Metaplectic geometrical optics Nicolas Alexander Lopez A Dissertation

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Metaplectic geometrical optics

Nicolas Alexander Lopez

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Astrophysical Sciences

Program in Plasma Physics

Adviser: Ilya Y. Dodin

November 2022

arXiv:2210.03188v1 [physics.optics] 6 Oct 2022

©Copyright by Nicolas Alexander Lopez, 2022.

Abstract

Ray optics is an intuitive and computationally eﬃcient model for wave propagation through nonuni-

form media. It is therefore the standard choice for multiphysics simulations that involve wave physics

in some capacity, such as designing and analyzing nuclear fusion experiments. However, the under-

lying geometrical-optics (GO) approximation of ray optics breaks down at caustics such as cutoﬀs

and focal points, erroneously predicting the wave intensity to be inﬁnite and thereby limiting the

predictive capabilities of these codes. Full-wave modeling can be used instead, but the added com-

putational cost brings its own set of tradeoﬀs. Developing cheaper, more eﬃcient caustic remedies

has therefore been an active area of research for the past few decades.

In this thesis, I present a new ray-based approach called ‘metaplectic geometrical optics’ (MGO)

that can be applied to any linear wave equation. Instead of evolving waves in the usual x(coordinate)

or k(spectral) representation, MGO uses a mixed X≡Ax+Bkrepresentation. By continuously

adjusting the matrix coeﬃcients Aand Balong the rays via sequenced metaplectic transforms (MTs)

of the waveﬁeld, corresponding to symplectic transformations of the ray phase space, one can ensure

that GO remains valid in the Xcoordinates without caustic singularities. The caustic-free result

is then mapped back onto the original xspace using metaplectic transforms, as demonstrated and

veriﬁed on a number of examples.

Besides outlining the basic theory of MGO, this thesis also presents specialized fast algorithms for

MGO. These algorithms focus on the MT, which is a unitary integral mapping used in MGO that can

be considered a generalization of the Fourier transform. First, a discrete representation of the MT is

developed that can be computed in linear time [O(Np) for Npsample points] when evaluated in the

near-identity limit; ﬁnite MTs can then be implemented as successive applications of K1 near-

identity MTs. Second, an algorithm based on Gauss–Freud quadrature is developed for eﬃciently

computing ﬁnite MTs along their steepest-descent curves, which may be useful in catastrophe-optics

applications beyond MGO. These algorithms lay the foundations for the development of an MGO-

based ray-tracing code.

iii

Contents

Abstract.............................................. iii

1 Introduction 1

1.1 Background......................................... 1

1.2 Outline ........................................... 4

1.3 Contributingpublications ................................. 6

2 Geometrical optics for scalar waves 7

2.1 Introduction......................................... 7

2.2 The geometrical-optics approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Caustics and catastrophes of geometrical optics . . . . . . . . . . . . . . . . . . . . . 13

2.4 Case study: caustics in paraxial propagation . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Cuspoidcaustics .................................. 18

2.4.2 Umbiliccaustics .................................. 20

2.5 Remedies to avoid caustics in ray-tracing simulations . . . . . . . . . . . . . . . . . . 23

2.5.1 Artiﬁcial intensity limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.2 Etalon integrals and uniform approximations . . . . . . . . . . . . . . . . . . 25

2.5.3 Phase-space rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Summary .......................................... 29

2.A Review of the Wigner–Weyl symbol calculus . . . . . . . . . . . . . . . . . . . . . . . 30

2.B Basic results from catastrophe theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Linear symplectic and metaplectic transforms 36

3.1 Introduction......................................... 36

3.2 Linear symplectic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Basic deﬁnitions and identities . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Symplectic submatrices in the SVD basis . . . . . . . . . . . . . . . . . . . . 39

3.2.3 Near-identity symplectic matrices . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.4 Linear orthosymplectic transformations . . . . . . . . . . . . . . . . . . . . . 43

3.3 Unitary metaplectic transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Deﬁnition, derivation, and integral form . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Singular metaplectic transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.3 Pseudo-diﬀerential representation of the metaplectic transform . . . . . . . . 51

3.3.4 Metaplectic transforms of some example functions . . . . . . . . . . . . . . . 55

3.4 Case study: time evolution of the quantum harmonic oscillator . . . . . . . . . . . . 57

3.5 Case study: paraxial propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.1 General expression and special cases . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.2 Connection to far-ﬁeld diﬀraction theory . . . . . . . . . . . . . . . . . . . . . 62

3.6 Summary .......................................... 66

3.A Metaplectic transforms in the mixed basis of conﬁguration and coherent states . . . 66

3.B Veriﬁcation of the operator MT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.C Deriving the MT from its pseudo-diﬀerential representation . . . . . . . . . . . . . . 69

4 Fast algorithm for near-identity metaplectic transforms 71

4.1 Introduction......................................... 71

4.2 Local fast near-identity metaplectic transform . . . . . . . . . . . . . . . . . . . . . . 71

4.2.1 Deﬁnitions ..................................... 71

4.2.2 Runtimeestimate.................................. 73

4.2.3 Stability....................................... 74

4.2.4 Examples ...................................... 79

4.3 Unitary fast near-identity metaplectic transform . . . . . . . . . . . . . . . . . . . . 82

4.3.1 Derivation...................................... 82

4.3.2 Unitarityveriﬁcation................................ 85

4.3.3 Runtimeestimate.................................. 86

4.3.4 Convergence and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.5 Examples ...................................... 89

4.4 Summary .......................................... 92

4.A Reducing the PMT to an envelope equation for eikonal functions . . . . . . . . . . . 94

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摘要：

MetaplecticgeometricalopticsNicolasAlexanderLopezADissertationPresentedtotheFacultyofPrincetonUniversityinCandidacyfortheDegreeofDoctorofPhilosophyRecommendedforAcceptancebytheDepartmentofAstrophysicalSciencesPrograminPlasmaPhysicsAdviser:IlyaY.DodinNovember2022©CopyrightbyNicolasAlexanderLopez,2022...

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