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

  • Vol. 19, Iss. 6 — Jun. 1, 2002
  • pp: 1191–1196

Fractional Fourier transformers through reflection

Kurt Bernardo Wolf and Guillermo Krötzsch  »View Author Affiliations


JOSA A, Vol. 19, Issue 6, pp. 1191-1196 (2002)
http://dx.doi.org/10.1364/JOSAA.19.001191


View Full Text Article

Enhanced HTML    Acrobat PDF (158 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We show that an arbitrary paraxial optical system, compounded with its reflection in an appropriately warped mirror, is a pure fractional Fourier transformer between coincident input and output planes. The geometric action of reflection on optical systems is introduced axiomatically and is developed in the paraxial regime. The correction of aberrations by warp of the mirror is briefly addressed.

© 2002 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(080.1010) Geometric optics : Aberrations (global)
(080.2720) Geometric optics : Mathematical methods (general)
(080.2730) Geometric optics : Matrix methods in paraxial optics

History
Original Manuscript: November 2, 2001
Revised Manuscript: December 6, 2001
Manuscript Accepted: December 6, 2001
Published: June 1, 2002

Citation
Kurt Bernardo Wolf and Guillermo Krötzsch, "Fractional Fourier transformers through reflection," J. Opt. Soc. Am. A 19, 1191-1196 (2002)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-19-6-1191


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. D. Mendlovic, H. M. Ozaktas, “Fourier transforms of fractional order and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993). [CrossRef]
  2. H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).
  3. M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986). [CrossRef]
  4. K. B. Wolf, G. Krötzsch, “La transformación raı́z de superficies refractantes y espejos,” Rev. Mex. Fı́s. 37, 540–554 (1991).
  5. R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000). [CrossRef]
  6. R. Simon, K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000). [CrossRef]
  7. M. Moshinsky, “Canonical transformations in quantum mechanics,” SIAM J. Appl. Math. 25, 193–212 (1973). [CrossRef]
  8. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1985). [CrossRef]
  9. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995). [CrossRef]
  10. E. J. Atzema, G. Krötzsch, K. B. Wolf, “Canonical transformations to warped surfaces: correction of aberrated optical images,” J. Phys. A 30, 5793–5803 (1997). [CrossRef]
  11. R. Simon, “Peres–Hodorecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000). [CrossRef] [PubMed]
  12. A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982). [CrossRef]
  13. S. Steinberg, “Lie series, Lie transformations, and their applications” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 3, pp. 45–102.
  14. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 4, pp. 105–158.
  15. O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 5, pp. 159–182.
  16. A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987). [CrossRef]
  17. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). [CrossRef]
  18. K. B. Wolf, G. Krötzsch, “El problema de las tres lentes,” Rev. Mex. Fı́s. 47, 291–298 (2001).
  19. K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988). [CrossRef]
  20. See, e.g., H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  21. K. B. Wolf, “Nonlinearity in aberration optics,” in Symmetries and Non-linear Phenomena, Proceedings of the International School on Applied Mathematics, Centro Internacional de Fı́sica, Paipa, Colombia, D. Levi, P. Winternitz eds., (World Scientific, Singapore, 1988), pp. 376–429.
  22. K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991). [CrossRef]
  23. K. B. Wolf, G. Krötzsch, “mexLIE 2, A set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS-UNAM No. 10 (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos 62251, México, 1995).
  24. K. B. Wolf, G. Krötzsch, “Metaxial correction of fractional Fourier transformers,” J. Opt. Soc. Am. A 16, 821–830 (1999). [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.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited