Abstract

Holographic optical elements that perform one-to-one coordinate transformations can be designed by using the stationary phase approximation. All coordinate transformations that are sufficiently differentiable can be decomposed into two transformations in series and therefore performed by using two holographic elements. Although the method presented here for designing the two required holographic elements is not explicit for the general case, a useful necessary condition for the decomposition is given. A simple explicit solution is given for an example for which the necessary equations for the first of the two serial transformations are separable.

© 1990 Optical Society of America

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References

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  1. J. N. Cederquist, A. M. Tai, “Computer-generated holograms for geometric transformations,” Appl. Opt. 23, 3099–3104 (1984).
    [CrossRef] [PubMed]
  2. J. N. Cederquist, M. T. Eismann, A. M. Tai, “Holographic polar formatting and real-time optical processing of synthetic aperture radar data,” Appl. Opt. 28, 4182–4189 (1989).
    [CrossRef] [PubMed]
  3. O. Bryngdahl, “Optical map transformations,” Opt. Commun 10, 164–168 (1974).
    [CrossRef]
  4. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.
  6. W. Kaplan, Advanced Calculus (Addison-Wesley, Reading Mass., 1959), pp. 243–249.
  7. W. Kaplan, Advanced Calculus (Addison-Wesley, Reading Mass., 1959), pp. 90–97.
  8. M. Spivak, Calculus on Manifolds (Benjamin, Menlo Park Calif., 1965), pp. 34–39.
  9. I. G. Petrovsky, Lectures on Partial Differential Equations (Interscience, New York, 1954), pp. 67–72.
  10. E. C. Zachmanoglou, D. W. Thoe, Introduction to Partia Differential Equations with Applications (Dover, New York1986), pp. 361–362.
  11. A. E. Taylor, Advanced Calculus (Ginn, Boston, 1955), pp. 267–271, 428–431.

1989 (1)

1984 (1)

1974 (2)

O. Bryngdahl, “Optical map transformations,” Opt. Commun 10, 164–168 (1974).
[CrossRef]

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.

Bryngdahl, O.

O. Bryngdahl, “Optical map transformations,” Opt. Commun 10, 164–168 (1974).
[CrossRef]

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

Cederquist, J. N.

Eismann, M. T.

Kaplan, W.

W. Kaplan, Advanced Calculus (Addison-Wesley, Reading Mass., 1959), pp. 243–249.

W. Kaplan, Advanced Calculus (Addison-Wesley, Reading Mass., 1959), pp. 90–97.

Petrovsky, I. G.

I. G. Petrovsky, Lectures on Partial Differential Equations (Interscience, New York, 1954), pp. 67–72.

Spivak, M.

M. Spivak, Calculus on Manifolds (Benjamin, Menlo Park Calif., 1965), pp. 34–39.

Tai, A. M.

Taylor, A. E.

A. E. Taylor, Advanced Calculus (Ginn, Boston, 1955), pp. 267–271, 428–431.

Thoe, D. W.

E. C. Zachmanoglou, D. W. Thoe, Introduction to Partia Differential Equations with Applications (Dover, New York1986), pp. 361–362.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.

Zachmanoglou, E. C.

E. C. Zachmanoglou, D. W. Thoe, Introduction to Partia Differential Equations with Applications (Dover, New York1986), pp. 361–362.

Appl. Opt. (2)

J. Opt Soc. Am. (1)

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

Opt. Commun (1)

O. Bryngdahl, “Optical map transformations,” Opt. Commun 10, 164–168 (1974).
[CrossRef]

Other (7)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 749–750.

W. Kaplan, Advanced Calculus (Addison-Wesley, Reading Mass., 1959), pp. 243–249.

W. Kaplan, Advanced Calculus (Addison-Wesley, Reading Mass., 1959), pp. 90–97.

M. Spivak, Calculus on Manifolds (Benjamin, Menlo Park Calif., 1965), pp. 34–39.

I. G. Petrovsky, Lectures on Partial Differential Equations (Interscience, New York, 1954), pp. 67–72.

E. C. Zachmanoglou, D. W. Thoe, Introduction to Partia Differential Equations with Applications (Dover, New York1986), pp. 361–362.

A. E. Taylor, Advanced Calculus (Ginn, Boston, 1955), pp. 267–271, 428–431.

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Figures (1)

Fig. 1
Fig. 1

Optical system for coordinate transformation. The lens L has focal length f and Fourier transforms plane P1 to plane P2.

Equations (32)

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a 2 ( s , t ) = i λ f a 1 ( x , y ) × exp [ i ϕ ( x , y ) i 2 π λ f ( x s + y t ) ] d x d y ,
[ x y ] [ s t ] = [ λ f 2 π ϕ ( x , y ) x λ f 2 π ϕ ( x , y ) y ] ,
[ x y ] [ s t ] = [ g ( x , y ) h ( x , y ) ] ,
λ f 2 π ϕ x = g ( x , y ) , λ f 2 π ϕ y = h ( x , y ) .
2 ϕ ( x , y ) y x = 2 ϕ ( x , y ) x y ,
g ( x , y ) y = h ( x , y ) x .
[ x y ] [ s t ] = [ s ( x , y ) t ( x , y ) ] ,
s ( x , y ) y t ( x , y ) x ,
[ x y ] [ u υ ] = [ u ( x , y ) υ ( x , y ) ] .
[ u υ ] [ s t ] = [ p ( u , υ ) q ( u , υ ) ] ,
u ( x , y ) y = υ ( x , y ) x ,
p ( u , υ ) υ = q ( u , υ ) u ,
s ( x , y ) = p [ u ( x , y ) , υ ( x , y ) ] , t ( x , y ) = q [ u ( x , y ) , υ ( x , y ) ] .
[ s x s y t x t y ] = [ p u p υ q u q υ ] [ u x u y υ x υ y ] .
D p υ = s y u x s x u y , D q u = t x υ y t y υ x ,
D = u x υ y u y υ x .
u y = υ x , t x υ y = t y υ x + s y u x s x u y ,
[ u y υ y ] + [ 0 1 ( t x ) 1 s y ( t x ) 1 ( s x t y ) ] [ u x υ x ] = [ 0 0 ] .
| λ 1 ( t x ) 1 s y ( t x ) 1 ( s x t y ) λ | = 0 .
λ ( x , y ) = ( 2 t x ) 1 { s x t y ± [ ( s x t y ) 2 + 4 s y t x ] 1 / 2 } .
( s x t y ) 2 + 4 s y t x = ( s x + t y ) 2 + 4 ( s y t x s x t y ) .
| s x s y t x t y | < 0 .
[ x y ] [ x y ] ,
[ x y ] [ s t ] = [ x cos y x sin y ] ,
s ( x , y ) = x cos y , t ( x , y ) = x sin y .
s ( x , y ) y = x sin y , t ( x , y ) x = sin y
s ( x , y ) = x cos y , t ( x , y ) = x sin y .
u y υ x = 0 ,
υ y + ( sin y ) 1 { x sin y u x + [ ( x + 1 ) cos y ] υ x } = 0 .
d h d y x d g d x = 0 .
[ x y ] [ u υ ] = [ c ln x + d c y + e ] ,
[ u υ ] [ s t ] = [ exp u cos υ exp u sin υ ] ,

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