# Development of a Computational Method to Calculate Resonances in 2D Dielectric Optical Cavities

I am attempting to develop a script for the computer algebra system Mathematica that will calculate the resonances in arbitrary collections of 2D optical cavities. The work will be based off of

```
J. Wiersig, "Boundary element method for resonances in dielectric microcavities," J. Opt. A Pure Appl. Opt. 5, 53-60
(2003).
```

My advisor is Professor Gary Wysin.
## Week-By-Week Progress

- Orientation, tours, etc. Lots of reading.
- More reading. Plan developed for program design.
The major design elements of this program are:
- Discretize the cavity boundary.
- Apply Wiersig's equation (37) until the complex wavevector k is found.
- Compute the resonant wave from k and the boundary.

- Developed code that will calculate discrete boundaries for a set of arbitrary polygonal cavities.
- Debugging code for equation (37) - It turns out that M
^{-1} can't be computed accurately in Mathematica, so I am using the determinant method instead.
- I produced a graph of the scattering cross section versus wave number. Unfortunately it was obviously wrong.
- I spent the whole week debugging scattering cross section code. Many bugs squashed, but some remain.
- I spent Saturday, Sunday, and Monday wondering why all the university's Mathematica licenses were taken. It turned out that the license server was malfunctioning.
This week I delved into some mathematical theory. My work is based on a theorem due to Weber,
*Mathematische Annalen* **1** (1869), 1-36.
I have also run loads of simulations trying to understand what is wrong with my program. Preliminary results show some improvement, but my work may still be flawed.
- Fixed more bugs - including errors in my previous fixes. It turns out the M
^{-1} problem was due to a bug in earlier code.
- Looks like I am very close - finding things wrong is becoming computationally hard. Also - I am almost out of time.
- Final Presentation (PDF warning)

If you would like to know anything about my experiences at K-State's REU program, email me - nlfrazer AT phys DOT ksu DOT edu