Toroidal circular dichroism

We demonstrate that the induced toroidal dipole, represented by currents flowing on the surface of a torus, makes a distinct and indispensable contribution to circular dichroism. We show that toroidal circular dichroism supplements the well-known mechanism involving electric dipole and magnetic dipole transitions. We illustrate this with rigorous analysis of the experimentally measured polarization-sensitive transmission spectra of an artificial metamaterial, constructed from elements of toroidal symmetry. We argue that toroidal circular dichroism will be found in large biomolecules with elements of toroidal symmetry and should be taken into account in the interpretation of circular dichroism spectra of organics.

Figure 1

(a) A schematic of the toroidal metamolecule and its orientation with respect to the metamaterial crystal lattice (indicated by dashed lines). The dimensions are $d=8,s=7.5,h=1.5,r=2.44,g=w=0.15$, and $a=1.8\mathrm{mm}$. Right circular polarization (RCP) (red) and left circular polarization (LCP) (blue) light propagates along the $z$ axis. The individual metamolecules are achiral as indicated by the mirror image. (b) The metamaterial slab is formed by translating the unit cell along the $x$ and $y$ axes, which imposes structural chirality on the sample. The mirror image shows the enantiomeric form of the metamaterial crystal. (c) Closeup photograph of the fabricated toroidal metamolecule. (d) Metamaterial sample after assembly.

Figure 2

Measured (left) and calculated (right) circular transmission responses of the toroidal metamaterial. (a) and (b) show the amplitudes of the circularly polarized transmission $t_{++}$ and $t_{--}$, whereas panels (c) and (d) show circular dichroism $\mathrm{\Delta }$. Results calculated using two different methods—finite-element simulation and multipole decomposition of currents are shown by solid and dashed curves, respectively. Pink and navy bands mark the resonances of corresponding conventional $\left({\nu }_1\right)$ and toroidal $\left({\nu }_2\right)$ circular dichroisms, respectively.

Figure 3

Amplitudes of the fields radiated by the four most dominant multipole components (electric dipole $\mathbf{p}$, magnetic dipole $\mathbf{m}$, toroidal dipole $\mathbf{T}$, and electric quadrupole ${\mathbf{Q}}_{\mathbf{e}})$ that contribute to the polarization conversion response of the metamaterial—(a) $t_{xy}$ and (b) $t_{yx}$. (c)–(f) give the field distributions $({log}_{10}\left|E\right|$ and ${log}_{10}\left|H\right|)$ around a metamolecule when excited by LCP light at (c) and (d) ${\nu }_1$ and (e) and (f) ${\nu }_2$. The black arrows give the corresponding $E$ and $H$ field directions. The green (blue) arrows show the orientation of the induced electric (magnetic) dipoles for the individual rings.

Figure 4

Ellipticity angles of the metamaterial eigenstates (left) and associated polarization ellipses at ${\nu }_1$ and ${\nu }_2$ (right), calculated based on the multipole expansion of microscopic charge-current excitations supported by toroidal metamolecules. (a) shows the results obtained using the complete multipole set. (b) and (c) show the results corresponding to pairs $p_x-m_x$ and $T_y-Q_e$ being removed from the microscopic response of the metamaterial (solid curves) and compared to the complete multipole response from the upper panel (dashed curves).