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

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[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]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228–230 (2008).

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

[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15, 2190–2203 (2007).

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).

[CrossRef]
[PubMed]

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]
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[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]

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]

A. A. Malyutin, “Complex-order fractional Fourier transforms in optical schemes with Gaussian apertures,” Quantum Electron. 34, 960–964 (2004).

[CrossRef]

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[CrossRef]

C. Wang and B. Lü, “Implementation of complex-order Fourier transforms in complex ABCD optical systems,” Opt. Commun. 203, 61–66 (2002).

[CrossRef]

D. J. Griffiths and C. A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137–154 (2001).

[CrossRef]

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]

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]

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[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).

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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).

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[CrossRef]

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

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

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J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15, 2190–2203 (2007).

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).

[CrossRef]
[PubMed]

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]

J. N. Bardsley, “Complex scaling: An introduction,” Int. J. Quantum Chem. 14, 343–352 (1978).

[CrossRef]

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, part II,” Commun. Pure Appl. Math. 20, 1–101 (1967).

[CrossRef]

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, part I,” Commun. Pure Appl. Math. 14, 187–214 (1961).

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[CrossRef]

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

[CrossRef]

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]

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).

[CrossRef]
[PubMed]

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

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M. J. Bastiaans, Applications of the Wigner Distribution Function in Optics, The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, 1997), pp. 375–426.

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[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]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15, 2190–2203 (2007).

[CrossRef]
[PubMed]

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

[CrossRef]

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

[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]

L. L. Sanchez-Soto, J. F. Carinena, A. G. Barriuso, and J. J. Monzon, “Vector-like representation of one-dimensional scattering,” Eur. J. Phys. 26, 469–480 (2005).

[CrossRef]

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[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]

E. Georgieva and Y. S. Kim, “Slide-rule-like property of Wigner’s little groups and cyclic S matrices for multilayer optics,” Phys. Rev. E 68, 026606 (2003).

[CrossRef]

D. J. Griffiths and C. A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137–154 (2001).

[CrossRef]

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

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228–230 (2008).

[CrossRef]
[PubMed]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).

[CrossRef]
[PubMed]

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]

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]

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).

[CrossRef]

C. Jung and H. Kruger, “Representation of quantum mechanical wavefunctions by complex valued extensions of classical canonical transformation generators,” J. Phys. A 15, 3509–3523 (1982).

[CrossRef]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).

[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]

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C. Jung and H. Kruger, “Representation of quantum mechanical wavefunctions by complex valued extensions of classical canonical transformation generators,” J. Phys. A 15, 3509–3523 (1982).

[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]

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

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

C. Wang and B. Lü, “Implementation of complex-order Fourier transforms in complex ABCD optical systems,” Opt. Commun. 203, 61–66 (2002).

[CrossRef]

A. A. Malyutin, “Complex-order fractional Fourier transforms in optical schemes with Gaussian apertures,” Quantum Electron. 34, 960–964 (2004).

[CrossRef]

D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61, 1118–1124 (1993).

[CrossRef]

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. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).

[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A Opt. 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]

L. L. Sanchez-Soto, J. F. Carinena, A. G. Barriuso, and J. J. Monzon, “Vector-like representation of one-dimensional scattering,” Eur. J. Phys. 26, 469–480 (2005).

[CrossRef]

M. Moshinsky, “Canonical transformations and quantum mechanics,” SIAM J. Appl. Math. 25, 193–212 (1973).

[CrossRef]

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

[CrossRef]

P. Kramer, M. Moshinsky, and T. H. Seligman, Complex Extensions of Canonical Transformations and Quantum Mechanics, Vol. 3 of Group Theory and Its Applications, E.M.Loebl, ed. (Academic, 1975), pp. 249–332.

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

[CrossRef]

L. Onural, M. F. Erden, and H. M. Ozaktas, “Extensions to common Laplace and Fourier transforms,” IEEE Signal Process. Lett. 4, 310–312 (1997).

[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]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).

[CrossRef]

F. S. 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]

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]

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]

L. Onural, M. F. Erden, and H. M. Ozaktas, “Extensions to common Laplace and Fourier transforms,” IEEE Signal Process. Lett. 4, 310–312 (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 Opt. 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]

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

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

[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15, 2190–2203 (2007).

[CrossRef]
[PubMed]

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

[CrossRef]

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]

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]

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