Quantum and Classical Optics of Dispersive and Absorptive Structured Media

This thesis presents a Hamiltonian formulation of the electromagnetic fields in structured (inhomogeneous) media of arbitrary dimensionality, with arbitrary material dispersion and absorption consistent with causality. The 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 macroscopic response of the medium appears in the derived eigenvalue equation for the polaritons. For both the discrete transparent-regime spectrum and the continuous absorptive-regime spectrum, standard codes for photonic modes in nonabsorptive systems can easily be leveraged to calculate polariton modes. Two applications of the theory are presented: pulse propagation and spontaneous parametric down-conversion (SPDC). In the propagation study, the dynamics of the nonfluctuating part of a classical-like pulse are expressed in terms of a Schr\"{o}dinger equation for a polariton effective field. The complex propagation parameters of that equation can be obtained from the same generalized dispersion surfaces typically used while neglecting absorption, without incurring additional computational complexity. As an example I characterize optical pulse propagation in an Au/MgF$_2$ metallodielectric stack, using the empirical response function, and elucidate the various roles of Bragg scattering, interband absorption and field expulsion. Further, I derive the Beer coefficient in causal structured media. The SPDC calculation is rigorous, captures the full 3D physics, and properly incorporates linear dispersion. I obtain an expression for the down-converted state, quantify pair-production properties, and characterize the scaling behavior of the SPDC energy. Dispersion affects the normalization of the polariton modes, and calculations of the down-conversion efficiency that neglect this can be off by 100$\%$ or more for common media regardless of geometry if the pump is near the band edge. Furthermore, I derive a 3D three-wave group velocity walkoff factor; due to the interplay of a topological property with a symmetry property, I show that even if down-conversion is into a narrow forward cone, neglect of the transverse walkoff can lead to an overestimate of the SPDC energy by orders of magnitude.Ph