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N. Boeuf and collaborators developed calculations of non-collinear phase matching in uniaxial and biaxial nonlinear crystals. The NIST-hosted 1999 draft extended calculations beyond cases where interacting waves all propagate along one line.^[ref]

For spontaneous parametric down-conversion, the pump, signal and idler satisfy ${\omega }_p={\omega }_s+{\omega }_i$ and ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. The second condition is a vector relation, so crystal anisotropy and propagation angles jointly constrain possible outputs. Solving only the frequency equation does not establish an efficient optical interaction.

In spontaneous parametric down-conversion, a strong pump drives a second-order nonlinear optical susceptibility, producing paired output photons in initially unseeded modes. The two outputs are conventionally called the signal photon and idler photon. They need not have equal frequencies. For a monochromatic pump, conservation of energy gives

Efficient bulk generation also requires phase matching: the nonlinear source and emitted waves must remain in phase. In the ideal plane-wave geometry this is ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. Different refractive indices and propagation directions determine which pairs satisfy it. A finite interaction length supplies a mismatch bandwidth rather than an infinitely sharp wavevector constraint. ^[ref]

At weak gain, the SPDC pair state contains a large vacuum component and a small pair amplitude. Detecting one output conditions the other, producing a heralded single-photon source. This is probabilistic: stronger pumping also increases unwanted multiple-pair events. Experiments collect two output modes with optical filters and single-photon detectors, comparing coincident detections with background counts. ^[ref]