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

NORTHWESTERN UNIVERSITY

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

for the degree

DOCTOR OF PHILOSOPHY

Field of Applied Mathematics

Matthew S. McCallum

EVANSTON, ILLINOIS

December 2008 2

c Copyright by Matthew S. McCallum 2008

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 mathematical 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 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. 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 structures, 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 theoretical 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 canonical 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 appropriate, 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, we exploit the cyclic property of our integral transforms of interest as the foundation of our new method. Second, many authors make no mention of the commuting relationship between differential and integral operators, which relationship will be explained and exploited more fully below. Like the cyclicity property, we show that this commutation property is fundamental in the development of our approach. Third, previous researchers such as Wolf have limited their analysis to first- and second-order differential operators, no doubt due to the fact that most problems in mathematical physics do not give the 11 researcher much compelling reason to look beyond the second order. We show that differential operators of arbitrary order can be used to generate eigenfunctions of the corresponding integral transform, even if these operators do not admit of immediately evident physical application.

Fourth, we apply our approach to cyclic operators with applications to optics always foremost in mind, pointing out connections to other fields when possible. Fifth, several previous researchers make no mention of the possibility of ”fractionalizing” their transforms. That is to say, some integral transforms with a finite cycle can be defined in terms of a parameter with bounded discrete integer values; this provides an isomorphism between the transform and a discrete cyclic group, with the various powers of the transform constituting elements in the group. When these parameter values are interpolated to non-integer values, a continuous range of operators can be defined, corresponding to so-called fractional integral transforms, some of which retain the cyclic group property.

Thus, underlying the set of discrete cyclic groups of a given transform is a continuous transformation group. We illuminate these fractionalization issues as well and show how they arise naturally in the context of cyclic transforms.

To be more concrete about the nature of our contributions to this area of research, we mention additional points contained in the present work. In this document we provide generalizations of previous formulas for generating eigenfunctions of cyclic transforms.

We give the most general form that an eigenfunction of a generic cyclic transform must have, and we explain how symmetry properties and transform eigenvalues must be related a priori for generic cyclic transforms; we specialize these results to yield previously published results for specific transforms such as the Fourier transform. Furthermore, we 12 define and introduce the concepts of transform of a differential operator and differential eigenoperator; these concepts are crucial in the development of this new method for generating eigenfuntions of cyclic transforms. We also indicate how to extend these concepts to two or more dimensions. In addition we show how the discrete Fourier transform can be used to decompose a function or operator into a superposition of basis functions or operators which are eigenobjects of any cyclic transform.

There are several points of contact between group theory and the formalism presented here. We point out multiple aspects of the group isomorphism between cyclic integral transforms and complex roots of unity, in particular showing how cyclic integral transforms can be viewed as roots of the identity transform, and we discuss various implications of this viewpoint. We also make comments indicating how the hyperdiﬀerential formalism can place the present anaylsis on a ﬁrmer theoretical foundation and we comment on connections with Lie group theory.

We present generalizations of previously published forms of the linear canonical transform (LCT) in 2-D which incorporate the previous forms as special cases. In particular,

I deﬁne and display the Fresnel-Hankel transform, representing Fresnel diﬀraction in 2-D with n-fold rotational symmetry.

We show how Fresnel and fractional Fourier transforms can be related via LCTs in both 1-D and 2-D. In particular, we show that the Fresnel and Fresnel-Hankel transforms can simply be deﬁned as the composition of the fractional Fourier transform or fractional

Hankel transform with another LCT. In a related vein, we show that the fractional Fourier and Hankel transforms oﬀer a more convenient and broader way to consider free-space diﬀraction in 1-D and 2-D, compared to their Fresnel transform counterparts. 13

We show how the eigenoperator formalism can be used to construct differential equations whose solutions represent eigenfunctions of the fractional Fourier transform in 1-D and the fractional Hankel transform in 2-D. Any such solutions that are finitely square- integrable represent self-similar diffracting beams with finite energy content. In particular, we show using our approach that Gauss-Hermite functions in 1-D and Gauss-Laguerre functions in 2-D represent finite energy, self-similar diffracting beams at all propagation distances. This reproduces known theoretical results about the propagation behavior of these waveforms. We also show that the scaling exhibited by these beams is not particular to them, but rather a property of all self-similarly diffracting profiles. We discuss these issues in further detail in later sections.

We discuss the conditions under which the eigenoperator formalism generates boundary- value problems whose eigenfunctions are potentially square-integrable and invariant under the action of a cyclic integral transform.

In the Appendix, we go into considerable technical detail to explore the existence and asymptotic nature of solutions to some of the eigenequations generated by eigenoperators of the Fourier transform. This is important for determining when solutions will actually be transformable, due to decay at inﬁnity, for example. We also prove two theorems on polynomial roots, which should prove helpful beyond the limits of this work for stability analysis.

All of the above points are new and not previously published, to the best that we can determine, with thorough searching and referencing.

In the opening sections that follow we lay out a new general procedure that may be used to determine eigenfunctions of cyclic integral transforms. A discussion of this 14 technique as applied to eigenfunctions of the Fourier transform – the so-called Fourier eigenfunctions (FEF’s) or self-Fourier functions (SFF’s) – can be found in an earlier paper by McCallum & Horikis[16]. We will also refer to the Fourier case in what follows to make the discussion concrete. As already mentioned, applications of SFF’s can be found in such diverse areas as optics[4, 21], quantum mechanics[13, 20], and analytic number theory[37, 10]. Recent developments focus on generalizations of known SFF’s and their applications to coherent laser design[29, 5, 6]. We shall see how the arena of eigenvalue problems (EVP’s), so amenable to treatment by the Fourier transform, can be used to generate and characterize eigenfunctions of not only that transform, but any cyclic integral transform. That is the main goal of the present body of work – to show how EVP’s (and their close cousin, boundary-value problems, or BVP’s) can be used to

ﬁnd eigenfunctions of any cyclic integral transform. Some of the material in the following sections is taken from a recent paper by the same authors[25]. Further details have been added that were not included in this original paper due to space constraints. 15

CHAPTER 2

Diﬀerential Eigenoperators

We define the integral transform Lˆ of a linear differential operator L to be that operator whose action on the transformed function fˆ results from transforming the action of L on f, i.e. Lfc ≡ Lˆfˆ. Then an eigenoperator is a linear differential operator which is formally invariant under an integral transform T ; namely, T [L] ≡ Lˆ = cL for some constant c, complex in general. We sometimes refer to c as a commuting constant, and we say that the transform and the operator commute to the constant c. Note that the choice of variable is irrelevant; whether we consider the operator to act on a function of x, t, k, or ω, for example, is immaterial. It is not the aspect of the transform that changes the domain of the input variable that interests us (e.g. from space to wavenumber or time to frequency in the case of the Fourier transform), but rather the formal structure of the differential operator itself. Hence whenever we apply a transform to an input operator or function, we take the output operator or function to have the same variable as the input. To clarify this point, we define the Fourier transform (FT) of the input function f(x) to be

∞ 1 Z 0 F{f(x0)} = fˆ(x) ≡ √ f(x0)eixx dx0 2π −∞ so that, for example, the FT of f(x) = e−x2/2 is fˆ(x) = e−x2/2. Strictly speaking, the exact form of the normalization constant is not really important for our purposes. 16

± m n n m m n With these preliminaries in mind, the operator Lmn ≡ x d /dx ± d /dx (x ·) is easily seen to be an eigenoperator of the Fourier transform:

m+n ˆ ˆ ˆ ˆ (2.1) L\mnf = ±(−i) Lmnf ≡ Lmnf ⇒ Lmn = cLmn where the commuting constant in this case is c = ±(−i)m+n. Sometimes for the sake of brevity we will suppress the superscripted ± and simply write Lmn. This result is easily veriﬁed using basic properties of the Fourier transform, whereby position and derivative operations are transformed into each other:

xdnf = (−i)ndn/dxnfˆ; dm\/dxmf = (−i)mxmfˆ

Denoting the Fourier transform by F, this result may also be written FLmn = cLmnF. At the moment, the successful identiﬁcation of this Fourier eigenoperator may seem rather ad hoc, like a lucky guess. We will show in the next section how this eigenoperator can be derived from ﬁrst principles.

For any integral transform, the general equation Lˆ = cL can be thought of as deﬁning the eigenoperator problem for the transform. Any operator satisfying this equation is an eigenoperator of the transform by virtue of commuting with the transform to the constant c; more importantly, we shall see that the eigenfunctions of L are potential eigenfunctions of the transform. The essential point is that the linear diﬀerential operators L that commute with T to a constant are precisely the eigenoperators of T . The operator

− 2 2 2 − L02 ≡ d /dx − x , from the eigenproblem for the Gauss-Hermite functions L02f = λf, is the most commonly, and typically only, cited example of such an eigenoperator for the Fourier transform, in works ranging over the decades from Wiener[40] to Ozaktas et 17 al. [32] In the language of our analysis, it is the classic self-Fourier operator, because

− 2 2 2 − Lc02 = −x + d /dx = L02, as can be seen by applying the basic Fourier theorems mentioned above. In this case, we have the commuting constant c = −(−i)0+2 = 1.

As such, the Fourier-transformable eigenfunctions of this operator – the Gauss-Hermite functions – are also Fourier eigenfunctions, a fact which has long been known. We discuss these eigenfunctions in more detail a little later.

Another way to think of the commuting relationship is by applying T −1 (we assume throughout that the generic transform is invertible) to both sides of T L = cLT :

T L = cLT ⇒ T LT −1 = cL which shows that up to the commuting constant c, the operator L is invariant under the similarity transformation deﬁned by T . These various interpretations of the eigenoperator relationship – commuting operators, invariance under action of the transform or its associated similarity transformation – are equivalent and each may be used as needed. 18

CHAPTER 3

General Theory – Eigenfunctions

With these definitions in mind, we will propose a general framework for finding the eigenfunctions of a periodic linear integral transform by finding the differential eigenoperators of this transform. Again, these are differential operators that are formally invariant under the action of the transform. A transform T is called N-cyclic (or N-periodic) if

N is the least natural number such that the application of T , N times in succession, to a function f reproduces that function. Several such transforms with optical applications were brieﬂy considered by Lohmann and Mendlovic[21, 22, 26], along with their eigenfunctions. However, before considering the construction of eigenoperators, it will be helpful to consider the construction of eigenfunctions of a cyclic transform. Once we un- derstand the general procedure for constructing transform eigenfunctions, we can apply it straight away to the construction of transform eigenoperators, with minor modiﬁcations.

We devote this section to the technical details of such constructions, then apply them to operators in the next section.

Consider the expression

N−1 X (3.1) f ≡ T ng n=0 where the function g is an arbitrary transformable input function; we refer to g and its various transforms T n[g] as generator functions. Then f is a self-transform function 19

(STF) or a transform eigenfunction (TEF) under the action of T since T [f] = f, due to cyclic permutation of the terms. That is, since T [(T n[g])] = T n+1[g], and T N = I, the identity transform (whose kernel is simply the Dirac delta function), the above expression is invariant under the action of T . Note that the transform sum in this equation is

PN−1 n basically the cyclotomic polynomial p(x) = n=0 x of degree N − 1 in the variable T . The action of this polynomial on the generator g generates the TEF f. Evidently, in order to generate a closed-form TEF from this scheme, one must evaluate the transform of multiple generator functions (since T n[g] = T [T n−1[g]]), which may or may not be easy to do. Also, the eigenequation for such a transform gives

T [f] = µf ⇒ T N [f] = µN f ⇒ µN = 1(3.2) since T N ≡ I, the identity transform. Hence the eigenvalues of T – which may be thought of as an N-th root of I – are precisely the complex Nth-roots of unity. We say more about transform roots of identity in later sections below.

We can take the Fourier transform case as a speciﬁc example. Since the fourth power of the Fourier transform yields the identity transform, the fourth power of the eigenvalues

µ must equal one:

F 4 = I ⇒ µ4 = 1

Thus the complete set of eigenvalues of the Fourier transform F, a fourth root of I, consists of the fourth roots of unity µ = ±1, ±i, corresponding to FEF’s called self-

Fourier, skew-Fourier, i-Fourier or skew-i-Fourier functions, respectively, in a previous paper by Horikis and McCallum[16]. 20

In theory, the complete set of SFF’s with µ = 1 is given by the generating formula given in Eqn. (3.1):

f(x) = g(x) +g ˆ(x) + g(−x) +g ˆ(−x)(3.3) where the parity property of the Fourier transform is used, which says that two successive

Fourier transforms of an input function reproduce the function with the sign of x changed:

F 2[f(x)] = f(−x). In practice, as discussed in their paper, the complete set of SFF’s

(µ = 1) is given by the generating formula presented by Horikis and McCallum:[16]

f(x) = g(x) +g ˆ(x)(3.4) where g(x) is an arbitrary, even function of x, complex in general. But this expression is missing terms compared to Eqn. (3.3) above. This is because one can argue that the expression in Eqn. (3.3) is simply the sum of the even parts of g andg ˆ. Therefore, f(x), being the sum of two even functions, is itself even. Writing ge = g(x) + g(−x) and gˆe =g ˆ(x) +g ˆ(−x) gives f(x) = ge(x) +g ˆe(x). But we may simply drop the subscripts ’e’ and consider the generator function g(x) to be intrinsically even to begin with; this gives

Eqn. (3.4) above. Alternatively, we may note that the above expression in Eqn. (3.4) does not include all terms which should be present from a strict reading of Eqn. (3.1); this is because the eigenvalue µ = 1 imposes symmetry requirements on the functions f and g, to wit:

fˆ = f ⇒ fbˆ = f(−x) = f(x) 21 by the parity property of the Fourier transform. Hence we ﬁnd that f(x) must be even.

This is the specialization of Eqn. (3.1) to the Fourier case. The “self-characteristic” functions with form h2(x) + hˆ ∗ hˆ(x) discussed by Nosratinia[30] in the context of probability theory are simply an instance of this more compact form, with g = h2 and ∗ denoting convolution. However one views the situation, the main complication at this point is that in order to generate an SFF, given a generator function g(x), one needs to evaluate its Fourier transform, a task which can be challenging. The eigenoperator approach will allow us to circumvent this potential diﬃculty.

We can see that the rest of the FEF’s corresponding to other Fourier eigenvalues

(µ 6= 1) can be generated in an entirely similar way. Without loss of generality we may consider the expression f = g + µ3gˆ + µ2g(−x) + µgˆ(−x) where µ is any Fourier eigenvalue; then fˆ =g ˆ+µ3g(−x)+µ2gˆ(−x)+µg = µf since µ4 = 1, so that the expression in Eqn. (3.1) can be specialized to generate FEF’s corresponding to whichever eigenvalue is desired. Although the x input looks one-dimensional on its face, these generating forms are valid for any number of dimensions in the argument. It seems reasonable to consider eigenfunction generation in two dimensions since most transverse intensity distributions of interest in optical applications are 2-D. As far as the diﬀerent approaches for generating

FEF’s are concerned, it is straightforward to extend to higher dimensions the generator function expressions for an FEF, but it is not immediately obvious how to vectorize the

Lmn eigenoperator approach described more fully below. Certainly the notion of partial rather than ordinary diﬀerential eigenoperators can be raised in conjunction with the latter approach. We comment on this and other aspects of higher-dimensional generalizations of the eigenoperator approach as we develop it in the sections that follow. 22

Rewriting the expression for f as f = g + µ2g(−x) + µ3(ˆg + µ2gˆ(−x)), we see that since µ2 = ±1, the strictly even or odd components of g andg ˆ will combine to result in

ﬁnal expressions like Eqn. (3.4) (where g in that equation represents the even part of the g function under discussion here). Hence we have, for µ = ±1, f = ge ± gˆe, while for

µ = ±i, f = go ∓ igˆo. The generated FEF f will inherit the same symmetry as that of the generator functions ge,o andg ˆe,o. This symmetry can also be discerned from the relation fbˆ = f(−x) = µ2f(x); hence µ = ±1 (resp. ±i) implies that g, gˆ and the resulting f will be even (resp. odd).

Further extensions of the above results can be made, relating the generator functions, their symmetries, and the transform eigenvalues. For example, we may inquire under what conditions the terms in the deﬁnition of the FT eigenoperator Lmn, considered

m (n) above, can be used to generate FEF’s. Again, this operator is deﬁned as Lmnf ≡ x f ±

(xnf)(m), for arbitrary transformable f(x). If we consider the general expression h = g +g ˆ with g ≡ xmf (n), where f is an arbitrary transformable function but not FEF, then gˆ = (−i)m+n(xnfˆ)(m), and hˆ = µh yields (−i)m+n(xnfˆ)(m) + (−1)m+nxm[f(−x)](n) =

µxmf (n) + µ(−i)m+n(xnfˆ)(m) and we see that we must have µ = 1, f(x) even, and m+n =

2k for equality to hold in general. Hence g, gˆ, and h will also be even; moreover, the

n ˆ (m) coeﬃcient on (x f) ing ˆ must be ±1, so the form of Lmn is reproduced. This shows the kind of condition that must hold to generate a SFF h(x) from the general recipe given in

Eqn. (3.4) with f now an arbitrary transformable function used to deﬁne g.

We can also consider the more general expression h = g + µ0gˆ in conjunction with an

FEF satisfying fˆ = αf, where µ0 and α are both Fourier eigenvalues (now f(x) is assumed to be FEF). Again taking g ≡ xmf (n) gives h = xmf (n) + µ0(−i)m+nα(xnf)(m) and hˆ = 23

(−i)m+nα(xnf)(m)+α2(−1)m+nµ0xmf (n). Requiring hˆ = µh gives µ = α2(−1)m+nµ0, µµ0 =

1 ⇒ 1 = α2(−1)m+nµ02 ⇒ µ0(−i)m+nα = ±1. This last product is exactly the coefficient on the (xnf)(m) term in the definition of the function h, so the final form for h becomes

m (n) n (m) h = x f ± (x f) which is exactly the expression for Lmnf. This is the form h must have if these particular summands are used to generate a new FEF (h) from a given FEF

(f). Note that the last relation above also couples the values of µ0 and α via the choice of (m, n); the value of µ is coupled to µ0, and hence to α, through µµ0 = 1. Hence all three Fourier eigenvalues present in the analysis are interrelated and cannot be chosen independently. Once one eigenvalue is specified, the other two are determined. For example, if the desired eigenvalue µ on h is specified, the values of the coefficient µ0 ong ˆ and the eigenvalue α on the input function f must satisfy the given constraints. Or, if the f-eigenvalue α is specified, then the h-eigenvalue µ is constrained.

This implies that a single eigenspace, so to speak, of the input f cannot span all four possible eigenspaces of the output h. Furthermore, if we now subtract an eigenterm κf in the expression for h, i.e. h = xmf (n) + µ0(−i)m+nα(xnf)(m) − κf, then the requirement hˆ = µh yields the same constraints as before with the additional constraints α = µ and m + n = 2k. So f and h must now be from the same eigenspace. Now, in this treatment the FEF f(x) is assumed to be generated by any method whatsoever; in particular it could be generated from a Fourier eigenoperator Lab in the manner previously described, by virtue of being an eigenfunction of Lab. Given an FEF f(x) satisfying Labf = λf

(including λ = 0), if we take h ≡ Lmnf 6= Labf then we have just seen that h will also be

FEF. If h ≡ Lmnf − κf, then h is again FEF with µ = α, and κ may be taken equal to 24

or distinct from λ. It could be the same or a diﬀerent eigenvalue of Lab, but really κ need not be an eigenvalue of any operator whatsoever. It is simply a proportionality constant.

To summarize the above proceedings, for a given FEF f(x), the generating expression h = Lmnf can be correlated to the various cases of the alternative generating expression

0 0 h = g + µ gˆ for the four possible values µ ∈ {±1, ±i}, via the critical feature L\mnf =

m+n ˆ ±(−i) Lmnf. Various symmetry properties and relations among the constants can be deduced from comparing these two generating procedures.

Generalizing the generator function treatment from the Fourier case F 4 = I considered previously to the case of general transform T N = I, we can write the general expression

PN−1 n f = n=0 µnT g, where the µn are eigenvalues of T in T f = µf and µ0 = µN = 1.

PN−1 n+1 PN−1 n N−n Then T f = n=0 µnT g = µf = n=0 µµnT g by design. So µn = µµn+1 = µ resulting in

N−1 X (3.5) f = µN−nT ng n=0 as the generating functional expression for TEF’s. To see the last equality in the µ-chain above, note that equality between T f and µf holds if 1 = µµ1, µ1 = µµ2, . . . , µN−1 = µ.

N−1 This implies µ1 = µ etc. and the general result follows.

The resulting function f(x) is a TEF with T -eigenvalue µ since T f = µf. This demonstrates the suﬃciency of this form for f to be TEF; the necessity of this form for f

1 n n to be TEF can be shown with a bit more work. If we take g = N f, then since T f = µ f, PN−1 N−n n PN−1 µN−nµn substituting this form for g into the expression n=0 µ T g gives n=0 N f = f PN−1 N−n n N N = f which demonstrates that f can be written in the form f = n=0 µ T g. This form is therefore both necessary and suﬃcient for f to be TEF. The choice of g in the 25

0 1 decomposition of f is far from unique, however. In general, let g = N f + h, where h is an PN−1 N−n n arbitrary but T -transformable function. Then this substitution gives n=0 µ T g =

PN−1 N−n n PN−1 N−n n f + n=0 µ T h. Thus if the added function h satisﬁes n=0 µ T h = 0, it may

1 0 be added to g = N f to produce another generator function g . This condition provides a symmetry-type constraint on h and its iterated transforms T nh. In the absence of more speciﬁc information about the transform T , this seems to be the most we can say about the general case. Depending on the nature of T , it may be that speciﬁc terms can be combined to simplify the constraint. For example, in the Fourier case, this constraint becomes h + µ3hˆ + µ2h(−x) + µhˆ(−x) = h + µ2h(−x) + µ3(hˆ + µ2hˆ(−x)) = 0, and the discussion now parallels the previous discussion of the generator g. A transformable odd function h will satisfy this constraint for µ = ±1 (h(x) = −h(−x) ⇒ hˆ(x) = ˆ −h(−x)), while even h will work for µ = ±i. In general, let h0, h1, . . . , hn all satisfy

PN−1 N−n n the general constraint n=0 µ T h = 0. Then an even more general expression for

1 PN−1 n N−n g is given by g = N f + n=0 µ T hn. It is straightforward to show by substitution PN−1 N−n n that n=0 µ T g = f, thereby generating the TEF f satisfying T f = µf. Although the analogy with ODE’s is not strictly exact, it is interesting to think of this general expression as representing a general solution to the functional equation Sg = f, with

PN−1 N−n n 1 S ≡ n=0 µ T . The solution is the sum of the particular solution gp = N f satisfying PN−1 n N−n Sgp = f and a homogeneous solution gh = n=0 µ T hn satisfying Sgh = 0. The results discussed in the above paragraphs represent new extensions and generalizations of previously published expressions considered in several of the references cited herein. 26

CHAPTER 4

General Theory – Eigenoperators

With a thorough understanding of eigenfunction generation under our belt, we may now turn to the generation of eigenoperators. Let A0 be a linear diﬀerential operator, and let A1,...,AN−1 be deﬁned by T Ak = cAk+1 for some complex constant c, with

k k 0 ≤ k ≤ N − 1; we take AN = A0. Then it follows that Ak = T A0/c . If we then deﬁne

N−1 N−1 k X X T A0 L ≡ γ A = γ k k k ck k=0 k=0 with weighting coeﬃcients γk, and consider the action of T on L, we have

N−1 N−1 k+1 N−1 N−1 X X T A0 X Ak+1 X T L = γ T A = γ = γ ck+1 = c γ A k k k ck k ck k k+1 k=0 k=0 k=0 k=0 since T is assumed to be linear. Requiring L and T to commute to the constant c gives the fundamental commutation relation T L = cLT (alt. Lˆ = cL). This relation holds if

γ0 = γ1 = ... = γN−1.

Now, since L is linear we may take γ0 = 1 without loss of generality. Hence the PN−1 operators L = k=0 Ak (distinguished by diﬀerent choices for the value of c) all commute with T to their respective constant c. These commuting operators L are precisely the self- transform operators (STO) or transform eigenoperators (TEO) of T . As for the possible values of the constant c, we apply T N−1 to both sides of the fundamental commutation 27 relation to obtain

T L = cLT ⇒ L = cT N−1(LT ) = cN L(T N ) = cN L ⇒ cN = 1

So the commuting constant c is also an Nth-root of unity and therefore an eigenvalue of the transform T . We may also note that since I = TT N−1 = T N−1T , the inverse of T is simply the (N − 1)-st power of T : T −1 = T N−1. Thus one could develop cyclic integral transform theory without ever making explicit use of the notion of inverse transform, if one so desired.

Precisely the same is true, in the arithmetical sense, of the eigenvalues µ of T . Since

µ−1 = 1/µ = µN /µ = µN−1, one could develop an arithmetical theory of so-called “cyclic numbers” satisfying µN = 1 without ever deﬁning a multiplicative inverse. Note, however, that if one omits the concept of the inverse transform or number, one is limited to discussing these transforms and their eigenvalues in the more abstract and general context of semi-groups, rather than groups. The possibility of exploiting the full machinery of group theory for understanding these structures is perhaps a good argument for deﬁning them as concretely as possible and not being too quick to disregard the notion of an inverse. In any case, we observe that the transform T exhibits behavior which in some respects is isomorphic to the behavior of its eigenvalues. This points to the underlying group structure of cyclic transforms, which we further discuss in a later section.

PN−1 We can apply the same argument to the operators Ak in the expression L = k=0 µkAk that we applied to the generator functions in Eqn. (3.5), to show that the generator form for a TEO must be identical. Actually it is slightly easier to consider L =

PN−1 k k=0 µkT A0 so we do not have to deal with the constant c ﬂoating around. Then 28

PN−1 k+1 T L = L = k=0 µkT A0. If the same relations as above are satisﬁed by the µk, then T L = µL with the T -eigenvalue µ. Then the operator

N−1 X N−k k (4.1) L = µ T A0 k=0 can be used as above to generate the TEO L, whose eigenfunctions give rise to potential

TEF’s f(x) satisfying T f = µf. That is to say, potential eigenfunctions of the integral transform T can be found by solving various instances of the eigenvalue problem Lf = λf.

As in the Fourier case, the expression A0f roughly corresponds to the generator function g(x) in the analysis for general T given above.

If we specialize the above discussion and results to the Fourier transform T = F, then

dn (−i)m+n dm N = 4, c4 = 1,A = xm ,A = (xn·), 0 dxn 1 c dxm (−i)m+n 2 (−i)m+n 2 A = A ,A = A 2 c 0 3 c 1

The value of c remains unchosen; we span the 4th-roots of unity with it and see what

m+n conclusions follow. Choosing c = ±(−i) yields A2 = A0,A3 = A1; ignoring duplicated

m n n m m n terms, the resulting eigenoperator L = A0 +A1 = x d /dx ±d /dx (x ·) is precisely the

Fourier eigenoperator Lmn that has been considered above and in previous work[16, 25].

m+n On the other hand, choosing c = ±i(−i) yields A2 = −A0, A3 = −A1 and all terms in L cancel. In this case no operator is formed. The crucial point is that, as we shall see momentarily, potential eigenfunctions of the Fourier transform can be found by solving various instances of the eigenvalue problem Lmnf = λf. 29

To further clarify the relationship between the eigenoperator L and eigenequation

Eqn. (3.1) for the transform T , we can consider the eigenequation for L as well. For clarity we take as a concrete example the operator Lmn and the Fourier transform. The eigenequation for the Fourier transform is

(4.2) F[f] = µf.

We refer to the set of functions satisfying Eq. (4.2) as Fourier eigenfunctions (FEFs).

Note that some authors use the term SFF (and its variants, e.g. SFrF for “self-fractional

Fourier”) to refer to FEFs satisfying Eq.(4.2) for any value of µ, while others take the term

SFF to refer strictly to FEFs with µ = 1. This can lead to some confusion when trying to read the various works in the literature with their various approaches to discussing eigenfunctions of the Fourier and related transforms. While we prefer to introduce a minimum of new terms and abbreviations, in order to accord with the conventions of the papers most directly antecedent to our work, we will try as much as possible to hew to the latter approach (SFF means µ = 1). On occasion however, the former approach may be more eﬃcient, and we respectfully request the reader to maintain a degree of ﬂexibility with regard to this terminology of SFF vs. FEF.

If we now consider the eigenvalue problem Lmnf = λf supplemented with appropriate boundary conditions (BC’s), the transformed equation yields

ˆ ˆ ˆ ˆ L\mnf = cLmnf = λf ⇒ Lmnf = (λ/c)f

+ BC’s. The boundary conditions are taken to be typical conditions that ensure transformability of the solution under the Fourier transform (e.g. decay to zero at inﬁnity, 30

absolute integrability, etc.) This shows that if f is an eigenfunction of Lmn with eigen- ˆ value λ, then f is also an eigenfunction of Lmn with eigenvalue λ/c. We would like to argue that f and fˆ must be proportional, because they satisfy formally identical BVP’s.

However, unless c = 1 and the eigenvalue λ is non-degenerate, we cannot invoke a proportionality argument relating f and fˆ; the BVP’s must be formally equivalent with the same simple eigenvalue for this to hold. Otherwise we could be considering eigenfunctions corresponding to diﬀerent eigenvalues, or diﬀerent eigenfunctions corresponding to the

+ − same eigenvalue, and no deﬁnite conclusion could be drawn. Thus for Lmn (resp. Lmn), we need m + n = 4k (resp. 4k + 2), k an integer, in order to make the additional claim that the eigenfunction f of Lmn is also a Fourier eigenfunction.

Furthermore, the Fourier eigenvalue µ can take on diﬀerent values for diﬀerent eigen-

− functions of Lmn. For example, the transformable eigenfunctions of L02 – that is, the

− solutions of L02f = λf, the Gauss-Hermite functions – are FEF’s with all four Fourier eigenvalues µ = ±1, ±i represented for various values of the L-eigenvalue λ. The present discussion provides an alternative explanation of this fact. As is the case with this operator, the speciﬁc value of λ for general Lmn determines which Fourier eigenvalue µ applies to the eigenfunction f. Alternatively, the symmetry of f will determine whether the

Fourier eigenvalue µ is ±1 (even) or ±i (odd).

In the case of general T and L, we must be sure to supplement the ordinary differential equation (ODE) Lf = λf with appropriate boundary conditions to ensure transformability of the solutions. The value of λ and any symmetry of the eigenfunction will help to determine the transform eigenvalue µ; we do not have a more precise statement at this 31 time to determine the specific value of µ for a particular transform T and an eigenoperator L in the absence of more specific information about their forms. But we observe that in physical applications where only the magnitudes or intensities (squared magnitudes) of f and fˆ matter, as opposed to their exact values including phase, the exact value of

µ becomes irrelevant, since fˆ = µf ⇒ |fˆ|2 = |f|2 for any value of the argument. This is true because an eigenvalue of the cyclic transform T has magnitude |µ| = 1 by virtue of being a root of unity, as previously discussed. An example of such an application in optics is provided by the search for non-diffracting beams – waveforms whose diffracted intensity profile equals their original intensity profile at any propagation distance through free space. We discuss this particular application in more detail as it becomes relevant in later sections.

Strictly speaking, we have not definitively proven that an operator L which commutes with F to a constant must necessarily be of the form Lmn. Indeed, in light of the general procedure described above there may be more complicated linear differential operators that can be shown to commute with F as well. However, in the absence of more sophis- ticated transformation theorems for the Fourier transform, we can certainly say that this is the only form that is built up from elementary properties and can be defined without the evaluation of any Fourier transforms.

For the next argument it will be convenient to represent the Fourier transform by the operator F. Given a Fourier eigenfunction f, if we apply an operator L (not necessarily diﬀerential in nature), which commutes with F to a constant c, to the equality F[f] = µf, 32 we have

1 F[Lf] = L(F[f]) = µ(Lf) ⇒ F[Lf] = cµ(Lf) ≡ µ0(Lf)(4.3) c

As above, we can conclude that if f is a Fourier eigenfunction with eigenvalue µ, then

Lf (assumed transformable) is a Fourier eigenfunction with eigenvalue cµ = µ0. Since a Fourier eigenvalue is necessarily a fourth root of unity, c itself must be a fourth root of unity for this argument to hold. Unfortunately, the Fourier eigenvalues are inﬁnitely degenerate, so we cannot use a similar line of reasoning as before to argue that Lf is proportional to f. In other words, we have proven that under the right conditions, an eigenfunction of L must be an eigenfunction of F, but we have not proven the converse.

We see that the mapping of sets of L-eigenfunctions to the set of F-eigenfunctions is into, but not necessarily onto. At most we can make the following general observations. If c is not a fourth root of unity, then Lf is not a Fourier eigenfunction. If c 6= 1, then Lf is also a Fourier eigenfunction but with different Fourier eigenvalue µ0 6= µ. If c = 1, then Lf is also a Fourier eigenfunction with the same Fourier eigenvalue µ. Thus, depending on the value of the commuting constant c, the operator L may carry f to another function in the same or a different Fourier eigenspace. If we assign more specific properties to L

m+n – namely, that it be a linear diﬀerential operator of the form Lmn – then c = ±(−i) , and at the very least, (m, n) must be such that c is a fourth root of unity in order for Lf to also be an FEF.

We can give a general discussion of the eigenfunctions of any two commuting operators

A and B, whatever their nature (diﬀerential, integral, etc.) If x is an eigenfunction of

A with simple eigenvalue λ, then Ax = λx. Now, if A and B commute, then A(Bx) = 33

B(Ax) = B(λx) = λBx, assuming B is linear. Hence Bx is an eigenfunction of A with the same eigenvalue λ; therefore Bx and x are proportional, so Bx = µx for some eigenvalue

µ of B. Conversely, we may start with Bx = µx for some simple eigenvalue µ and run the same argument in reverse (assuming linearity of A) to conclude that Ax = λx for some eigenvalue λ of A. Even if the eigenvalues are not simple, we can at least conclude that Bx is an eigenfunction of A with the same eigenvalue λ in the ﬁrst case, and that

Ax is an eigenfunction of B with the same eigenvalue µ in the second case. So in the worst case, the action of the second operator upon an eigenfunction of the ﬁrst operator yields another eigenfunction of the ﬁrst operator, for a particular value of λ. If we now take A = L and B = T in the context of our work here, then if Lf = λf for some simple λ, and L and T commute, then necessarily T f ≡ fˆ = µf. Even if λ is not simple, we see that fˆ must at worst equal (or be proportional to) some other eigenfunction of

L, so that candidate expressions for fˆ are determined without direct evaluation of the

T -transform. For instance, if λ has double multiplicity, then the two eigenfunctions of

L could transform into themselves or each other, or even into a linear superposition of them, under the action of T . But so far there is no guarantee in the degenerate case that any eigenfunction of L will also be an eigenfunction of T .

Such a guarantee is provided by a theorem on eigenfunctions of commuting operators, discussed in Friedman[11]. This theorem states that if two operators K and L commute, and the operator L has an eigenvalue λ of ﬁnite multiplicity (e.g. Ly = λy), then there is some function in the corresponding eigenspace that is simultaneously an eigenfunction of

K (Kx = λx and Lx = λx). We may set K = T and apply this theorem directly to our analysis to conclude that when L commutes with T , there will always be an eigenfunction 34 of L that is an eigenfunction of T as well. Without this theorem, we need λ to be simple in order to conclude that the corresponding L-eigenfunction f is also a T -eigenfunction.

This theorem generalizes the situation by removing the restriction on the simplicity of the eigenvalue, at the cost of guaranteeing a transform eigenfunction somewhere in the λ- eigenspace, rather than directly proportional to the single eigenfunction comprising it. We conclude that the commutation of two operators is a rarely satisﬁed and rather restrictive condition which, when satisﬁed, constrains the eigenfunction sets to overlap. It is more likely, of course, that a given pair of operators will not commute.

Returning to the case of general transform T , if we now consider the eigenproblem

Lf = λf with suitable boundary conditions ensuring that the T -transform of f exists, we may infer that under the right circumstances, this problem will generate solutions which are eigenfunctions of T , subject to requirements of transformability of f under

T as expressed in the boundary conditions (e.g. suitable decay at inﬁnity, integrability, etc.). For we ﬁnd that if f is an eigenfunction of L with eigenvalue λ, then T [f] is also an eigenfunction of L with eigenvalue λ/c:

Lf = λf ⇒ T [Lf] = cL(T [f]) = λT [f] ⇒ L(T [f]) = (λ/c)T [f]

Hence f and T [f] satisfy formally identical boundary-value problems. However, the value of the commuting constant is determined in part by the nature of the diﬀerential operator, and unless c = 1, we cannot conclude anything further. Only if c = 1, and the eigenvalue λ is simple, can we infer that f and T [f] are proportional, implying that the L-eigenfunction f is also a T -eigenfunction satisfying T [f] = µf. This is the eigenproblem for the transform T , which we saw in Eqn. (3.2) above. This reasoning explains why, for example, the 35

+ 00 2 1/2 2 ODE L02f ≡ f + x f = 0 generates the known FEF f(x) = D2(|x|) = |x| J1/4(x /2), the parabolic cylinder function (with λ = 0 the value c = −1 is irrelevant), but the sign

− on L must be changed in the equation L02f = λf to generate the Gauss-Hermite eigenfunctions (c = −(−i)2 = 1). The former result was discussed in previous work by this and other authors[16, 37]. On a technical note, the above function is not diﬀerentiable at the origin, so that strictly speaking it does not satisfy the eigen-ODE at all points. In such a case we can say that the ODE generates a candidate FEF, rather than rigorously arguing that the solution is FEF because it satisﬁes the ODE everywhere.

Conversely, if the eigenvalue µ of T also happens to be simple, then we may apply a similar line of reasoning to conclude that Lf = λf for some nonzero λ, so that a T - eigenfunction is also an L-eigenfunction. (This cannot be done in the Fourier case T = F because the eigenvalues µ = ±1, ±i are all inﬁnitely degenerate.) Both conclusions are consequences of the fundamental commutation relation T L = LT .

Finally, the FEO’s Lmn can give rise to self-adjoint operators. In fact, the two terms of Lmnf are virtually adjoints of each other to begin with, but for the exponents m and

† n n n m m n m m n; we have Lmn = (−1) d /dx (x ·) ± (−1) x d /dx . Since any operator plus its

† adjoint is self-adjoint, Lmn + Lmn is a self-adjoint operator for general (m, n). For m = 0

† and n even or vice-versa, L(0,2k) is self-adjoint (L(0,2k) = L(0,2k)); odd parity on n gives

+ † − + † n + − † n+1 − (L(0,2k+1)) = −L(0,2k+1). For m = n, (Lnn) = (−1) Lnn and (Lnn) = (−1) Lnn,

+ − so even n makes Lnn self-adjoint while odd n makes Lnn self-adjoint. Regardless of the sign choice in Lmn and the values of (m, n), we always have the deﬁning feature

m+n Ldmn = ±(−i) Lmn. 36

The self-adjointness of a BVP has implications for the integrability of its solutions.

In principle, self-adjoint eigenproblems can be constructed from the self-adjoint operators delineated above. In practice, BVP’s on infinite intervals are singular, and greater care must be taken than in the finite-interval case to analyze the nature of the limit at infinity in order to determine whether the BVP is truly self-adjoint. Further details on this can be found in Coddington & Levinson[3]. Standard results in Sturm-Liouville theory (see, for example, Stakgold[36], Ince[17], or Coddington & Levinson[3]) then would imply that for such a self-adjoint BVP, the eigenvalues of the operators are real; that the eigenfunctions are integrable functions in, say, L2(−∞, ∞); and that they form an orthonormal basis of expansion for functions in L2(−∞, ∞). Furthermore, the Riesz-Fischer theorem applied to these new bases indicates an isomorphism between integrable functions in L2 and sum- mable sequences of complex numbers in l2. The terms in such a sequence are simply the projection coefficients of the corresponding function onto the individual basis functions.

However, the most important of these consequences from a physical point of view is the integrability of solutions. For example, if the eigenoperator is derived from an integral transform representing free-space diffraction of a wave with transverse profile covering the entire doubly infinite real axis, and the eigenfunction is interpreted as an electric field, a square-integrable eigenfunction would correspond to a non-diffracting beam with infinite extent but finite total energy, rendering it physically realizable. We will say more about this in later sections below.

The results above can easily be specialized to the other transforms considered in some of the references cited herein[21, 22, 26]. Eigenfunctions of general cyclic T can therefore be determined without evaluating any T -transforms of generator functions like 37 those found above in Eqn. (3.1). All that is required is a straightforward transformation theorem for diﬀerential operators analogous to the position-derivative theorem for the

Fourier transform. Then the eigenproblem for T can be converted to an equivalent BVP.

We will point out below how the eigenproblem for T can also be represented as an integral equation. So a further advantage of the present approach is the relative ease of analysis aﬀorded by recasting the integral equation as an equivalent diﬀerential equation. 38

CHAPTER 5

Fourier Eigenoperators Lmn – Additional Comments & Examples

Here we oﬀer some further comments on the nature of FEF’s and the FEO’s Lmn which generate them. They are not crucial to the ﬂow of the general argument contained in other sections but rather serve to clarify and exemplify some secondary points about this approach.

In a recent article previously cited,[16] McCallum and Horikis presented new results on the theory of FEFs by considering linear diﬀerential operators that commute with the

Fourier transform. The analysis was based on the properties of the operator

dnf dm (5.1) L f ≡ xm ± (xnf) = xmf (n) ± (xnf)(m) mn dxn dxm with (m, n) an ordered pair of non-negative integers. Sometimes a ± superscript is used

± m (n) n (m) on Lmn to indicate a particular choice of sign in the expression: Lmn ≡ x f ±(x f) . The authors showed that these operators commute with the Fourier operator to a constant; that is,

± m+n ± F[Lmnf] = ±(−i) Lmn[Ff](5.2)

Essentially this occurs because the two terms in Lmn are transforms of each other, to a constant multiple. The main idea here is to exploit the fact that derivatives in x-space will

0 be transformed into powers of x in Fourier space and vice versa. Since the form of Lmn is 39 unchanged by the Fourier transform, we could refer to it as a self-Fourier operator (SFO), by analogy with the notion of SFF’s, or a Fourier eigenoperator (FEO), by analogy with the notion of FEF’s.

Thus, if an eigenvalue problem is formed by applying Lmn to an unknown function f(x) and imposing appropriate conditions of integrability or decay at inﬁnity, then in principle, solutions can be found which satisfy these conditions, which are even, and which are invariant under the Fourier transform. They correspond to eigenfunctions of the Fourier transform with eigenvalue µ = 1, for example; then fˆ = f. Odd solutions

π iπ/2 would be invariant with 2 phase rotation, corresponding to eigenvalue i = e , for example; then fˆ = if. We reiterate the previous observation that whatever the particular eigenvalue, the magnitudes or intensities of f and fˆ will always equal each other for any value of the argument, since |µ| = 1. In other words, |fˆ|2 = |f|2. So in eigenfunction applications where only the magnitudes or intensities of f and fˆ are important, rather than their exact values including phase, it is not crucial to even know the exact value of the eigenvalue µ. An example of such an application in optics is provided by the search for non-diffracting beams – waveforms whose diffracted intensity profile equals their original intensity profile. We discuss this particular application in more detail as it becomes relevant in later sections.

It bears mentioning here that the choice of sign in Lmn can inﬂuence the existence of these FEF solutions – we comment further on this point below. Thus the BVP’s

dnf dm(xnf) xm ± = λf, f(±∞) = 0(5.3) dxn dxm 40 can generate FEF’s. For the homogeneous case λ = 0, the generator function g(x) can now be deﬁned such that

(5.4) g(x) = xmf (n) or g(x) = ±(xnf)(m) where the other term will be proportional tog ˆ. This provides a method to deﬁne these generator functions from a given FEF f(x) without evaluating any Fourier integrals.

Provided m + n = 4k (resp. 2k), where k is an integer, the sum (resp. diﬀerence) of the above terms will generate another SFF h(x) from the recipe g+ˆg of Eqn. (3.4), assuming f does not satisfy Lmnf = 0. If f does satisfy Lmnf = 0, then in light of Eqns. (5.3,5.4), the operators Lmn represent various schemes for generating the trivial SFF h ≡ 0 according to Eqn. (3.4). If we want to generate the original function f, we need to consider the

L-eigenproblem Lf = λf for nonzero λ. Then we divide the terms in Eqn. (5.4) by

1 m (n) 1 m+n n (m) 1 n (m) λ so that g = λ x f , for example. Theng ˆ = λ (−i) (x f) = ± λ (x f) since fˆ = f and m + n must be even in order for g, g,ˆ and f to be even. Then g +g ˆ = f and

Eqn. (3.4) is recovered. An easy illustration of this can be seen in connection with the

Gauss-Hermite eigenfunction example mentioned previously.

To see an example of the generator decomposition for an Lmn-eigenfunction, consider

−x2/2 the Gauss-Hermite eigenfunction Gk(x) ≡ AkHk(x)e with k = 4. Here Ak is the appropriate normalization constant and Hk(x) is the Hermite polynomial of order k. − ˆ This eigenfunction satisﬁes L02f = λf and f = f with µ = 1. Then standard results

−x2/2 indicate that λ = −2k − 1 = −9, and the eigenfunction is f = G4(x) ≡ A4H4(x)e =

4 √ 1/2 4 2 −x2/2 00 1/(2 4! π) (16x − 48x + 12)e . We can now take either g(x) = −1/9 G4(x) or

2 2 00 g(x) = 1/9 x G4(x); theng ˆ(x) = 1/9 x G4(x) org ˆ(x) = −1/9 G4(x), accordingly, and 41

their sum yields G4(x). Comments analogous to the above hold for the other transform eigenvalues µ 6= 1 as well.

Note also that if Lmnf = 0 yields an FEF for particular sign choice, the individual terms in Eqn. (5.1) will also be multiples of another FEF derived from the solution f, with one term equal to ± the other term. For we have

(x\nf)(m) = (−i)mxmxdnf = (−i)m+nxmfˆ(n)

(5.5) = (−i)m+nxmf (n) = ∓(−i)m+n(xnf)(m) if f = fˆ and Lf = 0. Similarly we may also derive x\mf (n) = ∓(−i)m+nxmf (n). For a simple example, (m, n) = (0, 1) gives the Gaussian ODE f 0 + xf = 0, wherein the FEF

2 2 solution f = e−x /2 satisﬁes fb0 = −\xe−x2/2 = if 0 = −ixe−x /2, and xfc = xe\−x2/2 = ixf = ixe−x2/2. Of course these two results convey the same information and they correspond

−x2/2 to the known FEF behavior of the Gauss-Hermite function G1(x) ∝ xe .

Another example of the Lmn-operator approach is

+ 2 00 2 00 2 00 0 L22f ≡ x f + (x f) = 2x f + 4xf + 2f = 0(5.6)

√ with decaying linearly independent solutions |x|−1/2±i 3/2. Since n is even with + sign

2+2 + † 2 + + on L and c = (−i) = 1, this operator is self-adjoint [(L22) = (−1) L22 = L22] and it commutes with the Fourier operator. To ascertain whether these two functions transform into themselves or each other, we apply the theorem of Friedman discussed in a previous

+ section. Since L22 has the eigenvalue λ = 0 of ﬁnite multiplicity, there is some function √ √ −1/2+i 3/2 −1/2−i 3/2 f = af1 + bf2 = a|x| + b|x| in the corresponding eigenspace which is 42

FEF. It is straightforward to show via standard substitutions that the FT of |x|−s, for

0 < Re(s) < 1, is given by

r 2 πs |[x|−s = Γ(1 − s) sin |ω|s−1 π 2 √ This can also be found in standard transform tables[18]. For s = 1/2 ± i 3/2, evidently √ these two L-eigenfunctions transform into multiples of each other. Taking s = 1/2−i 3/2 yields √ √ r 2 √ π(1/2 − i 3/2) √ |x|−1/2+i 3/2 ↔ Γ(1/2 + i 3/2) sin |ω|−1/2−i 3/2 π 2 as the transform pair. Then the combination √ √ r 2 √ π(1/2 − i 3/2) √ f(x) = |x|−1/2+i 3/2 + Γ(1/2 + i 3/2) sin |x|−1/2−i 3/2 π 2 is an even FEF with µ = 1, in accordance with the generator expression f = g +g ˆ already discussed.

We can at least get a sense of the magnitude of the coeﬃcient on the second term as follows. From the integral deﬁnition of the gamma function we have Γ(s∗) = Γ∗(s), and

∗ π for Re(s) = 1/2 we have 1 − s = s . Hence for Im(s) = b, we have sin πs = Γ(s)Γ(1 − s) = Γ(1/2 − bi)Γ(1/2 + bi) = |Γ(s)|2 where the first equality is the reflection formula for Γ(s). q q q Therefore |Γ(1/2 ± bi)| = π = π = π√ in the present case. sin π(1/2±bi) cos (±πbi) cosh (π 3/2) √ √ √ √ √ Also, sin [π/2 · (1/2 − i 3/2)] = sin (π/4 − iπ 3/4) = 2/2(cosh π 3/4−i sinh π 3/4). √ √ √ √ q √ π(1/2−i 3/2) q cosh2 π 3/4+sinh2 π 3/4 q cosh π 3/2 Thus we have | 2 Γ(1/2+i 3/2) sin | = √ = √ = π 2 cosh π 3/2 cosh π 3/2 1 as the simplified magnitude of the coefficient, since cosh2 x + sinh2 x = cosh 2x. Thus the coefficient is simply a unit-magnitude complex number. In general we have shown 43

p π |Γ(s)| = cosh πb for s = 1/2 ± bi. Finally, in light of the comments in the previous paragraph, the functions x2f 00 and (x2f)00 = −x2f 00 must be FEF as well, although it is easy to see that in this case these expressions reduce to constant multiples of the original

FEF f(x).

A similar example is provided by the equation

− 3 000 3 000 2 00 0 L33f ≡ x f − (x f) = 3x f + 6xf + 2f = 0(5.7)

√ with decaying linearly independent solutions x−1/2±i 15/6. Since n is odd with - sign

3+3 − † 3+1 − − on L and c = −(−i) = 1, this operator is self-adjoint [(L33) = (−1) L33 = L33] and it commutes with the Fourier operator. The previous FEF f(x) with s replaced by √ s = 1/2 − i 15/6 will be FEF with µ = 1 for this example. Also, the functions x3f 000 and

(x3f)000 = −x3f 000 must be FEF as well. The preceding two examples generalize to complex

−1/2 + exponent the previously known SFF |x| , which itself satisﬁes L11f = 0. Even though these power functions under discussion are themselves not integrable over the whole line, the Fourier integrals converge for power function integrands |x|s with real part of the exponent s between -1 and 0.

We now take a moment to make some comments relating these diﬀerential eigen-

− operators to quantum mechanics. The operator L02 can be alternately viewed as the Hamiltonian for the quantum harmonic oscillator (QHO) in 1-D, with the position operator x and momentum operator p ≡ i d/dx, both raised to the second power. In other

− 2 2 words, L02 = −(x + p ) is symmetric in x and p. In light of this observation, we may say that the investigation of more Hamiltonians symmetric in the x and p operators would be one scheme for systematically generating additional FEF’s. Equating the actions of 44 these Hamiltonians on a wavefunction to an eigenterm proportional to that wavefunction – that is, setting up Schrodinger’s equation for these Hamiltonians – would produce an eigenproblem whose eigenvalues correspond to energy levels and whose eigenfunctions are potential FEF’s. Taking the zero energy level gives the homogeneous problem for nullspace functions of the Hamiltonian, which would also be potential FEF’s. Thus we may view the operators Lmn in some sense as generalized abstract Hamiltonians which also generate FEF’s by virtue of their symmetry.

Due to the superposition principle, we can form ﬁnite or inﬁnite sums of transform eigenoperators – which, incidentally, form the basis of a vector space – to generate new

TEO’s. For example, any linear combination of Lmn operators, with appropriate commuting constants, will also be an FEO. That is to say, the index sums must all be equal mod 4, since the commuting constant for general (m, n) is ±(−i)m+n. This assumes the same ± sign choice on all Lmn in the superposition. In other words, simple operators

Lmn may be superimposed to generate BVP’s with FEF solutions, though the individual values of m + n must be from the same equivalence class mod 4.

Speaking of equivalence classes, an equivalence relation can be constructed among the

Lmn operators in terms of their commuting constants. For speciﬁc index values (m, n), since −(−i)m+n = (−i)m+n+2 = (−i)m+n−2, we see that the commuting constants of

− + + + Lmn, Lm,n+2, and Lm,n−2 are all equal. But by the same token, any Lm0n0 whose indices (m0, n0) add to a number equivalent to m + n ± 2 mod 4 will have the same commuting constant. That is to say, for m0 + n0 ≡ m + n ± 2 ≡ m + n + 2 mod 4, the commuting constants are equal. (Here the ± sign choice is free and not to be connected to that of

+ + + − L.) Thus Lm±2,n, Lm±1,n±1, Lm±3,n∓1 ... is an inﬁnite equivalence class for Lmn. (The ± 45

− sign choices in the indices are connected.) To take the QHO operator L02 as an example,

− + + + + + + + + − we have L02 ≡ {L00,L04,L13,L31,L22,L40,L66,L53,...}. Of course, L operators whose index sum equals the original, modulo 4, can also be included in this class. Hence

− − − the preceding example also includes the operators {L20,L11,L24,...}. Four equivalence classes are thereby constructed, each corresponding to a distinct power of −i; for a given equivalence class, this power equals (−i)m+n+1∓1 mod 4, depending on whether the indices

(m, n) are read oﬀ a class member with + or − sign choice in the superscript. This allows us to relax the above restriction on the sign choices on the individual operators in a superposition and make the general statement that as long as the individual operators in a linear superposition are equivalent, they will all commute with F to the same constant.

Next we note in passing that the case of an inhomogeneous diﬀerential equation seems less interesting than the homogeneous or eigenvalue problems. If we consider

Lmnf = q where q(x) is a function of x satisfying all the conditions such that its T -transform exists, then q(x) must be a TEF in order to apply similar logic to this case. Then given a TEF q(x), if we can solve the homogeneous problem it is in principle straightforward to derive the solution of the inhomogeneous equation, by using generalized reduction of order, for example. Details on this procedure can be found in Ince[17]. The full solution will then be another TEF.

We have noted above the importance of the boundary conditions on the BVP’s in establishing the FEF character of solutions. It is interesting to consider whether certain combinations of the integrability and decay requirements may be violated by an FEF. In 46 particular we noted that the Riemann-Lebesgue lemma can be used to formulate a BVP for

FEF’s which are absolutely integrable. The lemma can be used to give a constraint on the class of functions which can be an FEF, for it implies that a function which is absolutely integrable but does not approach zero at inﬁnity cannot be an FEF, for the lemma requires the transform of this function to decay to zero. One may be tempted to think that there are no non-trivial examples of such functions, but in fact a straightforward example of one is discussed by Gelbaum & Olmsted[12]. It is essentially a decaying envelope with integrable

“spikes” superimposed at regular intervals, such that the resulting inﬁnite series of spike areas converges. One could even remove the decaying envelope and simply consider an inﬁnite series of integrable “spikes” along the x-axis. If the spike areas are proportional to

P∞ 1 successive terms of the Riemann zeta-function series ζ(2) = n=1 n2 , then the total area

π2 will be proportional to ζ(2) = 6 , so this series of spikes is integrable. However, a function may fail to satisfy the integrability condition and still be an FEF. For instance, the known

p + SFF f(x) = 1/ |x| ∈/ L1(−∞, ∞) satisﬁes L11f = 0. It is not absolutely integrable, but it does decay to zero at both ends of the real axis. It is Fourier-transformable and thus is a Fourier eigenfunction.

ix2/2 An even more exotic example is the known SFF e ∈/ L1(−∞, ∞), the so-called chirp function or imaginary Gaussian. We shall see below when we deﬁne FEO’s for

+ negative indices that this SFF satisfies L(−1,1)f = 0; however, it clearly does not satisfy conditions of absolute integrability or decay at infinity. Even a generalized function such P as the delta comb k δ(x − kT ), which is not absolutely Riemann-integrable and doesn’t decay to 0 at infinity, can be shown to be SFF for the proper choice of k. Thus we may have a self-Fourier function which is in the nullspace of an appropriate operator but does 47 not meet the requirement for integrability; we may even have a self-Fourier function - classical or generalized - in the nullspace of an appropriate operator while violating the requirements for integrability and decay. However, as mentioned above, an SFF cannot satisfy the integrability requirement while violating the decay requirement. Thus we have some qualitative information about the general characteristics of an SFF. Titchmarsh[37] discusses various function classes of self-Fourier and other self-transform functions. At any rate, the present BVP formulation with BC’s ensuring transformability (e.g. via decay and/or integrability) provides a sufficient but not necessary condition for a differentiable function f(x) to be eigen-Fourier. 48

CHAPTER 6

Lmn – Composition and sech mapping

We next point out that the commutation property of Fourier eigenoperators is preserved by composition:

a+b a+b+c+d ˆ F[Lab(Lcdf)] = (−i) La+bF[Lcdf] = (−i) Lab(Lcdf)(6.1) so that we may compose chains of arbitrary length of these individual operators in order to generate even more FEO’s. This feature is relevant to a discussion of the FEF f(x) = sech(x), which satisﬁes the ﬁrst-order ODE

(cosh x)f 0 + (sinh x)f = 0(6.2)

Clearly this ODE is not of a form generated by Lmn operators as it stands. Under the change of variable u = ex, this ODE becomes

u2 + 1 u − u−1 f 0(u) + f(u) = 0(6.3) 2 2 which can be written, after some algebraic manipulation, as

u2 + 1 u2 − 3 L f(u) + L f(u) = 0(6.4) 4 11 8 00 49 which can be written as a superposition of individual eigenoperators and their compositions as follows:

1 1 1 1 5 L (L f) + L f + L f − L (L f) − L f(u) = 0(6.5) 4 02 11 4 02 8 11 4 11 11 16 00

The calculations involved in deriving this result are tedious but straightforward. It is easy enough to recover the original ODE for f(x) from these ODE’s for f(u). In a similar fashion, the second-order ODE f 00 + (2sech2x − 1)f = 0 satisﬁed by sech(x) can be expressed as a superposition of compositions of basic operators if one wishes.

Although these constructions may seem a little artificial, it is hard to find a more direct way for incorporating the sech(x) function into our formalism for generating Fourier eigenfunctions. This can be seen as a reflection of the fact that the sech(x) function is typically expressed as a solution to the nonlinear homogeneous second-order differential equation f 00 + (2f 2 − 1)f = 0 (as shown above) and is typically associated with nonlinear processes such as soliton propagation in fiber optics. A deeper analysis of this function from the present perspective, then, would require extension of the linear TEO formalism to nonlinear operators, with the attendant increase in complexity which such an endeavor would entail, such as convolution terms and transforms of products. Certainly this is one possible and potentially quite fruitful direction for future exploration, but here we do not pursue the matter any further.

We close this section by pointing out that although the observations of these last two sections have been made within the context of self-Fourier functions and operators, such properties as superposition and composition of operators are of course not limited only to the Fourier transform, but can be developed for other speciﬁc transforms. They could 50 also be developed within the framework of the general cyclic transform T , just as we generalize other properties from the Fourier setting to general cyclic transform setting in other sections of this work. 51

CHAPTER 7

Lmn – Beyond whole-number (m, n)

Now we consider the possibility of negative integer values for the operator indices

(m, n). The decay requirement f(±∞) = 0 can be replaced by

Z ∞ |f(x)| dx < ∞ −∞ or Z ∞ f(x) dx < ∞ −∞ as appropriate, allowing functions that are simply bounded and in some sense integrable to be included in this formalism. We now show that the resulting expressions will reproduce the cases for positive (m, n) values via a one-to-one mapping between the regimes. To see this, assume (m, n) = (−M, −N) for some non-negative integers (M,N). The operator

L(−M,−N) generates the homogeneous equation

−M (−N) −N (−M) L(−M,−N)f ≡ x f ± (x f) = 0(7.1) where negative-order diﬀerentiation is interpreted as repeated integration. If we deﬁne Q to be either term in this expression, we can transform Eq. (7.1) into LNM Q = 0. For 52 example, substituting Q ≡ x−M f (−N) yields

−N M (N) (−M) L(−M,−N)f = Q ± (x (x Q) ) = 0 ⇒

N (M) M (N) x Q ± (x Q) ≡ LNM Q = 0

M (N) So L(−M,−N) maps to LNM . Thus Q and f = (x Q) will both be FEF solutions to these equations, or equivalently, both f and Q = x−M f (−N) will be FEF’s. Similarly we can start with the positive-index operator Lmn and transform it into L(−n,−m) with comparable results. Hence each operator Lmn can generate multiple FEFs, and solving

Lmnf = 0 in general is equivalent to solving L(−n,−m)f = 0. The indices m and n can also be of opposite sign in this argument, in which case both Lmn and L(−n,−m) will generate homogeneous integro-differential equations, which can easily be converted to differential equations via simple differentiation.

The case m = −n is somewhat special since it remains invariant under the Q-

−n (n) substitution. For example, taking Q = x f , the equation L(−n,n)f = 0 becomes

−n (n) 0 n (−n) L(−n,n)Q = 0. Hence f, Q = x f , and Q = (x f) , satisfy the same equation and are all FEF. If all three functions are proportional, then f (n) ∝ xnf. So the equation

L0nf = 0 may possibly be recovered in this case.

Further equivalences can be seen by manipulating the operators L(−n,n) and L(n,−n).

˜ −n If we substitute Q ≡ x f into L(n,−n)f = 0, the result is formally identical to the ODE

n ˜ (−n) L(−n,n)f = 0 multiplied through by x . On the other hand, upon substituting Q ≡ f , ˜ the result is formally identical to L(−n,n)f = 0 diﬀerentiated n times. In the ﬁrst case, Q is

n ˜ thus a FEF solution to L(−n,n)f = 0, and at the same time f = x Q is a FEF solution to

˜ ˜(n) L(n,−n)f = 0. In the second case, Q is a FEF solution to L(−n,n)f = 0, while f = Q is a 53

FEF solution to L(n,−n)f = 0. Hence solving these two ODE’s is in some sense equivalent, because the solutions of one can be simply related to the solutions of the other.

An example of the general equivalence with negative index is

Z Z + −2 L(0,−2)f ≡ fdx dx + x f = 0

+ + R R −2 which is equivalent to L20Q = L02Q = 0 under Q = fdx dx or Q = x f; the SFF nature of decaying solutions to this last operator was discussed above and in Horikis and

McCallum[16]. The equation

df Z L+ f ≡ x−1 + xf dx = 0 (−1,1) dx can be used to illustrate the special case m = −n. It is straightforward to verify that the chirp functions exp(±ix2/2) satisfy this equation; thus they are FEF. Note that these functions are integrable but not absolutely integrable, nor are they decaying. They also satisfy (d/dx ∓ ix)f = 0, although strictly speaking this is not an operator Lmn that

2 − commutes with F. The decaying Gaussian SFF exp(−x /2) satisﬁes L(−1,1)f = 0, as well

+ + 0 − 2 as L01f = L10f = 0, since f = −xf. The other solution to L(−1,1)f = 0 is exp(+x /2),

− − which also satisﬁes L01f = L10f = 0, but this solution grows without bound. So its transform does not exist and we discard it.

Moreover, through the deﬁnition of fractional-order derivatives, we can extend (m, n) to any real values. Even complex orders of diﬀerentation and integration can be de-

fined, but for the moment we content ourselves with real orders. The standard product- derivative transformation theorem for the Fourier transform can be used to define derivatives of non-integer order [32, 23], so that f (m)(x) is formally defined to be the inverse 54 transform of (−ix)mfˆ(x) for any real m. Thus Eqn. (5.2) really holds for any real ordered pair (m, n). An alternative definition of the fractional derivative generalizes the Cauchy rule for repeated integration to non-integer orders. This rule states

d−n 1 Z x (7.2) f = (x − t)n−1f(t)dt dx−n (n − 1)! for natural number n, but its generalization to non-integer n gives one deﬁnition of the fractional derivative of order n. With this deﬁnition of non-integer-order derivative, we

ﬁnd that interpretation of the expression Lmnf for non-integer values of (m, n) will lead to Volterra-type integral equations. For example, the operator L(0,1/2) yields the semidif- ferential equation

d1/2 L f ≡ f ± x1/2f = 0(7.3) (0,1/2) dx1/2 which is equivalent to the Abel-esque integral equation

1 d Z x f(t) √ 1/2 L(0,1/2)f ≡ 1/2 dt ± x f = 0(7.4) π dx 0 (x − t) on the positive half-line. The integral used in the above equation to define the semiderivative is known as the Riemann-Liouville definition of fractional differentiation. Eqn. (7.3) can be solved by generalized Frobenius-series methods in conjunction with standard theorems of fractional calculus, which are discussed by Oldham and Spanier[31]. With a little further manipulation, Eqn. (7.4) could also be solved by Neumann series methods. The even or odd extension of a suitably transformable (e.g. decaying or integrable) solution will be a SFF on the whole real line. 55

To illustrate the Frobenius approach, we assume that the solution to Eqn. (7.3) can be written in the form

∞ p X j/n (7.5) f = x ajx j=0 where p > −1 and the exponents on terms in the standard Frobenius series have been divided by the integer n. The restriction on p is necessary to ensure convergence of the integral used to deﬁne the fractional derivative, but otherwise the values of n and p both may be chosen as the analysis proceeds in order to facilitate a solution. We substitute this expansion into Eqn. (7.3) and use the result that for p > −1, the semiderivative d1/2 p Γ(p+1) p−1/2 dx1/2 x = Γ(p+1/2) x . This yields

∞ X Γ(p + j/n + 1) xp a xj/n[x−1/2 ± x1/2] = 0 j Γ(p + j/n + 1/2) j=0 which can be simpliﬁed considerably with the particular choice n = 2, p = −1/2. Hence

−1/2 P∞ j/2 f = x j=0 ajx , and the coeﬃcients aj are determined in the usual manner via the recurrence relation

Γ(1) Γ(3/2) ±a + a x−1/2 ± a x1/2 + a ± a x + ... = 0 0 1 Γ(1/2) 1 2 Γ(1) 2

−1 Γ(1/2) The very first term a0x Γ(0) in the full equation vanishes since Γ(0) is infinite; this provides the real motivation for the above choice of p. Hence we find a1 = a3 = ... = 0 56

Γ(3/2) Γ(5/2) and ±a0 + a2 Γ(1) = 0, ±a2 + a4 Γ(2) = 0,... so that f(x) takes the ﬁnal form

−1/2 X j/2 Γ(j/2) Γ(4/2) Γ(2/2) j−1 f = a [x + (∓1) ... x 2 ] 0 j+1 Γ(5/2) Γ(3/2) j even Γ( 2 ) ∞ X Γ(k) Γ(2) Γ(1) = a [x−1/2 + (∓1)k ... xk−1/2] 0 Γ(k + 1/2) Γ(5/2) Γ(3/2) k=1 2 8 ∼ a [x−1/2 ∓ √ x1/2 + x3/2 − ...](7.6) 0 π 3π

Convergence for all ﬁnite x can be established via the ratio test. The general coeﬃcient ak for k > 1 can be written

Qk−1 jk−j k j=1 ak = a0(∓1) k k(k+1) Q k+1−j 2 k/2 j=1(2j − 1) 2 π 2 k−2 k−1 k (k − 1)(k − 2) ... 2 · 1 = a0(∓1) k(k+1) (2k − 1)(2k − 3)2 ... 3k−1 · 1k · 2 2 πk/2 Qk−1 j! k j=1 = a0(∓1) k k(k+1) Q 2 k/2 j=1(2j − 1)!!2 π so applying the test gives

ak+1 k! Y k lim | | = |x|lim √ = |x|lim = 0 k→∞ k→∞ k+1 k→∞ ak π(2k + 1)!!2 2k + 1 k since the bounded monotone sequence of products in the last equality must converge.

The above infinite series of algebraic monomials, extended symmetrically or anti- symmetrically about the origin, is an FEF, as long as it is transformable. As already noted, the leading term is transformable, but the remaining terms in the sequence require further examination via the complex Fourier transform, with its bounds and conditions for transformability, to confirm the transformability, and hence FEF nature, of the full 57 solution. This provides another extension of the FEF concept from the simple algebraic monomial SFF |x|−1/2, which can be seen as the leading term in an infinite series comprising (potentially) an FEF. We also note that the choice of n seems free in the problem at hand, but it is straightforward to show that for any positive integer value of n, the

1 coeﬃcients a1 = ... = an−1 = 0 and an = ∓ Γ(3/2) a0, etc. as with n = 2, and so the same solution is always obtained up to index scaling. Negative values of n yield the trivial solution because a0 = 0.

In summary, the Lmn formalism can be extended from whole-number index values to integer and even non-integer values of the (m, n) indices. 58

CHAPTER 8

Decomposition into Eigenobjects

PN−1 N−n n PN−1 N−k k We have seen that the expressions f = n=0 µ T g and L = k=0 µ T A0 generate transform eigenfunctions (TEF’s) and transform eigenoperators (TEO’s), respectively. In both cases, the generator function or operator is ﬁxed and arbitrary (as long as it’s transformable), and the TEF or TEO with eigenvalue µ is generated by superimposing the action of each power of T on that ﬁxed generator. Thus an arbitrary input generator object leads to an output eigenobject with a particular eigenvalue. We can obtain a related result if we turn the situation around and ask: what if we express an arbitrary output object as a superposition of eigenobjects, one for each eigenvalue? That is to say, what

PN−1 PN−1 can be said about the expansions f = n=0 gn and L = k=0 Ak, where the subscripts now represent different eigenspaces of the basis objects? Are such expansions legitimate or even possible? It turns out that the answer is affirmative. Any T -transformable function or operator – if we assume a priori that it admits such a decomposition, in order to avoid subtle issues of completeness, etc. – can be decomposed into a particular superposition of TEF’s or TEO’s of the cyclic transform T . This is a point made in the abstract for the case of eigenfunctions by Marhic[24] and in the specific case of FEF’s by Cincotti et al.[2] Furthermore, if one considers the N = 2 case, corresponding to the parity transform

R ∞ 0 0 0 Pf ≡ −∞ δ(x + x )f(x )dx = f(−x), this is simply the statement that any function can be decomposed into even and odd parts, which are eigenfunctions of P with µ = 1 and

µ = −1, respectively. We extend these points by detailing the procedure for general cyclic 59

T and by pointing out that the procedure could equally well be applied to diﬀerential operators as to functions, just as the generation procedure for TEF’s has equally well been applied to the generation of TEO’s in this work. Also, we will see that any power of T applied to a decomposable input object can also be so decomposed.

For the sake of brevity we cast the process in terms of f and gn, where the terms can represent functions or operators. We begin with the initial decomposition f =

N−1 √1 P g , in which g represents an eigenobject – eigenfunction or eigenoperator N n=0 n n – of T with eigenvalue µn, where µ is some primitive N-th root of unity. That is,

n T gn ≡ gˆn = µ gn. Then all other eigenvalues of T can be expressed as powers of

µ. We may apply any transform power T k to the decomposition equation to obtain

N−1 T kf = √1 P µnkg . We may then cast the situation in the following matrix form: N n=0 n

      f 1 1 1 ... 1 g0              T f  1 µ µ2 . . . µN−1   g       1    1     (8.1)  2  = √  2 4 2(N−1)    T f  1 µ µ . . . µ   g2    N      .  .   .   .  .   .              N−1 N−1 2(N−1) (N−1)2 T f 1 µ µ . . . µ gN−1

Now, the above coeﬃcient matrix is symmetric and unitary. This can be seen from the

1 PN−1 nk −nk inner product of the k-th row with itself (0 ≤ k ≤ N − 1), which is N n=0 µ µ =

1 N N = 1, since the conjugate of a complex number of unit magnitude is simply its recip- 1 PN−1 nk −nj rocal. Also, the inner product of the k-th and j-th rows (j 6= k) gives N n=0 µ µ = 1 PN−1 n(k−j) 1 µN(k−j)−1 N N n=0 µ = N µk−j −1 = 0, since µ = 1. Therefore the system above can be 60 easily inverted by conjugating the coeﬃcient matrix, giving

      g0 1 1 1 ... 1 f              g  1 µ−1 µ−2 . . . µ−(N−1)   T f   1        1     (8.2)   = √  −2 −4 −2(N−1)  2   g2  1 µ µ . . . µ   T f    N      .  .   .   .  .   .              −(N−1) −2(N−1) −(N−1)2 N−1 gN−1 1 µ µ . . . µ T f

For consistency with previous results, one could express the negative eigenvalue powers µ−nk as µ(N−n)k if desired. Note that the above coefficient matrices are precisely those of the N-point discrete Fourier transform (DFT) and its inverse, regardless of the specific transform T in question. This is perhaps not as surprising as it might seem at first sight, given the fundamental role played by the N-th roots of unity in our analysis. We have

n now expressed the eigenobjects gn with diﬀerent eigenvalues µ as superpositions of the

n transform power objects T f; these eigenobjects gn are the basis objects for the decomposition in Eqn. (8.1) of a single output object f (not just an eigenobject anymore), as well as any transform power of f, into diﬀerent input eigenobjects from each eigenspace.

Note also how we have reproduced in Eqn. (8.2) the decomposition of output eigenob-

n n jects gn with diﬀerent eigenvalues µ (g ˆn = µ gn) into linear combinations of transform powers of a single arbitrary transformable input object f, where the weights are powers of eigenvalues. (Simply reverse the roles of f and g in the canonical decomposition

PN−1 N−n n k equation f = n=0 µ T g, and allow µ to be raised to any power µ before applying the various exponents. Then the resulting expression for g will satisfyg ˆ = µkg.) This latter decomposition has already been discussed from several perspectives in this work; 61 the former decomposition provides an inversion of this latter decomposition by projecting a given arbitrary object onto the various eigenspaces of the transform. The transform power objects are the DFT of the eigenobjects, and the eigenobjects are the inverse DFT of the transform power objects. More abstractly, the forward transform expresses each transform power as a decomposition into projectors onto each eigenspace, while the inverse transform expresses these eigenspace projectors as a decomposition into transform powers. From an engineering perspective, Marhic discusses ways to achieve these various decompositions optically[24], as well as their applications. 62

CHAPTER 9

Summary of One-Dimensional Results

To summarize the analysis thus far, we have generalized the method ﬁrst presented by Horikis and McCallum[16] to analyze and characterize eigenfunctions of the Fourier transform. The use of diﬀerential eigenoperators of positive, negative, and fractional order allows one to study these functions without having to evaluate any transform integrals.

We can find eigenfunctions of any general linear cyclic transform by finding eigenfunctions of those differential operators which commute with the transform – these are the eigenoperators. In principle this approach can be used with any integral transform, whether cyclic or not, so long as one can find or construct differential eigenoperators for that transform.

In the present cases under consideration, the structure of linear cyclic transforms makes it particularly easy to construct such commuting operators. However, a simple procedure for constructing eigenoperators of more general non-cyclic integral transforms is not as obvious.

Certainly the question of how to find and characterize such eigenfunctions is not a new one to ask; other approaches have been developed to analyzing these invariant functions for various forms of the integral transform operator. However most previous attempts to solve the eigenproblem for a particular cyclic integral transform offer debatable degrees of simplification over the problem in its original form. For example, perhaps the process requires the forward evaluation or inversion of some integral transform as part of the procedure; the various references discussing Fourier eigenfunctions show how a Fourier 63 transform[1, 4, 2], Laplace transform[38], or a Mellin transform[37] is required at one point or another in this approach. The simplest of these methods involves writing down a simple functional expression that can be used to generate Fourier eigenfunctions with a specific eigenvalue, for example. But it would be nice to find a procedure that does not require the evaluation of any transform integrals, since after all this is not usually an easy task in any given situation. Furthermore, the last of these approaches cited also requires the systematic solution of functional equations if any generality is to be gained from the proceedings.

An alternative approach might use a certain form for the normal modes of a particular transform and assume that any putative eigenfunctions can be expanded in these modes, deriving conditions and relationships that must be satisfied by the various mode indices in order for “eigen-ness” to hold, so to speak. An example of this can be found in the work of Piestun et al.[35] Such an approach might be powerful for that particular integral transform, but it does not necessarily illuminate the relationship between the structure of the operator and its eigenfunctions – one must still know the form of the normal eigenmodes corresponding to the transform in order to get off the ground. But how could one determine those eigenmodes from first principles? Nor does it suggest a straightforward means of generalizing to other operators. Thus a systematic framework possessing the triple virtues of simplicity, structural transparency, and generality is to be desired; we feel that we have presented such a framework here. 64

CHAPTER 10

Hyperdiﬀerential Operators

It is worth noting that some of the preceding analysis and discussion on commuting operators can be embedded within a larger theoretical framework that has been of interest to the mathematical physics community for some time. This framework, known as hyperdifferential operators, can be seen to encompass the commutation of differential and integral operators that is so fundamental to our analysis, as well as the fractionalization of various integral transforms discussed here and elsewhere. For example, Namias takes a hyperdifferential representation of the standard Fourier integral transform as the starting point in his development of the fractional Fourier transform [27]. The references by Wolf give the fullest account of this line of development. Here we content ourselves with merely making a handful of observations and comments, in order to describe this more abstract framework and to show how it can provide a firmer theoretical foundation for the analysis contained in this work.

The main thrust of this idea is an equivalence between a self-adjoint linear differential operator H and the associated hyperdifferential operator T ≡ eiτH formed by formal exponentiation of H. Here τ is an auxiliary continuous parameter that is manipulated or selected at will to whatever beneficial end is desired. Under the right circumstances it can be interpreted as the order of an integral transform associated with the hyperdifferential operator T . For a particular value τ often corresponds to a known standard transform, but when allowed to vary over a suitable range it provides a more general, fractionalized 65 transform. The hyperdifferential operator T can in theory be interpreted by expansion in Maclaurin series of the argument operator H, but in practice is typically shown to correspond to a particular integral transform T . For example, in terms of the operators

− Lmn considered in the present work, the Fourier transform arises from taking H ≡ L02 = d2/dx2 − x2; that is to say, it can be shown (see Wolf[44] or Namias[27], for example) that the action of this hyperdiﬀerential operator on a suitable function is equivalent to

iτL− iτ(d2/dx2−x2) the action of the FT on that function, so that literally Fτ = e 02 = e . The parameter τ then represents the angular order of the FT; the standard FT corresponds to

τ = π/2. We shall see below that allowing τ to vary continuously over the interval [0, 2π) gives the more general fractional Fourier transform. We discuss fractionalized transforms more fully in the later sections on linear canonical transforms (LCT’s).

Previous authors such as Wolf have typically only concerned themselves with second- order differential operators H. Furthermore, a rigorous justification of these proceedings requires the full machinery of Lie group theory, an area in which this author is not proficient. But we can give a heuristic argument for applying the same procedure to the operators of arbitrarily high order that we have encountered in this work. For many of the integral transforms considered herein, including the general LCT, this hyperdifferential operator T corresponds exactly to an equivalent integral transform T which commutes with H. Since any operator commutes with itself, the commuting of T and H can is easy to establish:

X (iτ)n X (iτ)n X (iτ)n TH = ( Hn)H = Hn+1 = H Hn = HT n! n! n! 66

Then it is a simple matter to see that eigenfunctions of H are also eigenfunctions of T :

Hf = λf ⇒

X (iτ)n X (iτ)n X (iτλ)n T f = eiτH f = Hnf = λnf = f = eiτλf ≡ µf n! n! n!

Hence T f = µf with µ = eiτλ.

The transforms T ≡ eiτH for various values of τ and fixed H form a continuous transformation group with respect to the parameter τ. For particular values of τ, the transform may be cyclic, with a corresponding discrete cyclic group structure underlying it. Through variation of τ, a standard cyclic transform with one particular τ value and associated period can be fractionalized to yield a more general form, the fractional transform. The parameter τ thus plays the role of a fractional order parameter for the transform; different choices of τ can yield related transforms with other periods. Roughly speaking, we may think of various transforms with various generators H as Lie groups with an associated Lie algebraic structure. This associated structure results from the parameter τ, which in the case of LCT’s, connects the transforms to particular subgroups of the GL(2, C) matrix group, described by a single parameter. For a particular T , a simple formal differentiation expresses H in terms of T via

∂T H = −i | ∂τ τ=0

This derivative can be thought of as a Lie derivative - a tangent space at the null value of a continuous parameter characterizing the transformation group. So in a way, for a given cyclic transform T (or T ), there are really two associated group structures operating 67 behind the scenes – the continuous Lie group that deﬁnes the transform, and the discrete cyclic group that describes its behavior or action under repeated application. In a later section we make some comments about the cyclic group structure of the transforms under consideration. The more general LCT’s are discussed in the eponymous section of this work.

We may now replace τ by the parameter α, interpreted as the angular order parameter for a fractional transform. From a more abstract perspective, the transform T can be deﬁned either via the hyperdiﬀerential expression T ≡ eiαH or via the integral trans-

R 0 0 form T ≡ Kα(x, x ) · dx . However, both definitions depend upon the parameter α, a fact reflected in their forms. We note also that the powers T n which occur repeatedly throughout this work can be regarded as defining iterated kernels, via

Z Z n 0 0 0 n−1 0 0 n−1 0 0 T f(x) ≡ Kn(x, x )f(x )dx = T [T f(x )] = K(x, x )T f(x )dx = Z Z Z 0 0 00 00 0 K(x, x ) K(x , x ) ... K(xn−1, xn)f(xn)dxn . . . dx dx = Z Z 0 0 00 00 0 ... K(x, x )K(x , x ) ...K(xn−1, xn)f(xn)dxn . . . dx dx = Z Z 0 0 00 0 00 ... K(x, x )K(x , x ) ...K(xn−1, xn)dx dx . . . f(xn)dxn

0 Swapping x ↔ xn gives, for the last line,

Z Z 00 0 00 0 0 ... = ... K(x, xn)K(xn, x ) ...K(xn−1, x )dxndx . . . f(x )dx

So the action of T n is represented by the kernel

Z Z 0 00 0 00 Kn(x, x ) = ... K(x, xn)K(xn, x ) ...K(xn−1, x )dxndx . . . dxn−1 68

We can also write

Z 0 0 00 00 00 0 00 Kn(x, x ) = [T Kn−1](x, x ) = T [Kn−1(x, x )] = Kn−1(x, x )K(x , x )dx

n−1 so that Kn = T Kn−1 = T K, with K1 = K. We have suppressed the subscript α throughout the above for clarity. Of course, this reiterates the fact that the eigenproblem for an integral transform can also be viewed as a homogeneous Fredholm integral equation of the second kind. For the equation T f = µf can also be written in the form

Z ∞ ˆ 0 0 0 T [f] ≡ f ≡ Kα(x, x )f(x )dx = µf(x)(10.1) −∞

0 where the speciﬁc form of the kernel Kα(x, x ) is determined by the speciﬁc transform

T . Trying to ﬁnd eigenfunctions of the transform can be equated to trying to solve this integral equation. What we are doing with the eigenoperator approach is essentially rendering this integral equation in the equivalent form of a BVP. Generally speaking, BVP’s are easier to analyze and solve than integral equations, so we are casting the problem into a simpler form that is more amenable to straightforward analysis and solution. This is another advantage of the eigenoperator approach – casting the problem into a simpler form. 69

CHAPTER 11

Transition to Two Dimensions

In this section we want to indicate how the conceptual framework presented and analyzed in some detail in one dimension can be extended to and formulated in two dimensions. We pointed out above that the optical applications of cyclic transform eigenfunctions provide suﬃcient motivation for considering the two-dimensional regime. The previously cited work by Piestun et al. [35] and other works by the same authors indicate one application of such 2-D eigenfunctions – e.g. the non-diﬀracting beams mentioned previously. So it behooves us to speak to these issues to some degree. We will show these results in terms of the FT, but they can often be shown in similar forms valid for the case of general transform T as well.

First of all, we can extend the one-dimensional diﬀerential eigenoperators L of the transform T to two dimensions. Again we take the Fourier case to illustrate. We start with the 2-D deﬁnition of the standard Fourier operator

∞ ∞ ∞ Z ¯ Z Z uˆ(k¯) = u(¯r)eik·r¯dr¯ = u(x, y)eik1xeik2ydx dy −∞ −∞ −∞

. If we assume the separable form u(x, y) = f(x)g(y), we have

Z ∞ Z ∞ Z ∞ Z ∞ uˆ = f(x)g(y)eik1xeik2ydxdy = f(x)eik1xdx g(y)eik2ydy −∞ −∞ −∞ −∞

ˆ = f(k1)ˆg(k2) = µ1f(k1)µ2g(k2) 70

if the functions f and g are FEF with respective eigenvalues µ1 and µ2. So the product fg is FEF with eigenvalue µ1µ2. We can write the 2-D Fourier operator used here as the

(x) (y) (y) (x) composition of successive 1-D operators in x and y, viz. F2D = F F = F F .

Again consider the separable function u(x, y) = f(x)g(y). If we deﬁne an eigenop-

(x) (y) erator L ≡ Lmn + Lm0n0 , where the superscripts denote the respective variable of the (x) (y) (x) (y) operator, then Lu = Lmn(fg) + Lm0n0 (fg) = gLmnf + fLm0n0 g. If we now assume that (x) (y) f(x) and g(y) are eigenfunctions of their respective operators Lmn and Lm0n0 , satisfying (x) (y) Lmnf = λ1f and Lm0n0 g = λ2g, then the last term in the preceding equality can be writ- (x) (y) ten gLmnf + fLm0n0 g = gλ1f + fλ2g = (λ1 + λ2)fg. We conclude that u = fg satisﬁes

Lu = (λ1 + λ2)u, so the product of eigenfunctions of two separate eigenoperators in x and y is an eigenfunction of the sum of the eigenoperators, with eigenvalue equal to the sum of the individual eigenvalues. No constraint has been imposed on the relative values of the ordered pairs (m, n) and (m0, n0). This result easily generalizes to any number of dimensions/variables, and is an alternative way to see that the product of two or more

FEF’s in 1-D is FEF in two or more dimensions.

(x,±) What can we say about the commuting behavior of this larger operator L ≡ Lmn +

(y,±) (x,±) Lm0n0 with the FT? We apply the 2-D FT to both sides of the BVP Lu = (Lmn + (y,±) Lm0n0 )u = λu + BC’s. Here we allow the form of u(x, y) to be unconstrained (i.e. not necessarily a separable product). Then it is straightforward to show that the ﬁnal result is

m+n (x,±) m0+n0 (y,±) Luc = [±(−i) Lmn ± (−i) Lm0n0 ]ˆu = λuˆ where the two ± sign choices in the operators are independent. For example, the transform

m n \m n (m+n) m n of the term x ∂x u is x ∂x u = (−i) ∂kx [kx uˆ(kx, ky)]. If we also have the two ± 71 sign choices and values of (m, n) and (m0, n0) such that the two imaginary commuting constants are equal (i.e for equal signs, m + n ≡ m0 + n0 mod 4, while for opposite signs, m + n ≡ m0 + n0 + 2), then we can factor this out front to obtain

(x,±) (y,±) Luc = c(Lmn + Lm0n0 )ˆu = cLuˆ = λuˆ so that L has commuting constant c. If λ = 0 is a valid simple eigenvalue of L, then it follows that a corresponding eigenfunction u(x, y) is FEF. For nonzero simple values of

λ, we must have c = 1, as discussed above, in order to conclude that an eigenfunction u(x, y) is FEF. This result can also be easily generalized to higher dimensions and higher numbers of variables.

Superpositions of 1-D diﬀerential eigenoperators are not the only 2-D extensions that commute with the FT. We can construct an additional form, with no 1-D analogue, by considering the 2-D Gaussian f(x, y) = e(−x2−y2)/2. Since this function can written as the separable product of Gaussians in x and y, its FT in 2-D can be found simply by multiplying the 1-D FT’s with respect to x and y, which, of course, are also Gaussians, since 1-D Gaussians are FEF. So the 2-D Gaussian is also FEF under the 2-D transform.

Note, by the way, that this is the only continuous radial function which can also be written as a separable product in x and y, i.e. f(x, y) = f(x2 +y2) = e−(x2+y2)/2 = e−r2/2 = f(r2).

If we consider f(r2) ≡ f(x2+y2) = f(x2)f(y2), the second equality is the deﬁning property of the exponential function f(z) = ekz. Choosing k = −1/2 gives the 2-D Gaussian FEF.

There are no other obvious examples of radial functions whose 2-D FT’s can be found from knowledge of the individual 1-D FT’s with respect to each variable. We make further comments about 2-D FT’s of radial functions below in connection with Hankel transforms. 72

It is easily veriﬁed that this 2-D Gaussian satisﬁes both fx = −xf and fy = −yf.

(x,+) (y,+) That is, L01 f = 0 and L01 f = 0. We may combine the two operators to form the

(x,+) (y,+) PDE [L01 + L01 ]f = fx + fy + (x + y)f = 0. Straightforward application of the theory of characteristics for this linear PDE gives dx/1 = dy/1 = df/[−(x + y)]. This

xy implies that ψ1 = y − x and ψ2 = fe are the two characterstics. Hence the general solution takes the form f(x, y) = e−xyh(y − x). Now, f(x, y) must be transformable and satisfy decay conditions (for example) in all directions. We can satisfy these criteria by choosing h(z) = e−z2/2. Then f(x, y) = e−xye−(y−x)2/2 = e(−x2−y2)/2, the 2-D Gaussian.

This demonstrates the FEF nature of this function via partial diﬀerential eigenoperators of the 2-D FT, in a way similar to the corresponding demonstration in 1-D. Note that other choices of h(z) could also lead to FEF solutions satisfying the necessary criteria; for example, h(z) = e−kz2n for positive k and n yields f(x, y) = e−xy−k(y−x)2n as an FEF solution. Note that these generalized forms for k 6= −1/2 and/or n > 1 will not be separable, however, and they are not strictly radial functions.

We can also see that yfx = xfy = −xyf for the 2-D Gaussian. Thus we arrive at the alternate PDE xfy − yfx = 0. We can deﬁne a new operator L ≡ x ∂/∂y − y ∂/∂x, and we thus have a new, curl-type eigenoperator in 2-D with no counterpart in 1-D.

The characteristics here are found from dx/(−y) = dy/x = df/0, which yield ψ1 =

2 2 2 2 2 1/2 (x + y ), ψ2 = f. So the solution takes the form f(x, y) = f[(x + y )/2] = f(r /2), a function with circular symmetry. Of course, f(z) must be such that f(r2/2) satisﬁes appropriate decay conditions and is transformable; then this function f is an eigenfunction of the 2-D FT. For example, f(z) = e−z yields the 2-D Gaussian f(r2/2) = e−r2/2. Other choices of f(z) could lead to other FEF’s f(r2/2), e.g. f(z) = 1/(1 + z). In light of the 73 circular symmetry of f, the 2-D FT actually becomes the zero-order Hankel transform with kernel J0(rρ), which is the circularly symmetric version of the 2-D FT. The eigen-

Hankel property of the radial Gaussian function f(r) = e−r2/2 is easily conﬁrmed by consulting a table of standard Hankel transforms (see, for example, Korn & Korn[18]).

We will say more about the Hankel transform and its eigenfunctions below in the context of two-dimensional linear canonical transforms.

Just as in the case of 1-D diﬀerential eigenoperators, the curl-type eigenoperator in

2-D can be generalized to higher powers and derivative orders. We consider the partial diﬀerential operator xm∂n/∂yn, applied to a generic transformable input function f(x, y).

If we apply the F2D transform to this expression, we ﬁnd

m n n (y) (x) m n n (y) n n (x) m F2D[x ∂ f/∂y ] = F F [x ∂ f/∂y ] = F [∂ /∂y (F [x f])] =

(−i)nynF (y)[(−i)m∂m/∂xm(F (x)[f])] = (−i)m+nyn∂m/∂xm(F (y)F (x)[f]) =

m+n n m m (−i) y ∂ /∂x (F2D[f])(11.1)

So we can say that in terms of operators, xm\∂n/∂yn = (−i)m+nym∂n/∂xm; the position and derivative operators switch exponents, so to speak. Similarly we can show that yp\∂q/∂xq = (−i)p+qxq∂p/∂yp. Hence if L = xm∂n/∂yn±yp∂q/∂xq is to be an eigenoperator of F2D, we must have (p, q) = (n, m). Then we can deﬁne the partial diﬀerential operator

(xy,±) m n n n m m Lmn ≡ x ∂ /∂y ± y ∂ /∂x , and the basic commuting property can be written

\(xy,±) m+n (xy,±) Lmn = ±(−i) Lmn , wherebrepresents the action of F2D. Higher-order PDE’s can be created from this generalized operator to yield FEF solutions in two dimensions, if so desired. 74

CHAPTER 12

Linear Canonical Transforms in One and Two Dimensions

The linear canonical transformation (LCT) is an integral transform with imaginary exponential kernel and several parameters that can be specialized to yield the standard

Fourier, fractional Fourier, and Fresnel transforms, in both one and two dimensions [32,

43, 44, 23]. First we develop the definition of the general transform in one and two dimensions, then we define the parameters. After some discussion and comments about the general forms, we then consider a collection of special cases of optical interest in subsequent sections. The main points to keep in mind for reference and use in later sections are the definitions of the forward transform with nonzero B parameter in 1-D and 2-D, as well as the equivalent matrix representation discussed below. This section contains a number of technical details and derivations which are not crucial to the content of later sections.

The general form of the LCT is

Z ∞ 0 i (Ax2−2xx0+Dx02) (12.1) T [f] = fˆ(x ) ≡ K f(x)e 2B dx −∞ in 1-D, and

Z ∞ ˆ i (Ar2+Dρ2) (12.2) T [f] = f(ρ) ≡ 2πK f(r)e 2B J0(rρ/B)r dr 0 75 in 2-D assuming circular symmetry with no angular dependence. A more general form exists in 2-D to allow for angular dependence of the input function f(r) as well, namely

T [f] = fˆ(ρ, ψ) ≡

Z ∞ Z 2π X im(ψ−π/2) −imθ i/2B(Ar2+Dρ2) (12.3) K e · f(r, θ)e e Jm(rρ/B)r dθ dr m 0 0 where the summation runs over all integers. This is derived from a transformation of the standard 2-D Fourier transform from Cartesian to polar co-ordinates, using a Fourier-

Bessel series expansion on the inner product term in the imaginary exponential, exchang- ing orders of summation and integration, and rearranging terms. If we now assume f(r, θ) = f(r)einθ for some integer n, so that f is 2π/n-periodic with n-fold rotational symmetry, then the θ-integrals all become zero except when m = n, so this is the only term that survives the summation. The ﬁnal result is

Z ∞ ˆ in(ψ−π/2) i/2B(Ar2+Dρ2) T [f] = f(ρ, ψ) = 2πKe · f(r)e Jn(rρ/B)r dr 0

≡ ein(ψ−π/2)fˆ(ρ)(12.4) of which Eqn. (12.2) above is clearly a special case with n = 0. It is evident that the transform fˆ(ρ, ψ) also possess n-fold rotational symmetry. This equation can be viewed as a generalized Hankel transform of order n of the function f(r, θ), and as such it is a generalization of the equation derived in Lohmann et al.[23] In optical problems it is often the magnitude or intensity (squared magnitude) of a wave form that is of interest, so we note that the magnitudes |f(r, θ)| and |fˆ(ρ, ψ)| will themselves possess full circular symmetry with no angular dependence. 76

In the above deﬁnitions, K is a normalization constant chosen to enforce the satisfac- tion of Parseval’s theorem for the unitary LCT T :

Z ∞ Z ∞ (12.5) |fˆ(x0)|2 dx0 = |f(x)|2 dx −∞ −∞

Physically this means that the transform is energy-preserving between input and output, which we shall desire for the transforms of interest to us. The constant K is typically √ taken to have the value K = 1/ 2πB or K = 1/p2π|B| in one dimension and K = 1/B in two dimensions[44, 32].

There is another representation of the LCT that offers a different perspective on the matter. The parameters A, B, and D are often included in the so-called ABCD parameter matrix with another parameter C. Thus the general LCT has the equivalent representa-   AB   tion in matrix form M =  . The parameter C only plays a role in kernels with CD B = 0, wherein the exponential kernel becomes a delta-function reproducing kernel. Such cases always correspond to simple chirp multiplication and/or magnification of the input function[44, 32]. In 1-D, for example, it is well known that a delta-approximating func- √ tion series can be constructed from the imaginary Gaussian or chirp function eix2/B/ B as B → 0. Then the LCT reduces to a simple geometric transformation, which in 1-D takes the form

√ 0 iC x2 0 T [f] = fˆ(x ) ≡ 1/ A e 2A f(x /A)(12.6) 77

  A 0   In this case the matrix M =  . The equality D = 1/A comes from the con- C 1/A straint det(M) = AD −BC = 1, which we discuss further momentarily. Although strictly speaking, C enters the picture only when B = 0, it is often illuminating to include it in the parameter matrix for speciﬁc instances of the LCT, as we shall see. In any case we shall not have much reason to consider transforms with B = 0.

The optical transforms of interest to us here typically involve real-parameter matrices, but a general theory also exists for more general complex-parameter matrices, some of which also lend themselves to physical interpretation and utility. For example, the matrix   1 2it   M =  , where t is time, gives rise to the so-called Gauss-Weierstrass transform, 0 1 which is essentially the convolution integral describing solutions to the diffusion or heat equation in terms of the Green’s function e−x2/4t (We say more on this below in connection with the Fresnel transform. Indeed, just as the convolution integral solution of the heat equation can be developed from the heat equation, so can the Fresnel convolution integral describing free-space wave propagation be developed as a solution to a propagation differential equation known as the parabolic wave equation, which is formally identical to the heat equation except for an imaginary coefficient on the first-derivative term. The end result is that the Fresnel transform has an imaginary instead of real Gaussian kernel, with a real value for the matrix parameter B, and the propagation distance replaces time in the Green’s function.)

The matrix M can be related to the ABCD ray-transfer matrix of geometric optics theory and the parameter matrix of operator optics theory[32, 23]. Indeed, these matrices 78 have identical forms for certain kinds of optical elements. For example, in the case of free- space propagation (chirp convolution), both kinds of matrices are upper-triangular with

A = D = 1, while passage through a thin lens (chirp multiplication) corresponds to a lower-triangular form for both kinds of matrices, again with A = D = 1. Also, M is always unimodular, so that the determinant det(M) = AD − BC = 1. Most importantly, it can be shown that the successive action of two or more LCT’s can be represented by the product of the respective ABCD-transformation matrices, so that the overall result is equivalent to the action of a single LCT whose parameter matrix is the ﬁnal product of the individual matrices. Hence the composition of two LCT’s is another LCT, and the set of LCT’s is closed under composition. Associativity of LCT composition can also be straightforwardly demonstrated, and we will discuss identity and inverse transforms as well in what follows. In this way, an isomorphism can be constructed between various groups of LCT’s and groups of unimodular 2 × 2 matrices, and the theory of LCT’s can in a way really be boiled down to matrix group theory – a point that Wolf makes quite convincingly in Chapter 9 of his book[44].

To give a concrete example, the standard Fourier transform is represented by the   0 −1   parameter matrix M =  . The action of two successive Fourier transforms on a 1 0  2 0 −1 2   given function f(x) is the LCT corresponding to the matrix product M =   = 1 0   −1 0     = P , the parameter matrix of the parity operator. It is straightforward to 0 −1 derive a delta-function kernel in the limit B → 0 as described above; this then leads 79 to the reproduction of the input function with a change in argument sign. In essence

− i (x+x0)2 p the exponential kernel becomes e 2B ; when combined with division by |B| in the deﬁnition of K, the overall kernel becomes δ(x + x0) upon passage to the limit.

This all represents the fact that the original function is recovered with a sign change in the argument after two successive Fourier transforms: F 2[f(x)] = P[f(x)] = f(−x).

Two more applications give M 4 = I, the identity matrix corresponding to the identity transform I, and so F 4[f(x)] = P2[f(x)] = I[f(x)] = f(x). Thus four successive Fourier transforms recover the original function f(x), a result well known. Furthermore, since   0 1 3   T −1 M =   = M = M , the transpose of M is its inverse, which means the −1 0 inverse transform is obtained simply from conjugating the complex exponential kernel of the original transform. Thus the Fourier transform is unitary, as is already known:

F −1 = F †. On a lighter note, we point out that in light of the relation F 2 = P, we can think of the Fourier transform as the square root of the parity transform! We say more about such “square-root” transforms a little later.

This unitarity property is true in general for any LCT with a real parameter matrix.

That is to say, any such LCT is unitary, meaning that (the kernel of) the inverse transform is the adjoint of (the kernel of) the original transform: T −1 = T †. Thus the inverse LCT’s corresponding to Eqns. (12.1) and (12.2) above are

Z ∞ (12.7) T −1[fˆ] = f(x) ≡ K∗ fˆ(x0)e−i/2B(Dx2−2xx0+Ax02) dx0 −∞ 80 in 1-D, and

T −1[fˆ] = f(r, θ) ≡ Z ∞ ∗ −in(θ−π/2) ˆ −i/2B(Aρ2+Dr2) −in(θ−π/2) (12.8) 2πK e · f(ρ)e Jn(ρr/B)ρ dρ ≡ e f(r) 0 in 2-D, assuming n-fold rotational symmetry of the input fˆ.

In both 1-D and 2-D cases, we can absorb the − sign in the complex exponential into the value of B and reverse the order of the terms in the exponent to see that the parameter   D −B 0   matrix of the inverse transform is M =  . But since M is unimodular, −CA M 0 = M −1, thus demonstrating that the parameter matrix of the inverse LCT is simply the inverse of the forward parameter matrix. A minor technical detail arises in the last equality of the 2-D case Eqn. (12.8), in light of the parity property of the Bessel function,

n Jn(−x) = (−1) Jn(x). If we negate B, then we must multiply the entire transform by a factor of (−1)n = einπ to compensate for this change. Then a comparison with Eqn.

(12.4) shows that in the phase constant after K∗, the resulting variable phase term is conjugated but the constant phase term e−inπ/2 is not. If we insist that the 2-D inverse transform be the full adjoint of the 2-D forward transform, we must take n to be an even integer, in which case a resulting power of e−inπ/2 = (−1)n/2 appears in both members of the transform pair, which can be absorbed into K and K∗ if desired. If we further desire this power to equal +1, we must take n to be a multiple of 4. Then this phase constant will merely equal einψ in the forward direction and e−inθ in reverse. Alternatively, we could keep n unconstrained and simply deﬁne the transform in both forward and reverse directions to be strictly that quantity which is multiplied by the phase constant. In any 81 case, since we are typically interested in the magnitude or intensity of output signals, these unit-magnitude phase factors are not of great signiﬁcance in the analysis.

Regarding the parameter C and the inverse matrix, we are free to negate C as well as

B since in most cases of interest to us, B is nonzero and C is irrelevant to the analysis. But in cases where it is relevant (B = 0), it is straightforward to show that the sign of C also changes for the inverse geometric transform; then the inverse transform simply undoes the scaling and chirp multiplication of the forward transform. In any case the values of

C and its inverse counterpart should always ensure that M and M −1 are unimodular.

The inverse transform of a real-parameter LCT is always the adjoint of the transform.

Analogously, the parameter matrix of the inverse transform is always the inverse – but not necessarily the ADJOINT – of the original parameter matrix. Note that it is very important not to confuse the unitarity of the LCT T (which is always true) with the unitarity of the parameter matrix M which corresponds to it (which is only sometimes   0 −1   −1 T true). In the Fourier case, the matrix M =   satisﬁes M = M , so M itself 1 0 is unitary (more precisely, since the parameters are real, M is orthogonal). But for other

LCT’s, M is not unitary. In other words, unitarity is not preserved in the isomorphism between the transforms and the matrices. A discrete matrix analog of the LCT integral operator, however, would of course always be unitary.

Having established some basic features of the general LCT, which can be found in standard references such as those cited herein, we now consider a collection of special cases with optical relevance in one and two dimensions. 82

CHAPTER 13

Standard Fourier Transform

The ﬁrst transform of interest in 1-D is the standard Fourier transform, which is   0 −1   an LCT with M =   as discussed above. The phase constant K can be taken 1 0 √ proportional to, or equal to, 1/ 2π, a common normalization constant for this transform.

The crucial point from an optical perspective is that the Fourier transform of an input electric field represents the far-field, infinite-distance diffraction pattern of that field (see, for example, Goodman[14]). Thus, Fourier eigenfunctions can be related to far-field diffraction-invariant beams. If such an eigenfunction is L2-integrable on (−∞, ∞), it potentially represents a far-field diffraction-invariant beam with finite total energy. We have already discussed the eigenfunctions and eigenoperator problem for this particular

− transform in great detail in other sections of this work. We remind the reader that L02

− is an eigenoperator of the FT; therefore the eigenfunctions of L02, the Gauss-Hermite functions, are simultaneously eigenfunctions of the FT. Interpreted in diffractive terms, these are FEF’s which are diffraction-invariant in the far field. We show later that these same functions are eigenfunctions of the fractional FT of arbitrary order, and discuss the diffractive implications.

Recall our discussion as well on Fourier eigenoperators which also happen to be self- adjoint. For m = 0 and n even or vice-versa, L(0,2k) is self-adjoint; for m = n even,

+ − Lnn is self-adjoint, and for m = n odd, Lnn is self-adjoint. Hence we already have a 83 number of self-adjoint Fourier eigenoperators at our disposal, and any eigenfunctions which satisfy decay conditions at ±∞ would be potentially square-integrable. Because

BVP’s on inﬁnite intervals are singular, a self-adjoint structure is not enough to conclude a priori that the solutions will be square-integrable. If the integrability of such solutions cannot be determined, then the subtle complication is to show that the singular BVP on the doubly inﬁnite interval is self-adjoint. This involves highly technical theorems on limit circles and limit points, which can yield certain criteria based on characterstics of the operator itself. A treatment of this issue is given by Coddington & Levinson[3].

For example, the previously discussed eigenproblem

00 2 L02f = f + x f = 0

f(−∞) = f(+∞) = 0 has the solutions

p 2 D2(x) = A |x|J±1/4(x /2)

The desired decay behavior for both choices of sign can be conﬁrmed via the asymptotic √ property Jp(x) ∼ cos(x − π/4 − pπ/2)/ x as x → ∞. Unfortunately, the asymptotic behavior also indicates that these solutions are not square-integrable; having inﬁnite energy content, they merit little further scrutiny for optical application.

In the Appendix, we go into considerable technical detail to explore the existence and asymptotic nature of solutions to some of these Fourier eigenproblems. 84

CHAPTER 14

Fresnel Transform

The next transform of optical interest in 1-D is the so-called Fresnel transform, which goes by several alternate names, including the chirp convolution and the diffraction integral. This transform relates the output transverse profile of a wave to the input profile, after propagation over a finite distance in free space. It results from applying a paraxial approximation to the Green’s function of the scalar wave equation in free space – see

Goodman[14], for example, for a discussion and derivation. It can also be derived from the parabolic wave equation in a fashion similar to the derivation of the Green’s function solution of the heat equation. The transform describes near-field diffraction at more mod- erate distances, as opposed to the far-field Fraunhofer diffraction described by the Fourier transform, as discussed above.

The parabolic wave equation with given initial condition is

i f + f = 0 xx 2λ z f(x, 0) = f in(x)(14.1)

2 The visual similarity with the homogeneous heat equation fxx − (1/α ) ft = 0 is readily apparent. The main differences are simply a dependence on propagation distance z rather than time t, and an imaginary coefficient on the first-derivative term. One can then modify the corresponding coefficients in the convolution solution of the heat equation to obtain 85 the convolution solution of the above equation; this solution is the diffraction integral of the input profile f in(x).

There are several various forms of this transform, involving diﬀerent dimensional or non-dimensional physical parameters. For simplicity we conﬁne our attention at the outset to the simplest, most generic form of the transform, which basically involves convolution of the input wave with the imaginary Gaussian/chirp function:

Z ∞ (14.2) T [f] ≡ fˆ(x0) = e(i/2)(x−x0)2 f(x)dx −∞ where x and x0 represent transverse distance along the proﬁle from a central axis of propagation. This can be obtained from the general LCT (ignoring the constant K) via   1 1   the parameter matrix M =  . The inverse transform is simply the Hermitian 0 1 conjugate, which for real f(x) is

Z ∞ (14.3) T −1[fˆ] ≡ f(x) = e(−i/2)(x−x0)2 f(x0)dx0 −∞   1 −1 −1   The parameter matrix of the inverse transform here is M =  . Note that 0 1 the pre-transform and post-transform variables here both represent distance, as opposed to the Fourier transform case, for example, where the two variables have distinct physical interpretations of distance and spatial frequency. A more general matrix arises from taking

B = λd, where λ is the wavelength of propagation in the z-direction; that is to say, the input is some transverse electric ﬁeld distribution in x multiplied by a plane wave eikz in the direction of propagation z. Physically the forward transform represents propagation of 86 a distance d in the forward z-direction, while the reverse transform represents backward propagation of the same distance. The variables d and z are used interchangeably by various authors to represent distance of propagation. This yields for the transform

√ Z ∞ (14.4) T [f] ≡ fˆ(x0) = 1/ λd e(i/2λd)(x−x0)2 f(x)dx −∞ which is a widely used standard deﬁnition of the Fresnel transform[32]; it corresponds to   1 λd   the 1-D LCT parameter matrix M =  . The preceding transform pair above 0 1 √ corresponds to λd = 1/2 (the 2 pre-factor can be absorbed into the deﬁnition of the transform). The decay of the output with distance is more apparent from this form; such decay is to be expected from the spherical wave point source which forms the starting point of the paraxial approximation implicit in Fresnel integrals. That the forward and reverse transforms undo each other can be seen by simply multiplying their parameter matrices together to yield the identity matrix. Note that the kernel of the original transform pair is the convolved chirp function e(±i/2)(x−x0)2 . Hence the transform is also described by some authors as a chirp convolution – that is, a convolution of the chirp function with the input wave.

The hyperdiﬀerential form of the generic transform is T = ei/2 d2/dx2 , as discussed in some references[32, 43, 44]. The basic transform properties of the position and diﬀeren-

1 tiation operators are easily shown to be T [d/dx] = d/dx[T·] and T [x] = (x − 2i d/dx)[T·] (see, for example, Ozaktas et al.[32]) These relations can also be written more compactly, using the carat symbol to represent the action of T , as d/dx[ = d/dx andx ˆ = (x− 1 d/dx), b 2i respectively. Note especially the invariance of the derivative operator d/dx under T , which 87 can be understood from the hyperdifferential form, which will clearly commute with a function of d/dx alone. Physically this means that the diffracted profile of the derivative of an input field is simply the derivative of the diffracted profile of the field, for any order of derivative. Note also that the operators x and d/dx do not transform into each other as they do in the standard Fourier case, nor does it appear that there is a simple expression involving x and d/dx that will be reproduced after a certain number of applications of T .

The transform T does not seem to be periodic as it stands; physically, there seems to be no natural period for straight-ahead diffractive propagation for a general input field. We may contrast this with the 4-periodic Fourier transform, where an arbitrary input field is reproduced after 4 successive applications.

This motivates us to modify the transform operation slightly so as to obtain a cyclic procedure. We deﬁne the self-transform expression L = A + Ac∗, where the transform of the conjugate of the input wave is taken as the second term. This is consistent with the self-Fresnel expressions deﬁned and discussed in Lohmann and Mendlovic[21], and it gives a period of N = 2. (The Fourier and Hankel transforms which we discuss herein do not need this conjugate construction because they are already periodic. If we use this conjugate in the case of the standard FT, for example, it is easy to show that the period decreases automatically from 4 to 2.) Physically this means we use the following procedure to construct a Fresnel-invariant wave f:

1) pick some input wave g;

2) diﬀract the conjugate g∗ of this wave in the forward direction;

3) add this output to the original wave g, giving f = g + gb∗; 88

Then it is straightforward to show that the wave f formed from this process is invariant under the modiﬁed transform. That is, if we diﬀract its conjugate f ∗ in the forward direction, the result is unchanged from the input. In other words, fb∗ = f, and the function f = g + gb∗ is self-transform under this construction. Since N = 2, the eigenvalues and commuting constants are ±1.

We can then construct diﬀerential eigenoperators as before, and use these to examine the possibility of eigenfunctions of the Fresnel transform considered here. These eigenfunctions would represent waveforms that are recreated after a certain propagation distance d = 1/2λ from the input plane, although their proﬁles will change at other distances.

Waveforms which are invariant at all propagation distances would be the so-called non- diﬀracting beams that have been of interest to researchers for some time in the ﬁeld of directed energy. Unfortunately, such functions remain the holy grail that does not yet seem achievable by this method.

An ODE which is invariant under this modiﬁed transform can be constructed via

L = A + Ac∗, with A = x2:

Lf = −1/4f 00 + ixf 0 + (i/2 + 2x2)f = λf

The operator L is easily seen to be self-adjoint. We can add the BC’s f(−∞) = f(+∞) =

0 to the ODE to form a BVP for decaying eigenfunctions. If we now set f(x) = eix2 g(x), it is straightforward to show that the resulting equation is

g00 − 4x2g = −4λg 89

√ Under the scalings u = 2x and λ0 = −2λ, this becomes

2 0 guu − u g = λ g which is the eigenvalue problem for the Gauss-Hermite functions with λ0 = −2n − 1 ⇒

λ = n + 1/2 in terms of the eigenvalue λ from the original equation. The solutions

−u2/2 here are gn(u) = Gn(u) ≡ e Hn(u), which when combined with the other substitu- √ ix2 −x2 tions gives fn(x) = e e Hn(x 2) as the eigenfunctions of the original problem. In other words, the eigenfunctions of this second-order self-Fresnel operator are the chirp- multiplied Gauss-Hermite eigenfunctions of the standard Fourier transform. In the present context, we conclude that these functions will reproduce their initial proﬁle at λd = 1/2.

We shall see later that the GH functions are actually eigenfunctions for any fractional order of the Fourier transform. This result points to a connection between the Fresnel transform and the fractional FT in the context of diﬀraction. We discuss in further detail in the next section how the Fresnel diﬀraction integral can be represented as a Fourier transform of fractional order. In other words, free-space propagation can really be viewed as a process of fractional Fourier transformation of continously varying order, with far-

field diffraction becoming the standard Fourier transform as the propagation distance approaches infinity – which is to say, the fractional transform approaches the standard

Fourier transform in this limit.

The previous result can actually be extended to the general second-order self-Fresnel operator. If we consider

L = ax2 + bxD + cD2 + dx + eD + fI 90 where D and I represent diﬀerentiation and identity operators, then we can use the operator relations discussed above to compute Lc∗. If we now require L to be self-transform

(i.e. Lc∗ = L), we arrive at the following constraints among the six coeﬃcients:

(14.5) a, d arbitrary real; Im(b) = a/2; Im(e) = d/4; Im(f) = a/4; Im(c) = Re(b)/4

We now apply this self-transform operator to a function F (x) and form the eigenvalue

2 problem LF = λF , in conjunction with the substitution F (x) = eik(x−x0) G(x), where k and x0 are constant parameters to be determined. Then it is straightforward to show that the coeﬃcient of the G0 term in the resulting equation is [Re(b) + a/2 i]x + [Re(c) +

Re(b)/4 i]4ik(x − x0) + [Re(e) + d/4 i]. Requiring this term to vanish yields the values

Re(b) + a/2 i k = Re(b) − 4iRe(c)

Re(e) + d/4 i x = 0 −kRe(b) + 4ikRe(c)

It is easy to verify that the speciﬁc example discussed previously results from the choices a = 2, b = i, c = −1/4 ⇒ k = 1; d = e = 0 ⇒ x0 = 0.

The remaining ODE for G has one term proportional to G00 and several terms multiplying G. Upon dividing through by the coeﬃcient (Re(c) + Re(b)/4 i) of G00, the eigenvalue 91

00 2 λ equation for G takes the form G + (Ax + Bx + C)G = Re(c)+Re(b)/4 i G, where

a + (Re(b) + a/2 i)(2ik) − 4k2(Re(c) + Re(b)/4 i) A = Re(c) + Re(b)/4 i (Re(b) + a/2 i)(−2ikx ) + d + 2ik(Re(e) + d/4 i) + (Re(c) + Re(b)/4 i)(8k2x ) B = 0 0 (Re(c) + Re(b)/4 i) −2ikx (Re(e) + d/4 i) + (Re(f) + a/4 i) + (Re(c) + Re(b)/4 i)(2ik − 4k2x2) C = 0 0 (Re(c) + Re(b)/4 i)

We now complete the square in x; this yields

G00 + A(x + B/2A)2G = λ0G

0 λ 2 where λ = Re(c)+Re(b)/4 i − C + B /4A. We now take u = x + B/2A which gives Guu + Au2G = λ0G; under the scaling v = αu this becomes

2 00 Gvv − v G = λ G

√ where λ00 = λ0/α2 and α = 4 −A. Thus we have recast the original ODE LF = λF into

−v2/2 −α2u2/2 the deﬁning ODE for the Gauss-Hermite functions G(v) = e Hn(v) = e Hn(αu).

In terms of the original function F (x), we have the full solution

2 2 2 2 ik(x−x0) ik(x−x0) −α /2(x+B/2A) F (x) = e G(x) = e e Hn[α(x + B/2A)]

It is straightforward to confirm that the previously discussed example results from the specific choices of operator coefficients when applied to the expressions for A, B, C, k, x0, α.

We have therefore shown that the eigenvalue problem for the general second-order self-

Fresnel operator can always be transformed into the deﬁning EVP for the Gauss-Hermite 92 functions; these are multiplied by a chirp factor to give the full eigenfunctions of the original operator. Thus the chirp-multiplied Gauss-Hermite functions are the transformable eigenfunctions of this general operator, and one must consider higher-order operators in order to explore potentially new functions whose initial input proﬁles are reproduced at

λd = 1/2. 93

CHAPTER 15

Fractional Fourier Transform – Deﬁnition and Basic Properties

The last transform of interest to us in 1-D is the fractional Fourier transform. This is a generalization of the standard Fourier transform that can be conceived of in several diﬀerent ways. The basic idea is to take a discrete-valued parameter that characterizes the standard Fourier transform and allow it to be continuous, thus interpolating a continuous spectrum of Fourier-type transformations between the identity transform and the standard

Fourier transform. Among other possibilities, this parameter can be related to the discrete eigenvalues einπ/2 of the standard Fourier transform, or to the discrete rotations that the standard Fourier transform effects upon a phase-space-type co-ordinate system whose axes correspond to the multiplication operator (multiplication by x) and the differentiation operator (application of d/dx). For example, the standard FT of angular order α = π/2 rotates these two axes by π/2 so that each becomes the other, i.e. multiplication in the time domain becomes differentation in the frequency domain and vice-versa. The fractional FT of general order α rotates the axes by some angle α so that each becomes a linear combination of both, rather than each strictly becoming the other. This is easily visualized by means of the classical continuous rotation matrix, as we shall see momentarily. In this sections we discuss technical details and theoretical properties of the fractional FT; we discuss its application to diffraction in the next section.

In both of the parametrizations described above, the integer parameter n is replaced by a real parameter a. In the eigenvalue framework, then, the discrete eigenvalue einπ/2 94 becomes the continuous eigenvalue eiaπ/2. In the rotation parametrization, the discrete rotation of nπ/2 eﬀected by the standard transform is replaced by a continuous rotation α ≡ aπ/2. By allowing the parameter a to take on any real value between 0 and

4 (or -2 and 2, equivalently), a continuum of eigenvalues or rotations is created, each corresponding to a transform of different fractional order. Different authors define the continous parameter in slightly different ways, but they differ at most by scaling factors, and the underlying concept is the same for all. We can also think of the fractional FT with parameter a as literally a fractional power F a of the standard FT. With a = 1/2, for example, the operator F 1/2 applied twice in succession gives F 1/2[F 1/2] = (F 1/2)2 = F, the standard Fourier operator. Thus we can speak of F 1/2 as the “square root” of the standard Fourier transform. In terms of rotations, the application of F 1/2 twice gives two successive rotations of α = π/4 applied to the multiplication-differentiation axes, for a total rotation of π/2, as effected by F.

The fractional FT can also be deﬁned from the general LCT via matrix parameters

A = D = cos(aπ/2),B = −C = − sin(aπ/2). When a = 1, it is easy to see that we recover the standard Fourier transform with A = D = 0 and B = −1. When a =

4k, giving B = C = 0 and A = D = 1, we obtain the identity matrix. From the integral-transform perspective, we take the limit of the exponential kernel to obtain the reproducing delta-function kernel δ(x − x0), as described previously; this simply means that a = 4k corresponds to an eigenvalue of 1, rotation of 2π, and the identity transform

F 4k = Ik = I. The inverse transform is given by the corresponding parameters of the inverse matrix, A = D = cos(aπ/2),B = −C = sin(aπ/2), which is simply the transpose of the original matrix, as in the standard Fourier case. Again, when a = 1, the inverse 95 matrix is simply the transpose of the forward matrix, yielding the inverse standard FT.

The transform pair thus takes the integral form

Z ∞ ˆ 0 −i/2(cot α x2−2 csc α xx0+cot α x02) (15.1) Fα[f] = f(x ) ≡ K f(x)e dx −∞ and

Z ∞ −1 ˆ ∗ ˆ 0 i/2(cot α x02−2 csc α xx0+cot α x2) 0 (15.2) Fα [f] = f(x) ≡ K f(x )e dx −∞ for the forward and inverse case respectively, where α = aπ/2 is the angular order parameter of the transform, which can be used interchangeably with the fractional order parameter a. The constant K can be taken proportional to, or simply equal to,

1/p|B| = 1/p| sin α|. For 0 < |α| < π, one common choice for the constant is

K = p1 − i cot(α) = e−i[π√sgn(α)/4−α/2] . This particular form of the constant comes from | sin(α)| a relation known as Mehler’s formula, which relates the spectral decomposition of the standard Fourier kernel to the fractional Fourier kernel; the details can be found in Oza- ktas et al.[32]. Clearly if α = π/2, then this constant equals 1, an alternative choice of normalization constant for the standard FT. This form of the constant is convenient for the study of diﬀraction, as we shall see below. Moreover, the transform as given is dimensionless; the variables x, x0, f, fˆ should all be considered dimensionless. While the fractional parameter varies continuously from 0 to 4 (or -2 to 2, etc.) mod 4, the angular parameter varies continuously from 0 to 2π (or −π to π) mod 2π. Thus the standard FT

F corresponds to a = 1 and α = π/2, while the square-root transform F 1/2 (a = 1/2) can   cos α − sin α   be written as Fπ/4 (α = π/4). We see that the parameter matrix M =   sin α cos α 96 is the continuous 2-D rotation matrix, as mentioned above. We can therefore think of the fractional FT as effecting a rotation of the multiplication and differentiation operator axes, so that the new post-transform operators are linear combinations of the original pre-transform operators. We return to this point a little later when we discuss the fractional transforms of operators. The preceding definitions and representations are the main points that a reader should keep in mind for future reference. In the following paragraphs we make some additional observations that clarify the mechanics and interpretation of this transform but are not essential for later material.

Note that we use superscripts to represent the non-angular order parameter a and subscripts to represent the angular order parameter α. But in either case the transform can be exponentiated, as though it were an algebraic symbol, to yield valid results. The order parameters of two diﬀerent transforms may be combined just as though they were exponents. Two fractional FT’s of diﬀerent orders, say F m/n and F a/b, can combine to yield a fractional FT of order F m/n+a/b, and similarly for the corresponding α values. This convenient isomorphism between order parameters and exponents is one reason for the use of the order parameter formalism in the study of fractional transforms. For example, we may write the square-root property of F 1/2 in terms of the angular order parameter

2 as (Fπ/4) = F2(π/4) = Fπ/2 = F. As already mentioned, it is straightforward to show that for α = 0, the transform reduces to F0 = I, the identity transform, by reducing the kernel to a delta function using standard identities.

Transforms with negative order parameters can be deﬁned analogously and can be equated with a corresponding positive-order transform modulo 4 or 2π. That is to say, transforms whose parameters have the same values modulo 4 or 2π are equivalent. So for 97 given order parameters α and a, we may deﬁne the inverse order parameters as simply

−α ≡ 2π − α mod 2π and −a ≡ 4 − a mod 4. If we solve a = 4 − a to get a = 2, we get a transform Fπ equal to its own inverse, but this is not terribly exciting. It is straightforward to show from the deﬁnition that α = ±π gives rise to the same transform

Fπ = F−π = P, the parity transform mentioned above; clearly this transform is its own inverse, since P2 = I and (−1)2 = 1. In any case, negative-order transforms can be thought of as inverse transforms if it is so desired, but this is not strictly necessary if the forward transform is cyclic; as discussed in a previous section of this work, the inverse of a cyclic transform T of period N is simply T −1 = T N−1, the (N − 1)-st power, or iterate, of the forward transform T . In any event, it is evident that the fractional Fourier transform with continuous order parameter α constitutes a one-parameter subgroup of the general group of LCT’s. This points to the notion of Lie groups – transfomation groups continuous in a single parameter – lurking in the shadows behind these integral transforms, a point we have mentioned previously in Section 10.

Moreover, the fractional Fourier transform of order α = 2π/N can be thought of as generating a cyclic group whose members are successive powers of this transform; the standard Fourier transform powers will always constitute a cyclic subgroup of this larger group. Fractionalization of an integral transform can thus be seen as a process of embedding the cyclic group deﬁned by the standard transform into a larger group which contains the original as a proper subgroup, with the members of that subgroup generated by powers of the fractional transform – although this is not always the case. The speciﬁc value α = π/3 is a 6-periodic fractionalization of the standard FT, but the FT itself and

2/3 its inverse are not in the group generated by Fπ/3 = F . However, the 2-cyclic group 98 consisting of the parity and identity transforms P and I = P2 is a subgroup; the transform

F 2/3 is a cube root of the parity transform P = F 2. We discuss group-theoretical issues more fully in Section 22.

In principle the order parameters a and α can take any real value, but if the transform order parameter a is such that 4/a is an integer, then the fractional transform will be cyclic with period N = 4/a, and we can then take advantage of the structure oﬀered by cyclic transform theory in our analysis. In terms of the angular order parameter α = aπ/2, if N is the least positive integer such that αN = 2π, then N = 2π/α = 2π/(aπ/2) = 4/a is the period. For example, the square root transform F 1/2 described earlier has a =

1/2 and thus a period of N = 4/a = 8. Hence the application of this transform eight times in succession to a transformable function will reproduce the original function, since

1/2 8 4 8 (F ) = F = I. In terms of α we can write (Fπ/4) = F(π/4)8 = F2π = I. Furthermore,

−1/2 the transform F = F−π/4 represents the square root of the inverse standard FT

−1 3 F = F−π/2, which can be thought of alternatively as F = F3π/2, so in terms of positive

−1/2 7/2 order parameters, we may write F = F = F−π/4 = F7π/4.

For the sake of completeness we should also point out that we could deﬁne two square roots for the forward and inverse standard FT’s. Since 2(5π/4) = 5π/2 ≡ π/2 mod 2π, and 2(3π/4) = 3π/2, we can say that F5π/4 = FπFπ/4 = PFπ/4 is the other square root of Fπ/2, and F3π/4 = FπF−π/4 = PF−π/4 is the other square root of F−π/2. In terms of

5/2 2 1/2 1/2 a values, F5π/4 = F = F F = PF is the other square root of F = Fπ/2, and so forth. In both cases, the two square roots are simply related by a factor of P, the parity transform; applying P to one square root gives the other. Thus the output of one square 99 root transform can be found by simply changing the sign of the output argument from the other square root transform.

The theory of LCT parameter matrices can be used to show easily how the Fresnel and fractional Fourier transforms are related. If we consider the matrix equation FX1 = M, where F is the Fresnel parameter matrix with the more general λd distance parameter, and M the fractional FT parameter matrix with order parameter α, we solve for X =   1 cos α − λd sin α −λd cos α − sin α −1   F M =  . Being the product of two unimodular sin α cos α matrices, X1 is also unimodular. This result indicates that the fractional Fourier transform of an input function can be thought of as the Fresnel transform applied to an intermediary

LCT of the input function, where the intermediary LCT has the parameter matrix given by X1. This LCT is also necessarily unitary and energy-preserving. If we instead solve

−1 X2F = M, we ﬁnd that X2 = MF is now the anti-transpose (transpose around the other diagonal) of the previous solution X1. Thus the fractional FT can be thought of as the LCT X2 of the Fresnel transform. Alternatively, we can turn things around and

−1 solve F = X3M to get X3 = FM , the inverse of X2. The Fresnel transform can thus be regarded abstractly, or even deﬁned, as the result of applying the LCT X3 for a particular order α to the fractional FT of the same order. Finally, we can solve F = MX4

−1 for X4 = M F , which is the inverse of X1 and the anti-transpose of X3. The Fresnel transform can be viewed as the successive application of the fractional FT of order α to the LCT X4 (of the same order) to a given function. In this way we see that the Fresnel and fractional Fourier transforms can each be regarded simply as an LCT of the other, without defining them or thinking of them as distinct transforms corresponding to distinct physical situations. Strictly speaking, there is no need to hold the Fresnel transform as 100 a separate entity with its own definition and physical interpretation; we can just define and interpret it in terms of the fractional FT. Because of the more flexible and systematic behavior displayed by cyclic transforms in the fractional family, we propose the fractional

FT as the more convenient approach for studying diﬀraction from an integral-transform perspective. We discuss the interplay between these two transforms further in the next section.

Moving in a different direction, we may inquire about an eigenvalue-symmetry relationship to generalize that of the FT discussed in earlier sections. We have already noted that in the absence of specific information about the form of the transform kernel, little can be said about specific symmetry properties of an eigenfunction f. Although we know

T kf = µkf, without knowing for certain that a particular power T k sends the function to its inverse image plane in the original variable (i.e. f(x) → f(−x)), we cannot say much. But if we take T to be the fractional Fourier transform, then assuming the transform order is a rational number a = m/n, we can set T ≡ F m/n. Then T N = I for some integer N such that T N = F Nm/n ≡ F 4 = I, where congruence modulo 4 has been used to represent the 4-cyclic property of the F operator. Equating the exponents gives

Nm/n = 4 ⇒ N = 4n/m; if this value of N is not an integer we may simply multiply by the denominator of the simpliﬁed fraction. Now, in general N may not be even; its parity depends on the order of the transform, which may be such that N is odd. In general the order of the transform is the continuous real parameter a ∈ [0, 4), which must necessarily be rational if T is to be cyclic, otherwise N = 4/a will be irrational. In any case, N must be even if some integral power of T (namely, N/2) is to reproduce the parity transform

P which merely switches the sign of the argument. 101

So if N is even, then it is a general property of the fractional FT that an application

N/2 times in succession negates the sign of the argument, sending the function to its x-inverted image in the original function space: T N/2f = F 2f = Pf = f(−x). The additional equality T N/2f = µN/2f(x) brings us to the desired symmetry relation, since

µN/2 = ±1 depending on the value of µ. This is true because µ = e2kπi/N ⇒ µN/2 = ekπi = ±1, with sign determined by the parity of the integer k. The result is µN/2f(x) = f(−x). So the eigenfunctions {f} fall into two broad classes of even symmetry or odd symmetry according to even or odd parity on k. The standard FT case corresponds to m/n = 1,N = 4 with µ2 = ±1 as outlined previously. Other symmetry properties may be derivable for other forms of T .

An example of an odd-cycle fractional Fourier transform with N = 3 is discussed by

Lohmann and Mendlovic[22]. There the authors state that most “sensible” cyclic optical transforms have even period, but this author feels that the existence of odd periods should not be surprising at all, nor should their sensibility be impugned, in light of the fractional

Fourier transform, whose period can be customized to any odd or even value via the order parameter, as we have just seen. In fact, the authors fail to recognize that they are considering an operator which is equivalent to a fractional Fourier transform; the nomenclature goes unused in their paper. So their operator is not quite as special or exceptional as it would seem. For N = 3 ⇒ a = 4/3, so this transform is a fractional

Fourier transform with fractional order parameter a = 4/3. Of course we may also describe a fractional transform in terms of the angular order parameter α = aπ/2. In the present

4/3 instance this yields α = 2π/3, so this transform can be expressed as T = F = F2π/3.

This relationship can also be derived from T 3 = I = P2 = F 4. Taking cube roots 102 gives T = I1/3 = P2/3 = F 4/3. This indicates that T could also be thought of, or even defined, as one of the cube roots of the identity transform I! Now, since T = P2/3, it follows that T 3/2 = P, so the parity transform is not obtainable from an integer power of the original transform. That is, P is not in the group orbit generated by T , as is to be expected from the odd parity of the period N. The eigenvalues of this transform T and its commuting constants with any differential eigenoperators are the cube roots of unity. If desired, eigenoperators of this transform could be constructed according to the general procedure developed herein. These eigenoperators could then be used to generate eigenfunctions of T , which would represent input wavefields which are invariant under the action of the transfrom or propagation through the optical setup corresponding to that action, as described by the authors in their paper.

We conclude this section by noting that we have made some observations of a group- theoretical nature in the preceding paragraphs. We comment further on group-theoretical issues in greater depth in a later section of this work. 103

CHAPTER 16

Fractional Fourier Transform – Application to Diﬀraction

We now turn our attention toward an important connection between the fractional

Fourier transform and the Fresnel transform of diﬀraction theory discussed previously.

First we will make some broad observations relating various physical phenomena, which are not essential to the following argument. Then we will focus more specifically on the concrete application of these transforms in diffraction theory. It turns out that these two transforms – Fresnel and fractional Fourier – can each be expressed in terms of the other through straightforward algebraic operations and variable changes. One important consequence of this is that Fresnel diffraction over all propagation distances from zero to infinity can be represented by a continual process of fractional Fourier transformation, where the order parameter is a continuously varying simple function of the distance of propagation. Thus, the two separate Fresnel and Fourier transforms, representing the two most typical approximation regimes for scalar diffraction theory – near-field/Fresnel diffraction and far-field/Fraunhofer diffraction – can be united into one fractional Fourier transform that covers all cases in 1-D. In this way the fractional Fourier transform subsumes not only the standard Fourier transform but also the Fresnel transform as well.

We seek to exploit this property to arrive at a deeper understanding of diﬀraction in

1-D and 2-D. In particular, we propose to unify 2-D diffraction transforms in exactly the same way, thereby introducing the fractional Hankel transform as a more general unified 104 framework for examining diffraction of rotationally symmetric 2-D wavefields. We will say more about this below in the discussion of specific 2-D LCT’s.

The situation here is reminiscent of the time-dependent Schrodinger equation for the quantum harmonic oscillator in 1-D. Namias[27] shows that the Green’s function for this problem is essentially the kernel of the fractional Fourier transform. Thus the time evolution of the solution to this problem is given by the continual fractional Fourier transform of the initial condition function, with the order parameter equal to the negative of the elapsed time. This is just one aspect of the broad analogy that is well-known between

Schrodinger problems in quantum physics and diffraction problems in optics. Indeed, the differential equation from which the Fresnel diffraction integral can be developed is formally identical to Schrodinger’s equation with a zero potential (cf. previous comments connecting the Fresnel integral to the heat equation), so we should not be too surprised that the solutions to problems from both areas can be viewed from similar perspectives via the fractional Fourier transform. In fact, the analogy here is really three-fold, because the same conceptual framework and results can be used to describe wave propagation in a quadratic graded-index (GRIN) medium[32]. These three disparate physical problems can be understood in parallel through the uniting formalism of the fractional FT.

In fact, a more general statement can be made in light of the discussion on hyperdifferential operators elsewhere in this work. Given Hψ +iψt = 0 for a particular differential operator H of x alone, the solution can be expressed compactly as ψ(s, t) = eitH ψ(s, 0) via the hyperdifferential operator eitH derived from H, just as if H were simply a function of r. But this hyperdifferential operator typically corresponds to a particular linear canonical transform, so the time evolution of the solution is essentially given by this 105 transform of the initial condition function. The Green’s function of the original problem is essentially the kernel of the transform. For example, if H = d2/dx2, then the equation becomes Schrodinger’s time-dependent equation in 1-D, with the hyperdifferential operator becoming the convolution transform that represents the solution via Green’s function.

Analogous comments can be made for distance evolution of a solution to a problem involving ψz, rather than ψt and time evolution, on the left-hand side. Then the same H now gives the parabolic wave equation discussed previously, with solution represented by the diﬀraction integral.

With these preliminary comments and observations out of the way, we now discuss more speciﬁcally how the Fresnel transform can be related to the fractional Fourier transform. Free-space propagation of a unit-amplitude monochromatic plane wave over a distance d, with wavenumber k = 2π/λ in the direction of propagation, can be represented by

Z ∞ ˆ ikd 1 i(x−x0)2/2λd ˆ 0 0 (16.1) f2(x) = e √ e f1(x )dx iλd −∞ which is simply Eqn. (14.4) multiplied by the plane-wave factor eikd in the direction of propagation. We are now using the hat symbol to indicate physical functions with dimensions, rather than the output of the transform; non-dimensional functions will be indicated by omitting this symbol. The subscripts 1 and 2 indicate input and output, respectively, and the variables x0, x represent physical input and output co-ordinates, respectively. Thus the above equation relates the physical diffracted output profile to the physical input profile. The extra factor of i is included in the square-root term to evoke the similarity with the heat equation, described above. Since we can take the principal 106

√ value i = eiπ/4, this factor simply has the eﬀect of multiplying the overall transform by a phase factor of e−iπ/4, which is irrelevant when magnitudes are taken.

The above transform relates the amplitude distribution on two hypothetical planar surfaces separated by a distance d. We also need to consider the amplitude distributions on spherical reference surfaces with these planar surfaces as tangent planes. The relations between the distributions on planar surfaces and spherical surfaces are given by

02 ˆ 0 ix /2λR1 ˆ 0 f1(x ) = e fsr1 (x )

2 ˆ ix /2λR2 ˆ f2(x) = e fsr2 (x)(16.2)

where R1 and R2 are the spherical radii (more properly in 1-D, cylindrical or circular radii) at the input and output plane, respectively.

0 0 We now introduce the variables u ≡ x /W0 and u ≡ x/W , which represent dimensionless input and output co-ordinates, respectively. The parameters W0 and W are scale parameters which correspond to the initial and ﬁnal sizes of the diﬀracting function.

The output size W will be found to depend on propagation distance d in a particular way. The more usual beam “waist” (width) parameter w is related to the size W via

W 2 = πw2. Note that the output waist value depends on propagation distance, as is to be expected for a diﬀracting waveform. Then we can relate dimensional and dimension- √ √ 0 ˆ 0 ˆ 0 less versions of the input and output proﬁles via fsr1 (u ) ≡ W0fsr1 (x ) = W0fsr1 (u W0) √ √ ˆ ˆ and fsr2 (u) ≡ W fsr2 (x) = W fsr2 (uW ). The corresponding analysis in 2-D entirely parallels the subsequent analysis in 1-D except that the scale pre-factors appear to the

ﬁrst power in 2-D, rather than being square-rooted as they are here in 1-D. 107

Using the above deﬁnitions and relations, we can cast Eqn. (16.1) in the form √ √ ikd Z ∞ e W0 W 2 2 0 2 02 i/2λd (g2W u −2WW0uu +g1W0 u ) 0 0 (16.3) fsr2 (u) = √ e fsr1 (u )du iλd −∞

where we have introduced g1 ≡ 1+d/R1 and g2 ≡ 1−d/R2. Now, in terms of dimensionless variables, the fractional Fourier transform can be written as

−i[π sgn(α)/4−α/2] 0 α e Fα{f(u )} ≡ f (u) ≡ · p| sin(α)| Z ∞ (16.4) f(u0)ei/2(cot α u2−2 csc α uu0+cot α u02) du0 −∞ with dimensionless input co-ordinate u0, dimensionless input function f(u0), dimensionless output co-ordinate u, and dimensionless output function fα(u). Comparing the preceding

2 2 two equations, we see that if we take g2W /λd = g1W0 /λd = cot(α),W0W/λd = csc(α),

0 then fsr2 (u) will be proportional to the fractional Fourier transform of fsr1 (u ): √ √ ikd i[π sgn(α)/4−α/2]p e W0 W e | sin(α)| α fsr (u) = √ f (u)(16.5) 2 iλd sr1

α α where fsr1 (u) represents the transform output fsr1 evaluated at the output argument u.

We need to specialize this relation by evaluating the various parameters. We take R1 → ∞

ˆ 0 ˆ 0 0 0 to represent a planar input proﬁle; then fsr1 (x ) = f1(x ) and fsr1 (u ) = f1(u ). Then g1 = 1, implying that cot(α) is positive and α lies in the interval [0, π/2]. Thus cot(α) =

2 2 W0 /λd ⇒ α = arctan(λd/W0 ). This equation is the key result; it relates the fractional order α of the Fourier transform to the diﬀractive propagation distance d in the Fresnel

p 2 4 transform. We then ﬁnd from W = λd/W0 csc(α) that W = W (d) = W0 1 + (λd) /W0 . 108

This gives the speciﬁc form of the functional dependence mentioned earlier of the so-

2 called “beam size” W on distance d. Finally from g2 = 1 − d/R2 = λd/W cot(α) we

4 2 ﬁnd R2 = R(d) = [1 + W0 /(λd) ]. The function R(d) is known as the phase radius of curvature; it describes the curvature of the phase fronts which constitute the propagating beam.

After a bit more simplifying algebra, the result of the above calculations is that the

Fresnel transform can be expressed as the fractional Fourier transform with additional multiplying factors, as follows:

ˆ 1 ikd −iα/2 ix2/2λR(d) α f2(x) = e e e · f (x/W (d))(16.6) pW (d) 1 where the fractional Fourier transform of angular order α is applied to the dimension- √ 0 ˆ 0 0 α less input function f1(u ) = W0f1(x = u W0), then the output f1 (u) of this transform is evaluated at the argument u = x/W (d). The term e−iα/2 is known as the Gouy phase or accumulated phase. The spherical phase factor eix2/2λR(d) can be cancelled out of the output by observing the output on a spherical reference surface of radius R2, or simply by observing the intensity of the output, since its magnitude is one. Again,

2 one important implication is that, due to the relation cot(α) = W0 /λd, diﬀractive free- space propagation to distance d corresponds to fractional Fourier transformation of order

λd λd a = 2/π arctan 2 and angular order α = aπ/2 = arctan 2 . So the continu- W0 W0 ous diffractive propagation of a wave through free space can be thought of equally as a process of continuous fractional Fourier transformation. As the propagation distance d increases from 0 to ∞, the angular transform order α increases from 0 to π/2, so the actual transform evolves from the identity transform, through the near-field Fresnel 109 regime with increasing fractional order, out to the far-field Fraunhofer regime, where it becomes a standard Fourier transform. Indeed, as d → ∞ in the above relations, we have a → 1, α → π/2,W (d) = λd/W0 → ∞,R(d) = d → ∞, and we recover the standard

Fourier transform. Another important observation is that the above scaling relationships involving the input and output variables and functions are fundamental and intrinsic to the representation of the Fresnel diffraction integral as a fractional Fourier transform. Al- though they enter the picture in the specific case of diffraction of Gauss-Hermite profiles, as we will see below, they are not a specific property only of Gauss-Hermite diffraction.

Any input proﬁle must diﬀract in accordance with these scaling relationships.

Turning our attention toward transform eigenfunctions, we conclude from the preceding reasoning that any dimensionless eigenfunction of the fractional FT of a particular order would then correspond, upon re-dimensionalization and conversion to the Fresnel output as above, not to an eigenfunction of the Fresnel transform per se, but rather to a self-similar diﬀracting proﬁle under the Fresnel transform at the corresponding distance of propagation. In other words, if f α = µf under the fractional FT, then assuming

|µ| = 1 (which is true for any cyclic fractional FT), the magnitude of the Fresnel output corresponding to input proﬁle f is given by

ˆ 1 (16.7) |f2(x)| = |f(x/W (d))| pW (d) which indicates that the Fresnel output has the same functional form as the input, but horizontally and vertically scaled by factors of 1/W (d) and 1/pW (d), respectively. The 110 profile retains the same basic functional shape, but spreads out and diminishes in amplitude as it diffracts. This interesting connection with diffraction motivates one to consider the problem of eigenfunctions of the fractional Fourier transform.

We can form diﬀerential eigenoperators in a straightforward fashion for cyclic orders of the fractional Fourier transform, following the procedure outlined in this work. These will be less concise but more general than eigenoperators of the standard Fourier transform.

±1/2 As an example, diﬀerential eigenoperators of either square-root transform F = F±π/4 can be constructed; these eigenoperators have commuting constants equal to the transform eigenvalues, which are the complex eighth-roots of unity, satisfying c8 = 1. As in the standard FT case, the eigenoperators for the fractional FT will be constructed from the transform relationships between the position and derivative operators, which are generalizations of the corresponding theorems in the standard case and have the forms derived by Namias[27]:

(16.8) Fα[x] = (x cos α − i sin α d/dx)Fα

(16.9) Fα[d/dx] = (−i sin α x + cos α d/dx)Fα for angular order parameter α. These equations could also be cast in the more compact formx ˆ = (x cos α − i sin α d/dx) and d/dx[ = (−i sin α x + cos α d/dx). Even more telling is to write

 \      x cos α − sin α x       (16.10)   =     i d/dx sin α cos α i d/dx 111 a form which makes the rotational interpretation of the fractional FT readily apparent in light of the classic unimodular rotation coeﬃcient matrix. While elegant, this form should not be taken to imply that a simple matrix power argument can be used to ﬁnd the transforms of operator powers xm and dm/dxm. These can instead be found by raising

Eqns. (16.8) and (16.9) to the m-th power.

To take a simple example, we apply these equations to construct eigenoperators for

m Fπ/4, starting from the operator A0 = x with c = 1. The general expression for the eigenoperator is

m m m m L = x + Fπ/4[x ] + Fπ/2[x ] + F3π/4[x ](16.11) keeping in mind the duplication or cancellation of terms when the full cycle of fractional transforms is included (cf. the standard FT case discussed in Section 4 of this work). For √ √ m = 1, the above transformation relation gives Fπ/4[x] = x/ 2 − i/ 2 d/dx. It is easy to show that when applied twice in succession, the standard FT result Fπ/2[x] = −i d/dx √ √ is obtained, and similarly for F3π/4[x] = −x/ 2 − i/ 2 d/dx. The third term can also √ √ be found from F3π/4[x] = Fπ/4[Fπ/2[x]] = −iFπ/4[d/dx] = −i(−i/ 2 x + 1/ 2) d/dx. √ Using these results in Eqn. (16.11) gives L = x − i(1 + 2) d/dx, which results in the eigenequation

i (16.12) f 0 + √ xf = λf 1 + 2

With λ = 0 and supplemental conditions of decay, integrability, etc., the solution is √ f = e−i/(1+ 2)x2/2, which is a scaled version of the FEF chirp function discussed previously. 112

For m = 2, it is straightforward to show that the eigenoperator becomes L = 2(x2 − d2/dx2), resulting in the eigenequation

(16.13) f 00 − x2f = λf

−x2/2 which is the deﬁning equation for the Gauss-Hermite eigenfunctions ψn(x) = AnHn(x)e of the standard FT, which we have discussed earlier; Hn represents the Hermite polynomials and An the normalization constants. Hence we have just shown that these are

1/2 eigenfunctions also of the square root Fourier transform Fπ/4 = F . In fact, we can easily show that the Gauss-Hermite functions are eigenfunctions for all fractional orders

− of the FT, using our eigenoperator formalism. To put it diﬀerently, the operator L02 is an eigenoperator for any fractional order of the FT. To see this, we simply need to take

− the fractional transform of L02, as follows:

2 2 Fα[x ] = (x cos α − i sin α d/dx) Fα =

2 2 2 2 2 (16.14) (x cos α − i cos α sin α d/dx x − i cos α sin α x d/dx − sin α d /dx )Fα and

2 2 2 Fα[d /dx ] = (−i sin α x + cos α d/dx) Fα =

2 2 2 2 2 (16.15) (−x sin α − i cos α sin α d/dx x − i cos α sin α x d/dx + cos α d /dx )Fα 113

Therefore

2 2 2 2 2 2 Fα[d /dx − x ] = Fα[d /dx ] − Fα[x ] =

2 2 2 2 2 2 2 2 2 2 (16.16) [(− sin α − cos α) x + (sin α + cos α) d /dx ]Fα = (d /dx − x )Fα

− − − In other words, Fα[L02] = L02Fα, which demonstrates that L02 is an eigenoperator for any fractional order α. Then the eigenfunctions of this operator, the Gauss-Hermite functions, are eigenfunctions for all fractional orders of the FT, as claimed above. Interpreted in terms of diffractive propagation, this implies that in one dimension, the Gauss-Hermite functions diffract in a self-similar manner under Fresnel propagation to any distance. Since these functions are also square-integrable, they constitute an entire family of finite-energy self-similarly diffracting beams.

What can be said about the general second-order operator which is invariant under the fractional FT? If we consider the operator

L = ax2 + bxD + cD2 + dx + eD + fI where D and I represent the diﬀerentiation and identity operators, then we can use the above operator relations to compute Lˆ. If we now require L to be self-transform (i.e. 114

Lˆ = L), we arrive at the following relations among the six coeﬃcients:

a(cos2 α − 1) − ib cos α sin α − c sin2 α = 0

(a + c)(−i sin 2α) + b(cos 2α − 1) = 0

−a sin2 α − ib cos α sin α + c(cos2 α − 1) = 0

d(cos α − 1) − ie sin α = 0

e(cos α − 1) − id sin α = 0

(a + c)(−i cos α sin α) − b sin2 α = 0

For arbitrary α, these equations imply that f is arbitrary, d = e = b = 0, and a = −c.

Therefore the general second-order self-transform operator must have the form L = a(D2− x2) + fI; it is clear that the eigenfunction problem for this operator is simply that of the

Gauss-Hermite functions with L = D2 − x2. Thus the Gauss-Hermite functions are the eigenfunctions of this general operator, and one must consider higher-order eigenoperators in order to explore potentially new fractional Fourier eigenfunctions.

We now return to Eqn. (16.6). If the input function f(u0) is an eigenfunction of the order-α fractional FT, then f α(u) = µf(u). We have established above via the eigenoperator approach that the Gauss-Hermite functions ψn(u) are eigenfunctions of the

inα order-α fractional FT for arbitrary α; this can be expressed via Fα[ψn(u)] = e ψn(u).

Some authors such as Namias[27] take this ansatz as the starting point for their derivation of the fractional FT; here we could derive it by formally raising both sides of the equation

inπ/2 Fπ/2[ψn(u)] = e ψn(u) with a = 1 – the Gauss-Hermite eigenfunction property of 115

0 0 the standard FT – to the arbitrary a power. In particular, if we take f(u ) = ψ0(u ) = e−u02/2, corresponding to a Gaussian input proﬁle, then f α(u) = f(u) = e−u2/2. Then the dimensional output of the Fresnel transform applied to this input can be written in accordance with Eqn. (16.6):

ˆ 1 ikd −iα/2 ikx2/2R(d) 1 −x2/2W (d)2 (16.17) f2(x) = p e e e · √ e W (d) W0

This can be written as

2 2 (x /2)W0 1 2 4 2 − 4 2 ˆ ikd −iα/2 iπx λd/[W0 +(λd) ] W +(λd) (16.18) f2(x) = e e e e 0 p4 4 2 W0 + (λd)

Now, the physical behavior of an input Gaussian proﬁle under Fresnel diﬀraction is a well-known standard result – see, for example, Pampaloni & Enderlein[33]. This behavior is given by

2 ikz− x 1 w2+ 2iz (16.19) g(z) = q · e 0 k 2 2iz w0 + k

2 2 in 1-D. In the above, z = d is the distance of propagation, and w0 = W0 /π as discussed above. Clearly when z = 0 in this expression, we have a Gaussian function for the input proﬁle. It is straightforward to show that the expression above is equal to the expected output expression from the Fresnel transform; we now proceed to do so.

If we write the ﬁrst factor in Eqn. (16.19) in polar form, we ﬁnd

s 2 2iz √ 1 w0 − π 2 k −i arctan(λz/W0 )/2 = 4 = e q 2iz 2 2 p4 4 2 2 w0 + 4z /k W0 + (λz) w0 + k 116 where we have used 2z/k = λz/π. But the exponential factor simply equals e−iα/2, since

2 α = arctan(λz/W0 ). Therefore the original expression takes the form

2 ikz− x 1 w2+ 2iz g(z) = q e 0 k 2 2iz w0 + k

√ πW 2 − 0 x2 2iz/k x2 π ikz −iα/2 4 2 4 2 2 = e e e W0 +(λz) e w0+4z /k p4 4 2 W0 + (λz)

2iz/k 2 iπλzx2 4 2 2 x 4 2 The last factor in the above expression can be written as e w0+4z /k = e W0 +(λz) . Thus we arrive at the ﬁnal form

2 ikz− x 1 w2+ 2iz g(z) = q e 0 k 2 2iz w0 + k

√ πW 2 2 − 0 x2 iπλzx π ikz −iα/2 4 2 4 2 (16.20) = e e e W0 +(λz) e W0 +(λz) p4 4 2 W0 + (λz)

This last expression can be matched term-for-term with the ﬁnal expression in Eqn.

(16.18) above. Except for factors of π or 2π in the coeﬃcients, these two expressions agree exactly. We have therefore shown the equivalence of this expression with expression

(16.19) above for the behavior of a diﬀracting Gaussian beam. Thus the known diﬀractive behavior of a 1-D Gaussian beam can be arrived at via eigenoperators of the fractional

Fourier transform.

In a very similar fashion, we can demonstrate the equivalence of our approach with the known propagation behavior of any higher-order Gauss-Hermite beam. It can be shown[33] that if the input proﬁle has the form of any Gauss-Hermite function, then the propagation behavior is represented by a simple modiﬁcation of the above expression 117

(16.20) for g(z):

√ √ ! πW 2 2 − 0 x2 iπλzx π ikz −iα/2 x πW0 W 4+(λz)2 W 4+(λz)2 inα (16.21)g(z) = e e Hn e 0 e 0 e p4 4 2 p 4 2 W0 + (λz) W0 + (λd) where Hn(·), the Hermite polynomial of order n, multiplies the Gaussian and an additional

Gouy phase accumulation of einα arises due to the higher-order functional proﬁle.

0 0 0 −u02/2 If we now take f(u ) = ψn(u ) = Hn(u )e , the order-n Gauss-Hermite function

α inα inα −u2/2 (without the An constant) as input proﬁle, then f (u) = e f(u) = e Hn(u)e .

Then the dimensional output of the Fresnel transform applied to this input can be written in accordance with Eqn. (16.6):

√ ˆ 1 ikd −iα/2 ikx2/2R(d) 1 −x2/2W (d)2 inα (16.22) f2(x) = p e e e · √ Hn(x π/W (d))e e W (d) W0

This can be written as

(16.23) √ ! 2 2 (x /2)W0 1 2 4 2 x πW0 − 4 2 ˆ ikd −iα/2 iπx λd/[W0 +(λd) ] W +(λd) inα f2(x) = e e e Hn e 0 e p4 4 2 p 4 2 W0 + (λd) W0 + (λd)

This last expression can be matched term-for-term with the ﬁnal expression in Eqn.

(16.21) above. Except for factors of π or 2π in the coefficients, these two expressions agree exactly. We have therefore shown the equivalence of the two expressions above for the propagation behavior of a diffracting Gauss-Hermite beam. Thus the known diffractive behavior of a 1-D Gauss-Hermite beam of any order can be arrived at via the eigenoperator formalism for the fractional Fourier transform. 118

More involved eigenoperators of Fα can certainly be constructed using concatenation of powers of Eqns. (16.8, 16.9) to develop eigenoperators from expressions such as xmdn/dxn and dm/dxm(xn·) that played a role in our analysis of the standard FT elsewhere in this work. The important point physically is that any self-adjoint BVP’s generated by

FrFEO’s (fractional Fourier eigenoperators), if the BVP’s are invariant for arbitrary order

α, would have solutions which represent ﬁnite-energy self-similarly diﬀracting beams in

1-D, by virtue of their square-integrability. This concludes our discussion of the interplay of Fresnel and fractional Fourier transforms modeling diﬀraction in one dimension. 119

CHAPTER 17

Standard Hankel Transform

We now turn our attention to speciﬁc LCT’s in two dimensions. The ﬁrst transform of optical interest is the standard Hankel transform, which is a rotationally symmetric version of the 2-D standard Fourier transform. The proof of this is standard and parallels our development of the 2-D LCT from the 1-D LCT given above. If we use in Eqn. (12.4)   0 −1   the same parameter matrix M =   that generates the FT, we obtain 1 0

Z ∞ ˆ in(ψ+π/2) H[f] = f(ρ, ψ) = 2πKe f(r)Jn(rρ)r dr 0

≡ ein(ψ+π/2)fˆ(ρ)(17.1) assuming n-fold rotational symmetry of the input. The normalization constant K now has unit magnitude since 1/B = −1. Since the inverse transform is the adjoint of the forward transform, and since the forward kernel consists of the real function Jn(rρ), which only involves the symmetric product of the forward and inverse transform variables, the inverse kernel is the same function. (It should be remembered that the r term in the transform is really a Jacobian factor that comes from converting Cartesian to polar coordinates for the integration.) The ﬁnal form of the phase constant in the forward direction comes from simplifying (−1)nein(ψ−π/2) = einπ+in(ψ−π/2) = ein(ψ+π/2). The (−1)n factor is a consequence of B = −1 and the previously mentioned parity property of Jn. 120

Thus the n-th order standard Hankel transform, as deﬁned here, is self-adjoint, and is its own inverse, except for the complex phase constants out front, which become

K∗e−in(θ−π/2). If we are only interested in the magnitude or intensity (squared magnitude) of the output, this phase factor is immaterial. In any case, the transform is essentially cyclic with period N = 2. Application of this transform twice in succession to a particular input function will reproduce a multiple of the input function. This is a well-known feature of the standard Hankel transform. Since N = 2, the eigenvalues and commuting constants of the HT are simply ±1. It should be noted that there are other deﬁnitions of the standard HT, which diﬀer by various powers of the variables in the integrand. We make further comments on the self-adjointness of this transform below.

Just as the 1-D FT may be physically interpreted as the far-field Fraunhofer diffraction pattern of an input wave, so the 2-D HT may be physically interpreted as the far-field

Fraunhofer diffraction pattern of an input wave with rotational symmetry. Thus we may say that eigenfunctions of the HT – the so-called Hankel eigenfunctions (HEF’s) or self-Hankel functions (SHF), as some authors call them – can be regarded as far-field diffraction-invariant beams in two dimensions – that is, waveforms which reproduce themselves in the far-field limit. If we can form self-adjoint BVP’s with differential eigenoperators of the Hankel transform, then their eigenfunctions will be finitely square-integrable, and hence could in principle be optically realizable as diffraction-invariant beams in the far-field. On a related note, the zero-order Bessel function J0(r) was determined to be a genuinely non-diffracting waveform by authors who undertook a first-principles analysis starting from the scalar wave equation in free space[7, 8]. That is to say, this function retains its original shape without diffractive modification at all propagation distances. 121

However, this function is not ﬁnitely square-integrable, so it is not physically realizable as a non-diﬀracting wave. In a later section we will discuss the connection between fractional

HEF’s and self-similarly diﬀracting beams.

Following the general approach presented in this work, we can ﬁnd HEF’s via differential eigenoperators of the standard HT. These can be developed from an operator transformation relationship between the position and derivative operators, just as in the

FT case. There is a variety of such relationships, but most of them do not preserve the order of the Bessel function in the kernel of the transform. The only one of any practical use to us is the relationship that preserves the transform order n on both sides of the equation:

(17.2) H[d2/dr2 + 1/r d/dr − n2/r2] = −r2H which can be found in standard treatments of the Hankel transform. Here H is the Hankel transform of Bessel function order n in the kernel. We refrain from subscripting the order on H to avoid confusion with the fractional case considered later. A second application of H to both sides of this relation yields the equivalent form

d2/dr2 + 1/r d/dr − n2/r2 = H[−r2](17.3)

2 ˆ 2 2 Deﬁning the operator A0 = −r , we can form the eigenoperator L = A0 + A0 = d /dr +

1/r d/dr − n2/r2 − r2, which is the radial operator component of Schrodinger’s equation for the quantum harmonic oscillator in 2-D. The eigenfunctions satisfying Lf = λf are well known to be the generalized two-dimensional Gauss-Laguerre functions ψ(r) =

|n| −kr2/2 |n| 2 |n| r e Lm (kr ), where the symbol Lm represents a generalized Laguerre polynomial 122 and is not to be confused with the diﬀerential eigenoperator L just constructed. (The parameter m is related to the order n and the eigenvalue λ, all of which are related to the quantum numbers for the system; k corresponds to the energy level of the system.)

Since L is an eigenoperator of the HT, the eigenfunctions of L – the Gauss-Laguerre functions – are simultaneously eigenfunctions of the standard HT. We show later that these same functions are eigenfunctions of the fractional HT of arbitrary order, and discuss the diﬀractive implications. More involved diﬀerential eigenoperators can be constructed by successive applications of the basic relation in Eqns. (17.2) or (17.3). 123

CHAPTER 18

Fresnel-Hankel Transform

Next is the 2-D version of the Fresnel diﬀraction integral in Eqn. (14.4). We use the   1 λd   matrix M =   from the 1-D case in the 2-D transform of Eqn. (12.4), which 0 1 gives

Z ∞ ˆ in(ψ−π/2) i/(2λd)(r2+ρ2) T [f] = f(ρ, ψ) = 2πKe · f(r)e Jn(rρ/λd)r dr 0

≡ ein(ψ−π/2)fˆ(ρ)(18.1) assuming n-fold rotational symmetry of the input, i.e. f(r, θ) = f(r)einθ. We have written the phase constant with the (−1)n factor to be formally consistent with the other Hankel transform pairs we discuss. The inverse transform is given by the adjoint of Eqn. (18.1),   1 −λd   which takes the parameter matrix M =  : 0 1

Z ∞ −1 ˆ ∗ −in(θ−π/2) n ˆ −i/(2λd)(ρ2+r2) T [f] = f(r, θ) ≡ 2πK e (−1) · f(ρ)e Jn(ρr/λd)ρ dρ 0

≡ e−in(θ+π/2)f(r)(18.2)

We refer to this pair as the forward and inverse Fresnel-Hankel transform of order n, respectively. It represents the process of free-space diffractive propagation to distance d, applied to a 2-D input waveform with n-fold rotational symmetry. The constant K may 124 be taken propotional to, or simply equal to, 1/λd, with any proportionality constant a simple phase factor. To the best of our knowledge, this is the first time that the complete form of the diffraction integral in two dimensions with general rotational symmetry has appeared in the literature. We include this transform here for completeness, but do not pursue it much further. Instead we reformulate the 2-D diffraction problem in terms of the fractional Hankel transform in the next section. We finally point out that this Fresnel transform in both 1-D and 2-D can be thought of as a one-parameter subgroup of the general LCT group, with continuous parameter λd. 125

CHAPTER 19

Fractional Hankel Transform – Deﬁnition and Basic Properties

Now we consider the 2-D version of the fractional Fourier transform with rotational symmetry, known as the fractional Hankel transform. As in 1-D, we start with technical details and theoretical properties in the present section, then discuss application to diﬀrac-   cos α − sin α   tion in the next section. We use the same parameter matrices M =   and sin α cos α M −1 = M T as in the 1-D case, to give the forward and inverse transform pair, assuming n-fold symmetry, as

Z ∞ ˆ in(ψ−π/2) n −i/2(cot α r2+cot α ρ2) Hα[f] = f(ρ, ψ) = 2πKe (−1) · f(r)e Jn(csc α rρ)r dr 0

≡ ein(ψ+π/2)fˆ(ρ)(19.1) and

Z ∞ −1 ˆ ∗ −in(θ−π/2) ˆ i/2(cot α ρ2+cot α r2) Hα [f] = f(r, θ) ≡ 2πK e · f(ρ)e Jn(csc α ρr)ρ dρ 0

≡ e−in(θ−π/2)f(r)(19.2)

n respectively. Again we have used Jn(−x) = (−1) Jn(x). Now K is proportional to, or simply equal to, 1/B = 1/ sin α. For α = π/2 we easily recover the standard HT Hπ/2 = H for unspeciﬁed order of the Bessel kernel. (Note that care must be taken to distinguish the fractional order of the transform from the order of the Bessel function in the kernel. 126

Furthermore, fractionalization of the order of the Bessel function from integers to real numbers is another way in which the standard Hankel transform can be fractionalized.

This approach is mentioned in Lohmann[23] where the authors lament its seeming lack of obvious optical applicability.)

Also, it is straightforward to show that α = 0 or π leads to a delta-function kernel which reproduces the input function, so H0 = Hπ = I, the identity transform; the former is consistent with the fractional Fourier case while the latter requires an angular parameter which is half of the corresponding fractional Fourier case. In contrast to the Fourier case, there is no value of α which produces the parity transform P. This is consistent with the nonnegativity of the radial co-ordinates r and ρ, which renders moot the issue of a parity transform.

The angular parameter α now varies from −π/2 to π/2 (or 0 to π), while a = (2/π)α ranges from -1 to 1 (or 0 to 2). Note that these intervals are half the size of the corresponding itervals for the fractional FT. As the unit circle with central angle α is traced from 0 to 2π, the range of fractional HT’s is spanned exactly twice while the range of fractional FT’s is spanned exactly once. In this sense, the set of fractional FT’s is isomorphic to a double cover of the set of fractional HT’s. This indicates that for a ﬁxed value of α giving rise to a cyclic transform, the corresponding fractional HT will have half the period of the corresponding fractional FT.

For a transform with given α and a values, the inverse transform corresponds to

−α and −a (or π − α and 2 − α). Transforms with the same parameters modulo π or 2 are identical. For even n we see that α = ±π/2 gives rise to the same transform

Hπ/2 = H−π/2 = H, the standard HT. Moreover, solving a = 2 − a to get a = 1 gives the 127

known result that the standard HT Hπ/2 is its own inverse, as discussed above. Just as in the Fourier case in 1-D, we see that the fractional Hankel transform with continuous order parameter α constitutes a one-parameter subgroup of the general group of LCT’s.

We note that the value of α here corresponds to half the value of α used by Namias[28] in his development.

As in the 1-D case, the order parameters may take any real value we wish, but the most advantageous values are those which render the fractional transform cyclic. If the transform order parameter a is such that 2/a is an integer, then the fractional transform will be cyclic with period N = 2/a. In terms of the angular order parameter α = aπ/2, if N is the least positive integer such that αN = π, then N = π/α = π/(aπ/2) = 2/a

1/2 is the period. For example, a “square root” Hankel transform Hπ/4 = H can also be deﬁned by analogy with the Fourier case. This transform is cyclic with period N = 4,

4 1/2 4 2 since (Hπ/4) = (H ) = Hπ = H = I. Also, the eigenvalues and commuting constants of this transform are the complex fourth-roots of unity. One can compare these comments with the analogous comments in the fractional FT case above.

The same relationships between the parameter matrices of the Fresnel and fractional

FT in 1-D automatically hold between the parameter matrices of the Fresnel-Hankel and fractional HT in 2-D. Thus we may regard the latter two transforms as being deﬁnable each in terms of the other, via the intermediary LCT’s with parameter matrices X1 − X4.

Just as in the 1-D case, there is no need to hold the Fresnel-Hankel transform in 2-D as a separate entity with its own definition and physical interpretation; we can just define and interpret it in terms of the fractional HT. Again, because of the more flexible and systematic behavior displayed by cyclic transforms in the fractional family, we propose 128 the fractional HT as the more convenient approach for studying diffraction of rotationally symmetric functions in 2-D from an integral-transform perspective.

We conclude by remarking that in light of the rotation matrix used in the fractional

FT and HT deﬁnitions, these transforms can be viewed as one-parameter subgroups of the general LCT group, with continuous parameter α. 129

CHAPTER 20

Fractional Hankel Transform – Application to Diﬀraction

The fractional Hankel transform of general order is the rotationally symmetric 2-D version of the fractional Fourier transform of general order. Just as with the fractional FT in 1-D, the fractional Hankel transform subsumes the standard HT and the 2-D rotationally symmetric diﬀraction integral given by the Fresnel-Hankel transform in the previous section. In light of our previous discussion relating the fractional Fourier transform to the

Fresnel transform and free-space propagation in 1-D, it follows that diﬀractive propagation in 2-D is equivalent to fractional Hankel transformation of continuously varying order applied to the input wave. Thus eigenfunctions of the fractional HT of a particular order would correspond to functions which diﬀract to a self-similar shape at the corresponding propagation distance.

Just as the Fresnel transform of an input function in 1-D can be expressed via the fractional Fourier transform, so can the diﬀracted output of a rotationally symmetric input function in 2-D be expressed via the fractional Hankel transform. The relation is entirely similar to Eqn. (16.6) above:

1 2 fˆ (r) = eikde−iαeikr /2R(d) · f α(r/W (d))(20.1) 2 W (d) 1

α where now f is the output of the transform Hα applied to the input f. The development in 2-D entirely parallels the development in 1-D except that the scale pre-factors 130 appear to the ﬁrst power in 2-D, rather than being square-rooted as they were in 1-D.

An important observation is that the above scaling relationships are fundamental and intrinsic to the Fresnel diffraction integral. Although they enter the picture in the specific case of diffraction of Gauss-Laguerre profiles, as we will see below, they are not a specific property only of Gauss-Laguerre diffraction.

Turning now to the matter of transform eigenfunctions, we infer that any dimensionless eigenfunction of the fractional HT of a particular order would then correspond, upon re- dimensionalization and conversion to the Fresnel output as above, not to an eigenfunction of the Fresnel transform per se, but rather to a self-similar diﬀracting proﬁle under the

Fresnel transform at the corresponding distance of propagation. In other words, if f α = µf under the fractional HT, then the magnitude of the Fresnel output corresponding to input proﬁle f is given by

1 (20.2) |fˆ (r)| = |f(r/W (d))| 2 W (d) which indicates that the Fresnel output has the same functional form as the input, but horizontally and vertically scaled by factors of 1/W (d). The profile retains the same basic functional shape, but spreads out and diminishes in amplitude as it diffracts. This interesting connection with diffraction motivates us to consider the question of eigenfunctions of the fractional Hankel transform, which we now do via transform eigenoperators.

Diﬀerential eigenoperators of the fractional HT can be developed from the fractional generalization of the standard HT relation in Eqn. (17.2) above. This is derived by 131

Namias [28] and can be written as

2 2 2 2 2 2 Hα[d /dr + 1/r d/dr − ν /r + cot α r ] =

2 2 (20.3) −[(1 − cot α)r + 2i cot α(1 + r d/dr)]Hα or alternatively as

2 Hα[r ] =

2 2 2 2 2 2 (20.4) csc α[−d /dr − 1/r d/dr + ν /r + r cot α − 2i cot α − 2ir cot α]Hα where the Bessel function of order ν is in the kernel. As an example, we can construct

2m a diﬀerential eigenoperator for the square root transform Hπ/4 by taking A0 = r and

α = π/4. The general expression for the eigenoperator L is then

2m 2m 2m 2m L = r + Hπ/4[r ] + Hπ/2[r ] + H3π/4[r ](20.5) where for m > 1 the relation in Eqn. (20.4) must be raised to the mth power, as in the fractional FT case. For m = 1 one can show that with proper application of (20.4) to

(20.5), the equation Lf = λf simpliﬁes to

(20.6) f 00 + f 0/r + (−ν2/r2 − r2)f = λf which is the deﬁning ODE for the Gauss-Laguerre eigenfunctions mentioned above for the standard HT. So these functions are seen to also be eigenfunctions of the square root

1/2 transform Hπ/4 = H , just as the Gauss-Hermite functions were shown above to also be

1/2 eigenfunctions of the square root FT Fπ/4 = F . 132

Just as with the standard HT, the above transform relation for the fractional HT is the only one that preserves the order of the Bessel function in the kernel of the transform.

So the operator

r2 + csc2 α[−d2/dr2 − 1/r d/dr + ν2/r2 + r2 cot α − 2i cot α − 2ir cot α] is the only 2nd-order self-transform operator. We can show via standard scalings and change of variables, similar to the Fresnel case discussed previously, that the Gauss-

Laguerrre functions are the eigenfunctions of this general operator. Thus one must consider higher-order eigenoperators in order to explore potentially new fractional Hankel eigenfunctions.

As in the one-dimensional case with Gauss-Hermite functions, we can use our eigenoperator formalism to show that in two dimensions, the Gauss-Laguerre functions are eigenfunctions of the Hankel transform of any fractional order. Rather than using the somewhat unwieldy and impractical operator transform relations, however, we can use a succinct hyperdiﬀerential argument. The fractional Hankel transform with order-n Bessel

iαL kernel has the hyperdiﬀerential form Hα = e , where α is the angular order and L is the standard Hankel diﬀerential operator, L = d2/dr2 + 1/r d/dr − n2/r2 − r2[28]. We have already discussed how the same is true of the fractional Fourier transform, which has the

iαL− representation Fα = e 02 . It is a simple matter to see that Hα and L commute:

X (iα)n X (iα)n X (iα)n (20.7) H L = ( Ln)L = Ln+1 = L Ln = LH α n! n! n! α

As in the Fourier case, this result is independent of α, so it implies that L is an eigenoperator for any fractional order of the HT. This in turn implies that the eigenfunctions of L – 133 the Gauss-Laguerre functions, discussed in more detail below – are eigenfunctions of the fractional HT of arbitrary order. Using our approach of commuting operators, we have given a concise alternative proof of this fact. Now, interpreted in terms of diffractive propagation, this result implies that in two dimensions, the Gauss-Laguerre functions diffract in a self-similar fashion under Fresnel propagation to any distance. Since these functions are also square-integrable, they constitute an entire family of finite-energy self-similarly diffracting beams.

ν −u2/2 (ν) 2 The Gauss-Laguerre functions ψn(u) = u e Ln (u ) are eigenfunctions of the order-α fractional HT with Bessel kernel of order ν for arbitrary α; this can be ex-

in2α pressed via Hα[ψn(u)] = e ψn(u). Some authors such as Namias[28] take this ansatz as the starting point for their derivation of the fractional HT; here we could derive it by

inπ formally raising both sides of the equation Hπ/2[ψn(u)] = e ψn(u) with a = 1 – the

Gauss-Laguerre eigenfunction property of the standard HT – to the arbitrary a power.

0 0 −u02/2 In particular, if we take f(u ) = ψ0(u ) = e , corresponding to a Gaussian input pro-

ﬁle, then f α(u) = f(u) = e−u2/2. Then the dimensional output of the Fresnel transform applied to this input can be written in accordance with Eqn. (20.1):

ˆ 1 ikd −iα ikr2/2R(d) 1 −r2/2W (d)2 (20.8) f2(r) = e e e · e W (d) W0

This can be written as

2 2 (r /2)W0 1 2 4 2 − 4 2 ˆ ikd −iα iπr λd/[W0 +(λd) ] W +(λd) (20.9) f2(r) = e e e e 0 p 4 2 W0 + (λd) 134

Now, the physical behavior of a diﬀracting Gaussian beam in two dimensions is a well-known standard result. This behavior is given by

2 ikz− r 1 w2+ 2iz 0 k (20.10) g(z) = 2 2iz · e w0 + k

2 2 2 2 2 In the above, r = x + y and z = d is the distance of propagation. Also, w0 = W0 /π as discussed above. It is straightforward to show that the expression above is equal to the expected output expression from the Fresnel transform; we now proceed to do so.

If we write the ﬁrst factor in Eqn. (20.10) in polar form, we ﬁnd

2 2iz 1 w0 − π −i arctan(λz/W 2) = k = e 0 2 2iz 4 2 2 p 4 2 w0 + k w0 + 4z /k W0 + (λz) where we have used 2z/k = λz/π. But the exponential factor simply equals e−iα, since

2 α = arctan(λz/W0 ). Therefore the original expression takes the form

2 ikz− r 1 w2+ 2iz 0 k g(z) = 2 2iz e w0 + k

πW 2 − 0 r2 2iz/k r2 π ikz −iα 4 2 4 2 2 = e e e W0 +(λz) e w0+4z /k p 4 2 W0 + (λz)

2iz/k 2 iπλzr2 4 2 2 r 4 2 The last factor in the above expression can be written as e w0+4z /k = e W0 +(λz) . Thus we arrive at the ﬁnal form

2 ikz− r 1 w2+ 2iz 0 k g(z) = 2 2iz e w0 + k

πW 2 2 − 0 r2 iπλzr π ikz −iα 4 2 4 2 (20.11) = e e e W0 +(λz) e W0 +(λz) p 4 2 W0 + (λz) 135

This last expression can be matched term-for-term with the ﬁnal expression in Eqn. (20.9) above. Except for factors of π or 2π in the coeﬃcients, these two expressions agree exactly.

We have therefore shown the equivalence of that expression with expression (20.10) above for the behavior of a diﬀracting Gaussian beam. Thus the known diﬀractive behavior of a 2-D Gaussian beam can be arrived at via eigenoperators of the fractional Hankel transform.

In a very similar fashion, we can demonstrate the equivalence of our approach with the known propagation behavior of any higher-order Gauss-Laguerre beam. It is straightforward to show that if the input proﬁle has the form of any Gauss-Laguerre function, then the propagation behavior is represented by a simple modiﬁcation of the above expression

(20.11) for g(z):

√ ! iπλzr2 π ikz −iα r πW0 W 4+(λz)2 in2α (20.12) g(z) = e e ψn e 0 e p 4 2 p 4 2 W0 + (λz) W0 + (λd) where ψn(·) represents the order-n Gauss-Laguerre function (we assume that the other index ν in these functions remains ﬁxed while n is allowed to vary). Similiar to 1-D, an additional Gouy phase accumulation of ein2α arises due to the higher-order functional proﬁle.

0 0 0ν −u02/2 (ν) 02 If we now take f(u ) = ψn(u ) = u e Ln (u ), the order-n Gauss-Laguerre func-

α in2α in2α tion, as input proﬁle, then f (u) = e f(u) = e ψn(u). Then the dimensional output of the diﬀraction transform applied to this input can be written in accordance with Eqn.

(16.6):

√ ˆ 1 ikd −iα ikr2/2R(d) 1 in2α (20.13) f2(r) = e e e · ψn(r π/W (d))e W (d) W0 136

This can be written as

√ ! 1 2 4 2 r πW0 ˆ ikd −iα iπr λd/[W0 +(λd) ] in2α (20.14) f2(r) = e e e ψn e p 4 2 p 4 2 W0 + (λd) W0 + (λd)

This last expression can be matched term-for-term with the ﬁnal expression in Eqn.

(20.12) above. Except for factors of π or 2π in the coefficients, these two expressions agree exactly. We have therefore shown the equivalence of the two expressions above for the propagation behavior of a diffracting Gauss-Laguerre beam. Thus the known diffractive behavior of a 2-D Gauss-Laguerre beam of any order can be arrived at via the eigenoperator formalism for the fractional Hankel transform.

More complicated eigenoperators can be constructed if one desires, starting from powers or combinations of the operator r2 and its transform, via the fundamental relation in

Eqn. (20.4). Just as in the 1-D case with the fractional Fourier transform, any self-adjoint boundary-value problems generated by FrHEO’s (fractional Hankel eigenoperators), would have eigenfunction solutions representing ﬁnite-energy self-similarly diﬀracting beams in

2-D with n = ν-fold rotational symmetry, by virtue of their square-integrability.

This concludes our discussion of the interplay of Fresnel and fractional Hankel transforms modeling diffraction in two dimensions. We have now shown how linear canonical transform theory can be used to illuminate the connection between diffraction integrals and fractional transforms in 1-D and 2-D. In particular, cyclic LCT’s afford a novel and systematic way for approaching the eigenfunction problem related to light propagation in one or two dimensions. 137

CHAPTER 21

Comments on Self-Adjoint Operators

The issue of self-adjointness is important in the eigenproblems under consideration here because the self-adjointness of a BVP implies the integrability, and hence finite energy content and physical realizability, of an eigenfunction solution satisying transformability conditions, such as decay boundary conditions at infinity. Here we make a few comments about this issue in relation to the operators and transforms under discussion. We have already seen that certain families of differential eigenoperators of a cyclic transform may turn out to be self-adjoint (cf. previous discussion of the standard FT in section 4 above); here we seek a more systematic understanding of the issue.

If the integral transform T is cyclic, then T −1 is also cyclic with the same period, so we can ask how the differential eigenoperators of T and T −1 are related. For some differential operator L and integral transform T which is both cyclic and unitary, we first observe that (T L)† = L†T † = L†T −1 since T † = T −1. If L happens to be self-adjoint, this becomes (T L)† = LT −1. On the other hand, if L is an eigenoperator of T with commuting constant c, then

L†T −1 = L†T † = (T L)† = (cLT )† = c∗T †L† = c∗T −1L†

This indicates that if L is self-transform under T with commuting constant c, then L† is self-transform under T −1 with commuting constant 1/c∗. Furthermore, if T is also 138 self-adjoint, so that T = T † = T −1, then L† is self-transform under T with commuting constant 1/c∗. But c = 1/c∗, so both L and L† are self-transform under T with commuting constant c. An example of such a transform is T = Hπ/2, the standard Hankel transform described above with commuting constants ±1. If L is an eigenoperator of Hπ/2 with commuting constant c, then L0 = L + L† is a self-adjoint diﬀerential operator which is also an eigenoperator of Hπ/2 with the same commuting constant:

T L0 = T (L + L†) = T L + T L† = cLT + cL†T = c(L + L†)T = cL0T

Decaying eigenfunctions of L0 will therefore be potential square-integrable eigenfunctions of Hπ/2, subject to the self-adjointness of the singular BVP on the inﬁnite interval, etc.

Finally, if L is self-adjoint but T is not, then L is self-transform under T −1 with commuting constant c. An example of this case is the standard FT, where a self-adjoint FEO such

± as L0n with n even commutes with c = ±1 with both the forward and inverse transforms. This is the most that can be said at this juncture about the issue of self-adjointness.

We have seen that it is straightforward to construct a differential operator which is either self-adjoint or self-transform under general cyclic T . The holy grail, so to speak, would be a similar systematic procedure for constructing differential operators which are both self-adjoint and self-transform under cyclic T – that is, self-adjoint TEO’s; these in turn should give rise to self-adjoint singular BVP’s on the infinite real line. The eigenfunctions of such operators would be simultaneously square-integrable and invariant under the transform. For the case of self-adjoint T , this is not hard to do, as we have seen in the previous paragraph. But because the operations of differential adjunction and integral transformation do not commute in general, it is not obvious what this procedure should 139 look like if T is unitary but not self-adjoint. Certainly this could be a fruitful area for further investigation. 140

CHAPTER 22

Group-theoretical Comments

At a number of places in the preceding sections we have made comments and observations of a group-theoretical nature in connection with the present work. For example, some of the generating expressions we have discussed for self-transform functions of T in previous sections can be seen as actions of the cyclotomic polynomial in T upon a generator g. This is a point we have previously mentioned. On a diﬀerent note, we have also seen that the general LCT theory is in a sense isomorphic to matrix group theory

– although the general LCT is not cyclic, just as the general matrix group GL(2) is not commutative. We feel that group theory offers a valuable additional perspective on, and further insight into, the theory of cyclic integral transforms. For this reason we devote a final section to a more detailed discussion of these issues. We start with a discussion of abstract details about transform power groups, exemplified by the Fourier transform, then we close with a key isomorphism relating the fractional Fourier transform to complex numbers.

The use of such terms as cyclic and generator to refer to the integral transforms discussed in these pages, along with the fundamental structural properties of these transforms, practically begs for connection to the notion of a cyclic group. The structural similarities are hardly coincidental, for the set of powers of the N-cyclic transform T comprises a group which is essentially isomorphic to a discrete cyclic group. Clearly two powers of T combine to give another power of T , so the set is closed under the action of 141 composition, and successive composition of powers of T is not only associative, but also commutative. Every power T k has the unique inverse T −k = T N−k, for k a residue mod

N, and clearly the power T 0 = I is the unique identity transform.

We can go further. In light of the cyclic nature of the multiplicative group formed by the complex N-th roots of unity – which we could call the cyclotomic group of order N, to stress its geometric ﬂavor in dividing the unit circle, or simply the cyclic group of order

N, since there exists one unique cyclic group of order N, up to isomorphism – the N-th roots of the identity transform I form a cyclic multiplicative group isomorphic to this group. A primitive transform T satisfying T N = I, with no smaller power producing the identity, is a generator of the transform group, just as any primitive N-th root of unity satisfying cN = 1 is a generator of the root group. Only powers k which are coprime to

N, when applied to one primitive root, will produce another primitive root. To put it succinctly, the eigenvalues of the primitive N-th roots of I are the N-th roots of 1, as we have seen. The present body of work could be seen in part as a study of primitive N-th roots of the transform I. The previously cited paper by Marhic[24] actually discusses abstract transform roots of identity and their optical implementation and realization.

Furthermore, just as a primitive M-th root and primitive N-th root of unity can be used to generate the group of R-th roots of unity, where R = lcm(M,N), so the same is true of primitive transform roots of identity. The order parameters of fractional transforms, which are cyclic for rational orders, combine just like powers of numbers, as we have seen.

We may consider the powers of a cyclic transform to be the powers of an element of a cyclic group whose order is the period of the transform. The sequence of individual powers constitutes the cycle or orbit of the group element corresponding to the transform. Any 142 transform in this orbit is a power of the original transform. If this cycle spans the entire sequence of possible transform powers, we call this transform a generator transform, by analogy with the corresponding group generator property. Moreover, any power relatively prime to the period, applied to a transform power with the same period, can be taken as the fundamental transform whose powers generate the group, just as any power relatively prime to the group order, when applied to a generator, creates another generator of the group. Only generator powers relatively prime to the period (order) will yield other generators of the group, and any generator can be expressed as some power of another generator.

In light of the theorem that a group of prime order is necessarily cyclic, we see that if an optical transform can be represented by a group-theoretical structure of prime order, that transform must be cyclic, hence any member of the transform group applied N times reproduces the original function. But again, only members with powers relatively prime to the order can generate the entire group; a power k which is not relatively prime to the group order corresponds to a transform that will reproduce the original function after application fewer than N times in succession, with the actual number N 0 given by the least residue satisfying kN 0 ≡ 0 mod N.

As for eigenfunction properties, an eigenfunction of any generator transform of the group is simultaneously an eigenfunction of every other power of that transform, since

T f = µf ⇒ T kf = µkf for T a generator transform. Furthermore, an eigenfunction of any power of a generator transform coprime to the group order is simultaneously an eigenfunction of every other power of that transform, since all other powers k mod N are members of the ﬁrst power’s residue orbit modulo the group order. That is to say, 143 if T nf = µf for (n, N) coprime, then T kf = (T n)af = µaf, for a satisfying na ≡ k ⇒ a ≡ kn−1 mod N. Finally, cyclic subgroups of the original transform group may be formed by taking powers of a transform generator which are not relatively prime to the period, just as with any cyclic group. By Lagrange’s theorem, any such subgroup has an order which evenly divides the original group order.

Again we can illustrate with the Fourier transform F with N = 4. Since 1 and 3 are relatively prime to 4, we have F and F 0 ≡ F −1 = F 3 as the two generator transforms of the transform group {F k}, with k any residue mod 4; the latter transform could also be taken as the generator transform with the former as its inverse and third power. This is consistent with the fact that the forward-inverse FT pair can be deﬁned in either direction. Furthermore, (F 0)3 = (F 3)3 = F 9 = F. The power F 2 does not generate the entire group, since 2 is not coprime to 4; hence F 2 reproduces an input function after N 0 = 2 applications, since 2(2) ≡ 0 mod 4. As already discussed, this power is simply the parity transform P, which merely changes the sign of the input argument.

Any eigenfunction of F or F 3 is an eigenfunction of all other powers of F, since all residues mod 4 can be expressed as multiples of 1 or 3. In particular, eigenfunctions of the forward FT F are also eigenfunctions of the inverse F 3, and vice-versa, with reciprocal eigenvalues, as is to be expected: fˆ = µf ⇒ 1/µ f = F −1[f], and µ3 = 1/µ.

Also, eigenfunctions of F are simultaneously eigenfunctions of F 2 = P, meaning FEF’s must possess either even or odd symmetry, depending on the eigenvalue µ, as previously discussed: fˆ = µf ⇒ F 2f = Pf = µ2f = ±f(x) = f(−x). Finally, since I = P2, we can say that the parity transform P = F 2 generates the 2-cyclic transform subgroup {Pk} with k = 0 or 1 and no other powers F k as members, since 2 divides 4. In a sense we 144 could say that the transform group {F k} is a fractionalization of the group {Pk}, with the indices of the former equal to half the indices of the latter. So, as discussed elsewhere in this work, the Fourier transform F could be thought of, or even deﬁned, as a square root of the parity transform, and a fourth root of the identity transform! In its turn, the standard FT group {F k} is isomorphic to the cyclotomic group of order 4, and is a cyclic subgroup of the more general cyclic groups of fractional FT’s with group order divisible by 4.

We can regard any order of fractional FT, say F m/n, with m/n in lowest terms, as

1/n 1 the m-th power of F = I 4n , an n-th root of the standard FT and a 4n-th root of the identity. Any transform root F 1/n can be taken as a generator for an entire group of 4n fractional FT’s with denominator n in their order parameter, which is isomorphic to the group of 4n-th roots of the identity. If km/n = 4, then the powers of F m/n form an order-k subgroup of this larger group. The order 4n of the group is the reciprocal of the identity root order 1/(4n). So the duplication of the term “order” in the transform and group contexts could be seen as more than coincidental; N-th complex roots of unity could be called cyclic roots of order 1/N, highlighting their membership in a group of reciprocal order.

All of the above points serve to demonstrate that cyclic transform theory is in a sense isomorphic to cyclic group theory in general, and cyclotomic group theory in particular.

Having made these general observations and clariﬁed with the standard FT, we now point out a key isomorphism between cyclic transforms and cyclic roots. We have discussed the fractional Fourier transform at some length in the eponymous sections above; we mentioned the interpretation of the transform of angular order α as a rotation through 145

α of the position-derivative operator vector (x id/dx)T . The application of a second transform of angular order β corresponds to an additional rotation of the operator vector through the angle β. The transforms of arbitrary order are commutative and associative, and they include an inverse of order −α for every order as well as the identity transform of order zero. Thus the group of arbitrary-order fractional FT’s is isomorphic to the continous group of rotational symmetries represented by the complex multiplier z = eiα, with continuous parameter α. The latter group of course can also be represented by   cos α − sin α   the rotation matrix   which makes the isomorphism transparent. The sin α cos α rotation multipliers can also be thought of simply as complex numbers, so the isomorphism is really three-fold, between fractional FT’s, continuous rotations, and unit-magnitude complex numbers. The group of SO(2) rotation matrices can be used to represent all three groups.

If we now fix the order at a particular value of α (which is to say, take α = β in successive rotations), the fractional FT is not necessarily cyclic; the choice α = 2πm/N for integer m, N renders the transform N-cyclic, with the resulting transform powers forming a discrete cyclic group with the properties that we have discussed. By the same token, the set of rotations will not constitute a finite-order group capturing the identity rotation in its orbit unless α = 2πm/N, with the discrete group of rotational symmetries through fixed angle α resulting. Similarly, for the complex number z = eiα, the set of powers {zk} will not necessarily be a cyclic group capturing unity; the choice α = 2πm/N renders the numbers N-cyclic, with the resulting powers forming the discrete cyclic group of N-th roots of unity. The powers zk and the successive rotations can be represented as powers of the rotation matrix with angle α. Thus the N-cyclic group of fractional FT’s 146 is isomorphic to the N-cyclic group of roots of unity and the group of N-fold discrete rotational symmetries, and the analogy between fractional Fourier transforms, rotations, and complex numbers is complete.

Likewise, we can describe an isomorphism between the Fresnel transform with param-   1 λd   eter matrix   and linear translation. A given value of the “order” d represents 0 1 multiplication of a reference vector by a factor of d in the positive direction, which can also be thought of as linear translation of the endpoint of the vector by d units. Successive values d1 and d2 correspond to successive multiplications or translations. The transforms of arbitrary order are commutative and associative, and they include an inverse of order

−d for every order as well as the identity transform of order d = 0. Thus the group of arbitrary-order Fresnel transforms is isomorphic to the continous group of translational symmetries represented by the real multiplier z = d, with continuous parameter d. Now we have a three-fold isomorphism between Fresnel transforms, continuous translations, and real numbers. 147

CHAPTER 23

Conclusion

In this work, we have developed and analyzed a new formalism for determining eigenfunctions of any cyclic integral transform. We have shown how the use of eigenoperators

– linear differential operators which commute with the transform – can accomplish this systematically. We have specialized these results and applied them to specific transforms of interest in the area of optical diffraction under the standard paraxial approximation.

In particular, we have gained new insight into the problem of self-similarly diﬀracting beams in one and two dimensions, showing the universal scalings that any such beams must exhibit.

We have also shown how linear canonical transform theory can be used to illuminate the connection between diffraction integrals and fractional transforms in 1-D and 2-D. To this end, we have shown the interrelation among 3 specific LCT’s in 1-D and their counterparts in 2-D, indicating how Fresnel-type diffraction integrals can be recast in terms of Fourier-type transforms. We used the new eigenoperator formalism to demonstrate that the Gauss-Hermite (1-D) and Gauss-Laguerre (2-D) functions are eigenfunctions of fractional orders of the Fourier and Hankel transforms, respectively. We also used the formalism to show that any eigenfunctions of the fractional Fourier and Hankel transforms correspond to self-similarly diffracting beams in 1-D and 2-D, respectively, with the same scalings exhibited by the Gaussian functions. We also connected our new approach with previously known results about the propagation of Gaussian beams in one and two 148 dimensions. Finally, we showed that for all the particular transforms of interest in this work, the EVP for the general second-order self-transform operator could be transformed into the defining EVP for known eigenfunctions, possibly with an additional chirp factor, in both 1-D and 2-D. This indicates that one must investigate higher-order eigenoperators in order to explore potentially new eigenfunctions of these transforms.

We submit that cyclic LCT’s aﬀord a novel and systematic way for approaching the eigenfunction problem related to diﬀractive wave propagation in one or two dimensions.

Of course, none of these results are intrinsically limited to optical wavelengths; they could be applied to the study of diﬀraction in other wavelength regimes as well, so long as the paraxial approximation and Fresnel theory remain valid.

One important suggestion for future work is the investigation of self-Fresnel-Hankel operators and their associated “eigenfunctions” – that is, functions whose initial 2-D pro-

ﬁles are reproduced at a certain diﬀractive propagation distance under the action of the

Fresnel-Hankel transform discussed in a previous section. The author intends to explore this avenue in more detail at the earliest opportunity. Other suggestions for further investigation include nonlinear extensions of the eigenoperator formalism, as well as the development of a systematic procedure to construct self-adjoint diﬀerential eigenoperators L of a cyclic transform T which is unitary but not self-adjoint. We saw especially how this latter point is fundamentally connected to the integrability, and hence physical realizability, of transform eigenfunctions.

One could also raise the question whether the eigenoperator approach offers any utility in working with non-cyclic transforms – to find their eigenfunctions, for example. If the transform T is not cyclic, then the generator series for an eigenobject (function or 149 operator) becomes infinite, rather than terminating after a finite number of terms. We could then define L via some generator operator A0 as

∞ X 1 L ≡ A + T A + T 2A + ... = T nA = A 0 0 0 0 I − T 0 n=0 and it is clear that the reciprocal transform to I − T becomes important to consider. Of course, this is a formal manipulation; a rigorous justification of this approach requires the theory of Banach spaces with its technical convergence requirements and so forth. Then the infinite series of transform powers converges to the final form above if the norm of the transform ||T || < 1. However, a complicating factor arises due to the fact that L is no longer a strict eigenoperator of T , since application of T to the infinite series above yields the same series without the first term, so that T L = L − A0:

∞ ∞ X n X n T L = T T A0 = T A0 = L − A0 n=0 n=1

So L is almost an eigenoperator, but not quite. Analogous comments and expressions can be given for the case of an almost-eigenfunction f in terms of a generator function g; then f satisﬁes T f = f − g. In any event, it is not immediately clear how best to modify the procedure to cope with a non-cyclic transform. This would be another area suggested for further investigation.

In a broader sense we could say that one result of the present body of work has been to illuminate a further aspect of the rich connection that has long been explored between boundary-value problems and integral equations. After all, what is the Green’s function of a BVP if not the kernel of an integral transform that can be used to solve the problem? Thus the posing of a seemingly innocuous question – why is the Gaussian 150 function an eigenfunction of the Fourier transform? – has led down paths of surprising richness and connection with classical results long known. This author is pleasantly surprised to discover that something new can still be said about this area of exploration. 151

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APPENDIX

Further Results on Fourier Eigenoperators

In this appendix we discuss further results of a more technical nature, such as singular points and asymptotic expansions of solutions, for the Fourier eigenoperators Lmn and the ODE’s Lmn = 0 associated with them. We focus primarily on the homogeneous forms

(λ = 0) for simplicity. We pay particular attention to equidimensional cases, and in this context we prove two theorems on polynomial roots, which are useful for stability investigations. We emphasize that our main search is for any decaying solutions of these equations. Although decay at inﬁnity in and of itself does not guarantee transformability, it is of interest to see just how much can be said a priori about the behavior of any solutions, given the form of the ODE’s they satisfy. Hence we make only partial claims and comments regarding transformability or square-integrability of solutions. However, we do show that for m < n there is always at least one solution which is integrable and potentially eigen-Fourier. 155

1. Singularities

If one gives a little thought to the form of the general ODE’s Lmnf = 0 and Lmnf = λf which have been shown useful for finding Fourier eigenfunctions, it becomes obvious that only for relatively few specific values of indices (m, n) can we expect to find closed-form expressions for the solutions f(x). We have seen some examples above of such expressions when m and n are integers of small absolute value, and for all integer m = n cases these ODE’s are equidimensional and thus readily solvable by standard methods; we discuss the equidimensional situation in further detail below. In the case of general real m 6= n however, we must resort to series methods if we want to say anything of substance about the solutions – for example, decay rates or asymptotic expansions. Moreover, in the physical applications we have in mind, it is helpful to know something about such issues as whether the solutions decay suitably quickly or whether they are integrable, as these properties are related to the transformability of solutions. So a consideration of such qualitative issues is not simply an empty exercise without merit. For example, a solution representing the transverse intensity distribution of a propagating lightwave must be square-integrable – corresponding to finite energy content – if it is to be physically realizable. In their turn, decay rates and asymptotic expansions are determined partly by the nature of any singular points of the ODE’s. Examining the solution behavior in the neighborhood of these singularities will give us some information about the leading-order terms in the relevant series expansions. So an analysis of the singular points of the ODE’s generated by the eigenoperators Lmn of the transform F seems like a natural place to begin this line of inquiry. 156

Consider the homogeneous singular BVP generated by L0n:

f (n) ± xnf = 0; f(±∞) = 0(.1)

where both ± signs at infinity are considered, regardless of the sign choice in Lmn. This equation has no singular points in the finite x-plane, since the leading coefficient is everywhere nonzero. This is a reflection of the standard theorem, discussed in Ince[17] and other references, that singular points of an ODE can only occur where the leading coefficient is zero, since dividing through by this leading coefficient produces terms which possibly grow without bound as the independent variable approaches this root. However,

1 there is an irregular or essential singularity at | x |= ∞. Under the substitution x → t , which is typically used to move a singular point from | x |= ∞ to t = 0 in an ODE, the

BVP becomes

d d d −t2 (−t2 (... (−t2 f) ...) ± t−nf(t) = 0; f(0±) = 0(.2) dt dt dt where the operator d/dx = −t2d/dt is applied n times in succession, and where the boundary condition requires that f approach 0 as t approaches 0 from either side; that is, f(t) is required to be continuous at 0. If we assume that all derivatives of f are bounded as t approaches 0, we may conclude from the diﬀerential equation that f(t) approaches 0. Hence f(x) approaches 0 as x approaches ±∞, and f(x) is potentially an FEF. However, this argument does not allow us to determine which choice of sign in the operator corresponds to FEF solutions. We shall see below that a normal-form assumption can be used to analyze the asymptotic behavior at inﬁnity of a solution to an

ODE with an irregular singularity at infinity (this mechanism is discussed in Ince[17]); 157 such an analysis sheds more light on the relationship between the choice of sign and the growth or decay of these solutions at infinity. However, before proceeding with lines of inquiry concerning asymptotics, let us first delve more deeply into the nature of the singularity at infinity.

In general, all coefficients in the first term of Eqn. (.2), upon expansion, will be simply monomials in t with decreasing positive degree. Hence the most singular term in the resulting ODE will originate from the term ±t−nf, the most singular term in the original ODE. The term proportional to f (n), the highest-order derivative, will have a coefficient of t2n, and so upon dividing through by this leading coefficient (which in this case happens to be the highest-degree monomial), the overall ODE will have a singularity of the form t−3n, the monomial now multiplying f. A well-known theorem of Fuchs

(discussed in standard references such as Ince[17] or Coddington & Levinson[3]) states that a necessary and suﬃcient condition for the origin to be a regular singular point of the ODE

(n) (n−1) 0 f + p1f + ... + pn−1f + pnf = 0(.3)

where the pk are functions of t, is that pk have a pole at the origin of order k at most;

−k that is, pk(t) = O(t ), k = 1, . . . , n. In the present case, this implies that the singularity at t = 0 (x = ∞) is irregular (3n > n for n > 0; the trivial case n = 0 is of no interest.)

For example, the operator L03 generates the ODE

−t6f¨˙ − 6t5f¨− 6t4f˙ ± t−3f = 0(.4) 158 which may be written

6 6 1 f¨˙ + f¨+ f˙ ∓ f = 0(.5) t t2 t9 indicating a 9th-degree irregular singularity at t = 0 (x = ∞).

If we now consider the more general operator Lmn, the BVP generated by this operator has the original form

m X m xmf (n) ± (xn)(k)f (m−k) = 0; f(±∞) = 0(.6) k k=0

1 in terms of x. This becomes, under the transformation x = t ,

d d d d d d t−m[−t2 (−t2 ... (−t2 f) ...] ± [−t2 (−t2 ... (−t2 (t−nf) ...] = 0; dt dt dt dt dt dt f(0±) = 0(.7)

2 d where the −t dt operator is applied n times in succession in the ﬁrst term, and m times in succession in the second term. We asssume for now that m ≤ n. The generic expansion of Eqn. (.7) will look like

n−1 m X −m+2n−q (n−q) X −n+2m−q (m−q) αqt f ± βqt f = 0(.8) q=0 q=0 where the αq and βq are proportionality constants whose exact values are immaterial. As

−m+2n in the L0n case, upon expansion of the first term, we find that the leading coefficient t

(the monomial multiplying f (n), the highest-order derivative) is the most positive-degree monomial in t. On the other hand, the most singular term in the resulting ODE, upon expansion of the second term and division by the leading coeﬃcient of the entire equation 159

−m+2n 2 d −n t , will originate from the monomial wherein all m operators −t dt act on t . This term – the coeﬃcient of f, the lowest-order derivative – will be proportional to t−n+m,

2 d the most negative-degree monomial in t (each action of −t dt raises the degree by one). Again the exponents on t decrease with decreasing derivative on f. (Note that in general, the coeﬃcients of intermediate-order derivatives may now be polynomials in t rather than monomials.) Thus the full ODE will ultimately have the generic form (ignoring constants of proportionality)

f (n) + ... + t−(3n−2m)f = 0(.9) and hence the overall degree of the singularity at t = 0 (x = ∞) will be 3n − 2m. Thus if m < n, we have an irregular singularity, since 3n − 2m > n. When m = 0, this conclusion reduces to the conclusion derived above for L0n. We can see further from Eqns. (.8,

.9) that if m = n, we have an equidimensional ODE with a regular singularity of order

3n − 2m = n.

−m+2n (n) As for the choice of sign in the operator Lmn, for m < n, the terms t f and t−n+mf will always be present, without possibility of their cancellation by other terms, so the choice of sign is immaterial to the foregoing analysis. For m = n, the + sign generates an equidimensional ODE with leading term 2tnf (n), while the − sign simply generates the corresponding equidimensional ODE of order n − 1 with leading term (−1)nn2tn−1f (n−1), so again the analysis is indiﬀerent to the choice of sign.

The case m > n requires slightly more care. Again we have the ODE

m X m xmf (n) ± (xn)(k)f (m−k) = 0(.10) k k=0 160 but now the sum will terminate when k = n < m since the (n + 1)-th derivative of xn is

0. So we have

m m xmf (n) ± [xnf (m) + nxn−1f (m−1) + n(n − 1)xn−2f (m−2) + ... 1 2 m (.11) + n!f (m−n)] = 0 n

Under change of variable, the term xmf (n) will transform (as before) into

n−1 X −m+2n−q (n−q) αqt f q=0

The leading term (highest derivative) of the full ODE becomes t−n+2mf (m), while the potentially most singular term in the full ODE is that involving f˙. That is to say, the term with the most negative-degree coefficient in t is proportional to t−m+n+1f˙. The exponents on t generally form a decreasing sequence from the f (m) term down to f˙. Division by the leading coefficient results in an ODE with singularity of degree 3m − 2n − 1. This singularity is necessarily irregular since the coefficient of f˙ must exhibit an O(t−(m−1)) singularity if that singularity is to be regular, but m > n ⇒ 3m − 2n − 1 > m − 1. As

−n+2m (m) for sign choice in the Lmn operator, the leading term will always be t f , and the term t−m+n+1f˙ always remains without cancellation in the ﬁnal ODE, so again sign choice is immaterial. Note that in converting the ODE from x to t, orders of derivative may be created which were not present in the original form in x. We also remark that some derivatives of a particular order in x may not occur in the converted ODE, due to internal cancellation of terms in the expansion and t-conversion of Eqn. (.10). 161

To clarify with examples, consider L12f = 0. Here m < n and the ODE simpliﬁes to

2 1 2 f¨+ f˙ ± (− f˙ + f) = 0(.12) t t3 t4 which exhibits an irregular singularity at inﬁnity of degree 3n − 2m = 4. The operator

+ L22f = 0 gives the equidimensional ODE

1 f¨+ f = 0(.13) t2

− with a regular singularity of degree 2, while L22f = 0 yields

1 f˙ − f = 0(.14) 2t which is equidimensional with a ﬁrst-degree regular singularity. Finally, the operator

L21f = 0 (now m > n) gives the ODE

1 f¨∓ f˙ = 0(.15) t3 which exhibits an irregular singularity of order 3m − 2n − 1 = 3. To take a more exotic example, L37 generates the ODE

−t11f (7) + ... ∓ (t−1f¨˙ + ... + 210t−4f) = 0(.16)

which exhibits an irregular singularity of order 3n − 2m = 15, while L73 yields

±(t11f (7) + ...) − t−1f¨˙ − 6t−2f¨− 6t−3f˙ = 0(.17) 162 which exhihibits an irregular singularity of order 3m − 2n − 1 = 14. A summary of the above results is given below in Table 1.

Certainly the ODE’s generated by Lmn for nonzero m will also have singular points at x = 0. In principle, this potential singularity might also aﬀect the integrability, and hence transformability, of the solution f(x). We now discuss the nature of this singular point.

In the case m < n, we can see from the form of Eqn. (.6) that the leading term of the

ODE is simply xmf (n), and the remaining terms will have coefficients which are monomials of x with decreasing consecutive positive degree. The greatest potential contribution to singularity will come from the lowest degree of these coefficients, which multiplies f and has degree n − m. Upon division by xm, this coefficient will have degree n − 2m, so if n − 2m ≥ 0, x = 0 will in fact be an ordinary point; the singularity at the origin is merely apparent, and all solutions f(x) will be non-singular there. On the other hand, if n − 2m < 0, then the singularity is real and of degree 2m − n, but since n > m, the coefficient xn−2m is clearly O(x−n), and all other coefficients will satisfy the sufficient conditions for regularity, so the singularity is regular. The choice of sign in the operator is immaterial in this case since no subsequent term of smaller derivative on f can cancel the leading term.

+ n (n) If m = n, then Lnn generates an equidimensional ODE with leading term 2x f , and whose subsequent terms are proportional to xn−kf (n−k), k = 1 . . . n (take m = n in Eqn. (.6) or (.10) above). Upon division by xn, these terms will clearly all satisfy the necessary and suﬃcient order conditions for regularity and the origin will thus be a

− regular singularity of degree n. The choice Lnn yields a similar result; the leading term is 163 now simply −n2xn−1f (n−1) rather than 2xnf (n), and all other terms remain, so again the conditions for a regular singularity (of degree n − 1 now) are satisﬁed.

In the case m > n, the leading term in the ODE will be ±xnf (m) (again see Eqn.

(.10) above); again the choice of sign is immaterial since this will always be the leading

m −n (m−n) n term. The term of lowest degree in x becomes n n!x f upon division by x . This and all preceding terms will clearly satisfy the conditions for regularity, regardless of the relative values of m and n (in contrast to the m < n case), since the term xmf (n) will become xm−nf (n) (which is non-singular) and all other terms will have coeﬃcients which are monomials of x with decreasing, consecutive nonpositive degree. Hence the origin will be a regular singularity of degree n. Also, we note that if n ≥ m − n, then two terms in f (n) arise in the expansion of Eqn. (.10) above: the beginning term xmf (n), and the

m n (m−n) (n) additional term ± m−n (x ) f .

For example, the L37 operator considered above generates the ODE

x3f (7) ± [x7f (3) + ... + 210x4f] = 0(.18)

wherein the origin is clearly an ordinary point. By contrast, L57 generates

x5f (7) ± [x7f (5) + ... + 12 · 210x2f] = 0(.19)

wherein the origin is a regular singularity of degree 2m − n = 3. The operator L73 generates

x7f (3) ± [x3f (7) + ... + 7 · 6 · 5f (4)] = 0(.20) 164 wherein the origin is a regular singularity of degree n = 3. Since n < m−n, no additional

(3) + term in f is generated. The operator L22f = 0 gives the equidimensional ODE

2x2f 00 + 4xf 0 + 2f = 0(.21)

− with a regular singularity of degree 2 at x = 0, while L22f = 0 yields

4xf 0 + 2f = 0(.22) which is equidimensional with a ﬁrst-degree regular singularity at x = 0. Note that

− + L22f = −2L11f; we comment further on this observation in the later section discussing equidimensional cases in more detail. Note also that the above examples illustrate the general lack of symmetry of the Lmn operator with respect to m and n.

To summarize the preceding analysis, the operator Lmn generates an ODE with an irregular singular point at x = ∞ and a regular singular point at x = 0 (the latter is an ordinary point if n ≥ 2m) unless m = n, in which case the ODE is equidimensional and both singular points are regular. These results are summarized in Table 1 below. In connection with these results we reiterate two corresponding theorems that were relevant in the preceding discussion. It is straightforward to show (see Coddington & Levinson[3], for example) that the point at inﬁnity is, at worst, a regular singular point of the ODE

Eqn. (.3) above iﬀ pk is analytic at x = ∞ and has a zero there of order at least k.

The theorem for x = 0 is essentially an equivalent statement and is quoted above as the 165 theorem of Fuchs.

Singular point n < 2m n ≥ 2m m = n

x = ∞ Irregular Irregular Regular

x = 0 Regular Ordinary (Apparent Singularity) Regular

TABLE 1 – Summary of Singular Points for Lmnf = 0

Since x = 0 is always an ordinary or regular singular point, standard Frobenius-type methods can be used to derive series expansions for the solutions to these ODE’s about x = 0. In the case of a regular singular point, for example, there will be at least one solution of the form f = xrV (x) where V (x) is a convergent power series and r may not be a whole number; other solutions may involve logarithmic factors and so forth, depending on the values of the exponent r. These expansions about x = 0 converge for all finite x, since the origin is the only finite singular point of these ODE’s. However, they offer little information about the behavior of the solutions at infinity, which in the general case is an irregular singular point. There may be some solutions which are regular

(i.e. products of algebraic and logarithmic terms) relative to the point at infinity, but typically there will be at least one solution with an essential singularity at infinity. Various necessary or sufficient conditions can be used to determine how many solutions may be regular at infinity (see Ince[17]). Similarly, other necessary or sufficient conditions can be used to demonstrate the existence and number of so-called normal solutions, which are not regular relative to the point at infinity. These normal solutions have the form

Q(x) r P −n f = e u(x), where Q is a polynomial and u has the form x cnx (a Frobenius-type modiﬁcation of a Laurent series). The essential singularity is therefore contained in the 166 so-called determining exponential factor. We would like to be able to analyze these normal solutions and determine the behavior of f(x) about the irregular singularity at inﬁnity, for we expect that a solution f(x) must decay properly if it is to be Fourier-transformable, let again eigen-Fourier. The general, complete analysis of the existence and nature of normal solutions is quite involved, so we will do our best to make some partial observations in the following pages. 167

2. Regular/Frobenius forms

We assume throughout this section that the indices (m, n) are nonnegative integers.

First we consider the singularity at the origin in more detail. What happens if we assume standard Frobenius-type forms for the solutions of Lmnf = 0, relative to the regular

r 2 singular point x = 0? That is, if we substitute f(x) = x (a0 + a1x + a2x + ...), we have

r−n+m r+1−n+m a0r(r − 1) ... (r − n + 1)x + a1(r + 1)r . . . (r − n + 2)x + ...

r+n−m ±[a0(r + n)(r + n − 1) ... (r + n − m + 1)x +

r+1+n−m a1(r + n + 1)(r + n) ... (r + n − m + 2)x + ...] = 0(.23)

The coeﬃcient of the dominant term (lowest degree in x), when equated to zero, provides the indicial, or characteristic, equation; its roots are the exponents relative to the singular point. For m < n the indicial equation is a0r(r − 1) ... (r − n + 1) = 0 ⇒ r = 0, . . . , n − 1, so the exponents relative to the singular point x = 0 are n distinct nonnegative integers.

(We point out that for a more compact notation of this and subsequent indicial equations to be discussed in this section, we could write the characteristic polynomials in terms

[n] Qn−1 of the so-called factorial polynomials r ≡ i=0 r − i; then the equation above reads

[n] a0r = 0. Whenever we see a polynomial of this form we may immediately conclude that its roots are the integers from 0 to n − 1 inclusive.)

Note that this result for the exponents is independent of the ± sign choice in Lmn, so the behavior of all n regular-form solutions as x → 0 is likewise independent of this sign choice. The solution corresponding to r = n − 1 (the greatest exponent) will be nonsingular, hence integrable, at the origin, but in general we can expect logarithmic 168 singularities to be present in the solution set because the exponents diﬀer by integers, unless all the conditions for an apparent singularity at x = 0 are satisﬁed (see Ince[17] for further details on this possibility). Evidently, the case n ≥ 2m is included in these conditions, because then the origin is merely an ordinary point of the ODE, and any singularities of a solution to an ODE must necessarily correspond to singular points of the ODE (although the converse is not true). Even solutions with logarithmic singularities will be integrable, since xk ln(x) is integrable at the origin for positive k. So in this case we expect all solutions to be integrable at the origin. Whether they will be integrable and/or

Fourier-transformable over the whole line (integrability implies transformability but not the converse) depends upon their behavior at the infinite ends of the line. However, the solutions f(x) could all be singular at infinity, which we know is an irregular singular point; only an analysis about the point at infinity can indicate the behavior of solutions there.

The fact that these ODE’s can be written in the so-called Hamburger form (discussed in the final section below) implies that these series solutions converge for all finite x, because the polynomial coefficients are entire.

For m = n, the equidimensional case, the indicial equation is a0[r(r−1) ... (r−n+1)±

(r+n)(r+n−1) ... (r+1)] = 0. In this case the series coeﬃcients after a0 will be zero, and all solutions will involve single powers of x, as we expect from an equidimensional equation.

There will be algebraic and/or logarithmic singularities at the origin unless the roots are all nonnegative and distinct, in which case the origin is an apparent singularity. (Again this can be connected to necessary vs. suﬃcient conditions for an apparent singularity.)

In any case the n solutions will be regular at worst. In general, some solutions (positive exponents) will be integrable at the origin but singular at the regular singular point 169 infinity, and others (negative exponents) will decay at infinity; only when all exponents are negative will all solutions decay at infinity, rendering infinity an apparent singularity.

Only solutions with negative exponents will be possibly integrable at inﬁnity, but of course the very monomials which are integrable at inﬁnity are non-integrable at the origin and vice-versa. But such monomials are at least transformable if their exponents lie between

0 and -1. The SFF f(x) = 1/p|x| = |x|−1/2 is an example of this. We note that in this equidimensional case, the choice of ± sign in Lnn enters into the characteristic polynomial, thereby demonstrating an inﬂuence of this sign choice on the exponents r and their associated solutions f(x).

For m > n, the indicial equation becomes ±a0(r +n)(r +n−1) ... (r +n−m+1) = 0.

The roots are the distinct integers r = −n, −(n − 1),..., −(n − m + 1). If n = m − 1, then the sequence will stop at zero and all m roots will be distinct nonpositive integers, while if n < m − 1, some roots will be positive integers. The solution corresponding to r = −(n − m + 1) will be nonsingular at the origin, and hence integrable there, but again we can expect logarithmic singularities to be present in the solution set, unless certain technical conditions are satisﬁed. Even in the absence of logarithmic terms, any solutions developed from negative exponents will be singular at x = 0, so the origin is undoubtedly a real singularity in this case. Only solutions with nonnegative exponents, possibly including logarithmic singularities, will be integrable at the origin. Again the point at inﬁnity must be analyzed separately to determine overall integrability or transformability of the solutions. Again note that these conclusions are independent of the ± sign choice in Lmn, so the behavior of all m regular-form solutions as x → 0 is again independent of this sign choice. 170

We can make observations of a similar nature about the point at inﬁnity, assuming a

r 2 Frobenius form f(t) = t (a0+a1t+a2t +...) for the solution near t = 0 (x = ∞). However,

Eqn. (.8) above shows that the expanded ODE in t is not as simple as the original ODE in x, so it will not be as easy to make definite conclusions about the exponents r. In the first place, we can see from the form of Eqn. (.8) that all terms in the first series are O(tr+n−m), while all terms in the second series are O(tr+m−n). Hence for m < n, the second series contains the dominant terms as t → 0, and the inidicial equation is the mth-degree polynomial β0r(r −1) ... (r −m+1)+β1r(r −1) ... (r −m+2)+...+βm = 0.

The m roots of this polynomial will be the exponents for the m possible regular solutions of the assumed form. We say possible regular solutions because in the case of an irregular singular point at the origin t = 0, regular/Frobenius expansions about the origin may not converge for any ﬁnite value of the argument, hence regular solutions may not exist. (Also possible is that the set of equations for the coeﬃcients may only have the trivial solution.)

Only when these expansions converge after a ﬁnite number of terms do they represent legitimate regular solutions. However, they may still represent legitimate asymptotic expansions of the solutions about the singular point, even if they are divergent. In any event, any exponents with positive real part will be singular, non-decaying, non-integrable, and non-transformable. Only exponents with negative real part represent non-singular decaying solutions; they may be transformable, but will only be integrable if the real part is between 0 and -1.

The best we can say at this point, then, is that for m < n, the number of distinct solutions regular at t = 0 is m at most. In general, when the degree of the characteristic polynomial is less than the order of the ODE, that degree provides an upper bound for 171 the number of possible regular solutions. A necessary and suﬃcient condition for the guaranteed existence of all m possible regular solutions may be formulated, involving the adjoint ODE. This is all in contrast to the equidimensional case, where a complete set of n regular solutions is guaranteed to exist because the indicial equation is an nth-degree polynomial and the origin is a regular singular point. A more complete discussion of these technical issues can be found in Ince[17].

As in the x = 0 case, the above result for t = 0 is independent of the ± sign choice in Lmn, so the behavior of any regular solutions as x → ∞ is likewise independent of this sign choice. It follows that any dependence on this sign choice can only be reﬂected in the behavior of the n − m possible irregular solutions. In our analysis of normal-form solutions in a later section we will see that this is indeed the case. In the extreme case with m = 0, there are no regular solutions, and all solutions must therefore be irregular in form. This is consistent with the fact that no indicial polynomial is generated at all, and hence no corresponding regular solution, when we substitute the Frobenius form into the t-version of L0nf = 0; all coeﬃcients ai are zero and we obtain the trivial solution.

Eqn. (.5) above provides an example of this situation.

Similar results hold for m > n. In this case, it is the ﬁrst series in Eqn. (.8) whose terms are dominant as t → 0, so the indicial equation is given by the nth-degree polynomial

α0r(r − 1) ... (r − n + 1) + α1r(r − 1) ... (r − n + 2) + ... + αn−1r = 0. In contrast with the preceding case, this polynomial contains no constant term because the summation in the

ﬁrst series ends with q = n − 1; that is, the lowest-order derivative on f is ﬁrst order, f˙.

The n roots of this polynomial will be the exponents for the n possible regular solutions.

(Again the degree of the characteristic polynomial is less than the order of the ODE, so 172 the degree n is merely an upper bound on the number of regular solutions that exist.) In particular, note that r = 0 is a root; thus there exists one regular solution of a standard power-series form as t → 0 (Laurent-series form as x → ∞), which approaches the constant value a0. Recall the continuity condition in Eqn. (.2) for a decaying solution. If we assume this solution is absolutely integrable, then since a0 is arbitrary, a nonzero value of a0 will not give a properly decaying solution. If a0 = 0, then we may apply the same

r+1 line of reasoning to the series expansion beginning with the a1t term to conclude that a1 must be zero as well to have a properly decaying solution. In this way we conclude that the trivial solution f ≡ 0 is the only possible decaying solution corresponding to r = 0. As with the m < n case, any exponents with positive real part will be singular, non-decaying, non-integrable, and non-transformable. Only exponents with negative real part represent non-singular decaying solutions; they may be transformable, but will only be integrable if the real part is between 0 and -1. Again note that, as with x = 0, these conclusions are independent of the ± sign choice in Lmn, so the m − n irregular solutions must reﬂect any inﬂuence of this sign choice on the limiting behavior of possible solutions.

Finally, in the equidimensional case m = n, all terms in both series are equally dominant, as they are all O(tr). The indicial equation in this case is the nth-degree polynomial

(α0±β0)r(r−1) ... (r−n+1)+(α1±β1)r(r−1) ... (r−n+2)+...+(αn−1±βn−1)r±βn = 0.

Because inﬁnity is now a regular singular point, we may expect a full set of n regular solutions, with exponents corresponding to the n roots of this polynomial. In fact, the

Frobenius expansions will terminate after the ﬁrst term, yielding the well-known monomial or monomial-logarithmic solutions of equidimensional ODE’s. It is straightforward to show that an ODE which is equidimensional in x remains equidimensional after the 173

1 transformation x → t , although of course the coefficients may change. This is consistent with the origin and infinity both being regular singular points for Lnnf(x) = 0. In addition, the lack of any other singular points of the ODE’s under investigation implies that regular solutions expanded about the regular singular point x = 0 converge for all finite x. These facts together imply that we need only consider the equidimensional case in terms of x to determine the form of the solutions and their limiting behavior as x → ∞.

Finally we note that in this case, as with x = 0, the choice of ± sign in Lnn enters into the characteristic polynomial, thereby demonstrating an inﬂuence of this sign choice on the exponents r and their associated solutions f(t).

The results of the above discussion are summarized in Table 2 below.

Singular point m < n m = n m > n

x = 0 n, all integrable n m, some integrable

x = ∞ ≤ m n ≤ n

TABLE 2 – Number of Regular Solutions for Lmnf = 0

In all cases, the continuity condition in Eqn. (.2) holds for a decaying solution. Thus only positive exponents r (which correspond to negative exponents in terms of x) could possibly provide a nontrivial decaying solution (assuming suitable extension to the negative real axis is made if r is nonintegral). The case r = 0 can only arise when m > n, because when m ≤ n, the constant βm in Eqn. (.8) is nonzero. We have seen above that this case provides the trivial solution.

When m < n, we may think of the terms in the first series in Eqn. (.8) as contributing regular singularities at the origin to the ODE. That is to say, upon division by the leading 174 coefficient, these terms will contain poles at t = 0 whose orders satisfy the conditions for a regular singular point. On the other hand, the terms in the second series contribute irregular singularities to the ODE; these terms will contain poles at t = 0 whose orders violate the conditions for a regular singular point. This is consistent with the fact that the second series contains the dominant terms in the Frobenius expansion as t → 0. The maximum number of regular solutions in the non-equidimensional cases m 6= n is related to the order of the derivative whose coefficient pole exhibits an irregular singularity at t = 0. For example, when m < n, the coefficient of f (m) in the second series will ultimately be a pole of O(t−(3n−3m)), whose order 3n − 3m exceeds the order n − m that is required for this term to contribute a regular singularity. Then it is straightforward to show (see

Ince[17]) that the singularity class ρ of the singular point t = 0 is given by ρ = n−m; that

(n−k) is, ρ is the lowest subscript k such that the coeﬃcient pk of f has a pole whose order exceeds k, the order required for regularity, by the maximum over all coeﬃcients. The class ρ tells us the minimum number of irregular solutions that must exist; furthermore, the degree of the indicial equation will be n − ρ.

For the ODE’s Lmn = 0, because the powers of t in the coefficients generally decrease consecutively, the singularities either do not exceed the required values at all (the first series), or they exceed it by the same amount (second series). Hence the first irregular singular term f (m) corresponds to the lowest such subscript k = n − m. Thus the class

ρ equals n − m. Then the ODE has a number of regular solutions which is at most n − ρ = m, the degree of the indicial polynomial. This is exactly the conclusion we reached above regarding the matter. When m > n, this situation and the corresponding argument are exactly reversed to yield ρ = m − n and maximum number of regular 175 solutions equal to m−ρ = n, the degree of the indicial polynomial. In the equidimensional case m = n, all terms in both series contribute regular singularities at the origin to the

ODE, the class ρ equals 0, the characteristic polynomial has maximum degree n, and all solutions are regular at the origin. We may summarize these conclusions by saying that for general (m, n), we expect at most min(m, n) solutions regular at t = 0 (x = ∞) to exist. Eqns. (.16, .17) above provide concrete examples to which these conclusions apply. In the extreme case m = 0, we ﬁnd ρ = n, there is no characteristic polynomial, and no solutions can be regular at t = 0. Eqn. (.5) oﬀers an example of this situation.

The conclusions in this paragraph were all reached above, but from a slightly diﬀerent standpoint. In general, the closer m is to n, the smaller ρ = n−m will be, the greater the degree m of the characteristic polynomial will be, and the greater the maximum possible number of regular solutions will be.

This last conclusion is a little counterintuitive, because in these ODE’s, the coefficients pk exhibit irregular singularity of the same order for m ≤ k ≤ n, due to the nature of the monomial coefficients in t. So the more terms we have displaying an irregular singularity in these ODE’s, the smaller ρ will be, and hence the greater the possible number of regular solutions will be. In the case of a general ODE, the lower the subscript k of maximal irregular singularity, the greater the possible number of regular solutions, regardless of the behavior of subsequent coefficients after pk. Thus a more irregular ODE, so to speak, can easily correspond to more possibility of regular solutions, not less! This is exactly the case with the ODE’s under discussion here, since more terms contribute irregular singularities as m approaches n, and the class ρ is lessened. In practice, as mentioned above, the actual number of existing, convergent regular solutions may be less than the maximum possible. 176

In the normal-form analysis of the ﬁnal section below, we will consider this issue of the singularity class from a diﬀerent perspective. 177

3. More on Equidimensional Cases

We now take a closer look at the homogeneous equidimensional ODE’s Lnnf = 0.

Such equations have characteristic polynomials and solutions that are well understood.

The expanded ODE for general n is

n n xnf (n) ± [xnf (n) + nxn−1f (n−1) + n(n − 1)xn−2f (n−2) + ... 1 2 n + n!f 0 + n!f] = 0(.24) 1

The ansatz f = xr leads to the characteristic polynomial

h(r) = 2r(r − 1) ... (r − n + 1) + n2r(r − 1) ... (r − n + 2)+

n2(n − 1)2 r(r − 1) ... (r − n + 3) + ... = 0(.25) 2

+ − for Lnn, and the same equation without the first term for Lnn. If we could affirm something about the sign of the roots of these polynomials, we could infer qualitative information about the growth or decay of f(x) at infinity. In particular, if we could affirm the existence of at least one root with negative real part, we could infer that there exists at least one solution f(x) that decays at infinity.

At this point the Routh-Hurwitz criterion for polynomials proves useful[18]. We will not cover the proof here, but the interested reader can ﬁnd it in Uspensky’s classic book Theory of Equations[39]. This criterion has several alternative equivalent for- mulations, one of which is the following. Let h(r) be the real nth-degree polynomial

Pn n−k 0 h(r) = k=0 akr , a0 > 0 by assumption. Deﬁne a new polynomial h (r) (NOT to be 178 confused with the derivative dh/dr !) of degree n − 1 to be

0 0 n−1 0 n−2 0 n−3 0 n−4 h (r) ≡ a0r + a1r + a2r + a3r + ...

n−1 a0 n−2 n−3 a0 n−4 n−5 (.26) ≡ a1r + (a2 − a3)r + a3r + (a4 − a5)r + a5r + ... = 0 a1 a1

Then the criterion states that all the roots of h(r) have negative real part iﬀ the same is true of h0(r). The construction may be applied repeatedly in succession to generate h00(r) from h0(r), h000(r) from h00(r), and so on. The criterion also states that the number of roots of the original polynomial h(r) with negative real part is exactly equal to the

(j) a0 number of negative multipliers − (j) encountered in successive stages of the construction. a1 (0) (0) Here j = 0, 1, . . . , n − 1 is the stage of the construction; a0 = a0 > 0, a1 = a1. (Extra

(j) care is required if one of the a1 vanishes.)

a0 Considering for a moment the ﬁrst multiplier − with j = 0, if a1 is positive, then a1 this multiplier will be negative. So at least one real root of the polynomial has negative real part. On the other hand, if a1 is negative, then this multiplier will be positive, so not all real roots of the polynomial can have negative real part. Therefore an immediate corollary of this criterion is the following

Theorem 1. If a1 is positive (negative), then h(r) has at least one root with negative

(positive) real part.

This concise result may not seem at all obvious a priori for a generic polynomial; however, it can be deduced from Vieta’s theorem, which states that the sum of the roots

a1 is equal to − . If a1 > 0, for example, then clearly at least one of the roots must have a0 179 negative real part, otherwise the sum of the roots would be positive. The reverse argument holds if a1 < 0.

(j) a0 In the present formulation, then, we need at least one of the multipliers − (j) derived a1 from our characteristic polynomial to be negative in order to infer that at least one root exists with negative real part, and hence that at least one solution of Lnnf = 0 will decay at inﬁnity. If we could compute all the successive multipliers and show that they are all negative, we could conclude that all solutions of Lnnf = 0 decay at inﬁnity.

The criterion can also be formulated to describe the number of polynomial roots with positive real part, as follows. The number of roots of h(r) with positive real part is equal to the number of sign changes (disregarding vanishing terms) in either one of the

T2 T3 Tn sequences T0,T1, , ,..., or T0,T1,T1T2,T2T3,...,Tn−2Tn−1, an, where the Tk are T1 T2 Tn−1 deﬁned as the determinants

a1 a0 0

a1 a0 T0 = a0 > 0,T1 = a1,T2 = ,T3 = a a a , 3 2 1 a3 a2

a5 a4 a3

a1 a0 0 0

a3 a2 a1 a0 (.27) T4 = ...

a5 a4 a3 a2

a7 a6 a5 a4

Hence all roots of h(r) have negative real parts iff all Tk are nonnegative. This in turn can be shown to be true iff all polynomial coefficients ai > 0 and either all T2k > 0 or all 180

T2k+1 > 0. In other words, if all ai > 0, then it may be true that all roots have negative real part; if not all ai > 0, then we have the following

Theorem 2. If at least one ai < 0, then it is certain that at least one root of h(r) has positive real part.

In other words, the a1 < 0 premise of the previous theorem need not be limited to the coefficient a1. (Again, this last result is easy to state but not at all obvious for a generic polynomial. It is more general than the preceding theorem, and the author has not been able to find it, or an immediate Vieta-like predecessor which implies it, anywhere in the literature.) So we need the number of sign changes in either of these sequences to be less than the degree of our characteristic polynomial in order to infer that at least one root exists with negative real part, and hence that at least one solution of Lnnf = 0 will decay at infinity. If we could show that all Tk are nonnegative for the characterisitic polynomial, we could conclude that all solutions of Lnnf = 0 decay at infinity.

Returning to the characteristic polynomials under discussion, we see that

(.28) h(r) ∼ 2rn − n(n − 1)rn−1 + n2rn−1 + ... = 2rn + nrn−1 + ... = 0 for positive choice of sign, and

n2(n − 1)(n − 2) n2(n − 1)2 h(r) ∼ −n2rn−1 + rn−2 − rn−2 + ... 2 2 n2(n − 1) = −n2rn−1 − rn−2 + ... = 0(.29) 2 for negative choice of sign, so the first theorem imples that for both choices of sign in the operator Lnn, there exists at least one root r with negative real part (In Eqn. (.29) we 181 may cancel the negative sign upon equating with zero, to yield positive signs for the first two coefficients). More extensive calculation is required to determine the exact number of

+ such roots. To take some concrete examples, L33f = 0 yields the characteristic equation

3 2 2r + 3r + 13r + 6 = 0. We have T0 = 2,T1 = 3,T2 = 27,T3 = a3T2 = 6T2, so all Tk > 0 and hence by the second formulation of the criterion, all roots have negative real part. If

− 2 we choose the minus sign, L33f = 0 yields the characteristic equation 3r + 3r + 2 = 0.

Now T0 = T1 = 3,T2 = a1a2 = 6 and again all roots have negative real part. Conﬁrming √ + this with the quadratic formula, we ﬁnd r = −1/2 ± i 15/6. The equation L22f = 0

2 yields the characteristic equation r +r+1 = 0. Now T0 = T1 = T2 = 1 and again all roots √ have negative real part; the quadratic formula gives r = −1/2 ± i 3/2. Therefore all members of the nullspaces of these three operators can be expected to decay at inﬁnity.

The latter two operators and their solutions were considered in the main body of this work.

In summary, when m = n in Lmn we can form homogeneous equidimensional differential equations with at least one decaying solution at infinity; further work is required to determine the transformability of such solutions. Such a solution will of course be singular at the origin and therefore integrable at either zero or infinity but not both. But if its exponent has real part between 0 and -1 it will at least be Fourier-transformable.

Clearly any solutions with positive exponents will be ﬁnite and integrable at the origin, but singular, non-decaying, and non-integrable at inﬁnity, and hence non-transformable

+ overall. The case m = n = 1 with positive sign choice yields L11f = 0, with the known SFF 1/p|x| as its solution. This function is transformable but non-integrable over the full line. This case is discussed in the main body of this work and serves to exemplify 182 these comments. Higher orders can yield solutions which were not previously known to be

FEF, if the associated characteristic polynomial has roots with negative real part between

-1 and 0, which is typically required for the conventional Fourier transform to exist.

We reiterate that to the best of our knowledge, the second of the two preceding theorems on polynomial roots does not appear elsewhere in the literature, so we claim it as a new result.

As a ﬁnal diversion we return to a minor observation from a previous section. Note that if the minus sign is taken in the deﬁnition of Lnn, an eigenoperator of order n − 1 will be reproduced. When n = 2, the two eigenoperators happen to be proportional, i.e.

− + L22 = −2L11 = −4xD − 2I, where D and I represent the diﬀerentiation and identity operators, respectively. Does this curious property hold for any larger n? The answer is negative. This is the only nontrivial case in which one eigenoperator is a constant multiple

− + of another eigenoperator one degree lower; for if Lnn = kLn−1,n−1, we see from Eqn. (.24) that −n! = k(n − 1)! ⇒ k = −n, but also −n2 = 2k, whence n = 2, k = −2 is the unique

− + solution. Only the trivial case L11 = −1/2 L00 = −I provides another example. Clearly there is no possibility of this occurring for unequal m and n, since the term xmDn will not be cancelled by a term from the expansion of Dmxn with negative sign choice on the

− latter. For higher values of n, it is certainly true that Lnn will be an equidimensional FEO of one lower order, whose form is easily found from Eqn. (.24). But it will not be expressible as a constant multiple of a positive-signed FEO of the next lower order.

− 2 2 + 2 2 For example, L33 = −9x D − 18xD − 6I and L22 = 2x D + 4xD + 2I, which are not proportional. 183

4. Irregular/Normal forms

Recall that as mentioned earlier, by assumption, a normal solution to an ODE with an irregular singular point has the form f = eQ(x)u(x), where Q(x) is a polynomial and

r P −n u(x) has the form u(x) = x cnx (a Frobenius-type modiﬁcation of a Laurent series).

Essentially we are modifying the regular Frobenius form about inﬁnity by multiplication with the exponential factor, sometimes called the determining factor since its growth or decay at inﬁnity will ultimately determine the growth or decay of the overall normal form.

The exploration of this or other irregular forms is necessitated by cases where the number of convergent regular Frobenius expansions is less than the order of the ODE, or when we have asymptotic rather than convergent regular expansions. In this section we quote a number of technical results without attempting to justify them; reference can be made to Ince[17] for full exposition of any details in this section.

In our normal-form analysis we will make use of two general forms for the homogeneous

ODE’s corresponding to Fourier eigenoperators. The ﬁrst is

(k) (k−1) 0 f + p1f + ... + pk−1f + pkf = 0(.30)

with leading coeﬃcient 1, where the coeﬃcients pv are assumed to be representable as

(possibly terminating) series of descending integral powers of x,

Kv −1 −2 pv(x) = x (av0 + av1x + av2x + ...)(.31) 184

This is similar in form to Eqn. (.3) above. The second useful form is the so-called

Hamburger form

k (k) k−1 (k−1) 0 x f + x p1f + ... + xpk−1f + pkf = 0(.32)

k with leading coeﬃcient x , where the coeﬃcients pv are entire functions of the complex variable x; in our case they will turn out to be polynomials. The general ODE Lmnf = 0 with m ≤ n can be written as

f (n) ± x−m(xnf)(m) = 0(.33) corresponding to the form in Eqn. (.30) upon expansion, and as

xnf (n) ± xn−m(xnf)(m) = 0(.34) corresponding to the form in Eqn. (.32) upon expansion. For m > n we have

±xm−nf (n) + x−n(xnf)(m) = 0(.35) corresponding to Eqn. (.30), and

±x2m−nf (n) + xm−n(xnf)(m) = 0(.36) corresponding to the form Eqn. (.32).

Let us ﬁrst consider Eqn. (.33) with m = 0, namely

(n) n L0nf ≡ f ± x f = 0(.37) 185

Generally speaking, if the point at inﬁnity is to be an irregular singular point of Eqn.

(.30), we must have Kv ≥ 1 − v for at least one index v. In the case of Eqn. (.37) we have

n p1, . . . , pn−1 ≡ 0; pn = x (±1 + 0 ...). Hence we can take Kn = n ≥ 1 − n, an0 = ±1, and all other Kv = 0. It can be shown that a necessary condition for the existence of a normal solution is that Kv ≥ 0 for at least one index v; here we have Kv ≥ 0 for all v, so the existence of such a solution is deﬁnitely possible. Furthermore, if a particular index

ρ between 1 and n inclusive may be found such that

Kv + v < Kρ + ρ, v < ρ

(.38) Kv + v ≤ Kρ + ρ, v > ρ then the degree of the characteristic equation relative to the point at infinity will be n − ρ, so there will be at most n − ρ solutions regular, and at least ρ solutions irregular, at infinity. The number ρ is called the class of the irregular singular point, just as in the case of the regular singular point at zero discussed in a previous section. In the present case, the only value of ρ satisfying the system of constraints is ρ = n; 1 ≤ ρ < n yields v < ρ when v < ρ and v ≤ ρ when v > ρ, while ρ = n yields v < 2n when v < n. This would imply that Eqn. (.37) has no solutions regular at infinity, and all solutions would therefore be of normal form or some other irregular form. This is exactly the conclusion we came to above in discussing regular Frobenius expansions about t = 0 for solutions of

L0nf = 0 at inﬁnity.

Now, in general, the degree s of the polynomial Q(x) (also called the grade of the

1 normal solution) must satisfy s ≤ g + 1, where g is the greatest value of v Kv. The number g + 1 is called the rank of the ODE relative to the singular point at inﬁnity; it 186 represents the maximum grade of the normal solution. Here g = n/n = 1, so s ≤ 2.

In fact, it is not diﬃcult to show that s = 2 in the present case. The details of this are somewhat technical, making use of something called a Puiseux diagram, and can be found in Ince[17], as the reader might expect. If we write Q(x) in the general form

Ps 1 v Q(x) = v=1 v Avx , then it also can be shown that As must be a root of the equation

ρ ρ−γ As + aγ0As + ... + aρ0 = 0(.39)

where γ is the least index v such that Kv = vg. Clearly γ = n in the present case, and since ρ = n, an0 = ±1, Eqn. (.39) simpliﬁes to

n A2 ± 1 = 0(.40)

If A2 is positive for some normal solution f(x) at a particular value of n, that solution will blow up at inﬁnity, while if A2 is negative, that solution will decay. Since the sign in Eqn. (.40) is determined ultimately by the choice of sign in L, we therefore can see a connection (ﬁnally!) between the choice of sign in the operator L and the growth or decay of solutions to Lf = 0. For if n = 1, then A2 = ∓1, and the normal form becomes

2 ± x simply the Gaussian function and its unbounded counterpart, e 2 . For n = 2, we have

− 2 L02 ⇒ A2 − 1 = 0 ⇒ A2 = ±1, where A2 = −1 yields the decaying Gauss-Hermite

+ 2 functions. On the other hand, L02 ⇒ A2 + 1 = 0 ⇒ A2 = ±i. This case corresponds to the cylinder function D2(x) discussed in the main body of this work; the asymptotics of

Bessel functions show decay of this solution at inﬁnity. Note that here the leading term in the exponent is purely imaginary, and in fact happens to be the FEF chirp function e±ix2/2 that we have encountered previously. In this particular case, the determining exponential 187 factor is merely bounded at inﬁnity, so it really isn’t “determining” of the behavior of the full normal form after all; the Frobenius term will really be the determining factor.

The problem for general n is simply the determination of the n-th roots of ∓1; some of these roots will have negative real part, corresponding to decaying normal solutions, and some will have positive real part, corresponding to growing normal solutions. For

+ n n odd in L0n, A2 = −1 will clearly be one solution of A2 + 1 = 0, giving an exact Gaussian term as the determining factor in one normal solution, with the possibility of other normal solutions with Gaussian-like terms with negative real part on A2. On the

− n other hand, in L0n, A2 − 1 = 0 will yield some Gaussian-like terms with negative real part on A2. For n even, the above statements are reversed with respect to the choice of sign. These results are summarized in Table 3 below. We thus find that in all cases there are likely to be one or more normal solutions to L0nf = 0 which decay like a Gaussian at infinity. Such solutions are therefore integrable at infinity. Now, we recall the previous analysis of the singularity at zero in the case m < n, when we concluded that all n regular solutions would be integrable at zero. In conjunction with this result here, we conclude that such solutions are integrable over the whole line and hence Fourier-transformable. In physical terms such solutions have finite energy content and represent potential Fourier eigenfunctions as well.

We have not analyzed the necessary/suﬃcient conditions in full technical detail for the existence of these normal solutions, but the above analysis provides a strong plausibility argument for their existence. The nth-roots of unity thus turn up once again as relevant to another aspect of Fourier eigenoperators. The connection between these concepts is clearly more than superﬁcial. 188

For general m > 0, it is straightforward to expand the ODE Lmnf = 0 in the forms of Eqns. (.30) and (.32); upon doing so we find that for m ≤ n both forms will have coefficients which are strict monomials of x, while for m > n both forms will have all monomial coefficients except for possibly one polynomial coefficient in x. As was pointed out in a previous section, this last possibility occurs if n ≥ m − n, so that two terms in f (n) arise in the expansion of Eqn. (.10) above: the beginning term xmf (n), and the

m n (m−n) (n) additional term ± m−n (x ) f . For example, (m, n) = (2, 3) with + sign choice for

Lmn yields the ODE

6 f 000 + xf 00 + 6f 0 + f = 0(.41) x in the form (.30) with regular singularity at 0 and irregular singularity at inﬁnity, while in the form (.32) it becomes

x3f 000 + x2(x2)f 00 + x(6x2)f 0 + 6x2f = 0(.42)

In both forms, all coeﬃcients are monomials in x. For (m, n) = (2, 2) we have

2 1 f 00 + f 0 + f = 0(.43) x x2 in form (.30) with regular singularities at both 0 and inﬁnity, and the equidimensional equation (as discussed above)

x2(2)f 00 + x(4)f 0 + 2f = 0(.44) 189 in form (.32); both forms again have all monomial coeﬃcients in x. Finally, for (m, n) =

(3, 2) we have

6 6 f 000 + (x + )f 00 + f 0 + 0f = 0(.45) x x2 in the form (.30) with regular singularity at 0 and irregular singularity at inﬁnity, while in the form (.32) we have

x3f 000 + x2(x2 + 6)f 00 + x(6)f 0 + 0f = 0(.46)

Now both forms have all monomial coeﬃcients, except for one polynomial coeﬃcient, since n ≥ m − n.

Consider now the speciﬁc case m = 1. The ODE L1nf = 0 becomes

f (n) ± (xn−1f 0 + nxn−2f) = 0(.47) in the form (.30). From previous discussion, we expect at most one solution regular at inﬁnity to exist. (Recall that for general (m, n), we expect at most min(m, n) solutions regular at inﬁnity to exist.) Now we have K1 = K2 = ... = Kn−2 = 0 and we can set

n−1 −1 −2 n −1 Kn−1 and Kn in diﬀerent ways. Since pn−1 = x (±1+0x +0x +...) = x (0±1x +

−2 0x +...), we can take Kn−1 = n−1, an−1,0 = ±1 or Kn−1 = n, an−1,0 = 0, an−1,1 = ±1.

n−2 −1 n−1 −1 n −1 −2 Similarly, pn = x (±n + 0x + ...) = x (0 ± nx + ...) = x (0 + 0x ± nx + ...), so we can take Kn = n − 2, an0 = ±n; or Kn = n − 1, an0 = 0, an1 = ±n; or

Kn = n, an0 = an1 = 0, an2 = ±n. Again note that Kv ≥ 0 ≥ 1 − v for at least one value of v. If we take Kn = n − 1 or n, we can again take ρ = n to satisfy Eqn. (.38) 190 above; it seems plausible to conjecture that ρ = n for general m ≥ 0 in our operators, implying that none of the solutions for general Lmn has a convergent regular form about inﬁnity. This would be consistent with the fact that these ODE’s can all be written in the Hamburger form (.32); one property of this form is that the point at inﬁnity is an essential/irregular singularity for every solution, as discussed by Ince.[17]

If we choose Kn−1 = n − 1 and Kn = n, the parameter g can be taken equal to 1 for all n, so the grade s ≤ 2. Again it can be shown in straightforward but technical fashion that s = 2 (again using a Puiseux diagram), and now γ = n − 1, so the leading coeﬃcient

A2 of the polynomial Q(x) satisﬁes

n A2 ± A2 = 0(.48)

and the same conclusions about the nature of A2 in Eqn. (.40) above hold here as well, with n replaced by n − 1 (we discount the trivial solution A2 = 0 since it contradicts s = 2). That is, we may simply apply the conclusions from the appropriate instance of

L0, n−1. Thus one or more normal solutions are likely to exist which decay like a Gaussian at inﬁnity for m = 1. Again we recall the previous analysis of the singularity at zero in the case m < n, when we concluded that all n regular solutions would be integrable at zero. Just as above with m = 0, we conclude that such solutions are therefore integrable over the whole line and hence Fourier-transformable. In physical terms such solutions have ﬁnite energy content and represent potential Fourier eigenfunctions as well. It seems plausible that similar conclusions will hold for m > 1 as well.

It is possible to formulate sets of conditions necessary and sufficient for the existence of one or more normal solutions to the general Hamburger equation (.32). The main 191 properties of this equation are the above-mentioned property regarding the singularity at infinity, and the property that the only other singular point is a regular singularity at x = 0. We should also note that in general, the existence of a normal-form solution depends on the convergence of the Frobenius-type factor u(x) corresponding to a particular determining factor eQ(x) in the expression. If this factor u(x) does not converge for finite x (that is to say, if the differential equation for u does not possess solutions which are regular at infinity), then the normal form does not exist. In this case some other irregular form must be explored for the solution. However, as noted above when analyzing regular forms about t = 0, divergent expansions for u(x) could still provide valid asymptotic expansions of the solutions about the singular point at infinity.

ODE n odd n even

+ L0nf = 0 A2 = −1 + other Re(A2) < 0 Some Re(A2) < 0

− L0nf = 0 Some Re(A2) < 0 A2 = −1 + other Re(A2) < 0

± ± ± L1nf = 0 Same as L0, n−1 Same as L0, n−1

TABLE 3 – Summary of Determining Factors for Lmnf = 0