Optical Möbius Symmetry in Metamaterials

We experimentally observed a new topological symmetry in optical composites, namely, metamaterials. While it is not found yet in nature materials, the electromagnetic Möbius symmetry discovered in metamaterials is equivalent to the structural symmetry of a Möbius strip, with the number of twists controlled by the sign change of the electromagnetic coupling between the meta-atoms. We further demonstrate that metamaterials with different coupling signs exhibit resonance frequencies that depend only on the number but not the locations of the “twists,” thus confirming its topological nature. The new topological symmetry found in metamaterials may enable unique functionalities in optical materials.

Figure 1

(a) Resonance frequency diagram of a single resonator, coupled two-resonator (dimer), and coupled three-resonator (trimer). In general, if there is a degenerate resonance in the trimer, its frequency should overlap with one of the two resonance frequencies of the constituent dimers. (b),(c) A metamolecular system representing a uniform negative coupling configuration in which the trimer can be symbolized as an equilateral triangle, or equivalently, an ordinary strip with $C_3$ symmetry. (d),(e) A metamolecular system displaying mixed signs of couplings in which the trimer can be symbolized as a Möbius strip with topological $C_3$ symmetry.

Figure 2

The transmission spectra of trimer ${\mathrm{C}}_3$ and its constituent dimer (dimer 13). The number in the parenthesis and the arrows denote the incident polarization angle. The polarizations are chosen only to reveal the resonance frequencies clearly. Trimer ${\mathrm{C}}_3$ spectrum shows a dips at 94 THz. Dimer 13 shows an antisymmetric mode at 87 THz and a symmetric mode at 107 THz; thus, ${\kappa }_{13}<0$. The shaded area highlights the region where trimer ${\mathrm{C}}_3$ and dimer 13 exhibit overlapping resonances, indicating that the degenerate resonance of trimer ${\mathrm{C}}_3$ occurs at 94 THz.

Figure 3

The transmission spectra of trimer ${\widehat{\mathrm{C}}}_3$ and its constituent dimers. The number in the parentheses and the arrows denote the incident polarization angle. The polarizations are chosen only to reveal the resonance frequencies clearly. (a) Trimer ${\widehat{\mathrm{C}}}_3$ spectrum shows two dips at 87 and 106 THz, representing two magnetic resonances. (b) Dimer 12 shows a symmetric mode at 96 THz and an antisymmetric mode at 108 THz; thus, ${\kappa }_{12}>0$. (c) Dimer 13 shows an antisymmetric mode at 87 THz and a symmetric mode at 107 THz; thus, ${\kappa }_{13}<0$. (d) Dimer 23 shows an antisymmetric mode at 96 THz and a symmetric mode (not optically active, obtained via simulation) at 106 THz; thus, ${\kappa }_{23}<0$. The shaded area highlights the region where trimer ${\widehat{\mathrm{C}}}_3$ and the dimers exhibit overlapping resonances, indicating that the degenerate resonance of trimer ${\widehat{\mathrm{C}}}_3$ occurs at 106 THz.

Figure 4

Upper row: Schematic graphs which highlight the signs of ${\kappa }_{ij}$’s of the hexamers. A solid line denotes a negative ${\kappa }_{ij}$ and a dashed line denotes a positive ${\kappa }_{ij}$. Lower row: Corresponding scanning electron microscopy images of the fabricated hexamers. Main panel: Two structurally distinct but topologically identical metamolecules (hexamer 2A and hexamer 2B) show nearly identical transmission spectra (shaded area) for a $y$-polarized incident light, which highlights a topological effect. Inset: The frequencies of the magnetic resonances for an $x$-polarized incident light displaying an oscillation with respect to odd or even number of positive ${\kappa }_{ij}$’s in the hexamers, which demonstrates another topological effect (see text). The experimental data and simulation results are shown for comparison.