Eigenfunctions and Eigenoperators of Cyclic Integral Transforms with Application to Gaussian Beam Propagation

Matthew McCallum

doi:https://doi.org/10.21985/N2FF1D

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Eigenfunctions and Eigenoperators of Cyclic Integral Transforms with Application to Gaussian Beam Propagation

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An integral transform which reproduces a transformable input function after a finite number $N$ of successive applications is known as a {\it cyclic} transform. Of course, such a transform will reproduce an arbitrary transformable input after $N$ applications, but it also admits eigenfunction inputs which will be reproduced after a single application of the transform. These transforms and their eigenfunctions appear in various applications, and the systematic determination of eigenfunctions of cyclic integral transforms has been a problem of interest to mathematicians since at least the early twentieth century. In this work we review the various approaches to this problem, providing generalizations of published expressions from previous approaches. We then develop a new formalism, {\it differential eigenoperators}, that reduces the eigenfunction problem for a cyclic transform to an eigenfunction problem for a corresponding ordinary differential equation. In this way we are able to relate eigenfunctions of integral equations to boundary-value problems, which are typically easier to analyze. We give extensive examples and discussion via the specific case of the Fourier transform. We also relate this approach to two formalisms that have been of interest to the mathematical physics community -- {\it hyperdifferential operators} and {\it linear canonical transforms}. We show how this new approach reproduces known results of Fourier optics regarding free-space diffractive propagation of Gaussian beams in both one and two dimensions. Finally we discuss the group-theoretical aspects of the formalism and describe an isomorphism between roots of the identity transform and complex roots of unity. In the appendix we derive several technical results related to integrability and transformability of solutions in the Fourier transform case, and we prove two theorems -- one of them new -- on polynomial roots. We conclude that the formalism offers a new and equally valuable perspective on an interesting eigenfunction problem with both pure and applied aspects; we also conclude that this approach offers specific advantages over previous approaches to the problem.

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