Amplification Effects on the Transmission and Reflexion Phases in 1D Periodic Systems

TL;DR

This paper studies how amplification influences the transmission and reflection phases in 1D periodic chains, revealing phase independence from length scale and oscillatory behavior near maximum transmission or reflection.

Contribution

It introduces the effect of amplification as an imaginary potential term and analyzes its impact on phase behavior in 1D periodic systems, highlighting new interference phenomena.

Findings

Phases become independent of length scale under amplification

Phases oscillate strongly near maximum transmission or reflection

Reflection phase remains below π/2 in the transmission band

Abstract

We investigate the localization observed recently for locally non-hermitian Hamiltonians by studying the effect of the amplification on the scaling behavior of the transmission and reflection phases in 1D periodic chains of $\delta$-potentials. The amplification here is represented by an imaginary term added to the on-site potential. It is found that both phases of the transmission and reflection amplitudes are strongly affected by the amplification term. In particular, the phases in the region of amplification become independent of the length scale while they oscillate strongly near the maximum transmission (or reflection). The interference effects on the phase in passive systems are used to interpret those observed in the presence of amplification. The phases of the transmission and reflection are found to oscillate in passive systems whith increasing periods in the allowed band for the transmission phase while for the reflection phase, its initial value is always less than $\pi /2$ in this band.