likely, as discussed in Ref. [14]. To achieve a ratchet ef- fect, we introduce the auxiliary lattice B$_{n}$ and assume that both arrays are periodically-curved with a bend- ing profile x 0 ( z ) made of a sequence of circular arcs of (paraxial) length z 0 and constant curvature with alter- nating sign ¨ x 0 ( z ) = ± 1 /R , as depicted in Fig.1(c). Such a bending profile corresponds to an ac square-wave force − n$_{s}$ ¨ x 0 ( z ), and to a triangular shape for the function Θ( z ) entering in Eqs.(2) and (3). Note that, since in this case the waveguide axis at the entrance plane z = 0 is tilted by an angle ˙ x 0 (0) = z 0 / (2 R ), the incidence angle θ should be now measured with respect to a plane tilted by an angle θ$_{c}$ = n$_{s}$z 0 / (2 R ) from the entrance facet [11], as shown in the inset of Fig.1(c). To achieve rectification of light refraction in the primary array over one undula- tion cycle, the radius of curvature R and spatial period z 0 of alternations are designed to satisfy the conditions ˙ Θ( z ) = ± σ and κ$_{ab}$z 0 = π/ 2, i.e. In the first semi-cycle of undulation, where ˙ Θ( z ) = σ , neglecting nonresonant terms Eqs.(2) and (3) reduce to i ˙ a$_{n}$ = − κ$_{eff}$ b$_{n}$ and i ˙ b$_{n}$ = − κ ∗ $_{eff}$a$_{n}$ , where κ$_{eff}$ = κ$_{ab}$ exp( iφ ) and φ = σz − Θ is a constant phase shift. This means that waveguide A$_{n}$ turns out to be coupled solely with waveguide B$_{n}$, and light transfer between them oc- curs like in an ordinary synchronous directional coupler. For a propagation length z 0 equal to one coupling length z 0 = π/ (2 κ$_{ab}$ ), light trapped in waveguide A$_{n}$ is thus fully transferred into waveguide B$_{n}$. At the successive propagation semi-cycle, where ˙ Θ( z ) = − σ , Eqs.(2) and (3) yield i ˙ a$_{n}$ +1 = − κ$_{eff}$ b$_{n}$ and i ˙ b$_{n}$ = κ ∗ $_{eff}$a$_{n}$ +1 with κ$_{eff}$ = κ$_{ab}$ exp( iφ ) and φ = σz + Θ, i.e. waveguide B$_{n}$ turns out to be coupled solely with waveguide A$_{n}$ +1 . For a propagation length z 0 equal to one coupling length, light trapped in waveguide B$_{n}$ is thus fully transferred into waveguide A$_{n}$ +1 . Therefore, after a full modulation cycle, an arbitrary light distribution in the primary ar- ray A$_{n}$ is shifted, without distortion, by one unit from the left to the right. The result is a net drift of refracted light, with a locked refraction angle θ$_{r}$ = d/ (2 z 0 ) inde- pendent of the initial incidence angle θ and beam shape , and a suppression of discrete diffraction. Similarly, an arbitrary light distribution in the secondary array B$_{n}$ is shifted by one unit but this time from the right to the left. As the ratchet effect results from an alternat- ing synchronization of couples of waveguides in the array ( A$_{n}$ → B$_{n}$ and B$_{n}$ → A$_{n}$ +1 ), different bending profiles x 0 ( z ), such as sequences of sinuosoidally-curved waveg- uides with alternating amplitudes mimicking the driving field of Ref. [16], do not generally generate a ratchet effect because of incomplete decoupling of the trans- port dynamics. As compared to other ratchet schemes in tight-binding lattices [14,16,18], our zigzag configuration offers the advantage of spatially separating a primary lat- tice (in which we want to realize a net transport) from an auxiliary one, which is transiently excited. The above analytical results are strictly valid within R = πn$_{s}$d/ ( λ | σ | ) , z 0 = π/ (2 κ$_{ab}$ ) . (4) Fig. 1. (Color online) (a) Schematic of a zigzag waveg- uide array, composed by a primary array A$_{n}$ and an aux- iliary array B$_{n}$. (b) Light refraction in a straight waveg- uide array. (c) Axis bending profile x 0 ( z ) for the obser- vation of light refraction rectification. The inset in (c) shows a schematic of the periodically-curved array. B$_{n}$ A$_{n}$ A$_{n}$$_{+1}$ B$_{n}$$_{+1}$ A$_{n-}$$_{1}$ B$_{n-}$$_{1}$ d v GLYPH<c107> aa GLYPH<c107> ab GLYPH<c107> bb X z x R R z$_{0}$ z$_{0}$ z$_{0}$ 0 (a) Y GLYPH<c113> GLYPH<c113>$_{r}$ (b) (c) d d X Z GLYPH<c113> X Z GLYPH<c113> c (b) 0 200 0 60 30 0 40 (a) (c) 0 200 0 200 z (mm) x ( m) GLYPH<c109> x ( m) GLYPH<c109> x ( m) GLYPH<c109> z (mm) Fig. 2. (Color online) Rectified refraction (pseudocolor map of integrated intensity distribution) in a 65-mm- long periodically-curved zigzag waveguide array (up- per images), and corresponding behavior expected for a straight zigzag array (bottom images), for an input ellip- tical beam exciting a few waveguides in the primary ar- ray at different tilting angles: (a) θ = 0, (b) θ = − 0 . 5 θ$_{B}$ , and (c) θ = 0 . 5 θ$_{B}$ . The values of other parameters are: λ = 633 nm, n$_{s}$ = 1 . 45, Δ n$_{A}$ = 0 . 003, Δ n$_{B}$ = 0 . 0032, w = 3 µ m, and d = 9 µ m. angle θ$_{B}$ = λ/ (2 d ) [see Fig.1(b)], will diffract and re- fract at an angle θ$_{r}$ ( θ ) = 2 κ$_{aa}$d sin( πθ/θ$_{B}$ ) as in an ordinary tight-binding linear lattice [4]. Rectification of beam refraction, i.e. a net transverse drift of discretized beams averaged over all possible incidence angles θ , is obviously absent because θ$_{r}$ ( − θ ) = − θ$_{r}$ ( θ ). Even for a single array A$_{n}$, curved to mimic a dc, ac or combined dc-ac driving fields, a net transport of wave packets - averaged over different initial conditions- is generally un- 2