Discrete diffraction and shape-invariant beams in optical waveguide arrays

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. \