Approaching the Tsirelson bound with a Sagnac source of polarization-entangled photons

High-fidelity polarization-entangled photons are a powerful resource for quantum communication, distributing entanglement and teleportation. The Bell-CHSH inequality $S\leq2$ is violated by bipartite only maximally entangled states can achieve $S=2\sqrt{2}$, the Tsirelson bound. Spontaneous parametric down-conversion sources produce with correlations close to Sagnac configurations offer intrinsic stability, compact footprint high collection efficiency, however, there often trade off between source brightness visibility. Here, we present $2\sqrt{2}-S=(5.65\pm0.57)\times10^{-3}$, on-par highest values recorded, while generating detecting $(4660\pm70)$ pairs/s/mW, which substantially higher than previously reported around two orders of magnitude brighter traditional cone $S$ parameter. Our records $0.9953\pm0.0003$ concurrence $0.99743\pm0.00014$ fidelity an ideal Bell state. By studying systematic errors in sources, identify that precision focal point inside crystal plays largest role reducing parameter our experiment. We provide pathway could enable recorded to-date maintaining very brightness.