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Fig.3(b)] brings the q parameter on the real axis, with a zero phase gradient σ = 0 and an enlarged beam size α = 2Δ d . The axis tilt thus plays a similar role of a collimating lens for a diverging Gaussian beam. Note however that, contrary to a conventional lens, the tilting angle θ to achieve beam collimation is independent of the distance d between source point (at z = 0) and the lens plane ( z = d ). Figure 4 shows an example of beam collimation in a 6-cm-long one-dimensional array as obtained by numerical analysis of the scalar wave equation for the electric field envelope E ( x, z ) propagating in the structure based on a standard beam propagation method. Figure 4(a) shows a pseudocolor map of the intensity beam evolution | E ( x, z ) | 2 along the structure when a single waveguide is excited in its fundamental mode at the input plane z = 0 and the waveguide axis is tilted at a distance d = 2 cm from the input plane [horizontal dotted curve in Fig.4(a)]. The refractive index profile n ( x ) used in the simulations is depicted in Fig.4(b), and the values of other parameters are λ = 1 . 55 µ m, n$_{s}$ = 1 . 52, and a = 11 µ m, corre- sponding to a tilting angle θ = λ/ (4 an$_{s}$ ) ≃ 23 . 2 mrad. For the sake of readability, the intensity distribution is plotted with the waveguide axis z unfolded along a straight line. Note that the numerical results provide a realistic behavior of beam propagation beyond the couple-mode equation approximation, accounting for radiation losses and coupling to higher-order bands due to axis bending. These latter effects, however, are very small for the parameter values adopted in the simu- lations, and the coupled-mode equation model works fine. 

A geometric interpretation of the self-imaging condition and polygonal Bloch oscillations . An array of length d shows a self-imaging property (also referred to as diffraction cancellation or dynamic localization) , when- ever | c$_{n}$ ( d ) | 2 = | c$_{n}$ (0) | 2 for any initial field distribution. The dynamic localization condition has a rather simple geometric interpretation in the complex q plane. In fact, if the array is excited in waveguide n = 0, q (0) = 0 and to achieve self-imaging after a propagation distance d one has necessarily to have q ( d ) = q (0) = 0, i.e the path described by the complex q parameter, starting from the origin O of the complex plane, should be closed [see Fig.2(b)]. Owing to the translational invariance of discrete Bessel beams [Eq.(73)], this condition is also sufficient. From Eq.(79), the closed-path condition q ( d ) = q (0) = 0 yields 

which is precisely the condition for dynamic localization derived originally by Dunlap and Kenkre in Ref.[33]. An application of the geometric condition of dynamic lo- calization is that of polygonal Bloch oscillations. Let us consider a waveguide array whose axis forms an (open) polygonal curve of large (mean) radius R made of a se- quence of straight segments of same length b and with tilt 

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angle θ , as shown in Fig.5(a). The function φ ( z ), defined by Eq.(80), is thus a staircase function, which increases in steps of γ = (2 π/λ ) aθn$_{s}$ [see Eq.(86)] at z = b, 2 b, 3 b, ... (the coordinate z is measured along the polygonal curve). After a propagation d = ( N + 1) b from the input z = 0 plane, where N is an integer number, it then follows that 

The sum of complex numbers (phasors) on the right hand side of Eq.(88) can be done analytically and has a well- known geometric interpretation; in particular, if γ satis- fies the condition γ = 2 π/ ( N + 1), i.e. if the tilt angle θ is given by 

the sum on the right hand side of Eq.(88) vanishes, and the condition for self-imaging is attained. An example of the self-imaging property of a polygonal waveguide array is shown in Fig.5(b) for the case N = 5. The figure depicts a characteristic breathing mode corresponding to a single waveguide excitation at the input plane. The waveguide array parameters are the same as in Fig.4, and a sequence of axis tilts are placed at distances b = 1 cm one to the next. The tilt angle θ , chosen according to Eq.(89), is θ ≃ 15 . 5 mrad, yielding a self-imaging plane at d = ( N + 1) b = 6 cm, as clearly shown in Fig.5(b). Note that, in the limit b → 0, N → ∞ and b/θ → R finite, the polygonal of Fig.5(a) approximates an arc of 

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