Physics · Electromagnetism · advanced

EM Wave Propagation

leapfrog E and B on a staggered grid

Light is E and B regenerating each other. A 1D Yee grid (FDTD) leapfrogs them in time, slows them in glass and blows up past the Courant limit.

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Interactive visual

Cyan is the electric field Ey (vertical), violet the magnetic field Hz (horizontal), launched from the orange source and absorbed at both ends; the translucent block is a dielectric slab. Raise εr to slow and shorten the wave inside the slab, and push the Courant number past 1 to watch the explicit scheme blow up.

What it is

An electromagnetic wave is a changing electric field that creates a changing magnetic field, which creates the electric field again. In a plane wave E and B are perpendicular to each other and to the direction of travel, in phase, and move at c = 1/√(με).

The finite-difference time-domain (FDTD) method solves Maxwell's curl equations directly. Kane Yee's trick is to store E and H on grids offset by half a cell in space and half a step in time, so every derivative is a centred difference. Each step updates H from the spatial change of E, then E from the spatial change of H: a leapfrog.

The scheme is explicit, so it has a stability limit. The Courant number S = cΔt/Δx must not exceed 1 in 1D (1/√2 in 2D, 1/√3 in 3D); above it, errors grow every step until the grid fills with garbage. Inside a dielectric with permittivity εr the wave slows by √εr, its wavelength shrinks, and part of it reflects at the boundary.

The same leapfrog pattern runs water ripples, audio propagation and sound waves in games; it is the grown-up version of the fixed timestep rule that stability depends on step size.

Key formulas

Maxwell's curl equations (1D, Ey and Hz): $ε \frac{\partial E _{y}}{\partial t} = - \frac{\partial H _{z}}{\partial x}, μ \frac{\partial H _{z}}{\partial t} = - \frac{\partial E _{y}}{\partial x}$
Yee update (normalised: c = 1, Δx = 1): $H_{i + \frac{1}{2}}^{n + \frac{1}{2}} = H_{i + \frac{1}{2}}^{n - \frac{1}{2}} - S (E_{i + 1}^{n} - E_{i}^{n}), E_{i}^{n + 1} = E_{i}^{n} - \frac{S}{ε _{r}} (H_{i + \frac{1}{2}}^{n + \frac{1}{2}} - H_{i - \frac{1}{2}}^{n + \frac{1}{2}})$
Courant stability: $S = \frac{c Δ t}{Δ x} \leq \frac{1}{d}$
d = number of spatial dimensions.
Speed and wavelength in a dielectric: $v = \frac{c}{ε _{r}}, λ = \frac{λ _{0}}{ε _{r}}$
Mur absorbing boundary (first order): $E_{0}^{n + 1} = E_{1}^{n} + \frac{S - 1}{S + 1} (E_{1}^{n + 1} - E_{0}^{n})$

Rules of thumb

Resolve the shortest wavelength (inside the densest material) with at least 10–20 cells, or numerical dispersion slows short waves.
Use S ≈ 0.9 of the limit; in 1D, S = 1 is the 'magic time step' where a pulse moves exactly one cell per step with zero dispersion.
Ramp a sinusoidal source over 1–2 periods; switching it on instantly launches a broadband spike.
A soft source (add to E) lets reflected waves pass through it; a hard source (overwrite E) reflects them.
Without an absorbing boundary the grid ends are perfect mirrors and you get standing waves.

Implementation

One 1D Yee step with a soft source and Mur boundaries:

export function step(ey: Float64Array, hz: Float64Array, eps: Float64Array,
                     S: number, src: number, value: number, mur: Float64Array) {
  const n = ey.length
  mur[0] = ey[0]; mur[1] = ey[1]; mur[2] = ey[n - 1]; mur[3] = ey[n - 2]
  for (let i = 0; i < n - 1; i++) hz[i] -= S * (ey[i + 1] - ey[i])
  for (let i = 1; i < n - 1; i++) ey[i] -= (S / eps[i]) * (hz[i] - hz[i - 1])
  ey[src] += value // soft source
  const m = (S - 1) / (S + 1)
  ey[0] = mur[1] + m * (ey[1] - mur[0])
  ey[n - 1] = mur[3] + m * (ey[n - 2] - mur[2])
}

Common mistakes

Storing E and H at the same grid points and times, which loses the second-order accuracy and the stability of the staggered scheme.
Picking Δt from the frame rate instead of from the Courant limit; a slow frame then makes S > 1 and the sim explodes.
Checking stability with the slow speed inside a dielectric. The limit is set by the fastest wave on the grid (vacuum).
Forgetting an absorbing boundary and mistaking the resulting standing wave for physics.

Prompt

Write a 1D FDTD (Yee) electromagnetic solver in TypeScript for a Three.js visual. Use normalised units (c = 1, Δx = 1) with Float64Array Ey on n nodes, Hz on n−1 half nodes and a per-node relative permittivity array. stepFdtd(grid, S, sourceIndex, source(t)) updates H then E, adds a soft source, applies first-order Mur boundaries at both ends and allocates nothing. Render Ey along y and Hz along z by writing into existing line geometries each frame, clamping non-finite values. Tests: at S = 1 a matched Gaussian pulse moves exactly one cell per step; at S = 0.5 the energy after the pulse exits is < 2% of the start (Mur absorbs); at S = 1.1 energy exceeds 1e6 within 400 steps.