P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).

[CrossRef]

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).

[CrossRef]

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

[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (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]

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

G. Situ, J. P. Ryle, U. Gopinathan, and J. T. Sheridan, “Generalized in-line digital holographic technique based on intensity measurements at two different planes,” Appl. Opt. 47, 711–717 (2008).

[CrossRef]

D.-Y. Wang, J. Zhao, F.-C. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47, D12–D20 (2008).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their application,” Inf. Sci. 49, 592–603 (2006).

[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).

[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]

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).

[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]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).

[CrossRef]

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

H. W. Turnbull and A. C. Aitken, An Introduction to the Theory of Canonical Matrices (Dover, 1961).

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

R. Bracewell, The Fourier Transform & Its Applications (McGraw-Hill, 1999), p. 261.

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their application,” Inf. Sci. 49, 592–603 (2006).

[CrossRef]

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

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

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).

[CrossRef]

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).

[CrossRef]

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–29 (2010).

[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]

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

M. T. Heath, Scientific Computing: An Introductory Survey (McGraw-Hill, 1997), Chap. 2.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).

[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).

[CrossRef]

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

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]

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, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (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]

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).

[CrossRef]

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).

[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).

[CrossRef]

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993).

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

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).

[CrossRef]

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 2009).

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).

[CrossRef]

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993).

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).

[CrossRef]

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 2009).

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–29 (2010).

[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]

G. Situ, J. P. Ryle, U. Gopinathan, and J. T. Sheridan, “Generalized in-line digital holographic technique based on intensity measurements at two different planes,” Appl. Opt. 47, 711–717 (2008).

[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, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (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]

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their application,” Inf. Sci. 49, 592–603 (2006).

[CrossRef]

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

H. W. Turnbull and A. C. Aitken, An Introduction to the Theory of Canonical Matrices (Dover, 1961).

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their application,” Inf. Sci. 49, 592–603 (2006).

[CrossRef]

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).

[CrossRef]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), p. 16.

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

D.-Y. Wang, J. Zhao, F.-C. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47, D12–D20 (2008).

[CrossRef]

F.-C. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).

[CrossRef]

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

D.-Y. Wang, J. Zhao, F.-C. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47, D12–D20 (2008).

[CrossRef]

G. Situ, J. P. Ryle, U. Gopinathan, and J. T. Sheridan, “Generalized in-line digital holographic technique based on intensity measurements at two different planes,” Appl. Opt. 47, 711–717 (2008).

[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).

[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their application,” Inf. Sci. 49, 592–603 (2006).

[CrossRef]

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).

[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]

Y. Zhang, B.-Z. Dong, B.-Y. Gu, and G.-Z. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).

[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. S. Oktem and H. M. Ozaktas, “Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space—bandwidth product,” J. Opt. Soc. Am. A 27, 1885–1895(2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–29 (2010).

[CrossRef]

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).

[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).

[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).

[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]

F.-C. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).

[CrossRef]

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).

[CrossRef]

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

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

[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).

[CrossRef]

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution-Theory and Applications in the Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, 1997).

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

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 2009).

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

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

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

H. W. Turnbull and A. C. Aitken, An Introduction to the Theory of Canonical Matrices (Dover, 1961).

M. T. Heath, Scientific Computing: An Introductory Survey (McGraw-Hill, 1997), Chap. 2.

While this statement is true it requires clarification. We note that no proof has been provided that unitary DLCTs form a group. In fact, the DLCTs do not (cannot) form a faithful representation of the group of 2×2 matrices M. What has been proven here is that the manifold of N×N matrices, i.e., defined by Eq. (4b), contains points where they are unitary.

http://mathworld.wolfram.com/RelativelyPrime.html .

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), p. 16.

R. Bracewell, The Fourier Transform & Its Applications (McGraw-Hill, 1999), p. 261.

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993).