content_image/391011.jpg content_image/391009.jpg content_image/390996.jpg content_image/390997.jpg content_image/390998.jpg content_image/390999.jpg
<img src='content_image/391011.jpg'>

a circumference of radius R , and the condition (89) for self-imaging is satisfied for a propagation distance 

which is the spatial period of Bloch oscillations on a curved waveguide array (radius of curvature R ) previ- ously considered in Refs.[9, 10]. The usual Bloch oscil- lations on a curved waveguide array may be therefore viewed as a limiting case of Bloch oscillations on a polyg- onal array. 

## B. Discrete Bessel beams in two-dimensional arrays 

A simple extension of the analysis of Sec.IV.A can be done for a two-dimensional rectangular-lattice waveguide array with nearest-neighboring coupling when the diago- nal coupling is neglected. This model is described by the 

<img src='content_image/391009.jpg'>

coupled mode equations 

<img src='content_image/390996.jpg'>

where f$_{x,y}$ ( z ) describe the rates of transverse index gra- dients induced by waveguide bending or lumped waveg- uide axis tilting along the x and y directions. Since Eqs. (91) admit of separable solutions c$_{n,m}$ ( z ) = c$_{n}$ ( z ) c$_{m}$ ( z ), with c$_{n}$ ( z ) and c$_{m}$ ( z ) solutions to the one-dimensional problem (72) with Δ = Δ$_{x,y}$ and f ( z ) = f$_{x,y}$ ( z ), a two- dimensional discrete Bessel beam has the form 

<img src='content_image/390997.jpg'>

The complex- q parameters of the beam along the x and y directions are defined by 

<img src='content_image/390998.jpg'>

and their evolution is ruled out by the equations 

which have a similar geometric interpretation as that dis- cussed in Sec.IV.A. The propagation properties of two- dimensional discrete Bessel beams in homogeneous ar- rays, across tilted axis regions or polygonal curves are the same as those investigated for one-dimensional beams, and are therefore not further discussed here. 

In this work, a comprehensive study of discrete diffrac- tion and linear propagation of light in homogeneous and curved waveguide arrays has been presented. In partic- ular, general laws describing beam spreading, beam de- cay and discrete far-field patterns in homogeneous arrays have been derived using the method of moments and the steepest descend method, and some remarks on the well- known self-collimation regime have been pointed out. In curved arrays and within the nearest neighboring cou- pling approximation, the method of moments has been extended to describe the evolution of global beam pa- rameters. This method provides an alternative means to algebraic operator techniques recently proposed in other physical contexts to study general properties of Bloch oscillations [28–30]. Finally, a family of shape-invariant discrete beams -referred to as discrete Bessel beams ow- ing to their functional form- has been introduced. It has been shown that propagation of such beams in curved waveguide arrays is simply described by the evolution of a complex q parameter, which plays a similar role to the complex q parameter used for Gaussian beams in contin- uous lensguide media. A few applications of the q param- eter formalism are discussed, including beam collimation via waveguide axis tilting, a geometric interpretation of the self-imaging effect in waveguide arrays, and optical Bloch oscillations on a polygonal array. 

<img src='content_image/390999.jpg'>

## V. CONCLUSIONS 

