Fourier theory of linear gain media

Abstract

The analysis of wave propagation in linear, passive media is usually done by considering a single real frequency (the monochromatic limit) and also often a single plane wave component (plane wave limit). For gain media, we demonstrate that these two limits generally do not commute; for example, one order may lead to a diverging field, while the other order leads to a finite field. Moreover, the plane wave limit may be dependent on whether it is realized with a finite support excitation or gaussian excitation, eventually of infinite widths. We consider wave propagation in gain media by a Fourier-Laplace integral in space and time, and demonstrate how the correct monochromatic limit or plane wave limit can be taken, by deforming the integration surface in complex frequency-complex wavenumber space. We also give the most general criterion for absolute instabilities. The general theory is applied in several cases, and is used to predict media with novel properties. In particular, we show the existence of isotropic media which in principle exhibit simultaneous refraction, meaning that they refract positively and negatively at the same time.

Minimal spaser threshold within electrodynamic framework: Shape, size and modes

“…An infinite gain medium may contain exponentially growing solutions. This is known to cause non‐trivial issues in Fresnel formulas, as well as in total internal reflection (see and refs. therein).…”

Negative index materials: at the frontier of macroscopic electromagnetism

“…In other words, one can conclude that the solution to the time-harmonic Maxwell's equations does not exist at the frequency ω 1 in the case of the perfect lens. Such situations where the time-harmonic Maxwell's equations have no solutions have been also uncountered with active (or gain) media [68][69][70].…”

Fresnel’s equations in statics and quasistatics

“…These waves will have their electric field perpendicular (TE) or parallel (TM) to the plane of incidence. Considering an incident TE wave, the electric field can be written 1 For gain media, there is a complication to identify the correct sign of the longitudinal wavevector based on causality [4][5][6][7].…”

“…For gain media, there is a complication to identify the correct sign of the longitudinal wavevector based on causality[4][5][6][7].…”