# Discrete diffraction and shape-invariant beams in optical waveguide arrays 

Dipartimento di Fisica and Istituto di Fotonica e Nanotecnologie del CNR, 

Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy 

Stefano Longhi 

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. 

Linear and nonlinear propagation of ’discretized’ light in arrays of evanescently-coupled optical waveguides has received a great and increasing interest in the past recent years (see, for instance, [1, 2] and references therein). As compared to diffraction or refraction in continuous (non-structured) media, discrete diffraction and refrac- tion in waveguide arrays show rather uncommon effects which result from the evanescent coupling among ad- jacent waveguides forming a one-dimensional or a two- dimensional lattice. For instance, linear propagation of light waves in homogeneous arrays may show diffrac- tion reversal and self-collimation effects [3, 4], anomalous refraction [4], the discrete Talbot effect [5], and quasi- incoherent propagation [6] to name a few. Remarkably, discrete diffraction can be tailored by properly introduc- ing inhomogeneities in the lattice or by varying its topol- ogy. In particular, since the first proposals and demon- strations of optical Bloch oscillations [7–9] and ’diffrac- tion management’ in zig-zag arrays [3], the use of waveg- uide arrays with curved optical axis has been extensively investigated both theoretically and experimentally, with the demonstration of diffraction suppression via Bloch oscillations [1, 7–10] or dynamic localization [12, 13], polychromatic diffraction management [14], astigmatic diffraction control [15], multicolor Talbot effect [14], and discrete soliton management [16]. Linear and nonlinear light propagation at the surface or at the interface of two waveguide lattices also exhibits a variety of inter- esting properties which have been investigated in several recent works (see, for instance, [2, 17–19] and references therein). In spite of such a great amount of works, some facets of discrete diffraction, even in the simplest linear propagation regime, have been overlooked. Though in the linear regime the impulse response (Green function) of the array may be rather generally calculated analyti- cally -either in straight or curved geometries and in pres- ence or not of boundaries- and its knowledge is enough 

PACS numbers: 42.82.Et, 42.79.Gn 

## I. INTRODUCTION 

to predict light evolution for any assigned initial exci- tation condition (see, for instance, [13, 17]), some gen- eral issues of discrete diffraction, which are well known for paraxial propagation of beams in continuous media, have not been comprehensively addressed, including: (i) a description of global beam parameter evolution in a closed analytical form; (ii) far-field discrete diffraction in homogeneous array (the analogue of Fraunhofer diffrac- tion in homogeneous continuous media); (iii) the gen- eral scaling law of beam broadening and beam decay, especially close to the self-collimation condition (also re- ferred to as sub-diffraction) which is commonplace to the more general class of photonic crystal structures (see, for instance, [20]); (iv) the existence of shape-invariant discretized beams, i.e. special families of field distribu- tions which -like Gaussian beams in continuous lensguide media- do propagate in straight or curved waveguide ar- rays maintaining their functional shape. It is the aim of this work to shed some light into such issues. In par- ticular, it is shown rather generally that: (i) the scaling law describing broadening of discretized light in homo- geneous arrays is the same as that of standard parax- ial diffraction theory of homogeneous continuous media (beam size asymptotically grows linearly with propaga- tion distance), independently of the precise array disper- sion curve and even along self-collimation directions; (ii) in a homogenous array, the discrete far-field pattern is not the (discrete) Fourier transform of the near-field dis- tribution, and the scaling law of beam decay may depend on the observation angle; (iii) special field distributions, which propagate in straight or curved waveguide arrays maintaining their functional shape and referred to as ’dis- crete Bessel beams’, can be introduced for simple tight- binding waveguide models; (iv) a discrete Bessel beam is defined by a complex q parameter, analogous to the one used for Gaussian beams in continuous lensguide media, and propagation of the q parameter along the array ad- mits of a simple geometric interpretation. 

The paper is organized as follows. In Sec.II general prop- 

