Abstract

We provide a general treatment of optical two-dimensional fractional Fourier transforming systems. We not only allow the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled. We further discuss systems that do not allow all these parameters to be controlled at the same time but are simpler and employ a fewer number of lenses. The variety of systems discussed and the design equations provided should be useful in practical applications for which an optical fractional Fourier transforming stage is to be employed.

© 1998 Optical Society of America

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    [CrossRef]
  2. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  3. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
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    [CrossRef]
  5. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  6. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  7. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  8. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [CrossRef]
  9. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  10. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
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  11. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
    [CrossRef]
  12. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
    [CrossRef] [PubMed]
  13. T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
    [CrossRef]
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    [CrossRef]
  15. M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
    [CrossRef]
  16. H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
    [CrossRef]
  17. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef] [PubMed]
  18. D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
    [CrossRef]
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  20. Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
    [CrossRef] [PubMed]
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  22. J. Garcia, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
    [CrossRef] [PubMed]
  23. B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
    [CrossRef]
  24. H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
    [CrossRef]
  25. H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Signal Process. 54, 81–84 (1996).
    [CrossRef]
  26. M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
    [CrossRef]
  27. D. Mendlovic, Y. Bitran, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
    [CrossRef] [PubMed]
  28. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
    [CrossRef]
  29. A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Ankara, Turkey, 1996).
  30. Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
    [CrossRef] [PubMed]
  31. L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  32. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1987).
    [CrossRef]
  33. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175.
  34. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  35. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
    [CrossRef]
  36. K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
  37. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  38. M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  39. M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
    [CrossRef]
  40. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  41. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

1997 (3)

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

1996 (7)

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

J. Garcia, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef] [PubMed]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

1995 (5)

1994 (11)

Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
[CrossRef] [PubMed]

L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

1993 (4)

1991 (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1987 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1987).
[CrossRef]

1979 (2)

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[CrossRef]

1978 (1)

M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Agullo-Lopez, F.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Alieva, T.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almedia, L. B.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Arikan, O.

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the 1995 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 12, pp. 937–940.
[CrossRef]

Barshan, B.

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Beck, M.

Bernardo, L. M.

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

Bitran, Y.

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Clarke, L.

Dorsch, R. G.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

Ferreira, C.

Garcia, J.

Karasik, Y. B.

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175.

Kutay, M. A.

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the 1995 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 12, pp. 937–940.
[CrossRef]

Lohmann, A.

Lohmann, A. W.

Lopez, V.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Mayer, A.

McAlister, D. F.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175.

Mendlovic, D.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

J. Garcia, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

D. Mendlovic, Y. Bitran, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
[CrossRef] [PubMed]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1987).
[CrossRef]

Onural, L.

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the 1995 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 12, pp. 937–940.
[CrossRef]

Ozaktas, H. M.

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

D. Mendlovic, Y. Bitran, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
[CrossRef] [PubMed]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the 1995 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 12, pp. 937–940.
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Raymer, M. G.

Sahin, A.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Ankara, Turkey, 1996).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Soares, O. D. D.

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Wolf, K. B.

K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

Zalevsky, Z.

J. Garcia, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Signal Process. (3)

L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175.

J. Inst. Math. Its Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1987).
[CrossRef]

J. Mod. Opt. (1)

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Commun. (10)

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

Opt. Lett. (4)

Optik (Stuttgart) (2)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

Signal Process. (2)

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

Other (4)

A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Ankara, Turkey, 1996).

K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the 1995 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 12, pp. 937–940.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Type 1 system that realizes the 1-D linear canonical transform.

Fig. 2
Fig. 2

Type 2 system that realizes the 1-D linear canonical transform.

Fig. 3
Fig. 3

Type 1 system that realizes 2-D linear canonical transforms.

Fig. 4
Fig. 4

Type 2 system that realizes the 2-D linear canonical transform.

Fig. 5
Fig. 5

Optical system that simulates anamorphic free-space propagation.

Fig. 6
Fig. 6

Optical setup with two cylindrical lenses and three sections of free space.

Fig. 7
Fig. 7

Sections A: Neither axis flipped. Sections B: x axis flipped. Sections C: y axis flipped. Section D: Both axes flipped.

Fig. 8
Fig. 8

Optical setup with four cylindrical lenses and two sections of free space.

Fig. 9
Fig. 9

Optical setup with four cylindrical lenses and three sections of free space.

Fig. 10
Fig. 10

Optical system with six lenses and three sections of free space.

Equations (90)

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a x , a y f x ,   y x ,   y = - -   B a x , a y x ,   y ;   x ,   y × f x ,   y d x d y ,
B a x , a y x ,   y ;   x ,   y = B a x x ,   x B a y y ,   y ,
B a x x ,   x = A ϕ x exp i π x 2 cot   ϕ x - 2 xx   csc   ϕ x + x 2 cot   ϕ x ,
B a y y ,   y = A ϕ y exp i π y 2 cot   ϕ y - 2 yy   csc   ϕ y + y 2 cot   ϕ y ,
A ϕ x = exp - i π ϕ ˆ x / 4 - ϕ x / 2 | sin   ϕ x | 1 / 2 , A ϕ y = exp - i π ϕ ˆ y / 4 - ϕ y / 2 | sin   ϕ y | 1 / 2 ,
f r r = -   A ϕ r exp i π r T C t r - 2 r T C s r + r T C t r f r d r ,
A ϕ r = A ϕ x A ϕ y ,     r = x   y T ,     r = x   y T , C t = cot   ϕ x 0 0 cot   ϕ y ,     C s = csc   ϕ x 0 0 csc   ϕ y .
a x 1 , a y 1 a x 2 , a y 2 f x ,   y = a x 1 + a x 2 , a y 1 + a y 2 f x ,   y .
a x , a y k   c k f x ,   y = k   c k a x , a y f x ,   y .
a x , a y f x ,   y = a x f x a y f y .
B a x , a y - 1 x ,   y ;   x ,   y = B - a x , - a y x ,   y ;   x ,   y .
B a x , a y - 1 x ,   y ;   x ,   y = B - a x , - a y x ,   y ;   x ,   y = B a x , a y * x ,   y ;   x ,   y ,
- -   f r * g r d r = - - a x , a y f r * × a x , a y g r d r ,
- -   | f r | 2 d r = - -   | a x , a y f r | 2 d r .
a x , a y f r - r 0 r = exp - i 2 π r s T r - 1 2 r c × a x , a y f r r - r c ,
r 0 = x 0   y 0 T ,     r c = x 0 cos   ϕ x   y 0 cos   ϕ y T , r s = x 0 sin   ϕ x   y 0 sin   ϕ y T .
a x , a y exp i 2 π m T r f r = exp i π m c T m s + 2 r × a x , a y f r r - m s ,
m = m x   m y T ,     c = m x cos   ϕ x   m y cos   ϕ y T , m s = m x sin   ϕ x   m y sin   ϕ y T .
a x , a y x m y n f x ,   y = x   cos   ϕ x + i π sin   ϕ x x m × y   cos   ϕ y + i π sin   ϕ y y n × a x , a y f x ,   y .
a x , a y m x m n y n   f x ,   y = i 2 π x   sin   ϕ x + cos   ϕ x x m × i 2 π y   sin   ϕ y + cos   ϕ y y n × a x , a y f x ,   y .
a x , a y f Kr r = C   exp i π r T Dr a x , a y f r K r ,
C = A ϕ x A ϕ y | k x k y | A ϕ x A ϕ y , K = k x 0 0 k y , ϕ x = arctan k x 2 tan   ϕ x ,     a x = 2 a ϕ x π , ϕ y = arctan k y 2 tan   ϕ y ,     a y = 2 a ϕ y π , D = cot   ϕ x k x 4 - 1 k x 4 + cot 2   ϕ x 0 0 cot   ϕ y k y 4 - 1 k y 4 + cot 2   ϕ y , K = sin   ϕ x k x sin   ϕ x 0 0 sin   ϕ y k y sin   ϕ y .
R = cos   θ sin   θ - sin   θ cos   θ ;
a f Rr r = a r Rr .
W g r ,   ν = W f Ar + B ν ,   Cr + D ν ,
r = x   y T ,     ν = ν x ν y T ,
A = cos   ϕ x 0 0 cos   ϕ y ,     B = - sin   ϕ x 0 0 - sin   ϕ y ,
C = sin   ϕ x 0 0 sin   ϕ y ,     D = cos   ϕ x 0 0 cos   ϕ y .
ϕ W x ,   ν = | a f x | 2 ,
ϕ y ϕ x W x ,   y ;   ν x ,   ν y = | a x , a y f x ,   y | 2 .
- -   B a x ,   a y x ,   y ;   x ,   y Ψ nm x ,   y d x d y = λ nm Ψ nm x ,   y ,
Ψ nm x ,   y = 2 1 / 2 2 n 2 m n ! m ! 1 / 2   H n 2 π x H m 2 π y × exp - π x 2 + y 2 ,
λ nm = exp - i π a x n / 2 exp - i π a y m / 2 .
g x ,   y = - -   h x ,   y ;   x ,   y f x ,   y d x d y , h x ,   y ;   x ,   y = exp - i π / 4 β x 1 / 2 × exp i π α x x 2 - 2 β x xx + γ x x 2 × exp - i π / 4 β y 1 / 2 exp i π α y y 2 - 2 β y yy + γ y y 2 ,
T A x 0 B x 0 0 A y 0 B y C x 0 D x 0 0 C y 0 D y γ x / β x 0 1 / β x 0 0 γ y / β y 0 1 / β y - β x + α x γ x / β x 0 α x / β x 0 0 - β y + α y γ y / β y 0 α y / β y ,
h f x ,   y ,   x ,   y = K f exp i π x - x 2 λ d + y - y 2 λ d ,
T f d = 1 0 λ d 0 0 1 0 λ d 0 0 1 0 0 0 0 1 .
h xl x ,   y ,   x ,   y = K xl δ x - x exp - i π x 2 / λ f x ,
T xl f x = 1 0 0 0 0 1 0 0 - 1 λ f x 0 1 0 0 0 0 1 ,
h yl x ,   y ,   x ,   y = K yl δ y - y exp - i π y 2 / λ f y ,
T yl f y = 1 0 0 0 0 1 0 0 0 0 1 0 0 - 1 λ f y 0 1 .
h xyl x ,   y ,   x ,   y = K xyl δ x - x ,   y - y × exp - i π x 2 λ f x + y 2 λ f y + xy λ f xy ,
T xyl f y = 0 1 0 0 0 1 0 0 - 1 λ f x - 1 2 λ f xy 1 0 - 1 2 λ f xy - 1 λ f y 0 1 .
B a x , a y x ,   y ;   x ,   y = A ϕ x exp i π x 2 p x exp i π x 2 s 2 2 cot   ϕ x - 2 xx s 1 s 2 csc   ϕ x + x 2 s 1 2 cot   ϕ x × A ϕ y exp i π y 2 p y exp i π y 2 s 2 2 cot   ϕ y - 2 yy s 1 s 2 csc   ϕ y + y 2 s 1 2 cot   ϕ y .
T A B C D ,
A = s 2 s 1 cos   ϕ x 0 0 s 2 s 1 cos   ϕ y ,
B = s 1 s 2 sin   ϕ x 0 0 s 1 s 2 sin   ϕ y ,
C = 1 s 1 s 2 p x cos   ϕ x - sin   ϕ x 0 0 1 s 1 s 2 p y cos   ϕ y - sin   ϕ y ,
D = s 1 s 2 sin   ϕ x p x + cot   ϕ x 0 0 s 1 s 2 sin   ϕ y p y + cot   ϕ y .
T = T f d 2 T xl f T f d 1 .
d 1 = β - α λ β 2 - γ α ,     d 2 = β - γ λ β 2 - γ α , f = β λ β 2 - γ α .
d 1 = s 1 s 2 - s 1 2 cos   ϕ λ   sin   ϕ ,     d 2 = s 1 s 2 - s 2 2 cos   ϕ λ   sin   ϕ , f = s 1 s 2 λ   sin   ϕ .
d = 1 λ β ,     f 1 = 1 λ β - γ ,     f 2 = 1 λ β - α .
d = s 1 s 2 sin   ϕ λ ,     f 1 = s 1 2 s 2 sin   ϕ s 1 - s 2 cos   ϕ , f 2 = s 1 s 2 2 sin   ϕ s 2 - s 1 cos   ϕ .
g r = -   h r ,   r f r d r ,
r = x   y T ,     r = x   y T .
h r ,   r = h x x ,   x h y y ,   y ,
g x x ,   y = -   h x x ,   x f x ,   y d x
f r = f x x f y y ,
g r = g x x g y y ,
g x x = -   h x x ,   x f x x d x .
g r = - -   h x x ,   x h y y ,   y f x x f y y d x d y .
d 1 x = β x - α x λ β x 2 - γ x α x ,     d 2 x = β x - γ x λ β x 2 - γ x α x , f x = β x λ β x 2 - γ x α x ,
d 1 y = β y - α y λ β y 2 - γ y α y ,     d 2 y = β y - γ y λ β y 2 - γ y α y , f y = β y λ β y 2 - γ y α y .
α x = cot   ϕ x / s 2 2 ,     γ x = cot   ϕ x / s 1 2 ,     β x = csc   ϕ x / s 1 s 2 ,
α y = cot   ϕ y / s 2 2 ,     γ y = cot   ϕ y / s 1 2 ,     β y = csc   ϕ y / s 1 s 2 .
d x = 1 λ β x ,     f 1 x = 1 λ β x - γ x ,     f 2 x = 1 λ β x - α x ,
d y = 1 λ β y ,     f 1 y = 1 λ β y - γ y ,     f 2 y = 1 λ β y - α y .
g x ,   y = C   - - exp i π x - x 2 / λ d x + y - y 2 / λ d y f x ,   y d x d y ,
d x = s 4 λ 2 f x ,     d y = s 4 λ 2 f y ,
d 1 x = d 1 y = d 1 = s 1 2 sin   ϕ y - sin   ϕ x λ cos   ϕ y - cos   ϕ x ,
d 2 x = d 2 y = d 2 = s 1 s 2 sin ϕ x - ϕ y λ cos   ϕ y - cos   ϕ x ,
f x = s 1 2 s 2 sin ϕ x - ϕ y λ s 1 - s 2 cos   ϕ x cos   ϕ y - cos   ϕ x ,
f y = s 1 2 s 2 sin ϕ x - ϕ y λ s 1 - s 2 cos   ϕ y cos   ϕ y - cos   ϕ x ,
p x = s 2 cos   ϕ y - cos   ϕ x + s 1 1 - cos ϕ y - ϕ x s 1 s 2 2 sin ϕ x - ϕ y ,
p y = s 2 cos   ϕ y - cos   ϕ x + s 1 cos ϕ y - ϕ x - 1 s 1 s 2 2 sin ϕ x - ϕ y .
d 1 x = d 1 y = d 1 = s 1 2 sin   ϕ y - sin   ϕ x λ cos   ϕ y - cos   ϕ x ,
d 2 x = d 2 y = d 2 = s 1 s 2 sin ϕ x - ϕ y λ cos   ϕ y - cos   ϕ x ,
f x 1 = s 1 2 s 2 sin ϕ x - ϕ y λ s 1 - s 2 cos   ϕ x cos   ϕ y - cos   ϕ x ,
f y 1 = s 1 2 s 2 sin ϕ x - ϕ y λ s 1 - s 2 cos   ϕ y cos   ϕ y - cos   ϕ x ,
f x 2 = s 1 s 2 2 sin ϕ x - ϕ y λ s 2 cos   ϕ y - cos   ϕ x + s 1 1 - cos ϕ y - ϕ x ,
f y 2 = s 1 s 2 2 sin ϕ x - ϕ y λ s 2 cos   ϕ y - cos   ϕ x + s 1 cos ϕ y - ϕ x - 1 .
f x 1 = s 1 s 2 d 2 sin   ϕ x / λ - s 2 / s 1 d 1 d 2 cos   ϕ x s 2 / s 1 d 1 + d 2 cos   ϕ x - s 1 s 2 sin   ϕ x / λ + d 3 ,
f y 1 = s 1 s 2 d 2 sin   ϕ y / λ - s 2 / s 1 d 1 d 2 cos   ϕ y s 2 / s 1 d 1 + d 2 cos   ϕ y - s 1 s 2 sin   ϕ y / λ + d 3 ,
f x 2 = d 2 d 3 s 2 / s 1 d 1 cos   ϕ x - s 1 s 2 sin   ϕ x / λ + d 2 + d 3 ,
f y 2 = d 2 d 3 s 2 / s 1 d 1 cos   ϕ y - s 1 s 2 sin   ϕ y / λ + d 2 + d 3 ,
p x = - cos   ϕ x + s 2 s 1 sin   ϕ x 1 - d 1 f x 1 - d 1 f x 2 - d 2 f x 2 + d 1 d 2 f x 1 f x 2 ,
p y = - cos   ϕ y + s 2 s 1 sin   ϕ y 1 - d 1 f y 1 - d 1 f y 2 - d 2 f y 2 + d 1 d 2 f y 1 f y 2 .
f x 3 = 1 λ p x ,
f y 3 = 1 λ p y .

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