Nonstandard FDTD Theory

4. Conductive Media

The conductive Maxwell's equations are defined by

. . . (1)

where σ, σ^* are the electric and magnetic conductivities, respectively. In conductive media, electromagnetic waves are exponentially decayed. Here we apply the standard and nonstandard FDTD algorithms and stabilities to conductive media.

Standard FDTD Algorithm for the Absorbing Wave Equation

Applying ∇× to both sides of (1) and using the vector identity, ∇×∇×E = ∇(∇⋅E) - ∇²E, we obtain an absorbing wave equation in homogeneous media,

. . . (2)

where

. . . (3)

The solution of the absorbing wave equation is

. . . (4)

where

. . . (5)

Using the S-FDTD algorithm, we obtain

. . . (6)

where α = aΔt, α^* = a^*Δt, and

. . . (7)

Let x = ih, y = jh, z = kh, t = nΔt (i,j,k,n are integers). Expanding the temoporal difference operators, we obtain a "S-FDTD algorithm for the absorbing wave equation,"

. . . (8)

where we write ψ_i,j,kⁿ = ψ(ih,jh,kh,nΔt).

Nonstandard FDTD Algorithm for the Absorbing Wave Equation

The NS-FDTD algorithm is derived by optimizing to the monochromatic wave (4). For the spatial derivative, we already have a high accuracy difference operator,

. . . (9)

Vanishing temporal difference errors, we require that

. . . (10)

we choose coefficients as (10) is exactly satisfied. The temporal-spatial difference operators give

. . . (11)

where

. . . (12)

Substituting (11) into (10) and requiring that the imaginary part vanishes, we find

. . . (13)

Setting the real part to zero, we obtain

. . . (14)

Expanding the temporal difference operators, we obtain a "NS-FDTD algorithm for the absorbing wave equation,"

. . . (15)

Standard Yee Algorithm for Conductive Maxwell's Equations

Using the S-FDTD algorithm in the same way as the 3D Yee cell, the conductive Maxwell's equations are discretized by

. . . (16)

Since H(r, t) and E(r, t+Δt) are not matched to the temporal arrangement in the Yee cell, we approximate

. . . (17)

Expanding the temporal differences, we obtain a "S-Yee algorithm for conductive Maxwell's equations,"

. . . (18)

where α = σΔt/(2ε) and α^* = σ^*Δt/(2μ).

Nonstandard Yee Algorithm for Conductive Maxwell's Equations

Based on the non-conductive NS-FDTD algorithm, we define

. . . (19)

where Z = (μ/ε)^1/2. (19) is a high accracy form optimized to monochromatic wave propagation, because it is equivalent to the NS-FDTD algorithm for the absorbing wave equation (15). Expanding the temporal differences and using (13), we obtain a "NS-Yee algorithm for conductive Maxwell's equations,"