Is there a wave equation describing electromagnetic waves in materials?

Question

Solutions to the usual wave equation for electromagnetic waves have the property that wave components of different wavelengths travel at the same phase speed $c$, so no dispersion occurs.

What I wonder is, is there a wave equation that describes electromagnetic waves propagating through a material? The equation would have to take into account dispersion so that $c = c(k)$ where $k$ is the wavenumber of the wave.

Is there anything sort of like this? If not, is there a reason why we should expect that no such equation exists?

Answer 1

The equations for EM propagation in materials (not necessarily waves) are still Maxwells equations.

• Electric Gauss Law: $\nabla D = \rho$
• Magnetic Gauss Law: $\nabla B = 0$
• Faraday Law: $\nabla \times E = -\frac{\partial B}{\partial t}$
• Ampere Law: $\nabla \times H = \frac{\partial E}{\partial t} + J$

However, these equations are not complete in the sense that we can readily solve them (given the appropriate bounday conditions).

What is missing is the relation between the $E$ and $D$ and $B$ and $H$

This is modeling exactly the material dependence.

In general one usually writes

• $D = \epsilon_0 E + P$
• $H = -\frac{1}{\mu_0} B + M$

with polarization $P = P(E, t)$ and magnetization $M = M(B,t)$. So one introduces these $P$ and $M$ terms to capture the deviation from behaviour of the vaccuum.

• For vaccuum these are ideally zero ($P = M = 0$).

• For materials there can be very complex relationships which might not even have a closed form solution.

• Think of e.g. the hystereis of a permanet magnet. There we have a dependency of $M$ on absolute time.

• Or consider a nonlinear material (which e.g. does optical frequency doubling). It is common to employ numerical quantum physical simulations to model the material behavior. There one calculates $P(E,t)$ which then serves as input to a equation solver for Maxwells equation.

• In the simplest case, one has a linear material which means that there is a convolution relationship between $P$ and $E$ and/or $M$ and $B$. In this case it is advantageous to transform Maxwells equation to frequency domain in which case they become linear coupled PDEs which can be solved easily (for each frequency point).

• For non-linear materials this transformation to frequency domain no longer possible/meaningful.

• For conductive materials ohms law $J = \sigma E$ for instance also has to be integrated in Maxwells equation. This is done by adding it into to the current term of Amperes Law. For frequency dependent behavior one again has a convolution relationship, which is best handled by transforming everything in the frequency domain first.

Note: A dispersion relation $c = c(k)$ is a result of all of these. It cannot be obtained (in a strict sense) for nonlinear material for instance.

Answer 2

On microscopic level propagation of EM waves in materials is described by the same vacuum Maxwell equations (from which the wave equation is trivially derived). However Macroscopic electrodynamics deals with the case where we are interested in length scales much bigger than inter-atomic distances: in this case one has to account for the polarization and magnetization, which are electric and magnetic field induced in the material averaged over a physically small volume - i.e., a volume that is much bigger than the inter-atomic space, but which is still very small on the scale of our problem. This is not unlike describing liquids or gases in terms of pressure and density, which is why macroscopic electrodynamics is sometimes called electrodynamics of continuous media.

One then reformulates Maxwell equations in terms of auxiliary fields: $$ \mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P},\\ \mathbf{H}=\frac{\mathbf{B}}{\mu_0}-\mathbf{M}, $$ where $\mathbf{P}$ and $\mathbf{M}$ are the polarization and the magnetization induced in the media. Maxwell equations are then incomplete - they need to be complemented by the material/constitutive relations that relate the auxiliary fields (or polarization and magnetization) to the true fields.
In the simplest case of isotropic and homogeneous medium, these relations are often taken to be $$ \mathbf{D}=\varepsilon\mathbf{E}, \mathbf{H}=\frac{\mathbf{B}}{\mu}. $$ However, one can consider more complicated cases, such as arbitrary isotropic linear medium, where we take $$ \mathbf{P}(\mathbf{r},t)=\varepsilon_0\int d^"\mathbf{r}'dt'\chi_e(\mathbf{r},t;\mathbf{r}',t')\mathbf{E}(\mathbf{r}',t'),\\ \mathbf{M}(\mathbf{r},t)=\frac{1}{\mu_0}\int d^"\mathbf{r}'dt'\chi_m(\mathbf{r},t;\mathbf{r}',t')\mathbf{B}(\mathbf{r}',t'). $$ Here $\chi_e,\chi_m$ are called electric and magnetic susceptibilities. This formulation already allows to include the dispersion as asked in the OP. (In a homogeneous medium $\chi_j(\mathbf{r},t;\mathbf{r}',t')=\chi_j(\mathbf{r}-\mathbf{r}',t-t')$ and the Maxwell equations greatly simplify after Fourier transform.)

Further levels of complexity may include anisotropy (by making the susceptibilities to be matrices), as well as non-linearity (by making the susceptibilities themselves dependent on the fields or by introducing higher-order non-linear susceptibilities.)