# Controlling the Focusing Ability of the Parabolic Reflector in Free Space at Microwave Frequencies
## 2 Formulation  
The problem geometry of a 2D parabolic reflector made of graphene is shown in Figure 1. It is illuminated by an H-polarized electromagnetic plane wave. It is polarized along z direction and coming from x direction. To obtain a regularized solution depending on RHP method, the open surface of the reflector part M should be converted to a periodic closed contour. Here we define a closed contour C in piecewise manner and so it consists of two parts. These are parabolic reflector part M and its complimentary part that is circular arc S. Combination of M and S constitutes the overall arc C. Also, these two curves connect to each other at edge points of reflector such that it has no discontinuity in the surface curvature even at the connection points. This closed contour C satisfies the periodic nature of the problem and so obtaining the regularized matrix equation of Fredholm second kind with RHP technique is possible [12]. The rigorous formulation of the presented BVP involves the Helmholtz equation, the Sommerfeld radiation condition for the observation points far from the reflector, the resistive boundary condition on reflector surface, and the edge condition around the reflector edge points. These conditions guarantee the uniqueness of the solution [22]. Scattering from thin penetrable layer is frequently studied in the literature, and one of the common methods for the numerical solution is the volume-integral-equation-based MoM procedure. However, for the very thin layers, some numerical deviations can be observed mainly due to the numerical evaluation of the finite difference derivatives [23,24]. Very thin penetrable sheet is a satisfactory model of a resistive boundary condition (BC). The presented geometry is smooth parabolic, and incident plane wave illuminates the reflector almost normally and makes a small angle with surface normal. In this case, it is close to the normal incidence and so the resistive BC works very well. Also, one of the advantages can be stated that there is no need to make meshing inside the thin layer because the inner field can be approximated by the average of the front and back field on the surface. The related equations of the two-side resistive BC for the H polarization case can be written as follows:  
\[\big{(}E_{\text{tan}}^{+}+E_{\text{tan}}^{-}\big{)}\hat{t}\ =\ 2Z_{s}\hat{n}\times(H_{z}^{+}-H_{z}^{-})\hat{z} \qquad\qquad\overset{\rightarrow}{r}\in M \tag{1}\]  
\[E_{\text{tan}}^{+}\ =\ E_{\text{tan}}^{-} \qquad\qquad\qquad\qquad\qquad\overset{\rightarrow}{r}\in M \tag{2}\]  
where the subscript \"tan\" indicates the tangential field component to reflector surface, the superscripts \"\(-\)\" and \"\(+\)\" show the locations of the points on the front and back faces of reflector, respectively. Additionally, \(\hat{n}\) is the unit vector normal to the front side of reflector, and \(\hat{t}\) is the tangential unit vector to the reflector surface. The electric surface-current density is defined as \(\overset{\rightarrow}{J}_{\text{tan}}=(H_{z}^{+}-H_{z}^{-})\hat{t}\), and \(Z_{s}\) is graphene's surface impedance [1; 2; 3; 4]. For graphene, the surface impedance is \(Z_{s}=Z_{n}Z_{0}\), where \(Z_{0}\) is the free space intrinsic impedance and \(Z_{n}\) is defined as the normalized surface impedance that is \(Z_{n}=1/\sigma Z_{0}\)  
Figure 1: Problem geometry. Here, \(\sigma\) is the surface conductivity of graphene, which can be modelled by using the Kubo formula as a sum of the intra-band and inter-band contributions [6]. Also, Kubo formula can be reduced to a simple Drude model if the condition \(u_{F}\gg h\omega\) is satisfied, where \(u_{F}\) is the Fermi level of graphene and \(\omega\) is the angular frequency. \[\sigma=\frac{2e^{2}\tau}{\pi h^{2}}k_{B}T\ln\left(2\cosh\frac{\mu_{c}}{2k_{B}T} \right)\frac{1}{1-j\omega\tau} \tag{3}\]  
where \(\tau\) is the relaxation time parameter in electron scattering process, \(T\) temperature in Kelvin, \(k_{B}\) is the Boltzmann constant, \(h\) is the Planck constant, and e is the unit electron charge.