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

M.J. Bastiaans, T. Alieva

Research output: Contribution to journalArticleAcademicpeer-review

28 Citations (Scopus)

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 twodimensional. 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
Original languageEnglish
Pages (from-to)1053-1062
Number of pages9
JournalJournal of the Optical Society of America A, Optics, Image Science and Vision
Volume24
Issue number4
DOIs
Publication statusPublished - 2007

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Linear transformations
Optical systems
Eigenvalues and eigenfunctions
nuclei
Lenses
magnification
Shearing
rays
eigenvectors
lenses
shearing
transformers
eigenvalues
operators
matrices

Cite this

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title = "Classification of lossless first-order optical systems and the linear canonical transformation",
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 twodimensional. 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. {\circledC} 2007 Optical Society of America",
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Classification of lossless first-order optical systems and the linear canonical transformation. / Bastiaans, M.J.; Alieva, T.

In: Journal of the Optical Society of America A, Optics, Image Science and Vision, Vol. 24, No. 4, 2007, p. 1053-1062.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Bastiaans, M.J.

AU - Alieva, T.

PY - 2007

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AB - 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 twodimensional. 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

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