) = - j exp ( 2 k j f ) 2 λ f Q [ 1 / 2 f ] { G 1 ( x / 2 λ f ) * C [ 1 / f ] g 2 ( 0 - x ) } ,
V [ 1 ; a ] { h ( x ) * f ( x ) } = a { h ( a x ) * f ( a x ) } ,
T g 1 ( x ) g 2 ( y ) = - j exp ( 2 j k f ) 2 λ f Q [ 1 / 2 f ] { G 2 ( y / 2 λ f ) * C [ 1 / f ] g 1 ( - y ) } .
u 1 ( x , y ) = exp ( 4 j k f ) 4 j f λ V [ 1 / 2 λ f ] F h ( x , y ) .
h ( x , y ) = h 1 ( x ) h 2 ( y ) ,
g 1 ( x ) exp ( 2 j k f ) 2 ( j λ f ) 1 / 2 V [ 1 ; 1 / 2 λ f ] H 1 ( x )
g 2 ( y ) exp ( 2 j k f ) 2 ( j λ f ) 1 / 2 V [ 1 ; 1 / 2 λ f ] H 2 ( y ) ,
u o ( x , y ) - exp ( 6 j k f ) 8 λ 2 f 2 Q [ 1 / 2 f ] ( { V [ 1 ; 1 / 2 λ f ] F [ 1 ] g 1 ( x ) V [ 1 ; 1 / 2 λ f ] H 1 ( x ) } = * { C [ 1 / f ] g 2 ( - x ) V [ 1 ; 1 / 2 λ f ] H 2 ( - x ) } ) .
u o ( x , y ) = - exp ( 6 j k f ) 4 λ f Q [ 1 / 2 f ] { G 1 ( x / 2 λ f ) * h 1 ( - x ) * C [ 1 / f ] g 2 ( - x ) H 2 ( - x / 2 λ f ) } ,
u o ( x , y ) = - exp ( 6 j k f ) 4 λ f Q [ 1 / 2 f ] { G 2 ( y / 2 λ f ) * h 2 ( - y ) * C [ 1 / f ] g 1 ( - y ) H 1 ( - y / 2 λ f ) } .
G 1 ( x / 2 λ f ) = 2 λ f δ ( x ) ;             H 2 = 1 , G 1 ( x 1 / 2 λ f ) * h 1 ( - x ) = 2 λ f h 1 ( - x ) .
u o ( x , 0 ) = exp ( 6 j k π f ) 2 Q [ 1 / 2 f ] { h 1 ( - x ) * g 2 ( - x ) } .
T 1 = R [ 2 f ] Q [ 1 ; - 1 / 2 f ] Q [ 2 ; - 1 / f ] R [ 2 f ] .
T 1 = - exp ( 4 j k f ) 2 ( j λ f ) 1 / 2 Q [ 2 ; 1 ] / f ] V [ 2 ; - 1 ] V [ 1 ; 1 / 2 λ f ] F [ 1 ] .
exp ( 2 j k f ) g 2 ( y ) Q [ 2 ; 1 / f ] h 2 ( - y ) .
u o ( x , y ) = exp ( 6 j k f ) 4 λ f Q [ 1 / 2 f ] { G 1 ( x / 2 λ f ) * h 1 ( - x ) * C [ 1 / f ] g 2 ( - x ) h 2 ( x ) } .
T = R [ d 2 ] Q [ - 1 / f ] Q [ - 1 / 2 f ] C [ - 1 / f ] R [ d 1 ] .
a = 1 / d 1 - 1 / f - 1 / 2 f + 1 / d 2 ,
T = exp [ j k ( d 1 + d 2 ) ] - λ 2 d 1 d 2 Q [ 1 / d 2 ] V [ 1 / λ d 2 ] F C [ - 1 / f ] × Q [ a ] V [ 1 / λ d 1 ] F Q [ 1 / d 1 ] .
g 1 ( x ) Q [ 1 ; a ] ν [ 1 ; 1 / λ d 1 ] F [ 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x )
G 1 ( x ) F [ 1 ] Q [ 1 ; a ] V [ 1 ; 1 / λ d 1 ] F [ 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x ) = ( j λ a ) 1 / 2 Q [ 1 ; - λ 2 / a ] * F [ 1 ] V [ 1 ; 1 / λ d 1 ] F [ 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x ) = ( j λ a ) 1 / 2 Q [ 1 ; - λ 2 / a ] * λ d 1 V [ 1 ; - λ d 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x ) ,
u ( x , y ) = 2 f λ 2 d 1 ( j λ a ) 1 / 2 Q [ 1 ; - λ 2 / a ] * V [ 1 ; - λ d 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x ) * V [ 1 ; - 2 λ f ] exp ( - 2 π j x y ) Q [ 1 ; a ] V [ 1 ; 1 / λ d 1 ] * F [ 1 ] Q [ 1 ; 1 / d 1 ] g 2 ( x ) .
G ( x ) V [ 1 ; 2 f / d 1 ] F [ 1 ] Q [ 1 ; 1 / d 1 ] g 2 ( x ) ,
u ( x , y ) = 2 f λ 2 d 1 ( j λ a ) 1 / 2 Q [ 1 ; - λ 2 / a ] * exp ( 4 π j λ f x y ) Q [ 1 ; 4 a λ 2 f 2 ] G ( x ) * V [ 1 ; - λ d 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x ) .
u ( x , y ) = 2 f λ 2 d 1 ( j λ a ) 1 / 2 Q [ 1 ; - λ 2 / a ] V [ 1 ; - λ / a ] F [ 1 ] Q [ 1 ; - λ 2 / a ] × exp ( 4 π j λ f x y ) Q [ 1 ; 4 a λ 2 f 2 ] G ( x ) * V [ 1 ; - λ d 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x ) .
a = 1 / 2 f ,
u ( x , y ) = 2 f λ 2 d 1 ( 2 j λ f ) 1 / 2 Q [ 1 ; - 2 λ 2 f ] 1 2 λ f F - 1 [ 1 ] × exp ( 2 π j x y ) G ( x / 2 λ f ) * V [ 1 ; - λ d 1 ] Q [ 1 ; 1 / d 1 ] g 1 ( x ) .
u ( x , y ) = 2 f λ 2 d 1 ( 2 j λ f ) 1 / 2 Q [ 1 ; - 2 λ 2 f ] g [ 2 λ f ( x - y ) ] * V [ 1 ; - λ d 1 ] Q [ 1 ; 1 / d i ] g 1 ( x ) ,
g ( x ) = F - 1 V [ 1 ; 2 f / d 1 ] F [ 1 ] g [ 1 ; 1 / d 1 ] g 2 ( x ) = d 1 2 f V [ 1 ; d 1 / 2 f ] Q [ 1 ; 1 / d 1 ] g 2 ( x ) , g [ 2 λ f ( x - y ) ] = d 1 2 f Q [ 1 ; λ 2 d 1 ] g 2 ( λ d 1 x ) x = x - y .
u ( x , y ) = λ 2 d 1 2 ( 2 j λ f ) 1 / 2 Q [ 1 ; - 2 λ 2 f ] { Q [ 1 ; λ 2 d 1 ] g 2 ( λ d 1 x ) x = x - y } * Q [ 1 ; λ 2 d 1 ] g 1 ( - λ d 1 x ) .
u ( x , y ) = λ 2 d 1 2 ( 2 j λ f ) 1 / 2 exp ( - j k 2 2 λ 2 f x 2 ) × exp [ j k 2 λ 2 d 1 ( x - y ) 2 ] g 2 [ λ d 1 ( x - y ) ] × exp [ j k 2 λ 2 d 1 ( x - x ) 2 ] g 1 [ - λ d 1 ( x - x ) ] d x .
d 1 = f ,
y = - x .
u ( x , y ) y = - x = λ 2 f 2 ( 2 j λ f ) 1 / 2 exp [ j k 2 λ 2 f ( x 2 + y 2 ) ] × g 2 [ λ f ( x + x ) ] g 1 [ - λ f ( x - x ) ] d x .
u ( x , y ) y = - x = λ f ( 2 j λ f ) 1 / 2 exp [ j k 2 λ 2 f ( x 2 + y 2 ) ] × g 2 ( ξ ) g 1 ( ξ - 2 λ f x ) d ξ .
u ( x , y ) y = - x = λ f ( 2 j λ f ) 1 / 2 Q [ λ 2 f ] V [ 1 ; 2 λ f ] { g 2 ( x ) * g 1 ( - x ) } .
d 2 = f .
T g 1 ( x ) g 2 ( y ) y = - x = - exp [ j k ( f + f ) ] λ f × ( 2 j λ f ) 1 / 2 Q [ 1 / f ] V [ 1 / λ f ] Q [ λ 2 f ] V [ 1 ; 2 λ f ] × [ g 2 ( x ) * g 1 ( - x ) ] y = - x , = - exp [ j k ( f + f ) ] ( 2 j f λ f 2 ) 1 / 2 Q [ 1 f + f f 2 ] × V [ 1 ; 2 f f ] [ g 2 ( x ) * g 1 ( - x ) ] y = - x .

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