Resonant enhancement of second harmonic generation in 2D nonlinear crystal integrated with meta-waveguide: analytical vs numerical approaches

TL;DR

This work addresses resonant second harmonic generation (SHG) in a hybrid structure where a two-dimensional nonlinear crystal is integrated with a dielectric meta-waveguide, focusing on enhancement at leaky modes and quasi-BICs and the limits set by nonradiative losses. It develops a microscopic analytical theory based on a Fourier-harmonic expansion of the metasurface polarizability $\alpha(x)=\sum_n \alpha_n e^{i n g x}$ in the meta-waveguide regime with a low-contrast grating, connecting the near-field distribution at the fundamental frequency to SHG through both local ($\chi$) and nonlocal ($\chi'$) nonlinearities. The theory describes SHG via two coupled near-field harmonics, captures the resonant contributions to forward and diffracted SHG beams, and provides expressions for resonant frequencies, linewidths, and their dependence on incidence angle and grating symmetry; numerical results validate the analytic predictions and reveal enhancements up to $\sim 10^7$ at MW resonances. The findings offer a framework for optimizing resonant nonlinear frequency conversion in 2D-material metasurface platforms and delineate fundamental limits imposed by inhomogeneous broadening and absorption, with maximal quasi-BIC enhancement scaling as $\big(\frac{\mathrm{Re}\alpha_0}{\mathrm{Im}\alpha_0}\big)^4$ at $\theta \sim \sqrt{\frac{\mathrm{Im}\alpha_0}{\mathrm{Re}\alpha_0}}$.

Abstract

We present an analytical theory of second harmonic generation (SHG) in hybrid structures combining a nonlinear 2D crystal with a dielectric metasurface waveguide. The theory describes the excitation spectrum and enhancement of SHG at both leaky mode and quasi-bound state in the continuum (quasi-BIC) resonances in terms of the material parameters. For low-loss systems, the SHG efficiency at leaky resonances is determined by their radiative broadening, governed by the relevant Fourier harmonics of the metasurface polarizability, whereas the SHG enhancement at quasi-BIC resonances is ultimately limited by inhomogeneous broadening and absorption in the system. We also describe the emergence and polarization properties of second harmonic diffracted beams. These beams appear even if both the 2D crystal and the meta-waveguide are centrosymmetric owing to the nonlocal mechanism of SHG. The developed framework provides a systematic theoretical basis for optimizing the resonant nonlinear frequency conversion in hybrid 2D-material-metasurface platforms and identifies the fundamental limitations of the SHG efficiency.

Figures (7)

• Figure 1: Second harmonic generation in 2D nonlinear crystal (NLC) integrated with dielectric metasurface waveguide (MW). Resonant excitation of bright (leaky) and dark (quasi-BIC) guided modes by the incident field at the fundamental frequency $\omega$ leads to the enhancement of the $2\omega$ emission. The $2\omega$ emission in the forward direction may include not only the collinear beam "$0_{2\omega}$" but also the diffracted beams "$\pm 1_{2\omega}$".
• Figure 2: Sketch of the major processes and the corresponding coupling coefficients $\alpha_n$ determining the widths and spectral positions of resonances in low-contrast meta-waveguides.
• Figure 3: Resonant enhancement of SHG by the meta-waveguide. (a) SHG enhancement as a function of the light frequency and the angle of incidence. (b) Spectral dependence of the SHG enhancement for three different angles of incidence. The figures are calculated for the symmetric MW with the parameters: $\alpha_0 g=0.2$, $\alpha_1 g=0.01$, $\alpha_2 g=0.005$, and the nonlinear susceptibility $\chi_{yyy}$.
• Figure 4: Spectral dependence of the SHG enhancement calculated numerically (solid curves) and plotted following analytical equations (dashed curves) for two different angles of incidence.
• Figure 5: (a) Intensity of the second harmonic diffracted beam "$-1_{2\omega}$" as a function of the frequency $\omega$ and the angle of incidence $\theta$. (b) Spectral dependence of the intensity for three different angles of incidence. The figures are calculated for the same parameters as Fig. \ref{['fig:SHG-Intens']}.