Saswatee Banerjee

Advances in finite-difference time-domain calculation methods

Although the finite-difference time-domain (FDTD) method was developed in the 1960s, beginning with Yee’s famous algorithm [1], and many advances have been made since then, FDTD is still an active field of research.

Simulation of arrays of metal nano-particles using a monochromatic recursive convolution finite-difference time-domain method

We developed a monochromatic version of recursive convolution finite-difference time-domain method and used the same to compute the extinction spectra of arrays of silver nanowires. Results show that the localized surface plasmon resonance properties depend on the separation between two neighboring nanowires.

Finite Difference Models of the Simple Harmonic Oscillator

The simple harmonic oscillator (SHO) is the basis of many physical models. Since it can be solved analytically, it is a useful vehicle to introduce the basic concepts of the FDTD methodology without being distracted by the technical details needed to implement FDTD for the wave equation and for Maxwell’s equations. In this chapter we derive exact FDTD algorithms from nonstandard FD models to solve the free, damped, and damped and driven SHO. The concepts developed in this chapter will be subsequently extended to develop nonstandard, high-precision versions of FDTD to solve the wave equation and Maxwell’s equations, which is one of the main purposes of this book. We first develop analytical solutions against which numerical solutions can be compared. Topics: • FDTD algorithms • Second-order versus fourth-order models • Nonstandard finite difference models • Numerical stability • Discrete Green’s function

Example Applications of FDTD

The basic development of the FDTD methodology is now complete, and we now proceed to set up a few example problems. This and the remaining chapters are devoted to applications and design problems, and to the extension of the FDTD methodology to dispersive materials.

The Yee Algorithm in One Dimension

The Yee algorithm is one realization (others are possible) of the FDTD methodology to solve Maxwell’s equations based on second-order central finite differences. In this chapter we develop the Yee algorithm for a linear nondispersive medium. Following our strategy of stepwise increasing complexity, we develop the Yee algorithm in one dimension to avoid the distracting technical details that arise in two and three. Nonetheless, there are many ‘devils in the details.’

Accuracy, Stability, and Convergence of Numerical Algorithms

Many real phenomena are too complicated to comprehend, but we can attempt to understand them by constructing models that contain their essential features while ignoring minor complications. (What is essential is not always obvious, and the choice may depend on one’s purpose.) Models can yield deep insight into the nature of reality. A good model not only explains a phenomenon, but also predicts things yet to be observed. The quark model was devised in the early 1960s to simply explain the puzzling multiplicity of hadrons (that were being discovered) in terms of a few simple (presumed to be imaginary) elementary particles called quarks. No one believed that quarks actually existed, but later it was found that they do exist. It is remarkable indeed that the best model seems to be the simplest one that accounts for the facts - the principle of Occam’s razor. The FDTD methodology, the main subject of this book, is appealing because of its simplicity. Methodologies such as Feynman diagrams can take on lives of their own and even indicate new phenomena. A finite difference model of the logistic equation, which gives the wrong solution, has been used to study chaos (see Section 2.6.1). It has even been proposed that reality itself is just a set of algorithms. Indeed, in this book we will encounter algorithms meant to solve one problem that lead to solutions and models of other problems that explain - at least heuristically - certain real physical phenomena. Before embarking on the main topic of this book, we introduce a few example algorithms and analyze their accuracy and numerical stability. Our purpose is not only to show what can go wrong in computer calculations and how to avert it, but also to illustrate the physical insights that some of these algorithms yield. Topics • Numerical instability and chaos • Accuracy • Nonlinear problems • Parallel versus serial programs • Matrix analysis of a relaxation algorithm • Noninteger-order integration and differentiation

FDTD for Dispersive Materials

Many nontransparent materials used in optics and optoelectronics devices show frequency-dependent complex electrical permittivity in the visible domain. The complex refractive index of bulk material is an important parameter that determines the optical behavior of a given material and in many cases any nanophotonic structure that might be fashioned from such a material. The complex refractive index for materials in the visible domain is given by the square root of the electrical permittivity [Eq. (8.36)]. The complex refractive index constitutes an important simulation parameter in optics and photonics. The imaginary parts of the refractive index and the electrical permittivity are associated with the absorption and the conductivity of the material, respectively. It is also important to understand that the implementation of FDTD depends on the actual relations among various components of the electric and magnetic fields, which in turn are determined by the nature of material properties. In this chapter we develop a simple dispersive algorithm that can be used when metal nanostructures need to be simulated in the visible domain.

Finite Difference Approximations

Let us begin by explaining the “finite difference” (FD) of the finite difference time domain (FDTD) methodology. We review the basic concepts and mathematical properties of different FD expressions, and introduce the basic concepts and the notation to be used throughout this book. Topics: • Elementary finite difference expressions • Nonstandard finite difference expressions • Computational molecules

The 1D Wave Equation

In a uniform medium, Maxwell’s equations reduce to the wave equation, and analytic solutions are derived by solving the wave equation subject to the appropriate boundary conditions. Although the 1D wave equation may seem simple, when boundary and initial conditions are specified and when sources are included, its analytical solutions are far from trivial and sometimes do not even exist. Before moving on to FDTD for the wave equation, we develop some analytical solutions to which numerical solutions can be compared. Topics: • Derivation and various forms of wave equation • Analytical solutions • Extra topics: relativity from the wave equation • Green’s function and associated mathematics

Optical Properties of a Film of Composite Medium with Random Distribution of Metal Nano-inclusions

We computed the transmission spectra due to a dielectric film impregnated with metal nano-inclusions. Effect of formation of aggregates is studied. A home-grown recursive-convolution finite-difference time-domain method is used for computation.