## First-order optical systems with unimodular eigenvalues

JOSA A, Vol. 23, Issue 8, pp. 1875-1883 (2006)

http://dx.doi.org/10.1364/JOSAA.23.001875

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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.

© 2006 Optical Society of America

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.4690) Fourier optics and signal processing : Morphological transformations

(080.2730) Geometric optics : Matrix methods in paraxial optics

(120.4820) Instrumentation, measurement, and metrology : Optical systems

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: November 10, 2005

Revised Manuscript: February 7, 2006

Manuscript Accepted: February 10, 2006

**Citation**

Martin J. Bastiaans and Tatiana Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Am. A **23**, 1875-1883 (2006)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-8-1875

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