Numerical Methods in Photonics

Andrei V. Lavrinenko, Jesper Lægsgaard, Niels Gregersen, Frank Schmidt, Thomas Søndergaard

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351832007
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Simulation and modeling using numerical methods is one of the key instruments in any scientific work. In the field of photonics, a wide range of numerical methods are used for studying both fundamental optics and applications such as design, development, and optimization of photonic components. Modeling is key for developing improved photonic devices and reducing development time and cost.

Choosing the appropriate computational method for a photonics modeling problem requires a clear understanding of the pros and cons of the available numerical methods. Numerical Methods in Photonics presents six of the most frequently used methods: FDTD, FDFD, 1+1D nonlinear propagation, modal method, Green's function, and FEM.

After an introductory chapter outlining the basics of Maxwell's equations, the book includes self-contained chapters that focus on each of the methods. Each method is accompanied by a review of the mathematical principles in which it is based, along with sample scripts, illustrative examples of characteristic problem solving, and exercises. MATLAB® is used throughout the text.

This book provides a solid basis to practice writing your own codes. The theoretical formulation is complemented by sets of exercises, which allow you to grasp the essence of the modeling tools.

1	Introduction

Photonics as a field embraces phenomena that happen on completely different spatial and temporal scales: from nanometres to kilometres and from subfemtoseconds to micro-and milliseconds. Often, such different scales clash in one modelling case. In, for example optical communication systems, the propagation of light through optical fibres with a length from a few metres to hundreds of kilometres is of interest. On the other hand, the size of the electromagnetic modes of such waveguides transverse to the fibre or waveguide axis should be measured in micrometres.

Scattering from micro-and nanostructures is also of interest. Here, near-field distributions of electromagnetic fields can have features in the nanoscale even if, for example visible wavelengths (400–700 nm) are used. Such fine details cannot be seen in the far-field as a consequence of the diffraction limit of light, but they can be seen in the near field using a scanning near-field optical microscope, where, for example a tapered fibre with or without metal coating is inserted into the near optical field and used to pick up light locally. The fibre introduced to pick up light will itself modify the near field. Naturally, theoretical modelling is very important to support and understand such experiments. Many more cases could be mentioned. Nevertheless, the unique property of all these phenomena in a classical description is that they are governed by Maxwell’s equations.

James Clerk Maxwell presented his well-known set of equations governing electromagnetic fields in 1861. These equations describe the interrelationship between electric and magnetic fields and electric charges and currents. Maxwell’s equations are a remarkable set of equations that can explain the behaviour (propagation and scattering) of electromagnetic fields in any kind of a medium, including vacuum, with the help of constitutive equations. The observation of electromagnetic waves was first demonstrated by Heinrich Hertz in 1884.

Maxwell’s equations in general have analytical solutions only for very simple cases. When assuming a linear material response, it is possible to some extent to find analytical solutions that describe propagation of light in infinitely long and perfectly straight waveguides with a perfectly circular cross section. However, if the cross section is instead rectangular, or if the waveguide is sharply bent, then this is already no longer possible, and numerical solutions are required. Similarly, scattering of light from a perfect sphere can also be obtained in terms of semi-analytical expressions. However, for other slightly less symmetric structures, then again numerical solutions are required. Such simple structures and related analytical solutions are very useful as test cases and for giving a basic physical insight. However, in modern photonics and nano-optics, one is very often interested in more complex geometries.

Therefore, numerical methods for solving Maxwell’s equations are needed. Unlike general numerical techniques for solving differential equations with full or partial derivatives like the Runge–Kutta methods, special routines are applied to solve, namely Maxwell’s equations. Such routines are named Maxwell’s solvers. Following the increasing growth of photonics, more and more computational procedures appear to address various specific needs, and the variety of available Maxwell’s solvers has recently been extended a lot. It has become important not only to know how to write a script in a programming language of choice but also to navigate between a variety of routines commercially available or elaborated by research groups regarding any particular concern in simulations. Thus, the objective of this book is to present methods for numerically solving Maxwell’s equations or equations derived from them such as the wave or Helmholtz equations. This book covers six numerical methods for solving Maxwell’s equations. We provide a short summary of each method in the following.

The finite-difference time-domain (FDTD) method is an example of a universal approach that can be applied to solve Maxwell’s equations directly as they are written in textbooks in the time domain without any further approximation or specification of the processes, like plane-wave solution and harmonic time dependence. Maxwell’s equations are solved on a grid, which is inserted in the region of interest. The most famous and widespread is the Yee grid (Yee mesh), which arranges six components of electric and magnetic fields in a special order finishing by having only tangential field components on the unit cell interfaces. All derivatives in Maxwell’s equations are approximated numerically by finite-difference schemes reducing differential equations to algebraic ones. This leads to an explicit algorithm for updating fields in time that has increased by a small time step. Thus, by applying the updating procedure, we are able to follow all peculiarities of the evolution of fields, which happen during propagation, scattering, absorption, and any other phenomena observed with classical optical waves. The FDTD method can be used to mimic real optical experiments and now very often serves as a reference for checking and validating results of characterization of different optical devices. Moreover, being implemented in the time domain, the FDTD method can be linked with physical phenomena other than the classical electromagnetism nature: atom or molecule kinetics, heat dissipation, diffusion, and so on. A drawback of the FDTD method is the necessity of updating the field components in all points in the numerical space including regions where nothing of interest is studied. However, this drawback is compensated by the linear scaling law of the method with the sizes, very simple meshing of the whole domain, and possibility of effective parallelization of the computation routine.

The finite-difference frequency-domain method can be used efficiently in the frequency domain to calculate eigenmodes of straight waveguides or micro-and nanocavities. The main advantage of the method is its generality and in particular its simplicity, which allows for very easy coding of fairly powerful algorithms in high-level languages such as MATLAB^®. In fact, such a code is distributed with this book. Implementation of periodic and absorbing boundary conditions is straightforward, although not exposed here. The main disadvantage of the method is that it is less efficient than more sophisticated approaches such as the finite element method (FEM).

The nonlinear split-step Fourier method can be used to effectively model non-linear propagation of short pulses in waveguides using a frequency-domain modal expansion to derive an effective 1+1D propagation equation. In its simplest form, this approach leads to the so-called nonlinear Schrödinger (NLS) equation for single-mode problems with second-order dispersion and a constant-mode profile, which lends itself to both analytic and numerical analyses and has been used to describe a wide range of important nonlinear phenomena, especially in fibre-optic waveguides. However, many generalizations of this equation are possible, accounting for effects of higher-order dispersion, delayed nonlinear response, mode profile variations, vectorial effects, and multimode behaviour. The only generalization which seriously affects numerical complexity is the inclusion of several waveguide modes and in some cases mode profile dispersion. In this book, it is shown how a very general propagation equation may be derived, specialized to the case of a singlemode waveguide, and eventually reduced to the NLS equation through various approximations.

The modal method (MM) is used in the frequency domain, where a harmonic time dependence at a single frequency is assumed. In the MM, the geometry under study is sliced into layers uniform along a propagation axis, usually the z-axis. Eigenmodes are computed in each layer assuming uniformity along the propagation axis, and the field is expanded on the eigenmodes of each layer. The z dependence of the eigenmodes is described analytically using propagation constants, and the thickness of a layer does not influence the computation time. The scattering at the interfaces between different layers is handled using a mode-matching technique leading to reflection and transmission matrices characterizing each interface. Advantages of the MM include direct access to the propagation constants of the modes as well as intermodal scattering coefficients, allowing for insight into, for example the reflection and transmission of specific optical modes of interest. Furthermore, the method naturally takes advantage not only of uniformity but also periodicity along the propagation axis using the Bloch mode formalism. The method is thus a natural choice for the analysis of gratings and photonic crystal geometries. The MM naturally supports closed and periodic boundary conditions in the plane lateral to the propagation axis; however, to mimic open geometries, the implementation of absorbing boundary conditions is required. Also, light sources such as dipole emitters or current distributions can be implemented in a straightforward way.

Green’s function integral equation methods (GFIEM) studied in this book solve Maxwell’s equations in the frequency domain. Instead of solving Maxwell’s equations on differential form, then equivalent integral equations are constructed, where the total field at any position is directly related to an overlap integral between a Green’s function and the field inside or on the surface of a scattering object in a reference geometry. Green’s function represents the field emitted by a point source in the reference geometry. The field inside or on the surface of the scattering object is obtained by solving self-consistent equations. The radiation coming from one part of a scatterer is driven by the total field, which includes not only the externally applied field but also contributions to the field due to emission from other parts of the scatterer. One of the strengths of these methods is that the numerical problem can be reduced to either the surface or the inside of the scatterer. Boundary conditions outside the scatterer are automatically taken care of via the choice of Green’s function, and there is no requirement for calculating the field in a region outside the scatterer nor to introduce perfectly matched or absorbing layers. Furthermore, if a scatterer placed on a layered structure is considered instead of the same scatterer in free space, then it is sufficient to replace the free-space Green’s function with the one for the layered structure, and then repeat the calculation. The size of the numerical problem is not increased, but additional work is needed to obtain Green’s function.

The FEM studied in this book solves Maxwell’s equations in the frequency domain. The realization of the FEM involves two basic steps. First, Maxwell’s equations are converted into a so-called variational form that involves integral expressions on the computational domain. Second, the solution space, which should contain a reasonable approximation to the exact solution, has to be constructed. This solution space is obtained by subdividing the computational domain into small geometric patches and by providing a number of polynomials on each patch for the approximation of the solution. The patches together with the local polynomials defined on them are called finite elements. The most common examples of finite elements are triangles and rectangles in 2D and tetrahedrons and cuboids in 3D together with constant, linear, quadratic, and cubic polynomials. These locally defined polynomial spaces have to be pieced together to ensure tangential continuity of the electric and magnetic field across the boundaries of neighbouring patches. The strengths of the method are the following: first, complex geometrical shapes can be treated without geometrical approximations, for example curved geometries can be well approximated; second, the finite element mesh can easily be adapted to the behaviour of the solution, for example to singularities at corners; and third, high-order approximations are available and ensure fast convergence of the numerical solution to the exact solution.

The main part of this book is organized in the following way: Chapter 2 introduces Maxwell’s equations in the time and frequency domains, the wave equations, equations for obtaining guided modes of waveguides, as well as the notation of the book. Specialized chapters for each of the aforementioned numerical methods then follow. The specialized chapters only refer to Chapter 2 and can thus be read independently from each other.

A very large number of numerical methods and variants of those methods exist for solving Maxwell’s equations, and it is not possible to consider all methods in one textbook. We have chosen the methods that we believe are predominantly used in photonics. Methods not covered here include, among others, the multiple multipole method, the beam propagation method, the finite integral and finite volume methods, the method of lines, and the plane-wave expansion method. However, we believe that familiarity with the material in this book will provide a good background for any available Maxwell’s solver.

2	Maxwell’s Equations

2.1 NOTATION

This book employs the following notations:

Object	Notation	Example
Vector in 3D space	Bold	r, E
Unit vector	Bold with hat	x̂, ŷ, ẑ
Vector of expansion coefficients	Plain, single bar	c̅
Matrix	Plain, double bar	O̅̅
Dyadic tensor	Bold	G
Scalar (dot, inner) product	Single dot	x̂ · E
Vector (cross) product	Single cross	x̂ × E
Tensor (dyadic, outer) product or dyad	No dot	RR

Furthermore, we...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Series Preface
• Preface
• Authors
• Acronyms
• Chapter 1 Introduction
• Chapter 2 Maxwell’s Equations
• Chapter 3 Finite-Difference Time-Domain Method
• Chapter 4 Finite-Difference Modelling of Straight Waveguides
• Chapter 5 Modelling of Nonlinear Propagation in Waveguides
• Chapter 6 The Modal Method
• Chapter 7 Green’s Function Integral Equation Methods for Electromagnetic Scattering Problems
• Chapter 8 Finite Element Method
• Index