Causality dictates that all physical media must be dispersive. (We will call a medium dispersive if its refractive index varies with frequency.) Ordinarily, strong dispersion is accompanied either by strong absorption or strong gain. However, over the past 15 years several groups have demonstrated that it is possible to have media that are both strongly dispersive and roughly transparent for some finite bandwidth. In these media, group and phase velocities may differ from each other by many orders of magnitude and even by sign. Relationships and intuitive models that are satisfactory when it is reasonable to neglect dispersion may then fail dramatically.
In this dissertation we analyze three such cases of failure. Before looking at the specific cases, we review some basic ideas relating to dispersion. We review some of the geometric meanings of group velocity, touch on the relationship between group velocity and causality, and give some examples of techniques by which the group velocity may be manipulated. We describe the interplay between group velocity and energy density for non-absorbing dispersive media. We discuss the ideas of temporary absorption and emission as dictated by an instantaneous spectrum. We then apply these concepts in three specific areas.
First, non-dispersive formulations for the momentum of light in a medium must be adjusted to account for dispersion. For over 100 years, there has been a gradual discussion of the proper form for the per-photon momentum. Two forms, each of which has experimental relevance in a ‘dispersionless’ medium, are the Abraham momentum, and the Minkowski momentum. If ħ is the angular frequency, n is the refractive index, ħ is Planck’s constant, and c is the speed of light, then these reduce in a dispersionless medium to per-photon momenta of ħω/(nc), and nħω/c respectively. A simple generalization of the two momenta to dispersive media entails multiplying each per-photon momentum by n/ng, where ng is the group refractive index. The resulting forms are experimentally relevant for the case of the Abraham momentum, but not for the Minkowski momentum. We show how dispersion modulates the displacement of a sphere embedded in a dispersive medium by a pulse.
Second, pulse transformation in a nonstationary medium is modulated by the presence of dispersion. Dispersion may enhance or mitigate the frequency response of a pulse to a changing refractive index, and if dispersion changes with time, the pulse bandwidth must change in a compensatory fashion. We introduce an explicit description of the kinetics of dispersive nonstationary inhomogeneous media. Using this description, we show how the group velocity can modulate the frequency response to a change in the refractive index and how Doppler shifts may become large in a dispersive medium as the velocity of the Doppler shifting surface approaches the group velocity. We explain a simple way to use existing technology to either compress or decompress a given pulse, changing its bandwidth and spatial extent by several orders of magnitude while otherwise preserving its envelope shape. We then introduce a dynamic descriptions of two simple media–one dispersive and one nondispersive. We compare the transformation of basic quantities like photon number, momentum density, and frequency by a temporal change in the refractive index in a specific non-dispersive medium to those wrought by a temporal change in the group refractive index in a specific dispersive medium. The differences between to media are fundamental and emphasize the salience of dispersion in the study of nonstationary media.
Finally, we note that the nature of a single optical cavity quasimode depends on intracavity dispersion. We show that the quantum field noise associated with a single cavity mode may be modulated by dispersion. For a well-chosen mode in a high-Q cavity, this can amount to either an increase or a decrease in total vacuum field energy by several orders of magnitude. We focus on the “white light cavity,” showing that the quantum noise of an ideal white light cavity diverges as the cavity finesse improves.