Snell-like and Fresnel-like formulas of the dual-phase-gradient metasurface

By patterning the metasurface of two phase gradients that are both space-orthogonal and polarization-orthogonal, we derived the three-dimensional (3D) Snell-like formula and the Fresnel-like formula of the proposed metasurface. Theoretically, the dual-phase-gradient metasurface resembles biaxial-like birefringence, i.e., decomposing any homogeneously polarized incident beam into two anomalously refracted beams whose polarizations vary as the incident beam. According to the Fresnel-like formula, the relative intensity between the two anomalously refracted beams not only depends on the incidence angle and the polarization ellipticity of the incident beam being similar to the biaxial crystals, but it also depends on the polarization ellipticity orientation even for a given incident polarization, which is an unique property absent in the biaxial crystals. All the theoretical analyses were numerically demonstrated. The 3D Snell-like and Fresnel-like formulas will make the design of functional devices based on the dual-phase-gradient metasurface much easier.