Nanophotonics, the study of light by materials structured on the wavelength scale, are important in a growing number of fields, from lasers to fibers to integrated optical devices. I became interested in this field through exposure to photonic crytals - a lattice of periodic dielectrics - while conducting my senior thesis in Professor Geoffrey Ozin's group in the Chemistry department at the University of Toronto. They are a fascinating and challenging area for research because unexpected, non-analytical solutions often arise in such complex media, which require the development of new theoretical and computational tools to design devices as well as novel experimental methods to fabricate and characterize them. I had a chance to gain insight into the computational methods used to study nanophotonics while a summer intern at the IBM Almaden Research Center with Dr. Geoffrey W. Burr prior to MIT.

My initial research has involved the finite-difference-time-domain (FDTD) method, a fundamental tool for computational electromagnetism, addressing two questions whose answers have been of great utility to the field: how to best capture small spatial features on a discrete grid, and how to continuously deform the structure into the optimal design for some task?

We have developed a new sub-pixel averaging scheme that significantly improves the accuracy of the standard FDTD method. We compared this new method to that of several previously published techniques and demonstrated that our method markedly improves the accuracy of the simulation in addition to restoring the underlying quadratic accuracy. Our results have also been incorporated into our own FDTD code that we have released open source to the nanophotonic community.

Work has also been done recently, in collaboration with Professor Stephen Boyd's group at Stanford University, to design waveguide tapers for coupling power between uniform and slow light periodic waveguides. The design has involved new optimization methods for designing robust tapers, which not only perform well under nominal conditions, but also over a given set of parameter variations owing to manufacturing and operating uncertainties.

An important yet unsolved problem that arose while designing robust waveguide tapers involved the truncation of the computational cell in the presence of inhomogeneous media. In the case of discontinuous periodic dielectric arrangements, the popular perfectly matched layer (PML) is no longer valid; thus to ensure infinitesimal reflections from the open boundaries, large PML regions must surround the computational cell that unnecessarily increase the size of the computation. My recent work is an attempt to design a spectrally convergent absorbing region for use in finite difference or boundary element methods having small size and reflection.

The study of nanophotonic materials and devices is inextricably linked with that of novel computational methods required to study them. Advances in this field depend on researchers having the necessary numerical tools to design devices as an alternative to design by nanofabrication which is wasteful. Computational electromagnetism is a powerful tool that has hitherto been little applied to the study of nanophotonics but holds promise for innovations in the field.

G. W. Burr and A. Farjadpour, ``Balancing accuracy against computation time: 3-D FDTD for nanophotonics device optimization,'' Proc. SPIE, vol. 5733, pp. 336-347, January 2005.

A. Farjadpour, David Roundy, Alejandro Rodriguez, M. Ibanescu, Peter Bermel, J. D. Joannopoulos, Steven G. Johnson, and G. W. Burr, ``Improving accuracy by subpixel smoothing in the finite-difference time domain,'' Opt. Lett. 31, 2972-2974 (2006).

A. Mutapcic, S. Boyd, A. Farjadpour, S. Johnson, Y. Avniel, ``Robust design of slow-light tapers in periodic waveguides'' (to be published in Engineering Optimization).

C. Kottke, A. Farjadpour, S.G. Johnson, ``Perturbation theory for anisotropic dielectric interfaces, and application to sub-pixel smoothing of discretized numerical methods,'' Phys. Rev. E 77, 036611 (2008).

K. K. Lee, A. Farjadpour, Y. Avniel, J. D. Joannopoulos, and S. G. Johnson, ``A tale of two limits: fundamental properties of photonic-crystal fibers,'' Proc. SPIE, vol. 6901, p. 69010K, January 2008. Invited paper

A.F. Oskooi, L. Zhang, Y. Avniel and S.G. Johnson, ``The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers,'' (accepted for publication)