OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 23, Iss. 8 — Aug. 1, 2006
  • pp: 1875–1883

First-order optical systems with unimodular eigenvalues

Martin J. Bastiaans and Tatiana Alieva  »View Author Affiliations


JOSA A, Vol. 23, Issue 8, pp. 1875-1883 (2006)
http://dx.doi.org/10.1364/JOSAA.23.001875


View Full Text Article

Enhanced HTML    Acrobat PDF (139 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the separable fractional Fourier transformer are related by means of a similarity transformation. Moreover, it is shown that the system that performs this similarity transformation is itself a lossless first-order optical system. Based on the fact that Hermite–Gauss functions are the eigenfunctions of a fractional Fourier transformer, the eigenfunctions of a unimodular first-order optical system can be formulated and belong to the recently introduced class of orthonormal Hermite–Gaussian-type modes. Two decompositions of a unimodular first-order optical system are considered, and one of them is used to derive an easy optical realization in more detail.

© 2006 Optical Society of America

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(070.4690) Fourier optics and signal processing : Morphological transformations
(080.2730) Geometric optics : Matrix methods in paraxial optics
(120.4820) Instrumentation, measurement, and metrology : Optical systems

ToC Category:
Fourier Optics and Optical Signal Processing

History
Original Manuscript: November 10, 2005
Revised Manuscript: February 7, 2006
Manuscript Accepted: February 10, 2006

Citation
Martin J. Bastiaans and Tatiana Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Am. A 23, 1875-1883 (2006)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-8-1875


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (North-Holland, 1998), pp. 263-342. [CrossRef]
  2. T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional transforms in optical information processing," EURASIP J. Appl. Signal Process. 2005, 1498-1519 (2005). [CrossRef]
  3. T. Alieva and M. L. Calvo, "Fractionalization of the linear cyclic transforms," J. Opt. Soc. Am. A 17, 2330-2338 (2000). [CrossRef]
  4. T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractional cyclic transforms in optics: theory and applications," in Recent Research and Developments in Optics, S.G.Pandalai, ed. (Research Signpost, 2001), Vol. 1, pp. 105-122.
  5. T. Alieva, M. J. Bastiaans, and M. L. Calvo, "Fractionalization of cyclic transformations in optics," in Proceedings of ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, December 12-15, 2005, Paper IT-OIP-2, pp. 1-15.
  6. S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970). [CrossRef]
  7. M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1780 (1971). [CrossRef]
  8. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).
  9. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  10. A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993). [CrossRef]
  11. D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation I," J. Opt. Soc. Am. A 10, 1875-1881 (1993). [CrossRef]
  12. H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transforms and their optical implementation II," J. Opt. Soc. Am. A 10, 2522-2531 (1993). [CrossRef]
  13. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  14. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (McGraw-Hill, 1968). [PubMed]
  15. K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).
  16. V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980). [CrossRef]
  17. A. C. McBride and F. H. Kerr, "On Namias' fractional Fourier transforms," IMA J. Appl. Math. 39, 159-175 (1987). [CrossRef]
  18. V. Bargmann, "On a Hilbert space of analytic functions and an associated integral transform, Part I," Commun. Pure Appl. Math. 14, 187-214 (1961). [CrossRef]
  19. I. E. Segal, "Distributions in Hilbert spaces and canonical systems of operators," Trans. Am. Math. Soc. 88, 12-42 (1958). [CrossRef]
  20. R. Simon and K. B. Wolf, "Structure of the set of paraxial optical systems," J. Opt. Soc. Am. A 17, 342-355 (2000). [CrossRef]
  21. M. J. Bastiaans and T. Alieva, "Generating function for Hermite-Gaussian modes propagating through first-order optical systems," J. Phys. A 38, L73-L78 (2005). [CrossRef]
  22. T. Alieva and M. J. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005). [CrossRef] [PubMed]
  23. R. Simon and N. Mukunda, "Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams," J. Opt. Soc. Am. A 15, 2146-2155 (1998). [CrossRef]
  24. T. Alieva and M. J. Bastiaans, "Alternative representation of the linear canonical integral transform," Opt. Lett. 30, 3302-3304 (2005). [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.

CrossCheck Deposited