Quantifying Polarization Aberration in Anisotropic GRIN Lenses via Stress-Birefringence Coupling and GPU-Accelerated Ray Tracing

Stress-induced birefringence in gradient-index (GRIN) lenses couples with the refractive index gradient to produce severe polarization aberrations, causing spatial separation between s- and p-polarized foci. Using a GPU-accelerated ray tracing framework (NVIDIA Warp), we systematically investigate this coupled effect and establish the effective-index model \(n_{\text{eff}}(r, p) = n_{\text{avg}}(r) + \kappa \Delta n_{\text{avg}} (r/R)^2 ((p \cdot a)^2 - 0.5)\). Our central discovery is an exact single-parameter scaling law: the entire spatial distribution of polarization aberration (PA) collapses onto a single scalar, \(\text{std} = 0.233 \cdot \text{split}\) with \(R^2 = 0.99997\), independent of GRIN strength and stress coupling over a \(40\times\) dynamic range. This hidden invariance is non-trivial because one quantity is an extreme-value statistic (maximum over rays) while the other is a distributional statistic (standard deviation); their strict proportionality implies a self-similar normalized aberration profile that survives even the strong-focusing regime (\(\Delta n_{\text{avg}} = 0.20\)). The focus split itself scales linearly with both parameters (\(\text{split} \propto \kappa^{0.99} \Delta n_{\text{avg}}^{0.96}\)), and the pupil-radius dependence is quantitatively captured by a Seidel spherical-aberration coefficient \(\alpha = 0.364\) (\(R^2 = 0.993\)). Our results provide a practical design rule: measuring one easily accessible scalar (focus split) instantly predicts the full worst-case polarization aberration without any polarization ray tracing.