Polarization-Multiplexed Spatial Differentiation and Filtering Driven by van der Waals Birefringence

We report that the biaxial birefringence of $\alpha$-MoO${}_3$ can activate two spectrally distinct quasi-bound states in the continuum (quasi-BICs) within a single symmetric TiO${}_2$ nanobar-pair metasurface, with each resonance governed by a different crystallographic axis pair of the van der Waals crystal. With a full 60,nm $\alpha$-MoO${}_3$ gap fill, a TE resonance at 883.9,nm ($Q=92$, Fano $q=0.090$) and a TM resonance at 923.2,nm ($Q=31$, Fano $q=0.393$) are obtained. The Q ratio follows the inverse-square permittivity contrast, $Q\propto (\mathrm{\Delta }\epsilon {)}^{-2}$, calibrated across both polarization channels. Oblique-incidence sweeps show that the TE channel acts as a dual-null spatial highpass filter with a broadband stopband ($\mathrm{\mid }H\mathrm{\mid }<0.13$ for $\mathrm{\mid }{k}_x\mathrm{\mid }\le 0.63\mu$m${}^{-1}$, $T_{\mathrm{b}\mathrm{g}}=0.962$), while the TM channel transfers as $\mathrm{\mid }H\mathrm{\mid }\propto \mathrm{\mid }{k}_x\mathrm{\mid }$ ($R^2=0.94$), consistent with first-order spatial differentiation. Both operations are verified on a USAF~1951 resolution chart processed in a simulated 4$f$ framework. Channel selection is purely by input polarization angle with no structural modification.