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Valley Hall edge solitons in a photonic graphene

Qian Tang, Boquan Ren, Victor O. Kompanets, Yaroslav V. Kartashov, Yongdong Li, and Yiqi Zhang

Open Access Open Access

Abstract

We predict the existence and study properties of the valley Hall edge solitons in a composite photonic graphene with a domain wall between two honeycomb lattices with broken inversion symmetry. Inversion symmetry in our system is broken due to detuning introduced into constituent sublattices of the honeycomb structure. We show that nonlinear valley Hall edge states with sufficiently high amplitude bifurcating from the linear valley Hall edge state supported by the domain wall, can split into sets of bright spots due to development of the modulational instability, and that such an instability is a precursor for the formation of topological bright valley Hall edge solitons localized due to nonlinear self-action and travelling along the domain wall over large distances. Topological protection of the valley Hall edge solitons is demonstrated by modeling their passage through sharp corners of the Ω-shaped domain wall.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Supplementary Material (1)

NameDescription
Visualization 1       The valley Hall edge soliton is cirvumventing the sharp corners of a O-shaped zigzag-type domain wall.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. Schematic configuration of honeycomb waveguide array with a domain wall. Two sublattices are distinguished by different colors. Domain wall between two honeycomb arrays with different detunings is highlighted by the red ellipse.

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Fig. 2.
Fig. 2. (a) Band structure of the photonic graphene with a domain wall. The black curves are the bulk states, while the red curve is the valley Hall edge state. (b) First-order $\beta '$ and second-order $\beta ''$ derivatives of the propagation constant of the valley Hall edge state. (c) Exemplary profiles of the valley Hall edge states with the Bloch momenta displayed in the right-bottom corner. These profiles correspond to the red and green dots in (a). The states are shown within the window $-20\le x \le 20$ and $-9.1\le y\le 9.1$.

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Fig. 3.
Fig. 3. (a) Peak amplitude (black curve; left vertical axis) and power (red curve; right vertical axis) of the nonlinear valley Hall edge state as a function of nonlinear propagation constant shift $\mu$. (b) Examples of profiles of the nonlinear valley Hall edge states for $\mu$ values displayed in the right-bottom corner and corresponding to the black and red dots in (a). The states are shown within the window $-20\le x \le 20$ and $-9.1\le y\le 9.1$. (c) Growth rate $\delta$ of the small perturbation added to the nonlinear edge state with $\mu =3.516$ versus frequency of the perturbation $\omega$.

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Fig. 4.
Fig. 4. (a) Amplitude profiles $|\psi |$ with $\mu =3.516$ of the perturbed propagating nonlinear valley Hall edge state at different propagation distances. (b) Peak amplitude $a$ of the state versus propagation distance. All states are shown within the window $-20\le x \le 20$ and $-121.2\le y\le 121.2$.

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Fig. 5.
Fig. 5. (a) Long-range stable propagation dynamics of the valley Hall edge quasi-soliton. (b) Profile for the same input after linear propagation at $z=500$. (c) Peak amplitude (right axis) for nonlinear $a_\textrm {nlin}$ (black curve) and linear $a_\textrm {lin}$ (blue curve) propagation regimes and integral soliton center position $y_c$ in nonlinear regime (left axis) versus propagation distance $z$. All amplitude distributions in (a) are shown within the window $-20\le x \le 20$ and $-121.2\le y\le 121.2$.

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Fig. 6.
Fig. 6. (a) Inversion-symmetry-broken honeycomb lattice with a $\Omega$-shaped domain wall (the blue channel). We indicate four corners of this structure with numbers. (b) Amplitude profiles at different propagation distances illustrating passage of soliton through all sharp bends of the domain wall. All states are shown within the window $-53\le x \le 53$ and $-121.2\le y\le 121.2$.

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Equations (2)

Equations on this page are rendered with MathJax. Learn more.

i∂ψ∂z=−12(∂2∂x2+∂2∂y2)ψ−R(x,y)ψ−|ψ|2ψ,
μu=12(∂2∂x2+∂2∂y2+2iky∂∂y−ky2)u+R(x,y)u+|u|2u,
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