Tunable Optical Spatial Differential Operations via Photonic Spin Hall Effect on Hexagonal Boron Nitride

We propose a photonic spin Hall spatial differentiator based on a natural hexagonal boron nitride (hBN) interface in the mid-infrared Reststrahlen band. The optical-axis orientation is set by the crystal cut and lies in the incidence plane, while the operating mode is selected through the incidence angle, input polarization, and analyzer orientation. Near the ENZ-assisted Brewster-like condition, co-polarized reflection is strongly suppressed, enabling a high-contrast X-differentiation channel, whereas the Y-differentiation channel originates from first-order angular-spectrum spin–orbit coupling and remains background-free. At ν=824.8cm−1 and fixed orientation ϕ=20∘, the effective Brewster-like minimum occurs at θBeff≈69.28∘ with |rp|2≈2.8×10−4. Numerical simulations demonstrate switching among intensity-like imaging, X differentiation, and background-free Y differentiation. These results identify fixed-orientation hBN as a natural-crystal platform for polarization-selective mid-infrared optical spatial differentiation without subwavelength patterning.