# Controlling the Focusing Ability of the Parabolic Reflector in Free Space at Microwave Frequencies
## 2 Formulation
The latter added ones can be inverted analytically, which helps the derivation of the dual series equations during the procedure of the semi-inversion regularization [12]:  
\[A(\varphi,\varphi^{\prime}) = H_{0}^{(1)}(k|\overset{\rightarrow}{r}(\varphi)-\overset{ \rightarrow}{r}^{\prime}(\varphi^{\prime})|)-H_{0}^{(1)}(2 ka\sin(\left| \varphi-\varphi^{\prime}\right|/2)) \tag{5}\] \[B(\varphi,\varphi^{\prime}) = \cos(\xi(\varphi)-\xi(\varphi^{\prime}))\beta(\varphi)\beta( \varphi^{\prime})H_{0}^{(1)}(k|\overset{\rightarrow}{r}(\varphi)-\overset{ \rightarrow}{r}^{\prime}(\varphi^{\prime})|)\] \[-\beta^{2}(\varphi)H_{0}^{(1)}(2 ka\sin((|\varphi-\varphi^{\prime }|/2))\]  
It can be stated that the functions A and B are continuous, and their first derivatives are also continuous, while their second derivatives with respect to \(\varphi\) and \(\varphi^{\prime}\) have only logarithmic singularities. This means that second derivatives belong to \(L_{2}\). Therefore, on the closed contour \(C\), their Fourier transforms can be computed numerically by using the efficient Fast Fourier Transform algorithm. Then, all the terms in the SIE (4) with the A and B functions are written in Fourier Series form. By this way, the SIE is discretized and with the zero current condition on the circular part of contour C that is arc S, it constitutes a dual series equation. The details of the derivation of dual series equation are given in [12]. Then, using the MAR method based on the semi-inversion procedure with the RHP technique [10; 11], we obtain an algebraic equation. This matrix equation has infinite dimension with the mathematical form of Fredholm second kind, hence the Fredholm theorems guarantee the existence of the unique solution, and also, the convergence of the approximate numerical solutions can be obtained by truncating the matrix with the increasing matrix orders. The far zone scattered field from the reflector has a mathematical form of cylindrical wave with components \(H_{z}^{sc}=(2/i\pi kr)^{1/2}e^{ikr}\phi(\varphi)\) and \(E_{\varphi}^{sc}\;=\;\eta_{0}H_{z}^{sc}\), where \(\phi(\varphi)\) is the angular scattering pattern. Then, total scattering cross-section (TSCS) is obtained by using the following equation:  
\[\sigma_{isc}=\frac{2}{\pi k}\int_{0}^{2\pi}\lvert\phi(\varphi)\rvert_{2}d\varphi \tag{7}\]  
As graphene is dominantly resistive at microwave range, it is a lossy material. In the modelling of the scattering performance of reflector, another important characteristic is absorption cross-section (ACS), which can be found from the optical theorem [7; 16]. \[\sigma_{abs}=-4k^{-1}\text{Re}\phi(0)-\sigma_{isc} \tag{8}\]  
One more parameter that should be studied is the focusing ability (FA), and it can be taken as the total H-field at the geometrical focus of the reflector. The FA is defined mathematically as follows:  
\[FA=\left\lvert H_{z}^{sc}(x,y)+H_{z}^{in}(x,y)\right\rvert/\left\lvert H_{z}^ {in}(x,y)\right\rvert\right\rvert\;\;\;at\;focus\] (9) \[\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; But if the reflector's electrical size becomes large, one should choose the smaller ones to simulate in reasonable time. I mean that in the range of 0-100 GHz, our findings work correctly. This was our claim in the paper. Here, by using this procedure, some numerical results are obtained and demonstrated in the following section