## Classification of lossless first-order optical systems and the linear canonical transformation

JOSA A, Vol. 24, Issue 4, pp. 1053-1062 (2007)

http://dx.doi.org/10.1364/JOSAA.24.001053

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### Abstract

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 first-order optical system.

© 2007 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: July 18, 2006

Revised Manuscript: October 27, 2006

Manuscript Accepted: November 10, 2006

Published: March 14, 2007

**Citation**

Martin J. Bastiaans and Tatiana Alieva, "Classification of lossless first-order optical systems and the linear canonical transformation," J. Opt. Soc. Am. A **24**, 1053-1062 (2007)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-4-1053

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