## Improved implementation algorithms of the two-dimensional nonseparable linear canonical transform |

JOSA A, Vol. 29, Issue 8, pp. 1615-1624 (2012)

http://dx.doi.org/10.1364/JOSAA.29.001615

Enhanced HTML Acrobat PDF (577 KB)

### Abstract

The two-dimensional nonseparable linear canonical transform (2D NSLCT), which is a generalization of the fractional Fourier transform and the linear canonical transform, is useful for analyzing optical systems. However, since the 2D NSLCT has 16 parameters and is very complicated, it is a great challenge to implement it in an efficient way. In this paper, we improved the previous work and propose an efficient way to implement the 2D NSLCT. The proposed algorithm can minimize the numerical error arising from interpolation operations and requires fewer chirp multiplications. The simulation results show that, compared with the existing algorithm, the proposed algorithms can implement the 2D NSLCT more accurately and the required computation time is also less.

© 2012 Optical Society of America

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(070.2590) Fourier optics and signal processing : ABCD transforms

(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

(080.2575) Geometric optics : Fractional Fourier transforms

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: April 2, 2012

Revised Manuscript: June 13, 2012

Manuscript Accepted: June 18, 2012

Published: July 20, 2012

**Citation**

Jian-Jiun Ding, Soo-Chang Pei, and Chun-Lin Liu, "Improved implementation algorithms of the two-dimensional nonseparable linear canonical transform," J. Opt. Soc. Am. A **29**, 1615-1624 (2012)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-8-1615

Sort: Year | Journal | Reset

### References

- H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).
- H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995). [CrossRef]
- K. B. Wolf, “Canonical Transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), pp. 381–416.
- L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996). [CrossRef]
- Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD—Bessel transformation,” Opt. Commun. 147, 39–41 (1998). [CrossRef]
- S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994). [CrossRef]
- G. B. Folland, Harmonic Analysis in Phase Space (Princeton University, 1989).
- 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]
- 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]
- J. J. Ding and S. C. Pei, “Eigenfunctions and self-imaging phenomena of the two dimensional nonseparable linear canonical transform,” J. Opt. Soc. Am. A 28, 82–95 (2011). [CrossRef]
- H. M. Ozaktas and O. Arikan, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150, (1996). [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]
- A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008). [CrossRef]
- D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006). [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 transform,” J. Opt. Soc. Am. A 27, 21–30 (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]
- X. Deng, B. Bihari, J. Gan, F. Zhou, and R. T. Chen, “Fast algorithm for chirp transforms with zooming–in ability and its applications,” J. Opt. Soc. Am. A 17, 762–771 (2000). [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]
- T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005). [CrossRef]
- K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).
- M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, 1990).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.