# Research

My research interests lie at the intersection of computational physics, numerical analysis, electromagnetics, and computational geometry. I am mostly focused on mathematical and theoretical aspects of practical nanophotonic problems. I frequently find myself frustrated by the overly-rigorous, abstract, and irrelevant approaches from the mathematics and applied math communities, while being equally frustrated by the physics community’s lack of rigor.

In undergrad, I was primarily interested in nano-fabrication of optoelectronics, and I began with Summer Undergraduate Resarch Fellowships in Axel Scherer’s lab working on surface plasmon enhancement of spontaneous emission and fabricating tiny III-V microdisk lasers. It was during this time that I first learned about the finite-difference time-domain (FDTD) method, and I coded up my own implementations of it. Also at the same time, I became interested in computational geometry, largely from being Matthew Fisher's roommate. I began by implementing several papers of Matthieu Desbrun.

In grad school, I decided to pursue theory and computation rather than experimental work. I worked with Shanhui Fan initially on resonant enhancement of optical forces in photonic crystal slabs. For this work, I implemented the Rigorous Coupled Wave Analysis (RCWA) method, which I have subsequently developed into a full simulation package called S4. Further development of RCWA, aka the Fourier Modal Method (FMM), is one of my main research interests.

In my spare time, I investigated the use of Dirichlet-to-Neumann maps for simulating photonic crystal structures. Thanks to some starting code from Prof. Ya Yan Lu, I managed to use the method for some extremely fast design and optimization of aperiodic structures like wavelength division multiplexers, mode converters, and multimode waveguide bends. It was only later that I discovered the related body of work on impedance maps, Poincare-Steklov operators, and domain decomposition methods. This area of research, the general simulation of large strongly scattering elements, is an ongoing research interest for me.

These days, I am always looking for new highly tailored or esoteric simulation methods that I can apply to modern engineering problems. In my work on implementing several numerical methods, I have had to dig down into low level code such as the Basic Linear Algebra Subroutines (BLAS), Lapack, and other close-to-the-metal routines. As a result, I have developed an appreciation for these low level details, and an interest in improving numerical algorithms at all levels of computation. In particular, I am interested in improved algorithms for the nonsymmetric eigenvalue problem, similar to the recent MRRR algorithm. I am also interested in the incorporation of user supplied shifts in general QR and QZ iterations, and more generally, other “warm-start” techniques.

## Notes

This is a collection of notes on various mathematical or physical (and esoteric) topics. I do not claim these to be accurate, finished, or adherent to any standards. Any additions, comments, corrections, and suggestions are welcome.

### Math reference

- Complex symmetric matrices - Spectral properties and orthogonality relations for complex symmetric matrices.
- Delta comb representation - Representation of a delta function by delta combs; essentially a derivation of the Poisson summation formula.
- FFT on non-orthogonal grids - A clarification to myself on how to apply FFTs when the basis vectors are not orthogonal.
- Trig identities - A collection of trigonometric identities.

### Physics notation

- Einstein and index notation - Some identities and reference formulas.
- Units - Unit conversions between SI and Gaussian units, as well as listing symbolic conversion factors and derived units.

### Math expositions

- Arc spline reference - A collection of representation and implementation notes on arc splines.
- Fourier transforms of polyhedra - Gives general explicit formulas for the Fourier transform of a polygon or a polyhedron given in boundary representation.
- Scattering and transfer matrices ( alternative convention ) - Explicit conversion formulas.
- Properties of the Redheffer star product - Explicit conversion formulas.
- Slab waveguide - A work in progress on completeness of slab waveguide modes.

### Physical expositions

- Dielectric slab - A collection of formulas for optical properties of a single dielectric slab.
- Maxwell equations - A work in progress book of electromagnetics.
- On the proper formulation of Maxwell’s equations for waveguide modes - Gives a linear formulation of the waveguide eigenvalue problem which largely eliminates spurious modes.
- van Hove singularities - Derivation of the singularities for various forms of band extrema.