Scattering theory of oscillator defects in an optical fiber

Abstract

We examine harmonic oscillator defects coupled to a photon field in the environs of an optical fiber. Using techniques borrowed or extended from the theory of two-dimensional quantum fields with boundaries and defects, we are able to compute exactly a number of interesting quantities. We calculate the scattering $S$ matrices (i.e., the reflection and transmission amplitudes) of photons off a single defect. We determine using techniques derived from thermodynamic Bethe ansatz the thermodynamic potentials of the interacting photon-defect system, and we compute several correlators of physical interest. We find the photon occupancy at finite temperature, the spontaneous emission spectrum from the decay of an excited state, and the correlation functions of the defect degrees of freedom. In an extension of the single defect theory, we find the photonic band structure that arises from a periodic array of harmonic oscillators. In another extension, we examine a continuous array of defects and exactly derive its dispersion relation. With some differences, the spectrum is similar to that found for electromagnetic wave propagation in covalent crystals. We then add to this continuum theory isolated defects, so as to obtain a more realistic model of defects embedded in a frequency-dependent dielectric medium. We do this both with a single isolated defect and with an array of isolated defects, and so compute how the $S$ matrices and the band structure change in a dynamic medium.