10 FIG. 3: (color online) (a) Schematic of a one-dimensional waveguide array with a tilt of waveguide axis at z = z 0 , and the transformation induced on the complex q parameter by the tilt (inset). (b) Principle of beam collimation via waveg- uide axis tilt [tilt angle θ = λ/ (4 an s )], and path followed by the complex q parameter for single waveguide input excitation from z = 0 to z = d + (inset). FIG. 2: (a) Geometric construction of the evolution of the complex q parameter for an infinitesimal propagation distance δz . The length of the segment AB is 2Δ δz , whereas the ro- tation angle is δγ = f ( z ) δz . The points A and C correspond to q ( z ) and q ( z + δz ), respectively. (b) Geometric representa- tion of a self-imaging array: the path followed by the complex parameter q ( z ), starting from the origin O , is closed. It should be noted that, as opposed to the case of Gaussian beams in free space -for which the Rayleigh range z$_{R}$ is proportional to the square of the spot size α$_{0}$ at the beam waist and the diffraction angle θ$_{d}$ is inversely proportional to α$_{0}$ - for discrete Bessel beams the Rayleigh range z$_{R}$ is proportional to the spot size α$_{0}$ at the beam waist whereas the divergence angle is independent of the beam spot size and always equal to the diffraction cone angle introduced in Sec.II.D. This peculiar property is closely related to the very general result, proven in Sec.II.D, that the far field of discrete diffraction in a homogenous waveguide array is peaked at the observation angles corresponding to the flattest points (self-collimation points) of the band dispersion curve. BC shown in Fig.2(a). The path AB, of length 2 δz Δ, accounts for discrete diffraction and corresponds to a change of q ( z ) along the imaginary q axis; the path BC is due to the transverse index gradient which produces a clockwise rotation by the angle δγ = f ( z ) δz around the origin O of the complex plane. It is interesting to note that, since J$_{n}$ (0) = δ$_{n,}$$_{0}$ , for q (0) = 0 the discrete Bessel beam (74) reduces to the well-known impulse response of a tight-binding array with nearest neighbor couplings (see, for instance, [33]). To appreciate the usefulness of the q -parameter descrip- tion and some properties of discrete Bessel beams, let us now discuss a few examples and applications. Propagation of discrete Bessel beams in homoge- neous arrays. For a homogeneous array ( f = 0), the propagation law of the complex- q parameter is simply given by Transformation of a discrete Bessel beam through a waveguide axis tilt. A tilt of the waveguide axis at z = z$_{0}$ by a (small) angle θ corresponds to impressing a phase shift between adjacent waveguides, where a is the waveguide spacing and n$_{s}$ the effective index of propagating modes [see Fig.3(a)]. Light propagation across the tilt can be thus modelled by assuming f ( z ) = γδ ( z − z$_{0}$ ) in Eq.(72), and its effect on the complex q parameter is to produce a rotation around the origin of the complex plane by an angle γ [see the inset of Fig.3(a)]. A tilt of the waveguide axis may be used to ’collimate’ a discrete beam, as schematically shown in Fig.3(b). Here a single waveguide is initially excited at the input plane, and after a propagation distance d the axis of the array is tilted by an angle θ = λ/ (4 an$_{s}$ ) such that γ = π/ 2. The 90 o rotation of the q parameter in the complex plane due to axis bending [see the inset in γ = 2 π λ aθn$_{s}$ (86) q ( z ) = q (0) − 2 i Δ z. (81) α ( z ) = α$_{0}$ √ 1 + ( 2Δ z α$_{0}$ ) 2 (82) σ ( z ) = − arctan 2Δ z α$_{0}$ . (83) z$_{R}$ = α$_{0}$ 2Δ (84) θ$_{d}$ = 2Δ . (85) From Eq.(82) we may introduce, as for Gaussian beams propagating in free space [27], the Rayleigh range z$_{R}$ and divergence angle θ$_{d}$ such that α ( z$_{R}$ ) = √$_{2}$$_{α}$$_{0}$ and θ$_{d}$ = lim$_{z}$$_{→∞}$ α ( z ) /z , i.e. If we assume, for the sake of definiteness, that at the input plane z = 0 the phase front of the beam is flat, i.e. q (0) = α (0) = α$_{0}$ real valued, the following propagation laws for beam size α and beam phase tilt σ are derived (a) (b) GLYPH<c113> a GLYPH<c113> x z d Re( ) q Im( ) q O GLYPH<c103> q z ( ) 0 q z ( ) 0 - + z z$_{0}$ Re( ) q Im( ) q O 0 (a) Re( ) q Im( ) q O A B C GLYPH<c100>GLYPH<c103> q z+ z ( ) GLYPH<c100> q z ( ) Re( ) q Im( ) q O (b)