12 coupled mode equations 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 i ˙ c$_{n,m}$ = − Δ$_{x}$( c$_{n}$$_{+1}$$_{,m}$ + c$_{n}$$_{−}$$_{1}$$_{,m}$ ) − Δ$_{y}$( c$_{n,m}$$_{+1}$ + c$_{n,m}$$_{−}$$_{1}$ ) − f$_{x}$ ( z ) nc$_{n,m}$ − f$_{y}$ ( z ) mc$_{n,m}$ (91) c$_{n,m}$ ( z ) = J$_{n}$ ( α$_{x}$ ) J$_{m}$ ( α$_{y}$ ) exp( − iσ$_{x}$n − σ$_{y}$m ) . (92) q$_{x}$ ( z ) = α$_{x}$ ( z ) exp[ iσ$_{x}$ ( z )] , q$_{y}$ ( z ) = α$_{y}$ ( z ) exp[ iσ$_{y}$ ( z )] (93) ˙ q$_{x,y}$ = − 2 i Δ$_{x,y}$ − if$_{x,y}$ ( z ) (94) The complex- q parameters of the beam along the x and y directions are defined by 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. 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 B. Discrete Bessel beams in two-dimensional arrays a circumference of radius R , and the condition (89) for self-imaging is satisfied for a propagation distance FIG. 5: (color online) (a) Schematic of a polygonal waveg- uide array for the observation of Bloch oscillations. (b) Pseu- docolor image of beam intensity propagation in a 8-cm-long polygonal array showing a Bloch oscillation breathing mode. The refractive index profile of the waveguide array used in the numerical simulations is the same as in Fig.4(b). The values of other parameters are given in the text. V. CONCLUSIONS d = ( N + 1) b → λR n$_{s}$a (90) 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. x ( m) GLYPH<c109> z (cm) 200 0 8 0 2 4 (b) R z GLYPH<c113> b b b (a) 6 -200