Nonlocal description of X waves in quadratic nonlinear materials

We study localized light bullets and X waves in quadratic media and show how the notion of nonlocality can provide an alternative simple physical picture of both types of multidimensional nonlinear waves. For X waves we show that a local cascading limit in terms of a nonlinear Schrödinger equation does not exist—one needs to use the nonlocal description, because the nonlocal response function does not converge toward a $\delta$ function. Also, we use the nonlocal theory to show that the coupling to the second harmonic is able to generate an X shape in the fundamental field despite having anomalous dispersion, in contrast to the predictions of the cascading limit.

Figure 1

(Color online) Linear and nonlinear length scales versus the degree of nonlocality, $\sigma$, for $A=5$ and $B=3$.

Figure 2

(Color online) Response function $R_{+}(x,t)$ for anomalous SH dispersion, given by Eq. (20). The degree of nonlocality is $\sigma =1$.

Figure 3

The effect of nonlocality $\sigma$ on the response function for both (a) anomalous ($d_2=1$, $R_{+}$) and (b) normal ($d_2=-1$, $R_{-}$) SH dispersion. Plots are made for constant $x=0$.

Figure 4

Intensity profile of the FW, $\mid E_1(x,t){\mid }^2$, at different propagation lengths for the modified X-wave case. Parameter values are $A=5,B=3$, $d_1=-1$, and $d_2=+1$. The NLS Eq. (3) solved for $\sigma =0.2$ (left). Full equations solved for $\sigma =0.2$ (center) and 10 (right). 15 contour lines from 0.001 logarithmically separated to the maximum intensity.

Figure 5

Intensity profiles at different propagation lengths for the modified X-wave case with nonlocality $\sigma =0.5$. Parameter values are $A=5,B=3$, $d_1=-1$, and $d_2=+1$. NLS FW Eq. (3) (left). Full equations (1, 2) solved for $\mid E_1(x,t){\mid }^2$ (center) and $\mid E_2(x,t){\mid }^2$ (right). 15 contour lines from 0.001 logarithmically separated to the maximum intensity.

Figure 6

Intensity profiles for the (left) FW, $\mid E_1{\mid }^2$, and (right) SH, $\mid E_2{\mid }^2$, at propagation lengths $z=$ (top) 4 and (bottom) 8. Parameter values are $A=5,B=3$, $d_1=-1$, $d_2=+1$, and $\sigma =10$, i.e., strongly nonlocal. 15 contour lines from 0.005 logarithmically separated to the maximum intensity.

Figure 7

(Color online) Response function in the Fourier domain, ${\tilde{R}}_{-}(k,\omega )$, for $d_2=-1$, $s_{\beta }=1$, and $\sigma =1$.

Figure 8

(Color online) The response function $R_{-}(x,t)$ [Eq. (24)] in the normally dispersive case $d_2=-1$ for $\sigma =1$. Compare with Fig. 2.

Figure 9

Intensity profile of the strongly nonlocal X-wave case, $\sigma =10$. The evolution of $\mid E_1{\mid }^2$ (left) and $\mid E_1{\mid }^2$(right). Parameter values are $A=5,B=3$, and $d_1=d_2=-1$. 15 contour lines from 0.005 logarithmically separated to the maximum intensity.

Figure 10

Comparison between $\mid \widetilde{E_2}{\mid }^2$ (a) and the nonlocal prediction $(1∕\beta {)}^2\mid \widetilde{E_1^2}{\mid }^2\mid \tilde{R}{\mid }^2$ (b) at $z=4$ with 25 logarithmically spaced contours between 0.0001 and 10. Parameter values are $A=5,B=3$, and $d_1=d_2=-1$. $\sigma =10$, i.e., strongly nonlocal. Dashed lines correspond to $\sigma \omega =\pm \sqrt{1+{\sigma }^2{k}^2}$.

Figure 11

FW and SH intensity profiles $\mid E_1{\mid }^2$ (left) and $\mid E_2{\mid }^2$ (right) at propagation lengths $z=0,4,$ and $8$. Parameter values are $A=5,B=3$, and $d_1=-d_2=1$. Strongly nonlocal case, $\sigma =10$. 25 contour lines from 0.005 logarithmically separated to the maximum intensity. The mesh size is $360$ on a $-50<x,t<50$ domain.