## 2 Solution to the Scattering Problem  
Let us consider a planar dielectric waveguide characterized by a width \(h\), with infinite graphene plane at \(z=-h\) and finite number of graphene strips at \(z=0\). The finite set of infinite graphene along the \(x\)-axis strips we denote as \(L=\underset{n=1}{\overset{N}{\cup}}L_{n}\), where \(N\) is the number of strips and \(L_{n}\) is the \(n\)th strip. The relative permittivity of the dielectric is \(\varepsilon\). The scattering geometry and notations are presented in Figure 1. The time dependence \(\exp(-i\omega t)\) is omitted, where \(\omega\) is the angular frequency.  
We consider the \(H\)-polarized incident natural waves of the waveguide, which propagate from the domain \(y<0\) to \(y=+\infty\). The total field is a superposition of the incident field, denoted as \(H^{i}\), and scattered field, denoted as \(H^{s}\):  
\[H^{t}=H^{i}+H^{s}. \tag{3}\]  
The field is the solution of the Helmholtz equation, with boundary conditions on and outside graphene (1) and (2), and  
Figure 1: Structure geometry.  
\[\Theta(\theta)=\exp(-ik_{1}h\gamma(\theta))k_{1}(\gamma(\theta)+\gamma(\theta) \gamma(\theta\sqrt{\epsilon})\sigma Z+\gamma(\theta\sqrt{\epsilon})\sqrt{ \epsilon}), \tag{16}\]  
\(\theta_{1}=\theta/\sqrt{\epsilon}\), \(C_{1}(\theta)\) is an unknown function and \(Z=120\pi\Omega\) is the impedance of free space. To find \(C_{1}(\theta)\), we reduce (6) and (7) to an SIE.  
We should note that \(\Gamma(\theta)\) possesses the following asymptotic relation: \(\Gamma(\theta)\sim i|\theta|-\frac{i}{2|\theta|}\frac{(1+\epsilon+\epsilon^{2 }+\epsilon^{2})}{\epsilon(1+\epsilon)^{2}}+O(|\theta|^{-3})\), if \(\theta\rightarrow\infty\). As one can see, the first term is increasing and all others are decreasing. To the increasing integrand in (6), we use the parametric representation of the Hilbert pseudo-differential operator [31] and reduce it to the singular integral. For all non-increasing terms, we collect to the regular part of the kernel-function \(K(y,\xi)\). As a result, the following SIE of the first kind with additional conditions can be obtained:  
\[\frac{1}{\pi}PV\int\limits_{L}K(y,\theta)G(\theta)d\xi=-\frac{(\epsilon+1)}{ \epsilon}\frac{\partial H^{inc}}{\partial z},\ \ \ y\in L, \tag{17}\]  
\[\frac{1}{\pi}\int\limits_{L_{m}}G(\theta)d\theta=0,\ \ \ m=1,\ \ 2,\ \ldots,\ N, \tag{18}\]  
where \(G(y)=ik_{1}\int\limits_{-\infty}^{\infty}\theta C_{1}(\xi)\exp(ik_{1}\theta y )d\xi\) is Fourier transform of \(C_{1}(\theta)\) with factor \(ik_{1}\theta\). \(F(y)\) is up to the constant the currents density on graphene strips. The kernel-function is  
\[K(y,\theta)=\frac{1}{\theta-y}+ik_{1}\int\limits_{0}^{\infty}\frac{(\eta+i \Gamma(\eta))\sin(k_{1}\eta(y-\theta))}{\eta}d\eta+\left\{\begin{array}{ll} ik(\epsilon+1)\pi(\sigma Z_{0})^{-1},\ \ \ if\ \ \ \xi\leq y,\\ 0,\ \ \ if\ \ \ \xi>y,\end{array}\right. \tag{19}\]  
and the first term in (19) is singular one. All other terms are regular. We should note that the second term in (7) equals \(G(y)\) (with the constant factor). However, we collect to the kernel-function as the third term in (19) to obtain an SIE of the first kind rather than the second kind.  
Based on the corresponding theorems, we can state that the solution of (17) and (18) is unique [32]. To carry out the discretization process, the Nystrom-type method is employed [26]. From the edge condition it follows that unknown function \(G(\theta)\) has inverse square root singularities at the edges of the strips. Thus, the appropriate quadrature rule with nodes at the zeros of the Chebyshev polynomials of the first kind is applied, while the collocation points are the zeros of the Chebyshev polynomials of the second kind. Theorems formulated in [32] confirm the convergence.