Time Domain Equation

The relations μH = ∇ × A and E = −∂A/∂t (using the gauge for which the scalar electric potential vanishes) make it possible to rewrite Maxwell-Ampère’s law using the magnetic potential.

This is the equation used by The Electromagnetic Waves, Transient Interface. It is suitable for the simulation of nonsinusoidal waveforms or nonlinear media.

Using the relation εr = n2, where n is the refractive index, the equations can alternatively be written

In 2D, different polarizations can be chosen by selecting to solve for a subset of the 3D vector components. When selecting all three components, the 3D equation applies with the addition that out-of-plane spatial derivatives are set to zero.

Solving for all three components in 2D is referred to as “hybrid-mode waves”. The equation form is formally the same as in 3D with the addition that out-of-plane spatial derivatives are set to zero.

The TM waves polarization has only one magnetic field component in the z direction, and thus the electric field and vector potential lie in the modeling plane. Hence it is obtained by solving only for the in-plane vector potential components. The equation is formally the same as in 3D, the only difference being that the out-of-plane vector potential component is zero everywhere and that out-of-plane spatial derivatives are set to zero.

As the field propagates in the modeling xy-plane a TE wave has only one nonzero vector potential component, namely in the z direction. The magnetic field lies in the modeling plane. Thus the equation in the time domain can be simplified to a scalar equation for Az:

Using the relation εr = n2, where n is the refractive index, the equation can alternatively be written

When using the refractive index, the assumption is that μr = 1 and σ = 0 and only the constitutive relations for linear materials can be used.

Solving for all three components in 2D is referred to as “hybrid-mode waves”. The equation form is formally the same as in 3D with the addition that spatial derivatives with respect to

are set to zero.

TM waves have a magnetic field with only a

component and thus an electric field and a magnetic vector potential with components in the rz-plane only. The equation is formally the same as in 3D, the only difference being that the

component is zero everywhere and that spatial derivatives with respect to

are set to zero.

A TE wave has only a vector potential component in the

direction, and the magnetic field lies in the modeling plane. Given these constraints, the 3D equation can be simplified to a scalar equation for

. To write the fields in this form, it is also required that εr and μr are nondiagonal only in the rz-plane. μr denotes a 2-by-2 tensor, while

and

are the relative permittivity and conductivity in the

direction.