Nonlinearity Selective Quasi Bound States in the Continuum via Symmetry Protected Decoupling in χ(2) Thin Films

Abstract

Second-harmonic generation in resonant structures is commonly evaluated in terms of intracavity field enhancement at the fundamental and harmonic frequencies. Here, we formulate nonlinear frequency conversion within a symmetry-resolved overlap framework that explicitly separates resonant field buildup from nonlinear mode projection. Using a simple and analytically tractable Fabry--Perot thin-film-on-substrate geometry, we show that, even in the presence of spectrally bright resonances at both $ω$ and $2ω$, the emitted second-harmonic signal can be strongly suppressed when the spatial parity of the pump-induced nonlinear polarization is incompatible with that of the radiating $2ω$ standing-wave mode. This mechanism gives rise to nonlinearity-selective quasi-bound states in the continuum. Beyond providing a compact interpretation of these nonlinear dark states, the framework unifies pump enhancement, harmonic enhancement, and symmetry-controlled modal overlap within a single predictive metric. More broadly, it identifies thickness regimes in which resonant buildup is accompanied by constructive nonlinear coupling, and distinguishes them from regimes in which apparently favorable resonance conditions remain conversion-inactive because the nonlinear source is orthogonal to the radiating harmonic mode.

Figures (5)

• Figure 1: Thickness--dispersion map of the normalized nonlinear overlap coefficient $|\beta(d)|^{2}$ [Eq. \ref{['eq:beta']}] in the $(d,n_{2\omega}/n_{\omega})$ plane. Bright ridges identify thickness--dispersion combinations for which the pump-induced nonlinear polarization $E_{\omega}^{2}(z)$ is symmetry-compatible with the radiating $2\omega$ standing wave.
• Figure 2: Design maps for doubly resonant $\chi^{(2)}$ thin films: (a) An RGB illustrative map in the $(d,\,n_{2\omega}/n_{\omega})$ plane. The red and green channels encode the depth-averaged intracavity build-up $\langle|E_{\omega}|^2\rangle_d$ and $\langle|E_{2\omega}|^2\rangle_d$, respectively, while the blue channel encodes the normalized nonlinear overlap $|\beta(d,n_{2\omega}/n_{\omega})|^2$. (b) Normalized logarithmic map of the optical overlap--buildup merit function $\mathcal{W}$ defined from Eq. 17 of the manuscript by isolating its field-dependent contribution. Contours indicate representative levels of $|\beta|^2$, highlighting overlap-limited valleys where large linear build-up does not translate into strong conversion.
• Figure 3: Apparent enhancement factor $\mathcal{E}_{\mathrm{app}}(d)$ that includes only the cavity build-up factors and explicitly neglects the $|\beta(d)|^{2}$ term by setting $|\beta(d)| \equiv 1$ in Eq. \ref{['eq:EnhancementFactor']}. This is representative of a conventional $Q$-driven cavity-enhanced SHG design.
• Figure 4: Enhancement factor $\mathcal{E}(d)$ and its constituents as a function of film thickness $d$. The solid black shows the total enhancement predicted by Eq. \ref{['eq:EnhancementFactor']}, while the orange and cyan curves represent the pump and second-harmonic build-up factors, respectively. The green curve tracks the purely geometric contribution $|\beta(d)|^{2}$. The vertical lines depict two representative thicknesses for local minima/maxima of $\mathcal{E}(d)$.
• Figure 5: Real-space structure of the nonlinear coupling channel, visualized via the normalized local overlap density $\rho(z,d)=\mathrm{Re}\{E_\omega^2(z;d)\, E_{2\omega}^\ast(z;d)\}$ [Eq. \ref{['eq:rho']}], plotted versus normalized depth $z/d$ and film thickness $d$. The vertical lines depict two representative cases; one with predominantly same-sign local overlap density and another with sign-alternating local overlap density.