Electromagnetic Simulation Using the FDTD Method with Python. Dennis M. Sullivan

1 ONE‐DIMENSIONAL SIMULATION WITH THE FDTD METHOD

      This chapter provides a step‐by‐step introduction to the finite‐difference time‐domain (FDTD) method, beginning with the simplest possible problem, the simulation of a pulse propagating in free space in one dimension. This example is used to illustrate the FDTD formulation. Subsequent sections lead to formulations for more complicated media.

1.1 ONE‐DIMENSIONAL FREE‐SPACE SIMULATION

      The time‐dependent Maxwell’s curl equations for free space are

(1.1a)

(1.1b)

E and H are vectors in three dimensions, so, in general, Eq. (1.1a) and (1.1b) represent three equations each. We will start with a simple one‐dimensional case using only E_x and H_y, so Eq. (1.1a) and (1.1b) become

      (1.2a)

      (1.2b)

      These are the equations of a plane wave traveling in the z direction with the electric field oriented in the x direction and the magnetic field oriented in the y direction.

      Taking the central difference approximations for both the temporal and spatial derivatives gives

(1.3a)

(1.3b)

In these two equations, time is specified by the superscripts, that is, n represents a time step, and the time t is t = Δt ⋅ n. Remember, we have to discretize everything for formulation into the computer. The term n + 1 means one time step later. The terms in parentheses represent distance, that is, k is used to calculate the distance z = Δx ⋅ k. (It might seem more sensible to use Δz as the incremental step because in this case we are going in the z direction. However, Δx is so commonly used for a spatial increment that we will use Δx.) The formulation of Eq. (1.3a) and (1.3b) assume that the E and H fields are interleaved in both space and time. H uses the arguments k + 1/2 and k − 1/2 to indicate that the H field values are assumed to be located between the E field values. This is illustrated in Fig. 1.1. Similarly, the n + 1/2 or n − 1/2 superscript indicates that it occurs slightly after or before n, respectively. Equations (1.3a) and (1.3b) can be rearranged in an iterative algorithm:

(1.4a)

(1.4b)

Notice that the calculations are interleaved in space and time. In Eq. (1.4a), for example, the new value of E_x is calculated from the previous value of E_x and the most recent values of H_y.

Figure 1.1 Interleaving of the E and H fields in space and time in the FDTD formulation. To calculate H_y, for instance, the neighboring values of E_x at k and k + 1 are needed. Similarly, to calculate E_x, the values of H_y at k + 1/2 and

      This is the fundamental paradigm of the FDTD method (1).

Equations (1.4a) and (1.4b) are very similar, but because ε₀ and μ₀ differ by several orders of magnitude, E_x and H_y will differ by several orders of magnitude. This is circumvented by making the following change of variables (2):

(1.5)

      Substituting this into Eq. (1.4a) and (1.4b) gives

(1.6a)

(1.6b)

      Once the cell size Δx is chosen, then the time step Δt is determined by

(1.7)

      where c₀ is the speed of light in free space. (The reason for this will be explained in Section 1.2.) Therefore, remembering

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