Computational electromagnetics

Case studies

The following pages describe the basics of computational electromagnetics, starting with relevant derivations and the basics of classical electromagnetism. The subpages include case studies with analytical solutions if they exist and numerical solutions.

• Triple dielectric step in 1D
• Photonic crystal in 1D
• Scattering from an infinite cylinder
• Point source near an anisotropic lens

Classical electromagnetism

Maxwell's equations in matter

Classical electrodynamics is historically one of the most eminent fields of physics as an extension of classical mechanics, it is very successful in explaining a plethora of phenomena. The dynamics of electric and magnetic fields are described with Maxwell's equations. As we will be studying the interaction of electromagnetic waves with different objects, we need Maxwell's equations in matter \( \newcommand{\dpar}[2]{\frac{\partial #1}{\partial #2}} \begin{align} &\nabla \times \b{E}(\b{r}, t) = - \dpar{\b{B}(\b{r}, t)}{t}, \label{eq:TFaraday} \\ &\nabla \times \b{H}(\b{r}, t) = \b{j}(\b{r}, t) + \dpar{\b{D}(\b{r}, t)}{t}, \label{eq:TMaxwell-Ampere} \\ &\nabla \cdot \b{D}(\b{r}, t) = \rho(\b{r}, t), \label{eq:TGaussE} \\ &\nabla \cdot \b{B}(\b{r}, t) = 0. \label{eq:TGaussM} \end{align} \)

The system of equations contains four fields. The electric field strength $\b E$ and density $\b D$ and the magnetic field strength $\b H$ and density $\b B$. The four fields are accompanied by the current density $\b j$ and the charge density $\rho$. For a full description of electromagnetic phenomena, we need to provide another two constitutive equations, that relate the strength and density of the fields \( \newcommand{\eps}{\varepsilon} \begin{align} \b{B} &= \mu_0 \mu \b{H}, \label{eq:constM} \\ \b{D} &= \eps_0 \eps \b{E} \label{eq:constE} \end{align} \) where $\varepsilon_0$ and $\mu_0$ are vacuum permittivity and permeability respectively. The dielectric function $\varepsilon$ and magnetic permeability $\mu$ are in general dependant on both $\b E$ and $\b B$ as well as the frequency $\omega$ and can be second order tensors in anisotropic materials. Equations \eqref{eq:constM} and \eqref{eq:constE} already assume linear material properties, generally the polarisation $\b P$ and magnetisation are defined as power series expansions of the electric field density and magnetic field density, as \( \begin{align} \label{eq:PMexpans} \b{P} &= \chi_E \b{D} + \mathcal{O}(D^2), \\ \b{M} &= \chi_M \b{H} + \mathcal{O}(H^2). \end{align} \) The material linearity assumption holds well for small external fields, meaning small $\b D$ and $\b H$. The treatment of nonlinear terms falls within the field of nonlinear optics and is not relevant for our discussion here.