Gradient catastrophe of nonlinear photonic valley-Hall edge pulses

We derive nonlinear wave equations describing the propagation of slowly varying wave packets formed by topological valley-Hall edge states. We show that edge pulses break up even in the absence of spatial dispersion due to nonlinear self-steepening. Self-steepening leads to the previously unattended effect of a gradient catastrophe, which develops in a finite time determined by the ratio between the pulse's nonlinear frequency shift and the size of the topological band gap. Taking the weak spatial dispersion into account results in the formation of stable edge quasisolitons. Our findings are generic to systems governed by Dirac-like Hamiltonians and validated by numerical modeling of pulse propagation along a valley-Hall domain wall in staggered honeycomb waveguide lattices with Kerr nonlinearity.

Figure 1

(a) Dimerized photonic graphene lattice with staggered sublattice potential. Here $\kappa$ denotes the tunneling coefficient between waveguides, $a_0$ is the distance between neighboring waveguides, and the round arrow $g$ illustrates the nonlinear self-action effect. (b) Transverse profile $\mathcal{I}\left(y\right)$ and (c) in-plane intensity distribution $\mathcal{I}(x,y)=|{\psi }_1{(x,y)|}^2+{\left|{\psi }_2(x,y)\right|}^2$ of the nonlinear edge wave propagating along the $x$ axis and bound to the domain wall located at $y=0$, where the effective mass $M\left(y\right)$ changes sign. Parameters are $M_0=1,g_{}=0.75$.

Figure 2

Gradient catastrophe development. The Gaussian pulse with $F_0=0.5$, ${\mathrm{\Lambda }}_0=3.5/\sqrt{2}$ is launched at $t=0$. Slices color code intensity distributions $\mathcal{I}(x,y)$ at the given moments: $t=20$ (bottom), $t=40$ (middle), $t=60$ (top). Cuts along the domain wall at $y=0$ show consistency of the numerical solution (blue curves) with the solution of the nonlinear simple wave equation (8) for the intensity (red dotted lines). Parameters are $M_0=1$, $g=0.75$, $g{F}_0=0.375M_0$. In Figs. 2 and 3, dashed lines trace the domain wall separating spatial domains with masses of the opposite sign as indicated by shading with different colors on the top surface. The theoretical estimate of breakdown time $t^{*}\approx 58$ according to formula (10) perfectly matches the numerical scenario.

Figure 3

Nonlinear pulse transformation in the domain-wall problem with dispersion. Snapshots show intensity distributions in-plane $\mathcal{I}(x,y)$ (top row) and along the domain wall $\mathcal{I}(x,y=0)$ (bottom row) at the given times: (a) $t=110$, (b) $t=66$, (c) $t=22$. Overlaid curves are the pulse envelopes calculated by using NLSE (green dashed) and Eq. (12) (red dotted). The Gaussian pulse with $F_0=0.18$, ${\mathrm{\Lambda }}_0=5/\sqrt{2}$ is launched at $t=0$. Parameters are $M_0=1$, $g=1$, $\eta =0.001$.

Figure 4

Formation of solitonic edge pulses. Upper row: Snapshots of the in-plane field distribution $|{\psi }_1{|}^2(x,y)$ at the given times: (a) $t=300$, (b) $t=180$, (c) $t=33$, (d) $t=22$, (e) $t=11$, (f) $t=0$. Lower row: Field profiles $|{\psi }_1{|}^2(x,y=0)$ along the domain wall located at $y=0$. The direction of propagation is to the left. Parameters are $M_0=1,g=1,\eta =0.05,F_0=0.5,{\mathrm{\Lambda }}_0=3.5/\sqrt{2}$.

Figure 5

Nonlinear dynamics of the optical beam at the valley-Hall domain wall at a zigzag interface in a honeycomb lattice of laser-written waveguides. The input beam has a Gaussian envelope along the domain wall with maximum intensity $I_0=2.5\times {10}^{15}$ W/${\mathrm{m}}^2$. (a), (b) Intensity distributions of the nonlinear beam (top panels, blue line) at propagation distances (a) $z=11$ mm (left column) and (b) $z=22$ mm (right column). For comparison, the linear beam evolution, i.e., with nonlinearity switched off, is shown in the bottom panels and with dotted red lines. Purple arrows points to the direction of motion. Dashed gray line depicts the domain wall. (c) Intensity distribution at the interface obtained in modeling of the paraxial equation (left) and the corresponding Dirac model (right). The purple dashed line traces a straight trajectory of the center of mass of the linear pulse. (d) Breakdown coordinate $z_{*}$ as a function of the input intensity $I_0$ estimated from the Dirac model (green curve) and paraxial modeling (cyan crosses). The dashed gray lines' intersection indicates the value of the input beam intensity used for (a), (b), and (c).

Figure 6

(a) Profiles of the nonlinear edge state's components ${\psi }_{1e}$ (solid black) and ${\psi }_{2e}$ (solid brown) confined to the domain wall at $y=0$. The transverse structure of the standing soliton (A1) shifted by $(-/+y_0)$ with components ${\psi }_{1s}$ (dashed/dotted violet) and ${\psi }_{2s}$ (dashed/dotted green) determines the structure of the nonlinear edge state in the positive/negative half-space. (b) Spin angle ${\alpha }_s\left(y\right)$ (pink curve) and intensity ${\varrho }_s\left({\alpha }_s\left(y\right)\right)$ (green curve) for the soliton shifted by $(-y_0)$ to the left of the domain wall. Purple line depicts the steplike mass distribution $M\left(y\right)$. Parameters are $M_0=1,{\mathcal{I}}_1\equiv |{\psi }_{1e}{\left(0\right)|}^2={\left|{\psi }_{2e}\left(0\right)\right|}^2=0.5,g=0.5,k=0$.

Figure 7

Edge soliton propagating along the domain wall and bypassing corners. Slices show snapshots of the intensity distribution $\mathcal{I}(x,y)$ at $t=50$ (bottom), $t=80$ (middle), $t=150$ (top). The shifted initial soliton envelope of maximum intensity 1.2 at ${\omega }_{\text{s}}=0.02$ is overlaid with red dots at (at the bottom slice). Parameters are $M_0=1$, $g=0.125$, $\eta =0.05$.

Figure 8

(a) Band structure $\beta \left(k\right)$ for a supercell of the staggered graphene lattice, composed of 20 dielectric elliptical waveguides with semiaxes $a_x,a_y$, as illustrated in (b). Normalization length parameters are $w_0=10\mu \mathrm{m}$, $z_0=2k_0{w}_0^2$. Dark green and light green colors in (b) correspond to two different perturbations of the refractive index $n_A,n_B$, respectively, in a bipartite lattice. (c) Profile of the stationary edge state $\mathcal{E}(x,y)$, with the wave number spotted with a red asterisk in (a). Gray dashed line in (c) locates the interfacial domain wall.