NORTHWESTERN UNIVERSITY Eigenfunctions and Eigenoperators of Cyclic Integral Transforms with Application to Gaussian Beam Propagation A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Applied Mathematics By Matthew S. McCallum EVANSTON, ILLINOIS December 2008 2 c Copyright by Matthew S. McCallum 2008 All Rights Reserved 3 ABSTRACT Eigenfunctions and Eigenoperators of Cyclic Integral Transforms with Application to Gaussian Beam Propagation Matthew S. McCallum An integral transform which reproduces a transformable input function after a finite number N of successive applications is known as a 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, 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, 4 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 math- ematical physics community – hyperdifferential operators and 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 iso- morphism 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. 5 Acknowledgements I would like to thank my mother Carol and my wife Hilary for their constant love and support over the years. I would also like to thank my son Max for arriving a month early and making the completion of this work very interesting, to say the least. I would also like to thank Theodoros Horikis for much fruitful discussion and collaboration in the early phases of this work. I would finally like to thank my advisor, William L. Kath, for his patient assistance, support, and encouragement over the course of my general graduate studies as well as this specific project. 6 Table of Contents ABSTRACT 3 Acknowledgements 5 Chapter 1. Introduction 9 Chapter 2. Differential Eigenoperators 16 Chapter 3. General Theory – Eigenfunctions 19 Chapter 4. General Theory – Eigenoperators 27 Chapter 5. Fourier Eigenoperators Lmn – Additional Comments & Examples 39 Chapter 6. Lmn – Composition and sech mapping 49 Chapter 7. Lmn – Beyond whole-number (m, n) 52 Chapter 8. Decomposition into Eigenobjects 59 Chapter 9. Summary of One-Dimensional Results 63 Chapter 10. Hyperdifferential Operators 65 Chapter 11. Transition to Two Dimensions 70 Chapter 12. Linear Canonical Transforms in One and Two Dimensions 75 7 Chapter 13. Standard Fourier Transform 83 Chapter 14. Fresnel Transform 85 Chapter 15. Fractional Fourier Transform – Definition and Basic Properties 94 Chapter 16. Fractional Fourier Transform – Application to Diffraction 104 Chapter 17. Standard Hankel Transform 120 Chapter 18. Fresnel-Hankel Transform 124 Chapter 19. Fractional Hankel Transform – Definition and Basic Properties 126 Chapter 20. Fractional Hankel Transform – Application to Diffraction 130 Chapter 21. Comments on Self-Adjoint Operators 138 Chapter 22. Group-theoretical Comments 141 Chapter 23. Conclusion 148 References 152 Appendix . Further Results on Fourier Eigenoperators 155 1. Singularities 156 2. Regular/Frobenius forms 168 3. More on Equidimensional Cases 178 4. Irregular/Normal forms 184 8 CHAPTER 1 Introduction The determination of the eigenfunctions of an integral transform is a problem of some importance in multiple areas of applied mathematics. Indeed one could argue that the “eigenproblem” is fundamental from a purely mathematical perspective, regardless of the specific context in which it arises. Given a process or transformation which maps one set of objects to another, it is natural and intriguing to pose the question whether a subset of the objects in the domain of the transformation remains invariant in some way under the action of that transformation. Such objects would be mapped to themselves, with possible attendant minor modifications in such things as scale, proportionality factors, etc. The transformation itself can be represented by any number of various mathematical struc- tures, depending on the nature of the transformation and the objects upon which it acts. Examples from different contexts include similarity transformations acting on members of a discrete or continuous group, which may or may not be a group of matrices; differential transformations acting on members of a continuous function space defined on a smooth manifold; and integral transformations or their discrete analog, matrix transformations, acting on suitably-behaved continuous functions or their discretizations. Of course the last two types can be closely developed in parallel via the Fredholm alternative theory. Here we take as our focus of interest a certain class of integral transforms as they are applied in the field of optics. In particular we are interested in transforms which are labeled cyclic or periodic; that is, after a finite number N of successive applications to a 9 particular input function in their respective domains, that input function is reproduced. Expressed in another way, the N-fold action of such an integral transform upon the input is equivalent to the identity transform. Of course, every function in the domain is an eigenfunction of the N-fold transform; we mean to consider here the eigenfunctions of a single action of the transform. The Fourier integral transform is probably the most well- known example of such a transform, but there are others. We shall see that a number of cyclic transforms have complex exponential kernels, which is suggestive of their periodic nature. The eigenfunctions of such transforms can be used to describe diverse phenomena of interest to researchers such as resonant laser modes, non-diffracting beams, and self- imaging intensity distributions within the field of optics, but their application is not limited exclusively to this field. Fundamental results in quantum mechanics and even in analytic number theory are related to eigenfunctions and so-called eigenoperators (to be defined below) of the Fourier transform. Eigenfunctions of the Mellin integral transform are also important in the latter area; although strictly speaking this transform is not cyclic on its face, a simple change of variables transforms it into the Fourier transform. The purpose of the present work is to develop a new formalism from first principles for finding and characterizing eigenfunctions of cyclic integral transforms. The theoret- ical groundwork for this approach to the problem was partially laid by K.B. Wolf from the standpoint of mathematical physics in a series of papers in the 1970’s[41, 42, 43]. In this work, Wolf showed how the eigenfunction problem for a certain class of integral transform operators with importance in quantum mechanics – the so-called linear canon- ical transforms, or LCT’s, which will be discussed in greater depth below – could be cast in terms of hyperdifferential operators. These are essentially differential operators 10 which can be exponentiated as though they were variables to give an equivalent form of the original integral transform. The crucial point is that via this exponentiation process, the eigenfunctions of the integral operator can be easily related to the eigenfunctions of the differential operator. Hence one can systematically determine eigenfunctions of the integral operator by examining eigenfunctions of the corresponding differential operator which is used to generate it. The eigenproblem for an integral transform – not an easy nut to crack in general – is thereby recast in the form of an ordinary differential equation, with all the vast and straightforward machinery of ODE theory now at one’s disposal for attacking the problem. Underlying this formalism is the machinery of Lie algebras and continuous transformation groups, which Wolf gives some discussion of; in the following pages we will make a few brief comments about these aspects of the problem when ap- propriate, but it is not the aim of this work to give a rigorous analysis of the theoretical underpinnings of this approach. Our approach provides extensions and generalizations of previous work in several ways. First,