Bastiaans, Martin J.Alieva Krasheninnikova, Tatiana2023-06-202023-06-202006-08-011084-752910.1364/JOSAA.23.001875https://hdl.handle.net/20.500.14352/51272© 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.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.engjournal articlehttp://dx.doi.org/10.1364/JOSAA.23.001875http://www.opticsinfobase.org/open access535Fractional fourier-transformsIntegral transformImplementationÓptica (Física)2209.19 Óptica Física