M. T. Homer Reid MIT Home Page
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December 09, 2010

My PhD Thesis:

Fluctuating Surface Currents:

A New Algorithm for Efficient Prediction of Casimir Interactions among Arbitrary Materials in Arbitrary Geometries

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.

September 15, 2006

How Does the Boundary-Element Method Work?

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 boundary-element 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 spatially-varying dielectric function that minimizes reflected power. The question is: How do you choose this spatially-varying 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 integral-equation technique and the Airy-function exact solution for the linear taper.
One could object that the integral-equation 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 brute-force discretized solve of the underlying differential equation. I like to think that the integral-equation 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, integral-equation method for arbitrary taper
taper2.pdf Tapering from uniform to periodic media: Exact solution for linear taper and extension of integral-equation method for arbitrary taper

April 11, 2006

Electrostatic Boundary Conditions at an Interface Between Dielectric and Electrolytic Media

What boundary conditions apply at the interface between an ordinary dielectric medium and an electrolyte containing Debye-Huckel 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.

March 21, 2006

Second-Order Self-Energy Diagrams

Here's a really short note in which I drew out and evaluated all six second-order irreducible self-energy diagrams that arise in perturbation theory with a two-particle interaction. I think I was originally going to use this for comparison purposes as a post-HF 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 self-energy evaluated through second order. However, I appear immediately thereafter to have lost interest.

August 2, 2005

Simulating Quantum Transport in Carbon Nanotube FETS

Here's a memo summarizing my initial round of inquiries into how to construct device models for carbon nanotube FETS. If not exactly earth-shattering 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.

Homer Reid's Research Memos Page, by Homer Reid
Last Modified: 11/16/16