# Controlling the Focusing Ability of the Parabolic Reflector in Free Space at Microwave Frequencies
## 3 Numerical Results
With the increase of \(\mu_{\rm c}\), the reflectivity of surface increases so the shape of reflector (here, f/d ratio) makes a change in FA plot. It is observed that the deeper reflectors with lower f/d ratio produce smaller focusing field at the focus point. The difference is important for larger \(\mu_{\rm c}\) and larger frequencies. This can be realized by considering the surface wave mechanisms on deeper dishes due to the curvature effects. Figure 4: Variation of relative error in ACS (**a**) and FA (**b**) with the increasing N\({}_{\rm tr}\) for two different chemical potential values given as \(\mu_{\rm c}=0.05\) (black line) and \(\mu_{\rm c}=1\) (blue line). Other parameters f = 30 GHz, \(\tau=0.135\) ps, and \(T=300\) K.  
In Figure 7, TSCS and ACS variation with frequency is presented in the microwave range. Increasing the \(\mu_{c}\) value increases the graphene surface conductivity, which reduces the surface impedance and so increases the reflector reflection. This can be seen in Figure 7a for various \(\mu_{c}\) values. In ACS variation, the maximum variation is observed around the chemical potential value of \(\mu_{c}\) = 0.4 eV. For chemical potentials higher than that value, ACS reduces, as seen in Figure 7b, where \(\mu_{c}\) = 1 eV. This observation is in agreement with Figure 3b. The oscillations observed in the low GHz range can be explained by the edge effects for the smaller sizes in terms of wavelength. Figure 5: Variation of the focusing ability parameter defined using total field (**a**) and scattered field (**b**) with the increasing frequency for three different chemical potentials given as \(\mu_{c}\) = 0.05 (black dashed), \(\mu_{c}\) = 0.4 (blue line), \(\mu_{c}\) = 1 (red line), and PEC case (black line). Other parameters d = 0.2 m, f/d = 0.7, \(\tau\) = 0.135 ps, and \(T\) = 300 K.  
Figure 6: Variation of the FA by the increase of the chemical potential for two different frequencies and two different f/d ratios. The black line: f/d = 0.7 and 20 GHz, black and dashed line: f/d = 0.4 and 20 GHz, the blue line: f/d = 0.7 and 5 GHz, and the blue-dashed line: f/d = 0.4 and 5 GHz. Other parameters are d = 0.2 m, \(\tau\) = 0.135 ps, and \(T\) = 300 K.  
Lowering the chemical potential \(\mu_{c}\) reduces the graphene conductivity, so the surface impedance increases, which makes reflector more transparent. Figure 8 shows that the field reduction is not only seen in the focusing ability but also present in almost all angular directions of the field pattern. For example, around 180\({}^{\circ}\), that is the backward direction, almost 8 dB field reduction can be observed compared to PEC case for \(\mu_{c}=0.05\) eV. If \(\mu_{c}\) increases towards unity, the scattered field level approaches the PEC surface reflection value, and it is also seen in Figure 8. This result implies that controlling the reflector by the chemical potential is possible, and it is valid for all directions. This is another important result obtained from the simulation of the presented geometry. Colored near-field patterns are obtained for the total H-field and shown in Figures 9 and 10 for two different chemical potential values. In Figure 9, \(\mu_{c}\) is taken as 1 eV, and normalized  
Figure 8: Variation of the radar cross-section field pattern for two frequencies (10 GHz (**a**) and 20 GHz (**b**)) with the angle for three different chemical potentials given as \(\mu_{c}=0.05\) (black dashed), \(\mu_{c}=0.4\) (blue line), \(\mu_{c}=1\) (red line), and PEC case (black line). Other parameters are d = 0.2 m, f/d = 0.7, \(\tau=0.135\) ps, and \(T=300\) K.  
Figure 7: Variation of the TSCS (**a**) and ACS (**b**) with the increasing frequency for three different chemical potentials given as \(\mu_{c}=0.05\) (black dashed), \(\mu_{c}=0.4\) (blue line), \(\mu_{c}=1\) (red line), and PEC case (black line). Other parameters are d = 0.2 m, f/d = 0.7, \(\tau=0.135\) ps and \(T=300\) K.  
surface impedance of reflector is smaller compared to the other values of chemical potential.