Number of propagating modes of a diffusive periodic waveguide in the semiclassical limit

TL;DR

This paper investigates how the number of propagating Bloch modes in a periodic waveguide scales with wavenumber, revealing different behaviors in ballistic versus diffusive regimes and confirming predictions through numerical analysis.

Contribution

It provides the first numerical verification of the semiclassical prediction that the number of propagating modes scales as sqrt(k) in diffusive systems, linking waveguide dynamics to spectral universality.

Findings

Number of propagating modes scales linearly with k in ballistic systems.

Number of propagating modes scales as sqrt(k) in diffusive systems.

Numerical results agree with semiclassical predictions for diffusive waveguides.

Abstract

We study the number of propagating Bloch modes N_B of an infinite periodic billiard chain. The asymptotic semiclassical behavior of this quantity depends on the phase-space dynamics of the unit cell, growing linearly with the wavenumber k in systems with a non-null measure of ballistic trajectories and going like ~ sqrt(k) in diffusive systems. We have calculated numerically N_B for a waveguide with cosine-shaped walls exhibiting strongly diffusive dynamics. The semiclassical prediction for diffusive systems is verified to good accuracy and a connection between this result and the universality of the parametric variation of energy levels is presented.