5 ψ$_{0}$ ( α ; z ) ∼ F ( k$_{0}$ ) | zg ( n $^{)}$( k$_{0}$ ) | 1 $_{/n}$( n !) 1 n Γ ( 1 n ) exp [ izg ( k$_{0}$ ) ± i π 2 n ] . (32) Therefore, the decay law for ψ$_{0}$ ( α ; z ) scales as ∼ z − 1 $^{/n}$, where n is the highest order of the saddle points of g ( k ), provided that F ( k$_{0}$ ) = 0. In the case of diffraction in a homogeneous continuous medium, the order of saddle point is always n = 2. To determine n for the discrete diffraction problem, let us note that the dispersion curve H$_{0}$ ( k ) admits of at least a couple of inflection points, say at k = ± k$_{0}$ , at which H ′′ $_{0}$( k$_{0}$ ) = 0. These points cor- respond to the self-collimation directions introduced in Sec.II.C. Since g $^{′}$( k ) = α − H ′ $_{0}$( k ), the inflection points k = ± k$_{0}$ turn out to be also saddle points when the ob- servation angle α is chosen equal to H ′ $_{0}$( ± k$_{0}$ ). Therefore, for the discrete diffraction problem the largest order n of saddle points is at least n = 3, and the decay law of ψ$_{0}$ ( α ; z ), at the two observation angles α = ± H ′ $_{0}$( k$_{0}$ ) corresponding to the self-collimation directions k$_{0}$ , is slowed down -as compared to continuous diffraction- to (at least) ∼ z − 1 / $^{3}$. More generally, if the dispersion curve H$_{0}$ ( k ) of the lattice is engineered to achieve a very flat behavior near a self-collimation point k = k$_{0}$ , with g $^{′′}$( k$_{0}$ ) = g $^{′′′}$( k$_{0}$ ) = ... = g ( n − $^{1)}$( k$_{0}$ ) = 0 and g ( n $^{)}$( k$_{0}$ ) = 0 ( n ≥ 3), the decay law of ψ$_{0}$ ( α ; z ) scales as ∼ z − 1 /n at the observation angle α = H $^{′}$( k$_{0}$ ). This scaling law of beam decaying in the discrete diffraction problem is therefore anomalous , in the sense that along the self- collimation directions the intensity decays slower that 1 /z , i.e. of the characteristic decay law that one might expect from power conservation arguments. This seem- ingly paradoxical circumstance may be solved by observ- ing that, for an observation angle α different than any of the self-collimation directions, the decay of ψ$_{0}$ ( α ; z ) may be either normal (i.e. ∼ 1 $_{/}$√$_{z}$$_{) or even}$ faster . More precisely, for a fixed value of α in modulus larger than α$_{max}$ = max$_{k}$ | H ′ $_{0}$( k ) | , the function g ( k ) given by Eq.(30) does not have saddle points on the real axis, and ψ$_{0}$ ( α ; z ) decays as ∼ 1 /z . Conversely, for | α | < α$_{max}$ the equation g $^{′}$( k ) = α − H ′ $_{0}$( k ) = 0 admits of at least one solution, to the Fourier spectrum of the input (near-field) distri- bution. In this case, g ( k ) = αk − k $^{2}$/ 2 has a unique saddle point at k = k$_{0}$ = α , with g $^{′′}$( k$_{0}$ ) = − 1 = 0; therefore, provided that the spectrum F ( k ) has a nonva- nishing component at k = k$_{0}$ and F ( k$_{0}$ ) does not diverge [25], according to the method of steepest descend one has as z → ∞ . From Eq.(31) we obtain the well-known result of paraxial diffraction theory that the amplitude ψ$_{0}$ ( α ; z ) of the beam decays as ∼ $_{1}$√$_{z}$ for any observation angle α [25], and that the far-field diffraction pattern is shaped as the Fourier spectrum F ( α ) of the near-field distribution. This scaling law may be referred to as the normal scaling law, in the sense that the beam intensity ψ$_{0}$ ( α ; z ) ∼ F ( α ) √ 2 π z exp[ izα $^{2}$/ 2 − iπ/ 4] (31) FIG. 1: (color online) Beam propagation in a one-dimensional tight binding lattice with nearest neighboring coupling terms, showing far-field properties of discrete diffraction. In (a) the intensity distributions | ψ ( x, z ) | 2 are plotted, in arbitrary units, for propagation distances z = 0, z = 5 / Δ, z = 10 / Δ, z = 15 / Δ, z = 20 / Δ, and z = 25 / Δ, where Δ is the coupling rate between adjacent waveguides. The inset in (a) shows the Gaussian spectrum F ( k ) of the beam ( w k = 1 . 5). In (c) the intensity distribution | ψ ( x, z ) | 2 is depicted for a propagation distance z = 20 / Δ as numerically calculated by Eq.(27) (solid curve) and by the approximate relation (33) (dotted curve, al- most overlapped with the solid one). In (c) the beam intensity is plotted at a propagation distance z = 500 / Δ, clearly show- ing the dominance of two peaks at the diffraction cone edges (self-collimation directions) and the onset of three different decay laws at | α | > 2Δ, α = ± 2Δ, and | α | < 2Δ. -100 -50 0 50 100 0 5 10 15 20 25 0 1 2 diffractionGLYPH<.notdef>cone (a) IntensityGLYPH<.notdef>(arb.GLYPH<.notdef>units) x z GLYPH<c68> -2000 0 2000 0 1 IntensityGLYPH<.notdef>(arb.GLYPH<.notdef>units) x z GLYPH<c68> =500 (b) (c) 0 0 1 GLYPH<c112> GLYPH<c45>GLYPH<c112> SpectrumGLYPH<.notdef>(arb.GLYPH<.notdef>units) k -60 -40 -20 0 20 40 60 0 0.05 0.1 0.15 x z GLYPH<c68> =20 I ∝ | ψ | 2 decays as ∼ 1 /z whereas the beam spot size w$_{x}$ increases asymptotically as ∼ z [see Eq.(17)], the product Iw$_{x}$ being constant according to the power conservation law. For the discrete diffraction problem, we prove now that the decay law is generally slower than ∼ 1 $_{/}$√$_{z}$ at the observation angles corresponding to self-collimation, and that the far-field pattern is peaked at such angles and does not reproduce the spectrum F of the near-field dis- tribution. To this aim, let us observe that, according to the steepest descend method, the slowest decay term in the integral of Eq.(29) comes from the saddle points g $^{′}$( k$_{0}$ ) = 0 of largest order. In particular, if k$_{0}$ is a saddle point of order n ≥ 2, i.e. g ( k ) ≃ g ( k$_{0}$ )+[ g ( n $^{)}$( k$_{0}$ ) /n !]( k − k$_{0}$ ) n for k close to k$_{0}$ ( g ( n $^{)}$( k$_{0}$ ) = 0), the contribution of the saddle point to the integral in Eq.(29) for large values of z is given by [24]