Resonance Problems in Photonics

An important area of photonics concerns the propagation and control of light in nonhomogeneous or nonlinear media. This is an area of great importance due to the range of physical effects to be exploited in the design of devices for communication and computing, as well as in the study of fundamental phenomena. Progress has involved a rich interplay of experiment, modeling, computation and analysis, which have impact beyond the specific motivating questions. Indeed, the models and analysis that we discuss are closely related those arising in the study of Bose-Einstein condensation in the regime governed by Gross-Pitaevskii type equations. The propagation of light in a medium is governed by Maxwell’s equations with appropriate linear and nonlinear constitutive relations. We consider a setting where there is a separation of scales in the electric field; the carrier wavelength is short compared to a wave envelope width; see Figure 1. An approximate description of the field evolution, in terms of its envelope is therefore very natural and effective. In this article we discuss various resonance phenomena leading to and playing a role in the dynamics of such envelope equations. We begin with the modeling of optical pulses traveling through a nonlinear and periodic medium. At Bragg resonance, when the carrier wavelength and medium periodicity satisfy a resonance relation, the wave envelope evolves according to a system of dispersive nonlinear partial differential equations (PDEs), the nonlinear coupled mode equations (NLCME). NLCME has gap soliton solutions. These are localized states with frequencies in a photonic band gap. Theory predicts that these can travel at any speed less than the speed of light. It is of fundamental and practical interest to understand whether one can “trap” such soliton pulses. This would have potential application to optical buffers and computing. We explore strategies for soliton capture via the insertion of suitably designed localized defects. The dynamics are governed by a variant of NLCME, but now with variable coefficient “potentials”, analogous to the nonlinear Schrödinger / GrossPitaevskii (NLS-GP) equation. Nonlinear scattering interactions of solitons and