We propose here to employ this formalism to point out the transmission properties of anisotropic metasurfaces. Consequently, we will define a criterion to evaluate the deviation of the transmitted beam properties from those of an incident plane wave. This is achieved through the determination of three parameters that will help to quantify the transmitted beam properties and are, as well, directly related to experimental considerations. These parameters are:  
(1) The transmission coefficient \(T\) defined as the ratio of the transmitted total power divided by the incident one. \[T=\dfrac{\iint\limits_{-\infty}^{+\infty}\vec{E}_{t}(x,y,z)\wedge\vec{H}_{t} (x,y,z)dxdy}{\iint\limits_{-\infty}^{+\infty}\vec{E}_{i}(x,y,z)\wedge\vec{H}_{ i}(x,y,z)dxdy} \tag{6}\]  
Figure 2: (**a**) Schema of a unit cell of the anisotropic metasurface. The rectangular apertures are engraved in a perfectly electric conductor layer of thickness \(h\). The two rectangle dimensions are different in order to induce artificial anisotropy. (**b**) Typical transmission spectra for the two orthogonal polarization states along \(OX\) and \(OY\) when \(\theta_{m}=0^{o}\) in the case of an incident plane wave. This parameter is obviously very important and must be maximized (100%) in order to build efficient optical components. (2) The average phase change \(\xi\) induced between the two transverse components of the transmitted beam. This parameter depends on the awaited metasurface properties. In our case, we are dealing with a \(\lambda/2\) (\(\lambda/4\)) plate, so we expect a target value of \(\xi=\pi\) (\(\xi=\pi/2\)) between the two transverse components of the electric field directed along the plate axes (\(Ox\) and \(Oy\)). Thus, we define this parameter as:  
\[\xi=\frac{1}{S}\iint\limits_{(S)}Arg\left(\frac{E_{tX^{\prime}}(X^{\prime},Y^ {\prime},Z^{\prime}=z_{0})}{E_{tY^{\prime}}(X^{\prime},Y^{\prime},Z^{\prime}= z_{0})}\right)dX^{\prime}dY^{\prime} \tag{7}\]  
where the integration is numerically restricted over a surface (\(S\)) corresponding to the set of points (\(X^{\prime},Y^{\prime}\)) in the plane \(Z^{\prime}=z_{0}\) such that \(I_{l}(X^{\prime},Y^{\prime},z_{0})\geq I_{max}/100\). For a Gaussian beam, this corresponds to 99.08% of the total energy. The frame \(OX^{\prime}Y^{\prime}Z^{\prime}\) is related to the transmitted field and is the same as \(OXYZ\) when \(\varepsilon_{1}=\varepsilon_{2}\). Otherwise, a frame change (rotation induced by the light refraction) is required. As defined by Equation (7), the numerical value of the phase shift is determined between \(-\pi\) and \(\pi\). Thus, we have to be careful when making the sum of all values by assuming a continuous spatial phase change (unwrapping the results before making the summation). (3) The polarization degree of the transmitted beam \(P\) quantified as the spatial average of the polarization degree of each plane wave and given by [15]:  
\[P=\left\langle\sqrt{1-\frac{4|J|}{(J_{xx}+J_{yy})^{2}}}\right\rangle_{x,y,z_{0}} \tag{8}\]  
where \(J\) is the polarization matrix or the coherency matrix defined by [16]:  
\[J=\left( \begin{array}{cc}J_{xx}&J_{xy}\\ J_{yx}&J_{yy}\end{array} \right)=\left( \begin{array}{cc}<E_{x}E_{x}^{*}>_{t}&< E_{x}E_{y}^{*}>_{t}\\ <E_{y}E_{x}^{*}>_{t}&<E_{y}E_{y}^{*}>_{t}\end{array} \right) \tag{9}\]  
\(|J|\) is the matrix \(J\) determinant. The value of \(P\) is real positive \(\in[0;1]\). \(P=0\) corresponds to a completely unpolarized beam, while the latter is entirely polarized (linear, circular or elliptical) when \(P=1\). Moreover, we are looking for an anisotropic metasurface that modifies the polarization direction (\(\lambda/2\) plate) or the polarization nature (\(\lambda/4\) plate). In both cases, the transmitted beam must be completely polarized whatever the polarization of the incident beam. This means that the value of \(P\) must be close to one.