Fourier–Laplace analysis and instabilities of a gainy slab

Abstract

The idealization of monochromatic plane waves leads to considerable simplifications in the analysis of electromagnetic systems. However, for active systems this idealization may be dangerous due to the presence of growing waves. Here we consider a gainy slab, and use a realistic incident beam, which is both causal and has finite width. This clarifies some apparent paradoxes arising from earlier analyses of this setup. In general it turns out to be necessary to involve complex frequencies ω and/or complex transversal wavenumbers kx. Simultaneously real ω and kx cannot describe amplified waves in a slab which is infinite in the transversal direction. We also show that the only possibility to have an absolute instability for a finite width beam, is if a normally incident plane wave would experience an instability.

Gain-induced scattering anomalies of diffractive metasurfaces

“…The main challenge in observing these singular features in experiments comes from the assumptions that the array and the incident plane wave are significantly extended in space and time, such that their frequency-momentum distributions are nearly Dirac deltas. In a real system, the singularities would be blurred due to finite size of the illuminated area [37,63], and due to finite time duration of the scattering events, especially in a pulsed pump-probe implementation [64,65]. Large amplification requires that each incoming photon is scattered and amplified many times by many nanoparticles, in a way resembling amplification in a resonant cavity [66], which may take a long time to build up.…”

Theoretical analysis of Fresnel reflection and transmission in the presence of gain media

“…x -axis in conjunction with a deformed -plane integration contour-a contour that moves away from the ′ -axis into the upper-half -plane, as described in Appendix B. Failure to construct a k x -plane contour C heralds the arrival of so-called absolute instabilities (e.g., onset of lasing), in contrast to "convective" instabilities, which have been the concern of the present paper [12,14,15].…”

“…Finally, it must be pointed out that, in the aforementioned step iii, one may not be able to choose an integration path that bypasses all the singular points in the k x -plane associated with the upper-half -plane. When this happens, the method fails and the system is said to exhibit an absolute instability-as opposed to a convective instability [12][13][14][15].…”

“…At this point in the analysis, the existence of branchpoints for k 2z (i.e., k z in medium 2) and/or poles associated with and in the upper-half -plane imposes a lower bound on Ω 0 that ensures the satisfaction of the all-important causality requirement [13][14][15] (causality decrees that the reflected and transmitted waves cannot reach the point (x, y, z) prior to t = |z|∕c ). Causality also fixes the signs of k 1z , k 2z , and k 3z so that, referring to Eq.…”

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].…”