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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 30, Iss. 10 — Oct. 1, 2013
  • pp: 2096–2110

Differential commuting operator and closed-form eigenfunctions for linear canonical transforms

Soo-Chang Pei and Chun-Lin Liu  »View Author Affiliations


JOSA A, Vol. 30, Issue 10, pp. 2096-2110 (2013)
http://dx.doi.org/10.1364/JOSAA.30.002096


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Abstract

The linear canonical transform (LCT) with a, b, c, d parameter plays an important role in quantum mechanics, optics, and signal processing. The eigenfunctions of the LCT are also important because they describe the self-imaging phenomenon in optical systems. However, the existing solutions for the eigenfunctions of the LCT are divided into many cases and they lack a systematic way to solve these eigenfunctions. In this paper, we find a linear, second-order, self-adjoint differential commuting operator that commutes with the LCT operator. Hence, the commuting operator and the LCT share the same eigenfunctions with different eigenvalues. The commuting operator is very general and simple when it is compared to the existing multiple-parameter differential equations. Then, the eigenfunctions can be derived systematically. The eigenvalues of the commuting operator have closed-form relationships with the eigenvalues of the LCT. We also simplify the eigenfunctions for |a+d|>2 and a+d=±2, b0 into the more compact closed form instead of the integral form. For |a+d|>2, the eigenfunctions are related to the parabolic cylinder functions.

© 2013 Optical Society of America

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(070.2590) Fourier optics and signal processing : ABCD transforms
(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: April 16, 2013
Revised Manuscript: August 28, 2013
Manuscript Accepted: August 29, 2013
Published: September 25, 2013

Citation
Soo-Chang Pei and Chun-Lin Liu, "Differential commuting operator and closed-form eigenfunctions for linear canonical transforms," J. Opt. Soc. Am. A 30, 2096-2110 (2013)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-10-2096


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