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