Discrete diffraction and shape-invariant beams in optical waveguide arrays

TL;DR

This paper explores the properties of discretized light in waveguide arrays, deriving laws for beam behavior, introducing shape-invariant discrete Bessel beams, and extending the $q$ parameter formalism for practical applications.

Contribution

It introduces discrete Bessel beams that maintain shape in curved arrays and extends the $q$ parameter formalism to describe their propagation.

Findings

Derived laws for beam spreading and decay in arrays

Introduced discrete Bessel beams with shape-invariant propagation

Extended $q$ parameter formalism for beam evolution

Abstract

General properties of linear propagation of discretized light in homogeneous and curved waveguide arrays are comprehensively investigated and compared to those of paraxial diffraction in continuous media. In particular, general laws describing beam spreading, beam decay and discrete far-field patterns in homogeneous arrays are derived using the method of moments and the steepest descend method. In curved arrays, the method of moments is extended to describe evolution of global beam parameters. A family of beams which propagate in curved arrays maintaining their functional shape -referred to as discrete Bessel beams- is also introduced. Propagation of discrete Bessel beams in waveguide arrays is simply described by the evolution of a complex $q$ parameter similar to the complex $q$ parameter used for Gaussian beams in continuous lensguide media. A few applications of the $q$ parameter formalism are discussed, including beam collimation and polygonal optical Bloch oscillations. \