3 distribution φ ( x, z$_{B}$ ) does not generally reproduce the input one φ ( x, 0). The output field is in turn given by the interference of the various wave packets φ$_{n}$ ( x, 0) with the appropriate phase delays γ$_{n}$ given by Eq.(10). This kind of interference is discussed in details in the next sec- tion and may lead to a Mach-Zehnder-like interferometry which does not require ZT. III. BEAM BREAK UP AND RECOMBINATION: CLASSICAL INTERFERENCE Let us consider a broad input beam, with narrow an- gular spectrum and near-field distribution G ( x ), imping- ing the array at the incidence angle θ , so that φ ( x, 0) = G ( x ) exp( iκx ) where κ = n$_{s}$θ/λ = ( π/a )( θ/θ$_{B}$ ) and is the Bragg angle [28]. We denote such an input field distribution as g ( l $^{)}$( x, 0; κ$_{0}$ ) and the corresponding propa- gated field as g ( l $^{)}$( x, z ; κ$_{0}$ ), where κ$_{0}$ and the integer l are defined such that κ$_{0}$ + 2 πl/a = κ and − π/a ≤ κ$_{0}$ < π/a . The normalization condition ∫ dx | g ( l $^{)}$( x, z ; κ$_{0}$ ) | 2 = 1 is assumed. In this case, following the analysis of Ref.[24], one can show that inside the arrayed structure one has θ$_{B}$ = λ 2 an$_{s}$ (11) where we have set | φ$_{n}$ ( x, z ) | 2 = ∣ ∣$^{θ$_{n,l}$}$$^{(}$$^{κ$_{0}$}$$^{)}$$^{ϕ$_{n}$}$ ( x, κ$_{0}$ + F z λ ) G ( x − x$_{n}$ ( z )) ∣ ∣$^{2}$ (12) φ ( x, z ) = ∑ n =1 , 2 , 3 ,.. φ$_{n}$ ( x, z ) . (5) B$_{n}$ ( k = κ + 2 πl/a ) = θ$_{n,l}$ ( κ ) (4) first Brillouin zone − π/a < κ ≤ π/a , one can introduce the set of functions which are defined over −∞ < k < ∞ . A typical ex- ample of refractive index profile and corresponding band diagram is shown in Figs.1(b) and 1(c) for an excitation wavelength λ = 633 nm, lattice period a = 11 µ m, bulk refractive index n$_{s}$ = 1 . 43 and maximum index change of the waveguides Δ n = 0 . 0016. The behavior of the Fourier coefficients | B$_{n}$ ( k ) | 2 of Bloch states is also de- picted in Fig.1(d). In presence of a transverse index gradient F , small enough to neglect interband transitions (i.e. ZT), the so- lution to Eq.(1) for an assigned beam distribution φ ( x, 0) at the z = 0 input plane can be written as a superposi- tion of wave packets φ$_{n}$ ( x, z ) belonging to the different bands of the arrays which propagate independently each other and undergo BOs, namely one can write [24] For negligible ZT, the evolution of the wave packet φ$_{n}$ can be calculated in a closed form by using the acceleration theorem [25] and reads [24] In Eq.(6), the coefficients c$_{n}$ ( κ ) are determined by the field distribution illuminating the array at the input plane according to φ$_{n}$ ( x, z ) = ∫ π/a − π/a dκc$_{n}$ ( κ ) ϕ$_{n}$ ( x, κ + F z/λ ) exp[ − iγ$_{n}$ ( κ, z )] . (6) c$_{n}$ ( κ ) = 〈 ϕ$_{n}$ ( x, κ ) | φ ( x, 0) 〉 = ∫ ∞ −∞ dxϕ ∗ $_{n}$( x, κ ) φ ( x, 0) (7) Equations (5) and (12) indicate that the injected beam φ ( x, 0) breaks into a superposition of wave packets φ$_{n}$ be- longing to the different bands of the array, with weight- ing factors | θ$_{n,l}$ ( κ$_{0}$ ) | $^{2}$, which undergo BOs along different paths x$_{n}$ ( z ) [see, for instance, Figs.3(a) and 4(a)-(d) to be discussed below]. According to Eq.(13), the path x$_{n}$ ( z ) followed by the wave packet φ$_{n}$ reproduces the shape of the band dispersion curve E$_{n}$ ( κ ), eventually corrected to include the Berry phase contribution. After a full BO cycle, i.e. at z = z$_{B}$ , the different wave packets φ$_{n}$ in- terference and the following scattering relations hold (see the Appendix) is the band dispersion curve, corrected to account for a possible topological (Berry phase) contribution [26, 27]. In particular, after a full BO cycle, i.e. for a propagation distance z = z$_{B}$ , one has φ$_{n}$ ( x, z$_{B}$ ) = φ$_{n}$ ( x, 0) exp( − iγ$_{n}$ ), where whereas the phase term γ$_{n}$ ( κ, z ) is given by γ$_{n}$ ( κ, z ) = 1 F ∫ Fz/λ 0 dκ $^{′}$E ′ $_{n}$( κ + κ $^{′}$) , (8) E ′ $_{n}$( κ ) = E$_{n}$ ( κ ) − F 2 π a ∫ a 0 dx iu$_{n}$ ( x, κ ) ∂u$_{n}$ ( x, κ ) ∂κ . (9) where g ( l $^{)}$( x, z$_{B}$ ; κ$_{0}$ ) = ∞ ∑ ρ = $_{−∞}$M $_{l,ρ}$g ( ρ $^{)}$( x, 0; κ$_{0}$ ) (14) where the scattering matrix M is given by M = A $^{†}$BA , A n,l = θ$_{n,l}$ ( κ$_{0}$ ) , B ρ,n = exp( − iγ$_{n}$ ) δ$_{ρ,n}$, (15) and γ$_{n}$ are given by Eq.(10). Note that, if the differ- ences of phase delays γ$_{n}$ − γ$_{ρ}$ were multiplies of 2 π , a full Since in the general case the phases γ$_{n}$ accumulated by the various wave packets φ$_{n}$ undergoing BOs are not the same or do not differ by multiplies of 2 π , the output field γ$_{n}$ = 1 F ∫ 2 π/a 0 dκE ′ $_{n}$( κ ) . (10) x$_{n}$ ( z ) = 1 F [ E ′ $_{n}$( κ$_{0}$ + F z/λ ) − E $^{′}$( κ$_{0}$ )] . (13)