Fig. 3. (Color online) Same as Fig.2(a), but for an ellip- tical Guassian input beam vertically displaced to excite waveguides of the auxiliary array. In (a) the arrays are straight, whereas in (b) they are periodically-curved. (b) 0 60 30 (a) 0 200 x ( m) GLYPH<c109> z (mm) 0 40 0 200 x ( m) GLYPH<c109> a tight-binding model of Eq.(1) and in the averaging limit, which neglects weak light transfer among waveg- uides due to non-resonant coupling terms. We checked the possibility of observing rectification of light refrac- tion beyond such approximations by a beam propa- gation analysis of Eq.(1). In the simulations, circular channel waveguides with a super-Gaussian index profile Δ n ( x, y ) = Δ n$_{A,B}$ exp {− [( x 2 + y $^{2}$) /w $^{2}$]$^{3}$} , with radius w and index changes Δ n$_{A}$ and Δ n$_{B}$ in the primary and auxiliary arrays, are assumed. The vertical distance d$_{v}$ between the two arrays is chosen to be d$_{v}$ = √$_{3}$$_{d/}$$_{2,}$ i.e. the zigzag is defined by an equilateral triangle. As an input beam, we typically assumed an elliptical Gaus- sian beam, with a tilting angle θ (in the x direction) smaller than the Bragg angle θ$_{B}$ to excite the lowest band of the array. The vertical beam position and size are adjusted to couple a few waveguides of the primary array. Examples of rectified light refraction, observed in the curved arrays for a few values of beam incidence an- gle θ , are shown in Fig.2. In the figure, the integrated beam intensity distribution ∫ | ψ ( x, y ) | $^{2}$dy versus propa- gation distance z is shown and compared to the behav- ior that one would observe if the arrays were straight, i.e. with usual discrete light refraction patterns. For the chosen array parameters and wavelength, the coupling rates and propagation mismatch are estimated to be κ$_{aa}$ = κ$_{bb}$ ≃ κ$_{ab}$ ≃ 1 . 939 cm $_{−}$1 and σ ≃ 16 . 19 cm $^{$_{−}$1}$, corresponding to a radius of curvature R = 4 cm and a semi-cycle period z 0 = 8 . 1 mm according to Eq.(4). The numerical results clearly show that, for curved arrays, light refraction always occurs at the angle θ$_{r}$ = d/ (2 z 0 ), regardless of the beam incidence angle θ , and discrete diffraction is well suppressed as a result of the controlled transport mechanism. Of course beam refraction is re- versed when the position of the elliptical input beam is shifted to excite waveguides of the auxiliary array. As an example, Fig.3 shows rectification of light refraction when the array B$_{n}$ is excited at the input by an ellip- tical Gaussian beam at normal incidence. Note that, as compared to Fig.2(a), the refraction angle of discretized light is now reversed. 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