We derive the theoretical limit of single-photon purity of heralded single-photon sources and accordingly demonstrate a bright, gigahertz-pulsed heralded source with the purity saturating the limit. Based on spontaneous four-wave mixing in a silicon spiral waveguide, this on-chip source is measured to have a coincidence rate exceeding 1.5MHz at a coincidence to accidental (CAR) ratio of 16.77. The single-photon purity, quantified by the auto-correlation function gh(2)(0), reaches the theoretical limit with the lowest value of 0.00094 ± 0.00002 obtained at a coincidence rate of 0.8kHz. We attribute our results to effective spectral filtering as well as the coherent pump condition helped by optical injection locking.
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