4 reconstruction of the input beam -apart from an unim- portant phase term- would be achieved since A $^{†}$A = I . In this case the beam leaving the array at z = z$_{B}$ would not split and would diffract at the same tilting angle θ as that of the incoming beam. However, the phase delays γ$_{n}$ do not generally satisfy the previous condition, and thus the beam leaving the array at z = z$_{B}$ breaks up into several beams which diffract at the different angles θ + 2 ρθ$_{B}$ ( ρ = 0 , ± 1 , ± 2 , ... ) according to Eq.(14). Af- ter some propagation distance, such beams are not over- lapped and can be thus spatially resolved. This general behavior is schematically illustrated in Fig.2(a). More generally, we can say that, if the array is simultaneously excited by a set of identical broad beams tilted at differ- ent angles θ + 2 lθ$_{B}$ ( l = 0 , ± 1 , ± 2 , ... ), the array behaves like a liner multiport optical system for the amplitudes of modes g ( l ) [see Fig.2(b)] with a scattering matrix M given by Eqs.(15). Here we have focused our analysis to the case where the length L of the array equals the BO cycle z$_{B}$ . Similar results are obtained by assuming -more generally- that the array length L comprises an integer number N of BO cycles, i.e. for L = N z$_{B}$ . In this case, the scattering matrix can be again factorized as M = A $^{†}$BA , where now B ρ,n = exp( − iN γ$_{n}$ ) δ$_{ρ,n}$ . As an important example, let us consider the case where only the two lowest bands of the array are involved in the dynamics, and the scattering relations (15) may be limited to two modes solely. An inspection of the curves E$_{n}$ ( κ ) and | B$_{n}$ ( k ) | 2 shown in Figs.1 (c) and (d) indicates that this condition is realized, to a good approximation, when the array is excited by two broad beams tilted the former at an angle θ$_{1}$ equal to or slightly smaller than the Bragg angle θ$_{B}$ , the latter at the angle θ$_{2}$ = θ$_{1}$ − 2 θ$_{B}$ ∼ − θ$_{B}$ [see Fig.3(a)]. If we use the simplified notations g$_{1}$ ( x, z ) ≡ g $^{(0)}$( x, z ; κ$_{0}$ ) and g$_{2}$ ( x, z ) ≡ g ( − $^{1)}$( x, z ; κ$_{0}$ ) for the two modes scattered by the array in one BO cycle, where κ$_{0}$ ∼ π/a , the input-output relations [Eq.(14)] re- duce to FIG. 2: (color online) (a) Schematic of BO motion in the multiband regime. A broad beam illuminates the array at the incidence angle θ and excites several wave packets φ 1 , φ 2 , φ 3 belonging to the different bands of the array. The wave packets undergo BOs along distinct paths which reproduce in real space the spectral shapes of the band dispersion curves. After one BO cycle, owing to different phase delays γ n accu- mulated by the wave packets, several beams, diffracted at the angles θ , θ ± 2 θ B , θ ± 4 θ B , ...., are produced. (b) Schematic of a linear-optic multiport system described by a scattering matrix M . absolute phases of u$_{1}$ ( x, κ$_{0}$ ) and u$_{2}$ ( x, κ$_{0}$ ). After setting ρ$_{11}$ = ρ$_{22}$ = √$_{T}$ and R = 1 − T , where T = | θ$_{1}$$_{,}$$_{0}$ ( κ$_{0}$ ) | 2 = | θ$_{2}$$_{,}$$_{−}$$_{1}$ ( κ$_{0}$ ) | $^{2}$, we can then write ( ρ$_{11}$ ρ$_{12}$ ρ$_{21}$ ρ$_{22}$ ) = ( √$_{T}$ √$_{R}$$_{exp(}$$_{iα}$$_{)}$ − √$_{R}$$_{exp(}$ − iα ) √$_{T}$ ) (20) g$_{1}$ ( x, z$_{B}$ ) = S$_{11}$g$_{1}$ ( x, 0) + S$_{12}$g$_{2}$ ( x, 0) (16) g$_{2}$ ( x, z$_{B}$ ) = S$_{21}$g$_{1}$ ( x, 0) + S$_{22}$g$_{2}$ ( x, 0) (17) where we have set S$_{11}$ = M 0 , $_{0}$, S$_{12}$ = M 0 , − $_{1}$, S$_{21}$ = M − 1 , 0 and S$_{22}$ = M − 1 , − $_{1}$. According to Eqs.(15), the 2 × 2 scattering matrix S entering in Eqs.(16) and (17) is given by where α is the phase of ρ$_{12}$ . Note that Eq.(20) is analo- gous to the scattering matrix of a lossless beam split- ter with transmittance T = 1 − R (see, for instance, [29]). Physically, such a transformation corresponds to the mixing of the incoming beams g$_{1}$ and g$_{2}$ into the two wave packets φ$_{1}$ and φ$_{2}$ belonging to the two lowest- order bands of the array [see Fig.3(a)]. At exact Bragg incidence, i.e. for θ$_{1}$ = − θ$_{2}$ = θ$_{B}$ , one has T = 1 / 2, i.e. the equivalent beam splitter is balanced. Note also that the full transformation (18) is analogous to that of a two- port Mach-Zehnder interferometer in which two waves g$_{1}$ and g$_{2}$ are mixed by a fist beam splitter BS1, undergo different phase delays γ$_{1}$ and γ$_{2}$ in the two arms of the in- terferometer, and are then recombined by a second beam splitter BS2 [see Fig.3(b)]. The transfer matrix S of the equivalent Mach-Zehnder interferometer reads explicitly Since the matrix ρ$_{ik}$ is unitary, one has | ρ$_{11}$ | = | ρ$_{22}$ | , | ρ$_{12}$ | = | ρ$_{21}$ | , | ρ$_{11}$ | 2 + | ρ$_{12}$ | 2 = 1 and ρ$_{11}$ρ ∗ 12 + ρ$_{21}$ρ ∗ 22 = 0. ( ρ$_{11}$ ρ$_{12}$ ρ$_{21}$ ρ$_{22}$ ) = ( θ$_{1}$$_{,}$$_{0}$ ( κ$_{0}$ ) θ$_{1}$$_{,}$$_{−}$$_{1}$ ( κ$_{0}$ ) θ$_{2}$$_{,}$$_{0}$ ( κ$_{0}$ ) θ$_{2}$$_{,}$$_{−}$$_{1}$ ( κ$_{0}$ ) ) . (19) S = ( ρ ∗ 11 ρ ∗ 21 ρ ∗ 12 ρ ∗ 22 ) × ( exp( − iγ$_{1}$ ) 0 0 exp( − iγ$_{2}$ ) ) × ( ρ$_{11}$ ρ$_{12}$ ρ$_{21}$ ρ$_{22}$ ) (18) where F /F=0.93 0 F /F=1 0 (a) (c) x z inputGLYPH<.notdef>plane (z=0) L GLYPH<c102>$_{1}$ GLYPH<c102>$_{2}$ GLYPH<c113> InputGLYPH<.notdef>beam scatteredGLYPH<.notdef>beams (b) array outputGLYPH<.notdef>plane (z=L) GLYPH<c102>$_{3}$ g (0) g (-1) g (-2) g (1) scattering matrix GLYPH<c77> g (0) g (1) g (-1) g (-2) Without loss of generality, we may assume ρ$_{11}$ = ρ$_{22}$ to be real-valued and positive by a suitable choice of the