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

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 $\omega$ and $2\omega$, 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 $2\omega$ 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.