## 4 Numerical Results
### Scattering of the Natural Waves  
Let us consider the scattering of the natural wave \(P_{1}\). The maximum of the fields of plasmon wave is concentrated near the graphene plane. The amplitude vanishes exponentially when moving away from the graphene plane. To obtain effective coupling of this wave with graphene strips, the width of the dielectric waveguide should be minimal. The graphene is able to absorb the electromagnetic field. The amplitude of the natural waves of the waveguide decreases if \(y\) increases. We suppose that incident natural wave has unit power at the edge of the first graphene strip. Let us denote the chemical potential of the strips as \(\mu_{C}\). The chemical potential of the graphene plane is, as before, \(\mu_{C}\). Figures 5-7 show dependences of the radiated power on the frequency in the case of \(P_{1}\) natural wave incidence. The number of strips is taken so that the incident wave is almost attenuated along the length of the structure at the frequency band of interest, \(N=5\). In periodic gratings with dielectric substrate, the grating-mode (or lattice-mode) resonances can arise [18; 23; 24; 25; 26]. Their frequency depends on material parameters of the substance of the waveguide and the period. Near these resonances, minima of the radiation are observed. The current distribution is in-phase on the strips. The maximum of radiation is at \(\varphi=90^{\circ}\). At the frequencies higher than the frequency of the grating-mode resonance, the second lobe appears. We denote the first such resonance  
Figure 3: Dependences of the propagation constant of the natural wave \(H_{1}\) on the frequency, \(\varepsilon=2.25\). (a) Real part; (b) Imaginary part. as \(G_{1}\). Graphene strips are capable of maintaining plasmon resonances within the THz range. Their frequency is influenced by the parameters of graphene strips, including width and conductivity. Near plasmon resonances, the maxima of the radiation are observed. As one can see in Figures 5-7, in the case of simultaneous excitation of the plasmon and \(G_{1}\) resonances, the radiation level still significantly decreases. Thus, the preferable frequency band is near the plasmon resonances but up to \(G_{1}\). If we change the value of the chemical potential of the infinite graphene plane, the propagation constant and, as a result, the resonance frequency of \(G_{1}\) is also changed. With the decrease in the chemical potential of infinite graphene plane, the attenuation constant of plasmon mode and the absorption of the transferred power increases. The chemical  
Figure 5: Dependences of the radiated power on the frequency for \(h=20\) μm, \(d=10\) μm, \(l=50\) μm, \(N=5\), \(\varepsilon=2.25\) and \(P_{1}\) natural wave incidence, (**a**) \(\mu_{c}=1\) eV; (**b**) \(\mu_{c}=0.6\) eV. The circles show the value of the frequency for which the radiation patters will be presented. In Figures 8-10, for \(0^{\circ}<\varphi<180^{\circ}\) (in the domain above the waveguide), the increase in the chemical potential of graphene plane leads to a leftward shift of the left peak. For \(180^{\circ}<\varphi<360^{\circ}\) (in the domain below the waveguide), the increase in the chemical potential leads to a rightward shift of the peak. It can be explained if one considers \(\text{Re}(\beta)\) at a fixed frequency for different values of the chemical potential of the graphene plane (see Figure 3a). \(\text{Re}(\beta)\) decreases monotonically as \(\mu_{c}\) increases. Additionally, as one can observe from Figures 5-7, the position of the grating-mode resonance, for which the current distribution is in-phase on the strips and the elevation angle of the main lobe is \(90^{\circ}\), tends to be at the higher values of the frequency. Rotation angle value can be increased if we consider a frequency greater than the frequency of the resonance \(G_{1}\).