Second-harmonic generation via double topological valley-Hall kink modes in all-dielectric photonic crystals

Letter

Second-harmonic generation via double topological valley-Hall kink modes in all-dielectric photonic crystals

Zhihao Lan, Jian Wei You, Qun Ren, Wei E. I. Sha, and Nicolae C. Panoiu

Phys. Rev. A 103, L041502 – Published 21 April 2021

Abstract

Nonlinear topological photonics, which explores topics common to the fields of topological phases and nonlinear optics, is expected to open up a new paradigm in topological photonics. Here, we demonstrate second-harmonic generation (SHG) via nonlinear interaction of double topological valley-Hall kink modes in all-dielectric photonic crystals (PhCs). We first show that two topological frequency band gaps can be created around a pair of frequencies, $ω_{0}$ and $2 ω_{0}$ , by gapping out the corresponding Dirac points in two-dimensional honeycomb PhCs. Valley-Hall kink modes along a kink-type domain wall interface between two PhCs placed together in a mirror-symmetric manner are generated within the two frequency band gaps. Importantly, through full-wave simulations and mode dispersion analysis, we demonstrate that tunable, bidirectional phase-matched SHG via nonlinear interaction of the valley-Hall kink modes inside the two band gaps can be achieved. In particular, by using Stokes parameters associated with the magnetic part of the valley-Hall kink modes, we introduce the concept of SHG directional dichroism, which is employed to characterize optical probes for sensing chiral molecules. Our work opens up avenues toward topologically protected nonlinear frequency mixing and active photonic devices implemented in all-dielectric material platforms.

Received 8 July 2020
Accepted 29 March 2021

DOI:https://doi.org/10.1103/PhysRevA.103.L041502

Physics Subject Headings (PhySH)

Nonlinear optics Photonic crystals Second order nonlinear optical processes Topological effects in photonic systems

Atomic, Molecular & Optical

Authors & Affiliations

Zhihao Lan ¹, Jian Wei You ¹, Qun Ren ^1,2, Wei E. I. Sha^1,3, and Nicolae C. Panoiu¹

¹Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom
²School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
³State Key Laboratory of Modern Optical Instrumentation, College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China

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Vol. 103, Iss. 4 — April 2021

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Images

Figure 1
Emergence of two valley gaps for SHG. (a) Schematic of the system. (b) Unit cell of the PhC, containing two cylinders of nonlinear material with radii $r_{1}$ and $r_{2}$ , dielectric constant $ε = 12$ , and $χ^{(2)} = 10^{- 21}$ ${CV}^{- 2}$ . (c) First Brillouin zone of the PhC. (d) The existence of double Dirac points (marked by red and blue dots) of the PhC at $r_{1} = r_{2} = 0.2 a$ . (e) The same as in (d), but calculated for $r_{1} = 0.18 a$ and $r_{2} = 0.22 a$ . (f) The frequencies of the two Dirac points in (d) vs $r_{0}$ when $r_{1} = r_{2} = r_{0}$ . (g) The width of the two valley gaps in (e) vs $δ$ , defined as $r_{1} = r_{0} - δ$ and $r_{2} = r_{0} + δ$ with $r_{0} = 0.2 a$ .
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Figure 2
Topological properties of the two valley gaps. (a) Phase winding behaviors of $E_{z}$ at $K$ and $K^{'}$ for the four bands around the Dirac points in Fig. 1. (b) Berry curvature distributions of bands 2 and 5 (Berry curvatures of bands 1 and 4 around $K, K^{'}$ show behaviors opposite to bands 2 and 5 and thus are not shown). The peak of Berry curvature at the $Γ$ point of band 5 is due to the band degeneracy between bands 5 and 6 at $Γ$ . In the simulations, a small cylinder difference is used to gap out the Dirac points, and we have checked the integral of the Berry curvature around $K$ and $K^{'}$ gives $\pm π$ .
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Figure 3
Emergence of valley-Hall kink modes in the two valley gaps for SHG. (a) Band structure of a kink-type domain-wall interface separating two PhCs (with $r_{1} = 0.2 a, r_{2} = 0.23 a$ ) that are mirror symmetric to each other (see the insets). Note the red (light gray) and blue (dark gray) kink modes correspond to the interface with bigger cylinders (red, left inset) and the interface with smaller cylinders (blue, right inset), respectively. (b) Tuning the kink mode dispersion to achieve phase matching for SHG, where the fundamental modes $ω_{0}$ are shown in the second-harmonic gap by applying the transformation $(ω_{0}, k_{f}) \mapsto (2 ω_{0}, 2 k_{f})$ and $Ω_{2}$ refers to the second-harmonic modes. Along the arrow direction, $r_{1}$ varies from $0.23 a$ to $0.18 a$ with a step of $0.01 a$ , whereas $r_{2} = 0.24 a$ .
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Figure 4
SHG via double valley-Hall kink states. (a) Dispersion curves (left) used for the SHG [from the bottom panel of Fig. 3 at $r_{1} = 0.19 a, r_{2} = 0.24 a$ ], where the shaded dark gray area is occupied by the bulk modes, whereas the light blue (light gray) indicates the frequency-matching window and chirality maps (right) of the valley modes at $k_{f}$ and $k_{s}^{+}$ . (b) Simulated field intensities of the fundamental ( $E_{1}$ ) and second-harmonic ( $E_{2}$ ) waves at the frequency marked by the blue (dark gray) line in (a). The fundamental wave is excited unidirectionally via a chiral source located at the extremum of the $k_{f}$ chirality map, which is realized by six dipoles with phase winding in the simulations. (c) and (d) Fourier transform of fields $E_{1}$ and $E_{2}$ in (b) (both are normalized by the maximum of $| {\tilde{E}}_{1}^{k} |$ ). (e) The forward-to-backward ratio $η_{⇄}$ of the generated second-harmonic waves vs frequency determined from the amplitudes of the peaks at ${\bar{k}}_{s}^{-}$ and ${\bar{k}}_{s}^{+}$ in (d). (f) Extracted dispersion curves from the peaks of $| {\tilde{E}}_{1}^{k} |$ and $| {\tilde{E}}_{2}^{k} |$ [green (medium gray) dots for the fundamental wave and blue (dark gray) dots for the harmonic wave) in (c) and (d) compared to the dispersion curves of (a). (g) Extracted $| Δ \bar{k} |$ from the peaks of $| {\tilde{E}}_{2}^{k} |$ in (d) compared to that obtained from the dispersion curves of (a).
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Figure 5
Direction-tunable SHG and nonlinear directional dichroism. (a) The propagation direction of the SHG could be switched from rightward to leftward by changing the source location from $A$ to $C$ as labeled in the chirality map. (b) The SHG-DD, defined as $(η_{\to} - η_{\leftarrow}) / (η_{\to} + η_{\leftarrow})$ , determined for $k_{f} > 0$ ( $k_{f} < 0$ ), corresponding to case A (C) in (a).
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