Electromagnetic wave equation
The electromagnetic wave equation describes
the propagation of light through various media.
Since an electromagnetic (light) wave consists of
an electric field
and a magnetic field,
we need Maxwell’s equations
in order to derive the wave equation.
We will use all of Maxwell’s equations,
but we start with Ampère’s circuital law for the “free” fields and ,
in the absence of a free current :
We assume that the medium is isotropic, linear,
and uniform in all of space, such that:
Which, upon insertion into Ampère’s law,
yields an equation relating and .
This may seem to contradict Ampère’s “total” law,
but keep in mind that here:
Now we take the curl, rearrange,
and substitute according to Faraday’s law:
Using a vector identity, we rewrite the leftmost expression,
which can then be reduced thanks to Gauss’ law for magnetism :
This describes .
Next, we repeat the process for :
taking the curl of Faraday’s law yields:
Which can be rewritten using same vector identity as before,
and then reduced by assuming that there is no net charge density
in Gauss’ law, such that :
We thus arrive at the following two (implicitly coupled)
wave equations for and ,
where we have defined the phase velocity :
Traditionally, it is said that the solutions are as follows,
where the wavenumber :
In fact, thanks to linearity, these plane waves can be treated as
terms in a Fourier series, meaning that virtually
any function is a valid solution.
Keep in mind that in reality and are real,
so although it is mathematically convenient to use plane waves,
in the end you will need to take the real part.
A useful generalization is to allow spatial change
in the relative permittivity
and the relative permeability .
We still assume that the medium is linear and isotropic, so:
Inserting these expressions into Faraday’s and Ampère’s laws
We then divide Ampère’s law by ,
take the curl, and substitute Faraday’s law, giving:
Next, we exploit linearity by decomposing and
into Fourier series, with terms given by:
By inserting this ansatz into the equation,
we can remove the explicit time dependence:
Dividing out ,
we arrive at an eigenvalue problem for ,
Compared to a uniform medium, is often not arbitrary here:
there are discrete eigenvalues ,
corresponding to discrete modes .
Next, we go through the same process to find an equation for .
Starting from Faraday’s law, we divide by ,
take the curl, and insert Ampère’s law:
Then, by replacing with our plane-wave ansatz,
we remove the time dependence:
Which, after dividing out ,
yields an analogous eigenvalue problem with :
Usually, it is a reasonable approximation
to say ,
in which case the equation for
becomes a Hermitian eigenvalue problem,
and is thus easier to solve than for .
Keep in mind, however, that in any case,
the solutions and/or
must satisfy the two Maxwell’s equations that were not explicitly used:
This is equivalent to demanding that the resulting waves are transverse,
or in other words,
the wavevector must be perpendicular to
the amplitudes and .
- J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade,
Photonic crystals: molding the flow of light,
2nd edition, Princeton.