Fast Algorithms for the Digital Computation of Linear Canonical Transforms

FAST ALGORITHMS FOR THE DIGITAL COMPUTATION OF LINEAR CANONICAL TRANSFORMS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Aykut Koc¸ March 2011

This dissertation is online at: http://purl.stanford.edu/fq782pt6225

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Lambertus Hesselink, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Shanhui Fan

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

R Pease

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii iv Abstract

LTHOUGH IT IS straightforward to determine the relationship between the in-focus A image and the object of a simple optical system such as a lens, it is far more challenging to compute the input/output relationships of general first-order and astigmatic optical systems. Such optical systems are known as quadratic-phase systems (QPS) and they include the Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic, astigmatic, nonorthogonal elements. Such computation is accomplished by representing the physical system with a general mathematical framework of integrations against kernels and then distilling the entire system into one input-output relationship that can be represented by a linear integral transform. The underlying mathematical integral transforms can be applied to a wider field of signal processing where they are known as the linear canonical transform (LCT) of a signal. Conventional numerical integration methods have a computational complexity of O(N 2) where N is the space-bandwidth product of the sampling scheme, e.g. the number of pixels in the field for an optical system. The algorithms described here yield a complexity of only O(N log N). The key is the use of different decompositions (or factorizations) of a given input/output relationship into simpler ones. Instead of following the general physical subparts in cascaded systems and computing input-output relations separately, these algorithms use the simplest possible decompositions to represent the entire system in terms of least possible number of steps. The algorithms are Fast Fourier Transform (FFT) based methods and the only essential

v deviation from exactness arises from approximating a continuous Fourier transform (FT) with the discrete Fourier transform (DFT). Thus the algorithms work with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy. Unlike conventional techniques these algorithms also track and control the space-bandwidth products, in order to achieve information that is theoretically sufﬁcient but not wastefully redundant.

vi to my late grandmother, Fatma

vii viii Acknowledgments

WAITED FOR writing the acknowledgments section until I got all three signatures and I hence made my Ph.D thesis de facto approved. I did not realize that this very small part of the thesis would end up being by far the hardest one to write, even after the relief of knowing that my Ph.D story has come to a happy ending. Five and a half years have passed at Stanford with their joys and sorrows. During these years, I worked on my Ph.D, a study which is in general deceptively seen as an individual accomplishment. However, after all these years, I believe that I did not do it alone but with the contributions from several people. Therefore, now I will do my best to deliver the rights to those who collectively made this thesis possible. I am proud of having them in my life more than I am proud of earning a Ph.D at Stanford. First of all, I want to send my thanks to my advisor Professor Lambertus ‘Bert’ Hes- selink. I am indebted most of my Stanford experience to him and I cannot express the full extent of my gratitude to him. I sincerely think that he is a truly ideal advisor in all aspects. Professionally, academically and personally, he has thought me lots. His guidance, experience, support and vast knowledge were always there for me. Lastly, a ﬁnal and special thank you goes to Bert for the hospitality he showed during the great weekend retreats at his estate in Tahoe that we enjoyed as a research group. Those were wonderful times that I will always remember. My oral defense and thesis reading committee members, Professor Shanhui Fan and Professor R. Fabian Pease helped me through the hardships of my orals exam and thesis

ix writing stages. I thank them for their feedback, insights and inputs that contributed to my thesis. I would also like to thank Professor Martin M. Fejer for agreeing to serve as the chair of my oral defense committee. I would like to thank Professor Haldun M. Ozaktas of Bilkent University for all his contributions to my academic development. I started research in an undergraduate senior year project under his supervision. With his passion for research and his vast knowledge, he was a role model for me and greatly affected my decision to pursue a Ph.D. I also need to thank Professor Peter Peumans for his support, especially during the early hard times. My Master’s Program Advisor, Professor Stephen Boyd should also be acknowledged for his help during my early years of studies. Dr. Yuzuru Takashima and Ludwig Galambos, senior researchers in our department, have also contributed to my development at Stanford. I would like to especially acknowledge Dr. Yuzuru Takashima for his support, guidance and help in the projects that I was involved in. I would like to thank for all the help, companionship and discussions provided by my past and present research group members: Paul, Yao-Te, Yin, Xiaobo, Brian, Yuxin, Eu- gene and Toan. Particularly, I cannot forget the friendships of Paul and Yao-Te. Thank you guys, for listening my absurd business ideas during lunches we enjoyed. Our research group’s administrative assistants Ms. Lilyan Sequeira and Ms. Ann Guerra deserve to be acknowledged for the administrative issues that have superbly been taken care of for our group. I also send my thanks and gratitude to Ms. Natasha Newson, the Student Services Ofﬁcer of Electrical Engineering Department, for her helpful attitude that have made lots of ofﬁcial issues easy. I couldn’t complete this long process without the support and enjoyment provided by great companions and friends. I had the chance to befriend great people during my years at Stanford. I have countless good moments and memoirs with them. I am grateful for their support and friendship. Although I cannot list every friend of mine, I still want to

x name at least some of them: Berk Atiko˘glu, Duygu Ozuysal,¨ Yusuf Ozuysal,¨ Ays¸e Türker Çınar, Murat Çınar, Bihter Padak, Erdem S¸as¸maz, Gürer Kıratlı, Türev Dara Acar, Hakan Baba, Emel Tas¸yürek, Ozge¨ Is¸legen,˙ Onur Kılıç, Uygar Sümbül, Ali Ozer¨ Ercan, Murat Aksoy, Emine UlküSarıtas¸,¨ Tolga Çukur, Ipek˙ Kasımo˘glu, Ismail˙ Kasımo˘glu, Vinay Ma- jjigi, Maryam Etezadi-Amoli, Michelle Hewlett, Thomas Sushil John, and Zuley Rivera- Alvidrez. I owe my deepest gratitude to my mother for raising me and being there always at all costs. I am indebted everything to her, who gave me the first key steps of my education. Her dedication to education was, in my opinion, key to my future educational success. I would not be here without her unconditional love and support. I am also grateful to my family, especially to my grandparents who helped raise me while I was a little kid and who gave me their love. Finally, I want to acknowledge my greatest support since the last three years, Ozlem.¨ Without her love, I would not finish, even dare to continue my studies. She is always happy, cheerful, optimistic and every such good thing that one can imagine. Most importantly, she has offered these to me generously and I have found the necessary encouragement even when I am feeling ‘down’. Thank you very much my love...

March 2011 Stanford, CA, USA.

xi Don't be proud of your knowledge, Consult the ignorant and the wise; The limits of art are not reached, No artist's skills are perfect. †

†the Vizier Ptahhotep, from the Papyrus Prisse, dating back the Middle Kingdom of Ancient Egypt

xii Contents

Abstract v

Acknowledgments ix

1 Introduction 1 1.1 MotivationandPreviousWork ...... 9 1.2 OrganizationoftheThesis ...... 15

2 Fundamentals 17 2.1 Preliminaries ...... 17 2.1.1 Quadratic-PhaseSystemsandMatrixOptics...... 17 2.1.2 LinearCanonicalTransforms-1D ...... 18 2.1.3 LinearCanonicalTransforms-2D ...... 19 2.1.4 LinearCanonicalTransforms-Complex...... 23 2.1.5 RelationofLCTstotheWignerdistribution ...... 24 2.2 SpecialLinearCanonicalTransforms...... 26 2.2.1 Scaling ...... 26 2.2.2 FourierTransformation...... 26 2.2.3 ChirpMultiplication ...... 27 2.2.4 ChirpConvolution ...... 28

xiii 2.2.5 FractionalFouriertransformation ...... 30

3 The Algorithm for 1D Quadratic-Phase Systems 33 3.1 Introduction...... 33 3.2 AnalysisofDecompositionsandAlgorithmI ...... 35 3.3 AlgorithmII...... 43 3.4 ResultsandVeriﬁcationoftheAlgorithms ...... 46 3.5 ConcludingRemarks ...... 52

4 The Algorithm for 2D Quadratic-Phase Systems 55 4.1 Introduction...... 55 4.2 Preliminaries ...... 57 4.2.1 A 3-Sphere for Space-Bandwidth Control, Wigner Distribution and DimensionalNormalizationfor2DFunctions ...... 57 4.3 TheAlgorithm ...... 60 4.4 Space-BandwidthandSamplingRateControl ...... 66 4.4.1 TheFirstCoordinateRotator ...... 67 4.4.2 2D Separable Fractional Fourier Transform ...... 69 4.4.3 TheSecondCoordinateRotator ...... 70 4.4.4 2DScalingOperation...... 70 4.4.5 2DChirpMultiplication ...... 73 4.4.6 SummaryoftheAlgorithm...... 74 4.5 NumericalResults...... 75 4.6 ConcludingRemarks ...... 85

5 The Algorithm for Complex Quadratic-Phase Systems 87 5.1 Introduction...... 87 5.2 Preliminaries ...... 91

xiv 5.2.1 WignerDistributions ...... 91 5.2.2 CLCTsinOpticsandSpecialCLCTs ...... 93 5.2.2.1 ComplexScaling(Magniﬁcation) ...... 93 5.2.2.2 Gaussian Apertures (Complex Chirp Multiplication)... 94 5.2.2.3 Gauss-WeierstrassTransform ...... 95 5.2.2.4 Complex-ordered Fractional Fourier Transform . . . .. 97 5.3 TheAlgorithm ...... 98 5.3.1 b =0 case: ...... 98 5.3.2 b =0 case: ...... 99 6 5.4 NumericalExamples ...... 105 5.5 ConcludingRemarks ...... 108

6 Application to the Beam Propagation Method 113 6.1 BasicsofBPM ...... 115 6.2 TheRelationbetweenBPMandABCD-Systems ...... 118 6.3 NumericalTests...... 121 6.4 ConcludingRemarks ...... 123

7 Conclusion 129 7.1 FutureWork...... 130

A Simpson’s Rule for 1D and 2D Functions 133

Bibliography 135

xv xvi List of Tables

3.1 Restrictionsandoversamplingfactors ...... 40 3.2 Computational Complexities. I(x, y) stands for the cost to interpolate x samples by a factor of y to obtain xy samples and S(z) stands for the cost of the scaling operation on z samples...... 41 3.3 Percentage errors for different functions F, transformsT,andalgorithmsA. 48

4.1 Percentage errors for different functions F and transformsT...... 77 4.2 Percentage errors for different interpolation methods and functions F for T1. 84 4.3 Percentage errors for different interpolation methods and functions F for T2. 85

5.1 Summary of the conditions to have bounded, R R CLCTs ...... 103 → 5.2 Percentage errors for different functions F and transformsT...... 107

xvii xviii List of Figures

1.1 Asimpleopticalsystem...... 1 1.2 Freespacepropagation ...... 2 1.3 Passagethroughthinlens ...... 3 1.4 PropagationthroughQGRIN ...... 4 1.5 Exampleinputandkernel...... 11 1.6 Anexamplegeneralsystem...... 14

2.1 EffectofscalingontheWignerdistribution...... 27 2.2 Effect of Fourier transformation on the Wigner distribution...... 28 2.3 Effect of chirp multiplication on the Wigner distribution...... 29 2.4 Effect of fractional Fourier transformation on the Wignerdistribution. . . . 31

3.1 Sequence of geometrical distortions for the decomposition in Eq. 3.1. The parallelogram in (c) is obtained by shearing the dashed rectangle in (b) in ordertocovertheworstcase...... 38 3.2 Sequence of geometrical distortions for the decompositioninEq.3.18. . . . 44 3.3 ExamplefunctionF4 ...... 48 3.4 Transforms (T1) of F1, F2, F3, F4. The results obtained with Methods I and II and the reference result have been plotted with dotted, dashed, and solid lines respectively. However, in most cases these lines are indistin- guishablesincetheresultsareveryclose...... 49

xix 3.5 Percentage errors versus N forselectedfunctionsandtransforms...... 50

4.1 ExamplefunctionF4 ...... 76 4.2 T1ofF1(ouralgorithmandreference) ...... 78 4.3 T1ofF2(ouralgorithmandreference) ...... 79 4.4 T2ofF3(ouralgorithmandreference) ...... 80 4.5 T1ofF4(ouralgorithmandreference) ...... 81 4.6 T2ofF4(ouralgorithmandreference) ...... 82

5.1 TheeffectoftheCCMoperationontheWD...... 96 5.2 ExamplefunctionF4 ...... 107 5.3 ExamplefunctionF5 ...... 108 5.4 Transform (T1) of F1, F2, F3, F4, F5. The results obtained with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indis- tinguishablesincetheresultsareveryclose...... 109 5.5 Transform (T3) of F1, F2, F3, F4, F5. The results obtained with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indis- tinguishablesincetheresultsareveryclose...... 110 5.6 CFRT with order 0.8 i0.2 of F1, F2, F3, F4, F5. The results obtained − with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indistinguishable since the results are very close...... 111

6.1 TestSystem1 ...... 122 6.2 AmplitudesforTestSystem1...... 123 6.3 PhasesforTestSystem1 ...... 124

xx 6.4 IndexdistributionforTestSystem2 ...... 125 6.5 AverageindexforTestSystem2 ...... 126 6.6 AmplitudesforTestSystem2...... 126 6.7 PhasesforTestSystem2 ...... 127

xxi xxii Chapter 1

Introduction

IMULATION AND COMPUTATIONAL study of optical and signal processing systems S are of prominent importance as our information age give way to bigger and bigger systems with enormous numbers of parameters to adjust and study. Therefore, computational methods and fast/efficient algorithms to digitize and digitally compute the optical systems are needed. In Fig. 1.1, a very simple optical system model is shown where we are interested in relating the samples of the input field to the samples of the output field.

Figure 1.1: A simple optical system

There are several well-known computational methods for modeling these input-output

1 2 CHAPTER 1. INTRODUCTION

relations such as Finite-Difference Time-Domain (FDTD), Rigorous Coupled-Wave Analy- sis (RCWA), Beam Propagation Method (BPM), etc. Each of these tools and methods have their advantages and disadvantages as well as certain situations and systems that they are suitable for. A broad class of optical systems including Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these can also be modeled and represented by Quadratic-Phase Systems (QPS). To get a better understanding about the QPSs, one can start studying very basic cases. First of all, consider the input-output relation of a free-space propagation as given in Fig. 1.2.

yi y

xi x

z U Ui o Figure 1.2: Free space propagation

When the incoming light as described by complex ﬁeld function Ui(xi,yi) propagates a distance z in free space under the paraxial approximation, the output complex ﬁeld function

Uo(x, y) is obtained. The relation between Ui and Uo is given by Eq. 1.1, which is also called the Fresnel transform. Eq. 1.1 is an integration against a kernel, where the kernel in this case represents the Fresnel free-space propagation.

ikz ∞ ∞ i e i k [(x−x )2+(y−y )2] U (x, y)= e 2z i i U (x ,y )dx dy (1.1) o −λ z i i i i i Z−∞ Z−∞

Secondly, consider a light beam passing through a thin lens as given in Fig. 1.3. The 3

z f

Ui Uo

Figure 1.3: Passage through thin lens

input-output relation of this simple system is given by

k 2 2 −i f (x +y ) Uo(x, y)= e 2 Ui(xi,yi). (1.2)

Eq. 1.2 is an equation of multiplication with a quadratic-phase and it can be seen as a special case of an integration against a kernel by using impulse functions, though there is not an explicit integration.

Lastly, consider a light beam propagates through a Quadratic Graded-Index media (QGRIN) as shown in Fig. 1.4. A QGRIN is a media whose transverse index proﬁle, n(x), is given by the quadratic equation n(x) = n2(1 (n /n )x2), where n and n are the 1 − 2 1 1 2 parameters of the media. As light propagatesp through this media, the light rays bend and

for a normalized distance d0 = n1/n2, the light is focused. For any distance d before focus, the input-output relation ofp the light ﬁelds are given by

∞ 2 2 iπ(cot θx −2 csc θxxi+cot θx ) Ud(x)= e i Ui(xi)dxi. (1.3) Z−∞ 4 CHAPTER 1. INTRODUCTION

Ui Uo Figure 1.4: Propagation through QGRIN

where θ = dπ/do2.

General optical systems that are any arbitrary combination of these three basic systems, namely propagation, thin lens effect and Quadratic-GRIN media propagation are known as Quadratic-Phase systems. A Quadratic-Phase system (QPS) is a unitary system, with parameter matrix M, whose output g(u) is related to its input f(u) through a quadratic- phase integral:

∞ g(u)= β e−iπ/4 exp iπ(αu2 2βuu′ + γu′2) f(u′) du′, (1.4) −∞ − p Z h i where α, β, γ are real parameters. The parameter matrix M (also called ABCD-matrix), which is deﬁned for the general case as

A B γ/β 1/β M = = , (1.5)  CD   β + αγ/β α/β  −     5

is another way of representing the QPSs in phase-space. M operates on the vectors representing the space and bandwidth pairs for the input ﬁelds. Then the output space/bandwidth pair can be simply found by a simple matrix multiplication. This is also analogous to the well-known ray-transfer matrices. For the aforementioned examples, the free-space propagation has the matrix representation

A B 1 λz M = = (1.6)  CD   0 1      and the passage through the thin lens has the matrix

A B 1 0 M = = (1.7)  CD   1/λf 1  −     where λ is the wavelength and f is the focal length.

When we study cascades of these system, we can simply use the matrix multiplication method to ﬁnd the overall input-output relation. To do this, consider a cascade of systems

S1,2,...,m, each represented by a matrix M1,2,...,m. If these systems are cascaded, the overall matrix is simply M = MmMm−1...M2M1. For example, consider a system consists of propagation of z1, passage through a lens of focal length f and again a propagation of z2. These three steps have the following matrices, respectively:

1 λz1 M1 = (1.8)  0 1    1 0 M2 = (1.9)  1/λf 1  −   6 CHAPTER 1. INTRODUCTION

1 λz2 M3 = (1.10)  0 1    When we cascade them, we get:

γ/β 1/β (f z2)/f λz1(f z2)/f + λz2 M = = M3M2M1 = − −  β + αγ/β α/β   1/λf z /f +1  − − − 1    (1.11) Eq. 1.11 can be solved to ﬁnd the QPS parameters α, β and γ so that we can ﬁnd the resulting integral equation that links the input to the output instead of manipulating cascades of integrals.

QPSs are identical to the Linear Canonical Transforms (LCTs). LCT is the name given to the same input-output relationships used in signal processing. LCTs supply a general framework and mathematical model to study QPSs and other application areas in signal processing. Since the literature that uses the notation of LCTs is larger than that of QPSs and LCTs allow us to study QPSs and other systems in a more streamlined way, we will use the name LCT from now on to refer to QPSs. Linear Canonical Transforms (LCTs), which are commonly referred to as quadratic-phase integrals or quadratic-phase systems in optics [1], have also been referred to by different names such as generalized Huygens integrals [2], generalized Fresnel transforms [3, 4], special afﬁne Fourier transforms [5, 6], extended fractional Fourier transforms [7], and Moshinsky-Quesne transforms [8], among other names. More importantly, the ABCD systems widely used in optics, [9], is also represented by linear canonical transforms or quadratic-phase systems.

There are four main classes of LCTs: one dimensional LCTs (1D-LCTs), two dimensional separable LCTs (2D-S-LCTs), two-dimensional non-separable LCTs (2D-NS-LCTs) and complex LCTs (CLCTs).

The class of 1D-LCTs [8,10] is a three-parameter class of linear integral transformations [1,11,12] which includes among its many special cases, the one-parameter subclasses 7

of fractional Fourier transforms (FRTs)1, scaling operations, and chirp multiplication (CM) and chirp convolution (CC) operations, the latter also known as Fresnel transforms.

The class of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs) is the class of linear integral transforms [1,11,12] that includes among its several special cases non-separable two-dimensional fractional Fourier transforms (2D-NS-FRTs) [13], two-dimensional versions of chirp multiplication (2D-CM) and chirp convolution (2D- CC) operations, the two-dimensional Fourier transform (2D-FT), and generalized astigmatic scaling (magnification) operations, as well as their separable special cases. The class of non-separable transforms is significantly more general than 2D separable linear canonical transforms (2D-S-LCTs) since it can represent a wide variety of anamorphic/astigmatic/nonorthogonal systems as well. The systems these integrals represent are also known as ABCD systems, which are also known as lossless first-order optical systems [8,14–20]. Classification of first-order optical systems and their representation through linear canonical transforms are studied in [21] and [17,22–24] for one-dimensional and two-dimensional cases, respectively.

Two-dimensional separable LCTs or symmetrical transforms that do not include the general non-separable case are addressed in [8, 10, 15, 25–28]. The most special case possible are the isotropic 2D-LCTs in which the system is fully symmetric, orthogonal and the parameters for both of the dimensions are identical. This case can be represented by only three parameters as in a 1D-LCT [22]. When the system is still orthogonal but the parameters for the orthogonal dimensions differ, the system becomes a 2D-S-LCT, which is represented by six parameters [22]. This case is also termed as axially symmetric [24]. The separable 2D transforms do not pose much difﬁculty because the separable transform is essentially two independent one-dimensional transforms along the two dimensions and the dimensions can be treated independently. However, the non-separable

1Not to confuse with the famous fast Fourier transform’s abbreviation FFT, the general convention is to use FRT or frFT for fractional Fourier transforms instead of using again FFT. 8 CHAPTER 1. INTRODUCTION

transform (2D-NS-LCT) is the most general case of this class of integrals where the two dimensions are coupled to each other by four additional cross-parameters, increasing the total number of parameters to ten. This general case is non-separable, non-axially symmetric, non-orthogonal, and anamorphic/astigmatic [2, 19, 22, 24, 29]. 2D-NS-LCTs are able to represent not only systems involving anamorphic/astigmatic components and reference surfaces, but other interesting systems such as optical mode convertors and resonators since they can represent the coupling between the dimensions [22,30–32]. Another prominent feature of 2D-NS-LCTs is their ability to represent systems with rotations between any arbitrary planes in phase-space, like rotations and gyrations [22,24]. These systems are collected under the general name of gyrators and are useful in two-dimensional image processing, signal processing, mode transformation, etc. [24, 33–36]. The efﬁcient and accurate digital computation of 2D-NS-LCTs is of importance in many areas of optics, optical signal processing and general digital image processing.

Finally, Bilateral Laplace transforms, Bargmann transforms, Gauss-Weierstrass transforms, [8, 37, 38], fractional Laplace transforms, [39, 40], and complex-ordered fractional Fourier transforms [41–44] are all special cases of complex linear canonical transforms (CLCTs).

An important special case of the CLCTs is the family of complex-ordered fractional Fourier transforms (CFRTs). The CFRT is the generalization of the fractional Fourier transform (FRT) where the order of the transformation is allowed to be a complex number, and consequently the abcd matrix elements are in general complex. The optical interpretation of the CFRT, its properties and optical realizations can be found in [41–45]. An interesting property of CFRTs is that, in some restricted cases, they can be optically realized by real LCTs [46].

To avoid confusion, we note that a number of publications have used the term complex Fractional Fourier transformation to refer to a particular generalization of the FRT [47,48], which is not a complex FRT in the sense of the order parameter being a complex number. 1.1. MOTIVATION AND PREVIOUS WORK 9

The entity referred to as a complex FRT in these publications is distinct from what we refer to as a complex FRT, and is actually a special case of real two-dimensional (2D) non-separable (NS) or non-symmetrical FRTs. Since such transforms are a special case of two-dimensional non-separable LCTs, their digital computation is covered by the algorithm proposed in [49]. To avoid confusion with this important but distinct entity, we will use the term complex-ordered to refer to complex FRTs belonging to the class of CLCTs. By developing a general algorithm for CLCTs, we also obtain an algorithm for the important special case of CFRTs. Linear canonical transforms (LCTs) appear widely in optics [2, 10, 11], electromagnetics, classical and quantum mechanics [8,50,51], as well as in computational and applied mathematics [52]. The application areas of LCTs include, among others, the study of scattering from periodic potentials [53–55], laser cavities [2,56,57], and multilayered structures in optics and electromagnetics [58]. They can also be used for fast and efﬁcient realization of ﬁltering in linear canonical transform domains [59].

1.1 Motivation and Previous Work

These integral transforms are of great importance in electromagnetic, acoustic, and other wave propagation problems since they represent the solution of the wave equation under a variety of circumstances. At optical frequencies, LCTs can model a broad class of optical systems including thin lenses, sections of free space in the Fresnel approximation, sections of quadratic graded-index media, and arbitrary concatenations of any number of these, sometimes referred to as ﬁrst-order optical systems or QPSs [1, 5, 6, 10, 12]. Therefore, given its ubiquitous nature and numerous applications, the discrezitation, sampling and the fast/efﬁcient digital computation of LCTs is of considerable interest. Their fast and accurate digital computation is of vital importance to utilize these tools in applications in a digital domain. Many works have addressed the problem of sampling of real and continuous 10 CHAPTER 1. INTRODUCTION

LCTs and some computation issues, using both decomposition-based and discrete-LCT- based methods [60–68]. The recent advances in the fractional Fourier transform and linear canonical transform areas can be found in the review [69]. A review on real one-dimensional (1D) and real symmetrical (separable) two-dimensional (2D) LCTs are given in [70]. When we want to ﬁnd the input-output relation of LCTs in a digital domain, we need to start with the general integral transform that takes the input function and transforms it to the output function by performing an integration against a kernel as the following:

∞ g(u) = β e−jπ/4 exp iπ(αu2 2βuu′ + γu′2) f(u′) du′, (1.12) −∞ − output Z input p h kernel i |{z} | {z } The conventional method to solve the| problem of digitally{z compute} this integral is the brute force numerical integration. First, one need to take the samples of the input and those of the kernel and then to ﬁnd each sample of the output, one need to multiply and then sum through the input and kernel samples. This kind of computation obviously takes O(N 2) time where N is the biggest of the number of samples necessary to properly sample the input and the kernel. This method cannot be practically applied to the large problems because of two reasons. Firstly, the algorithm has O(N 2) computational cost, which means that the computation time increases with the number of samples very rapidly. Secondly, the value of N should generally be taken extremely larger than the value necessary for representing the input function because of the highly oscillatory kernel forces much bigger values for N. For instance, we can consider a simple propagation of a Gaussian (represented by exp( πx2)) − in free-space with the following system parameters: wavelength λ = 500nm, propagation distance z = 100mm. We sample the input Gaussian with N = 256 samples and plot in Fig. 1.5. On the other hand, a sample value of the kernel is also plotted in Fig. 1.5 but proper sampling of the kernel can only be achieved by using N = 64 256 samples due × to its oscillatory nature. Therefore, if one uses this brute force method, the biggest of these 1.1. MOTIVATION AND PREVIOUS WORK 11

1 1

0.8 0.5 0.6 0 0.4

-0.5 0.2

0 -10 -5 0 5 10 -0.5 0 0.5 1 1.5 2 Input Kernel Figure 1.5: Example input and kernel

N values should be taken, which increases the computation cost dramatically.

In this thesis, we develop algorithms for digitally computing continuous LCTs with careful attention to sampling issues. There has been a certain amount of work on defining discrete/finite fractional Fourier transforms and, to a much lesser degree, discrete/finite linear canonical transforms [71–90]. While definitions of the discrete fractional Fourier transform (DFRT) may be considered satisfactory and well recognized [80, 88, 90], definition of the DLCT is far from being established. Further work on the definition and fast computation of discrete transforms, and their relationship to their continuous counterparts is desirable.

The Fourier transform is the most popular and prominent special case of LCTs and the algorithm for its fast computation, namely the fast Fourier transform (FFT) algorithm [91] is a breakthrough in science and applied mathematics because it allows the application of FT by using digital tools in O(N log N) time in parallel with the developments in computer technology that makes the computational power cheaper and easier to use. Later, Fourier transform has been generalized to fractional Fourier transform (FRT), which is also another important special case of LCTs. A fast algorithm for the digital computation of FRT is developed in [62]. Computation of the Fresnel diffraction integral, which is a 12 CHAPTER 1. INTRODUCTION

special case of LCTs has also received the greatest attention since it describes the propagation of light in free space (see Refs. [92] and [93] and the references therein). Since the input-output relationship represented by the Fresnel integral is time-invariant and takes the form of a convolution, it can be computed in O(N log N) time. The algorithms we present can compute the most general case of LCTs in O(N log N) time, despite the fact that the relationship represented by the more general LCTs is not time-invariant and is not a convolution.

A fast and accurate algorithm for numerical computation of two-dimensional non- separable linear canonical transforms (2D-NS-LCTs) is also developed in this thesis. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional (1D) case and the separable two-dimensional case. The algorithm takes O(N log N) time where N is the two-dimensional space-bandwidth product of the signal. the family of complex linear canonical transforms (CLCTs), which represent the e e e input-output relationship of complex quadratic-phase systems (CQPS). Allowing the linear canonical transform (LCT) parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts or complex graded-index media (CGRIN), as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm.

In other words, these fast algorithms opens the path of general usage and easy application of this general class of integral transform in a same way that fast Fourier transform (FFT) algorithm opens the way for FT/DFT to become widely-used tools.

It is also important to underline that here N is chosen close to the time-bandwidth product of the set of input signals, which is usually the smallest possible value of N that can 1.1. MOTIVATION AND PREVIOUS WORK 13

be chosen in terms of information-theoretic considerations. Therefore, the presented algorithms are highly efficient. Indeed, the distinguishing feature of the present approach is the care with which sampling and space-bandwidth product issues are handled. Straight- forward use of conventional numerical methods can result in inefficiencies either because their complexity is larger than O(N log N) and/or because the highly oscillatory quadratic- phase kernel in these systems forces N to be chosen much larger than the time-bandwidth product of the signals. In other words, the straightforward method of sampling the input field and the kernel, and then calculating the output field is not suitable for several reasons. First of all, due to the highly oscillatory nature of the integral kernel, a naive application of the Nyquist sampling theorem to determine the sampling rate would result in an excessively large number of samples and inefficient computation. On the other hand, ignoring the oscillations of the kernel and determining the sampling rate according to the input field alone may cause under-representation of the output field in the Nyquist-Shannon sense. This unacceptable situation arises due to the fact that the particular 2D-LCT that we are calculating may increase the space-bandwidth product in one or both of the dimensions. If we do not increase the number of samples that we are working with so as to compensate for this increase, there will be information loss and we will not be able to recover the true transformed output from our computed samples.

The methods developed in this thesis, in general, uses different decompositions (or factorizations) of the given LCT into other simpler LCTs with the purpose of fast and accurate calculation of the LCT integral. Different decompositions may be advantageous for LCTs with different parameters. The use of matrices will greatly facilitate our study of different decompositions, since dealing directly with the corresponding integral expressions is quite cumbersome. We will study all these decompositions in later Chapters of the thesis.

If we turn back to the example of our initial ﬁrst-order optics scheme, we can practically exemplify and summarize the general methods and algorithms developed in this thesis as 14 CHAPTER 1. INTRODUCTION

the following. Consider a general cascade of several propagations under Fresnel approximation, passage through thin lenses and propagation through QGRIN media as shown in

Fig. 1.6. To ﬁnd the relation between the input U1 to the overall output UK, if the con-

L QGRIN L L L 1 2 3 T y yi Input (Ȝ) x Output xi … z

U1 U2 U3 U4 U5 U6 U7 U8 U9 UK-2 UK-1 UK

Figure 1.6: An example general system ventional methods are used, one need to work through the input-output relations of every physical subpart of the system, do necessary calculations and operations to handle sampling issues and need to work through cumbersome integral equation within other integral equations. This way is both computationally inefﬁcient and information-theoretically in- correct if one cannot follow the space-bandwidth of the light ﬁelds along the way. Instead of doing this, one can:

1. Calculate Msystem by simple matrix multiplications

2. Do not follow the actual physical system steps but decompose Msystem into simplest possible decomposition

3. Solve for the parameters of the decomposition in terms of Msystem

4. Carry out the decomposition steps by paying attention to the space-bandwidth product issues. Start by the minimum amount of N required for the input ﬁeld and increase the sampling rate only when necessary for each step

5. Computation reduces to O(NlogN) time 1.2. ORGANIZATION OF THE THESIS 15

1.2 Organization of the Thesis

The thesis is organized as follows.

In Chapter 2 the fundamentals, mathematical preliminaries and background information for the general topic of LCTs are given. First, the deﬁnitions of the kinds of LCTs are given. The one-dimensional, two-dimensional non-separable and complex LCTs are discussed. Secondly, the ABCD-system in geometrical optics are brieﬂy explained. Thirdly, the Wigner distribution (WD), which is a method of space-frequency distribution to track down the signal’s space-frequency extents, is introduced and its relationship to the LCTs is given. Then the important special cases of the LCTs that arise in several application areas are summarized. Finally, the summary of some applications of LCTs are covered.

In Chapter 3, one-dimensional LCTs (1D-LCTs) and the fast computation algorithms for their digital computation are addressed. 1D-LCTs are important because they serve as the basic and most fundamental class of LCTs. If the systems that needed to be analyzed and computed digitally are one-dimensional or two-dimensional with the two-dimensions are symmetric to each other, then the 1D-LCTs are sufﬁcient to model and compute these systems. Additionally, there may be systems of interest in which the modeling can be made only in one-dimension, because the system may simply be one-dimensional or it may be a two-dimensional system with perfect symmetry so that it can be modeled by a one- dimensional system. Two algorithms are presented, compared and tested with respect to brute force reference calculations. The ﬁrst is based on decomposition of the LCT into chirp multiplication, Fourier transformation, and scaling operations. The second is based on decomposition of the LCT into a fractional Fourier transform followed by scaling and chirp multiplication. Then, we present the characterization and tests performed on the algorithms to verify their accuracy and performance. Finally, the concluding remarks are given.

In Chapter 4, two-dimensional LCTs (2D-LCTs) are addressed. An algorithm for their 16 CHAPTER 1. INTRODUCTION

fast and accurate digital computation is designed and presented. Some tools that are used in the development of this particular algorithm is also explained in this Chapter. Similarly, the numerical test are performed and presented to demonstrate the proper functioning of the algorithm. Chapter 5 covers the complex LCTs in detail. This class of LCTs is the mostgeneral and sophisticated of the entire family and their mathematical foundations are more involved. There are some restrictions on the transform parameters to ensure stability of the systems that are represented by these transforms. All these issues addressed and conditions are derived. Then a fast algorithm is designed for CLCTs. Numerical tests are presented and the Chapter conclusion is given. In Chapter 6, the relationship between the so-called Beam Propagation Method (BPM) and LCTs are studied. Usually, BPM is used to model system in inhomegenous media and LCTs are capable of representing light propagation in homogeneous media as well as some special inhomogeneous media. We investigated the possible ways to ﬁnd a link between these two tools in an effort to make the implementation of BPM faster for certain systems. First, the conditions and required properties of systems are found and presented to make such a link possible. Then, we present how a BPM can be implemented in a series of cascades of LCT systems under certain conditions. We present some test cases, characterize and draw the limitations of such a link. Finally, Chapter 7 gives the conclusions. Chapter 2

Fundamentals

N THIS CHAPTER, the preliminaries and the details of the fundamentals of LCTs will I be given. These include the main deﬁnitions and properties of different types of LCTs, important special cases of LCTs, and some other mathematical tools that will be used to develop the algorithms.

2.1 Preliminaries

2.1.1 Quadratic-Phase Systems and Matrix Optics

A quadratic-phase system (QPS) is a unitary system, with parameter matrix M, whose

output fM(u) is related to its input f(u) through a quadratic-phase integral:

∞ −jπ/4 2 ′ ′2 ′ ′ fM(u)= β e exp iπ(αu 2βuu + γu ) f(u ) du , (2.1) −∞ − p Z h i where α, β, γ are real parameters. This relationship is also known under other names including linear canonical transforms and ABCD-systems. [1,6,8,10,12,14]. The 2 2 matrix M, whose elements are A, B, C, D, represents the same information × 17 18 CHAPTER 2. FUNDAMENTALS

as the three parameters α, β, and γ which uniquely deﬁne the QPS:

−1 A B γ/β 1/β α/β 1/β M = = = − (2.2)  CD   β + αγ/β α/β   β αγ/β γ/β  − −       The unit-determinant matrix M is in the class of unimodular matrices. More on the group- theoretical structure of QPSs may be found in [8,10]. The result of repeated application (concatenation) of QPSs can be handled easily with the above-deﬁned matrix. When two or more QPSs are cascaded, the resulting system is again a QPS whose matrix is given by multiplying the matrix of each QPS in the cascade

structure. That is, if two QPSs with matrices M1 and M2 operate in a successive manner,

then the equivalent system is a QPS with the matrix M3 = M2M1. QPSs are not commutative. The matrix of the inverse of an QPS is simply another QPS whose matrix is the inverse of the matrix of the original QPS [8,10].

2.1.2 Linear Canonical Transforms - 1D

The one-dimensional linear canonical transform (1D-LCT) of f(u) with parameter matrix

M is denoted as fM(u)=( Mf)(u): C

∞ −iπ/4 2 ′ ′2 ′ ′ ( Mf)(u)= βe exp iπ(αu 2βuu + γu ) f(u ) du , (2.3) C − Z−∞ p ′ where α, β, and γ are real parameters independent of u and u and where M is the LCT C operator. The transform is unitary. The 2 2 matrix M whose elements are A, B, C, D, × represents the same information as the three parameters α, β, and γ which uniquely deﬁne the LCT:

−1 A B γ/β 1/β α/β 1/β M = = = − (2.4)  CD   β + αγ/β α/β   β αγ/β γ/β  − −       2.1. PRELIMINARIES 19

The unit-determinant matrix M belongs to the class of unimodular matrices. More on the group-theoretical structure of LCTs may be found in [8,10]. The result of repeated application (concatenation) of LCTs can be handled easily with the above-deﬁned matrix. When two or more LCTs are cascaded, the resulting transform is again an LCT whose matrix is given by multiplying the matrix of each LCT in the cascade

structure. That is, if two LCTs with matrices M1 and M2 operate in a successive manner, then the equivalent transform is an LCT with the matrix M3 = M2M1. LCTs are not commutative. The matrix of the inverse of an LCT is simply another LCT whose matrix is the inverse of the matrix of the original LCT [8,10].

2.1.3 Linear Canonical Transforms - 2D

Two-dimensional non-separable linear canonical transforms (2D-NS-LCTs) are the most general possible case of LCTs. In this section, the definition of 2D-NS-LCTs is given. An explicit-kernel definition with the least possible number of independent variables is provided and the forward and backward relations between the parameters of this definition and the parameters of conventional ABCD-matrices are derived. This derivation is important because it allows the usage of LCTs with more ease by explicitly exposing the parameters of the transform. It also makes the link between two main definitions of LCTs by giving the forwards- and backwards-relations between the two different, but equivalent, sets of transform parameters. The 2D-NS-LCT with parameter matrix M, of an input function f(u), can be denoted and defined as [94, 95]

∞ ∞ 1 ′T −1 ′ ′T −1 g(u)= fM(u)=( Mf)(u) = exp[iπ(u B Au 2u B u C √det iB −∞ −∞ − Z Z +uTDB−1u)]f(u′) du′ (2.5)

T ′ ′ ′ T where u = [ux uy] , u = [ux uy] with T denoting the transpose operation. A, B, C, D 20 CHAPTER 2. FUNDAMENTALS

are 2 2 submatrices deﬁning the transformation matrix M of the system that represents × the 2D-LCT, B being nonsingular. The matrix M, which is given as

A B M = , (2.6)  C D    is real and symplectic so that the following hold (I stands for the 2 2 identity matrix) × [23,95]:

ABT = BAT, CDT = DCT, ADT BCT = I, − ATC = CTA, BTD = DTB, ATD CTB = I. (2.7) −

From a group-theoretical point of view, 2D-NS-LCTs form the ten-parameter symplectic group Sp(4, R). (M has 16 parameters with six constraints leaving 10 independent parameters). More on group-theoretical properties of LCTs can be found in [8,10].

We will write the integral relationship between the input function f(ux,uy) and the

output function g(ux,uy) more explicitly as

∞ ∞ g(u ,u ) = e−iπ/2 β β η η K(u ,u ,u′ ,u′ )f(u′ ,u′ ) du′ du′ , x y x y − x y x y x y x y x y Z−∞ Z−∞ K(u ,u ,u′ ,u′ ) = exp[iπp(α u2 2β u u′ +2η u u′ + η u u + γ u′ 2 x y x y x x − x x x x x y α x y x x +α u2 2β u u′ +2η u′ u + η u′ u′ + γ u′ 2)] y y − y y y y x y γ x y y y (2.8)

where αx, βx, γx, αy, βy, γy, ηα, ηx, ηy, ηγ are the 10 independent parameters defining the 2D-NS-LCT (we will refer to them as the “scalar parameters”). These parameters also uniquely define the LCT. We will use this set of parameters for two reasons. First, although the definition using matrices gives us a compact and streamlined representation, the kernel and coefficients are not seen easily and explicitly in this case. When one needs to restrict the 2.1. PRELIMINARIES 21

parameters to obtain the kernel of any desired particular subclass of 2D-LCTs, it is not easy to derive the elements of the ABCD matrices directly, whereas this is straightforward with Eq. 2.8. Secondly and more importantly, when the ABCD submatrices are used directly, we need to manipulate 16 parameters (four 2 2 matrices with four elements each), despite × the fact that only 10 of them are independent. However, with the explicit deﬁnition we use the least number of required parameters, namely 10, and match the corresponding ten- parameter symplectic group with exactly these 10 parameters. In other words, we have given the simple relations between the two different representations of the same LCT. This is an issue of only convention in giving the deﬁnitions of the LCTs.

It is easy to convert from one set of parameters to the other. The 10 scalar parameters are given in terms of the elements of A, B, D as follows (only three of the submatrices are independent): D B D B α = 11 22 − 12 21 (2.9) x det B B β = 22 (2.10) x det B B η = 21 (2.11) x det B D B + D B D B D B η = 12 11 21 22 − 11 12 − 22 21 (2.12) α det B B A B A γ = 22 11 − 12 21 (2.13) x det B D B D B α = 22 11 − 21 12 (2.14) y det B B β = 11 (2.15) y det B B η = 12 (2.16) y det B A B + A B A B B A η = 21 11 12 22 − 11 21 − 12 22 (2.17) γ det B 22 CHAPTER 2. FUNDAMENTALS

B A A B γ = 11 22 − 12 21 (2.18) y det B

If we wish to obtain the submatrices A, B, D in terms of the scalar parameters, we can use the following reverse formulas:

1 ηyηγ +2βyγx ηγ βy +2ηyγy A = (2.19) 2(β β η η )   x y − x y ηγβx +2ηxγx ηxηγ +2βxγy  

1 βy ηy B = (2.20) β β η η   x y − x y ηx βx   1 ηxηα +2βyαx ηαβx +2ηyαx D = (2.21) 2(β β η η )   x y − x y ηαβy +2ηxαy ηyηα +2βxαy   As noted earlier, submatrix C is not independent and can be expressed in terms of A, B, D:

C = (A D B + A D B B B A D B A D )/ det B 11 11 11 22 12 12 22 − 22 − 12 21 11 − 12 22 12 C = (A D + A D B C )/B 21 12 22 11 21 − 12 22 11 C = (A D + A D B C )/B 12 21 11 22 12 − 21 11 22 C = (A D B + A D B B B A D B A D )/ det B 22 22 22 11 21 21 11 − 11 − 21 12 22 − 21 11 21 (2.22)

Eq. 2.22, along with the corresponding entries in Eqs. 2.19, 2.20, 2.21, deﬁnes C in terms of the scalar parameters. (Because the ﬁnal expressions for C are cumbersome we do not write them here explicitly.) Note that when we set the “cross” parameters ηα, ηx, ηy, ηγ to zero, the generalized 2D-NS transformation matrix M will reduce to the transformation matrix of the 2D separable case studied in [25]. Also note that A, B, C, D as given in Eqs. 2.19, 2.20, 2.21, 2.22 satisfy the required properties given in Eq. 2.7. 2.1. PRELIMINARIES 23

The utility of giving the relations between these two different but equivalent definitions of the LCTs is that one can obtain an arbitrary LCT given in one of the two definitions, Eq. 2.5 and Eq. 2.8. Since the 2D LCTs are cumbersome with lots of parameters, it is not easy to directly transfer one definition to the other. So the above bidirectional relations are useful to make the link between the two conventions.

2.1.4 Linear Canonical Transforms - Complex

The CLCT of f(u) with complex parameter matrix M is denoted as fM (u)=( M f)(u): C C C C

∞ ′ ′ ′ ( M f)(u)= K (u,u )f(u ) du , C C C Z−∞ K (u,u′)= e−iπ/4 β exp iπ(αu2 2βuu′ + γu′2) , (2.23) C − q ′ where α, β, γ are complex parameters independent of u and u and where M is the CLCT C C operator. MC again has unit-determinant and is given by

a b ar + iac br + ibc γ/β 1/β MC = = = (2.24)  c d   c + ic d + id   β + αγ/β α/β  r c r c −      

where ar, ac, br, bc, cr, cc, dr, dc are real numbers. The overline over the parameters α, β, γ is to emphasize that these parameters are now complex, corresponding to a total of 6 real

parameters: α = αr + iαc, β = βr + iβc, γ = γr + iγc. In terms of these parameters the kernel KC can be rewritten as

2 ′ ′2 2 ′ ′2 ′ −iπ/4 iπ(αru −2βruu +γru ) −π(αcu −2βcuu +γcu ) KC(u,u )= e βr + iβc e e . (2.25) p 24 CHAPTER 2. FUNDAMENTALS

2.1.5 Relation of LCTs to the Wigner distribution

Here we will review the relationship between LCTs and the Wigner distribution, which will aid us in understanding the effects of the elementary blocks used in our decompositions. The relationship between the ﬁrst-order optical systems (quadratic-phase systems or linear canonical transforms) and the Wigner distributions are studied in [1,11,12,96]. The Wigner

distribution Wf (u,µ) of a signal f(u) can be deﬁned as follows [97, 98]:

∞ ′ W (u,µ)= f(u + u′/2)f ∗(u u′/2)e−2πiµu du′. (2.26) f − Z−∞ Roughly speaking, W (u,µ) is a function which gives the distribution of signal energy over ∞ ∞ time and frequency. Its integral over time and frequency, −∞ −∞ W (u,µ) dudµ, gives the signal energy. R R

Let f denote a signal and fM be its LCT with parameter matrix M. Then, the Wigner

distribution (WD) of fM can be expressed in terms of the WD of f as [10]

W (u,µ)= W (Du Bµ, Cu + Aµ). (2.27) fM f − −

This means that the WD of the transformed signal is a linearly distorted version of the original distribution. The Jacobian of this coordinate transformation is equal to the determinant of the matrix M, which is unity. Therefore this coordinate transformation does not change the support area of the Wigner distribution. (A precise definition of the support area is not necessary for the purpose of this paper; it may be defined as the area of the region where the values of the Wigner distribution are non-negligible, or the area of a region containing a certain high percentage of the total energy.) The invariance of support area means that LCTs do not concentrate or deconcentrate energy, ie. they do not carry energy in or out of the defined support area, keeping the total energy with the support area constant. The support area of the Wigner distribution can also be approximately interpreted as the number 2.1. PRELIMINARIES 25

of degrees of freedom of the signal. Therefore, the number of samples needed to represent the signal does not change after an LCT operation.

It is well known that a non-zero function and its FT cannot both be confined to finite intervals. However, in practice we always work with samples of finite duration signals. We assume that a large percentage of the signal energy, as represented by the WD, is confined to an ellipse with diameters ∆T in the time dimension and ∆B in the frequency dimension, which can be ensured by choosing ∆T and ∆B suitably. This implies that the time-domain representation is approximately confined to the interval [ ∆T/2, ∆T/2] and − that the frequency-domain representation is approximately confined to [ ∆B/2, ∆B/2]. − We then define the time-bandwidth product ∆T ∆B, which is always 1, because of ≥ the uncertainty relation. Let us now introduce the scaling parameter s and scaled coordinates, such that the time- and frequency-domain representations are confined to intervals of length ∆T/s and ∆Bs. Let s = ∆T/∆B so that the lengths of both intervals become equal to the dimensionless quantityp √∆B∆T which we denote by ∆u, and the ellipse becomes a circle with diameter ∆u. In the new coordinates, signals can be represented in both domains with ∆u2 samples spaced ∆u−1 apart. We will assume that this dimensional normalization has been performed and that the coordinates u and µ are dimensionless. The space-bandwidth product concept and its importance for the optical signals and systems are given in more detail in [99]. Those interested in its further generalization to bicanonical width products may see [100].

For a signal with rectangular space-frequency support, the space-bandwidth product is equal to the number of degrees of freedom. This is not true for signals with other support shapes. While we haveobserved that LCTs do not change the number of degrees of freedom of a signal, they may change its time-bandwidth product. This will be illustrated in what follows. 26 CHAPTER 2. FUNDAMENTALS

2.2 Special Linear Canonical Transforms

Here we discuss the effects of certain operations, all special cases of LCTs, on the Wigner distribution of a signal. These are of interest since we will decompose general LCTs in terms of these operations.

2.2.1 Scaling

The scaling operation is a special case of the LCT deﬁned as:

M f(u)= f(u)= 1/M f(u/M), (2.28) C M MM p −1 M 0 1/M 0 MM = = . (2.29)  0 1/M   0 M      Its effect on the WD is given by

WMM f (u,µ)= Wf (u/M,Mµ), (2.30) where M > 0. The scaling operation does not change the support area, time-bandwidth product, or required number of samples (Fig. 2.1), but it changes the sampling intervals in both the time and frequency domains by factors of M and 1/M respectively.

2.2.2 Fourier Transformation

The ordinary Fourier transform operation is also a special case of the LCT:

∞ −iπ/4 ′ −i2πuu′ ′ F f(u)= f(u)= e f(u )e du , (2.31) C lc Flc Z−∞ 2.2. SPECIAL LINEAR CANONICAL TRANSFORMS 27

µ µ

d2 d2/M

u u

Md1 d1 (a) Before scaling operation (b) After scaling operation (parameter M)

Figure 2.1: Effect of scaling on the Wigner distribution.

−1 0 1 0 1 Flc = = − . (2.32)  1 0   1 0  −     The subscript “lc” reminds us that the deﬁnition of the Fourier transform as a special case of LCTs differs from the conventional deﬁnition by the factor e−iπ/4. Readers wishing to understand the technical reason behind this inconsequential discrepancy may consult [8, 10]. The effect of Fourier transformation on the WD is given by

W (u,µ)= W ( µ,u), (2.33) Flcf f −

which is a rotation of π/2 in the clockwise direction (Fig. 2.2), which again does not change the time-bandwidth product.

2.2.3 Chirp Multiplication

The chirp multiplication (CM) operation is another special case of the LCT:

−iπqu2 Q f(u)= f(u)= e f(u), (2.34) C q Qq 28 CHAPTER 2. FUNDAMENTALS

µ µ

d2 d1

u u

d2 d1 (a) Before Fourier transformation (b) After Fourier transformation

Figure 2.2: Effect of Fourier transformation on the Wigner distribution.

−1 1 0 1 0 Qq = = . (2.35)  q 1   q 1  −     Its effect on the WD is

WQqf (u,µ)= Wf (u,µ + qu). (2.36)

Although the support area and therefore the number of degrees of freedom are pre- served after chirp multiplication, the time-bandwidth product increases. This is due to the increase in signal bandwidth as a result of vertical shearing of the WD (Fig. 2.3). The new time-bandwidth product is d (d + q d ). If we wish a signal with such a support to be 1 2 | | 1 recoverable from its samples in the conventional manner, this is the number of samples we need. (The sampling interval in the time domain is 1/(d + q d ) and that in the frequency 2 | | 1 domain is 1/d1.)

2.2.4 Chirp Convolution

The chirp convolution (CC) operation is indeed the Fresnel propagation integral and it is the dual of the chirp multiplication and corresponds to a horizontal, rather than a vertical 2.2. SPECIAL LINEAR CANONICAL TRANSFORMS 29

µ µ

d + q d 2 | | 1 d2

u u

d1 (a) Before chirp multiplication (b) After chirp multiplication

Figure 2.3: Effect of chirp multiplication on the Wigner distribution.

shear in the time-frequency plane:

−iπ/4 2 R f(u)= f(u)= f(u) e 1/r exp(iπu /r), (2.37) C r Rr ∗ p −1 1 r 1 r Rr = = − . (2.38)  0 1   0 1      Its effect on the WD is W (u,µ)= W (u rµ,µ). (2.39) Rrf f − The Fresnel propagation integral for a distance z along the propagation direction and for a wavelength of λ is denoted by F(z,λ) and is given by

iπz/λ ∞ ′ e i π (x′−x)2 F(z,λ)(x )= e λz f(x) dx (2.40) √iλz −∞ Z Its ABCD matrix (the propagation matrix) is given by

1 zλ 2π . (2.41)  0 1    30 CHAPTER 2. FUNDAMENTALS

Therefore, Fresnel propagation is exactly a chirp convolution with parameter r = zλ/2π. In terms of α-β-γ parameters of LCTs, the Fresnel propagation special case can be obtained by setting α = β = γ =2π/zλ.

2.2.5 Fractional Fourier transformation

The ath order fractional Fourier transform (FRT) [10,101–107] of a function f(u), denoted

fa(u), is most commonly deﬁned as

∞ af(u)= f (u)= K (u,u′)f(u′) du′, (2.42) F a a Z−∞ K (u,u′)= A exp iπ(cot θu2 2 csc θ uu′ + cot θu′2) , a θ − aπ A = √1 i cot θ, θ = θ − 2

when a = 2j and K (u,u′) = δ(u u′) when a = 4j and K (u,u′) = δ(u + u′) when 6 a − a a =4j 2, where j is an integer. The square root is deﬁned such that the argument of the ± result lies in the interval ( π/2,π/2]. For 0 < a < 2 (0 < θ < π), A can be rewritten − | | | | θ without ambiguity as e−i[πsgn(θ)/4−θ/2] Aθ = , (2.43) sin θ | | where sgn( ) is the sign function. When apis outside the interval 0 a 2, we need · ≤ | | ≤ simply replace a by its modulo 4 equivalent lying in this interval and use this value in Eq. (2.43).

The FRT is also a special case of the LCT with matrix

−1

a cos(aπ/2) sin(aπ/2) cos(aπ/2) sin(aπ/2) Flc = = − , (2.44)  sin(aπ/2) cos(aπ/2)   sin(aπ/2) cos(aπ/2)  −     2.2. SPECIAL LINEAR CANONICAL TRANSFORMS 31

differing only by the factor e−iaπ/4:

a −iaπ/4 a Fa f(u)= f(u)= e f(u) (2.45) C lc Flc F

Again the subscript “lc” denotes this inconsequential discrepancy between the deﬁnition of the FRT given by Eq. 2.42 and the FRT deﬁned as a special case of the LCT [8,10]. The FRT rotates the WD in the clockwise direction with an angle of θ = aπ/2, as shown in Fig. 2.4:

WF a f = Wf [cos(aπ/2)u sin(aπ/2)µ, sin(aπ/2)u + cos(aπ/2)µ] . (2.46) lc −

µ µ d2

u u

d1 (a) Before FRT operation (b) After FRT operation

Figure 2.4: Effect of fractional Fourier transformation on the Wigner distribution.

QPSs/LCTs are capable of exactly representing quadratic-graded index (GRIN) media, [108], in which the inhomogeneous refractive index distribution is given by n2(x)= n2[1 1 − 2 (n2/n1)x ] where n1 and n2 are the medium parameters and x is the transverse coordinate.

The parameters n1 and n2 are assumed to be constants along the propagation direction. FRTs are directly giving the propagation through these GRIN media. When the propagation distance through a GRIN media with parameters n1 and n2 is d, and if the order of FRT, a, 32 CHAPTER 2. FUNDAMENTALS

is set to a = d/d , where d = π n1 , then the result of FRT gives the output ﬁeld after 0 0 2 n2 GRIN propagation. q Chapter 3

The Algorithm for 1D Quadratic-Phase Systems

3.1 Introduction1

N THIS CHAPTER we discuss two approaches for the digital computation of LCTs. I The first algorithm decomposes an LCT with arbitrary transform parameters into some combinations of three simpler operations: scaling, Fourier transformation, and chirp multiplication. The second method decomposes the LCT into fractional Fourier transformation, chirp multiplication and scaling. Both are fast algorithms which take O(N log N) time, where N is the time-bandwidth product of the input signal. Despite the highly oscillatory nature of the integral kernel, special care is taken to carefully manage the sampling rate so as to ensure that the number of samples N is not chosen much larger than the time(space)- bandwidth product of the input signal, so that the algorithms are as efficient as possible. A naive application of the Nyquist sampling theorem to determine the sampling rate, on the other hand, would result in an excessively large value of N and inefficient computation. The only deviation from exactness arises from the approximation of a continuous Fourier

33 34 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

transform with the discrete Fourier transform (DFT). Thus the algorithms compute LCT integrals with a performance similar to that of the fast Fourier transform (FFT) algorithm in digitally computing the continuous Fourier transform (FT), both in terms of speed and accuracy.

The proposed algorithms use matrix factorizations to decompose LCTs into cascade combinations of the elementary LCT blocks discussed above. Since each stage can be computed in O(N log N) time, the overall LCT can also be. Numerous such decompositions are possible [10,29], but they are not equally suited for numerical purposes. For instance, direct naive application of the decomposition of chirp multiplication, Fourier transformation, scaling (magnification), and again chirp multiplication, which suggests itself upon inspec- tion of Eq. (2.3) will in general lead to very high sampling rates. This is because the early appearance of the chirp multiplication in the cascade and the lack of sampling rate management that controls the first chirp multiplication. If one directly uses this decomposition for every parameter set, there may be two large increases in the number of samples because of the two chirp multiplications. These combined increases in the number of samples results in unnecessary high sampling rates. We have carried out a systematic exhaustive analysis of all possible decompositions of arbitrary LCTs into the three basic operations of scaling, chirp multiplication (CM), and Fourier transformation (FT). We have considered all possible decompositions with three, four, and five cascade blocks. Every permutation has been checked to see if that decomposition is capable of expressing an LCT with arbitrary parameters. We have also considered the required sampling rates for each decomposition and pick the ones that require the least possible number of samples.

This chapter will start by studying the different decompositions (or factorizations) of the given LCT into other LCTs with the purpose of fast and accurate calculation of the LCT integral. Different decompositions may be advantageous for LCTs with different parameters. The use of matrices will greatly facilitate our study of different decompositions, since dealing directly with the corresponding integral expressions is quite cumbersome. 3.2. ANALYSIS OF DECOMPOSITIONS AND ALGORITHM I 35

The chapter is organized as follows: Section 3.2 presents a systematic analysis of all possible decompositions of an LCT into the three basic operations of scaling, Fourier transformation and chirp multiplication. Based on this, the ﬁrst algorithm is also presented in Section 3.2. The second approach involving the fractional Fourier transform is given in Section 3.3. Numerical results, veriﬁcation of the performance of the developed algorithms and the comparison of the two approaches are presented in Section 3.4. In Section 3.5, the conclusion is given. In this section, by deriving the 1D algorithms, we also show the main essence of the algorithms developed by demonstrating the general approach taken in the fundamental 1D case. Then, the same main approach of divide and conquer will be applied to more general cases of two-dimensional LCTs and complex LCTs in later chapters of the thesis.

3.2 Analysis of Decompositions and Algorithm I

It is well known that arbitrary LCTs can be decomposed in either the form or Qq1 RrQq2 , where and are the chirp multiplication and convolution operations re- Rr1 QqRr2 Qq Rr spectively [10]. Chirp convolutions can be realized as a Fourier transform followed by a chirp multiplication followed by an inverse Fourier transform (the Fourier transform of a chirp is also a chirp). Since we already consider all permutations involving chirp multiplication and Fourier transformation, approaches involving chirp convolution are also included in our development. While generating all possible decompositions through permutations, duplicate decompositions arise. For example, two consecutive scaling or two consecutive chirp multiplication operations can both be collapsed into one. Or, for instance, in the case of ﬁve-stage decompositions with more free parameters than the three free parameters of LCTs, we have the freedom of choosing the additional parameters. When we select the scaling parameter as 1, this reduces to an equivalent four stage decomposition. In other words, some 36 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

decompositions turn out to be equivalent to others, reducing the total number of possible decompositions.

Careful consideration shows that there are a total of sixteen distinct decompositions. Four of these have 4 stages and twelve of them have 5 stages; decompositions with 3 stages are not flexible enough to match arbitrary LCTs. However, some decompositions among these sixteen are equivalent in implementation. For example a scaling and FT cascade can be replaced with a FT and inverse scaling cascade, a trivial and inconsequential difference. When such trivial replacements are deducted from the set of sixteen decompositions, we end up with twelve decompositions. We will immediately eliminate two of these decompositions. These two decompositions involve significantly more computational load than the others. As will be seen, the CM stages, especially when they occur early in the cascade, require us to increase the sampling rate and thus the complexity. Generally speaking it is desirable to have as few CM stages as possible and to have them appear as late in the cascade as possible. Therefore we eliminate the two decompositions having three CM stages. Finally, our exhaustive consideration and careful sorting out of all decompositions with three, four, and five stages leaves us with ten distinct decompositions to consider:

1/β 0 1 0 0 1 1 0 M = (3.1)  0 β   α/β2 1   1 0   γ 1  −         1 0 0 1 1 0 β 0 M = (3.2)  α 1   1 0   γ/β2 1   0 1/β  −         1 0 1/β 0 0 1 1 0 M = (3.3)  α 1   0 β   1 0   γ 1  −         1 0 γ/β 0 0 1 1 0 0 1 M = − (3.4)  α β2/γ 1   0 β/γ   1 0   1/γ 1   1 0  − − − − −           1 0 0 1 1 0 0 1 γ/β 0 M = − (3.5)  α β2/γ 1   1 0   γ/β2 1   1 0   0 β/γ  − − − − −           3.2. ANALYSIS OF DECOMPOSITIONS AND ALGORITHM I 37

γ/β 0 1 0 0 1 1 0 0 1 M = − (3.6)  0 β/γ   γ + αγ2/β2 1   1 0   1/γ 1   1 0  − − − − −           0 1 α/β 0 1 0 0 1 1 0 M = − (3.7)  1 0   0 β/α   α/β2 1   1 0   γ β2/α 1  − − − − −           0 1 1 0 0 1 β/α 0 1 0 M = − (3.8)  1 0   1/α 1   1 0   0 α/β   γ β2/α 1  − − − − −           0 1 1 0 0 1 1 0 β/α 0 M = − (3.9)  1 0   1/α 1   1 0   α + γα2/β2 1   0 α/β  − − − − −           0 1 1 0 α/β 0 0 1 1 0 M = − (3.10)  1 0   1/α 1   0 β/α   1 0   γ β2/α 1  − − − − −          

Recall that the diagonal matrices correspond to scaling, the skew-diagonal ones to FT, and the lower-triangular ones to CM. The parameters α,β,γ of these operations are chosen in terms of the elements A, B, C, D of the matrix M so as to equate the left and right hands of the above equations (see Eq. (2.4)).

Scaling and FT do not require an increase in the number of samples. However, since CM will change the time-bandwidth product, we introduce 2 oversampling when con- × fronted with the ﬁrst CM operation, to allow us room to maneuver. We will however try to avoid oversampling beyond this, as much as possible. Speciﬁcally, we will impose any necessary restrictions so as to avoid further oversampling until the stage that involves the very last CM.

We will make sure that after each stage, the number of samples is sufﬁcient (in the Nyquist sense) for recovery of the continuous signal. In order to illustrate how we deal with each decomposition, we consider the decomposition given in Eq. 3.1 as an example (Fig. 3.1). For graphical purposes, here we assumed that the initial time-frequency support is a square of edge length ∆u rather than a circle. Since we require that 2N samples be 38 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

µ µ

∆u(1 + γ ) | | ∆u

u u

∆u(1 + γ ) ∆u | |

(a) After the ﬁrst stage: CM I (b) After the second stage: FT

µ µ ∆u[1+(1+ γ ) |α| ] | | β2 ∆u β [1+(1+ γ ) |α| ] | | | | β2

u u

∆u(1 + γ )/ β | | | | ∆u(1 + γ ) | |

Figure 3.1: Sequence of geometrical distortions for the decomposition in Eq. 3.1. The parallelogram in (c) is obtained by shearing the dashed rectangle in (b) in order to cover the worst case.

sufﬁcient, we must ensure that the time-bandwidth product does not exceed 2N:

∆u ∆u(1+ γ ) 2N = 2∆u2 γ 1 1 γ 1. (3.11) × | | ≤ ⇒ | | ≤ ⇒ − ≤ ≤

Therefore, we obtain the restriction 1 γ 1 for the parameter γ appearing in the ﬁrst − ≤ ≤ chirp multiplication operation. This restriction ensures that oversampling by two is sufﬁ- cient, by ensuring that the bandwidth following the geometric distortion has not increased by more than a factor of two. After the FT, the second chirp multiplication operation, and 3.2. ANALYSIS OF DECOMPOSITIONS AND ALGORITHM I 39

the scaling operation, the time-bandwidth product becomes:

α ∆u (1 + γ ) ∆u 1+ | | (1 + γ ) . (3.12) | | β2 | | Note that the parallelogram in Fig. 3.1(c) is obtained by shearing the dashed rectangle in Fig. 3.1(b) in order to cover the worst case. Is the above expression greater than 2N = 2∆u2 and if so, how much greater is it? Let k/2 denote the additional oversampling factor required. Equating the above expression to the number of samples k∆u2 corresponding to a total of k oversampling, leads us to the minimum value of k as

α (1 + γ )2 k 1+ γ + | | | | . (3.13) ≥ | | β2

If the right hand side of this expression turns out to be 2, that means that we do not need ≤ any additional oversampling, in which case we simply take k = 2. Before continuing, we also note that the scaling operation merely changes the sampling interval in the sense of reinterpretation of the same samples with a scaled sampling interval, in a manner which corresponds to scaling of the underlying continuous signal. Thus, by carefully following the evolution of the time-frequency support region through each stage of the decomposition, we have obtained (i) any necessary restrictions on the parameters of the stages appearing in the decomposition, so that oversampling beyond 2 × is not required until the last CM stage, (ii) the additional oversampling factor k/2 which may be needed before the last CM stage to fully represent the continuous output signal. We underline that these considerations are guided by our goal to accommodate arbitrary input signals. In those cases where there exists some a priori knowledge of the input signal or the signal is restricted to a particular class, it may be possible to customize the approach here with beneﬁt. The same procedure has been repeated for the decompositions given in Eqs. 3.2–3.10 and the results are given in Table 3.1. It will be convenient to choose k as the smallest 40 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

integer 2 satisfying the applicable inequality. ≥ We have also added together the computational complexity of each stage to obtain the overall computational complexity of each decomposition and presented these in Table 3.2.

Decomposition Restriction Oversampling factor |α|(1+|γ|)2 Eq. 3.1 γ 1 k 1+ γ + β2 | | ≤ ≥ | | |α|(1+|γ|)2 Eq. 3.2 γ 1 k 1+ γ + β2 | | ≤ ≥ | | |α|(1+|γ|)2 Eq. 3.3 γ 1 k 1+ γ + β2 | | ≤ ≥ |1 | (1+|γ|)2 β2 Eq. 3.4 γ 1 k 1+ |γ| + β2 α γ | | ≥ ≥ 2 | − | 1 (1+|γ|) β2 Eq. 3.5 γ 1 k 1+ |γ| + β2 α γ | | ≥ ≥ 2 | − 2 | Eq. 3.6 γ 1 k 1+ 1 + (1+|γ|) α β | | ≥ ≥ |γ| β2 | − γ | 2 2 2 2 Eq. 3.7 γ β 1 k 1+ γ β + α 1+ γ β | − α | ≤ ≥ | − α | | β2 | | − α | 2 2 2 2 Eq. 3.8 γ β 1 k 1+ γ β + α h1+ γ β i | − α | ≤ ≥ | − α | | β2 | | − α | 2 2 2 2 Eq. 3.9 γ β 1 k 1+ γ β + α h1+ γ β i | − α | ≤ ≥ | − α | | β2 | | − α | 2 2 2 2 Eq. 3.10 γ β 1 k 1+ γ β + α h1+ γ β i | − α | ≤ ≥ | − α | | β2 | | − α | h i Table 3.1: Restrictions and oversampling factors

Observation of Table 3.1 reveals a natural grouping of the ten decompositions. The decompositions given in Eqs. 3.1, 3.2, 3.3 exhibit the same restriction and oversampling factors. Likewise, the decompositions given in Eqs. 3.4, 3.5, 3.6 share the same restriction and oversampling factors. Moreover, the restrictions of these two groups are complementary, the ﬁrst three can be used for γ 1 and the last three can be used for γ 1, | | ≤ | | ≥ spanning the whole parameter space of LCTs. Finally, we observe that the computational complexity of the third and last group of four decompositions is larger than the others. This group has a term with complexity (Nk)log(Nk) due to the second CM operation being followed by a DFT operation. There- fore we discard the last four decompositions belonging to the third group. It should be noted that our elimination procedure is primarily based on complexity as opposed to error. However, as we will see in Table 3.3, the algorithm obtained produces results with errors as 3.2. ANALYSIS OF DECOMPOSITIONS AND ALGORITHM I 41

Decomposition Computational Complexity Eq. 3.1 2N + (2N) log(2N)+ Nk + I(N, 2)+ I(2N,k/2) + S(Nk) Eq. 3.2 2N + (2N) log(2N)+ Nk + I(N, 2)+ I(2N,k/2) + S(N) Eq. 3.3 2N + (2N) log(2N)+ Nk + I(N, 2)+ I(2N,k/2) + S(2N) Eq. 3.4 2N + (2N) log(2N)+ N log N + Nk + I(N, 2) + I(2N,k/2) +S(2N) Eq. 3.5 2N + (2N) log(2N)+ N log N + Nk + I(N, 2) + I(2N,k/2) +S(N) Eq. 3.6 2N + (2N) log(2N)+ N log N + Nk + I(N, 2) + I(2N,k/2) +S(Nk) Eq. 3.7 2N + (2N) log(2N)+(Nk)log(Nk)+ Nk + I(N, 2) +I(2N,k/2)+ S(Nk) Eq. 3.8 2N + (2N) log(2N)+(Nk)log(Nk)+ Nk + I(N, 2) +I(2N,k/2)+ S(2N) Eq. 3.9 2N + (2N) log(2N)+(Nk)log(Nk)+ Nk + I(N, 2) +I(2N,k/2)+ S(N) Eq. 3.10 2N + (2N) log(2N)+(Nk)log(Nk)+ Nk + I(N, 2) +I(2N,k/2)+ S(2N)

Table 3.2: Computational Complexities. I(x, y) stands for the cost to interpolate x samples by a factor of y to obtain xy samples and S(z) stands for the cost of the scaling operation on z samples.

small as can be reasonably expected; therefore we have not really lost anything by basing our procedure primarily on complexity.

Use of the information summarized in the two tables finally leads us to choose to work with two decompositions, one from the first and one from the second group of three. To ensure that oversampling by two is sufficient, we ended up using two complementary decompositions for different regions of the parameter space. Since they have a slightly lower complexity, we will prefer to work with the second decomposition in both groups, whose advantage arises primarily from the relative positions of the scaling and CM operations. The algorithm can now be outlined as follows: 42 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

If γ 1, use the decomposition: • | | ≤

1 0 0 1 1 0 β 0 M = .  α 1   1 0   γ/β2 1   0 1/β  −         In operator notation:

M = J 2 J , (3.14) C Q−α k/2 Flc Q−γ/β 2 Mβ

where J represents the x oversampling operation. The minimum value of k is: x × α (1 + γ )2 k 1+ γ + | | | | . (3.15) ≥ | | β2

If γ > 1, use the decomposition: • | |

1 0 0 1 1 0 0 1 γ/β 0 M = − .  α β2/γ 1   1 0   γ/β2 1   1 0   0 β/γ  − − − − −           In operator notation:

M = 2 J 2 J , (3.16) C Q−α+β /γ k/2 Flc Qγ/β 2 Flc M−γ/β

The minimum value of k is:

1 (1 + γ )2 β2 k 1+ + | | α (3.17) ≥ γ β2 | − γ | | |

We have chosen to avoid unnecessary increases in the time-bandwidth product in the early stages to avoid increasing the number of samples until the last CM stage, where the major and unavoidable increase in sampling rate occurs. 3.3. ALGORITHM II 43

3.3 Algorithm II

The second approach we discuss is based on the following decomposition involving the FRT, scaling, and chirp multiplication:

A B 1 0 M 0 cos θ sin θ M = = . (3.18)  CD   q 1   0 1/M   sin θ cos θ  − −         Here θ = aπ/2 where a is the order of the FRT, q is the chirp multiplication parameter, and M is the scaling factor. As we will see, these three parameters are sufﬁcient to satisfy the above equality for arbitrary ABCD matrices, so that this decomposition is capable of representing arbitrary LCTs. Since the fast method proposed in [62] can be used for the computation of the FRT, this decomposition leads to a fast algorithm for LCTs. This decomposition was inspired by the optical interpretation in Ref. [107] and is also a special case of the widely-known Iwasawa decomposition [19,23,95]. It was also proposed later in [61,63]. Fig. 3.2 illustrates the sequence of geometrical distortions corresponding to this decomposition, where the initial time-frequency support is a circle of diameter ∆u.

If we multiply out the right hand side of Eq. 3.18 and replace the matrix entries A, B, C, D with α,β,γ, we obtain:

γ/β 1/β M cos θ M sin θ = , (3.19)  β + αγ/β α/β   qM cos θ sin θ/M qM sin θ + cos θ/M  − − − −     which is equivalent to four equations which we can solve for a, q, M:

a = (2/π)arccotγ, (3.20) 1+ γ2/β, γ 0, M = ≥ (3.21)  p1+ γ2/β, γ< 0,  − q = γβ2/(1p + γ2) α. (3.22)  − 44 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

µ θ = aπ/2 = cot−1(γ)

∆u u

∆u

(a) After the ﬁrst stage: FRT

∆u/M

M∆u

(b) After the second stage: scaling

µ ∆u/M + q M∆u | |

M∆u

Figure 3.2: Sequence of geometrical distortions for the decomposition in Eq. 3.18. 3.3. ALGORITHM II 45

The ranges of the square root and the arccotangent both lie in ( π/2,π/2]. Figure 3.2 − shows the geometric effect of the decomposition stages on the WD of a function, which is rotation, scaling, and shearing, respectively. In operator notation this algorithm can be expressed as:

a M = J (3.23) C Qq k MM Flc

In this method, the first operation is a FRT, whose fast computation in O(N log N) time is presented in Refs. [62,109]. (Other works dealing with fast computation of the FRT include [110,111].) The algorithm presented in Ref. [62] was based on decomposing the FRT into a CM followed by a CC followed by a final CM, and computed the samples of the continuous FRT in terms of the samples of the original signal. Just as in the present work, care was taken to ensure that the output samples represented the continuous FRT in the Nyquist-Shannon sense. The presently discussed algorithm employs the algorithm in Ref. [62] as a subroutine. The only approximation in this subroutine comes from the step involving chirp convolution in which a DFT/FFT is used to approximate the samples of the continuous FT. No other approximation need be made, either in this subroutine or in any of the other operations that we employ. Thus the only source of approximation can be traced to the evaluation of a continuous FT by use of a DFT (implemented with a FFT), which is a consequence of the fundamental fact that the signal energy cannot be confined to finite intervals in both domains. The second operation in this method is scaling, which only involves a reinterpretation of the same samples with a scaled sampling interval. The final operation is CM which takes O(N) time, leading to an overall complexity of O(N log N). (Detailed expressions will be given in Section 3.4.) As in the first method, it is again necessary to ensure that the output samples are sufficient to represent the transformed signal in the Nyquist-Shannon sense. Since LCTs distort the original time-frequency support, both the time and frequency extent of the signal, as well as its time-bandwidth product may increase, despite the fact that the area of the support remains the same. Therefore, a greater 46 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

number of samples than ∆u2 may be needed to represent the transformed signal as we have showed in an example in Fig. 3.1. Delaying confrontation with the necessity to deal with this greater number of samples until the very last step is a significant advantage of this method. Since the FRT corresponds to rotation and scaling only to reinterpretation of the samples, these steps do not require us to increase the number of samples. At the last CM step, if we multiply the samples of the intermediate result with the samples of the chirp, the samples obtained will be good approximations of the true samples of the transformed signal at that sampling interval. If these samples are sufficient for our purposes, nothing further needs be done. However, in general these samples will be below the Nyquist rate for the transformed signal and will not be sufficient for full recovery of the continuous function. To obtain a sufficient number of samples that will allow full recovery, we must interpolate the intermediate result at least by a factor k corresponding to the increase in time-bandwidth product:

k 1+ γ α(1 + γ2)/β2 . (3.24) ≥ −

Again, for convenience we choose k to be the smallest integer satisfying this inequality.

3.4 Results and Veriﬁcation of the Algorithms

We have considered several examples to illustrate and compare the presented methods. We refer to the ﬁrst algorithm involving Fourier transformation, scaling, and chirp multiplication as A1, and the second algorithm involving the fractional Fourier transform as A2. We consider the chirped pulse function exp( πu2 iπu2), denoted F1, and the trapezoidal − − function 1.5tri(u/3) 0.5tri(u), denoted F2 (tri(u) = rect(u) rect(u)). Since these two − ∗ functions are well conﬁned to a circle with diameter ∆u = 8 we take N = 82. We also consider the binary sequence 01101010 occupying [ 8, 8] with each bit 2 units in length, − 3.4. RESULTS AND VERIFICATION OF THE ALGORITHMS 47

so that N = 162. This binary sequence is denoted by F3 and the function shown in Fig. 5.2 is denoted by F4, again with N = 162. These choices for ∆u result in 0 %, 0.0002 %, ∼ 0.47 %, 0.03 % of the energies of F1, F2, F3, F4 respectively, to fall outside the chosen frequency extents. The chosen time extents include all of the energies of F2, F3, F4 and virtually all of the energy of F1. We consider two transforms, the ﬁrst (T1) with parameters (α,β,γ)=( 3, 2, 1), and the second (T2) with parameters ( 4/5, 1, 2). The LCTs − − − − T1 and T2 of the functions F1, F2, F3, F4 have been computed both by the presented fast methods A1 and A2 and by a highly inefﬁcient brute force numerical approach based on composite Simpson’s rule with extensive number of intervals that can handle highly oscillatory functions which is here taken as a reference. The details of the Simpson’s method is given in Appendix A.

The results for all functions (F1, F2, F3, F4) and both algorithms (A1, A2) are plotted in Figure 3.4 for transform T1, and tabulated in Table 3.3 for both transforms (T1, T2). Also shown are the errors that arise when using the DFT in approximating the FT of the same functions, which serves as a reference. (The error is deﬁned as the energy of the difference normalized by the energy of the reference, expressed as a percentage.)

The key observations that can be made from this table are as follows. The errors obtained depend on the function, since different functions have different amounts of energy contained in their tails which fall outside the chosen time and frequency extents (or assumed time-frequency region). For those cases in which the error is large, such as F3, this means that we have determined the time-bandwidth product less conservatively than the other examples, and the error can be reduced by increasing N. Generally speaking, the errors obtained depend very little on the transform parameters or which method we use, and are comparable to the error arising when we use the DFT to approximate the FT. Since a DFT lies at the heart of both methods, this is the smallest error we could have hoped to achieve to begin with.

Fig. 3.5 shows the error versus number of sample points N for selected functions and 48 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

0 −8 −6 −4 −2 0 2 4 6 8

Figure 3.3: Example function F4

A1T1 A1T2 A2T1 A2T2 DFT F1 3.2 10−22 9.5 10−22 2.7 10−17 6.6 10−17 2.0 10−21 × × × × × F2 7.8 10−4 8.1 10−4 11 10−4 9.9 10−4 6.2 10−4 F3 1.5× 1.6× 1.4× 1.5× 1.2 × F4 9.7 10−2 11 10−2 8.9 10−2 9.9 10−2 8.3 10−2 × × × × × Table 3.3: Percentage errors for different functions F, transforms T, and algorithms A. transforms. We observe that the error decreases steeply at ﬁrst with increasing N as expected, but saturates when we approach and exceed the time-bandwidth product of the signals (here 64). This demonstrates that the number of samples N can be chosen comparable to the time-bandwidth product, which is the smallest number we can expect to work with, and need not be chosen larger. A2 was used to obtain this plot for illustration purposes but 3.4. RESULTS AND VERIFICATION OF THE ALGORITHMS 49

Re of transform of F1 Im of transform of F1 1 1

0.5 0.5

0 0

−0.5 −0.5 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3

Re of transform of F2 Im of transform of F2 1.5 1.5

1 1

0.5 0.5

0 0

−0.5 −0.5

−1 −1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3

Re of transform of F3 Im of transform of F3 2 2

1 1

0 0

−1 −1

−2 −2 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

Re of transform of F4 Im of transform of F4 3 3

2 2

1 1

0 0

−1 −1

−2 −2 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

Figure 3.4: Transforms (T1) of F1, F2, F3, F4. The results obtained with Methods I and II and the reference result have been plotted with dotted, dashed, and solid lines respectively. However, in most cases these lines are indistinguishable since the results are very close. 50 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

similar results can also be obtained when we use A1.

20 F1 T1 18 F1 T2 F2 T1 16 F2 T2

10 Error % 8

0 0 1 2 3 10 10 10 10 N

Figure 3.5: Percentage errors versus N for selected functions and transforms.

We now turn our attention to discussing the complexity (cost) of the algorithms as a function of N, the number of sample points. Based on the preceding paragraph, N can be chosen comparable to the time-bandwidth product so that the expressions given below can also be interpreted as functions of time-bandwidth product. The computational complexity of the ﬁrst method is given by either of the following expressions, depending on which decomposition is used (which was determined by whether γ 1 or not, respectively): | | ≤

2N +2N log2N + Nk + I(N, 2)+ I(2N,k/2) + S(N) •

2N +2N log2N + N log N + Nk + I(N, 2) + I(2N,k/2)+ S(N) • 3.4. RESULTS AND VERIFICATION OF THE ALGORITHMS 51

On the other hand, for the second method, one of the following applies depending on the branch of the FRT algorithm used, which depends on whether 0.5 a 1 or a < 0.5, ≤| | ≤ | | respectively. Not surprisingly, this turns out to be the same as the condition γ 1 or | | ≤ γ > 1, respectively: | |

6N +6N log2N + Nk + I(N, 2)+ I(2N,k/2)+ S(2N) •

6N +6N log2N + N log N + Nk + I(N, 2) + I(2N,k/2)+ S(2N) •

The above expressions are derived by calculating the complexity of the FRT algorithm of [62] in its most efﬁcient implementation (6N +6N log2N + I(N, 2) [+N log N]), and adding the cost of the other operations. Although included for completeness, the cost S( ) of the scaling operation is minimal · and not of much consequence, since it amounts only to a reinterpretation of samples. In the above expressions, I(x, y) stands for the cost to interpolate x samples by a factor of y to obtain xy samples. There are several efﬁcient approximate interpolation methods which have complexities of order O(xy) [112]. We will write this cost as cxy where c is a constant. Taking the difference of the costs of the two methods, Method II will have lower cost if

(1 + c)(k k ) > 4[1+ log2N], (3.25) I − II

where kI and kII are associated with Method I and Method II respectively. If we go back to the steps of each method, we observe that it is usually possible to choose kII much more tightly than kI . Numerical simulations also confirm that kII is usually smaller than kI. Therefore, either method may turn out to have lower cost depending on the values of α,β,γ,N and it is not possible in general to declare one method superior over the other in this respect. If need be, both methods can be incorporated in the same code and the more efficient one invoked based on the parameters, but in many cases the difference may not be very significant. However, apart from its effect on cost of computation, having the lowest 52 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

oversampling factor is desirable in itself since it produces an output represented with the least number of samples. This is a plus for Method II, which we also favor for its elegant construction. Finally, we compare Method I and Method II, with the direct use of the well-known CM-CC-CM decomposition [2] without the kind of sampling rate management undertaken in this paper (“Method III”):

1 0 1 1/β 1 0 M = , (3.26)  α β 1   0 1   γ β 1  − −       which is valid as long as β =0. The CC stage has been implemented in the Fourier domain 6 as a CM operation. At each step of the process, the sampling rate has been chosen as the minimum compatible with the shear-induced increases in the time and frequency extents. Considering T1, the minimum oversampling rate k is 5, 2, 8 for Methods I, II, and III, respectively. With the more demanding T2, the minimum value of k is 14, 7, 24, respectively. Larger values of k and thus larger total numbers of samples not only result in representa- tional redundancy, they translate into greater computational time. For instance, for F4 and T2, the time of computation in seconds is 0.0910, 0.0396, 0.347 for Methods I, II, and III, respectively (obtained with MATLAB running on a personal computer), demonstrating that Method III is signiﬁcantly slower. The corresponding percentage errors are comparable as expected: 0.11, 0.099, 0.10, respectively.

3.5 Concluding Remarks

In this chapter, two algorithms for the computation of linear canonical transforms (LCTs) from the N samples of the input signal in O(N log N) time are discussed. Our approach is based on concepts from signal analysis and processing rather than conventional numerical analysis. With careful consideration of sampling issues, N can be chosen very close to the 3.5. CONCLUDING REMARKS 53

time-bandwidth product of the signals, and need not be much larger. The transform output may have a higher time-bandwidth product due to the nature of the transform family. This is accomplished by doing a careful study of the deformations of the time-bandwidth products of the signals along the steps of the algorithm. As can be seen in Fig. 3.1, the effects of the every step of a decomposition that is used in the algorithm is carefully studied and the time and frequency extents of the signals and the resulting time-bandwidth products are calculated. From these calculation, one can judge the minimum required number of samples that should be taken to represent the signal in an information theoretically proper way. As a result, the number of samples N is kept at the minimum possible values to match the resulting time-bandwidth product of the output signal.

Both algorithmsrelate the samples of the inputfunction to the samples of the continuous LCT of this function in the same sense that the fast Fourier transform (FFT) implementation of the discrete Fourier transform (DFT) computes the samples of the continuous FT of a function. Since the sampling rates are carefully controlled, the output samples obtained are accurate approximations to the true ones and the continuous LCT can be recovered via interpolation of these samples. The only inevitable source of deviation from exactness arises from the fundamental fact that a signal and its transform cannot both be of ﬁnite extent. This is the same source of deviation encountered when using the DFT/FFT to compute the continuous FT. Thus the algorithms compute LCTs with a performance similar to the DFT/FFT in computing the Fourier transform, both in terms of speed and accuracy.

The fact that the two methods, although being arrived at from considerably different starting points, both exhibit similar limits in performance, strongly suggests that the performance achieved is close to the best achievable. Indeed, as already noted, it is difﬁcult to expect an accuracy which is better than that of the DFT in approximating the FT, and a cost which is less than O(N log N) with N being close to the time-bandwidth product. Despite the different decompositions employed, an interesting structural similarity emerges between the two methods in their optimized forms. Both methods have two branches, the 54 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

first one determined by whether γ > 1 or not, and the second one determined by whether | | a < 0.5 or not. If a is expressed in terms of the LCT parameters α, β, γ, we see that | | the two conditions are the same. Therefore, the separation of the LCT parameter space into two regions is most likely not a characteristic of the algorithm chosen, but an intrinsic structural property of the LCT parameter space. Compared to earlier approaches, these algorithms not only handle a much more general family of integrals, but also effectively address certain difficulties, limitations, or tradeoffs that arise in other approaches to computing the Fresnel integral, which is of importance in the theory of diffraction (see Ref. [61] for a systematic comparison of several approaches). These algorithms can also be used for efficient realization of filtering in linear canonical transform domains [59]. Chapter 4

The Algorithm for 2D Quadratic-Phase Systems

4.1 Introduction1

HIS CHAPTER PRESENTS the fast algorithms for the two-dimensional QPSs, or T 2D-LCTs. Given an algorithm for efﬁciently computing 1D-LCTs [60,63,64], the efﬁcient computation of separable 2D transforms is straightforward because the kernel can be separated and the 2D transform can be reduced to two successive 1D-LCTs. Much work has been done on 1D and 2D separable LCTs in terms of sampling issues and fast algorithms for their digital computation, [65–68]. On the other hand, in the non-separable case, the two dimensions are coupled. Handling this case requires special attention and to the best of our knowledge has not been addressed before. The current established LCT computation algorithms presented in Chapter 3 are not able to compute 2D-NS-LCTs. An alternative representation of LCTs is presented and studied in [95]. This decomposition is based on the well-known Iwasawa decomposition [113]. In [95], the authors

55 56 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

further decompose the first matrix of the Iwasawa decomposition into a 2D separable fractional Fourier transform that is sandwiched between two coordinate rotators. We had earlier employed a 1D Iwasawa decomposition to develop a fast and efficient algorithm for 1D- LCTs [63,64]. In this chapter, we use the 2D version of this Iwasawa-type decomposition to derive our efficient algorithm. As in the 1D case, the distinguishing feature of our approach is the way our algorithm carefully addresses sampling and space-bandwidth product issues from an information-theoretical perspective. Special care is taken to ensure that the output samples represent the continuous transform in the Nyquist-Shannon sense during every stage of the algorithm, so that the continuous transform can be fully recovered from the samples.

To our knowledge, there is no algorithm in the literature that efficiently calculates 2D- NS-LCTs. Furthermore, despite the highly oscillatory nature of the integral kernel, the sampling rate is managed so as to ensure that the number of samples used is sufficient, but not much larger than the space-bandwidth product of the input signal, so that the algorithms are as efficient as possible. Straightforward sampled numerical integral computation takes O(N 2) time where N = MN for a 2D signal sampled on a M N grid. In contrast, × the complexity of our algorithm is O(N log N). This efficiency is even more crucial in e e the 2D case than in the 1D case since the number of points is much larger. By choosing e e the number of samples N equal to the 2D space-bandwidth product of the signals, we ensure that the efficiency is near the best that is theoretically possible. More generally, e through each stage of the algorithm, we carefully manage the sampling rate to maintain the information theoretically sufficient, but not wastefully redundant, sampling required for reconstruction of the underlying continuous functions at any stage of the algorithm.

In Chapter 2, the definition of 2D-NS-LCTs is given. An explicit-kernel definition with the least possible number of independent variables is provided and the forward and backward relations between the parameters of this definition and the parameters of conventional ABCD-matrices are derived. Section 5.2 provides the preliminary mathematical 4.2. PRELIMINARIES 57

background and the tools that we use in the algorithm. In Section 4.3, our algorithm is presented. Section 4.4 addresses the issue of sampling rate and space-bandwidth product control in order to ensure the necessary sampling rates sufﬁcient for proper reconstruction in the Nyquist-Shannon sense at each step of the algorithm. Next, numerical results are reported in Section 4.5. We conclude in Section 5.5.

4.2 Preliminaries

4.2.1 A 3-Sphere for Space-Bandwidth Control, Wigner Distribution and Dimensional Normalization for 2D Functions

When we study one-dimensional input functions and one-dimensional LCTs, the corresponding Wigner distribution is two-dimensional. One dimension represents the space extent and the other represents the spatial frequency extent of the signal. However, for two- dimensional signals, there exist two space extents and two corresponding spatial frequency extents resulting in a four-dimensional Wigner distribution. The deﬁnition of Wigner distribution and its effects on linear canonical transforms for one-dimensional signals were given in 2.1.5. The corresponding Wigner distribution Wf (ux,uy,µx,µy) of a two-dimensional signal f(ux,uy) can be deﬁned as follows [1]:

∞ ∞ ′ ′ ∗ ′ ′ Wf (ux,uy,µx,µy)= f(ux + ux/2,uy + uy/2)f (ux ux/2,uy uy/2) −∞ −∞ − − Z Z ′ ′ e−2πi(µxux+µyuy) du′ du′ . (4.1) × x y

We call this four-dimensional Wigner distribution the “4D Wigner distribution” whereas the usual two-dimensional Wigner distribution used for one-dimensional functions will be referred to as the “2D Wigner distribution”. Its integral over time and frequency, ∞ ∞ −∞ −∞ W (u,µ) dudµ, gives the signal energy. Let f denote a function and fM be its R R 58 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

2D-LCT with parameter matrix M. Then, the relation between the Wigner distribution

(WD) of fM and the WD of f can be expressed as [1]:

WfM (Ms)= Wf (s), (4.2)

T where the vector s = [ux uy µx µy] is used for the sake of notational simplicity. An example of the use of the Wigner distribution in sampling issues from another perspective can be found in [114].

In Chapter 3, we used 2D Wigner distributions for tracking and control of the space- bandwidth products of signals through the stages of our algorithms. 2D Wigner distributions are easy to visualize and therefore easy to understand. However, for two-dimensional signals we cannot graphically show the Wigner distribution because it is a four-dimensional function. Therefore we will develop and use a more abstract and rigorous approach to space-bandwidth tracking and control in order to achieve the information-theoretic minimum sampling rate for lossless reconstruction of the continuous output function from the output samples.

First, we need to recall the geometrical object known as a “3-sphere”. In general, for a natural number n, an “n-sphere” is the generalization of the ordinary “2-sphere” in common three-dimensional Euclidian space to any dimension. Explicitly, an n-sphere, denoted as Sn and centered at the origin, is the analogue of a sphere in (n +1)-dimensional Euclidian space and is deﬁned as: Sn = x Rn+1 : x = r (4.3) { ∈ k k } where the positive real number r is the radius of the n-sphere and Rn+1 is an (n + 1)- dimensional vector space over R. More on n-spheres can be found in [115,116]. Then, we can provide the generic deﬁnition of the 3-sphere, S3, centered at the origin explicitly as:

S3 = (x , x , x , x ) R4 : x2 + x2 + x2 + x2 = r2 . (4.4) { 1 2 3 4 ∈ 1 2 3 4 } 4.2. PRELIMINARIES 59

Now, let us turn our attention to our two-dimensional input functions. It is well known that a non-zero function and its Fourier transform (FT) cannot both be confined to finite regions. However, in practice, we always work with samples of finite extent functions by assuming that the energy of the signal falling outside of some region is negligible. In general, the signal will exhibit some distribution of energy in the two-dimensional space-frequency hypervolume (which is four dimensional). We will assume that a finite hyperellipsoidal boundary in R4 is chosen so as to confine most of the energy of the signal. This hyperellipsoidal boundary will imply finite extents in the two space dimensions and the two spatial-frequency dimensions. The intervals of confinement thus defined will be denoted by [ ∆S /2, ∆S /2] and [ ∆S /2, ∆S /2] in the space dimensions, and − x x − y y [ ∆B /2, ∆B /2] and [ ∆B /2, ∆B /2] in the spatial-frequency dimensions. The space − x x − y y and spatial-frequency representations of the signal will be approximately confined within these intervals. Given these, it also follows that both space-domain extents are confined within the worst case interval [ ∆S /2, ∆S /2], where ∆S = max ∆S , ∆S , − max max max { x y} and both frequency-domain extents are confined within the interval [ ∆B /2, ∆B /2], − max max where ∆B = max ∆B , ∆B . Under these conditions, the Wigner distribution of the max { x y} function is confined within the boundary O in R4 (note that this is not defining a 3-Sphere yet):

s2 b2 O R4 x x = (sx,sy, bx, by) : 2 + 2 { ∈ (∆Smax/2) (∆Bmax/2) s2 b2 y y (4.5) + 2 + 2 =1 (∆Smax/2) (∆Bmax/2) } where sx and sy are temporary space variables and bx and by are temporary spatial-frequency variables of the Wigner distribution of the signal.

Let us now introduce the scaling parameter P and scaled dimensionless coordinates

ux = sx/P, 60 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

uy = sy/P,

µx = bxP,

µy = byP, (4.6) such that the two space- and two frequency-domain representations are conﬁned to intervals of length ∆Smax/P and ∆BmaxP , respectively. Let P = ∆Smax/∆Bmax so that the lengths of all the four intervals become equal to the dimensionlessp quantity

√∆Smax∆Bmax , which we denote by ∆u. Expressed in dimensionless coordinates, the boundary O deﬁned in Eq. 4.5 reduces to the desired 3-sphere, denoted by Osp:

2 2 4 2 2 2 2 √∆Smax∆Bmax ∆u Osp = (u ,u ,µ ,µ ) R : u + u + µ + µ = = . { x y x y ∈ x y x y 2 2 } (4.7) To summarize, after the dimensional normalization procedure given above has been performed, the 4D Wigner distribution of our two-dimensional input function can be assumed

to be conﬁned within a 3-sphere Osp of diameter ∆u.

4.3 The Algorithm

As noted before, one of the most important features of our method is to control the sampling rate of the function with the goal of having enough samples to be able to reconstruct the continuous function without information loss, and at the same time without needlessly increasing the number of samples to maintain efﬁciency. In this Section, we present our algorithm, discuss the stages in the decomposition and derive the parameters of each stage from the parameters of the 2D-NS-LCT that is being computed. The effects of each stage of the decomposition on the Wigner distribution of our function (thus on the space-bandwidth products) and associated sampling rate issues will be addressed in Section 4.4. 4.3. THE ALGORITHM 61

The Iwasawa decomposition is the core of our algorithm. After the dimensional normalization explained in Section 5.2.4.2.1, any transformation matrix M can be written in the following Iwasawa form [95, 113]:

A B I 0 S 0 XY M = = (4.8)  C D   G I   0 S−1   YX  − −         where

G = (CAT + DBT)(AAT + BBT)−1 (4.9) − S =(AAT + BBT)1/2 (4.10)

X =(AAT + BBT)−1/2A (4.11)

Y =(AAT + BBT)−1/2B (4.12)

Given the 4 4 matrix M, we can determine 2 2 matrices G, S, X, Y by using Eqs. × × 4.9, 4.10, 4.11, 4.12. If we are able to develop a fast algorithm to compute the three stages in O(N log N) time, the overall transform can also be calculated in O(N log N) time. In this decomposition, the first operation is an orthosymplectic system, followed by a scaling e e e e (magnification) system, finally followed by a two-dimensional chirp multiplication (2D- CM). (Note that each of the stages of the algorithm are special cases of 2D-NS-LCTs.)

We begin with the ﬁrst and the most sophisticated stage of the decomposition, the orthosymplectic system. This stage of the decomposition can be further decomposed into a two-dimensional separable fractional Fourier transform (2D-S-FRT) that is sandwiched between two coordinate rotators [95]:

XY = Rr2 Fax,ay Rr1 (4.13)  YX  −   62 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

where the 4 4 matrices Rr , Fa a , Rr are deﬁned as: × 1 x, y 1

cos(r1) sin(r1) 0 0   sin(r1) cos(r1)0 0 R = − (4.14) r1    0 0 cos(r1) sin(r1)       0 0 sin(r1) cos(r1)   −   

cos(r2) sin(r2) 0 0   sin(r2) cos(r2)0 0 R = − (4.15) r2    0 0 cos(r2) sin(r2)       0 0 sin(r2) cos(r2)   −    cos(axπ/2) 0 sin(axπ/2) 0   0 cos(ayπ/2) 0 sin(ayπ/2) F = (4.16) ax,ay    sin(axπ/2) 0 cos(axπ/2) 0   −     0 sin(ayπ/2) 0 cos(ayπ/2)   −    Rr1 and Rr2 are rotation matrices that impose rotations of angles r1 and r2, respectively, through the spatial variables (ux,uy) and through their frequency variables (µx,µy). Un- like these traditional rotators which rotate within space and spatial frequency separately, the fractional Fourier transform rotates within the space-frequency planes of each dimension.

Fax,ay stands for 2D-S-FRT that makes separable rotations of angle axπ/2 over the variables (ux,µx) and of angle ayπ/2 over the variables (uy,µy). Since this two-dimensional FRT operation is separable, it corresponds to two one-dimensional fractional Fourier transformation operations performed over each of the dimensions. Explicitly, this means ﬁrst performing 1D-FRTs with the fractional order ax for each of the rows (or columns) and then performing 1D-FRTs with the fractional order ay for each of the columns (or rows) of the sampling grid. It is this observation that enables us to implement this stage of the decomposition efﬁciently in O(N log N) time. There are fast and established algorithms to

e e 4.3. THE ALGORITHM 63

compute one-dimensional fractional Fourier transforms [61–64], so that this stage can be calculated in O(N log N) time easily.

The interpretatione ofe the coordinate rotators requires care. When we are working with sampled functions, we know the value and coordinates (the location where the particular sample is taken) of all the samples we have. A coordinate rotation can be interpreted in this situation as a rotation of the locations of the samples resulting in a new sampling grid, rather than a change in the sample values. If we assume we start with a regular rectangular grid, after the coordinate rotation, the grid would no longer coincide with the original grid unless the rotation is an integer multiple of π/2. Unfortunately, in order to perform FRT operations along the horizontal and vertical directions, we need the samples to be on a regular rectangular grid in order to employ available fast algorithms. Therefore, we must carry out an interpolation operation to determine the values of the function on a regular rectangular grid. There are several techniques and algorithms to perform this interpolation efﬁciently. We have chosen to use in our numerical simulations fast and standard implementations of nearest neighbor, bilinear and cubic interpolations, [117,118], but any other efﬁcient method may also be used. This interpolation step and its performance can be a major source of error in our algorithm, as we will further discuss later.

We now turn our attention to determining the coordinate rotation angles r1 and r2, and the FRT fractional orders ax and ay. When we plug Eqs. 4.14, 4.15, 4.16 in Eq. 4.13, carry out the matrix multiplications and equate the entries of both sides of Eq. 4.13, we get the following equations in the four unknowns r1, r2, ax and ay:

X = cos r cos r cos(a π/2) sin r sin r cos(a π/2) 11 1 2 x − 1 2 y

X12 = sin r1 cos r2 cos(axπ/2) + cos r1 sin r2 cos(ayπ/2) X = cos r sin r cos(a π/2) sin r cos r cos(a π/2) 21 − 1 2 x − 1 2 y X = sin r sin r cos(a π/2) + cos r cos r cos(a π/2) 22 − 1 2 x 1 2 y 64 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

Y = cos r cos r sin(a π/2) sin r sin r sin(a π/2) 11 1 2 x − 1 2 y

Y12 = sin r1 cos r2 sin(axπ/2) + cos r1 sin r2 sin(ayπ/2) Y = cos r sin r sin(a π/2) sin r cos r sin(a π/2) 21 − 1 2 x − 1 2 y Y = sin r sin r sin(a π/2) + cos r cos r sin(a π/2) (4.17) 22 − 1 2 x 1 2 y

These equations are sufficient to solve for and unambiguously determine the rotation and fractional Fourier angles of the decomposition in a straightforward manner, provided one pays proper attention to sign considerations when inverting the trigonometric functions. To summarize, the first stage of our algorithm involves determining the angles from the above equations, performing the first coordinate rotation, following this by two 1D-FRTs over each of the dimensions, and then finishing with the second coordinate rotation. All these steps can be calculated in O(N log N) time. The second stage is the scaling operation and it seems to be the simplest of the three e e stages. It is not, however, as trivial as in the one-dimensional case [64]. In one dimension, it corresponds to only a reinterpretation of the spacing between the samples. The sampling interval scales with the scaling parameter. Intuitively, it squeezes in or stretches out the total number of samples as the word scaling implies. This means there is no change in the total number of samples and thus no need to oversample the input samples. The analogue of the one-dimensional scalar scaling parameter in the two-dimensional case is the matrix S. When S is diagonal, which means there is no coupling between the two dimensions of the function for scaling purposes, the scaling is separable. Due to this separability, this situation does not impose an increase in the space-bandwidth products and thus does not require oversampling, just as in the one-dimensional case. But when the off-diagonal elements of S are non-zero, the scaling operation is no longer so trivial. Although the total number of degrees of freedom of the signal remains the same, the space-bandwidth products may increase and an oversampling to match this increase may be necessary. Readers wishing to better understand how the space-bandwidth product may increase despite the fact that the 4.3. THE ALGORITHM 65

number of degrees of freedom remains the same are referred to [64], where these issues are studied graphically for the 1D case. An analogous, though not visually demonstrable, situation exists for 2D signals. The sampling rate control mechanism for such 2D scaling operations will be developed in detail in Section 4.4. At this point, we note that in those cases where the number of samples needs to be increased, the oversampling should be performed ﬁrst, prior to scaling. Afterwards, scaling is achieved by mere reinterpretation of the locations of the samples without changing the samples themselves (other than a constant multiplicative factor). Computationally, such a scaling operation amounts to modifying the information that tells us which coordinates the samples belong to. Since it requires only the reinterpretation of the coordinates of the samples plus a possible oversampling, it does not impose much computational load. Eq. 4.10 gives us the scaling parameters. The matrix S can be easily used to determine the output samples by using the input-output relation of the scaling operation:

1 −1 fsc(u)= f(S u) (4.18) √det S

T where f is the function to be scaled and fsc is the scaled function, and u = [ux uy] .

The last stage of our main Iwasawa decomposition is the 2D-CM operation whose parameters are given by the matrix G as deﬁned in Eq. 4.9. The input-output relation of this 2D-CM is given as:

2 2 −iπ(G11x +(G12+G21)xy+G22y ) fch(u)= e f(u) (4.19)

where fch stands for the chirp-multiplied function. The 2D-CM operation is the stage that is mainly burdened with any shears inherent in the 2D-NS-LCT to be computed. Such shears may considerably increase the space-bandwidth products of the function. Thus, before the 2D-CM operation, the space-bandwidth products of the function should be calculated carefully and any necessary oversampling should be performed. This CM operation may turn out to be non-separable or separable for particular 2D-NS-LCTs but regardless, it 66 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

requires only one multiplication for each sample, resulting in O(N) time computation. As a result, we see that we can perform all of the stages of the decomposition in e O(N log N) time or faster, which makes the computational complexity (cost) of our overall algorithm O(N log N). e e e e 4.4 Space-Bandwidth and Sampling Rate Control

In this section, we develop a method to track the space-bandwidth products of our functions as we perform the consecutive operations in our decomposition and focus on how to control the number of samples efﬁciently. We need to calculate the necessary sampling intervals and sampling rates for both dimensions that are necessary to represent the continuous signal without information loss for each of the stages. Oversampling should be undertaken prior to any stage that increases either of the space-bandwidth products. As given in Section 5.2.4.2.1, the Wigner distribution of the input signal is assumed to be conﬁned within a 3-sphere with radius ∆u/2, which also means that the signal is assumed to be almost space- and band-limited in both dimensions. The 4D Wigner representation gives us two space extents, two spatial-frequency extents and two space-bandwidth products, one for each dimension of the function. Let us denote the space-bandwidth prod-

uct along the ux direction by Nx and that along the uy direction by Ny. These extents deﬁne the minimum required number of samples along the corresponding direction, with the total number of samples being N N . Since the Wigner distribution is conﬁned within a x × y 3-sphere of diameter ∆u, all the extents of the function (space and spatial-frequency) are equal to ∆u at the beginning. Thus, the function should be sampled on a N N grid, x × y where the u -coordinate of the function spans the interval u = ( ∆u/2, ∆u/2) and the x x − u -coordinate spans the interval u = ( ∆u/2, ∆u/2). The distance between two adja- y y − cent samples is equal to ∆u−1 along both dimensions. As a result, the space-bandwidth

2 products are initially Nx = Ny = ∆u . 4.4. SPACE-BANDWIDTH AND SAMPLING RATE CONTROL 67

We will track the effects of each stage in our algorithm to the Wigner distribution boundary to which the original function is confined, and calculate the extents and the two space-bandwidth products and eventually the required structure of the sampling grid before each stage is performed. We will address each of the three stages (the first with 3 steps) given in Section 4.3 in sequence. To start, we first write down the 3-sphere boundary in

hyper-spherical coordinates. The 3-sphere Osp given in Eq. 4.7 can be transformed to the equivalent hyper-spherical coordinates with the following coordinate transformation:

ux cos φ1     uy ∆u sin φ1 cos φ2 O = = (4.20) sp   2    µx   sin φ1 sin φ2 cos φ3           µy   sin φ1 sin φ2 sin φ3          where angular hyperspherical coordinates φ1 and φ2 range over [0, π], and angular hyper-

spherical coordinate φ3 ranges over [0, 2π]. (Note that this coordinate system transformation is not unique.) The sum of the squares of the elements of the vector on the right-hand side of Eq. 4.20 again equals (∆u/2)2 as expected. Eq. 4.2 allows us to calculate the new

boundary sout of the Wigner distribution after any operation from the boundary sin before the operation. Just as the old boundary confined most of the energy of the signal represented by the Wigner distribution, so does the new boundary. This is because the mapping in Eq. 4.2 merely maps values of the Wigner distribution to new space-frequency points, and values which were confined within the old boundary remain confined within the new boundary.

4.4.1 The First Coordinate Rotator

At the very beginning of the algorithm, we start with the boundary vector sin = Osp.

In other words, the input boundary vector sin before the ﬁrst coordinate rotator is given 68 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

by Eq. 4.20. Then sout is found by multiplying sin with the transformation matrix of the coordinate rotator as

sout = Rr1 sin

cos(r1) sin(r1) 0 0 cos φ1     ∆u sin(r1) cos(r1)0 0 sin φ1 cos φ2 = − 2      0 0 cos(r1) sin(r1)   sin φ1 sin φ2 cos φ3           0 0 sin(r1) cos(r1)   sin φ1 sin φ2 sin φ3   −        cos(r1) cos φ1 + sin(r1) sin φ1 cos φ2   ∆u sin(r1) cos φ1 + cos(r1) sin φ1 cos φ2 = − (4.21) 2    cos(r1) sin φ1 sin φ2 cos φ3 + sin(r1) sin φ1 sin φ2 sin φ3       sin(r1) sin φ1 sin φ2 cos φ3 + cos(r1) sin φ1 sin φ2 sin φ3   −   

With φ1 and φ2 ranging over [0, π], and φ3 ranging over [0, 2π], sout represents the boundary of the output Wigner distribution. We can show that this boundary remains a 3-sphere of

∆u radius 2 by writing:

∆u u2 + u2 + µ2 + µ2 = ( )2[(cos(r ) cos φ + sin(r ) sin φ cos φ )2 x y x y 2 1 1 1 1 2 +( sin(r ) cos φ + cos(r ) sin φ cos φ )2 − 1 1 1 1 2 2 +(cos(r1) sin φ1 sin φ2 cos φ3 + sin(r1) sin φ1 sin φ2 sin φ3) +( sin(r ) sin φ sin φ cos φ + cos(r ) sin φ sin φ sin φ )2] − 1 1 2 3 1 1 2 3 ∆u = ( )2 (4.22) 2 as can be veriﬁed after some algebra with trigonometric functions. This result means that the coordinate rotation operation does not change the 3-sphere nature of the conﬁning boundary of the Wigner distribution, and since rotating an n-sphere (just like an ordinary sphere) does not change its extent along any direction, does not have any effect on 4.4. SPACE-BANDWIDTH AND SAMPLING RATE CONTROL 69

the space-bandwidth products. Therefore, no matter what the angles are, the coordinate rotation operation does not require a change in the number of samples and the sampling grid.

4.4.2 2D Separable Fractional Fourier Transform

Since the previous rotation operation left the Wigner distribution conﬁned within the original 3-sphere, and since we are interested only in the worst-case boundary, sin (sout of the preceding operation) can still be expressed as in Eq. 4.20. Then, new sout is found as:

sout = Fax,ay sin

cos(axπ/2) 0 sin(axπ/2) 0   ∆u 0 cos(ayπ/2) 0 sin(ayπ/2) = 2    sin(axπ/2) 0 cos(axπ/2) 0   −     0 sin(ayπ/2) 0 cos(ayπ/2)   −    cos φ1   sin φ1 cos φ2 ×    sin φ1 sin φ2 cos φ3       sin φ1 sin φ2 sin φ3      cos(axπ/2) cos φ1 + sin(axπ/2) sin φ1 sin φ2 cos φ3   ∆u cos(ayπ/2) sin φ1 cos φ2 + sin(ayπ/2) sin φ1 sin φ2 sin φ3 = (4.23) 2    sin(axπ/2) cos φ1 + cos(axπ/2) sin φ1 sin φ2 cos φ3   −     sin(ayπ/2) sin φ1 cos φ2 + cos(ayπ/2) sin φ1 sin φ2 sin φ3   −   

As in the coordinate rotator step, sout again deﬁnes the boundary of the output Wigner 2 2 2 2 2 distribution. Once again it deﬁnes a 3-sphere since ux + uy + µx + µy = (∆u/2) . This too can be easily shown by using simple algebra and trigonometric function properties. This is an expected result since FRT corresponds to rotation in joint space-frequency; if 70 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

the original conﬁnement region is an n-sphere, it remains an n-sphere after a 2D-S-FRT. Therefore, we again need not change the number of samples and sampling grid before the FRT step.

4.4.3 The Second Coordinate Rotator

Similar considerations as with the ﬁrst coordinate rotation apply so that an increase in the number of samples or a change in the sampling grid is not needed.

4.4.4 2D Scaling Operation

Up to the scaling stage, we do not have to worry at all about the sampling rate. The three steps which constitute the first stage have the effect of rotating the original 3-sphere, and the extent of the 4D Wigner distribution remains unchanged in all directions. We are able to track the confinement boundary through the steps precisely since we are able to write down the entire boundary parametrically by using hyper-spherical coordinates and since after each step, the transformed points still form a 3-sphere. In fact, the Wigner distribution of the signal is confined within the same 3-sphere as at the beginning. However, the scaling operation does not preserve the 3-sphere and thus it is very difficult to track all the points on the boundary since they may not constitute an easily trackable geometrical object by analytical and parametric means. Therefore, instead of tracking the infinite number of boundary points of our 3-sphere, we will use a tesseract (a 4-cube), which is basically the counterpart of an ordinary cube in R4, just as the 3-sphere is the counterpart of the ordinary sphere [115]. The unit tesseract is defined as

(x , x , x , x ) R4 : 1 x 1 . (4.24) { 1 2 3 4 ∈ − ≤ i ≤ } 4.4. SPACE-BANDWIDTH AND SAMPLING RATE CONTROL 71

It has 16 vertices and we will use these 16 points to track the Wigner distribution after the scaling operation. We take the smallest tesseract that contains the 3-Sphere within itself and use its 16 vertices to find the 16 vertices of the output. These 16 vertices define the maximum extents of the distribution and by employing them, we can safely define the worst-case boundary confining the Wigner distribution after the operation. Then the two space-bandwidth products can be calculated by finding, separately for each of the four coordinates, the maximum distances between the corresponding coordinates of the 16 vertices. Readers wishing to find a simpler example of such a streamlined procedure in a one-dimensional setting can refer to [61].

Let us represent, in R4, the coordinates of the 16 vertices of the tesseract of edge length ∆u (which is the smallest one conﬁning the 3-sphere with diameter ∆u) with columns of the matrix V:

11111111 1 1 1 1 1 1 1 1 − − − − − − − − ∆u  1 1 1 1 1 1 1 11 1 1 1 1 1 1 1  V = − − − − − − − − . 2    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   − − − − − − − −     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   − − − − − − − −   (4.25)

After the scaling operation is performed, these coordinates of the 16 input vertices are mapped to 16 new vertices, which we will hold in the columns of V as follows:

S 0 V = [v1 v2 v3 ... v15 v16]= V (4.26)  0 S−1    4 where vi (i =1, 2, ..., 16) are vectors in R that hold the coordinates of the scaled vertices. Then, we need to ﬁnd the coordinate-wise distances for every possible combination of pairs of vertices, for each of the four coordinates separately. There are 120 possible combinations of pairs out of 16 vectors. We calculate the distances between their coordinates and denote 72 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

this with di,j as

vi(1) vj(1) | − |   vi(2) vj(2) d = | − | (4.27) i,j    vi(3) vj(3)   | − |     vi(4) vj(4)   | − |    and then construct the 4 120 distance matrix D ×

D = [d1,2 d1,3 ... d1,16 d2,3 d2,4 .... d2,16 .... d15,16] (4.28) where i = 1, 2, ..., 15 and j = i +1, ...., 16. By using D, we can deﬁne the sampling grid and sampling rates that are necessary to represent the function without any information loss. The extent of the function along ux and uy should be max(D1,1,D1,2,D1,3, ..., D1,120) and max(D2,1,D2,2,D2,3, ..., D2,120), respectively. On the intervals along ux and uy given above, the samples should be taken with intersample spacings of (max(D3,1,D3,2,D3,3, −1 −1 ..., D3,120)) and (max(D4,1,D4,2,D4,3, ..., D4,120)) , respectively. The corresponding space-bandwidth products are then equal to

N = max(D ,D ,D , ..., D ) max(D ,D ,D , ..., D )(4.29) Sx 1,1 1,2 1,3 1,120 × 3,1 3,2 3,3 3,120 N = max(D ,D ,D , ..., D ) max(D ,D ,D , ..., D )(4.30) Sy 2,1 2,2 2,3 2,120 × 4,1 4,2 4,3 4,120

and the total necessary number of samples after the scaling is given by N = NSxNSy. Remember that the number of samples should be increased to this number N = N N e Sx Sy before the scaling operation is performed (the minimum appropriate integer number of e samples greater than the calculated values may be used for simplicity). The determined number of samples should be uniformly spread so as to snugly ﬁt the original extents (thus they will be spaced closer than the original samples). After the scaling operation is performed by using the matrix S, these samples are transformed to new and extended positions 4.4. SPACE-BANDWIDTH AND SAMPLING RATE CONTROL 73

as predicted by the above calculations. Finally, though not a necessity, a similar simple interpolation may be employed (as done after the coordinate rotations) to carry the samples from these transformed locations to the regular grid within the predicted extents. This may facilitate implementation of the next operation.

4.4.5 2D Chirp Multiplication

The 2D-CM operation is the stage that is mainly burdened with any shears which may be inherent in the 2D-NS-LCT to be computed. Such shears may considerably increase the space-bandwidth products of the function. These increases are unavoidable if these elongating distortions in space-frequency are part of the 2D-NS-LCT which we wish to compute. This will in turn require an increase in the number of samples if we wish to be able to reconstruct the continuous output function without any information loss. Therefore, as in the previous subsection, we must increase the number of samples prior to the chirp multiplication operation. The vertices obtained as a result of the scaling operation are taken as the starting vertices for the 2D-CM operation. We begin with the coordinates of these vertices, denoted by V, and determine what happens to them as a result of the 2D-CM operation, and calculate the new difference matrix D by using the following equation along with Eqs. 4.27, 4.28, 4.29, and 4.30:

I 0 V = [v1 v2 v3 ... v15 v16]= V. (4.31)  G I  −   Finally, the sampling extents, rates, and locations can be determined similarly as in the scaling stage. After the number of samples has been increased, the 2D-CM stage can be safely performed to complete the entire transformation. 74 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

4.4.6 Summary of the Algorithm

Having explained all the stages in detail, we summarize the entire algorithm stage by stage. The algorithm can be compactly stated in operator notation as follows:

M = G K S K J (4.32) C Q G M S Rr2 Fax,ay Rr1

where the operators G, S, , , respectively represent: 2D-CM with pa- Q M Rr2 Fax,ay Rr1 rameter matrix G, 2D scaling with parameter matrix S, coordinate rotation with angle r2,

2D-S-FRT with orders ax and ay, and coordinate rotation with angle r1. J stands for a simple interpolation without oversampling that is performed to obtain the function on a regular rectangular grid from the rotated samples. KS and KG stand for the interpolation operations before the scaling and chirp multiplication operations, respectively. Beyond the task of J, these also increase the number of samples as explained in Section 4.4.4.4.4 and Section 4.4.4.4.5. The algorithm can be summarized as follows:

1. Normalize the input ﬁeld (function) as explained in Section 5.2.4.2.1 and obtain the input samples.

2. Given the transform matrix M, obtain the chirp multiplication (G) and scaling (S)

matrices, the coordinate rotation angles (r1 and r2) and FRT orders (ax and ay) by using Eqs. 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, and 4.17.

3. Perform the ﬁrst coordinate rotation and obtain the samples on a regular grid by simple interpolation.

4. Use the fast algorithm for 1D-FRTs to implement the 2D-S-FRT by successively applying 1D-FRTs along the two dimensions. 4.5. NUMERICAL RESULTS 75

5. Perform the second coordinate rotation and obtain the samples on a regular grid by simple interpolation.

6. Use the method given in Section 4.4.4.4.4 to obtain the necessary number of samples before the 2D scaling operation, perform the oversampling and then apply the scaling operation. Optionally, go back to a regular rectangular grid by simple interpolation after scaling has changed the locations of the samples to a non-rectangular grid.

7. Use the method given in Section 4.4.4.4.5 to obtain the necessary number of samples before the 2D chirp multiplication operation, perform the oversampling and then apply the chirp multiplication operation.

4.5 Numerical Results

Here we report numerical results for some example functions and transforms in order to demonstrate the performance and accuracy of our algorithm. We also discuss sources of error in our algorithm and the effect of interpolation methods on the error. As example input functions, we consider the 2D Gaussian field exp( π(x2 + y2)) and denote it with F1, the − 2D Chirped-Gaussian field exp( π(x2 + y2)) exp( iπ(x2 + y2)) and denote it with F2, − × − a 2D Non-Symmetric Chirped Gaussian field exp( π(3x2 + y2)) exp( iπ(x2 +2y2)) − × − and denote it with F3. All these first three input fields are sampled on a 64 64 grid. × Additionally, we also consider a more challenging function exhibiting discontinuities and larger frequency extents depicted in Fig. 4.1. This S-shaped function is denoted with F4, and is sampled on a 256 256 grid. We consider two different arbitrarily chosen 2D- × NS-LCTs, the first one (T1) has a parameter set (αx, βx,γx, αy, βy,γy, ηx, ηy, ηα, ηγ) = ( 3, 2, 1, 2, 3, 4, 0.1, 0.2, 1, 0.1) and the second one (T2) has a parameter set (1, 2, 3, − − − − 2, 1, 0.8, 0.6, 0.5, 0.3, 0.4). As a result of the space-bandwidth and sampling rate − − − − − control procedure presented in Section 4.4 and for the given number of initial samples, the 76 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

output ﬁelds are obtained by the algorithm on 141 166 and 740 211 sampling grids for × × T1 and T2 respectively, for the input functions F1, F2, and F3. For the input function F4, the output grids are 563 663 and 2958 842 for T1 and T2, respectively. T1 is of such × × a nature that it requires a relatively small amount of oversampling whereas T2 is of such a nature that it requires a relatively large amount of oversampling. These oversamplings are necessary to be able to recover the continuous output from the output samples produced by the algorithm.

8 1

0.9 6 0.8 4 0.7 2 0.6

0 0.5

0.4 −2 0.3 −4 0.2 −6 0.1

0 −8 −6 −4 −2 0 2 4 6

Figure 4.1: Example function F4

The 2D-NS-LCTs (T1 and T2) of the functions F1, F2, F3, F4 have been calculated both by the presented fast algorithm and by an extremely ﬁnely tuned and inefﬁcient brute force numerical approach based on the 2D Simpson’s method [119] which we use as an accurate reference. See Appendix A for more details of the Simpson’s method. The results for 4.5. NUMERICAL RESULTS 77

T1 T2 DFT F1 2.25 10−3 3.82 10−4 2.12 10−23 F2 1.12 × 10−2 1.09 × 10−3 2.02 × 10−21 F3 7.17 × 10−2 3.21 × 10−3 2.58 × 10−8 × × × F4 2.07 1.92 1.05 Table 4.1: Percentage errors for different functions F and transforms T.

T1 of (F1, F2, F4) and T2 of (F3, F4) along with the corresponding brute force reference results are plotted in Figures 4.2 - 4.6. The error percentages for all functions (F1, F2, F3, F4) are tabulated in Table 4.1, for both transforms T1 and T2. There are no visible differences for F1, F2, F3 and a very small visible difference for F4. We define the error as the energy of the difference of the two results normalized by the energy of the reference, expressed as a percentage. The tabulated error percentages show that the presented fast algorithm is very accurate. Another important observation from Table 4.1 is that the error does not depend so much on the transform parameter set as it does on the transformed function; the error percentages for T1 and T2 are close to each other. In general, our algorithm maintains approximately the same performance over different transforms. A similar conclusion was reached for the 1D case [63,64]. To the best of our knowledge the presented algorithm is the first fast and accurate algorithm that is capable of computing the very general class of 2D-NS-LCTs and the first generalization of the one-dimensional fast algorithms for LCTs to two dimensions. Moreover, it also deals with the space-bandwidth and sampling rate issues very carefully so that the output samples—indeed the samples at any stage—are sufficient to accurately reconstruct the underlying continuous function, but are not wastefully redundant either. Therefore our algorithm is able to effectively obtain a continuous transform from a continuous input function. In Table 4.1, we also show the errors that arise when the DFT is used to approximately compute the ordinary 2D Fourier transform (2D-FT) of the same functions. (The DFT would most likely be implemented with the FFT algorithm but how the DFT is implemented does not effect the error comparison.) The same reference method that we use in 78 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

Re of T1 of F1 − Our Algorithm Re of T1 of F1 − Reference 0.9 0.9

1 0.8 1 0.8

0.7 0.7

0.5 0.6 0.5 0.6

0.5 0.5

y 0 0.4 y 0 0.4

0.3 0.3

−0.5 0.2 −0.5 0.2

0.1 0.1

−1 0 −1 0

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x (a) Real part (fast algorithm) (b) Real part (reference)

Im of T1 of F1 − Our Algorithm Im of T1 of F1 − Reference

1 0.2 1 0.2

0.1 0.1 0.5 0.5

0 0

y 0 y 0

−0.1 −0.1 −0.5 −0.5

−0.2 −0.2

−1 −1 −0.3 −0.3 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x (c) Imaginary part (fast algorithm)) (d) Imaginary part (reference)

Figure 4.2: T1 of F1 (our algorithm and reference) 4.5. NUMERICAL RESULTS 79

Re of T1 of F2 − Our Algorithm Re of T1 of F2 − Reference 1.5 1.5

0.8 0.8

1 0.7 1 0.7

0.6 0.6

0.5 0.5 0.5 0.5

0.4 0.4

y 0 0.3 y 0 0.3

0.2 0.2 −0.5 −0.5 0.1 0.1

0 0 −1 −1 −0.1 −0.1

−0.2 −0.2 −1.5 −1.5 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x x (a) Real part (fast algorithm) (b) Real part (reference)

Im of T1 of F2 − Our Algorithm Im of T1 of F2 − Reference 1.5 0.1 1.5 0.1

1 0 1 0

0.5 −0.1 0.5 −0.1

y 0 −0.2 y 0 −0.2

−0.5 −0.3 −0.5 −0.3

−1 −0.4 −1 −0.4

−1.5 −1.5 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x x (c) Imaginary part (fast algorithm) (d) Imaginary part (reference)

Figure 4.3: T1 of F2 (our algorithm and reference) 80 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

Re of T2 of F3 − Our Algorithm Re of T2 of F3 − Reference 2 0.3 2 0.3

1.5 1.5 0.2 0.2

1 1 0.1 0.1 0.5 0.5 0 0

y 0 y 0

−0.5 −0.1 −0.5 −0.1

−1 −0.2 −1 −0.2

−1.5 −1.5 −0.3 −0.3 −2 −2 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x (a) Real part (fast algorithm) (b) Real part (reference)

Im of T2 of F3 − Our Algorithm Im of T2 of F3 − Reference 2 0.3 2 0.3

1.5 1.5 0.2 0.2

1 1 0.1 0.1 0.5 0.5 0 0

y 0 y 0

−0.1 −0.1 −0.5 −0.5

−1 −0.2 −1 −0.2

−1.5 −0.3 −1.5 −0.3

−2 −2 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x (c) Imaginary part (fast algorithm) (d) Imaginary part (reference)

Figure 4.4: T2 of F3 (our algorithm and reference) 4.5. NUMERICAL RESULTS 81

Abs of T1 of F4 − Our Algorithm Abs of T1 of F4 − Reference 1.5 1.5 10 10

5 5 1 1

y 0 y 0

0.5 0.5 −5 −5

−10 −10 0 0 −5 0 5 −5 0 5 x x (a) Magnitude (fast algorithm) (b) Magnitude (reference)

Phase of T1 of F4 − Our Algorithm Phase of T1 of F4 − Reference 3 3 10 10 2 2 5 5 1 1

y 0 0 y 0 0

−1 −1 −5 −5 −2 −2 −10 −10 −3 −3 −5 0 5 −5 0 5 x x (c) Phase (fast algorithm) (d) Phase (reference)

Figure 4.5: T1 of F4 (our algorithm and reference) 82 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

Abs of T2 of F4 − Our Algorithm Abs of T2 of F4 − Reference 15 1.5 15 1.5

10 10

5 1 5 1

y 0 y 0

−5 0.5 −5 0.5

−10 −10

−15 0 −15 0 −10 −5 0 5 10 −10 −5 0 5 10 x x (a) Magnitude (fast algorithm) (b) Magnitude (reference)

Phase of T2 of F4 − Our Algorithm Phase of T2 of F4 − Reference 15 3 15 3

10 2 10 2

5 1 5 1

y 0 0 y 0 0

−5 −1 −5 −1

−10 −2 −10 −2

−15 −3 −15 −3 −10 −5 0 5 10 −10 −5 0 5 10 x x (c) Phase (fast algorithm) (d) Phase (reference)

Figure 4.6: T2 of F4 (our algorithm and reference) 4.5. NUMERICAL RESULTS 83

calculating the error percentages for our algorithm is used to numerically calculate “exact” continuous Fourier transforms of the example functions. The DFT serves as an ultimate benchmark for comparing our results. Theoretically, our algorithm cannot reduce the error below that value which results from computing a Fourier transform with the DFT because they share the same inevitable source of error that arises from the fundamental fact that a signal and its transform cannot both be of ﬁnite extent. In the one-dimensional version of our algorithm, as well as the separable two-dimensional case, it is possible to achieve errors which approach that for the DFT, and which are thus the best which one may ever hope to obtain [63,64]. Unfortunately, the necessity of interpolation in the two-dimensional case does not allowthis, but still it is possibleto achieve very low errors that would be acceptable in most applications.

The key observations that can be made from this table are as follows. The resulting errors depend strongly on the function and the assumed space and spatial-frequency extents. Indeed, this is the main determinant of the error for a given interpolation method. Different functions have differing degrees of decay rates of their tails, and for given assumed extents, different amounts of energy left out of the extents. Since a function cannot be made to contain 100% of its energy in both the space and spatial-frequency domains, a compro- mise between error and computational complexity is necessary. If we choose the extents within which we assume the function and its Fourier transform to be mostly contained in a conservative manner, the extents will be relatively large and the number of samples will be relatively large. If we economize on the extents and the number of samples, a relatively large fraction of the energy will be left outside and the resulting error will be large. Among our examples, F4 is an example where the space-bandwidth product has been chosen less conservatively than the other examples, and therefore the error is relatively large around 2%. The error can be reduced by increasing the number of samples taken.

From a fundamental perspective, our algorithm is supposed to compute 2D-NS-LCTs with a performance similar to the DFT in computing the Fourier transform. As noted, this 84 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

F1 F2 F3 F4 Nearest 3.4 10−3 1.75 10−1 6.18 10−1 11.3 Bilinear 1.02× 10−2 5.04 × 10−2 2.66 × 10−1 4.33 Cubic 2.25 × 10−3 1.12 × 10−2 7.17 × 10−2 2.07 × × × Table 4.2: Percentage errors for different interpolation methods and functions F for T1.

is achieved for separable transforms which reduce to one-dimensional transforms. How- ever, in the general non-separable case, although our algorithm is quite accurate, for the ﬁrst three functions, its accuracy is quite below that for the DFT. This degradation is due to the complex and challenging nature of non-separable LCTs which forces us to employ interpolation operations from irregular grids to regular grids. This is an important source of error. If needed, to reduce this interpolation error, one can use and implement more advanced, sophisticated and accurate interpolation methods. However, this is totally different area of research and beyond the scope of this thesis. For F4, our algorithm has a comparable accuracy with the DFT. This is due to the fact that in this case, the error is a result of the signiﬁcant amount of signal energy that lies outside the assumed space and spatial-frequency extents. This source of error, which affects both our algorithm and the DFT in the same way, dominates the error arising from interpolation (which affects only the non-separable LCT computation), so that the results are similar. On the other hand, for the other functions, the interpolation error (which does not affect the DFT) results in higher errors for the LCT computations as compared to the DFT.

To be more conﬁdent in the above claims, we also studied the effects of the method of interpolation on our algorithm and studied how they change the accuracy of the algorithm. We employed in our algorithm nearest neighbor, bilinear, and cubic interpolation methods because they are among the most standard, mainstream, and efﬁcient methods, [117,118]. Different versions of the algorithm have been implemented by using each of the above methods. The error percentages resulting from the use of different interpolation methods are tabulated in Tables 4.2 and 4.3 for T1 and T2, respectively. 4.6. CONCLUDING REMARKS 85

F1 F2 F3 F4 Nearest 1.72 10−1 3.28 10−1 4.59 10−1 11.24 Bilinear 1.78 × 10−2 5.58 × 10−2 8.4 ×10−2 6.24 Cubic 3.82 × 10−4 1.09 × 10−3 3.21× 10−3 1.92 × × × Table 4.3: Percentage errors for different interpolation methods and functions F for T2.

As can be seen from the tabulated data, the error values are affected considerably by the interpolation method chosen. The best results are obtained when we use the cubic interpolation method, which is the most advanced among the three. Since there are essentially two sources of error, the one that is fundamental equally affecting LCTs and the DFT, we are not surprised to observe that as the quality of the interpolation is increased, the accuracy of the algorithm improves and approaches to the DFT benchmark. The results of our fast algorithm were obtained within a couple of seconds by using MATLAB code running on a standard personal computer. The calculation of the brute force reference results took several days.

4.6 Concluding Remarks

We presented an algorithm for the fast digital computation of the most general family of two-dimensional non-separable linear canonical transforms. This family of transform integrals represents a quite general class of two-dimensional quadratic-phase systems in optics. Our approach is based on concepts from signal analysis and processing rather than conventional numerical analysis. With careful consideration of sampling issues, the number of samples M N of the sampling grid can be chosen very close to the space-bandwidth × product of the functions. A naive approach based on examination of the frequency content of the integral kernels would, on the other hand, result in an unnecessarily high number of samples being taken due to the highly oscillatory nature of the kernels, which would not only be representationally inefﬁcient but also increase computation time and storage 86 CHAPTER 4. THE ALGORITHM FOR 2D QUADRATIC-PHASE SYSTEMS

requirements. The transform output may have a higher space-bandwidth product than the input due to the nature of the transform family. Through careful space-bandwidth tracking and control, we can assure that the output samples obtained are accurate approximations to the true ones and that they are sufficient (but not unnecessarily redundant) in the Nyquist- Shannon sense, allowing full reconstruction of the underlying continuous function. The algorithm takes the samples of the input function and maps them to the samples of the continuous2D-NS-LCT of this function in the same sense that the fast Fourier transform (FFT) implementation of the discrete Fourier transform (DFT) computes the samples of the continuous FT of a function. A second source of error which was not of substantial impact in the 1D case or the separable 2D case but which is significant in the non-separable 2D case arises from the necessity to carry out interpolations to revert samples on rotated grids to the original rectangular grid. This error depends on how accurately the interpolation operation is handled. We have used well-established and standard methods for interpolation that are readily available, since advancing methods of interpolation is beyond the scope of this thesis. While we believe the levels of accuracy attained with these interpolation methods will be sufficient for most applications, in those cases where they are not, more efficient and customized interpolation methods for non-rectangular grids can be utilized to further improve accuracy. Chapter 5

The Algorithm for Complex Quadratic-Phase Systems

5.1 Introduction1

INEARCANONICALTRANSFORMS with real parameters have received consider- L ably more attention than LCTs with complex parameters [8]. Real linear canonical transforms (RLCTs) are unitary mappings between the elements of Hilbert space of square integrable functions of a variable in R. RLCTs are represented by 2 2 unimodular real × matrices a b MR = , (5.1)  c d    with determinant equal to 1, where a,b,c,d are real. The parameter matrices MR form the real symplectic group Sp(2, R) with three independent parameters, [26]. RLCTs are of great importance in electromagnetic, acoustic, and other wave propagation problems since they represent the solutionof the wave equation under a variety of circumstances. At optical frequencies, RLCTs can model a broad class of lossless optical systems including thin

87 88CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

lenses, sections of free space in the Fresnel approximation, sections of quadratic graded- index media, and arbitrary concatenations of any number of these, sometimes referred to as ﬁrst-order optical systems or quadratic-phase systems (QPS) [1,5,6,10,12,17,19,20,22, 23,25,33,34].

Extension of RLCTs to complex linear canonical transforms (CLCTs) is rather involved [8,37,121,122]. The extension is very brieﬂy summarized as follows. When we let the entries of the unimodular transform matrices be complex numbers, we obtain the unit determinant matrices a b MC = , (5.2)  c d    where a,b,c,d are complex parameters. The matrices MC form the complex symplectic group Sp(2, C) with six independent parameters [122]. However, CLCTs represented by this symplectic group can no longer be established as a unitary mapping between the Hilbert space of square integrable functions in R. Instead, we have a mapping from the Hilbert space of square integrable functions of a real variable, to analytic functions of a complex variable on the complex plane in the Bargmann-Hilbert space of square integrable functions, [123,124], as established in [8,37,121,122]. The CLCTs are required to be bounded but not necessarily unitary, in which case we need to represent CLCTs with a semigroup HSp(2, C) within the group Sp(2, C). More on the mathematical foundations and theory of CLCTs can be found in [8,37,121,122].

The distinguishing feature of our approach is the way our algorithm carefully addresses sampling and space-bandwidth product issues from an information-theoretical perspective. Special care is taken to ensure that the output samples represent the continuous transform in the Nyquist-Shannon sense during every stage of the algorithm, so that the continuous transform can be fully recovered from the samples. Despite the highly oscillatory nature of the integral kernel, we carefully manage the sampling rate so as to ensure that the number of samples used is sufﬁcient, but not much larger than the space-bandwidth product of 5.1. INTRODUCTION 89

the input signal, so that the algorithms are as efficient as possible. The straightforward method of sampling the input field and the kernel, and then calculating the output field is not suitable for several reasons. First of all, due to the highly oscillatory nature of the integral kernel, a naive application of the Nyquist sampling theorem to determine the sampling rate would result in an excessively large number of samples and inefficient computation. On the other hand, ignoring the oscillations of the kernel and determining the sampling rate according to the input field alone may cause under-representation of the output field in the Nyquist-Shannon sense. This unacceptable situation arises due to the fact that the particular 2D-LCT that we are calculating may increase the space-bandwidth product in one or both of the dimensions. If we do not increase the number of samples that we are working with so as to compensate for this increase, there will be information loss and we will not be able to recover the true transformed output from our computed samples. The computation of complex LCTs involves a number of issues which do not arise in the case of real LCTs. The decompositions employ complex chirp multiplications whose effect on the Wigner distribution must be clarified to ensure proper space-bandwidth tracking and control.

Complex-parametered LCTs allow several kinds of optical systems to be represented, including lossy as well as lossless ones. When complex parameters are involved, LCTs may no longer be unitary and boundedness issues may arise. The decomposition of general CLCTs into Fourier transforms and real and imaginary chirp multiplications allows us to derive conditions on the transform parameters that ensure boundedness.

We also need to ﬁnd the conditions under which HSp(2, C) can be constructed as a mapping from R R. This is because we are interested in optical applications of CLCTs → where the inputs and outputs are functions of real spatial variables. Such CLCTs which map functions over Hilbert spaces from the real line to the real line are called passive CLCTs in [37], whereas CLCTs that map functions from the real line to analytic functions on complex Bargmann-Hilbert spaces are called active CLCTs. Thus the eligible parameters 90CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

also depend on how the HSp(2, C) semigroup is constructed for R R. Wolf derived → the parameter spaces for which the CLCTs represent a mapping from R R and the → output is bounded [37]. However, this construction excludes some optically important special cases like Gaussian apertures. The decompositions we use allow the derivation of conditions which do not exclude Gaussian apertures. Speciﬁcation of such conditions were not necessary in the RLCT case.

The chapter is organized as follows. Section 5.2 presents some mathematical preliminaries that we use in the derivation of our algorithm for the complex case and review some special and important CLCTs in optics. In Section 5.3, we present a careful analysis of every possible case the complex transform matrix may assume and present fast algorithms based on decompositions into generalized chirp multiplications, real scalings, and FTs. We also determine whether a given CLCT represents an optically possible bounded input- output relationship from the real line to the real line. Numerical examples to demonstrate the accuracy of the algorithm are given in Section 5.4. Finally we conclude in Section 5.5.

The CLCT of f(u) with complex parameter matrix MC is denoted as fMC (u) =

( M f)(u): C C

∞ ′ ′ ′ ( M f)(u)= K (u,u )f(u ) du , C C C Z−∞ K (u,u′)= e−iπ/4 β exp iπ(αu2 2βuu′ + γu′2) , (5.3) C − q ′ where α, β, γ are complex parameters independent of u and u and where M is the CLCT C C operator. MC again has unit-determinant and is given by

a b ar + iac br + ibc γ/β 1/β MC = = = (5.4)  c d   c + ic d + id   β + αγ/β α/β  r c r c −       where ar, ac, br, bc, cr, cc, dr, dc are real numbers. The overline over the parameters α, β, 5.2. PRELIMINARIES 91

γ is to emphasize that these parameters are now complex, corresponding to a total of 6 real

parameters: α = αr + iαc, β = βr + iβc, γ = γr + iγc. In terms of these parameters the kernel KC can be rewritten as

2 ′ ′2 2 ′ ′2 ′ −iπ/4 iπ(αru −2βruu +γru ) −π(αcu −2βcuu +γcu ) KC(u,u )= e βr + iβc e e . (5.5) p The bidirectional relationship between the α, β, γ parameters and the matrix entries are given as follows:

drbr + dcbc dcbr drbc αr = 2 2 , αc = 2 − 2 br + bc br + bc br bc βr = 2 2 , βc = 2− 2 br + bc br + bc a b + a b a b a b r r c c c r r c (5.6) γr = 2 2 , γc = 2 − 2 br + bc br + bc

βcγc + βrγr βrγc βcγr ar = 2 2 , ac = 2 − 2 βr + βc βr + βc βr βc br = 2 2 , bc = 2− 2 βr + βc βr + βc β α + β α α β α β c c r r c r r c (5.7) dr = 2 2 , dc = 2 − 2 βr + βc βr + βc

5.2 Preliminaries

5.2.1 Wigner Distributions

The Wigner distribution Wf (u,µ) of a signal f(u) can be deﬁned as follows [97, 98]:

∞ ′ W (u,µ)= f(u + u′/2)f ∗(u u′/2)e−2πiµu du′. (5.8) f − Z−∞ Roughly speaking, W (u,µ) is a function which gives the distribution of signal energy over ∞ ∞ space and frequency. Its integral over space and frequency, −∞ −∞ W (u,µ) dudµ, gives the signal energy. R R

Let f denote a signal and fM be its LCT with parameter matrix M. Then, the Wigner distribution (WD) of fM can be expressed in terms of the WD of f as [10]

W (u,µ)= W (du bµ, cu + aµ). (5.9) fM f − −

This means that the WD of the transformed signal is a linearly distorted version of the original distribution. The Jacobian of this coordinate transformation is equal to the determinant of the matrix M, which is unity. Therefore this coordinate transformation does not change the support area of the Wigner distribution. The invariance of the support area means that LCTs do not concentrate or deconcentrate energy. The support area of the Wigner distribution can also be approximately interpreted as the number of degrees of freedom of the signal. Therefore, the number of samples needed to represent the signal does not change after a real LCT operation.

For the purpose of space-bandwidth tracking as employed in our algorithm, we do not require a full characterization of the effects of CLCTs on the WD. However, we do need to know the effect of multiplying a function with another function on the WD to derive a space-bandwidth product tracking method for CLCTs. This multiplication property is not required in deriving our previous algorithms for real LCTs, [49,64] but will be necessary in our CLCT algorithm. The WD has the following multiplication property [10]: let h(u) and f(u) be two functions and let Wh(u,µ) and Wf (u,µ) be their corresponding WDs. Then 5.2. PRELIMINARIES 93

h(u)f(u) has the WD given by

W (u,µ µ′)W (u,µ′) dµ′. (5.10) h − f Z In other words, when two functions are multiplied, the WD of the resulting function is given by the convolution of the WDs of the initial two functions along the frequency dimension.

5.2.2 CLCTs in Optics and Special CLCTs

Magniﬁcation (scaling), Fourier transformation (FT), real fractional Fourier transformation (RFRT), real chirp multiplication (CM), complex chirp multiplication (CCM), Gauss- Weierstrass Transform, complex-ordered fractional Fourier transformation (CFRT) are all special cases of CLCTs that have optical realizations. Scaling, FT, RFRT, and CM, which have real parameters, belong to the narrower class of RLCTs and have been reviewed in [64]. In this Section, we only review complex parametered cases that are essential for our development.

5.2.2.1 Complex Scaling (Magniﬁcation)

Simple scaling with a real parameter M is an operation which corresponds to optical mag- niﬁcation [64]. If the parameter M is allowed to be complex, we obtain the complex scaling operation, which is a special case of CLCTs. With complex scaling, the real axis on which the input function is deﬁned is mapped to a straight line in the complex plane passing through the origin and making an angle arg(M) with the real axis. The mapping becomes R C. The interpretation of complex scaling in quantum mechanics has been discussed → in [125,126], while an interpretation from a signal processing perspective can be found in [127]. However, we are not aware of the realization and application of complex scaling from a purely optical point of view. 94CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

5.2.2.2 Gaussian Apertures (Complex Chirp Multiplication)

Gaussian apertures, also called soft apertures, are a special case of CLCTs. They are actually chirp multiplications with a complex parameter and are the complex counterparts of CM operations. We will hereafter refer to them as complex chirp multiplication (CCM). The deﬁnition of CCM is similar to the deﬁnition of real CM, where we replace the chirp parameter with a purely imaginary complex parameter:

πqu2 Q f(u)= f(u)= e f(u), (5.11) C iq Qiq

−1 1 0 1 0 Qiq = = . (5.12)  iq 1   iq 1  −     where q is real and so iq is a purely imaginary parameter. It essentially behaves like a multiplicative ﬁlter where the transmission is dependent on the transverse dimension quadratic- exponentially. To exclude the unbounded case we require q 0. ≤ We now discuss the effect of complex chirp multiplication (CCM) on the WD. We need this result in order to track and control the space-bandwidth product of CLCTs. This result is not needed in algorithms for RLCTs because there are no CCM stages in RLCTs, and is of a considerably different nature than the operations employed there. We use the property

πqu2 given in Eq. 5.10 with h(u)= e . TheWD of h(u), denoted Wh(u,µ), can be obtained by directly using the deﬁnition of the WD (Eq. 5.8):

2 2 2π µ2 W (u,µ)= e2πqu e q , q< 0. (5.13) h q r− The WD of the Gaussian function is a 2D Gaussian function in the space-frequency plane. Since q < 0, this function decays with increasing u and µ. Therefore, we can specify a rectangular region which contains almost all of the energy of the function. We will choose the extents of this rectangle to correspond to plus/minus four standard deviations of 5.2. PRELIMINARIES 95

the Gaussian in both the space and frequency dimensions, which deﬁnes a rectangle with extents

g = 16/π q 1 | | g = p16 q /π (5.14) 2 | | p in the space dimension and the frequency dimension, respectively. When the WD of the input function and the WD of the Gaussian function are convolved along the µ direction to ﬁnd the resulting WD of the output function (as illustrated in Fig. 5.1), the resulting space extent of the support of the output WD will be given by min(d1,g1) and the resulting frequency extent will be given by d2 + g2 where d1 and d2 are the space and spatial-frequency extents of the input function.

5.2.2.3 Gauss-Weierstrass Transform

The Gauss-Weierstrass transform with parameter t is given by the integral transform [8],

∞ − ′ 2 − π(u u ) ′ ′ tf(u)= 1/t e t f(u ) du . (5.15) G −∞ p Z It gives the solution of the heat equation. The complex chirp convolution (CCC) operation, which is a special case of CLCTs, is represented by the transform matrix

1 ir Rir = (5.16)  0 1    and is equivalent to convolution by a Gaussian function:

iπ/4 2 R f(u)= f(u)= f(u) e 1/r exp(πu /r). (5.17) C ir Rir ∗ p 96CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

(a) The input function (b) Rectangle containing WD of Gaussian function

Figure 5.1: The effect of the CCM operation on the WD.

We observe that CCC is the same as the Gauss-Weierstrass transform when we choose the CCC parameter r = t. As in thecase of theFT, there is again the inconsequentialconstant − phase factor e−iπ/4 difference between the two deﬁnitions. CCC operations are covered by our algorithm since they are a special case of CLTs. CCC operations are most conveniently calculated by expressing them as an FT followed by a CCM operation followed by an 5.2. PRELIMINARIES 97

inverse FT.

The combined effect of two CCM (or CCC) operations following each other is again a CCM (or CCC) operation, whose parameter is found by summing the parameters of the two constituent operations. If two CCM operations with real or complex parameters q1 and q2 follow each other, the equivalent operation is a new CCM operation with parameter q1 +q2.

If two CCC operations with real or complex parameters r1 and r2 follow each other, the equivalent operation is a new CCC operation with parameter r1 + r2.

5.2.2.4 Complex-ordered Fractional Fourier Transform

The ath order real fractional Fourier transform (RFRT, or simply FRT) is well studied in the literature, [10,13,27, 101–107]. Complex FRTs are FRTs whose order parameter is complex [41–45].

When the order is an imaginary number ib, then we obtain the following special case of CLCTs with the transform matrix

ib cosh(bπ/2) i sinh(bπ/2) Flc = , (5.18)  i sinh(bπ/2) cosh(bπ/2)  −   which again differs only by the factor ebπ/4 from fractional Fourier transforms as commonly deﬁned:

ib bπ/4 ib Fib f(u)= lc f(u)= e f(u). (5.19) C lc F F Since FRTs are additive in index, a real-ordered and a purely imaginary-ordered FRT can be combined as a+ib = a ib to yield a general complex-ordered FRT, where the complex F F F order may be denoted by ac = a + ib. CFRTs can be optically realized by using thin lenses, free-space propagations, and Gaussian apertures, or by combination of Gauss-Weierstrass transforms with Gaussian apertures [41]. 98CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

5.3 The Algorithm

We now show how given ABCD matrices can be decomposed in a manner that leads to a fast algorithm for computation of CLCTs. In the most general case, the matrix MC is composed of the four complex parameters a,b,c,d, whose real and imaginary parts add up to a total of 8 parameters. These 8 parameters are restricted by the unimodularity condition on MC, which requires the real part of the determinant to be 1 and the imaginary part to be 0. Because of these two equations, the total number of independent parameters of a general CLCT is 6. These 6 parameters correspond to the 6 parameters of the group HSp(2, C), which is a 6 parameter semigroup of the complex symplectic group Sp(2, C). Before giving the main decomposition which covers the general case, we start with a special case whose treatment is straightforward:

5.3.1 b =0 case:

When b = 0, the unimodularity requirement requires a = 0 and the transform output can 6 be written as 1 jcy2 ( M f)(u)= e 2a f(y/a). (5.20) C C √a In this case, the output is given by a scaling operation with parameter a followed by a chirp multiplication operation with parameter c/2a. We will restrict ourselves to the − case where a is real, since only in this case will the scaling operation result in a R R → mapping. The case where a is complex produces complex scaling operations and therefore causes mappings from functions on the real line to functions on the complex plane. This case would require special treatment, which we do not attempt since we are not aware of any optical realization or application of such transforms. Also necessary is the condition Im(c/a), which is necessary to ensure boundedness. In order to have a bounded and R R → mapping, it becomes necessary for a to be real and Im(c/a) 0. Together with the unit ≥ 5.3. THE ALGORITHM 99

determinant condition, these conditions can be explicitly summarized as follows:

a = 0 r 6 d = 1/a

ac = 0

a c 0 (5.21) r c ≥

where the first two are intrinsically required to define any LCT (det MC = 1) and the last two are required to obtain a bounded R R mapping. When the conditions in 5.21 are → satisfied, the matrix MC can be decomposed as:

ar 0 M = C   c 1/ar   1 0 ar 0 =     c/ar 1 0 1/ar     1 0 1 0 ar 0 = . (5.22)       cr/ar 1 icc/ar 1 0 1/ar       The above decomposition can be used for the fast computation of the special case b =0.

5.3.2 b =0 case: 6

Now, we turn our attention to the more general case in which the following decomposition will be the basis of our fast algorithm:

1 0 1 0 0 1 1 0 1 0 MC = −  q 1   iq 1   1 0   q 1   iq 1  − 3r − 3c − 2r − 2c           100CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

0 1 1 0 1 0 . (5.23) ×  1 0   q 1   iq 1  − − 1r − 1c       This decomposition consists of three imaginary CM and real CM pairs with Fourier/Inverse- Fourier transform operations in between. The imaginary CM and real CM pairs can also be viewed as complex CM (CCM) operations:

1 0 0 1 1 0 MC = −  (q + iq ) 1   1 0   (q + iq ) 1  − 3r 3c − 2r 2c       0 1 1 0 . (5.24) ×  1 0   (q + iq ) 1  − − 1r 1c     The three matrices in the center can also be expressed as a CCC operation:

1 0 1 (q2r + iq2c) 1 0 MC = (5.25)  (q + iq ) 1   0 1   (q + iq ) 1  − 3r 3c − 1r 1c       which is nothing but the complex version of the well-known CM-CC-CM decomposition [10]. When we multiply out the matrices on the right-hand side of Eq. 5.23, equate the result to the general CLCT matrix given in Eq. 5.4 and solve for our decomposition parameters in terms of the CLCT parameters, we get the following:

br brar acbc q1r = − 2 −2 br + bc bcar bc brac q1c = −2 −2 br + bc q2r = br

q2c = bc

br brdr dcbc q3r = − 2 −2 br + bc 5.3. THE ALGORITHM 101

b d b b d c r c r c (5.26) q3c = −2 −2 br + bc

Thus, all 6 parameters of our decomposition have been expressed in terms of the 6 parameters oftheCLCT which wedesiretocompute. By usingEq.5.7, we can also easily calculate the decomposition parameters in terms of the complex α,β,γ parameters if needed.

The decomposition in Eq. 5.23 can also be expressed in operator notation:

−1 M = (5.27) C Qq3r Qiq3c Flc Qq2r Qiq2c Flc Qq1r Qiq1c

We now discuss the various cases that arise depending on the values of the parameters. When b = 0, separate treatment is required depending on whether a is zero or not. First, 6 consider the case when a =0. The decomposition parameters given in Eq. 5.26 become

br q1r = 2 2 br + bc bc q1c = 2− 2 br + bc q2r = br

q2c = bc

br brdr dcbc q3r = − 2 −2 br + bc b d b b d c r c r c (5.28) q3c = −2 −2 br + bc

As discussed in Section 5.2. 5.2.2. 5.2.2.2, the CCM parameters q , q , q should be 0 1c 2c 3c ≤ leading to the conditions

b c (5.29) 2− 2 0 br + bc ≤ b 0 (5.30) c ≤ b d b b d c r c r c (5.31) −2 −2 0 br + bc ≤ 102CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

Eqs. 5.29 and 5.30 imply b = 0 and Eq. 5.31 becomes b d 0. When we set b = 0 in c r c ≥ c Eq. 5.28, we obtain the decomposition parameters:

q1r = 1/br

q1c = 0

q2r = br

q2c = 0

q = (1 d )/b2 3r − r r q = d /b (5.32) 3c − c r with the condition b d 0. The decomposition we should use in this case therefore can r c ≥ be expressed as

1 0 1 0 0 1 MC = −  (1 d )/b 1   id /b 1   1 0  − − r r c r       1 0 0 1 1 0 . (5.33) ×  b 1   1 0   1/b 1  − r − − r      

We now turn our attention to the case b = 0 and a = 0. The decomposition given in 6 6 Eq. 5.23 and the decomposition parameters given in Eq. 5.26 are applicable. The below three conditions should be satisﬁed to have a bounded and R R mapping: →

b 0 c ≤ b a b a b c r − r c ≤ c b d b d b (5.34) c r − r c ≤ c 5.3. THE ALGORITHM 103

which can be equivalently expressed in terms of the α,β,γ parameters:

β 0 c ≥ α β c ≥ c γ β (5.35) c ≥ c

which depends only on the imaginary parts. This is expected since real LCTs are always bounded and unitary, and it is the imaginary parts that are involved in issues of boundedness. These conditions are derived by restricting the parameters of the Gaussian aperture steps in the CLCT decompositions we employ. There are no such conditions required for RLCTs. However, these constraints are crucial for computation of CLCTs. To better illustrate these conditions, we summarize them in Table 5.1.

Case 1 Case 2 Case 3 b=0 b =0 & a =0 b =0 & a =0 6 6 6 a =0 b =0 b 0 r 6 c c ≤ d =1/a b d 0 b a b a b r c ≥ c r − r c ≤ c ac =0 bcdr brdc bc a c 0 − ≤ r c ≥ Table 5.1: Summary of the conditions to have bounded, R R CLCTs →

The special case b = 0 requires the computation of only real and complex chirp multiplications and a real scaling operation. The decomposition for the general case includes chirp multiplications and Fourier transformations. Chirp multiplications require only N multiplications and can be done in O(N) time. The FT and inverse FT can be computed in O(N log N) time by using the fast Fourier transform algorithm (FFT). We also note that the scaling operation merely changes the sampling interval in the sense of reinterpretation of the same samples with a scaled sampling interval, in a manner which corresponds to scaling of the underlying continuous signal. Thus the cost of the scaling operation is minimal 104CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

and not of consequence, since it amounts only to a reinterpretation of the samples. There- fore, the overall CLCT can be computed in O(N log N) time. To ensure that the number of samples required to represent the function is sufficient in the Nyquist-Shannon sense at each step of the decomposition, we will track the space-bandwidth representation of the function by using the WD and increase the sampling rate when necessary. We will do this by the help of the procedures summarized in [64] for real steps and by the help of Fig. 5.1 and the discussion given in Sec. 5.2. 5.2.2. 5.2.2.2 for the complex components. Since the FRT corresponds to rotation and the scaling operation only to a reinterpretation of the samples, these steps never require us to increase the number of samples. Chirp multiplications, however, require careful handling of the space-bandwidth and sampling issues. Finally, we summarize our algorithm and the associated space-bandwidth product tracking and sampling control methodology for the most general case. (For the b = 0 and b =0, a =0 special cases, this procedure can be easily simplified to correspond to the sim- 6 pler decompositions in Eq. 5.22 and Eq. 5.33, respectively.) Whenever the current number of samples will not be sufficient to fully represent the operated-on signal in the Nyquist- Shannon sense, an increase in the number of samples is required prior to performing the operation.

1. We will use Es and Ef to denote the spatial and frequency extents of the function as we go through the stages of the algorithm. We assume that the initial space-frequency

support is a square of edge length ∆u so that at the beginning Es = Ef = ∆u, and 2 the signal can be represented with EsEf = ∆u samples.

2. The ﬁrst step of the decomposition is the ﬁrst CCM with parameter q1c. We use Eq. 5.14 to obtain the space and frequency extents of the Gaussian function, which we denote by G and G , respectively. E and E are changed according to E s1 f1 s f s → min(E ,G ) and E E + G . The required number of samples then becomes s s1 f → s f1 E E which are taken in the interval [ ∆E /2, ∆E /2] with a spacing of 1/E s × f − s s f 5.4. NUMERICAL EXAMPLES 105

apart from each other. This may or may not require an increase in the number of samples depending on whether the new E E product is bigger than the starting s × f number of samples ∆u2. If an increase in the number of samples is required, we oversample the signal using an appropriate interpolation scheme and then the CCM operation is performed on the input samples.

3. The second step is a CM operation with parameter q1r. We see that the extents must now become E E and E E + q E . The number of samples required s → s f → f | 1r| s becomes E (E + q E ) which will require oversampling with a factor k =1+ s × f | 1r| s q E /E . After this oversampling is performed, the CM operation is performed. | 1r| s f 4. We now take the FT of the samples by using the FFT algorithm. We have E E s → f and E E since FT only switches the spatial variable and its spatial-frequency f → s variable. The FT operation does not change the space-bandwidth product of the signal, so oversampling is not required at this stage.

5. Repeat Steps 2 and 3 with the parameters q2c and q2r corresponding to subsequent stages of the decomposition.

6. Repeat Step 4, this time with an inverse FT operation instead of a forward FT operation.

7. Repeat Steps 2 and 3 with the parameters q3c and q3r corresponding to the ﬁnal stages of the decomposition, to get the ﬁnal output samples.

5.4 Numerical Examples

We have considered several examples to illustrate the performance of the presented algorithm. We consider the chirped pulse function exp( πu2 iπu2), denoted F1, and − − the trapezoidal function 1.5tri(u/3) 0.5tri(u), denoted F2 (tri(u) = rect(u) rect(u)). − ∗ 106CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

Since these two functions are well confined to a circle in the space-frequency plane with diameter ∆u = 8, we take N = 82. We also consider the binary sequence 01101010 occupying [ 8, 8] with each bit 2 units in length, so that N = 162. This binary se- − quence is denoted by F3 and the function shown in Fig. 5.2 is denoted by F4, again with N = 162. F4 is chosen to test the algorithm with an information-theoretically challenging function who has several sharp rises and falls at arbitrary positions. Additionally, we also test the example function given in Fig. 5.3, with N = 82, that has complex values (ie. amplitude and phase). These choices for ∆u result in 0 %, 0.0002 %, 0.47 %, ∼ 0.03 %, 0.25 % of the energies of F1, F2, F3, F4, F5 respectively, to fall outside the chosen frequency extents. The chosen space extents include all of the energies of F2, F3, F4, F5 and virtually all of the energy of F1. We consider three transforms, the first (T1) with parameters (α , β ,γ ; α , β ,γ )=( 2, 1.2, 0.9;0.04, 0.02, 0.12), the second (T2) r r r c c c − − with parameters (1.15, 0.14, 0.1;0.003, 0.001, 0.002), and the third (T3) with parame- − − ters ( 1.2, 0.3, 0.1;0.6, 0.5, 1). The CLCTs T1, T2 and T3 of the functions F1, F2, F3, − − F4, F5 have been computed both by the presented fast algorithm and by a highly inefficient brute force numerical approach based on Simpson’s numerical integration, which is here taken as a reference.(See Appendix A.)

The results for all functions (F1, F2, F3, F4, F5) are plotted in Fig. 5.4 and Fig. 5.5 for transform T1, and T2, respectively and tabulated in Table 5.2 for all transforms T1, T2 and T3. Also shown are the errors that arise when using the DFT in approximating the FT of the same functions, which serves as a reference. (The error is deﬁned as the energy of the difference normalized by the energy of the reference, expressed as a percentage.)

We also tested our algorithm for the CFRT, which is an important special case of CLCTs and the complex extension of the real-parametered FRT. A CFRT with order 0.8 − i0.2 is calculated with our algorithm and the reference method and the results are plotted for all functions (F1, F2, F3, F4, F5) in Fig. 5.6, and again tabulated in Table 5.2. The CFRT order 0.8 i0.2 corresponds to CLCT parameters (α , β ,γ ; α , β ,γ ) = − r r r c c c 5.4. NUMERICAL EXAMPLES 107

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

0 −8 −6 −4 −2 0 2 4 6 8

Figure 5.2: Example function F4

(0.292, 0.9919, 0.292;0.3331, 0.098, 0.3331).

T1 T2 T3 CFRT DFT F1 4.12 10−6 2.19 10−6 2.8 10−3 1.24 10−5 2.0 10−21 F2 3.73 × 10−4 7.1 ×10−3 1.4 × 10−3 1.2 ×10−3 6.2 × 10−4 F3 0.53× 0.35× 0.26× 0.22× 1.2 × F4 1.2 10−3 4.96 10−2 2.0 10−3 2.2 10−3 7.1 10−2 F5 0.11× 0.2× 8.0 × 10−3 6.4 × 10−3 1.7 × × × Table 5.2: Percentage errors for different functions F and transforms T.

Examination of the table shows that our algorithm can accurately compute CLCTs for a variety of transforms and functions. We observe that the main determinant of the error is not the transform, but the function, and more speciﬁcally the energy of the function lying outside of the assumed extents. If we require the error to be further reduced, we can reduce the excluded energy by increasing the extents and the number of samples involved. 108CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

Amplitude 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 −8 −6 −4 −2 0 2 4 6 8

Phase 1.5

0.5

−0.5

−1

−1.5 −8 −6 −4 −2 0 2 4 6 8

Figure 5.3: Example function F5

5.5 Concluding Remarks

We presented an algorithm for the fast and accurate digital computation of the general family of complex-parametered linear canonical transforms. This family of transform integrals can represent a general class of complex quadratic-phase systems in optics. Our approach is based on concepts from signal analysis and processing rather than conventional numerical analysis. With careful consideration of sampling issues, the number of samples N can be chosen very close to the space-bandwidth product of the functions. A naive approach based on examination of the frequency content of the integral kernels would, on the other 5.5. CONCLUDING REMARKS 109

Re of transform of F1 Im of transform of F1 1 1

0.5 0.5

0 0

−0.5 −0.5

−1 −1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3

Re of transform of F2 Im of transform of F2 0.5 0.5

0 0

−0.5 −0.5

−1 −1

−2 −1 0 1 2 −2 −1 0 1 2

Re of transform of F3 Im of transform of F3 0.5 0.5

0 0

−0.5 −0.5

−1 −1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3

Re of transform of F4 Im of transform of F4 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 −1 0 1 2 −2 −1 0 1 2

Re of transform of F5 Im of transform of F5

0 0

−0.5 −0.5

−1 −1 −2 −1 0 1 2 −2 −1 0 1 2

Figure 5.4: Transform (T1) of F1, F2, F3, F4, F5. The results obtained with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indistinguishable since the results are very close. 110CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

Re of transform of F1 Im of transform of F1

0.5 0.5

0 0

−0.5 −0.5 −2 −1 0 1 2 −2 −1 0 1 2

Re of transform of F2 Im of transform of F2

0.5 0.5

0 0

−0.5 −0.5 −2 −1 0 1 2 −2 −1 0 1 2

Re of transform of F3 Im of transform of F3 0.5 0.5

0 0

−0.5 −0.5 −2 −1 0 1 2 −2 −1 0 1 2

Re of transform of F4 Im of transform of F4 1 1

0.5 0.5

0 0

−0.5 −0.5 −2 −1 0 1 2 −2 −1 0 1 2

Re of transform of F5 Im of transform of F5

0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 −2 −1 0 1 2 −2 −1 0 1 2

Figure 5.5: Transform (T3) of F1, F2, F3, F4, F5. The results obtained with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indistinguishable since the results are very close. 5.5. CONCLUDING REMARKS 111

Re of CFRT of F1 Im of CFRT of F1

0 0.4 −0.2

0.2 −0.4 −0.6 0 −0.8 −2 −1 0 1 2 −2 −1 0 1 2

Re of CFRT of F2 Im of CFRT of F2 1.5 0.2 1 0 −0.2 0.5 −0.4 0 −0.6 −0.8 −2 −1 0 1 2 −2 −1 0 1 2

Re of CFRT of F3 Im of CFRT of F3 0.5 0.6 0.4 0 0.2 −0.5 0 −0.2 −1 −2 −1 0 1 2 −2 −1 0 1 2

Re of CFRT of F4 Im of CFRT of F4 2 0.5

1.5 0 1 −0.5 0.5 0 −1 −0.5 −1.5 −2 −1 0 1 2 −2 −1 0 1 2

Re of CFRT of F5 Im of CFRT of F5 1 0.2

0.5 0

−0.2 0 −0.4 −0.5 −2 −1 0 1 2 −2 −1 0 1 2

Figure 5.6: CFRT with order 0.8 i0.2 of F1, F2, F3, F4, F5. The results obtained with the presented algorithm and the reference− result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indistinguishable since the results are very close. 112CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

hand, result in an unnecessarily high number of samples being taken due to the highly oscillatory nature of the kernels, which would not only be representationally inefﬁcient but also increase computation time and storage requirements. The transform output may have a higher space-bandwidth product than the input due to the nature of the transform family. Through careful space-bandwidth tracking and control, we can assure that the output samples obtained are accurate approximations to the true ones and that they are sufﬁcient (but not unnecessarily redundant) in the Nyquist-Shannon sense, allowing full reconstruction of the underlying continuous output functions. The algorithm takes the samples of the input function and maps them to the samples of the continuous CLCT of this function in the same sense that the fast Fourier transform (FFT) implementation of the discrete Fourier transform (DFT) computes the samples of the continuous FT of a function. Chapter 6

Application to the Beam Propagation Method

T HE BEAM PROPAGATION METHOD (BPM) is an important computational method in electromagnetics to solve the time-harmonic Helmholtz equation under the slowly varying envelope approximation (SVEA), [128–131]. It predicts the propagation of beams in inhomogeneous media in which the refractive index changes are small relative to the average index such that the SVEA can hold. It is mostly used to simulate and study optical waveguides and other optical devices with an inhomogeneous refractive index distribution. It has also been modiﬁed to be used in analysis of diffraction gratings and in anisotropic media, [132,133]. On the other hand, quadratic-phase systems (QPSs) can model a broad class of optical systems including thin lenses, sections of free space in the Fresnel approximation, sections of quadratic graded-index media, and arbitrary concatenations of any number of these, sometimes referred to as ﬁrst-order optical systems [10,12]. Fractional Fourier transforms (FRTs) [101,102], scaling operations, and chirp multiplication (CM) and chirp convolution (CC) operations, the latter also known as Fresnel transforms, are special cases of QPSs, [10].

113 114 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

The QPSs are capable of exactly representing so-called quadratic-graded index media (GRIN), [108], in which the inhomogeneous refractive index distribution is given by n2(x)= n2[1 (n /n )x2] where n and n are the medium parameters and x is the trans- 1 − 2 1 1 2 verse coordinate. However, in GRIN the parameters are assumed to be the same along the propagation coordinate and the change along the transverse direction is limited to a quadratic dependence. Therefore, this index distribution is currently the most inhomogeneous case QPSs are capable of representing to the best of our knowledge. In this chapter, we study the link of QPSs to BPM in an attempt to extent this limit in the degree of inho- mogeneity further. Also, our contribution allows BPM to become extremely efﬁcient under certain conditions.

In our method, each layer of BPM is represented by a cascade of two fundamental ABCD-steps. One is the chirp convolution and the other is the chirp multiplication. The former stands for the propagation in an homogenous medium (Fresnel transform), which is the first step of the corresponding single BPM layer, and the latter gives the approximated phase correction, which is the second step of the BPM layer. Our idea is to find, for every BPM layer, the corresponding chirp convolution sub-systems for each BPM layer, approximate the phase correction with chirp multiplication sub-systems, cascade all of these sub-systems to find a single equivalent QPS. In other words, the overall ABCD matrix is the multiplication of the matrices of the subsystems.

The chapter is organized as follows. In Sections 6.1 and 2.1.1, we summarize the basics of BPM and QPS/ABCD-Systems, respectively. In Section 6.2, we give the main procedure to derive the relation between BPM and QPSs. In Section 6.3, we present the examples of our method and Section 6.4 is devoted to the concluding remarks. 6.1. BASICS OF BPM 115

6.1 BasicsofBPM

In this section, we summarize the basic structure of BPM. We assume the 2D case with propagation in the z direction and an inhomogeneous medium with refractive index distribution n(x, z), x being the transverse direction. We start with a time-harmonic monochro- matic wave ﬁeld E(x, z, t) = Re U(x, z)e−jωt that propagates in an inhomogeneous { } medium. U(x, z) is the complex amplitude and ω is the angular frequency. Then the scalar wave equation is given by ∂2E c2 = 2E (6.1) ∂t2 n2 ∇ where c is the speed of light in free-space and 2 = ∂2/∂x2 + ∂2/∂z2. By using the ∇ time-harmonic assumption, the wave-equation is simpliﬁed to the Helmholtz Equation in an inhomogeneous medium:

( 2 + n2k2)U(x, z)=0 (6.2) ∇ 0

where k0 =2π/λ0 and λ0 is the free-space wavelength.

Next, we make the ﬁrst key assumptionof BPM that we have a refractive index distribution in the form n(x, z)= n + ∆n(x, z), where ∆n n meaning that the inhomogeneous ≪ medium is such that the refractive index varies within a small neighborhood of an average value n, ie. a weak index modulation. Additionally, when we make the paraxial approximation and assume that the wave-propagation is along directions making very small angles with the z direction, U(x, z) can be written as

U(x, z)= U(x, z)e−jnk0z (6.3) where U(x, z) part is a slowly varying function of z. This so-called slowly-varying envelope approximation (SVEA) is justiﬁed by BPM’s assumptions of weak index modulation 116 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

and of paraxial propagation. By substituting Eq. 6.3 into Eq. 6.2 we get

∂ ( 2 2jnk + n2 n2)U =0 (6.4) ∇ − 0 ∂z −

By SVEA, we can neglect the second order partial derivative of U with respect to z in Eq. 6.4 to get the ﬁnal “paraxial wave equation for inhomogeneous media”, [128],

∂ j ∂2U 2 2 (6.5) U = − [ 2 +(n n )k0U] ∂z 2nk0 ∂x −

Eq. 6.5 is the fundamental BPM equation that provides its theoretical foundation. This equation can be solved through iteration and starting from an initial value. The implementation of Eq. 6.5 is done by dividing the z axis into N slices of length ∆z and by treating each slice as a combination of propagation of ∆z in a homogeneous medium of refractive index ni where i =1, 2, ..., N are the average refractive index values along the corresponding slice and a virtual lens effect for phase correction. The lens effect is the multiplication of the phase exp( j(n(x, z ) n )k ∆z) due to the slowly varying index distribution on − i − i 0 top of the average value, which models the inhomogeneous medium. Note that zi = i∆z

and z0 is the input plane. Then, starting from an initial wave field at z = 0, the BPM method iterates this initial value for each slice and uses the output of i’th slice as the input of (i+1)’th. By letting N be large and ∆z be very small, the above assumptions in the derivation of Eq. 6.5 are justified. This way the slowly-varying inhomogeneous index vari- ation is separated from the homogenous propagation by virtue of Eq. 6.5 and after iterating the entire set of slices, one can get the output wave field.

For the homogeneous propagation steps, the common method to use is the Angular Spectrum Method (ASM) that relies on the Angular Spectrum of Planes Waves, [16, 130]. If ASM is used to implement the homogenous propagation part one can get the iteration

equation for one slice between zi and zi+1 as: 6.1. BASICS OF BPM 117

2 −1 ni 2 U(x, zi+1) = U(x, zi) exp j2π∆z 2 fx F (F sλ0 − !!) exp ( j (n(x, z ) n ) k ∆z) (6.6) × − i − i 0 where z = z + ∆z and n is the average refractive index for i’th slice. stands for the i+1 i i F Fourier transformation and fx is the spatial-frequency of variable x. Also note that, Eq. 6.6

assumes fx < ni/λ0, which means that evanescent waves are ignored.

We can write this in operator notation to be used later as

U(x, z )= U(x, z ) (6.7) i+1 Ci Pni i where stands for the free-space propagation in a homogeneous region of length ∆z and Pni with refractive index n and where is the operator for the multiplicative phase compen- i Ci sation of C = exp ( j (n(x, z ) n ) k ∆z) for the slice i. i − i − i 0

The advantages of BPM is that it is a relatively simple, straightforward and fast method. On the other hand, it has some drawbacks that it cannot handle reﬂections and non-paraxial propagation problems as well as works only for small variations in the refractive index. Reﬂections cannot be handled because we ignored the effect of the second partial derivative with respect to z in the above derivation. However, more advanced BPM algorithms have been developed to work in non-paraxial propagation cases as well. The version of BPM that we consider in this chapter is a version of the BPM as given in [130,134]. We also consider two-dimensional BPM (2D-BPM) where we have one transverse dimension, x and the propagation dimension, z. Generalization of BPM to three dimensions is straightforward. 118 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

6.2 The Relation between BPM and ABCD-Systems

As given in Section 6.1, BPM should run the ASM method or any other method for homogeneous propagation and then make a phase compensation as a multiplicative filter for each slice. Typically, BPM is implemented with large amounts of slices to better satisfy the assumptions made in its derivation. Although faster than some other computational tools, it still carries significant amounts of burden due to its iterative nature and the repetitive calculations on the large number of data points of the wavefunction. In this chapter, we show that, under a quadratic-phase approximation, each step of BPM can be written as a QPS. By representing each and as a QPS defined by ABCD matrices and then combining Pni Ci them by using the cascade rule for QPSs as given in Section 2.1.1, we can obtain a single QPS. This overall QPS, which is an approximation to BPM, is defined by a single ABCD matrix. Given the input wave field, there are fast algorithms to digitally calculate the output field in O(N log N) time, [60,63,64]. Thus, the iterative nature of BPM can be by-passed and computational savings by several factors can be achieved.

The homogenous propagation part of BPM, , can also be implemented by using Pni Fresnel transform/propagation instead of ASM, [16]. The Fresnel transform gives the output ﬁeld, E(x, z), after a propagation of length z in an homogenous medium of refractive index n from the input ﬁeld E(x, 0) with the relation:

jn 2π z λ jnπ ′ e 0 ′ (x−x )2 ′ E(x, z)= E(x , 0)e λ0z dx (6.8) √jλz Z

It is a special case of QPS systems and the corresponding ABCD matrix, denoted by MF , is given by

λ0z 1 n MF = (6.9)  0 1    where α = β = γ = n . If one needs to care about the constant phase terms preceding the λ0z 6.2. THE RELATION BETWEEN BPM AND ABCD-SYSTEMS 119

above equation, it should be taken into account that the ABCD matrix formalism for the jn 2π z Fresnel transform does not take into account the preceding constant phase-term of e λ0 , which should be added to the resulting ﬁeld at the end.

So far, we did not make any approximation to the BPM. The crucial operation to relate the BPM and QPSs is the implementation of the multiplicative phase correction . Since Ci this phase correction operates as a multiplicative filter in the BPM implementation, it pre- vents us from obtaining an overall calculation but requires us to stick to iteration. If we approximate this multiplicative phase filter with a quadratic-phase filter we can represent this step as a special case of QPSs as well. The quadratic-phase filter is actually a chirpmultiplication operation given by E′(x, z) = e−jπqx2 E(x, z). The corresponding ABCD matrix is 1 0 MC = . (6.10)  q 1  −   Recall the phase correction term

C = exp ( j (n(x, z ) n ) k ∆z) = exp ( j∆n(x, z )k ∆z) (6.11) i − i − i 0 − i 0

in Eq. 6.6. We want to approximate it with a form exp ( j(ax2 + b)k ∆z). To ﬁnd the best − 0 approximation we use the Mean-squared error (MSE) approximation such that we solve the following optimization problems:

2 2 min(aix + bi ∆n(x, zi)) (6.12) ai,bi −

to get the a and b parameters of the quadratic approximation for each layer, where (i = 1, 2, ..., N) represents the BPM slices along z. Because we work on digital computational problems, a discritization and then practical solution of the above optimization problem 120 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

can be carried out as follows. We discritize the problem in Eq. 6.12 such that

Nx 2 2 min (aixk + bi ∆n(xk, zi)) (6.13) ai,bi − Xk=1 where (k =1, 2, ..., Nx) represents the samples taken along the transverse direction. Solv- ing this optimization problem yields the following two dimensional system for each i:

−1 Nx 4 Nx 2 Nx 2 ai x x x ∆n(xk, zi) = k=1 k k=1 k k=1 k . (6.14)    Nx 2   Nx  bi Pk=1 xk PNx P k=1 ∆n(xk, zi)    P   P  Once these optimization parameters are determined, we can write the ABCD matrix representation Si of a single BPM iteration step as follows:

1 0 1 λ0∆z/ni Si = (6.15)  2a ∆z/λ 1   0 1  − i 0     2π with the constant phase residue exp j ∆z(ni bi) . Note that the bi terms in this con- λ0 − stant phase residue are due to the quadratic approximation and ni terms are due to the constant phase compensation of the Fresnel transform. Then the overall ABCD matrix that approximates the inhomogeneous media with a QPS can be obtained by simple matrix multiplication requiring manipulation of only 2 2 matrices. The overall matrix M is × overall given by M = S S S S (6.16) overall N N−1 ··· 2 1 and the constant phase residues can be combined to add at the very end altogether as

2π exp j ∆z (ni bi) . Once the Moverall is calculated, it can be computed fast by al- λ0 i − gorithms givenP in [60,63,64] for only one single wave ﬁeld given at the input side, instead of iterating the intermediate wavefunctions in the classical BPM. The cost of optimization, which only includes a matrix multiplication and summations, and the cost of calculating 6.3. NUMERICAL TESTS 121

2 2 matrix M are considerably smaller than the savings done by not performing the × overall BPM iterations on the waveﬁeld that are typically sampled at large amounts of data points.

6.3 Numerical Tests

To test and characterize the reported relation between BPM and QPS, we set up some example systems. We implemented both BPM and the QPS implementation of it, calculate the output fields from both methods and then compare them. The first system we consider is the propagation along a medium with an arbitrary refractive index distribution. The system can be seen in Fig. 6.1. Here the input field propagates along the positive z-axis. The media is also divided into N = 25 layers of ∆z width along the propagation direction. Each layer has a different average refractive index ni (where i =1, 2, ..., N) chosen randomly between 2.5 and 3.5. Each layer is also divided into 4 sub-regions with arbitrarily chosen transverse widths as shown in Fig. 6.1. Each of these subregions has a random index modulation where the maximum deviation from the average is 2.5 10−3. These variations are also × different from those of the other layers. As ∆n n, this ensures the validity of the ≪ SVEA of BPM. A square-wave input field, which is defined from 10 mm to 10 mm with − a support of 2 mm width, is propagated along the z-axis for 0.1 mm. The input field is sampled with 256 samples and 10,000 BPM layers are used. The wavelength is 650 nm. The error between the output fields of the conventional BPM and our QPS-based BPM is 0.9%. The error is defined as the energy of the difference normalized by the energy of the reference (conventional BPM), expressed as a percentage. In addition to this quite high accuracy, our QPS-based BPM is 72 times faster than conventional BPM. The computational times are measured using MATLAB's tic and toc methods. The fields at the output of the system are plotted in Figs. 6.2 and 6.3. As a second example system, we consider the generalization of GRIN systems to show the extension of the limits of QPS to represent more general GRIN media. We assume the 122 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

Figure 6.1: Test System 1

second test media has GRIN distribution along the transverse direction but a different GRIN proﬁle is present for slices of every 10 nm. Basically, for each of these slices, we have different parameters of GRIN. In this way, the media obtains some kind of inhomogeneous distribution. The index distribution of this medium is given in Fig. 6.4 and the average index distribution along the z direction is plotted in Fig. 6.5. The wavelength of the light, sampling extensions and input function are same as those of the ﬁrst test system.

The error between the two methods for this second test case is 1.64% and the speed im- provement is now by a factor of 74. The results are plotted in Figs. 6.6 and 6.7. This second example particularly shows the extension of the ability of QPSs to represent generalized GRIN media. 6.4. CONCLUDING REMARKS 123

BPM and QPS−BPM outputs (magnitudes) 1.4 QPS−BPM BPM 1.2

0.8

0.6

0.4

0.2

0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 x in meters x 10

Figure 6.2: Amplitudes for Test System 1

6.4 Concluding Remarks

Our aim is to make the link between two important topics in optics and to give new insights to both of these important methods. By showing the conditions in which the BPM can be represented as a QPS and then can be computed in a more efﬁcient way, we can contribute to both of these techniques. For BPM, exploiting this relationship can lead to faster implementations for certain systems. For the QPS point of view, this relationship shows that the limits of the capabilities of QPSs can be stretched to the point that they can, in a certain extent, represent the wave-propagation in some inhomogeneous medium satisfying the aforementioned conditions. Especially, QPSs are shown to be capable of 124 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

BPM and QPS−BPM outputs (phases) 3 QPS−BPM BPM 2

Phase in radians −1

−2

−3 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 x in meters x 10

Figure 6.3: Phases for Test System 1

representing a generalized GRIN medium, which has a more inhomogeneous index proﬁle than that of GRIN media. This deﬁnitely expands the theory of QPS and ABCD-systems. On top of these, we think that the demonstration of such a relationship is itself interesting and can make two different important areas of research converge towards each other for likely future developments and contributions.

BPM is generally used to ﬁnd only the intensity within the structure. This also strength- ens the applicability of our method since its main source of error comes from the phase approximations. Also, even if there are some sub-parts of the entire region, in which our 6.4. CONCLUDING REMARKS 125

−3 Index distribution for Test System 2 x 10 −10

−8 1.95

−6 1.9 1.85 −4 1.8 −2 1.75 0 1.7 x in meters 2 1.65 4 1.6 6 1.55 8 1.5

0 2 4 6 8 10 −5 z in meters x 10

Figure 6.4: Index distribution for Test System 2

QPS-based BPM method’s performance degrades because the quadratic-phase approximation does not hold, conventional BPM can be used for these sub-parts and for the other sub-parts our fast method can be used instead of iteration. Using this hybrid method instead of using BPM for the entire region, the speed of the calculation can still be partially increased. 126 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

Average index along the z axis for Test System 2 2

1.9

1.8

n 1.7

1.6

1.5

1.4 0 0.2 0.4 0.6 0.8 1 −4 z in meters x 10

Figure 6.5: Average index for Test System 2

BPM and QPS−BPM outputs (magnitudes) 1.4 QPS−BPM BPM 1.2

0.8

0.6

0.4

0.2

0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 x in meters x 10

Figure 6.6: Amplitudes for Test System 2 6.4. CONCLUDING REMARKS 127

BPM and QPS−BPM outputs (phases) 4 QPS−BPM BPM 3

−1 Phase in radians

−2

−3

−4 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 x in meters x 10

Figure 6.7: Phases for Test System 2 128 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD Chapter 7

Conclusion

N THIS THESIS, four algorithms for the computation of linear canonical transforms I (LCTs) from the N samples of the input signal in O(N log N) time are discussed. Our approach is based on concepts from signal analysis and processing rather than conventional numerical analysis. With careful consideration of sampling issues, N can be chosen very close to the time-bandwidth product of the signals, and need not be much larger. The transform output may have a higher space-bandwidth product due to the nature of the transform family. All algorithms relate the samples of the input function to the samples of the continuous LCT of this function in the same sense that the fast Fourier transform (FFT) implementation of the discrete Fourier transform (DFT) computes the samples of the continuous FT of a function. Since the sampling rates are carefully controlled, the output samples obtained are accurate approximations to the true ones and the continuous LCT can be recovered via interpolation of these samples. The only inevitable source of deviation from exactness arises from the fundamental fact that a signal and its transform cannot both be of ﬁnite extent. This is the same source of deviation encountered when using the DFT/FFT to compute the continuous FT. Thus the algorithms compute LCTs with a performance similar to the DFT/FFT in computing the Fourier transform, both in terms of speed and accuracy.

129 130 CHAPTER 7. CONCLUSION

This limitation affects not only the separable and one-dimensional versions of the algorithm reported earlier, but also the computation of Fourier transforms using the DFT. Thus this is a source of error we cannot hope to overcome. Compared to earlier approaches, these algorithms not only handle a much more general family of integrals, but also effectively address certain difficulties, limitations, or tradeoffs that arise in other approaches to computing the Fresnel integral, which is of importance in the theory of diffraction. We have also developed the link between the compact, matrix-based 16-parameter definition of two-dimensional non-separable LCTs and the 10-parameter explicit kernel definition. Complex-parametered LCTs allow several kinds of optical systems to be represented, including lossy as well as lossless ones. When complex parameters are involved, LCTs may no longer be unitary and boundedness issues may arise. We have identified the conditions for a CLCT to constitute a bounded map from functions on the real axis to functions on the real axis. As a special case of our general CLCT algorithm, we have also obtained an efficient and accurate algorithm for complex-ordered fractional Fourier transforms.

7.1 Future Work

In this part, several possible future extensions and research directions for the algorithms and approaches developed in this thesis will be summarized. In this thesis algorithms up to 2D-LCTs are covered. 2D-LCTs are sufﬁcient to represent a realistic 3D optical system where we have two transverse directions and a direction of propagation. However, from a more theoretical and abstract point of view, there are LCTs with more than two dimensions. The fundamental deﬁnitions of the LCTs can be extended straightforwardly to several dimensions. Then, the fast algorithms and other issues can be studied in these high dimensional domains. 7.1. FUTURE WORK 131

As the name implies, LCTs are ’linear’ transforms. In addition to the linear versions of them, the other related class is the Radial canonical transforms, [39,121]. Radial transforms are defined in multidimensions and assume spherical symmetry. For example, the Hankel transform, which is defined in terms of the famous Bessel functions of the first kind and which is important in telecommunications, is a prominent member of the radial canonical transforms. The question of whether similar fast and efficient algorithms for computation of these radial transforms can be found is another future extension of the research in this thesis. LCTs are also closely associated to the underlying mathematical theory and operations like the Radon transform and the Projection-Slice Theorem used in medical imaging, tomography and magnetic resonance, [135–138]. Researching the ways to extend these algorithms and/or the key decomposition based approaches to the concepts used in medical imaging can be done to see whether the decomposition approaches can improve the image processing algorithms in those fields. The relationship of the LCTs to the Beam Propagation Method (BPM) is a very interesting finding. Currently, the relationship derived in Chapter 6 of the thesis has some restrictions and it is revealed that the relationship holds subject to some conditions about the refractive index distribution of the medium. More research can be carried out to further investigate this relationship and to look for possible potential new findings and methods to enlarge the types of systems that can be modeled by using LCTs (or equivalently QPSs). Another future extension to the thesis may be the development of new simulation and modeling software based on the developed algorithms with all graphical user interfaces and modeling interfaces. Polishing the code and developing high-level software additions to allow users to define their optical systems and to set parameters in a graphical tool can be done. Then the computational fast algorithms are used underneath to quickly simulate the system and obtain the output. Ultimately, it may be possible to release this as some stand alone simulation software for first-order optical system design and analysis. 132 CHAPTER 7. CONCLUSION Appendix A

Simpson’s Rule for 1D and 2D Functions

In this Section, Simpson’s rule will be summarized. Simpson’s rule is a very-well known method for numerical integration in numerical analysis, [139,140]. The Simpson’s rule is credited to the famous mathematician Thomas Simpson of the 18th Century. Simpson’s rule approximates the deﬁnite integral of a function f by using quadratic polynomials. For one-dimensional functions the basic approximation equation is the following:

b b a a + b f(x) dx − f(a)+4f + f(b) (A.1) ≈ 6 2 Za where the corresponding error term E is given by

1 E = (b a)5f (4)(x) (A.2) 90 − where x is an arbitrary value in the interval [a, b]. The accuracy of the Simpson’s method is quite high if the function to be approximated is smooth in the interval [a, b]. To satisfy this, usually a composite Simpson’s rule should be used. In this method, the main interval [a, b] is divided into m number of subintervals, where m is even. Then, let us divide [a, b] into m subintervals denoted by [x , x ] m/2 where the length of each subinterval is given by { k−1 k }k=1 133 134 APPENDIX A. SIMPSON’S RULE FOR 1D AND 2D FUNCTIONS

(b a)/m. Finally, −

m/2−1 m/2 b b a 2(b a) 4(b a) f(x) dx − (f(a)+f(b))+ − f(x2k)+ − f(x2k−1) (A.3) a ≈ 3 3 3 Z Xk=1 Xk=1 The error term for the composite Simpson’s rule is

(b a)5f (4)(x) E = − (A.4) − 180m4 where x is an arbitrary value in the interval [a, b]. By taking m very large enough, one can obtain very short intervals in which the function f is almost constant. Therefore, the composite Simpson’s method can be used to ﬁnd very accurate results at the cost of computational time. For two-dimensional functions, f(x, y), there are also Simpson’s rule based methods, [119]. The generalization to two dimensions are as follows: The double integral that needs to be numerically calculated is:

d b f(x, y) dxdy (A.5) Zc Za The main interval along x-direction [a, b] is divided by n number of subintervals and along y-direction [c,d] is divided by m number of subintervals where n and m are even. Then the two-dimensional integral can be approximated as

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