## Self-affinity in phase space

JOSA A, Vol. 17, Issue 4, pp. 756-761 (2000)

http://dx.doi.org/10.1364/JOSAA.17.000756

Enhanced HTML Acrobat PDF (145 KB)

### Abstract

The expression for the Wigner distribution (WD) in polar coordinates was derived, based on the decomposition of coherent and partially coherent fields on the orthogonal sets of Hermite–Gauss modes. This representation allows one to analyze easily the structure of the WD and to describe the field propagation through first-order optical systems, including the self-imaging phenomenon.

© 2000 Optical Society of America

**OCIS Codes**

(070.1170) Fourier optics and signal processing : Analog optical signal processing

(110.1650) Imaging systems : Coherence imaging

(110.6980) Imaging systems : Transforms

**History**

Original Manuscript: June 21, 1999

Manuscript Accepted: October 5, 1999

Published: April 1, 2000

**Citation**

Tatiana Alieva and Martin J. Bastiaans, "Self-affinity in phase space," J. Opt. Soc. Am. A **17**, 756-761 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-4-756

Sort: Year | Journal | Reset

### References

- R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1966).
- K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
- M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982). [CrossRef]
- C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).
- V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]
- D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993). [CrossRef]
- D. F. V. James, G. S. Agarwal, “The generalised Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996). [CrossRef]
- G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992). [CrossRef]
- T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997). [CrossRef]
- T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998). [CrossRef]
- T. Alieva, M. J. Bastiaans, “Self-imaging in first-order optical systems,” in Optics and Optoelectronics: Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, pp. 126–131.
- T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999). [CrossRef]
- T. Alieva, A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999). [CrossRef]
- M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
- M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution, W. Mecklenbräuker, F. Hlawatsch, eds. (Elsevier, Amsterdam, 1997).
- T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998). [CrossRef]
- I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 838.
- S. Y. Lee, H. H. Szu, “Fractional Fourier transforms, wavelet transforms, and adaptive neural networks,” Opt. Eng. 33, 2326–2329 (1994). [CrossRef]

## 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.