8 size w$_{x}$ ( z ) are thus ruled by Eqs.(47), (53) and (56). Note that beam evolution depend on the input beam parame- ters q$_{1}$ , q$_{2}$ and q$_{3}$ -defined by Eqs.(50),(54) and (55)- and by the complex amplitude Ω( z ), defined by Eqs.(48-49) and accounting for bending of waveguide axis. Note also that for straight arrays Ω( z ) = Δ z , and one thus retrieves the results of discrete diffraction derived in Sec.II.B, in particular the linear asymptotic increase of w$_{x}$ with z . The condition for diffraction suppression, i.e. a non- secular growth of w$_{x}$ ( z ) with z , is that Ω( z ) remains a limited function of z as z increases. This condition is always satisfied for a constant value of E $_{x}$, which corre- sponds to circularly-curved waveguides and to the onset of the optical analogue of Bloch oscillations [9]. Similarly, for periodic axis bending with spatial period Λ, E $_{x}$( z ) is a periodic function of z , and the condition of boundness of Ω( z ) is given by which is precisely the condition of ’dynamic localization’ previously investigated in Refs.[12, 13]. ∫ Λ 0 dξ exp[ − iφ ( ξ )] = 0 , (57) B. Two-dimensional array For the two-dimensional waveguide array model (38), the moment equations turn out to decouple into two set of equations, similar to Eqs.(42-46), separately acting onto the x and y directions. The evolution equations for the beam center of mass 〈 x 〉 , 〈 y 〉 are then given by Equation (43) can be readily integrated, yielding the fol- lowing evolution equation for the beam center of mass where we have set Similarly, integration of Eqs.(45) and (46) yields Successive application of the Ehrenfest equation (9) yields the following equations for 〈 x 〉 and 〈 θ 〉 and the following coupled equations for 〈 x $^{2}$〉 , 〈 ρ 〉 and 〈 σ 〉 Taking into account that Δ exp[ − iφ ( z )] = d Ω /dz and that 2Re {∫ z 0 dξ Ω $^{∗}$( ξ )( d Ω /dξ ) } = | Ω( z ) | $^{2}$, substitution of Eq.(52) into Eq.(44) yields Similarly, the beam sizes w$_{x}$ and w$_{y}$ , defined as are calculated using Eqs.(58-59) and the following evolu- tion equations for 〈 x $^{2}$( z ) 〉 and 〈 y $^{2}$( z ) 〉 For a given field distribution c$_{n}$ (0) at the input plane, the evolution of the beam center of mass 〈 x ( z ) 〉 and beam The beam size w$_{x}$ is then given by where we have set and where we have set 〈 x ( z ) 〉 = 〈 x (0) 〉 + 2Im { q$_{0}$ Ω $^{∗}$( z ) } (47) Ω( z ) ≡ ∫ z 0 dξ Δ exp[ − iφ ( ξ )] , (48) z φ ( z ) ≡ 0 dξ E $_{x}$( ξ ) , (49) q$_{0}$ ≡ ∑ n c ∗ $_{n}$(0) c$_{n}$$_{+1}$ (0) . (50) d 〈 x $^{2}$〉 dz = 2ΔRe ( 〈 σ 〉 ) (44) d 〈 ρ 〉 dz = − 2 i E $_{x}$( z ) 〈 ρ 〉 + i E $_{x}$( z ) (45) d 〈 σ 〉 dz = 4Δ 〈 ρ 〉 − i E $_{x}$( z ) 〈 σ 〉 . (46) d 〈 x 〉 dz = 2ΔIm ( 〈 θ 〉 ) (42) d 〈 θ 〉 dz = i E $_{x}$( z ) 〈 θ 〉 , (43) 〈 x ( z ) 〉 = 〈 x (0) 〉 + 2Im { q$_{0}$$_{x}$ Ω ∗ $_{x}$( z ) } (58) y ( z ) = y (0) + 2Im q$_{0}$$_{y}$ Ω $_{y}$( z ) (59) Ω$_{x,y}$( z ) = ∫ z 0 dξ Δ$_{x,y}$ exp[ − iφ$_{x,y}$ ( ξ )] (60) z φ$_{x,y}$ ( z ) = 0 dξ E $_{x,y}$( ξ ) (61) q$_{0}$$_{x}$ = ∑ n,m c ∗ $_{n,m}$(0) c$_{n}$$_{+1}$$_{,m}$ (0) (62) q$_{0}$$_{y}$ = n,m c ∗ $_{n,m}$(0) c$_{n,m}$$_{+1}$ (0) . (63) q$_{1}$ ≡ i ∑ n (2 n − 1) c ∗ $_{n}$(0) c$_{n}$$_{−}$$_{1}$ (0) (54) q$_{2}$ ≡ n c ∗ $_{n}$(0) c$_{n}$$_{−}$$_{2}$ (0) . (55) 〈 x $^{2}$( z ) 〉 = 〈 x $^{2}$(0) 〉 + 2 | Ω( z ) | 2 + 2Re { q$_{1}$ Ω( z ) − q$_{2}$ Ω $^{2}$( z ) } , (53) 〈 ρ ( z ) 〉 = exp[ − 2 iφ ( z )] × × { 〈 ρ (0) 〉 + 1 2 exp[2 iφ ( z )] − 1 2 } (51) 〈 σ ( z ) 〉 = exp[ − iφ ( z )] × × { 〈 σ (0) 〉 + 4Ω( z ) [ 〈 ρ (0) 〉 − 1 2 ] + 2Ω $^{∗}$( z ) } (52) w$_{x}$ ( z ) = √ 〈 x $^{2}$( z ) 〉 − 〈 x ( z ) 〉 $^{2}$. (56) 〈 x $^{2}$( z ) 〉 = 〈 x $^{2}$(0) 〉 + 2 | Ω$_{x}$ | $^{2}$+ 2Re { q$_{1}$$_{x}$ Ω$_{x}$ − q$_{2}$$_{x}$ Ω 2 x } (66) w$_{x}$ ( z ) = √ 〈 x $^{2}$( z ) 〉 − 〈 x ( z ) 〉 2 (64) w$_{y}$ ( z ) = y $^{2}$( z ) y ( z ) $^{2}$, (65)