Maxwell's Equations

We now take a brief look at Maxwell's equations, the (3+1)D system of PDEs which describes the time evolution of electromagnetic fields. This system was the original motivation behind the development of FDTD [184,214], and MDWD network methods for Maxwell's system were explored early on in [50]. In the interest of solidifying the link between these two types of methods, we show how a passive circuit representation yields a DWN, which is no more than a scattering form of FDTD.

Maxwell's Equations, for a linear isotropic (though not necessarily spatially homogeneous) medium, are usually written in vector form as

$$\epsilon \frac{\partial {\bf E}}{\partial t} = \nabla \times {\bf H}\hspace{0.5in}\mu \frac{\partial {\bf H}}{\partial t} = -\nabla \times {\bf E}$$ (4.127)

where ${\bf E} = [E_{x}, E_{y}, E_{z}]^{T}$ and ${\bf H} = [H_{x}, H_{y}, H_{z}]^{T}$ are, respectively, the electric and magnetic field vectors, $\epsilon(x,y,z)$ and $\mu(x,y,z)$ are the dielectric constant and magnetic permeability of the medium, assumed positive and bounded away from 0. (We have left out losses here.) This system has the form of (3.1), with ${\bf w} = [{\bf E}^{T}, {\bf H}^{T}]^{T}$, and

$${\bf P} = \begin{bmatrix}\epsilon {\bf I}_{3}&\cdot\\ \cdot&\mu{\... ...A}_{j\times}\\ {\bf A}_{j\times}^{T}&\cdot\\ \end{bmatrix}\hspace{0.2in}j=1,2,3$$

where ${\bf I}_{3}$ is the $3\times 3$ identity matrix, $\cdot$ stands for zero entries, and where we also have

$${\bf A}_{1\times} = \begin{bmatrix}0&0&0\\ 0&0&1\\ 0&-1&0\\ \end{... ...0.4in}{\bf A}_{3\times} = \begin{bmatrix}0&1&0\\ -1&0&0\\ 0&0&0\\ \end{bmatrix}$$