8 V. CONCLUSIONS Appendix A: Derivation of the scattering matrix Repeating the previous analysis to each of the plane waves entering in the integral on the right hand side of Eq.(A7) and using the superposition principle yields which are precisely the scattering relations between in- put and output waves given in the text [Eq.(14)], with a scattering matrix M = M ( κ$_{0}$ ) given by M = A $^{†}$BA . Such relations can be extended to the case where the input waves g ( l $^{)}$( x, 0; κ$_{0}$ ) are not strictly plane waves, rather tilted broad beams with a narrow angular spec- trum ˆ G (Δ κ ) = (2 π ) − 1 / 2 ∫ dxG ( x ) exp( − iκx ) and near- field distribution G ( x ). In this case we may write g ( l $^{)}$( x, z$_{B}$ ; κ$_{0}$ ) = ∑ ρ =0 , ± 1 , ± 2 ,... ( A $^{†}$BA )$_{l,ρ}$ g ( ρ $^{)}$( x, 0; κ$_{0}$ ) (A6) Substitution of Eq.(A5) into Eq.(A4) finally yields g ( l $^{)}$( x, 0; κ$_{0}$ ) = 1 √$_{2}$$_{π}$ ∫ d Δ κ ˆ G (Δ κ ) × × exp ( iκ$_{0}$x + i Δ κx + i 2 π a lx ) . (A7) ( l = 0 , ± 1 , ± 2 , ... ). Owing to the completeness of Bloch states ϕ$_{n}$ ( x, κ ), one can write Since ϕ$_{n}$ ( x, κ ) = u$_{n}$ ( x, κ ) exp( iκx ) and taking into ac- count the Fourier decomposition of u$_{n}$ ( x, κ ) [Eq.(3)], one g ( l $^{)}$( x, 0; κ$_{0}$ ) = ∑ n =1 , 2 , 3 ,... ∫ π/a − π/a dκ 〈 ϕ$_{n}$ | g ( l $^{)}$〉 ϕ$_{n}$ ( x, κ ) . (A2) g ( l $^{)}$( x, 0; κ$_{0}$ ) = 1 √$_{2}$$_{π}$ exp ( iκ$_{0}$x + i 2 π a lx ) (A1) In this appendix we derive the scattering relations given in the text [Eqs.(14) and (15)] for the amplitudes g ( l ) of tilted beams at the input and output planes of the array, i.e. after one BO cycle. To this aim, let us first prove Eqs.(14) and (15) when the incident beams are tilted plane waves, i.e. let us first assume In this work interference phenomena for classical and non-classical light propagating in arrays of coupled waveguides and undergoing multiband optical Bloch os- cillations with negligible Zener tunneling have been the- oretically investigated. A wave scattering analysis of multiband BOs shows that Mach-Zehnder-like interfer- ence effects spontaneously arise owing to beam splitting and subsequent beam recombination occurring at each BO cycle. A noteworthy example of quantum inter- ference is provided by the doubling of the interference fringes in photon counting rates for a correlated pho- ton pair undergoing two-band BOs. This phenomenon is analogous to the one observed in a Mach-Zehnder inter- ferometer excited by pairs of correlated photons and is a manifestation of the so-called Broglie wavelength of the two-photon entangled state produced after the first beam splitter of the interferometer and probed by the second one [22]. It is envisaged that the present results may stimulate further theoretical and experimental investiga- tions of classical and quantum interference phenomena of light propagating in complex periodic, quasi-periodic or disordered photonic lattices. where we have introduced the matrix A n,l ≡ θ$_{n,l}$ ( κ$_{0}$ ) and A † is the adjoint of A , i.e. A † l,n = ∗ $_{n,l}$. After one BO cycle, the Bloch state ϕ$_{n}$ ( x, κ$_{0}$ ) accumulates a phase shift exp( − iγ$_{n}$ ), where γ$_{n}$ is given by Eq.(10). After the in- troduction of the diagonal matrix B ρ,n = exp( − iγ$_{n}$ ) δ$_{n,ρ}$ , we can thus write readily finds 〈 ϕ$_{n}$ ( x, κ ) | g ( l $^{)}$( x, 0; κ ) 〉 = θ ∗ $_{n,l}$( κ$_{0}$ ) δ ( κ − κ$_{0}$ ), so that Eq.(A2) yields g ( l $^{)}$( x, 0; κ$_{0}$ ) = ∑ n =1 , 2 , 3 $_{,...}$A † $_{l,n}$ϕ$_{n}$ ( x, κ$_{0}$ ) (A3) g ( l $^{)}$( x, z$_{B}$ ; κ$_{0}$ ) = ∑ n =1 , 2 , 3 $_{,...}$A † $_{l,n}$exp( − iγ$_{n}$ ) ϕ$_{n}$ ( x, κ$_{0}$ ) = n =1 , 2 , 3 ,... ( A $^{†}$B )$_{l,n}$ ϕ$_{n}$ ( x, κ$_{0}$ ) . (A4) Since the matrix A is unitary, A − 1 = A † and Eq.(A3) can be inverted yielding ϕ$_{n}$ ( x, κ$_{0}$ ) = ∑ l =0 , ± 1 , ± 2 $_{,...}$A $_{n,l}$g ( l $^{)}$( x, 0; κ$_{0}$ ) . (A5) g ( l $^{)}$( x, z$_{B}$ ; κ$_{0}$ ) = ∑ ρ =0 , ± 1 , ± 2 ,... ∫ d Δ κ 1 √$_{2}$$_{π}$ ˆ G (Δ κ ) M $_{l,ρ}$( κ$_{0}$ + Δ κ ) exp ( iκ$_{0}$x + i Δ κx + i 2 π a ρx ) . (A8) spectral extension of ˆ G (Δ κ ), we may take M $_{l,ρ}$( κ$_{0}$ + Δ κ ) ≃ M $_{l,ρ}$( κ$_{0}$ ) out of the integral in Eq.(A8). The re- If the Fourier coefficients θ$_{n,l}$ ( κ ) of Bloch states - and hence the transfer matrix M ( κ )- vary slowly over the