Vectorial diffraction field calculation between arbitrary parallel planes in free space

Precise calculation of vectorial optical fields is the cornerstone of modern photonics, yet a unified theoretical framework governing propagation between arbitrary planes remains elusive. Existing models enforce a trade-off between physical rigor and geometric flexibility: the vectorial Debye integral (VDB) is the gold standard for polarization coupling but is structurally restricted to the focal configuration; conversely, the vectorial Rayleigh–Sommerfeld integral (VRS) allows flexible propagation between parallel planes but fails to capture the intrinsic rotational coupling of the electric field vector in high-NA regimes. Here, we resolve this fundamental dichotomy by establishing the spherical vectorial diffraction (SVD) framework. Note that the abbreviation SVD in this work refers exclusively to spherical vectorial diffraction and should not be confused with the singular value decomposition commonly used in linear algebra. Deriving from the rigorous Franz formula, we reveal a unified spectral transfer matrix [H] that governs full-vectorial evolution across arbitrary parallel planes. We mathematically demonstrate that VDB’s geometric projection matrix is the exact mathematical solution of [H] at the focal limit, whereas [H] remains valid at any propagation distance with full vectorial fidelity. Crucially, SVD rigorously predicts cross-polarization generation and vectorial coupling effects—phenomena systematically underestimated by the VRS—without compromising the algorithmic scalability of FFT-based angular spectrum methods. Validated by pinhole diffraction experiments, this framework reframes polarization coupling as an intrinsic diffractive operator, providing the necessary theoretical foundation for the systematic exploration of photonic phenomena in regimes where classical approximations break down.