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with g $^{′′}$( k ) ̸ = 0 for a second-order saddle point. In this case, according to the method of stationary phase the asymptotic behavior of ψ$_{0}$ ( α ; z ) for large values of z fol- lows the normal law ∼ 1 $_{/}$√$_{z}$$_{.}$ To summarize, ψ$_{0}$ ( α ; z ) scales: as ∼ F ( k$_{0}$ ) z − 1 /n at a self collimation direction α , where H ′ $_{0}$( k$_{0}$ ) = α and k$_{0}$ is a saddle point of order n ≥ 3; as ∼ F ( k$_{0}$ ) z − 1 for an observation angle α outside the ’diffraction cone’ | α | > α$_{max}$ ; as ∼ F ( k$_{0}$ ) z − 1 / 2 in- side the diffraction cone region | α | < α$_{max}$ but far from a self collimation direction. The far-field pattern of dis- crete diffraction tends therefore to confine light inside the diffraction cone | α | ≤ α$_{max}$ with asymptotic peaks at the propagation directions corresponding to the angles of self-collimation. 

This very general behavior may be illustrated more in details for the case of a tight-binding lattice in the 

nearest-neighbor approximation considered in Sec.II.C, for which H$_{0}$ ( k ) = − 2Δ cos k . In this lattice model one has g $^{′}$( k ) = α − 2Δ sin k , g $^{′′}$( k ) = − 2Δ cos k , so that the angle of diffraction cone is given by α$_{max}$ = 2Δ. Two saddle points of second-order are found at k = k$_{0}$ = ± π/ 2 for the observation angles α = ± α$_{max}$ , i.e at the edge of the diffraction cone, at which the far-field discrete diffrac- tion pattern is thus expected to show two peaks. For an observation angle α strictly inside the diffraction cone ( | α | < 2Δ), the equation g $^{′}$( k ) = 0 has two solutions which are saddle points of first order since g $^{′′}$( k ) ̸ = 0. The main contribution to the integral on the right hand side of Eq.(29) comes from these two saddle points, and can be calculated by the method of stationary phase, yielding explicitly 

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where k$_{0}$ is the solution to the equation sin k$_{0}$ = | α/ 2Δ | in the interval 0 ≤ k$_{0}$ < π/ 2. It should be noted that the far-field discrete diffraction pattern given by Eq.(33) holds for | α | < 2Δ. As | α | approaches 2Δ from below, the two saddle points of second order coalesce into a sin- gle saddle point of third order, and this explain the di- vergence of Eq.(33) as | α | → 2Δ, i.e at the self colli- mation directions, where the decay is slower and scales as ∼ z − 1 / $^{3}$. For | α | > 2Δ, i.e. outside the diffraction cone, there are not saddle points on the real axis and the decay is faster and scales as ∼ 1 /z . An example of far-field discrete diffraction for a beam with a Gaus- sian spectrum F ( k ) ∼ exp[ − ( k/w$_{k}$ ) $^{2}$] ( − π ≤ k < π ) is shown in Fig.1. In Fig.1(a) the intensity distribution | ψ ( x, z ) | $^{2}$, as obtained by accurate numerical computa- tion of the integral entering in Eq.(27), is plotted for a few propagation distances z . For the sake of readabil- ity, at each propagation distance z the field intensity has been normalized to its peak value. The diffraction cone and the emergence of two intensity peaks at the self- collimation directions α = ± α$_{max}$ are clearly visible just after a propagation distance z of ∼ 10 − 20 times the cou- pling length 1 / Δ [Fig.1(a)]. Inside the diffraction cone, the intensity distribution at such propagation distances is very well fitted by the analytical far-field distribution given by Eq.(33), as shown in Fig.1(b). At much longer propagation distances, the self-collimation peaks become 

dominant, and the appearance of three different scaling laws of beam decay (fast decay outside the diffraction cone | x | > 2Δ z ; normal decay inside the diffraction cone | x | < 2Δ z ; slower decay at the self-collimation directions x = ± 2Δ z ) is very clearly visible, as shown in Fig.1(c). 

## III. BEAM PROPAGATION IN CURVED WAVEGUIDE ARRAYS 

Discrete diffraction of light waves in linear optical waveguide arrays can be controlled by introducing trans- verse index gradients or local phase slips, which may produce a kind of refocusing or re-imaging of beam dis- tributions along the propagation distances (see, for in- stance, [3, 9, 12, 13, 26]) similarly to what happens to light propagating in continuous lensguide media. In par- ticular, waveguide arrays with a curved axis provide a particularly interesting set up to manage discrete diffrac- tion for both monochromatic and polychromatic light [9, 12–15]. It is therefore of major interest to have gen- eral laws describing the global behavior of beam prop- agation in curved waveguide arrays. In addition, it is well known that for the problem of paraxial diffraction in homogeneous continuous media or, more generally, of paraxial propagation in elementary optical systems and lensguides, one can introduce special families of field dis- 

