Formal tilt invariance of the nonlocal curvature approximation and its connection to the integral equation method

Tilt invariance is a stringent constraint that second-order scattering models such as the integral equation method (IEM) should satisfy in order to expand their domain of applicability. Moreover, second-order scattering models must reproduce elementary limits such as the small perturbation method (SPM) and the high-frequency Kirchhoff approximation. Tilt invariance is met if and only if a scattering model yields the same asymptotic limit whether the scattering surface is tilted before or after the limiting process. In particular, the tilted SPM coefficients are well determined by simply tilting the reference frame. If it is tilt invariant, a second-order scattering model will reproduce these tilted coefficients by simply tilting the surface explicitly present in the expression of the scattering model before reducing it to the SPM limit. In this letter, we demonstrate that our nonlocal curvature approximation (NLCA) is formally tilt invariant up to first order in the tilting vector. Satisfying the tilt invariance property can extend the applicability of scattering models to account, for example, for scattering from multiscale surfaces and polarization mixing due to out-of-plane tilting. It is also suggested that replacing the field coefficients of IEM by the curvature kernel of NLCA introduces a promising alternative technique that includes multiple scattering up to double reflections from the rough surface, while remaining analytically compact and formally tilt invariant.