On the limit of spectral gap formation via Bragg's reflection

TL;DR

This paper investigates the minimal conditions for spectral gap formation in periodic media, revealing a scaling law for the noise level of the gap and discovering a new parabolic frequency shift behavior with increased modulation depth.

Contribution

It provides a quantitative scaling law for spectral gap clarity based on the number and depth of periodic modulations, and reports a novel parabolic frequency shift phenomenon.

Findings

The bottom-edge ratio scales inversely with the square of the number and depth of magnetic mirrors.

A threshold of 7 mirrors and 0.4 depth is needed for clear spectral gaps.

A new parabolic shape of the spectral gap center frequency shift is observed.

Abstract

Although it is well recognized that waves propagating in periodic media have forbidden gaps in the continuous spectrum, it is still yet unknown about the minimum number and depth of periodic modulations for clear spectral formation. This limit is explored by investigating shear Alfv\'{e}n waves propagating through magnetic mirror array in a low temperature plasma cylinder, referring to an existing experiment [Zhang et al., Phys. Plasmas 15, 012103 (2008)]. It is found that the bottom-edge ratio, a defined parameter characterizing how "noisy" the formed spectral gap is, scales inversely with the square of the number and depth of magnetic mirrors, and they should be bigger than $7$ and $0.4$ respectively to form a clear spectral gap for the plasma conditions employed. This scaling is consistent with a pervious analysis [Kryuchkyan and Hatsagortsyan, Phys. Rev. Lett. 107, 053604 (2011)], and also applicable to other fields such as optics and semiconductors. Moreover, it shows that the center of spectral gap experiences a parabolic shape of descending frequency shift when the modulation depth is increased, which is believed to be a new discovery.