Abstract

In this paper, we consider the properties of the nonsymmetrical Fourier transformation which is space-variant in both rectangular and polar coordinates. A coherent optical processor composed of two nonsymmetrical Fourier transformers is introduced. This processor allows rotation-variant linear filtering operations and matched filtering. Two configurations for such a processor are proposed. For certain parameters of both nonsymmetrical Fourier transformers it is possible to obtain a space-invariant processor with both lateral magnifications equal to unity. However, introducing any filter operation results in a rotation-variant performance.

© 1985 Optical Society of America

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References

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  1. W. T. Rhodes, “Space-Variant Optical Systems and Processing,” in Applications of Optical Fourier Transform,H. Stark, Ed. (Academic, New York, 1982), p. 333.
    [CrossRef]
  2. J. W. Goodman, “Linear Space-Variant Optical Data Processing,” in Optical Information Processing,S. H. Lee, Ed. (Springer, Berlin, 1981), p. 235.
    [CrossRef]
  3. J. F. Walkup, “Space-Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
    [CrossRef]
  4. L. Farrenc, “An Apparatus and Method for the Production of Photographs and Stereoscopic Portraits and Pictures,” British Pat.1453 (1862).
  5. E. Abbe, P. Rudolph, “Anamorphotisches Linsensystem,” German Pat.99722 (1897).
  6. P. Culmann, “The Formation of Optical Image,” in Geometrical Investigation of the Formation of Images in Optical Instruments,M. von Rohr, Ed. (H. M. Stationery Office, London, 1920), p. 125.
  7. A. W. Lohmann, D. P. Paris, “Space-Variant Image Formation,” J. Opt. Soc. Am. 55, 1007 (1965).
  8. A. W. Lohmann, N. Streibl, “Map Transformation by Optical Anamorphic Processing,” Appl. Opt. 22, 780 (1983).
    [CrossRef] [PubMed]
  9. M. Montel, “Classification des Aberrations Geometriques des Systemes Optiques sans Symetrie de Revolution,” Rev. Opt. 32, 585 (1953).
  10. C. G. Wynne, “The Primary Aberrations of Anamorphotic Lens Systems,” Proc. Phys. Soc. London Sect B. 67, 529 (1954).
    [CrossRef]
  11. R. Barakat, A. Houston, “The Aberrations of Non-rotationally Symmetric Systems and Their Diffraction Effects,” Opt. Acta 13, 1 (1966).
    [CrossRef]
  12. H. H. Arsenault, “A Matrix Representation for Non-symmetrical Optical Systems,” J. Opt. Paris 11, 87 (1980).
    [CrossRef]
  13. B. Macukow, H. H. Arsenault, “Matrix Decompositions for Nonsymmetrical Optical Systems,” J. Opt. Soc. Am. 73, 1360 (1983).
    [CrossRef]
  14. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical Data Processing and Filtering Systems,” IRE Trans. Inf. Theory IT-6, 386 (1960).
    [CrossRef]
  15. T. Szoplik, W. Kosek, “Directional Analysis of Optical Fourier Spectra in an Anamorphic System,” in Abstracts, 1983 European Optical Conference, Rydzyna, Poland1983), p. 88.
  16. T. Szoplik, W. Kosek, C. Ferreira, “Nonsymmetric Fourier Transforming with an Anamorphic System,” Appl. Opt. 23, 905 (1984).
    [CrossRef] [PubMed]
  17. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 308.

1984 (1)

1983 (2)

1980 (2)

J. F. Walkup, “Space-Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

H. H. Arsenault, “A Matrix Representation for Non-symmetrical Optical Systems,” J. Opt. Paris 11, 87 (1980).
[CrossRef]

1966 (1)

R. Barakat, A. Houston, “The Aberrations of Non-rotationally Symmetric Systems and Their Diffraction Effects,” Opt. Acta 13, 1 (1966).
[CrossRef]

1965 (1)

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical Data Processing and Filtering Systems,” IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

1954 (1)

C. G. Wynne, “The Primary Aberrations of Anamorphotic Lens Systems,” Proc. Phys. Soc. London Sect B. 67, 529 (1954).
[CrossRef]

1953 (1)

M. Montel, “Classification des Aberrations Geometriques des Systemes Optiques sans Symetrie de Revolution,” Rev. Opt. 32, 585 (1953).

Abbe, E.

E. Abbe, P. Rudolph, “Anamorphotisches Linsensystem,” German Pat.99722 (1897).

Arsenault, H. H.

B. Macukow, H. H. Arsenault, “Matrix Decompositions for Nonsymmetrical Optical Systems,” J. Opt. Soc. Am. 73, 1360 (1983).
[CrossRef]

H. H. Arsenault, “A Matrix Representation for Non-symmetrical Optical Systems,” J. Opt. Paris 11, 87 (1980).
[CrossRef]

Barakat, R.

R. Barakat, A. Houston, “The Aberrations of Non-rotationally Symmetric Systems and Their Diffraction Effects,” Opt. Acta 13, 1 (1966).
[CrossRef]

Culmann, P.

P. Culmann, “The Formation of Optical Image,” in Geometrical Investigation of the Formation of Images in Optical Instruments,M. von Rohr, Ed. (H. M. Stationery Office, London, 1920), p. 125.

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical Data Processing and Filtering Systems,” IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Farrenc, L.

L. Farrenc, “An Apparatus and Method for the Production of Photographs and Stereoscopic Portraits and Pictures,” British Pat.1453 (1862).

Ferreira, C.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 308.

Goodman, J. W.

J. W. Goodman, “Linear Space-Variant Optical Data Processing,” in Optical Information Processing,S. H. Lee, Ed. (Springer, Berlin, 1981), p. 235.
[CrossRef]

Houston, A.

R. Barakat, A. Houston, “The Aberrations of Non-rotationally Symmetric Systems and Their Diffraction Effects,” Opt. Acta 13, 1 (1966).
[CrossRef]

Kosek, W.

T. Szoplik, W. Kosek, C. Ferreira, “Nonsymmetric Fourier Transforming with an Anamorphic System,” Appl. Opt. 23, 905 (1984).
[CrossRef] [PubMed]

T. Szoplik, W. Kosek, “Directional Analysis of Optical Fourier Spectra in an Anamorphic System,” in Abstracts, 1983 European Optical Conference, Rydzyna, Poland1983), p. 88.

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical Data Processing and Filtering Systems,” IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Lohmann, A. W.

Macukow, B.

Montel, M.

M. Montel, “Classification des Aberrations Geometriques des Systemes Optiques sans Symetrie de Revolution,” Rev. Opt. 32, 585 (1953).

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical Data Processing and Filtering Systems,” IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Paris, D. P.

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical Data Processing and Filtering Systems,” IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, “Space-Variant Optical Systems and Processing,” in Applications of Optical Fourier Transform,H. Stark, Ed. (Academic, New York, 1982), p. 333.
[CrossRef]

Rudolph, P.

E. Abbe, P. Rudolph, “Anamorphotisches Linsensystem,” German Pat.99722 (1897).

Streibl, N.

Szoplik, T.

T. Szoplik, W. Kosek, C. Ferreira, “Nonsymmetric Fourier Transforming with an Anamorphic System,” Appl. Opt. 23, 905 (1984).
[CrossRef] [PubMed]

T. Szoplik, W. Kosek, “Directional Analysis of Optical Fourier Spectra in an Anamorphic System,” in Abstracts, 1983 European Optical Conference, Rydzyna, Poland1983), p. 88.

Walkup, J. F.

J. F. Walkup, “Space-Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

Wynne, C. G.

C. G. Wynne, “The Primary Aberrations of Anamorphotic Lens Systems,” Proc. Phys. Soc. London Sect B. 67, 529 (1954).
[CrossRef]

Appl. Opt. (2)

IRE Trans. Inf. Theory (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical Data Processing and Filtering Systems,” IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

J. Opt. Paris (1)

H. H. Arsenault, “A Matrix Representation for Non-symmetrical Optical Systems,” J. Opt. Paris 11, 87 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Acta (1)

R. Barakat, A. Houston, “The Aberrations of Non-rotationally Symmetric Systems and Their Diffraction Effects,” Opt. Acta 13, 1 (1966).
[CrossRef]

Opt. Eng. (1)

J. F. Walkup, “Space-Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

Proc. Phys. Soc. London Sect B. (1)

C. G. Wynne, “The Primary Aberrations of Anamorphotic Lens Systems,” Proc. Phys. Soc. London Sect B. 67, 529 (1954).
[CrossRef]

Rev. Opt. (1)

M. Montel, “Classification des Aberrations Geometriques des Systemes Optiques sans Symetrie de Revolution,” Rev. Opt. 32, 585 (1953).

Other (7)

W. T. Rhodes, “Space-Variant Optical Systems and Processing,” in Applications of Optical Fourier Transform,H. Stark, Ed. (Academic, New York, 1982), p. 333.
[CrossRef]

J. W. Goodman, “Linear Space-Variant Optical Data Processing,” in Optical Information Processing,S. H. Lee, Ed. (Springer, Berlin, 1981), p. 235.
[CrossRef]

L. Farrenc, “An Apparatus and Method for the Production of Photographs and Stereoscopic Portraits and Pictures,” British Pat.1453 (1862).

E. Abbe, P. Rudolph, “Anamorphotisches Linsensystem,” German Pat.99722 (1897).

P. Culmann, “The Formation of Optical Image,” in Geometrical Investigation of the Formation of Images in Optical Instruments,M. von Rohr, Ed. (H. M. Stationery Office, London, 1920), p. 125.

T. Szoplik, W. Kosek, “Directional Analysis of Optical Fourier Spectra in an Anamorphic System,” in Abstracts, 1983 European Optical Conference, Rydzyna, Poland1983), p. 88.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 308.

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Figures (3)

Fig. 1
Fig. 1

Anamorphic system providing a 2-D Fourier transformation. The lens in the x5y5 plane transforms the horizontal dimension of an object placed in the input x1y1 plane into the x8y8 plane. The telescopic system of lenses in the x2y2 and x4y4 planes images the vertical dimension of an object in the x6y6 plane. The lens in the x7y7 plane transforms the image into the common output x8y8 plane.

Fig. 2
Fig. 2

Configuration of the NFT–NFT type for 2-D anamorphic coherent processors where one nonsymmetrical transformer is followed by the other.

Fig. 3
Fig. 3

Configuration of the NFT–TFN type for 2-D anamorphic coherent processors where the second NFT is rotated by an angle π/2 with respect to the first, as indicated by the reversed abbreviation.

Tables (1)

Tables Icon

Table I Lateral Magnification Factors Attainable with the NFT–NFT and NFT–TFN Processors; Parameters of the Second NFT are Indicated by the Superscript

Equations (27)

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G ( x 8 , y 8 ) = C / i λ f y 2 / f y 1 f y f x exp ( i k 2 f x ) g ( x 1 , y 1 ) exp [ 2 π i ( x 1 u + y 1 υ ) ] d x 1 d y 1 ,
{ a f ( x , y ) + b g ( x , y ) } = a F ( u , υ ) + b G ( u , υ ) .
f ( x , y ) exp [ 2 π i ( x u + y υ ) ] dxdy = F ( u , υ ) ,
f ( x a , y b ) exp [ 2 π i ( x u + y υ ) ] dxdy = F ( u , υ ) exp [ 2 π i ( a u + b υ ) ] .
F ( η , ξ ) = f ( x , y ) exp [ 2 π i ( x η + y ξ ) ] dxdy ,
exp [ 2 π i r ρ cos ( θ ϕ ) ] = h ( ρ , θ ϕ ; r ) ,
G ( ρ , ϕ ) = 00 2 π g ( r , θ ) exp [ 2 π i r ρ ( cos θ cos ϕ + M sin θ sin ϕ ) ] rdrd θ ,
g ( x , y ) exp [ 2 π i ( x u + y υ ) ] dxdy = g ( x cos α + y sin α , x sin α + y cos α ) × exp [ 2 π i ( x u + y υ ) ] dxdy = G ( u cos α + υ sin α , u sin α + υ cos α ) ,
F ( u , υ ) = f ( x , y ) h n ( u , υ ; x , y ) dxdy ,
F ( u , υ ) = F ( η , ξ ) ĥ n ( u , υ ; η , ξ ) d η d ξ ,
ĥ n ( u , υ ; η , ξ ) = h n ( u , υ ; x , y ) exp [ 2 π i ( x η + y ξ ) ] dxdy
ĥ n ( u , υ ; η , ξ ) = exp { 2 π i [ x ( u η ) + y ( υ ξ ) ] } dxdy = ĥ n ( u η , υ ξ ; η , ξ ) .
M y = f y f y f y 2 f y 1 f y 1 f y 2 .
g 2 ( x i , y i ) = g 1 ( x 1 , y 1 ) d x 1 d y 1 F ( x F , y F ) exp { 2 π i [ x F ( x 1 λ f x + x i λ f x ) + y F ( y 1 λ f y f y 1 / f y 2 + y i λ f y f y 1 / f y 2 ) ] } d x F d y F ,
g 2 ( x i , y i ) = g 1 ( M x x 1 , M y y 1 ) h ( x i M x x 1 , y i M y y 1 ) d x 1 d y 1 ,
I ( r i , θ i ) = | 0 0 2 π G ( ρ , ϕ ) F ( ρ , ϕ ) exp [ 2 π i r i ρ ( cos θ i cos ϕ + M sin θ i sin ϕ ) ] ρ d ρ d ϕ | 2 ,
I ( r i , θ i ) = | g ( r i , θ i ) h ( r i , θ i ) | 2 ,
M y = f y 2 f y 1 f y 1 f y f y f y 2
M y = f y 2 f y 1 f f y
M x = f y 1 f x f y f y 2
M x = f y f x
M x = f y 1 f x f y f y 2
M x = f y f x
M y = f y 2 f y 1 f x f y
M y = f x f y
M y = f y 2 f y 1 f x f y
M y = f x f y

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