Effective dissipation and nonlocality induced by nonparaxiality

We investigate beam diffraction and spatial modulation instability of coherent light beams propagating in the nonparaxial regime in a nonlinear Kerr medium. We study the instability of plane-wave solutions in terms of the degree of nonparaxiality, beyond the regime of validity of the nonlinear Schroedinger equation. We also numerically analyze the way nonparaxial terms break the integrability and affect the periodical evolution of higher-order soliton solutions. We discuss nonlocality and effective dissipation introduced by the nonlinear coupling with evanescent waves.

Figure 1

(a) Comparison of the numerical simulations of the linear UPPE (5) with the paraxial theory, given by Eq. (14) with $A$ approximated by the first term in the asymptotic expansion (15). The figure shows that the theory is correct up to an uncertainty $\delta$ of $17\%$ for $\epsilon =0.1$. (b) As in (a) going beyond the paraxial approximation, i.e., including higher-order corrections in $\epsilon$ (we use the first six terms in the series expansion); in this case the error made is always lower than $1\%$.

Figure 2

(a) Comparison of the theoretical gain functions from NLS and modified NLS. The choice of the scale for the axes in the figure is such that the gain function in the paraxial case is independent of the value of $\gamma$ while in the nonparaxial case it depends only on the product $\epsilon \gamma$. In panels (b) and (c) we show the comparison between the theoretical predictions and the numerical results for the gain function, finding a good agreement for the range of the amplified wave numbers in the paraxial (b) and nonparaxial model (c). The simulations were performed by a split step Fourier method. The vertical dashed black lines separate the propagating Fourier components from the evanescent spectrum. The modified NLS equation takes correctly into account the nonpropagating nature of these components; indeed in (c) the noise beyond the black lines, present in (b), vanishes.

Figure 3

We simulate the filamentation process using the NLS equation (a) and (b), and the modified NLS equation (c) and (d). The initial condition is a constant of unitary amplitude perturbed by a small noise. The parameters used are $\epsilon ={10}^{-4}$ and $\gamma ={10}^3$. The filaments described by the modified NLS equation in panel (c) have wider size and less intensity in the region of focusing, with respect to the NLS equation in panel (a). This is due to the exponential decay of the evanescent components of the spectrum, which limits the narrowing of the filaments. The NLS equation preserves the power of the beam (b) while the modified NLS equation is a dissipative model (d) and hence it is valid until the relative loss of power can be considered negligible.

Figure 4

(a) Gain functions relative to the NLS, modified NLS and UPPE equations for $\epsilon \gamma =0.1$. (b) Comparison between the theoretical trends and the gain function obtained from the numerical simulation of the UPPE.

Figure 5

(a) Filamentation process with the UPPE model. The filaments present the same structure as those in Fig. 3. The introduction of small nonlocal interactions by the modification of the nonlinear term increases the instability and this enhances the loss of power during propagation.

Figure 6

(a) The periodicity of a third-order bright soliton ($\gamma =9$) is destroyed when including higher-order corrections in the linear (b) and nonlinear term (c) of the NLS equation. The initial condition is $\psi (x,0)=\text{sech}\left(x\right)$ and the $\epsilon$ parameter is chosen as $\epsilon =0.01$ such that $\epsilon \gamma \sim 0.1$.