Plane wave solutions of Maxwell’s equation using Geometric Algebra
Posted by peeterjoot on September 3, 2012
Study of reflection and transmission of radiation in isotropic, charge and current free, linear matter utilizes the plane wave solutions to Maxwell’s equations. These have the structure of phasor equations, with some specific constraints on the components and the exponents.
These constraints are usually derived starting with the plain old vector form of Maxwell’s equations, and it is natural to wonder how this is done directly using Geometric Algebra.  provides one such derivation, using the covariant form of Maxwell’s equations. Here’s a slightly more pedestrian way of doing the same.
Maxwell’s equations in media
We start with Maxwell’s equations for linear matter as found in 
We merge these using the geometric identity
where is the 3D pseudoscalar , to find
We want dimensions of for the derivative operator on the RHS of 1.2.3b, so we divide through by for
This can now be added to 1.2.3a for
This is Maxwell’s equation in linear isotropic charge and current free matter in Geometric Algebra form.
We write the electromagnetic field as
so that for vacuum where we have the usual . Assuming a phasor solution of
where is allowed to be complex, and the actual field is obtained by taking the real part
Note carefully that we are using a scalar imaginary , as well as the multivector (pseudoscalar) , despite the fact that both have the square to scalar minus one property.
We now seek the constraints on , , and that allow this to be a solution to 1.2.5
As usual in the non-geometric algebra treatment, we observe that any such solution to Maxwell’s equation is also a wave equation solution. In GA we can do so by right multiplying an operator that has a conjugate form,
where is the speed of the wave described by this solution.
Inserting the exponential form of our assumed solution 1.3.7 we find
which implies that the wave number vector and the angular frequency are related by
Our assumed solution must also satisfy the first order system 1.3.9
The constraints on must then be given by
we must then have all grades of the multivector equation equal to zero
Writing out all the geometric products, noting that commutes with all of , , and and employing the identity we have
This and 1.3.12 describe all the constraints on our phasor that are required for it to be a solution. Note that only one of the two cross product equations in are required because the two are not independent. This can be shown by crossing with 1.3.18b and using the identity
One can find easily that 1.3.18b and 1.3.18c provide the same relationship between the and components of . Writing out the complete expression for we have
Since , this is
Had we been clever enough this could have been deduced directly from the 1.3.14 directly, since we require a product that is killed by left multiplication with . Our complete plane wave solution to Maxwell’s equation is therefore given by
 C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.
 D.J. Griffith. Introduction to Electrodynamics. Prentice-Hall, 1981.