Vectorial diffraction field calculation between arbitrary parallel plane in free space

Vectorial diffraction calculation is fundamental to modern photonics, yet existing models suffer from mutually exclusive limitations: the Vectorial Debye integral (VDB), while accurate for high-NA focusing, is geometrically restricted to focal configurations and fails entirely at arbitrary propagation distances; the Vectorial Rayleigh-Sommerfeld integral (VRS), though applicable to arbitrary plane, systematically underestimates cross-polarized field components due to its scalar-like treatment of transverse propagation. Here we present a Spherical Vectorial Diffraction (SVD) framework that resolves the longstanding challenge of vectorial field calculation between arbitrary parallel planes in free space. Through systematic derivation from the Franz formula, we obtain a unified transfer matrix [H] for spherical wave propagation and demonstrate that VDB's geometric projection matrix M is the special case of [H] in the focal limit, while [H] extends seamlessly to arbitrary parellel planes with full vectorial fidelity. Crucially, SVD achieves VDB-level accuracy in high-NA focusing (mean deviation approx 1*10^-3) without additional computational cost, maintaining the efficiency of FFT-based angular spectrum methods. At arbitrary plane, SVD eliminates the order-of magnitude error amplification in longitudinal fields that plagues VRS's divergence-based reconstruction. Pinhole diffraction experiments confirm that only SVD correctly predicts cross-polarized field distributions—the signature of vectorial coupling inaccessible to either classical model. This unification reframes polarization coupling as an intrinsic diffractive phenomenon rather than a high-NA correction, establishing a general foundation for vectorial wavefront engineering.