We present a formalism for optical pulse propagation in nonlinear photonic crystals of arbitrary dimensionality. Using a multiple-scale analysis, we derive the dynamical nonlinear Schrödinger equation obeyed by the envelope function modulating an underlying Bloch function. Effective coefficients appear in that equation characterizing the effects of Kerr nonlinearity, linear gain or loss, and material dispersion. They depend on how the underlying Bloch function "samples" these effects in the photonic crystal, and require for their calculation a specification of these effects throughout the photonic crystal, and the calculated bandstructure of the photonic crystals in the linear, nondispersive limit. We show that wave packets from different bands can experience significantly modified effective material properties.
We introduce a Hamiltonian formulation of electromagnetic fields in dispersive and absorptive structured media of arbitrary dimensionality; the Kramers-Kronig relations are satisfied by construction. Our method is based on an identification of the photonic component of the polariton modes of the system. Although the medium degrees of freedom are introduced in an oscillator model, only the susceptibility of the medium appears in the derived eigenvalue equation for the polaritons; the theory is applicable to both classical and quantum optics. A discrete polariton spectrum is obtained in the transparent regime below the absorption cutoff frequency, and the normalization condition contains the material dispersion in a simple way. In the absorptive regime, a continuous polariton spectrum is obtained. The expressions for the full electromagnetic field of the system can be written in terms of the modes of a limiting, nondispersive, nonabsorptive system, so the theory is well suited to studying the effect of dispersion or absorption on photonic dispersion relations and mode structure. Available codes for dispersive photonic modes can easily be leveraged to obtain polariton modes in both the transparent and absorptive regimes.
We introduce an effective field theory for the nonlinear optics of photonic crystals of arbitrary dimensionality. Based on a canonical Hamiltonian formulation of Maxwell's equations, canonical effective fields are introduced to describe the electromagnetic field. Conserved quantities are easily constructed and their physical significance identified; the formalism can be easily quantized. We illustrate the approach by considering a periodic Kerr medium, and show how the nonlinear coupled mode and nonlinear Schrödinger equations emerge. We extend the latter to treat optical shock effects, and compare our canonical formulation with earlier treatments.
We derive effective photon modes that facilitate an intuitive and convenient picture of photon dynamics in a structured Kramers-Kronig dielectric in the limit of weak absorption. Each mode is associated with a mode field distribution that includes the effects of both material and structural dispersion, and an effective line-width that determines the temporal decay rate of the photon. These results are then applied to obtain an expression for the Beer-Lambert-Bouguer law absorption coefficient for unidirectional propagation in structured media consisting of dispersive, weakly absorptive dielectric materials.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.