Bright nonlocal quadratic solitons induced by boundary confinement

Under the Dirichlet boundary conditions, a family of bright quadratic solitons exists in the regime where the second harmonic can be regarded as the refractive index of the fundamental wave with an oscillatory nonlocal response. By simplifying the governing equations into the Snyder-Mitchell mode, the approximate analytical solutions are obtained. Taking them as the initial guess and using a numerical code, we found two branches of bright solitons, of which the beam width increases (branch I) and decreases (branch II) with the increase of the sample size, respectively. If the nonlocality is fixed and the sample size is varied, the soliton width varies piecewise and approximately periodically. In each period, solitons only exist in a small range of sample size. Single-hump fundamental wave solitons with the same beam width in narrower samples can be, if the second harmonics are connected smoothly, jointed to be a multihump soliton in a wider sample whose size is the sum of those for the narrower ones. The dynamical simulation shows that the found solitons are unstable.

Figure 1

Sketch of the bright nonlocal quadratic soliton induced by boundary confinement in the setup composed of parametric material and air. Red solid and blue dashed lines represent the FW and the SH, respectively.

Figure 2

The existence domain for the two branches of symmetric single-hump bright solitons in one period (the sample size varies from $1.5T_s$ to $2.5T_s$) for various degree of nonlocality (the nonlocal degree becomes weaker with the increase of $\sqrt{\alpha }$).

Figure 3

(a) The rms width $w$ for ${\varphi }_1$ of branch I (blue solid lines) and branch II (red dashed lines) solitons versus sample size $L$, when $\sqrt{\alpha }=0.3$. (b)–(h) Soliton profiles of branch I [blue solid lines (FW) and blue solid lines with circle symbol (SH), left ordinate] and branch II [red dashed lines (FW) and red dashed lines with circle symbol (SH), right ordinate] for different sample sizes corresponding to points 1–9 in panel (a). Panels (f)–(h) all correspond to points 8 and 9, but the FWs are located at different positions of the sample.

Figure 4

(a) The single-hump branch I [blue solid lines (FW) and blue solid lines with circle symbol (SH), left ordinate] and branch II [red dashed lines (FW) and red dashed lines with circle symbol (SH), right ordinate] solitons in the sample for $L=2.054T_s$ and $\sqrt{\alpha }=0.3$. (b) The double-hump solitons in the sample with the same parameters. (c) The triple-hump solitons in the sample with the same parameters.

Figure 5

(a) The single-hump branch I [blue solid lines (FW) and blue solid lines with circle symbol (SH), left ordinate] and branch II [red dashed lines (FW) and red dashed lines with circle symbol (SH), right ordinate] solitons in the sample with the size $L_1=1.05T_s$. (b) The single-hump branch I and branch II solitons in the sample with the size $L_2=1.554T_s$. (c) The double-hump branch I and branch II solitons in the sample with the size $L=L_1+L_2$. (d) The double-hump branch I and branch II solitons in the sample with the size $L=2L_2$. In panels (a)–(d) $\sqrt{\alpha }=0.3$.

Figure 6

Upper panels: Intensity evolution of FW (left) and SH (right) of the branch I soliton for $L=1.054T_s$ and $\sqrt{\alpha }=0.3$. Lower panels: Evolution of FW (left) and SH (right) for the branch II soliton; the parameters are the same as those for the upper panels. The simulation used $d_1=2, d_2=1, \lambda =-2$, and $N_x=339$ (for branch I) and $N_x=259$ (for branch II) points in $x$ and $N_z=5\times {10}^4$ steps in $z$ (corresponding to that the step length along $z$ is $\mathrm{\Delta }z=L_d/{10}^4$, where $L_D=w^2/\left(2\right|d_1\left|\right)$ is the Rayleigh distance of the FW and $w$ is the rms width of the FW.)

Figure 7

An example for the evolution of FW (red solid line) and SH (blue dashed line) for the case that only the FW beam (with Gaussian profile) is input. At the entrance plane the width of the FW equals that of the FW of the branch II soliton for $L=1.054T_s$ and $\sqrt{\alpha }=0.3$, and the power equals the total power of the FW and the SH of the branch II soliton. Other parameters are the same as those in Fig. 6.