According to these parameters (\(T,\xi\) and \(P\)) and for anisotropic plates, we define a Figure-Of-Merit (_FOM_) function that corresponds to the total deviation between the response of a plane wave from that of the Gaussian beam by:  
\[FOM=\Delta T+\Delta P+\Delta(\frac{\xi}{\Psi}) \tag{10}\]  
where \(\Psi\) is the phase change expected to be introduced by the anisotropic metasurface and the operator \(\Delta\) denotes the deviation with respect to the plane wave case. Consequently, we assume that a maximum value of \(FOM_{max}=10^{-2}\) is allowed to recover the plane wave behavior. This value can be seen as an onerous condition. Nevertheless, we will see in the following that this criterion can be easily fulfilled in the case of lossless materials (perfect conductors for instance). As mentioned earlier, our application deals with the illumination by a finite-size beam of the two anisotropic metasurfaces studied in Figure 4a,b of [14] corresponding to a half-wave and a quarter-wave plate, respectively. The general schema of such an anisotropic metasurface is depicted in Figure 2 and consists of a bi-periodic grating (period \(p\) along both \(x\)- and \(y\)-directions) of two rectangular apertures engraved within an \(h\)-thick layer of a perfect electric conductor. All the geometrical parameters are explicitly given in the caption and in the insets of Figure 4a,b of [14]. Nonetheless, they will be reminded below for each case. The incident beam is set to be a plain Gaussian beam, which is commonly used in experiments. For the three fundamental polarization states (circular, elliptical or linear), its angular spectrum expansion can be expressed in the \(Oxyz\) reference frame related to the metasurface (see Figure 1) by:  
\[F_{x}(k_{x},k_{y}) = A_{inc}(k_{x}\cos\theta_{m}+k_{z}\sin\theta_{m},k_{y})\left\{ \alpha-\frac{k_{y}\sin\theta_{m}}{k_{z}}\beta\right\}\] \[F_{y}(k_{x},k_{y}) = A_{inc}(k_{x}\cos\theta_{m}+k_{z}\sin\theta_{m},k_{y})\left\{ \beta\left(\cos\theta_{m}+\frac{k_{x}\sin\theta_{m}}{k_{z}}\right)\right\} \tag{11}\] \[F_{z}(k_{x},k_{y}) = -A_{inc}(k_{x}\cos\theta_{m}+k_{z}\sin\theta_{m},k_{y})\left\{ \frac{k_{x}\alpha+\beta k_{y}\cos\theta_{m}}{k_{z}}\right\}\]  
where \(A_{inc}(k_{X},k_{Y})=\pi W_{0}^{2}E_{0}\exp\left[-W_{0}^{2}(k_{X}^{2}+k_{Y}^{2 })/4\right]\) is the Fourier amplitude of the plane wave characterized by its \((k_{X},k_{Y})\) transverse wave vector components expressed in the proper frame of the Gaussian beam (\(OXYZ\)). \(k_{x}\), \(k_{y}\) and \(k_{z}\) are the same components in the frame of the anisotropic metasurface; \(E_{0}\) is the maximum amplitude of the electric incident field; and \(W_{0}\) is the beam-waist defined as the beam width at \(1/e\) of its amplitude \(E_{0}\). \(\alpha\) and \(\beta\) are given by:  
\[\alpha = cos\phi \tag{12}\] \[\beta = a\sin\phi \tag{13}\]  
where \(\phi\) and \(a\) are two parameters that define the polarization state of the incident beam as:  
* linear directed along the angle \(\phi\) measured from the \(X\)-axis with \(a=1\) (TEfor \(\phi=\pi/2\) and TMfor \(\phi=0\))
* circular with \(\phi=\pi/4\) and \(a=\pm\sqrt{-1}\)
* elliptical with the major to minor axis ratio equal to \(|\beta/\alpha|=tan\phi\) and \(a=\pm\sqrt{-1}\) (ellipse axes along the \(x\)- and \(y\)-directions). This formulation is general and can be applied to any illumination direction provided that the plan of incidence is parallel to \(xOz\). Otherwise, a simple change of variables (rotation around the \(Oz\) axis) is needed (see Figure 1). The first case consists of a half-wave plate where \(a_{x}=0.75p,b_{x}=0.1p,c_{x}=0.15p\), \(a_{y}=0.5818p\), \(b_{y}=0.2p,c_{y}=0.6p\) and \(h=0.8p\) (see Figure 2). For an illumination by a plane wave at normal incidence, the transmission coefficient, the phase change and the polarization degree are \(T_{\mathrm{pw}}=0.9256\), \(\xi_{\mathrm{pw}}=1.01\pi\) and \(P_{\mathrm{pw}}=1\), respectively, at the operation wavelength \(\lambda_{0}=1.194p\).