Finite-Difference Time-Domain Methods for Electrodynamics

Abstract

The FDTD method belongs in the general class of grid-based differential numerical modeling methods (finite-difference methods). The time-dependent Maxwell’s equations (in partial differential form) are discretized using central-difference approximations to the space and time partial derivatives. The resulting finite-difference equations are solved in either software or hardware in a leapfrog manner: the electric field vector components in a volume of space are solved at a given instant in time; then the magnetic field vector components in the same spatial volume are solved at the next instant in time; and the process is repeated over and over again until the desired transient or steady-state electromagnetic field behavior is fully evolved.”

Appendices

7.19 Appendix: The Yee Scheme Is Exact in 1D for the “Magic” Time Step

For easy reference, let us repeat the curl-E Yee equation (7.12) in 1D:

$$\begin{aligned} \zeta \, \frac{E_{n+1, m} - E_{n,m}}{\mathrm {\Delta }x} \,=\, -\mu \, \frac{H_{n+1/2,m+1/2} - H_{n+1/2,m-1/2}}{\mathrm {\Delta }t} \end{aligned}$$

(7.158)

and consider the “magic” case $\mathrm {\Delta }x = v_p \mathrm {\Delta }t$. The Yee equation then becomes

$$\begin{aligned} \zeta \, (E_{n+1, m} - E_{n,m}) \,=\, -\mu v_p \, (H_{n+1/2,m+1/2} - H_{n+1/2,m-1/2}) \end{aligned}$$

(7.159)

We intend to show that this FD equation is in fact exact for any traveling wave (TW)—for definiteness, moving in the $+x$ direction:

$$\begin{aligned} E_{TW}(x, t) \,=\, E_0 f(x - v_p t); ~~~ H_{TW}(x, t) \,=\, H_0 f(x - v_p t) \end{aligned}$$

(7.160)

$$ Z H_0 = E_0, ~~ Z = \sqrt{\frac{\mu }{\epsilon }} $$

here f is an arbitrary differentiable function, Z is the intrinsic impedance of the medium, and $E_0$ is an arbitrary amplitude. Substituting this wave into the left-hand side of (7.159), we have

$$ \mathrm {l.h.s.} = \zeta \, E_0 \, \left[ f(x_n + \mathrm {\Delta }x - v_p t_m) - f(x_n - v_p t_m) \right] $$

Similarly, the right-hand side is

$$ \mathrm {r.h.s.} = -\mu v_p H_0 \, \left[ f(x_n + \frac{1}{2} \mathrm {\Delta }x - v_p (t_m + \frac{1}{2} \mathrm {\Delta }t)) - f(x_n + \frac{1}{2} \mathrm {\Delta }x - v_p (t_m - \frac{1}{2} \mathrm {\Delta }t) \right] $$

which for the “magic” step simplifies to

$$ \mathrm {r.h.s.} = \mu v_p H_0 \, \left[ f(x_n + \mathrm {\Delta }x - v_p t_m) - f(x_n - v_p t_m) \right] $$

Since $v_p = \zeta /\sqrt{\mu \epsilon }$ and $H_0 = E_0 \sqrt{\epsilon /\mu }$, we see that indeed r.h.s. = l.h.s.    $\square$

7.20 Appendix: Green’s Functions for Maxwell’s Equations

7.1.1 7.20.1 Green’s Functions for the Helmholtz Equation

By definition, Green’s function $g(\mathbf {r})$ for the Helmholtz equation satisfies, in n dimensions, the equation

$$\begin{aligned} -\nabla ^2 g(\mathbf {r}) - k^2 g(\mathbf {r}) \,=\, \delta (\mathbf {r}) \end{aligned}$$

(7.161)

and the Sommerfeld radiation condition

$$\begin{aligned} \lim _{r \rightarrow \infty } r^{\frac{1}{2}(n-1)} (\partial _r g \pm ik g) \,=\, 0 \quad \mathrm {for~the} ~\exp (\pm i \omega t) ~ \mathrm {phasor~convention} \end{aligned}$$

(7.162)

As usual, $\delta (\mathbf {r})$ is the Dirac delta function, and $k > 0$ is a fixed wavenumber. The negative sign in the left-hand side of (7.161) is introduced as just a matter of convenience, to eliminate the respective negative sign in the expressions for Green’s functions. The radiation condition reflects the fact that under the “electrical engineering” phasor convention $\exp (+i \omega t)$ [$\equiv \exp (+j \omega t)$], the phase of an outgoing wave is asymptotically $\exp (-ikr)$. Under the “physics” phasor convention $\exp (-i \omega t)$, the signs are opposite.

Expressions  for Green’s functions in 1D, 2D and 3D are widely available from various sources and are given below without derivation. They can be verified by direct substitution in the Helmholtz equation, provided that the singularity at the origin is treated with care in the sense of distributions.

$$\begin{aligned} g_{1D}(x) \,=\, \frac{1}{2ik} \exp (\mp i k |x|) \quad \mathrm {for} ~\exp (\pm i \omega t) \end{aligned}$$

(7.163)

$$\begin{aligned} g_{2D}(r) \,=\, \frac{i}{4} h_0^{(\gamma )} (kr) \end{aligned}$$

(7.164)

where $h_0^{(\gamma )}$ is the Hankel function of type $\gamma = 1$ for the $\exp (-i \omega t)$ phasor convention, and of type $\gamma = 2$ for $\exp (+i \omega t)$. Hankel functions are numerically expensive to compute, so their far-field ($kr \rightarrow \infty$) asymptotic expansions are ordinarily used:

$$ h_0^{(\gamma )} (z) \, \sim \, \sqrt{\frac{2}{\pi z}} \, \exp \left( -i\left( z - \frac{\pi }{4} \right) \right) $$

Finally,

$$\begin{aligned} g_{3D}(r) \,=\, \frac{\exp (\mp ikr)}{4\pi r} \quad \mathrm {for} ~\exp (\pm i \omega t) \end{aligned}$$

(7.165)

7.1.2 7.20.2 Maxwell’s Equations

Maxwell’s equations with a nonzero right-hand side, in the frequency domain, in a homogeneous linear isotropic medium filling the whole 3D space, are written in the SI system under the $\exp (i \omega t)$ phasor convention as

$$\begin{aligned} \nabla \times \mathbf {E}\,=\, -i\omega \mathbf {B}\,+\, \mathbf {J}_m \end{aligned}$$

(7.166)

$$\begin{aligned} \nabla \times \mathbf {H}\,=\, i\omega \mathbf {D}\,+\, \mathbf {J}_e \end{aligned}$$

(7.167)

$$\begin{aligned} \mathbf {D}= \epsilon \mathbf {E}, ~~~ \mathbf {B}= \mu \mathbf {H}\end{aligned}$$

(7.168)

where $\epsilon$ is the dielectric permittivity, and $\mu$ is the magnetic permeability; $\mathbf {J}_m$ and $\mathbf {J}_e$ are magnetic and electric current densities, respectively. These currents may be either auxiliary quantities introduced for mathematical analysis or, in the case of $\mathbf {J}_e$, actual electric currents (however, see Sect. 10.2). Assume further that the whole space is filled with a linear isotropic homogeneous medium, with a permittivity $\epsilon$ and a permeability $\mu$.

By applying the divergence operator to (7.166), (7.167), one immediately obtains

$$\begin{aligned} \nabla \cdot \mathbf {B}~=~ -\frac{i}{\omega } \, \nabla \cdot \mathbf {J}_m \end{aligned}$$

(7.169)

$$\begin{aligned} \nabla \cdot \mathbf {D}~=~ \frac{i}{\omega } \, \nabla \cdot \mathbf {J}_e \end{aligned}$$

(7.170)

It is convenient to consider the general excitation as a superposition of electromagnetic fields induced by the electric current $\mathbf {J}_e$ alone, and by the “magnetic current” $\mathbf {J}_m$ alone.

7.1.3 7.20.3 Subcase $J_m = 0$

First, let us consider the fields induced by $\mathbf {J}_e$, with $J_m = 0$. Then, from (7.166), $\nabla \cdot \mathbf {B}= 0$ and $\mathbf {B}= \nabla \times \mathbf {A}$, where $\mathbf {A}$ is the magnetic vector potential; hence, also from from (7.166),

$$\begin{aligned} \mathbf {E}\,=\, -i\omega \mathbf {A}- \nabla \phi \end{aligned}$$

(7.171)

where $\phi$ is an electric scalar potential. Then (7.167) becomes

$$\begin{aligned} \nabla \times \mathbf {H}\,=\, i\omega \epsilon (-i\omega \mathbf {A}- \nabla \phi ) \,+\, \mathbf {J}_e \end{aligned}$$

(7.172)

or

$$\begin{aligned} \nabla \times \nabla \times \mathbf {A}\,=\, \mu \epsilon \left( \omega ^2 \mathbf {A}- i\omega \nabla \phi \right) \,+\, \mu \mathbf {J}_e \end{aligned}$$

(7.173)

Using the standard calculus identity $\nabla \times \nabla = \nabla \nabla \cdot - \nabla ^2$, one rewrites (7.173) in terms of the Laplace operator:

$$\begin{aligned} - \nabla ^2 \mathbf {A}\,=\, \mu \epsilon \left( \omega ^2 \mathbf {A}- i\omega \nabla \phi \right) \,-\, \nabla \nabla \cdot \mathbf {A}\,+\, \mu \mathbf {J}_e \end{aligned}$$

(7.174)

So far only the curl of $\mathbf {A}$ has been defined (it is equal to $\mathbf {B}$), and therefore the magnetic vector potential is not unique. One can choose the divergence of $\mathbf {A}$ arbitrarily. A convenient choice simplifying (7.174) is the Lorenz gauge

$$\begin{aligned} \nabla \cdot \mathbf {A}\,=\, -i\omega \mu \epsilon \phi \end{aligned}$$

(7.175)

This gauge decouples $\mathbf {A}$ and $\phi$ in (7.174):

$$\begin{aligned} -(\nabla ^2 \mathbf {A}+ k^2 \mathbf {A}) \,=\, \mu \mathbf {J}_e, ~~~ k^2 = \omega ^2 \mu \epsilon \end{aligned}$$

(7.176)

The respective equation for $\phi$ can be obtained from (7.170), (7.171), with $\mathbf {D}= \epsilon \mathbf {E}$:

$$ -i\omega \epsilon \nabla \cdot \mathbf {A}- \epsilon \nabla ^2 \phi = \frac{i}{\omega } \nabla \cdot \mathbf {J}_e $$

With the Lorenz gauge, this simplifies to

$$ i\omega (i\omega \mu \epsilon \phi ) - \nabla ^2 \phi = \frac{i}{\omega \epsilon } \nabla \cdot \mathbf {J}_e $$

or

$$\begin{aligned} -(\nabla ^2 + k^2) \phi = \frac{i}{\omega \epsilon } \nabla \cdot \mathbf {J}_e \end{aligned}$$

(7.177)

Let $g(\mathbf {r})$ be Green’s function of the scalar Helmholtz operator $\nabla ^2 + k^2$ in the left-hand side of (7.177). Explicit expressions for $g(\mathbf {r})$, with relevant references, are given in Sect. 7.20.1. Solutions of Eqs. (7.176), (7.177) can then be written as

$$\begin{aligned} \mathbf {A}= \mu \mathbf {J}_e * g \end{aligned}$$

(7.178)

$$\begin{aligned} \phi = \left( \frac{i}{\omega \epsilon } \nabla \cdot \mathbf {J}_e \right) * g \end{aligned}$$

(7.179)

where “*” denotes the standard convolution integral over the whole space. From these expressions for the potentials, one obtains the electric field (7.171)

$$\begin{aligned} \mathbf {E}\,=\, -i\omega \mu \mathbf {J}_e * g - \nabla \frac{i}{\omega \epsilon } \nabla \cdot \mathbf {J}_e * g = -i\omega \mu \mathbf {J}_e * g - \frac{i}{\omega \epsilon } \nabla \nabla \cdot \mathbf {J}_e * g \end{aligned}$$

(7.180)

or

$$\begin{aligned} \mathbf {E}\, = \, \left( -i\omega \mu - \frac{i}{\omega \epsilon } \nabla \nabla \right) g * \mathbf {J}_e \end{aligned}$$

(7.181)

The last transformation relies on the fact that differentiation (in this case, the $\nabla$ operator) can be applied to either term of a convolution. In general, care must be exercised when dealing with the singularity of Green’s functions at the origin; however, in far-field analysis this issue does not arise, since all Green’s functions in the far field are smooth.

Readers familiar with the notion of dyadic Green’s functions will recognize the first term in the convolution (7.181) as exactly that; but such readers are likely to be familiar with the material of this section anyway. Dyadic Green’s functions are not used anywhere else in this book, so (7.180) is sufficient for our needs.

With the electric field determined by the convolution integral (7.180) or, equivalently, (7.181), the magnetic field is found from the Maxwell curl equation as

$$\begin{aligned} \mathbf {H}= \frac{i}{\omega \mu } \nabla \times \mathbf {E}\end{aligned}$$

(7.182)

7.1.4 7.20.4 Subcase $J_e = 0$

The analysis of fields induced by $\mathbf {J}_m$ completely similar to the analysis for $\mathbf {J}_e$ of the previous section, so only a brief summary is needed. To avoid cumbersome notation, the fields in this section are still denoted generically as $\mathbf {E}$ and $\mathbf {H}$, but one should bear in mind that these fields are induced by different sources and hence differ from the fields in the previous section.

If $J_e = 0$, then, from (7.167), $\nabla \cdot \mathbf {D}= 0$ and $\mathbf {D}= \nabla \times \mathbf {F}$, where $\mathbf {F}$ is the electric vector potential; hence

$$\begin{aligned} \mathbf {H}\,=\, i\omega \mathbf {F}- \nabla \psi \end{aligned}$$

(7.183)

Here $\psi$ is the magnetic scalar potential. Then (7.166) becomes

$$\begin{aligned} \nabla \times \mathbf {E}\,=\, -i\omega \mu (i\omega \mathbf {F}- \nabla \psi ) \,+\, \mathbf {J}_m \end{aligned}$$

(7.184)

or

$$\begin{aligned} \nabla \times (\epsilon ^{-1} \nabla \times \mathbf {F}) \,=\, -i\omega \mu (i\omega \mathbf {F}- \nabla \psi ) \,+\, \mathbf {J}_m \end{aligned}$$

(7.185)

Then,

$$\begin{aligned} -\nabla ^2 \mathbf {F}+ \nabla \nabla \cdot \mathbf {F}\,=\, \omega ^2 \mu \epsilon \mathbf {F}+ i\omega \mu \epsilon \nabla \psi \,+\, \epsilon \mathbf {J}_m \end{aligned}$$

(7.186)

The Lorenz-like gauge

$$\begin{aligned} \nabla \cdot \mathbf {F}\,=\, i\omega \mu \epsilon \psi \end{aligned}$$

(7.187)

decouples $\mathbf {F}$ and $\psi$ in (7.186):

$$\begin{aligned} -(\nabla ^2 \mathbf {F}+ k^2 \mathbf {F}) \,=\, \epsilon \, \mathbf {J}_m \end{aligned}$$

(7.188)

For the magnetic scalar potential $\psi$, one obtains, from (7.169) (7.183) with $\mathbf {B}= \mu \mathbf {H}$:

$$ \nabla \cdot \mu (i\omega \mathbf {F}- \nabla \psi ) \,=\, -\frac{i}{\omega } \, \nabla \cdot \mathbf {J}_m $$

$$ i\omega \mu \nabla \cdot \mathbf {F}- \mu \nabla ^2 \psi = -\frac{i}{\omega } \nabla \cdot \mathbf {J}_m $$

With the gauge (7.187), this simplifies to

$$\begin{aligned} -(\nabla ^2 + k^2) \psi = -\frac{i}{\omega \mu } \nabla \cdot \mathbf {J}_m \end{aligned}$$

(7.189)

Solutions of Eqs. (7.188), (7.189) can be written as convolutions with Green’s function

$$\begin{aligned} \mathbf {F}= \epsilon \mathbf {J}_m * g \end{aligned}$$

(7.190)

$$\begin{aligned} \psi = -\left( \frac{i}{\omega \mu } \nabla \cdot \mathbf {J}_m \right) * g \end{aligned}$$

(7.191)

The magnetic field (7.183)

$$\begin{aligned} \mathbf {H}\,=\, i\omega \mathbf {F}- \nabla \psi = i\omega \epsilon \mathbf {J}_m * g + \nabla \left( \frac{i}{\omega \mu } \nabla \cdot \mathbf {J}_m \right) * g \end{aligned}$$

(7.192)

or

$$\begin{aligned} \mathbf {H}\, = \, \left( i\omega \epsilon + \frac{i}{\omega \mu } \nabla \nabla \right) g * \mathbf {J}_m \end{aligned}$$

(7.193)

$$\begin{aligned} \mathbf {E}= -\frac{i}{\omega \epsilon } \nabla \times \mathbf {H}\end{aligned}$$

(7.194)

7.1.5 7.20.5 Excitation by both Electric and “Magnetic” Currents

In general, when both $\mathbf {J}_e$ and $\mathbf {J}_m$ are nonzero, the electromagnetic field is just a superposition of fields induced by $\mathbf {J}_e$ and $\mathbf {J}_m$ separately—that is, the sum of the final results of the previous two sections. For the electric field, the relevant equations are (7.180) or (7.181) and (7.182); for the magnetic field—(7.192) or (7.193) and (7.194).

7.1.6 7.20.6 Summary: Near-to-Far-Field Transformation

The key ideas of near-to-far-field transformation in the frequency domain can thus be summarized as follows.

• In a typical wave scattering problem, sources and inhomogeneities (e.g. scattering bodies) are confined within a bounded domain $\mathrm {\Omega }$, the exterior of which is homogeneous (usually just air). The field inside $\mathrm {\Omega }$ can be simulated using FDTD, with proper absorbing conditions or PML on the boundary $\partial \mathrm {\Omega }$.

• The far-field pattern, many wavelengths away from $\mathrm {\Omega }$, is very often of great practical interest. The objective of near-to-far-field transformation is to derive that pattern from the near field, which has been computed within $\mathrm {\Omega }$.

• To this end, it is convenient to consider an auxiliary field, equal to the actual field in the exterior domain but artificially set to zero inside $\mathrm {\Omega }$. It is clear that this auxiliary field satisfies Maxwell’s equations everywhere except for the boundary $\partial \mathrm {\Omega }$, where jumps in the tangential components of the $\mathbf {E}$ and $\mathbf {H}$ fields are conceptually attributable to surface currents—electric and magnetic.

• These surface currents can then be viewed as sources of the far field. Since the exterior domain is homogeneous, the far field can be expressed as a convolution of the surface currents with Green’s function (Sect. 7.20). In 2D, Green’s function is a computationally expensive Hankel function, but large-argument asymptotic expansions can be used for the far field. Convolutions with 3D Green’s functions are described above.

In the time domain, the near-to-far-field transformation is, naturally, much more involved. The interested reader is referred to papers by T. B. Hansen & A. D. Yaghjian, S. González García et al., A. Shlivinski & A. Boag, J. Li & B. Shanker, C.-C. Oetting & L. Klinkenbusch [HY94b,HY94a,GGG00, OK05, SB09, LS15].