Abstract

The linear canonical transform may be used to simulate the effect of paraxial optical systems on wave fields. Using a recent definition of the discrete linear canonical transform, phase space diagram analyses of the sampling requirements of the direct method of calculating the Fresnel and other linear canonical transforms are more favorable than previously thought. Thus the direct method of calculating these transforms may be used with fewer samples than previously reported simply by making use of an appropriate reconstruction filter on the samples output by the algorithm.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]

2010 (1)

2009 (4)

J. J. Healy and J. T. Sheridan, Signal Process. 89, 641 (2009).
[CrossRef]

F. Oktem and H. M. Ozaktas, IEEE Tran. Sig. Process. Lett. 16, 727 (2009).
[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. G. Voelz and M. C. Roggemann, Appl. Opt. 48, 6132 (2009).
[CrossRef] [PubMed]

2008 (2)

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, Opt. Lett. 33, 2599 (2008).
[CrossRef] [PubMed]

2007 (1)

L. Bing-Zhao, R. Tao, and Y. Wang, Signal Process. 87, 983 (2007).
[CrossRef]

2006 (3)

A. Stern, Signal Process. 86, 1421 (2006).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, Sci. China Ser. F Inf. Sci. 49, 592 (2006).
[CrossRef]

A. Stern, in Proceedings of the 5th International Workshop on Information Optics, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (Springer, 2006), pp. 225-234.

2005 (3)

2001 (1)

J.-J. Ding, Ph.D. thesis (National Taiwan University, 2001).

2000 (2)

S.-C. Pei and J.-J. Ding, IEEE Trans. Signal Process. 48, 1338 (2000).
[CrossRef]

X. Deng, B. Bihari, J. Gan, F. Zhao, and R. T. Chen, J. Opt. Soc. Am. A 17, 762 (2000).
[CrossRef]

1949 (1)

C. E. Shannon, Proc. IRE 37, 10 (1949).
[CrossRef]

Bihari, B.

Bing-Zhao, L.

L. Bing-Zhao, R. Tao, and Y. Wang, Signal Process. 87, 983 (2007).
[CrossRef]

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

Chen, R. T.

Deng, B.

B. Deng, R. Tao, and Y. Wang, Sci. China Ser. F Inf. Sci. 49, 592 (2006).
[CrossRef]

Deng, X.

Ding, J.-J.

J.-J. Ding, Ph.D. thesis (National Taiwan University, 2001).

S.-C. Pei and J.-J. Ding, IEEE Trans. Signal Process. 48, 1338 (2000).
[CrossRef]

Gan, J.

Healy, J. J.

Hennelly, B. M.

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, Opt. Eng. 48, 095801 (2009).
[CrossRef]

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry, Optical and Digital Methods (Wiley-VCH, 2005).

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

Naughton, T. J.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, Opt. Eng. 48, 095801 (2009).
[CrossRef]

Oktem, F.

F. Oktem and H. M. Ozaktas, IEEE Tran. Sig. Process. Lett. 16, 727 (2009).
[CrossRef]

Ozaktas, H. M.

F. Oktem and H. M. Ozaktas, IEEE Tran. Sig. Process. Lett. 16, 727 (2009).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

Pandey, N.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, Opt. Eng. 48, 095801 (2009).
[CrossRef]

Pei, S.-C.

S.-C. Pei and J.-J. Ding, IEEE Trans. Signal Process. 48, 1338 (2000).
[CrossRef]

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, Opt. Eng. 48, 095801 (2009).
[CrossRef]

Roggemann, M. C.

Shannon, C. E.

C. E. Shannon, Proc. IRE 37, 10 (1949).
[CrossRef]

Sheridan, J. T.

Stern, A.

A. Stern, Signal Process. 86, 1421 (2006).
[CrossRef]

A. Stern, in Proceedings of the 5th International Workshop on Information Optics, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (Springer, 2006), pp. 225-234.

Tao, R.

L. Bing-Zhao, R. Tao, and Y. Wang, Signal Process. 87, 983 (2007).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, Sci. China Ser. F Inf. Sci. 49, 592 (2006).
[CrossRef]

Voelz, D. G.

Wang, Y.

L. Bing-Zhao, R. Tao, and Y. Wang, Signal Process. 87, 983 (2007).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, Sci. China Ser. F Inf. Sci. 49, 592 (2006).
[CrossRef]

Zhao, F.

Appl. Opt. (1)

IEEE Tran. Sig. Process. Lett. (1)

F. Oktem and H. M. Ozaktas, IEEE Tran. Sig. Process. Lett. 16, 727 (2009).
[CrossRef]

IEEE Trans. Signal Process. (2)

S.-C. Pei and J.-J. Ding, IEEE Trans. Signal Process. 48, 1338 (2000).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, Opt. Eng. 48, 095801 (2009).
[CrossRef]

Opt. Lett. (1)

Proc. IRE (1)

C. E. Shannon, Proc. IRE 37, 10 (1949).
[CrossRef]

Sci. China Ser. F Inf. Sci. (1)

B. Deng, R. Tao, and Y. Wang, Sci. China Ser. F Inf. Sci. 49, 592 (2006).
[CrossRef]

Signal Process. (3)

L. Bing-Zhao, R. Tao, and Y. Wang, Signal Process. 87, 983 (2007).
[CrossRef]

A. Stern, Signal Process. 86, 1421 (2006).
[CrossRef]

J. J. Healy and J. T. Sheridan, Signal Process. 89, 641 (2009).
[CrossRef]

Other (3)

T. Kreis, Handbook of Holographic Interferometry, Optical and Digital Methods (Wiley-VCH, 2005).

A. Stern, in Proceedings of the 5th International Workshop on Information Optics, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (Springer, 2006), pp. 225-234.

J.-J. Ding, Ph.D. thesis (National Taiwan University, 2001).

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Figures (2)

Fig. 1
Fig. 1

Tracing the PSD of a signal through a general LCT system decomposed into a chirp, scaled Fourier transform, and a second chirp. (a) Initial PSD, assuming width W and output width W T . The black dot in the upper right corner is to aid the reader in following the transformations. (b) After the first chirp multiplication. (c) After the Fourier transform. (d) After the second chirp multiplication.

Fig. 2
Fig. 2

Tracing the PSD of a signal through a numerical LCT using the DM. (a) Initial PSD, assuming width W and output width W ̂ . (b) First sampling operation. The ellipses indicate infinite, periodic replication. (c) First chirp multiplication. (d) Discrete-space Fourier transform. (e) Second sampling operation. Note that (d) and (e) are achieved in one step using an FFT. (f) Second chirp multiplication.

Tables (1)

Tables Icon

Table 1 Comparison of Sampling Rates from [4] with Those of this Letter for an Example Input and Various Propagation Distances, z

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