Abstract

We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non-orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case. The algorithm takes Ñ  log  Ñ time, where Ñ is the two-dimensional space-bandwidth product of the signal. Our method properly tracks and controls the space-bandwidth products in two dimensions, in order to achieve information theoretically sufficient, but not wastefully redundant, sampling required for the reconstruction of the underlying continuous functions at any stage of the algorithm. Additionally, we provide an alternative definition of general 2D-NS-LCTs that shows its kernel explicitly in terms of its ten parameters, and relate these parameters bidirectionally to conventional ABCD matrix parameters.

© 2010 Optical Society of America

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2010 (1)

2009 (2)

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

2008 (4)

2007 (5)

2006 (3)

2005 (4)

2004 (4)

J. C. Brégains, I. C. Coleman, F. Ares, and E. Moreno, “Calculating directivities with the two-dimensional Simpson’s rule,” IEEE Antennas Propag. Mag. 46, 106–112 (2004).
[CrossRef]

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

A. Stern and B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

2003 (1)

2002 (2)

S. C. Pei and J. J. Ding, “Eigenfunction of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

B. Jahne, Digital Image Processing, 5th ed. (Springer, 2002).

2001 (2)

D. W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, 2nd ed. (Pearson, 2001).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

2000 (1)

1998 (2)

1997 (6)

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).
[CrossRef]

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

M. J. Bastiaans, “Applications of the Wigner distribution function in optics,” in The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, 1997), pp. 375–426.

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

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

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

1996 (2)

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

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

1995 (2)

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

1994 (4)

H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994); Corrigenda in pp. 7937–7938.
[CrossRef]

S. Abe and J. T. Sheridan, “Optical operations on wavefunctions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

1993 (3)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1989 (1)

G. B. Folland, Harmonic Analysis in Phase Space (Princeton U. Press, 1989).

1986 (1)

A. E. Siegman, Lasers (University Science Books, 1986).

1982 (1)

1981 (1)

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. 29, 1153–1160 (1981).
[CrossRef]

1979 (2)

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

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

1978 (1)

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

1973 (1)

H. S. M. Coxeter, Regular Polytopes (Dover, 1973).

1971 (1)

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

1970 (1)

1966 (1)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

Abe, S.

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994); Corrigenda in pp. 7937–7938.
[CrossRef]

S. Abe and J. T. Sheridan, “Optical operations on wavefunctions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

Agarwal, G. S.

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Alieva, T.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Almeida, L. B.

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

Ares, F.

J. C. Brégains, I. C. Coleman, F. Ares, and E. Moreno, “Calculating directivities with the two-dimensional Simpson’s rule,” IEEE Antennas Propag. Mag. 46, 106–112 (2004).
[CrossRef]

Arikan, O.

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

Bagini, V.

Barshan, B.

Bastiaans, M.

Bastiaans, M. J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Bozdagi, G.

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

Brégains, J. C.

J. C. Brégains, I. C. Coleman, F. Ares, and E. Moreno, “Calculating directivities with the two-dimensional Simpson’s rule,” IEEE Antennas Propag. Mag. 46, 106–112 (2004).
[CrossRef]

Calvo, M.

Calvo, M. L.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Coleman, I. C.

J. C. Brégains, I. C. Coleman, F. Ares, and E. Moreno, “Calculating directivities with the two-dimensional Simpson’s rule,” IEEE Antennas Propag. Mag. 46, 106–112 (2004).
[CrossRef]

Collins, S. A.

Coxeter, H. S. M.

H. S. M. Coxeter, Regular Polytopes (Dover, 1973).

Ding, J. J.

S. C. Pei and J. J. Ding, “Eigenfunction of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

Erden, M. F.

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

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

Folland, G. B.

G. B. Folland, Harmonic Analysis in Phase Space (Princeton U. Press, 1989).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Healy, J.

Healy, J. J.

Henderson, D. W.

D. W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, 2nd ed. (Pearson, 2001).

Hennelly, B. M.

Hua, J.

Jahne, B.

B. Jahne, Digital Image Processing, 5th ed. (Springer, 2002).

James, D. F. V.

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Javidi, B.

Keys, R. G.

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. 29, 1153–1160 (1981).
[CrossRef]

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
[CrossRef]

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

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

Li, G.

Liu, L.

Luisa Calvo, M.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

Mendlovic, D.

Moreno, E.

J. C. Brégains, I. C. Coleman, F. Ares, and E. Moreno, “Calculating directivities with the two-dimensional Simpson’s rule,” IEEE Antennas Propag. Mag. 46, 106–112 (2004).
[CrossRef]

Moshinsky, M.

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

Nazarathy, M.

Oktem, F.

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

Onural, L.

Ozaktas, H. M.

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

U. Sümbül and H. M. Ozaktas, “Fractional free space, fractional lenses, and fractional imaging systems,” J. Opt. Soc. Am. A 20, 2033–2040 (2003).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
[CrossRef]

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

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

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

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

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

A. Sahin, H. M. Ozaktas, and 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, and 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]

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

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

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

Palma, C.

Pei, S. C.

S. C. Pei and J. J. Ding, “Eigenfunction of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

Quesne, C.

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

Rodrigo, J.

Rodrigo, J. A.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Sahin, A.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
[CrossRef]

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

A. Sahin, H. M. Ozaktas, and 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]

Sari, I.

Shamir, J.

Sheridan, J. T.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, R.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Stern, A.

Sümbül, U.

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wolf, K.

Wolf, K. B.

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

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

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Appl. Opt. (2)

IEEE Antennas Propag. Mag. (1)

J. C. Brégains, I. C. Coleman, F. Ares, and E. Moreno, “Calculating directivities with the two-dimensional Simpson’s rule,” IEEE Antennas Propag. Mag. 46, 106–112 (2004).
[CrossRef]

IEEE Signal Process. Lett. (1)

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. 29, 1153–1160 (1981).
[CrossRef]

IEEE Trans. Signal Process. (4)

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

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

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

S. C. Pei and J. J. Ding, “Eigenfunction of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

J. Math. Phys. (1)

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (17)

M. J. Bastiaans and T. Alieva, “Classification of lossless first-order optical systems and the linear canonical transformation,” J. Opt. Soc. Am. A 24, 1053–1062 (2007).
[CrossRef]

U. Sümbül and H. M. Ozaktas, “Fractional free space, fractional lenses, and fractional imaging systems,” J. Opt. Soc. Am. A 20, 2033–2040 (2003).
[CrossRef]

J. Rodrigo, T. Alieva, and M. Luisa Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A 23, 2494–2500 (2006).
[CrossRef]

R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

T. Alieva and M. J. Bastiaans, “Properties of the canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658–3665 (2007).
[CrossRef]

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937 (2005).
[CrossRef]

J. Rodrigo, T. Alieva, and M. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24, 3135–3139 (2007).
[CrossRef]

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917–927 (2005).
[CrossRef]

K. Wolf and T. Alieva, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365–370 (2008).
[CrossRef]

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

A. Stern and B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
[CrossRef]

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

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

H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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]

J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).
[CrossRef]

J. Phys. A (1)

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994); Corrigenda in pp. 7937–7938.
[CrossRef]

Opt. Commun. (8)

A. Sahin, H. M. Ozaktas, and 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]

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

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

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

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

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

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Signal Process. (1)

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
[CrossRef]

Other (11)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. E. Siegman, Lasers (University Science Books, 1986).

M. J. Bastiaans, “Applications of the Wigner distribution function in optics,” in The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, 1997), pp. 375–426.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

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

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

G. B. Folland, Harmonic Analysis in Phase Space (Princeton U. Press, 1989).

B. Jahne, Digital Image Processing, 5th ed. (Springer, 2002).

H. S. M. Coxeter, Regular Polytopes (Dover, 1973).

D. W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, 2nd ed. (Pearson, 2001).

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Figures (6)

Fig. 1
Fig. 1

Example function F 4 .

Fig. 2
Fig. 2

T 1 of F 1 (our algorithm and reference).

Fig. 3
Fig. 3

T 1 of F 2 (our algorithm and reference).

Fig. 4
Fig. 4

T 2 of F 3 (our algorithm and reference).

Fig. 5
Fig. 5

T 1 of F 4 (our algorithm and reference).

Fig. 6
Fig. 6

T 2 of F 4 (our algorithm and reference).

Tables (3)

Tables Icon

Table 1 Percentage Errors for Different Functions F and Transforms T

Tables Icon

Table 2 Percentage Errors for Different Interpolation Methods and Functions F for T 1

Tables Icon

Table 3 Percentage Errors for Different Interpolation Methods and Functions F for T 2

Equations (69)

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g ( u ) = f M ( u ) = ( C M f ) ( u ) = 1 det   i B exp [ i π ( u Τ B 1 A u 2 u Τ B 1 u + u T D B 1 u ) ] f ( u ) d u ,
M = [ A B C D ] ,
A B T = B A T ,     C D T = D C T ,     A D T B C T = I ,
A T C = C T A ,     B T D = D T B ,     A T D C T B = I .
g ( u x , u y ) = e i π / 2 β x β y η x η y K ( u x , u y , u x , u y ) f ( u x , u y ) d u x d u y ,
K ( u x , u y , u x , u y ) = exp [ i π α x u x 2 2 β x u x u x + 2 η x u x u y + η α u x u y + γ x u x 2 + α y u y 2 2 β y u y u y + 2 η y u x u y + η γ u x u y + γ y u y 2 ] ,
α x = D 11 B 22 D 12 B 21 det   B ,
β x = B 22 det   B ,
η x = B 21 det   B ,
η α = D 12 B 11 + D 21 B 22 D 11 B 12 D 22 B 21 det   B ,
γ x = B 22 A 11 B 12 A 21 det   B ,
α y = D 22 B 11 D 21 B 12 det   B ,
β y = B 11 det   B ,
η y = B 12 det   B ,
η γ = A 21 B 11 + A 12 B 22 A 11 B 21 B 12 A 22 det   B ,
γ y = B 11 A 22 A 12 B 21 det   B .
A = 1 2 ( β x β y η x η y ) [ η y η γ + 2 β y γ x η γ β y + 2 η y γ y η γ β x + 2 η x γ x η x η γ + 2 β x γ y ] ,
B = 1 β x β y η x η y [ β y η y η x β x ] ,
D = 1 2 ( β x β y η x η y ) [ η x η α + 2 β y α x η α β x + 2 η y α x η α β y + 2 η x α y η y η α + 2 β x α y ] .
C 11 = ( A 11 D 11 B 22 + A 12 D 12 B 22 B 22 B 12 A 21 D 11 B 12 A 22 D 12 ) / det   B ,
C 21 = ( A 12 D 22 + A 11 D 21 B 12 C 22 ) / B 11 ,
C 12 = ( A 21 D 11 + A 22 D 12 B 21 C 11 ) / B 22 ,
C 22 = ( A 22 D 22 B 11 + A 21 D 21 B 11 B 11 B 21 A 12 D 22 B 21 A 11 D 21 ) / det   B .
W f ( u x , u y , μ x , μ y ) = f ( u x + u x / 2 , u y + u y / 2 ) f ( u x u x / 2 , u y u y / 2 ) e 2 π i ( μ x u x + μ y u y ) d u x d u y .
W f M ( M s ) = W f ( s ) ,
F a f ( u ) = f a ( u ) = K a ( u , u ) f ( u ) d u ,
K a ( u , u ) = A θ   exp [ i π ( cot   θ u 2 2   csc   θ u u + cot   θ u 2 ) ] ,
A θ = 1 i   cot   θ ,     θ = a π 2
A θ = e i [ π   sgn ( θ ) / 4 θ / 2 ] | sin   θ | ,
S n = { x R n + 1 : x = r } ,
S 3 = { ( x 1 , x 2 , x 3 , x 4 ) R 4 : x 1 2 + x 2 2 + x 3 2 + x 4 2 = r 2 } .
O = { ( s x , s y , b x , b y ) R 4 : s x 2 ( Δ S max / 2 ) 2 + b x 2 ( Δ B max / 2 ) 2 + s y 2 ( Δ S max / 2 ) 2 + b y 2 ( Δ B max / 2 ) 2 = 1 } ,
u x = s x / P ,
u y = s y / P ,
μ x = b x P ,
μ y = b y P ,
O s p = { ( u x , u y , μ x , μ y ) R 4 : u x 2 + u y 2 + μ x 2 + μ y 2 = ( Δ S max Δ B max 2 ) 2 = ( Δ u 2 ) 2 } .
M = [ A B C D ] = [ I 0 G I ] [ S 0 0 S 1 ] [ X Y Y X ] ,
G = ( C A T + D B T ) ( A A T + B B T ) 1 ,
S = ( A A T + B B T ) 1 / 2 ,
X = ( A A T + B B T ) 1 / 2 A ,
Y = ( A A T + B B T ) 1 / 2 B .
[ X Y Y X ] = R r 2 F a x , a y R r 1 ,
R r 1 = [ cos ( r 1 ) sin ( r 1 ) 0 0 sin ( r 1 ) cos ( r 1 ) 0 0 0 0 cos ( r 1 ) sin ( r 1 ) 0 0 sin ( r 1 ) cos ( r 1 ) ] ,
R r 2 = [ cos ( r 2 ) sin ( r 2 ) 0 0 sin ( r 2 ) cos ( r 2 ) 0 0 0 0 cos ( r 2 ) sin ( r 2 ) 0 0 sin ( r 2 ) cos ( r 2 ) ] ,
F a x , a y = [ cos ( a x π / 2 ) 0 sin ( a x π / 2 ) 0 0 cos ( a y π / 2 ) 0 sin ( a y π / 2 ) sin ( a x π / 2 ) 0 cos ( a x π / 2 ) 0 0 sin ( a y π / 2 ) 0 cos ( a y π / 2 ) ] ,
X 11 = cos   r 1   cos   r 2   cos ( a x π / 2 ) sin   r 1   sin   r 2   cos ( a y π / 2 ) ,
X 12 = sin   r 1   cos   r 2   cos ( a x π / 2 ) + cos   r 1   sin   r 2   cos ( a y π / 2 ) ,
X 21 = cos   r 1   sin   r 2   cos ( a x π / 2 ) sin   r 1   cos   r 2   cos ( a y π / 2 ) ,
X 22 = sin   r 1   sin   r 2   cos ( a x π / 2 ) + cos   r 1   cos   r 2   cos ( a y π / 2 ) ,
Y 11 = cos   r 1   cos   r 2   sin ( a x π / 2 ) sin   r 1   sin   r 2   sin ( a y π / 2 ) ,
Y 12 = sin   r 1   cos   r 2   sin ( a x π / 2 ) + cos   r 1   sin   r 2   sin ( a y π / 2 ) ,
Y 21 = cos   r 1   sin   r 2   sin ( a x π / 2 ) sin   r 1   cos   r 2   sin ( a y π / 2 ) ,
Y 22 = sin   r 1   sin   r 2   sin ( a x π / 2 ) + cos   r 1   cos   r 2   sin ( a y π / 2 ) .
f s c ( u ) = 1 det   S f ( S 1 u ) ,
f c h ( u ) = e i π ( G 11 x 2 + ( G 12 + G 21 ) x y + G 22 y 2 ) f ( u ) ,
O s p = [ u x u y μ x μ y ] = Δ u 2 [ cos   ϕ 1 sin   ϕ 1   cos   ϕ 2 sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 ] ,
s out = R r 1 s in = Δ u 2 [ cos ( r 1 ) sin ( r 1 ) 0 0 sin ( r 1 ) cos ( r 1 ) 0 0 0 0 cos ( r 1 ) sin ( r 1 ) 0 0 sin ( r 1 ) cos ( r 1 ) ] [ cos   ϕ 1 sin   ϕ 1   cos   ϕ 2 sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 ] = Δ u 2 [ cos ( r 1 ) cos   ϕ 1 + sin ( r 1 ) sin   ϕ 1   cos   ϕ 2 sin ( r 1 ) cos   ϕ 1 + cos ( r 1 ) sin   ϕ 1   cos   ϕ 2 cos ( r 1 ) sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 + sin ( r 1 ) sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 sin ( r 1 ) sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 + cos ( r 1 ) sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 ] ,
u x 2 + u y 2 + μ x 2 + μ y 2 = ( Δ u 2 ) 2 [ ( cos ( r 1 ) cos   ϕ 1 + sin ( r 1 ) sin   ϕ 1   cos   ϕ 2 ) 2 + ( sin ( r 1 ) cos   ϕ 1 + cos ( r 1 ) sin   ϕ 1   cos   ϕ 2 ) 2 + ( cos ( r 1 ) sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 + sin ( r 1 ) sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 ) 2 + ( sin ( r 1 ) sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 + cos ( r 1 ) sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 ) 2 ] = ( Δ u 2 ) 2 ,
s out = F a x , a y s in = Δ u 2 [ cos ( a x π / 2 ) 0 sin ( a x π / 2 ) 0 0 cos ( a y π / 2 ) 0 sin ( a y π / 2 ) sin ( a x π / 2 ) 0 cos ( a x π / 2 ) 0 0 sin ( a y π / 2 ) 0 cos ( a y π / 2 ) ] [ cos   ϕ 1 sin   ϕ 1   cos   ϕ 2 sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 ] = Δ u 2 [ cos ( a x π / 2 ) cos   ϕ 1 + sin ( a x π / 2 ) sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 cos ( a y π / 2 ) sin   ϕ 1   cos   ϕ 2 + sin ( a y π / 2 ) sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 sin ( a x π / 2 ) cos   ϕ 1 + cos ( a x π / 2 ) sin   ϕ 1   sin   ϕ 2   cos   ϕ 3 sin ( a y π / 2 ) sin   ϕ 1   cos   ϕ 2 + cos ( a y π / 2 ) sin   ϕ 1   sin   ϕ 2   sin   ϕ 3 ] .
{ ( x 1 , x 2 , x 3 , x 4 ) R 4 : 1 x i 1 } .
V = Δ u 2 [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] .
V ¯ = [ v 1 v 2 v 3 v 1 5 v 1 6 ] = [ S 0 0 S 1 ] V ,
d i , j = [ | v i ( 1 ) v j ( 1 ) | | v i ( 2 ) v j ( 2 ) | | v i ( 3 ) v j ( 3 ) | | v i ( 4 ) v j ( 4 ) | ] ,
D = [ d 1 , 2 d 1 , 3 d 1 , 1 6 d 2 , 3 d 2 , 4 d 2 , 1 6 d 1 5 , 1 6 ]
N S x = max ( D 1 , 1 , D 1 , 2 , D 1 , 3 , , D 1 , 120 ) max ( D 3 , 1 , D 3 , 2 , D 3 , 3 , , D 3 , 120 ) ,
N S y = max ( D 2 , 1 , D 2 , 2 , D 2 , 3 , , D 2 , 120 ) max ( D 4 , 1 , D 4 , 2 , D 4 , 3 , , D 4 , 120 ) ,
V ̿ = [ v 1 v 2 v 3 v 1 5 v 1 6 ] = [ I 0 G I ] V ¯ .
C M = Q G K G M S K S R r 2 F a x , a y J R r 1 ,

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