Exact Bloch oscillations in optical waveguide arrays with arbitrary long-range coupling

TL;DR

This paper derives exact solutions for Bloch oscillations in curved optical waveguide arrays with arbitrary long-range coupling, revealing how higher-order interactions influence wave dynamics and frequency doubling phenomena.

Contribution

It provides the first exact analytical description of Bloch oscillations in waveguides with arbitrary long-range coupling, including the effects of second-order interactions.

Findings

Exact eigenvalues form Wannier-Stark ladder independent of second-order coupling

Broad pulses exhibit Bloch oscillations with frequency doubling at finite second-order coupling

Single waveguide excitation results in a breathing mode, not Bloch oscillations

Abstract

We find the exact Bloch oscillations in zigzag arrays of curved optical waveguides under the influence of arbitrary long-range coupling. The curvature induces a linear transverse potential gradient in the equations of the light evolution. In the case of arrays with second-order coupling, steady states can be obtained as linear combinations of Bessel functions of integer index. The corresponding eigenvalues are equally spaced and form the well-known Wannier-Stark ladder, the spacing being independent of the second-order coupling. We also solve exactly the wave packet dynamics and compare it with experimental results. Accordingly we find that a broad optical pulse performs Bloch oscillations. Frequency doubling of the fundamental Bloch frequency sets up at finite values of the second-order coupling. On the contrary when a single waveguide is initially excited, a breathing mode is activated with no signature of Bloch oscillations. We present a generalization of our results to waveguide arrays subject to long-range coupling. In the general case the centroid of the wave packet shows the occurrence of multiples of the Bloch frequency up to the order of the interaction.