content_image/395122.jpg content_image/395134.jpg content_image/395124.jpg content_image/395125.jpg content_image/395126.jpg content_image/395127.jpg content_image/395135.jpg content_image/395128.jpg
hold for any functions f ( p ) and g ( r ). For A = r , one obtains 

i.e. 

which is the evolution equation of the beam center of mass. Note that the path followed by any beam is al- ways straight, regardless of the specific form of H$_{0}$ or initial field distribution which just determine the trans- verse drift velocity 〈∇ $_{p}$H$_{0}$ 〉 of the beam. To determine the evolution equation of the beam spot size w$_{x}$ , let us as- sume A = x $^{2}$, so that the following cascade of Eherenfest equations [Eq.(9)] is obtained 

After integration, one obtains 

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where the expectation values on the right hand side of Eq.(16) are calculated at z = 0, i.e. for the initial beam distribution. From Eqs.(6), (12) and (16) the following evolution equation for the beam spot size w$_{x}$ is then ob- tained 

where we have set 

and the expectation values are calculated at z = 0. A similar expression for w$_{y}$ ( z ) can be obtained by replac- ing x with y in Eqs.(17), (18) and (19). A major re- sult expressed by Eq.(17) is that, regardless of the par- ticular form of H$_{0}$ , w$_{x}$ ( z ) (and similarly w$_{y}$ ( z )) asymp- totically grows with z linearly, with a diffraction length given by ∼ 1 /β$_{x}$ . Therefore -and this one of the ma- jor result of this section- the broadening law of a spatial beam due to diffraction does not differ for discrete or continuous diffraction. In addition, for a beam carry- ing a finite power and admitting of finite moments 〈 x $^{2}$〉 and 〈 y $^{2}$〉 , the coefficient β 2 x given by Eq.(19) is always 

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strictly positive and does not vanish. This can be gen- erally proven by observing that β 2 x is the variance of the operator ( ∂H$_{0}$/∂p$_{x}$ ), which is always positive and van- ishes solely when the initial field distribution ψ ( x, y, 0) is an eigenfunction of ( ∂H$_{0}$/∂p$_{x}$ ), i.e. of p$_{x}$ = − i∂$_{x}$ . Since such eigenfunctions are delocalized plane waves, it then follows that the variance of ( ∂H$_{0}$/∂p$_{x}$ ) is strictly positive for any initial beam distribution carrying a finite power, regardless of the specific form of H$_{0}$ . 

Beam self-collimation (also referred to as beam sub- diffraction) is a well known phenomenon occuring in homogeneous arrays and, more generally, in photonic crystal structures with engineered band structure H$_{0}$ ( p ) showing points of local flatness (see, for instance, [20]). The simplest example of sub diffraction is the ’arrest’ of beam spreading in a one-dimensional tight-binding lat- tice with nearest-neighboring couplings, which was ob- served in Ref.[4] using relatively broad Gaussian beams at an incidence angle set in correspondence of an inflex- ion point of the band dispersion curve. Though it is well understood that in such a regime diffraction is cancelled solely at low orders, it was perhaps overlooked the fact that self-collimation does not modify the beam broaden- ing scaling law [Eq.(17)]. In other words, self-collimation will correspond to a reduction of the coefficient β 2 x for special initial field distributions, but not to a change of the scaling law describing beam broadening. If we con- sider, for the sake of simplicity, a one-dimensional lat- tice and assume that the spectrum F ( k ) of the exciting beam, defined as F ( k ) = 1 / (2 π ) ∫ dxψ ( x, 0) exp( − ikx ), is narrow at around its mean k$_{0}$ , the value of β 2 $_{x}$, as given by Eq.(19), can be expanded in series of moments I$_{l}$ = ∫ dk ( k − k$_{0}$ ) $^{l}$| F ( k ) | 2 ( l = 2 , 3 , 4 , ... ) as 

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where b$_{l}$ is the value of the derivative ( ∂ $^{l}$H$_{0}$ /∂k $^{l}$) evalu- ated at k = k$_{0}$ . As I$_{l}$ rapidly goes to zero as the order l increases, Eq.(20) shows that at the points k$_{0}$ of the dis- persion curve where b$_{2}$ (and possibly b$_{3}$ , b$_{4}$ , ...) vanishes beam broadening is reduced. We will refer to such points, where the dispersion curve H$_{0}$ ( k ) is locally flat, to as self- collimation points [note that the condition H ′ $_{0}$( k$_{0}$ ) = 0 is not requested]. 

As an example, let us consider the simplest one- dimensional waveguide array in the nearest-neighboring approximation, considered in Ref.[4] to demonstrate self- collimation effects. The Hamiltonian H$_{0}$ has the form H$_{0}$ = − 2Δ cos( p ), and the self-collimation points are lo- 

## C. Self-collimation regime 

