Entanglement Certification in Bulk Nonlinear Crystal for Degenerate and Non-degenerate SPDC for Quantum Imaging Application

Abstract: Quantum imaging with entangled photon pairs promises performance beyond classical limits, yet phase-matching, nonlinear crystal properties, and pump size jointly constrain its ultimate spatial resolution. We develop a unified model that relates these factors to the transverse correlations observed in both near and far-field planes, treating both degenerate and non-degenerate Type-I SPDC processes equally. By explicitly incorporating crystal length, pump beam waist, and spectral filtering into the biphoton amplitude, we demonstrate that narrowband signal filtering influences frequency-angle mixing. This approach minimizes conditional position uncertainty, particularly in non-degenerate SPDC scenarios, which enhances spatial resolution while maintaining the necessary multimode structure for imaging. We further analyze birefringent walk-off in bulk crystals and demonstrate that its apparent degradation of entanglement, such as weakened transverse anti-correlations and inflated Reid products, can be corrected. This correction follows frequency non-degeneracy and walk-off-aware reconstruction, recovering the correct correlation ridge and improving entanglement strength. The framework provides quantitative design rules that link filter bandwidth, crystal length, and pump waist to achievable resolution. Our results offer practical guidance for optimizing quantum microscopy and ghost imaging setups, where achieving high spatial resolution and robust entanglement certification simultaneously is crucial.