Direct $S$-matrix calculation for diffractive structures and metasurfaces

The paper presents a derivation of analytical components of $S$ matrices for arbitrary planar diffractive structures and metasurfaces in the Fourier domain. The attained general formulas for $S$-matrix components can be applied within both formulations in the Cartesian and curvilinear metric. A numerical method based on these results can benefit from all previous improvements of the Fourier domain methods. In addition, we provide expressions for $S$-matrix calculation in the case of periodically corrugated layers of two-dimensional materials, which are valid for arbitrary corrugation depth-to-period ratios. As an example, the derived equations are used to simulate resonant grating excitation of graphene plasmons and the impact of a silica interlayer on corresponding reflection curves.

Figure 1

Convergence of the method based on the direct $S$-matrix calculation [Eqs. (22) and (23)] and the comparison with $S$ matrices calculated by the Fourier modal method. The maximum absolute difference between corresponding $S$-matrix components is plotted against the inverse number of Fourier harmonics. The grating parameters are the period-to-wavelength ratio $\mathrm{\Lambda }/\lambda =1.5$, the depth-to-wavelength ratio $h/\lambda =0.5$, and $\alpha =0.5$. The substrate and cover refractive indices are 1.5 and 1, respectively. The grating permittivity ${\varepsilon }_1=6.25$.

Figure 2

Same as in Fig. 1, but for metallic grating of permittivity ${\varepsilon }_1=-9.6+1.1i$.

Figure 3

Illustration of the curvilinear coordinate generalized source method idea: Maxwell's equations are written in a curvilinear coordinate system $(z^1,z^2,z^3)$ such that one of its coordinate planes exactly fits the corrugation profile and these coordinates continuously become Cartesian $(x_1,x_2,x_3)$ in the region in the vicinity of the grating. Dashed lines demonstrate coordinate planes $z^3=\text{const}$.

Figure 4

Dependence of the zeroth-order reflection efficiency on the frequency for the normally incident plane wave for a corrugated graphene sheet placed on top of the silicon substrate with a corrugation period of $0.8\mu \text{m}$ and varying corrugation depth.

Figure 5

Similar to Fig. 4, but the graphene sheet is supposed to be in contact with an intermediate silica layer of varying thickness $h_{{\text{SiO}}_2}$.