# Controlling the Focusing Ability of the Parabolic Reflector in Free Space at Microwave Frequencies
## 2 Formulation
Equation (3) contains the dominant intra-band contribution, and it is satisfied at microwave frequencies and up to mm-waves with the practical statement that the chemical potential \(\mu_{c}>0.05\) eV. The normalized surface impedance using graphene conductivity given in (3) is shown in Figure 2 in the microwave range. The graphene conductivity and related surface impedance is almost frequency independent at microwaves and so it exhibits completely different behavior with respect to the THz and optics. Therefore, one can say that even if it is not shown in the plot of Figure 2, it is almost the same for larger frequencies in microwave range. The other important property is seen in that the surface impedance is dominantly resistive, and the reactive part can be neglected as it is smaller than 2.5% compared to real part. In addition, it can be stated that controlling the chemical potential \(\mu_{c}\) with electrostatic biasing, one can control the scattering level and focusing ability of the reflector. Numerically, it can be said that if \(0.05<\mu_{c}<1\), then \(Z_{n}\) can change between 4 \(Z_{o}\) and 0.1 \(Z_{o}\). This is a practically realizable range where graphene surface varies from very low conductivity (almost invisible) to higher conductivity (approaching good conducting surface). More specifically, starting from Equations (1) and (2) using auxiliary scalar and vector potentials, the electric field integral equation (EFIE) can be derived in the following form (See [25] for more details):  
\[ \begin{array}{ll}Z_{n}J_{\text{tan}}&=-\frac{i}{\pi}\frac{\partial H_{\text{ tan}}^{\mu}}{\partial t}+\frac{i}{\pi}\frac{\partial}{\partial t}\int_{M}\left[ \frac{\partial}{\partial t}J_{\text{tan}}\left(\overset{\rightarrow}{r}^{ \prime}\right)\right]G(\overset{\rightarrow}{r},\overset{\rightarrow}{r}^{ \prime})dl^{\prime}\\ &+ik\int_{M}J_{\text{tan}}\left(\overset{\rightarrow}{r}^{\prime}\right) \cos\left[\xi(\overset{\rightarrow}{r})-\xi(\overset{\rightarrow}{r}^{ \prime})\right]G(\overset{\rightarrow}{r},\overset{\rightarrow}{r}^{ \prime})dl^{\prime}\end{array}  \tag{4}\]  
Figure 2: Variation of the normalized surface impedance of graphene with the chemical potential \(\mu_{c}\) where (**a**) real part (**b**) imaginary part. Other parameters are \(\tau=0.135\) ps, \(T=300\) K, f = 30 GHz. where \(G\) is the two-dimensional Green's function satisfying the radiation condition, i.e., \(G(\overset{\rightarrow}{r},\overset{\rightarrow}{r})=(i/4)H_{0}^{(1)}(k| \overset{\rightarrow}{r}-\overset{\rightarrow}{r}^{\prime}|\)), and the angle \(\xi(\varphi)\) is between the normal on \(M\) and the \(x\)-direction. The curve \(M\) can be represented mathematically by the parametric equations given as \(x=x(\varphi),\;y=y(\varphi)\), where \((x,y)\;\in\;M\). In addition, we define the differential length in the tangential direction along \(M\) as \(\partial l=a\beta(\varphi)\partial\varphi\), where \(\beta(\varphi)=r(\varphi)/[a\cos(\varphi/2)]\). To apply the MAR method to the given problem, we should add and subtract the similar functions from the integral kernels in (4). The subtracted ones with the original kernel produce the following similar functions having smooth behavior without singularity.