# Controlling the Focusing Ability of the Parabolic Reflector in Free Space at Microwave Frequencies
## 1 Introduction  
Graphene is a very thin material, in principle, one atomic layer, therefore, a zero thickness model can be used in the simulations. It has electrical and mechanical properties like high conductivity, mechanical strength, and optical transparency [1; 2]. Furthermore, graphene can be used within a wide range of frequencies, from infrared for nano-size samples to THz for micro-size samples. In the H-polarization case, it supports the surface plasmon (SP) wave [1], and this SP wave produces standing waves on finite-size scatterers. This effect is called Fabry-Perot type resonances, established mainly due to the reflection from the edges. For a graphene sample, the edge effects can be ignored if its size is larger than 100 nm. On the other hand, one of the best features of graphene is that its electron conductivity can be controlled by applying an external electrostatic biasing, which can change the graphene's chemical potential. Various applications of graphene exist already such as in waveguides [3], tunable nano sensor devices [4], and antennas [5]. For the purpose of modelling, the frequency-dependent conductivity of the graphene can be modelled using the Kubo formula [6]. Then, the scattering from a flat or curved graphene strip can be simulated using the electromagnetic boundary-value problem (BVP) with the resistive-sheet boundary condition. The method of moments (MoM) can be used to solve the singular integral equations (SIEs) obtained from the boundary conditions. However, the accuracy of conventional MoM is limited to only 2-3 digits. Due to the limited accuracy and convergence trouble for a denser meshing, only medium-size strips of up to 10 wavelengths can be simulatedeven with such accuracy [7; 8]. For larger geometries, the MoM matrix grows quickly, and the condition number of the matrix increases. The consequence of this situation is a huge computation time with degrading accuracy. To alleviate this trouble, certain special iterative algorithms are applied to the MoM procedure like fast-multipole technique and numerical preconditioning. However, this is not easy and convergence is still not guaranteed mathematically, and accuracy suffers as well. To obtain a more reliable solution of the scattering from a graphene strip, Nystrom method can be used by applying the special quadrature formulas to handle the singularity of the kernels [9]. Another important approach is the method of analytical regularization (MAR) [10]. Here, the SIE kernel is separated into two parts, the most singular (usually static) and the remainder. The most singular part is analytically inverted using special techniques like the Riemann-Hilbert Problem (RHP) method [11; 12]. The remainder produces a Fredholm second-kind matrix equation, and in this case, the numerical solution is convergent. The SIE-MAR technique enables an accurate and economical solution of the electromagnetic scattering from even quasi-optical size scatterers. By following this procedure in [13; 14] the graphene strip, disc, and infinite strip grating problems are simulated with a high level of accuracy using Nystrom method and MAR. The focusing effect of a graphene reflector in free space is studied in [15; 16] by using MAR method based on the RHP technique. Then, in [17], a graphene reflector with a dielectric substrate case is simulated as a more realistic geometry. Also, in [18], a dielectric strip sandwiched between graphene sheets is modelled by the Nystrom method. Another research area is the use of graphene at microwave range. The graphene conductivity and related surface impedance are almost frequency independent at microwaves and so it presents completely different behavior with respect to THz and optics. At microwaves, the surface impedance becomes dominantly resistive and the reactive part can be neglected, and this resistance can be controlled by electrostatic biasing through the change of the chemical potential of the graphene, as explained in [19].