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

Abstract

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.function of the first kind, and  

Electromagnetic toroidal excitations in matter and free space

“…Importantly, since toroidal and electric multipoles are identical in terms of far-field scattering and their difference in the near field (due to non-vanishing radial electric field) is not recognized by the boundary conditions, their contributions in a l,m are mixed together and cannot be separated without the knowledge of the actual charge-current distribution, yielding the so-called renormalized electric multipoles 50 . Thus, unlike the multipole expansion, Mie theory offers merely a mathematical description of the scattering problem providing little physical insight.…”

“…A metamaterial of toroidal topology can be engineered in such a way that the lower order electric and magnetic multipoles cannot be excited due to symmetry, allowing thus the toroidal dipole to become the leading term in the multipole expansion and to contribute strongly to the electromagnetic properties of the metamaterial 10 . When scattering contributions of the conventional multipoles are dominant, toroidal response of the nanostructure might still be detected through optical forces 50 . Even for small, deeply sub-wavelength systems, such as atoms and molecules, one can expect that toroidal dipole excitations should be observable, in principle, given that effects of the same order, for instance magnetic quadrupole transitions 52 , are also experimentally accessible.…”

“…This paradox has been recently brought to light by Miroshnichenko and co-workers, who experimentally studied light scattering by Si nanodisks in the visible part of the spectrum 49 . Ignoring the role of the toroidal multipoles in the physical picture may also have implications for understanding the optical force as a consequence of external fields interacting with nanostructures 50 …”

“…Dynamic toroidal dipoles, in contrast to their static counterparts, interact with oscillating electromagnetic fields, and thus contribute to the optical properties of materials across the entire electromagnetic spectrum 3 . However, the inclusion or omission of the dynamic toroidal multipoles from the multipole expansion has been the subject of an on going discussion [46][47][48] , in particular with respect to equivalent descriptions, such as Mie theory 49,50 . We address this in detail in a dedicated section (see tutorial inset II), where we demonstrate that although both the vector spherical harmonic Mie expansion and the (full, including toroidal contributions) charge-current multipole expansion are complete, the toroidal multipoles appear as distinct entities only in the latter.…”

Tuning toroidal dipole resonances in dielectric metamolecules by an additional electric dipolar response

“…Besides, TD can support many other intriguing phenomenon, such as the ideal magnetic dipole [9], optical activity [10], optical transparency [11,12], and ultra-high energy density [13,14]. Meanwhile, TD has been widely implemented to enhance light-matter interactions like all-optical Hall effect [15], non-linear laser generation [16], and optical force [17]. Thus, TD mode is of great importance in the thriving fields of metamaterials and nanophotonics.…”