Re: ? Maxwell eqn in inhomo media


Cheng Cosine wrote:
Let's put in this way: when domain properties are all constants Maxwell eqns
can be

reduced into a vector wave equations. Are there other cases that one can
reduce

Maxwell equations into a set of wave eqns or a set of wave eqns with some
extra term.

More specifically, what are the conditions to reduce Maxwell eqns into a
form that one

can use Green's function of wave eqns to express its soln?

Thanks,
by Cheng Cosine
Jun/06/2k6 NC

If you are working with vector equations, inhomogeneous EM wave
equations are complicated, with coupling between field components. So,
there two approaches that I know.

First, scalar equations are fine for most of optics, and only have to
work with the electric field. Fresnel diffraction is paraxial, and
Rayleigh-Sommerfeld is non-paraxial. Both are convolutions, which can
be done efficiently using FFTs. Analytic solutions are rare. The
thing you are convolving the electric field with is either a Fresnel
approximated spherical wave or, for R-S diffraction, a spherical wave
with a cos(angle) factor on it (some additional terms too). These are
called the kernel, or the impulse response, or the Green's function.

Second, you can work with the vector equations. I will spell it out,
but you won't like it. You have to look into equivalent surface source
currents, since these are the drive terms for the vector and scalar
potentials. You have to include magnetic charge and current, since
they are required to directly generate normal magnetic and tangential
electric fields. Then, look into Hertz vector potentials. Hertz
vectors allow the vector and scalar potential to be written as just one
equation, consistent with the Lorenz (sometimes mistakenly called
Lorentz) gauge condition. The drive terms for these equations are
dipole vectors, which link current and charge. The Hertz vectors each
have a simple single vector drive term, so there is no coupling between
vector components, i.e., three scalar equations for each of the two
vector equations. The drive terms are the dipole vectors, so a Hertz
vector solution is the 3-D Green's function convolved with the drive
vector. Once you have the Hertz vectors, you calculate the scalar and
vector potentials, then the electric and magnetic field.

Working with the vector equations are a pain in the you know what.
Stick with Fresnel diffraction for most work, or, if required, use
Rayleigh-Sommerfeld diffraction. If you are working with very small
apertures or objects, you have to start considering what is going on on
the surface of the object. Boundary value problems are quite a bit
more complicated, since induced surface currents radiate. One part of
a surface radiates and affects another part. The vector solution
discussed earlier is for free space, or for a uniform isotropic medium.

.