Entanglement certification in bulk nonlinear crystals for degenerate and non-degenerate SPDC: spectral filter effects on transverse spatial correlations

Spatial correlations of photon pairs from spontaneous parametric down-conversion (SPDC) underpin quantum imaging and entanglement certification. We present the first systematic study of spectral filter bandwidth effects on transverse spatial correlations in bulk Type-I BBO for degenerate and non-degenerate configurations. In the far field, the degenerate conditional momentum width is pump-limited and filter-invariant, while non-degenerate configurations exhibit monotonic growth in both marginal and conditional momentum widths – with the walk-off axis $\approx 100$ times more sensitive than the non-walk-off axis. In the near field, we identify a previously unreported flat-dip-rise profile: the conditional position width narrows by $\approx 10\mathrm{\%}$ at an optimal bandwidth ${\mathrm{\Delta }}_{\mathrm{d}\mathrm{i}\mathrm{p}}\approx 1.35\mathrm{\Delta }{\lambda }_{\mathrm{S}\mathrm{P}\mathrm{D}\mathrm{C}}$ before rising due to geometric displacement. When the filter is placed on the idler arm, the dip shifts by the exact factor $({\lambda }_i\mathrm{/}{\lambda }_s{)}^2$. Both results are universal for any non-degenerate SPDC source, requiring only a finite crystal length, $d\theta \mathrm{/}d\lambda \mathrm{\ne }0$, and incoherent spectral averaging. The Reid EPR uncertainty product is consistently smaller on the walk-off axis – a structural advantage of bulk birefringent geometry absent in quasi-phase-matched sources. The optimal filter bandwidth ${\mathrm{\Delta }}_F={\mathrm{\Delta }}_{\mathrm{d}\mathrm{i}\mathrm{p}}$ is determined entirely by the intrinsic phase-matching bandwidth of the crystal and is directly readable from the X-entanglement spectral width of the source.