The problem geometry of a 2-D parabolic graphene reflector frontally illuminated by a plane wave is presented in Fig. 1. Reflector’s contour M is defined as a finite parabolic profile. An auxiliary closed contour denoted as C is the contour M completed with the circular arc S, which must have the same curvature as the reflector at the latter’s edge points. Such a smooth contour C is necessary for obtaining the regularized (i.e. Fredholm second kind) matrix equation - see [9, 12].
The rigorous formulation of the considered BVP involves the Helmholtz equation, the Sommerfeld radiation condition far from the reflector, the resistive boundary condition on M, and an edge condition such that the field power is limited in any finite domain around the reflector edge. Collectively, these conditions guarantee the uniqueness of the problem solution.
The resistive boundary condition on a zero-thickness sheet is a well-established model of a thin penetrable sheet, e.g. a metal thinner than skin depth or a very thin dielectric layer. In view of “atomic” thickness of graphene, the same boundary condition can also be used for a flat or curved graphene surface, avoiding introduction the thickness of graphene of 2-3 nm that generates meshing troubles in the use of purely numerical codes like COMSOL. It can be written as the following two equations valid at \( \overrightarrow{r}\in M \):
$$ \left({\overrightarrow{E}}_{\tan}^{+}+{\overrightarrow{E}}_{\tan}^{-}\right)/2= Z\kern0.1em \overrightarrow{n}\times \left({\overrightarrow{H}}_{\tan}^{+}-{\overrightarrow{H}}_{\tan}^{-}\right),\kern0.6em {\overrightarrow{E}}_{\tan}^{+}={\overrightarrow{E}}_{\tan}^{-}, $$
(1)
where the subscript “tan” indicates the tangential field, the superscripts “- “ and “+” relate to the front and back faces of reflector, respectively, and \( \overrightarrow{n} \) is understood as the unit vector normal to the concave side of reflector. The jump in the tangential magnetic field, \( \overrightarrow{J}={\overrightarrow{H}}_{\tan}^{+}-{\overrightarrow{H}}_{\tan}^{-} \), is unknown function of the electric surface-current density, and the coefficient Z is graphene’s surface impedance [1,2,3,4,5].
Note also that the surface impedance is related to the graphene surface electron conductivity σ as Z = 1/σ, and the conductivity can be found as the Kubo sum of intraband and interband contributions [1,2,3,4,5,6]. As condition (1) was derived for infinite planar layer, in the modeling of the wave-scattering by finite surfaces it must be combined with the edge condition to provide the uniqueness of the BVP solution.
In the H-wave case, on using the boundary condition (1) we obtain a hyper-singular SIE for the surface current J
t on the reflector. On integrating by parts, it can be cast to the following form:
$$ Z\;{J}_{\mathrm{t}}-\frac{i{Z}_0}{k}\;\frac{\partial }{\partial l}\;{\displaystyle \underset{M}{\int}\left[\frac{\partial }{\partial l\hbox{'}}\;{J}_t\left(\overrightarrow{r}\hbox{'}\right)\right]}\; G\left(\overrightarrow{r},\overrightarrow{r}\hbox{'}\right) dl\hbox{'}+ i k{Z}_0\;{\displaystyle {\int}_M{J}_t\left(\overrightarrow{r}\hbox{'}\right) \cos \left[\xi \left(\overrightarrow{r}\right)-\xi \left(\overrightarrow{r}\hbox{'}\right)\right] G\left(\overrightarrow{r},\overrightarrow{r}\hbox{'}\right)\; dl\hbox{'}}=\frac{i{Z}_0}{k}\;\frac{\partial {H}_z^{i n}}{\partial n}, $$
(2)
where the 2-D Green’s function G is a Hankel function of zero order and first kind satisfying the radiation condition, i.e. \( G\left(\overrightarrow{r},\overrightarrow{r}\hbox{'}\right)=\left( i/4\right){H}_0^{(1)}\left({k}_o R\right) \), \( R=\left|\overrightarrow{r}-\overrightarrow{r}\hbox{'}\right| \), and the angle ξ(φ) is between the normal on M and the x-direction.
Now, we assume that the curve M can be characterized with the aid of the parametric equations x = x(φ), y = y(φ), where 0 ≤ |φ| ≤ θ, in terms of the polar angle, φ. Besides, we denote the differential length in the tangential direction at any point on M as ∂l = aβ(φ)∂φ. We introduce also a function β(φ) = r(φ)/[a cos γ(φ)], where γ(φ) is the angle between the normal on M and the radial direction. Then we extend the surface-current density J
t
with zero value to arc S and cast IE (2) to a dual equation on the arcs S and M [9].
To continue with the MAR, we add and subtract, from the integral kernels in (2), similar functions at a full circular contour of the same radius as S. The latter operators can be inverted analytically while the remaining ones have smooth kernels,
$$ A\left(\varphi, \varphi \hbox{'}\right)={H}_0^{(1)}(kR)-{H}_0^{(1)}\left[2 ka \sin \left(\left|\varphi -\varphi \hbox{'}\right|/2\right)\right], $$
(3)
$$ B\left(\varphi, \varphi \hbox{'}\right)= \cos \left[\xi \left(\varphi \right)-\xi \left(\varphi \hbox{'}\right)\right]\beta \left(\varphi \right)\beta \left(\varphi \hbox{'}\right){H}_0^{(1)}\left[ k\left|\overrightarrow{r}\left(\varphi \right)-\overrightarrow{r}\hbox{'}\left(\varphi \hbox{'}\right)\right|\right]-{\beta}^2\left(\varphi \right){H}_0^{(1)}\left[2 ka \sin \left(\left|\varphi -\varphi \hbox{'}\right|/2\right)\right] $$
(4)
For the inversion of the singular operators, all functions including the incident field should be expanded in terms of the Fourier series in φ. Note that the functions A and B are continuous and have also continuous first derivatives, while their second derivatives with respect to φ and φ’ have only logarithmic singularities and hence belong to L
2. Therefore on the curve C their Fourier coefficients in φ decay fast enough with larger indices and hence can be efficiently computed by the Fast Fourier Transform algorithm. Then the discretized version of the SIE and the zero current condition on the aperture S give us a dual series equation. Its semi-inversion, based on the MAR approach using the RHP technique [7, 10], finally produces an algebraic equation set [9]. This infinite matrix equation is of the Fredholm second kind hence the Fredholm theorems guarantee the existence of the unique solution and also the convergence of the approximate numerical solutions when truncating the set with progressively larger orders.
In the E-wave case, on using the boundary condition (1) we obtain the following log-singular IE for the surface current J
z
on the reflector:
$$ Z\;{J}_z- i k{Z}_0\;{\displaystyle {\int}_M{J}_z\left(\overrightarrow{r}\hbox{'}\right)\; G\left(\overrightarrow{r},\overrightarrow{r}\hbox{'}\right)\; dl\hbox{'}}={E}_z^{in}, $$
(5)
As mentioned, convergence of usual discretizations of this equation is guaranteed by its Fredholm second-kind nature. Therefore we apply the projection to the set of entire-domain angular exponents [12]. In either polarization case we adapt the matrix truncation number to provide the 4-digit or better accuracy of computations.
The scattered electromagnetic field in the far zone of reflector is a cylindrical wave with functions \( {H}_z^{sc} \) or \( {E}_z^{sc} \) (depending on the polarization) reduced to (2/iπkr)1/2 e
ikr
ϕ(φ), where ϕ(φ) is the angular scattering pattern. Then TSCS can be obtained by using the following expression:
$$ {\sigma}_{tsc}=\frac{2}{\pi k}{\displaystyle {\int}_0^{2\pi}\left|\;\phi \left(\varphi \right)\right|{}^2\; d\varphi}, $$
(6)
and ACS of a lossy graphene reflector can be found from the optical theorem,
$$ {\sigma}_{abs}=-\frac{4}{k}\mathrm{R}\mathrm{e}\;\phi (0)-{\sigma}_{tsc} $$
(7)
