# Controlling the Focusing Ability of the Parabolic Reflector in Free Space at Microwave Frequencies
## 3 Numerical Results  
In this section, we give the results obtained from the simulation of the problem. These results are produced based on the formulation explained in the previous section, and the figures here show the verification of the method. These results are computed using an available laptop PC with the Intel i7 processor of the 7 th generation and 32 GB RAM working on the Windows 10 platform. From the observation of the plots given in Figure 2, the normalized surface impedance of the graphene reduces with the increasing chemical potential \(\mu_{c}\). Eventually, it approaches the small values of the limit case of the PEC surface. In Figure 3, the reflectivity (R), transmissivity (T), and absorptivity (A) are demonstrated versus the chemical potential \(\mu_{c}\) for the plane wave normal incidence to infinite graphene sheet. As expected, the reflected power level increases from small values to the unity with increasing \(\mu_{c}\). Therefore, the graphene as a thin resistive layer turns from invisible one to an almost PEC surface. The transmitted and absorbed power levels are also seen to vary according to the power conservation law. Therefore, it can be said that this behavior is almost frequency independent in the microwave range. Then, we extend our study to a graphene parabolic reflector as a 2D cylindrical structure. Variation of the surface impedance of the graphene with the control of \(\mu_{c}\) will change the radar cross section and focusing ability of the reflector. However, before that, we should demonstrate the convergent behavior of the presented MAR method. In Figure 4, the relative errors in ACS and FA are demonstrated with the increasing truncation number N\({}_{tr}\). A consistent reduction in the relative error is observed, and 4-5-digit accuracy can be achieved in the given range of N\({}_{tr}\). Figure 3: Variation of the reflectivity, transmissivity (**a**), and absorptivity (**b**) depending on the chemical potential for a plane graphene sheet. The black line for R and blue line for T in part (**a**). The problem parameters are f = 30 GHz, \(\tau\) = 0.135 ps, \(T\) = 300 K.  
In Figure 5, the focusing ability parameter is presented in two different ways, as a ratio of the total field to the incident field at the geometrical focus in Figure 5a and its scattered field counterpart in Figure 5b. The plots are demonstrated for three different \(\mu_{\rm c}\) values. Increasing chemical potential \(\mu_{\rm c}\) reduces the surface impedance of the graphene reflector and accordingly, the surface reflectivity increases, together with FA. Sharp drop in the graphene's surface impedance approaches reflector surface to PEC. The presented findings show that 3 times bigger FA can be possible at focus point at most for \(\mu_{\rm c}=1\) eV. In the case of higher chemical potential possibilities, the results will more closely approach PEC, and higher range of control will be possible. This is a limitation of the graphene material. For a very low chemical potential, the surface becomes almost transparent which results in very low surface reflection. These results are also compatible with Figure 3. As we see in Figure 5b, the scattered field level varies from 0.8 to 4.3 at 30 GHz, so we can conclude about a controlling mechanism that is possible for the scattered field or FA by adjusting the chemical potential via electrical biasing. The oscillations in part (a) of the figure can be explained by the interference of the scattered field and incident field. The variation of FA with chemical potential is seen in Figure 6. As it is visible from the figure, the variation of FA is larger at higher frequency. At 20 GHz, FA varies from 1.5 to 3.7 and closer to PEC case as we see in Figure 5. It also approaches PEC result at 5 GHz since the size is smaller. Figure 6 also demonstrates the FA variation for two different f/d values. For a lower \(\mu_{\rm c}\) value, the reflector is more transparent, so the surface shape has a lesser effect on FA.