Multiperiodicity in plasmonic multilayers: General description and diversity of topologies

We introduce multiperiodicity in periodic metal-dielectric multilayers by stacking more than two types of metal and/or dielectric layers into the unit cell. A simple way to characterize arbitrary multiperiodic multilayers using permutation vectors is suggested and employed. Effects of multiperiodicity up to its fourth order are investigated. We demonstrate that various topologies of multiple-sheet isofrequency and dispersion surfaces exist for such plasmonic multilayers, including a photonic realization of nontrivial isolated Dirac cones.

Figure 1

Diagram of a multiperiodic multilayer formation by means of $U^{\left(a\right)}$ and $U^{\left(b\right)}$. Layers of different materials are marked with different colors, with 1 and 2 denoting plasmonic interfaces of two kinds. Period of the structure is $\mathfrak{d}=2(d_d+d_m)$.

Figure 2

List of generative $U$'s. Different kinds of plasmonic interfaces are highlighted with different dashing.

Figure 3

Asymptotic dispersion diagrams and isofrequency surfaces (in arbitrary units) typical for plasmonic multilayers in the long-wavelength limit. Structures are designed to embody the multiperiodicity of (a) the second order (bi-periodicity), (b) the third order, and (c) the fourth order. Cross sections of the surfaces (the isofrequency contours) are marked with green.

Figure 4

Band diagrams showing $k_x$ zero eigenwaves of the plasmonic multilayer in (a) $U_{1<}$ and (b) $U_{1>}$ configurations, as per Eq. (35). Frequency is normalized according to Eq. (32), and wave numbers are normalized by a factor of $\pi /\mathfrak{d}$. Roman numerals (I–IV) number the modes; highlighted regions numbered as 1–3 correspond to characteristic cases depicted in Fig. 6. Blue and red dots denote mode crossings.

Figure 5

Band diagrams for the plasmonic multilayer in (a) $U_{3<}$ and (b) $U_{3>}$ configurations, as per Eq. (36). Highlighted regions 1 and 2 correspond to characteristic cases shown in Fig. 7. The rest is the same as in Fig. 4

Figure 6

Dispersion surfaces corresponding to regions 1–3 of Fig. 4, with $k_x$ spanning the first Brillouin zone. Blue arrows mark the asymptotic behavior of the dispersion surfaces as ${\tilde{k}}_y\to \infty$, tending towards $\tilde{W}=0$ or 1. Blue dots indicate touching points between the surfaces at the crossing between modes I and II [see Fig. 4]. Red dots indicate Dirac-type points, where outer dispersion surfaces collapse.

Figure 7

Dispersion surfaces corresponding to regions 1 and 2 of band diagrams of Fig. 5. Red dots indicate isolated Dirac-type points. The rest is the same as in Fig. 6.

Figure 8

(a) An enlarged view of an isolated Dirac-type cone of Fig. 7(2), with its cross section marked by the dashed line. (b) Zero photon density of states corresponding to the Dirac-type point (in arbitrary units).