11 -20 -15 -10 -5 0 5 10 15 20 0 0.005 0.01 x ( m) GLYPH<c109> x ( m) GLYPH<c109> z (cm) -300 300 0 6 0 2 4 n x -n ( ) s (b) (a) FIG. 4: (color online) (a) Pseudocolor map of beam intensity propagation in a waveguide array with one axis tilting as ob- tained by numerical simulations, showing beam collimation. (b) Refractive index profile of the waveguide array used in the numerical simulations. The values of other parameters are given in the text. 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 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 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 θ = λ an$_{s}$ ( N + 1) (89) ∫ d 0 dz exp[ iφ ( z )] = b N ∑ n =0 exp( iγn ) . (88) ∫ d 0 dz exp[ iφ ( z )] = 0 (87) 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