Optical force on toroidal nanostructures: Toroidal dipole versus renormalized electric dipole

We study the optical forces acting on toroidal nanostructures. A great enhancement of optical force is unambiguously identified as originating from the toroidal dipole resonance based on the source representation, where the distribution of the induced charges and currents is characterized by the three families of electric, magnetic, and toroidal multipoles. On the other hand, the resonant optical force can also be completely attributed to an electric dipole resonance in the alternative field representation, where the electromagnetic fields in the source-free region are expressed by two sets of electric and magnetic multipole fields based on symmetry. The confusion is resolved by conceptually introducing the irreducible electric dipole, toroidal dipole, and renormalized electric dipole. We demonstrate that the optical force is a powerful tool to identify toroidal response even when its scattering intensity is dwarfed by the conventional electric and magnetic multipoles.

Figure 1

(a) Schematic diagram of a gold helix that supports a magnetic dipole resonance in the optical regime. Structural parameters are inner radius $r=30\mathrm{nm}$, outer radius $R=100\mathrm{nm}$, and pitch $p=200\mathrm{nm}$. (b) Calculated scattering cross sections and (c) $z$ component of optical forces acting on the irreducible dipoles as well as the toroidal dipole in the source representation [Eq. (4)] for the gold helix.

Figure 2

(a) Schematic diagram of the designed nanostructure that supports a toroidal dipole resonance. (b) The intensity distribution of the current density associated with magnetic field lines (black lines) for the toroidal dipole resonance at $\sim 71\mathrm{THz}$. (c) Calculated scattering cross sections and (d) $z$ component of optical forces based on the source representation. The toroidal response manifests itself through a resonant peak in the total optical force at $\sim 71\mathrm{THz}$.

Figure 3

Calculated $z$ component of the optical forces in the optical regime acting on the irreducible dipoles and the toroidal dipole based on the source representation. The inset of the figure illustrates the studied toroidal structure with a different selection of the helical handedness compared to the one in Fig. 2.

Figure 4

(a) Calculated scattering cross sections and (b) $z$ component of optical forces acting on the renormalized dipoles for the toroidal nanostructure, according to the field representation.

Figure 5

Calculated $z$ component of the optical forces in the microwave regime acting on the toroidal structure (see the inset of Fig. 3) based on (a) source representation and (b) field representation. The parameters are inner radius $r=0.3\mathrm{mm}$, outer radius $R=1\mathrm{mm}$, and pitch $p=2\mathrm{mm}$. (c) Calculated scattering cross sections for the irreducible dipoles, the toroidal dipole, and the renormalized dipoles.

Figure 6

Calculated $z$ component of the optical forces acting on the $\mathsf{U}$-shaped SRRs based toroidal structure based on (a) source representation and (b) field representation. The parameters of each SRR are $L=300\mathrm{nm}, h_1=w_1=w_2=50\mathrm{nm}, h_2=200\mathrm{nm}$, and the distance between the structure center and the center of each SRR is 300 nm. The structure is shown in the inset of the top panel. The arrows show the direction of the $k$ vector of the incident plane wave and its polarization. The resonant optical force at slightly above 160 THz can be attributed mainly to a toroidal dipole moment (green line, upper panel) or to the renormalized electric dipole moment (red line, lower panel) depending on how we choose to do the multipole expansion.

Figure 7

Calculated scattering cross sections for the structure in Fig. 2 due to all dipolar components based on source (solid lines) and field (dashed lines) representations.