Publication: First-order optical systems with unimodular eigenvalues
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2006-08-01
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Optical Society of America
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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.
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© 2006 Optical Society of America. The Spanish Ministry of Education and Science is acknowledged for financial support: Ramon y Cajal grant and project TIC 2002-01846 (T. Alieva) and SAB2004-0018 (M. J. Bastiaans). Stimulating discussions with K. Bernardo Wolf (Universidad Nacional Autónoma de México, Cuernavaca) are gratefully acknowledged. The authors’ e-mail addresses are m.j.bastiaans@tue.nl and talieva@fis.ucm.es.
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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.
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).
3. T. Alieva and M. L. Calvo, “Fractionalization of the linear cyclic transforms,” J. Opt. Soc. Am. A 17, 2330–2338 (2000).
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).
7. M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
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).
11. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
12. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
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).
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).
17. A. C. McBride and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
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).
19. I. E. Segal, “Distributions in Hilbert spaces and canonical systems of operators,” Trans. Am. Math. Soc. 88, 12–42 (1958).
20. R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
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).
22. T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. 30, 1461–1463 (2005).
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).
24. T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005).