4 D. Discrete far-field pattern and anomalous beam decay In spite of the fact that the asymptotic law describing beam broadening due to diffraction is the same for dis- crete and continuous media, a deep difference is found when analyzing the decay behavior of the field intensity versus propagation distance and the far-field diffraction patterns. For the sake of simplicity, we will consider the diffraction problem in one dimension, though the results may be generalized to the two-dimensional diffraction problem. We then rewrite Eq.(2) as For a bell-shaped (e.g. Gaussian-shaped) and flat beam incident onto the array at a given tilting angle θ (nor- malized to the Bragg angle), we may write c$_{n}$ = | c$_{n}$ | exp( − iπθn ), and one obtains where p = p$_{x}$ = − i∂$_{x}$ . For the usual paraxial one- dimensional diffraction problem in a homogeneous con- tinuous medium, one has H$_{0}$ ( p ) = p $^{2}$/ 2, whereas for dis- crete diffraction in a one-dimensional waveguide array one has H$_{0}$ ( − p ) = H$_{0}$ ( p ) ( − π ≤ p < π ) and H ′ $_{0}$( p ) = 0 at p = 0 and at the band edges p = ± π . The most general solution to Eq.(26) can be written as i∂$_{z}$ψ ( x, z ) = H$_{0}$ ( p ) ψ ( x, z ) , (26) ψ ( x, z ) = ∫ dkF ( k ) exp[ ikx − iH$_{0}$ ( k ) z ] (27) where the spectrum F ( k ) is determined by the beam dis- tribution at the input plane ψ ( x, 0) F ( k ) = 1 2 π ∫ dxψ ( x, 0) exp( − ikx ) (28) ( ∫ dx → ∑ $_{n}$, x → n and ψ ( x = n ) → c$_{n}$ in the discrete diffraction problem). Our aim is to calculate the decay behavior of the field amplitude ψ ( x, z ) as the propaga- tion distance z increases, at either a constant x position (for instance at x = 0) or along a line x = αz , where α is a constant parameter defining the ’observation angle’ of the diffracted pattern. Note that, as the observation angle α is varied, the function ψ$_{0}$ ( α ; z ) = ψ ( x = αz, z ) corresponds, for large values of z , to the far field diffrac- tion pattern. We need thus to calculate the asymptotic behavior of the integral ψ$_{0}$ ( α ; z ) = ∫ dkF ( k ) exp[ izg ( k )] (29) g ( k ) = αk − H$_{0}$ ( k ) . (30) for z → ∞ , where we have set For this purpose, we may use the methods of station- ary phase or steepest descend (see, for instance, [24]), which predict that the asymptotic behavior of ψ$_{0}$ ( α ; z ) as z → ∞ depends on the existence and of the order of stationary points of the phase g ( k ) inside the integration domain. Let us first consider the continuous diffraction problem, H$_{0}$ ( p ) = p $^{2}$/ 2, and re-derive the well-known result that the amplitude ψ$_{0}$ ( α ; z ) decays as ∼ $_{1}$√$_{z}$ for any obser- vation angle α and the far-field pattern is proportional The ratio Γ between the diffraction parameters at subd- iffractive ( θ = 1 / 2) and normal incidence ( θ = 0) regimes takes then the form [see Eq.(24)] Γ = [(1 − α $^{2}$) / (1 + α $^{2}$)]1 / $^{2}$. Note that, for a very broad beam excitation ( α → 1), both κ$_{1}$ and κ$_{2}$ gets close to 1, β$_{x}$ tends to vanish [see Eq.(22)], and the diffraction length ∼ 1 /β$_{x}$ diverges independently of beam tilting angle θ , as ex- pected for a very broad input beam. However, in this case the ratio of diffraction lengths in the normal ( θ = 0) and subdiffractive ( θ = 1 / 2) regimes, which scales as ∼ Γ, tends to vanish as Γ ∼ (1 − α ) 1 / $^{2}$. Conversely, for a very narrow input beam ( α → 0), both κ$_{1}$ and κ$_{2}$ vanish and the diffraction length ∼ 1 /β$_{x}$ turns out to be inde- pendent of tilting angle and given by ∼ 1 / $_{(}$√$_{2Δ) [see}$ Eq.(22)] as expected for single waveguide excitation. may get very small for a broad input beam. To illustrate this point, let us consider as an example the following beam distribution at the input plane : | c$_{n}$ | = N α | n $^{|}$, where the parameter α (0 < α < 1) determines the spot size of the input beam ( α → 0 for single waveguide ex- citation, and α → 1 for a plane wave excitation), and N = [(1 − α $^{2}$) / (1 + α $^{2}$)]1 / 2 is a normalization factor. For such a field distribution, the values of coefficients κ$_{1}$ and κ$_{2}$ can be evaluated in a closed form, and read κ$_{1}$ = 2 α 1 + α 2 , κ$_{2}$ = α $^{2}$(3 − α $^{2}$) 1 + α 2 . (25) Γ = β$_{x}$ ( θ = 1 / 2) β$_{x}$ ( θ = 0) = √ 1 + κ$_{2}$ − 2 κ 2 1 1 − κ$_{2}$ , (24) κ$_{1}$ = ∑ n | c$_{n}$c$_{n}$$_{+1}$ | , κ$_{2}$ = ∑ n | c$_{n}$c$_{n}$$_{+2}$ | . (23) β 2 x = 2Δ 2 [ 1 − Re ( ∑ n c ∗ $_{n}$c$_{n}$$_{+2}$ )] + + Δ 2 [ ∑ n c ∗ $_{n}$( c$_{n}$$_{+1}$ − c$_{n}$$_{−}$$_{1}$ ) ] 2 . (21) cated at p = ± π/ 2. From Eq.(19) one obtains β 2 $_{x}$( θ ) = 2Δ 2 [ 1 − κ 2 1 + ( κ 2 1 − κ$_{2}$ ) cos(2 πθ ) ] (22) where κ$_{1}$ and κ$_{2}$ are defined by Generally, it turns out that κ 2 1 > κ$_{2}$ , so that the minimum of β$_{x}$ is attained at θ = ± 1 / 2, i.e. at the self-collimation points as expected from Eq.(20). Conversely, the max- imal diffraction (maximum value of β$_{x}$ ) is attained at normal incidence ( θ = 0). The ratio between the mini- mum and maximum values of β$_{x}$ , given by