Natural Science Forum / Physics / Optics / August 2008

I don't recall seeing this done with cylindrical harmonics. With spherical
harmonics, plenty of times, both in the context of scattering by an object
near a plane interface, or just calculating the reflection of vector
spherical waves from a plane interface.

The usual approach is to take the Fourier transform of the spherical
harmonic to find its plane wave spectrum, find the reflected
and transmitted plane waves by the usual Fresnel formulae, and then use
the usual plane wave -> spherical harmonic expansion to convert back to
find the spherical harmonic spectra of the reflected and transmitted
waves.

If you want, I can point you to examples of this, but I don't recall any
especially readable or clear ones.

Perhaps the infinite power of each cylindrical harmonic - they're not
localised in the same way as spherical harmonics - would cause problems in
the cylindrical case. But OTOH, cylindrical harmonics are basically just
vector Bessel beams, which, given the usual explanation of how they're
generated (approximately) using an axicon, might have a nicely behaved
and simple plane wave spectrum. So maybe the same thing works easily. I
don't know offhand if there's an analytical formula for the cylindrical
wave spectrum of a plane wave, but even if there isn't, at worst one can
find it numerically.

And the similar problem of the reflection of a Gaussian beam (2D or 3D) is
done along the same lines. Often, maybe usually, one doesn't bother
finding the spectrum of HG modes of the reflected/transmitted waves, but
just calculates the fields directly from the plane wave spectrum. Maybe
it's done, but the number of computational/theoretical papers on the
Goos-Haenchen shift etc far exceeds the number I've read in any detail.

Since once you're not dealing with a cylindrical geometry, the cylindrical
harmonics are the Mini Moke of basis functions for the Helmholtz equation,
combining the worst features of plane waves and spherical harmonics (the
Moke combines the worst features of car and motorcycle), the only real
applied motivation I can see for doing it are to look at Goos-Haenchen
shifts of Bessel beams. So, perhaps it hasn't been done? All the spherical
wave/Gaussian beam stuff I've seen has been application-motivated.

"AES" <siegman@stanford.edu> wrote in message

snip

(Not sure this will help or even be relevant, maybe vaguely entertaining.)

Perhaps in the old "Handbuch der Physik", i.e. the pre WW2 variety.

In the various optics volumes of the series.

I am sure Stanford has them in one or some of its libraries.

I know for example that in an unrelated topic (polarization) they
had things there you could find nowhere else.

All this stuff had to be laboriously "re-discovered" (for some
photolithographic application) since:

a. nobody reads scientific german anymore

b. people look mostly at the new post WW2 version (which
skips a lot of the old material to make room for the new).

Well actually looking at the new version might not be a waste
of time either.  :-)

BTW why can't the problem be worked out from assuming that
your spherical wave is made up of a large number of small plane
waves and then go to the limit?

Stupid? Too complicated?

I have from good intelligence that Fresnel who worked out his
famous equations from prison (true) wanted to treat precisely
your case but was released before and forgot all about it.

Or maybe he did but he left some pages of his manuscript behind
in his cell... or in the stage coach that was taking him back to Paris.