Lithium niobate~(LN) is a birefringent material, where the strong birefringence thermo-optic effect is promising for the generation of quantum photon source with widely tunable wavelength. Here, we demonstrate birefringent phase-matching in a 20-mm-long waveguide fabricated on 5~m-thick x-cut lithium niobate on insulator. The waveguide is deviated from the optical axis of LN by an angle of 53.5, enabling the phase matching between telecom and visible wavelengths. The phase-matching wavelength of this device can be thermally tuned with rate of 0.617~nm/K. We demonstrate the type-1 spontaneous parametric down-conversion to generate photon pairs with brightness of 2.2~MHz/mW and coincidence-to-accidental ratio up to . Furthermore, the heralded single photon is obtained from the photon pair with efficiency of 13.8% and count rate up to 37.8~kHz.

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