Publication: Classification of lossless first-order optical systems and the linear canonical transformation
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Optical Society of America
Based on the eigenvalues of the ray transformation matrix, a classification of ABCD systems is proposed and some nuclei (i.e., elementary members) in each class are described. In the one-dimensional case, possible nuclei are the magnifier, the lens, and the fractional Fourier transformer. In the two-dimensional case we have-in addition to the obvious concatenations of one-dimensional nuclei-the four combinations of a magnifier or a lens with a rotator or a shearing operator, where the rotator and the shearer are obviously inherently two-dimensional. Any ABCD system belongs to one of the classes described in this paper and is similar (in the sense of matrix similarity of the ray transformation matrices) to the corresponding nucleus. Knowledge of a nucleus may be helpful in finding eigenfunctions of the corresponding class of first-order optical systems: one only has to find eigenfunctions of the nucleus and to determine how these functions propagate through a firstorder optical system.
© 2007 Optical Society of America. M. J. Bastiaans appreciates the hospitality at Universidad Complutense de Madrid. T. Alieva acknowledges the Spanish Ministry of Education and Science (project TEC 2005-02180/MIC). M. J. Bastiaans’ e-mail address is firstname.lastname@example.org. T. Alieva’s e-mail address is email@example.com.
1. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1966). 2. A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data-processing systems,” Proc. IEEE 54, 1055–1063 (1966). 3. H. J. Butterweck, “General theory of linear, coherent optical data-processing systems,” J. Opt. Soc. Am. 67, 60–70 (1977). 4. M. Nazarathy and J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982). 5. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996). 6. K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004). 7. S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). 8. M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971). 9. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979). 10. S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002). 11. V. Namias, “The fractional order Fourier transform and its applications to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980). 12. A. C. McBride and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987). 13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). 14. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875– 881 (1993). 15. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993). 16. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001). 17. F. R. Gantmacher, The Theory of Matrices (Chelsea, 1974–1977). 18. M. J. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A 23, 1875–1883 (2006). 19. W. W. Lin, V. Mehrmann, and H. Xu, “Canonical forms for Hamiltonian and symplectic matrices and pencils,” Linear Algebr. Appl. 302–303, 469–533 (1999). 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. 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). 22. T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005). 23. 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). 24. M. J. Bastiaans and T. Alieva, “First-order optical systems with real eigenvalues,” Opt. Commun. 272, 52–55 (2007).