

This is not exactly a "memo," but in some ways it represents the
culmination of all the groping, searching, striving (and, yes,
from time to time, failing) described in the memos below, so
I'll put it here for lack of a more obvious destination.
This is a brief memo I wrote out to help myself in debugging a
BEM solver. The idea is to solve a simple electrostatics problem
analytically, and then solve it again using the boundaryelement
method. This helps elucidate the relative magnitudes and signs
of the various terms appearing in the BEM equations, and gives some
feel for how to expect these terms to behave in more complicated
problems.


August 8, 2006
Tapering Between Disparate Photonic Media in 1D

Suppose you have light propagating through a photonic medium
(vacuum, say), and you need to couple this light into a different
photonic medium (maybe a photonic bandgap material with a
periodically varying index of refraction). If you just connect
one medium to another, you will lose a lot of power due to the
"impedance" mismatch at the abrupt interface. To remedy this, you
would like to insert the optical analogue of an
impedance matching network, consisting
of an intermediate region between the two media (the "taper" region)
with some sort of spatiallyvarying dielectric function that
minimizes reflected power. The question is: How do you choose this
spatiallyvarying function? A necessary tool for this interesting
engineering problem is an algorithm for computing the reflection
coefficient for a given taper function.
The case of practical relevance is that in which the two media
are optical waveguides, confining the propagating light in one
or more dimensions. I got interested in a toy version of this
problem, in which we consider just the 1D case, and spent a couple
of days goofing around with it. The main conclusions are

For the special case of a linear taper, in which
the dielectric constant in the taper region simply rises
linearly with distance from one end to the other, the reflection
coefficient may be computed exactly in terms of Airy
functions. This is true regardless of the nature of
the photonic media between which we are tapering; in particular,
it holds true for periodic media, i.e. layered photonic
bandgap materials. The exact solution in the case of a
sinusoidally varying photonic crystal is particularly
interesting and involves Mathieu functions.

For more general, i.e. nonlinear, taper functions,
the reflection coefficient may be computed to arbitrary
accuracy using an integralequation technique and the
Airyfunction exact solution for the linear taper.
One could object that the integralequation method isn't
particularly useful here, since at the end of the day it still
yields only a numerical solution, which could just as easily be
obtained from a bruteforce discretized solve of the underlying
differential equation. I like to think that the integralequation
method is helpful in providing some alternative insight into
the problem, and I found the process of working it out to be a
useful exercise in Green's function methods, but other than
that I can't really refute the criticism.
Memo 
Description 
taper1.pdf

Tapering between uniform media:
Exact solution for linear taper,
integralequation method for arbitrary taper

taper2.pdf

Tapering from uniform to periodic media:
Exact solution for linear taper and extension of
integralequation method for arbitrary taper

What boundary conditions apply at the interface between an ordinary
dielectric medium and an electrolyte containing DebyeHuckel screening
ions? Most of the literature seems simply to use the boundary conditions
that would apply if the electrolyte were just a dielectric itself.
But this is only accurate to the extent that we can ignore the buildup
of electrolytic screening charge at the interface. Can we? I spent
a brief, tortured period trying to figure this out before concluding that
it wasn't really the right question to be asking (official international
code lingo for "I didn't really know what I was talking about")
and wandering off to misunderstand other things. However, in the process
I did manage to remind myself of several obvious points, which for me
must be declared something of a minor victory. If anyone can shed some
actual light on this matter, I would be very grateful.
Here's a really short note in which I drew out and evaluated all six
secondorder irreducible selfenergy diagrams that arise in perturbation
theory with a twoparticle interaction. I think I was originally going to
use this for comparison purposes as a postHF energy calculation
scheme in quantum chemistry problems, to see how good an approximation
to the exact energy we could get from a numerical solution of the
matrix Dyson's equation with the selfenergy evaluated through
second order. However, I appear immediately thereafter to have lost
interest.
Here's a memo summarizing my initial round of inquiries into
how to construct device models for carbon nanotube FETS.
If not exactly earthshattering in content, this memo was nonetheless
of personal significance for me, as it represented my return from
the precipice of utter demoralization after my abortive first
attempt at grad school.
