364 Pages 96 B/W Illustrations
    by CRC Press

    364 Pages 96 B/W Illustrations
    by CRC Press

    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.

    Introduction

    Maxwell’s Equations

    Notation

    Maxwell’s Equations

    Material Equations

    Frequency Domain

    1D and 2D Maxwell’s Equations

    Wave Equations

    Waveguides and Eigenmodes

    FDTD

    Introduction

    Numerical Dispersion and Stability Analysis of the FDTD Method

    Making Your Own 1D FDTD

    Absorbing Boundary Conditions

    FDTD Method for Materials with Frequency Dispersion

    FDTD Method for Nonlinear Materials, Materials with Gain and Lasing

    Conclusion

    Exercises

    References

    Finite-Difference Modeling of Straight Waveguides

    Introduction

    General Considerations

    Modified Finite-Difference Operators

    Numerical Linear Algebra in MATLAB®

    Two-Dimensional Waveguides and the Yee Mesh

    Exercises

    Modeling of Nonlinear Propagation in Waveguides

    Introduction

    Formalism

    Nonlinear Polarization

    The Nonlinear Schrödinger Equation

    Numerical Implementation

    Exercises

    The Modal Method

    Introduction

    Eigenmodes

    The 1D Geometry

    The 2D Geometry

    Periodic Structures

    Current Sources

    Exercises

    References

    Green’s Function Integral Equation Methods for Electromagnetic Scattering Problems

    Introduction

    Theoretical Foundation

    Green’s Function Area Integral Equation Method

    Green’s Function Volume Integral Equation Method

    Green’s Function Surface Integral Equation Method (2D)

    Construction of Two-Dimensional Green’s Functions for Layered Structures

    Construction of the Periodic Green’s Function

    Reflection from a Periodic Surface Microstructure

    Iterative Solution Scheme Taking Advantage of the Fast Fourier Transform

    Further Reading

    Exercises

    References

    Finite Element Method

    Introduction: Helmholtz Equation in 1D

    General Scattering Problem in 1D

    Mathematical Background: Maxwell and Helmholtz Scattering Problems and Their Variational Forms

    FEM for Helmholtz Scattering in 2D and 3D

    FEM for Maxwell Scattering in 2D and 3D

    Exercises

    Biography

    Andrei V. Lavrinenko, Technical University of Denmark, Kongens Lyngby
    Jesper Lægsgaard, Technical University of Denmark, Kongens Lyngby
    Niels Gregersen, Technical University of Denmark, Kongens Lyngby
    Frank Schmidt, Zuse Institute, Berlin, Germany
    Thomas Søndergaard, Aalborg University, Denmark

    "… useful to students and researchers who want to have a deeper understanding of the methods commonly used in computational electromagnetics. After addressing the basic principles, this book provides the readers with the details and mathematical/numerical framework of commonly used methods including FDTD, finite element, Green’s function, and modal. It then goes on to more advanced topics such as modelling nonlinear materials and materials with gain. This book is a useful addition to the library of any research university."
    —C T Chan, Hong Kong University of Science and Technology