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

## Nicholas Laszlo Frazer, Research Experiences for Undergraduates at Kansas State UniversityDepartment of Physics, 2007

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

1. Orientation, tours, etc. Lots of reading.
2. More reading. Plan developed for program design. The major design elements of this program are:
1. Discretize the cavity boundary.
2. Apply Wiersig's equation (37) until the complex wavevector k is found.
3. Compute the resonant wave from k and the boundary.
3. Developed code that will calculate discrete boundaries for a set of arbitrary polygonal cavities.
4. 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.
5. I produced a graph of the scattering cross section versus wave number. Unfortunately it was obviously wrong.
6. I spent the whole week debugging scattering cross section code. Many bugs squashed, but some remain.
7. 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.
8. Fixed more bugs - including errors in my previous fixes. It turns out the M-1 problem was due to a bug in earlier code.
9. Looks like I am very close - finding things wrong is becoming computationally hard. Also - I am almost out of time.
10. 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