Steven Golowich

Photonic microstructures are media with micro- or nano-scale features with contrasting refractive indices, individually shaped and collectively distributed so as to finely tune light propagation properties. We study a class of optical waveguides with novel transverse microstructure, so-called microstructured or "holey" optical fibers. In the scalar approximation, the optical properties of such waveguides are governed by a Schrodinger equation with a high contrast and oscillatory potential. Energy escapes transversally from these photonic structures due to a combination of propagation and tunneling. We outline a scattering resonance perturbation theory for this non-selfadjoint eigenvalue problem. The imaginary parts of the scattering resonance "eigenvalues" correspond to the radiative leakage rates from these photonic waveguides. We implement a multiple scale approach to the computation of these eigenvalues. The leading order theory agrees with classical homogenization theory, describing an effective homogeneous medium with dielectric properties given by an appropriate averaging of the refractive index profile. We compute the first non-trivial correction, which takes into account the microstructure, and find that the higher order homogenization expansion gives very good agreement with direct numerical simulations by Fourier and multipole methods. The higher order expansion is crucial for estimation of the leakage rates; in various examples of physical interest the leading order (homogenization) gives a substantial underestimation. (Joint work with Michael Weinstein.)