## 4 Scattering by Coplanar Multilevel Ordered Strip System
### Scattering by the Simplest System of Electrically Narrow Strips  
Let us return to the mathematical modeling of the plane \(E\)-polarized electromagnetic wave scattering from non-classical coplanar systems of electrically conducting flat strips using the Carleman regularization technique. It is interesting to consider the asymptotic mathematical models of the plane wave scattering for the simplest, sparsely filled coplanar system with a finite number of electrically narrow impedance strips. Figure 3 presents such coplanar strips systems: the first strips system deals with the first stage of a five-part perfect set (see Figure 2b). Here, we have two equal electrically conducting flat strips (see Figure 3a). Normalized input parameters of corresponding \(E\)-polarized electromagnetic wave scattering problem have to be under the assumptions: \(\rho=k\sqrt{b^{2}+c^{2}}\)= O(1), \(\alpha=ka<<1\). Figure 3: (**a,b**) The simplest gratings dealing with irregular Cantor functions. For that simple electrically conducting flat strips' system determinant of linear algebraic equations set (20) of the second order (\(N=2\)) is defined by the simplest formula:  
\[D_{2}(\alpha,\rho)=\left[\ln(\gamma\alpha/4i)\right]^{2}+\left[\pi\cdot H_{0}^{( 1)}(2\rho)/2\right]^{2}. \tag{21}\]  
Due to the simplest Cramer's rule the main constants \(j_{n0}\) (which define output variables of that asymptotic mathematical model) are determined by the expressions:  
\[j_{n0}=\frac{d_{n0}[\ln(\gamma\alpha/4i)]-\pi d_{(3-n)0}\cdot H_{0}^{(1)}(2\rho )/2i}{D_{2}(\alpha,\rho)\cdot(q_{2}\mu-1)},\quad n=1,2. \tag{22}\]  
Here, new constants are presented the formula: \(d_{n0}=2\pi\cdot\exp(ik(-1)^{n-1}(q_{1}b+q_{2}c))\). Thus, the plane \(E\)-polarized electromagnetic wave scattering problem is completely solved for the simplest sparsely filled coplanar system dealing with irregular Cantor functions in the main approximation. Those two levels of coplanar grating consists of two electrically narrow impedance strips. The second coplanar strips system, presented in Figure 3b, deals with the second stage of a three-part perfect set and we have three unequal strips: the central strip is not equal to two others (see Figure 2a). In that more complicated case, we ought to consider three normalized input parameters. The first one deals with distances between strips, so, it will be under the same assumption (as for two levels previous coplanar gratings):\(\rho=k\sqrt{b^{2}+c^{2}}=\mathrm{O}(1)\). Two other normalized input parameters, dealing with electrically narrow strips, are under similar assumptions. For the main (central) strip, we have the same assumption: \(\alpha=ka<<1\); for two other equal strips, similar assumptions are true automatically due to the three-part perfect set's construction [5,7]. So, two principal normalized parameters \(\alpha\) and \(\rho\) determine input variables of geometric structure of grating. For that, the simplest electrically conducting flat strips grating determinant of linear algebraic equations set (20) of the third order (\(N=3\)) is defined by a rather unwieldy formula:  
\[ \begin{array}{l}D_{3}(\alpha,\rho)=\left[\ln(\gamma\alpha_{2}/4i)\right]^{2 }\cdot\ln(\gamma\alpha_{1}/2i)+i\pi^{3}[H_{0}^{(1)}(\rho)]^{2}H_{0}^{(1)}(2 \rho)/4+\\ +\pi^{2}\ln(\gamma\alpha_{1}/4i)\cdot[H_{0}^{(1)}(2\rho)]^{2}/4+\pi^{2}\ln( \gamma\alpha_{2}/4i)\cdot[H_{0}^{(1)}(\rho)]^{2}/2.\end{array}  \tag{23}\]  
The constants \(j_{n0}\) (which define output variables of that asymptotic mathematical model) can be determined by the simplest Cramer's rule as well. Let us perform some computer simulation to show simplicity just presented asymptotic scattering model. The initial normalized parameters: \(\alpha=0.1\), \(kb=2\), \(kc=1\). The direction vector \(\overrightarrow{q}=(1,\sqrt{3})/2\) and the dimensionless impedance parameter \(\mu=(1+i)/2\). Easy calculation gives us numerical results: \(j=(0.14-2.4i\), \(2.34+0.36i\), \(-0.36+1.68i\)). Due to above presented assumptions the calculation error is 0.01.