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.

中文翻译：

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双相梯度超表面的Snell式和Fresnel式公式

通过图案化空间正交和极化正交的两个相位梯度的超颖表面，我们推导了拟议超颖表面的三维（3D）Snell式公式和Fresnel式公式。从理论上讲，双相梯度超表面类似于双轴样双折射，即，将任何均匀偏振的入射光束分解为两个偏振态随入射光束而变化的异常折射光束。根据类菲涅耳公式，两个异常折射光束之间的相对强度不仅取决于入射角和入射光束的偏振椭圆率（类似于双轴晶体），还取决于偏振椭圆率的取向，即使对于给定的入射极化 这是双轴晶体所没有的独特特性。所有的理论分析都得到了数值证明。类似于3D Snell和类似Fresnel的公式将使基于双相梯度超表面的功能设备的设计更加容易。