First-order optical systems with unimodular eigenvalues.

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.

[1]  Haldun M. Özaktas,et al.  The fractional fourier transform , 2001, 2001 European Control Conference (ECC).

[2]  Fractionalization of cyclic transformations in optics , 2005 .

[3]  Mj Martin Bastiaans,et al.  Fractional cyclic transforms in optics: theory and applications , 2001 .

[4]  Tatiana Alieva,et al.  Generating function for Hermite-Gaussian modes propagating through first-order optical systems , 2005 .

[5]  F. H. Kerr,et al.  On Namias's fractional Fourier transforms , 1987 .

[6]  R. K. Luneburg,et al.  Mathematical Theory of Optics , 1966 .

[7]  Joseph W. Goodman,et al.  Introduction to Fourier Optics; Second Edition , 1996 .

[8]  B. M. Fulk.  MATH , 1992 .

[9]  Tatiana Alieva,et al.  Mode mapping in paraxial lossless optics. , 2005, Optics letters.

[10]  V. Namias.  The Fractional Order Fourier Transform and its Application to Quantum Mechanics , 1980 .

[11]  K. Wolf,et al.  Structure of the set of paraxial optical systems. , 2000, Journal of the Optical Society of America. A, Optics, image science, and vision.

[12]  A. Lohmann.  Image rotation, Wigner rotation, and the fractional Fourier transform , 1993 .

[13]  Martin J. Bastiaans,et al.  Fractional Transforms in Optical Information Processing , 2005, EURASIP J. Adv. Signal Process..

[14]  T. Alieva,et al.  Fractionalization of the linear cyclic transforms. , 2000, Journal of the Optical Society of America. A, Optics, image science, and vision.

[15]  A. Lohmann,et al.  IV: Fractional Transformations in Optics , 1998 .

[16]  I. Segal.  Distributions in Hilbert space and canonical systems of operators , 1958 .

[17]  Granino A. Korn,et al.  Mathematical handbook for scientists and engineers. Definitions, theorems, and formulas for reference and review , 1968 .

[18]  Kurt Bernardo Wolf,et al.  Geometric Optics on Phase Space , 2004 .

[19]  N. Mukunda,et al.  Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams. , 1998 .

[20]  Z. Zalevsky,et al.  The Fractional Fourier Transform: with Applications in Optics and Signal Processing , 2001 .

[21]  Christiane Quesne,et al.  Linear Canonical Transformations and Their Unitary Representations , 1971 .

[22]  V. Bargmann.  On a Hilbert space of analytic functions and an associated integral transform part I , 1961 .

[23]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[24]  S. A. Collins.  Lens-System Diffraction Integral Written in Terms of Matrix Optics , 1970 .

[25]  Tatiana Alieva,et al.  Alternative representation of the linear canonical integral transform. , 2005, Optics letters.

[26]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .