Transmission and reflection in a perfectly amplifying and absorbing medium

TL;DR

This paper analytically investigates transmission and reflection in perfect amplifying and absorbing media, revealing oscillatory behaviors, crossover length scales, and asymptotic properties depending on the medium's strength and type.

Contribution

It provides new analytical expressions for transmittance and reflectance in amplifying and absorbing media, highlighting the role of sample length and amplification strength.

Findings

Oscillatory behavior of transmittance and reflectance for even and odd sample lengths.

Existence of a crossover length scale in weak amplifying media.

Exponential decay of transmission in strong amplifying/absorbing media.

Abstract

We study transmission and reflection properties of a perfectly amplifying as well as absorbing medium analytically by using the tight binding equation. Different expressions for transmittance and reflectance are obtained for even and odd values of the sample length which is the origin of their oscillatory behavior. In a weak amplifying medium, a cross-over length scale exists below which transmittance and reflectance increase exponentially and above which transmittance decays exponentially while the reflectance gets saturated. Depending on amplification transmittance and reflectance show singular behavior at the cross-over length. In a weak absorbing medium we do not find any cross-over length scale. The transmission coefficient behaves similar to that in an amplifying medium in the asymptotic limit. In a strong amplifying/absorbing medium, the transmission coefficient decays exponentially in the large length limit. In both weak as well as strong absorbing media the logarithm of the reflection coefficient shows the same behavior as obtained in an amplifying medium but with opposite sign.