Generation and dynamics of quadratic birefringent spatial gap solitons

A method is proposed to generate and study the dynamics of spatial light solitons in a birefringent medium with quadratic nonlinearity. Although no analytical expression for propagating solitons has been obtained, our numerical simulations show the existence of stable localized spatial solitons in the frequency forbidden band gap of the medium. The dynamics of these objects is quite rich and manifests for instance elastic reflections, or inelastic collisions where two solitons merge and propagate as a single solitary wave. We derive the dynamics of the slowly varying envelopes of the three fields (second harmonic pump and two-component signal) and study this new system theoretically. We show that it does present a threshold for nonlinear supratransmission that can be calculated from a series expansion approach with a very high accuracy. Specific physical implications of our theoretical predictions are illustrated on LiGaTe${}_2$ (LGT) crystals. Once irradiated by a cw laser beam of $10\hspace{0.5em}\mu$m wavelength, at an incidence beyond the extinction angle, such crystals will transmit light, in the form of spatial solitons generated in the nonlinear regime above the nonlinear supratransmission threshold.

Figure 1

A 2D birefringent crystal, represented in the principal crystal plane, is lighted on its surface $\mathit{xOy}$ from a material of index $n_1$, greater than the ordinary optical index $n_o(\omega )$ of the crystal at given irradiation frequency $\omega$. The incident angle ${\alpha }_1$ is chosen slightly greater than the extinction angle, thus $\alpha =\pi /2$ and the light is evanescent in the direction $\mathit{Ox}$. The angle $\theta$ is chosen such as to obtain perfect phase matching for waves propagating in the direction $\mathit{Oz}$.

Figure 2

Three-component soliton generated in the system (27) with boundary values (29) for $B=2.4$ and $C=2.7$ with a reflecting end (31) and for $\nu =1$. The boundary amplitudes are settled smoothly from $B=C=0$ on the interval $\zeta \in [0,30]$ such as to avoid shock because of vanishing initial data $\varphi (\xi ,0)=0$ and $\phi (\xi ,0)=0$. The maximum amplitude values have been measured to be $\left|\psi \right|{}_m~4$, $\left|\varphi \right|{}_m~4$ and $\left|\phi \right|{}_m~5$.

Figure 3

Examples of dynamics of the SHG soliton $|\psi (\xi ,\zeta )|$ generated in the system (27) with $B=C=2.53$ and $\nu =1$. Left: boundary values (29) applied on a finite domain $\zeta \in [20,40]$ in $\xi =0$ to show the stability of the BGS soliton. Right: inelastic interaction of two identical solitons generated with the same boundary values but both in $\xi =0$ and $\xi =30$. The noise generated by the interaction is seen to perturb the trajectory of the emerging soliton.

Figure 4

Threshold curves in the $(B,C)$ plane for the system (27) submitted to boundary values (32) with $\nu =1$, for $A=0$ (up) and $A=1$ (down). Numerical results (squares) are checked against analytical ones (solid curves) obtained by solving (36) using the truncated series method at order $N=10$.

Figure 5

Basic definitions for refraction laws in a birefringent medium.