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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.

The notions of toroidal multipoles have been widely used in different disciplines of physics [1][2][3][4][5][6][7]. The simplest model for a toroidal dipole can be visualized as current flowing on the surface of a torus along its meridians. Such complex current distributions are difficult to excite but thanks to recent advances in material fabrication [8][9][10], the toroidal metamaterials were theoretically predicted [11] and experimentally demonstrated at microwave frequencies [12], followed by various studies on their intriguing properties [13][14][15][16][17][18][19]. The need for introducing toroidal multipoles is due to the fact that a general type of the charge/current distribution, either free or induced, cannot be represented exclusively by the standard electric and magnetic multipoles as defined in textbooks. From the mathematical point of view, the current is a vector with three independent components, while the charge is a scalar.
With the continuity equation, one needs three functionals to characterize a general type of source, corresponding to three families of multipoles. This constitutes the source-representation, so termed as it focuses on parametrizing the source, either free or induced charge/current. On the other hand, a divergenceless vector function can be expanded in terms of two sets of multipole fields, termed electric and magnetic multipole fields described by the vector spherical functions [20], with the expansion coefficients designated as electric and magnetic multipole coefficients (see, e.g., [21]).
This forms the field-representation since it focuses on the radiated fields. The electromagnetic fields in the source-free region can be defined in the field-representation based on electric and magnetic multipoles, without resorting to the third type of toroidal multipoles [20][21][22][23]. This naturally leads to a question as to whether toroidal excitation is indispensable to characterize the electromagnetic fields. This issue has been largely ignored since the toroidal excitation is typically obscured by much stronger electric or magnetic multipole excitations. But with more and more "toroidal excitations" being reported in the literature, it is highly desirable that we can reconcile the source-representation and the field-representation.
We clarify this issue by studying the optical forces acting on nanostructures that support toroidal excitations. On one hand, the optical force can produce a measurable physical consequence that has been extensively employed to manipulate particles [24][25][26][27][28][29][30][31][32]. On the other hand, it can be theoretically calculated either, in the source-representation, by the interaction between the external fields and the three families of multipoles induced on the object or, in the field-representation, through the coupling between the incident fields and the two types of multipoles excited on the scatter. Hence, it serves as a good platform to address the aforementioned confusion.
While the toroidal dipole is necessary for the description of optical forces in the source-representation and may overwhelmingly dominate the optical force when toroidal dipole resonance is excited, its role is entirely attributed to the renormalized electric dipole resonance in the field-representation. More importantly, we will show that optical force can serve as a definitive quantity for the detection of toroidal response in nanostructures even when its scattering intensity is overwhelmed by the conventional electric and magnetic multipoles.
We start by considering an object illuminated by a time-harmonic incident wave.
The time-averaged optical force can be written as an integral over a closed spherical surface S∞ with radius RS → ∞ [27,30], where c is the speed of light in vacuum,  (2) which are the Cartesian components of the primitive electric dipole moment (p), magnetic dipole moment (m), electric quadrupole moment (q (e) ), and magnetic quadrupole moment (q (m) ), respectively. These primitive multipoles are actually not widely used because they are non-traceless and asymmetric. One usually extracts the irreducible parts from the primitive multipoles, and attributes the remaining terms to the toroidal multipoles, leading to the irreducible electric and magnetic multipoles, with a few lowest order moments given by i p and i m in Eq. (2), and With the multipole moments determined by the induced current density, Eq. (1) can be written as a sum [27,30] of the incident forces F p , F m , and F t , due to the direct interaction between the incident fields and the irreducible electric, magnetic and toroidal dipoles (p, m, t), and the recoil force F int [30] Re + Im , 12 12 where k and ε0 are, respectively, the wave vector and the free space permittivity. Here, we are only concerned with the optical force in the k-direction and we focus only on the dipole terms in Eq. (4) because they dominate the force acting on a sub-wavelength nanostructure.
In the field-representation, one starts by expanding the scattered field directly in terms of the outgoing and divergenceless vector spherical functions      They are therefore termed "renormalized dipoles".
Inserting the scattered fields into Eq. (1), we can similarly decompose the optical force into incident forces p F and m F , arising from the direct interaction between incident field and the renormalized dipoles, and the recoil force int F , due to the interference of the renormalized dipoles, with its k-(z-) component written as [32,35]   We therefore see that the optical force acting on sub-wavelength nanostructures can be written as a summation of forces acting on the toroidal dipole as well as the irreducible electric and magnetic dipoles, or alternatively, interpreted as a summation of forces exerting exclusively on the renormalized electric and magnetic dipoles. We will identify the relationship between the irreducible and renormalized dipoles by performing a numerical study on toroidal nanostructures.
We consider a toroidal nanostructure shown in Fig. 1 , and they will collectively support a toroidal dipole resonance. The incident wave is a linearly x-polarized plane wave propagating in the z-direction. In the optical regime, the permittivity of gold is described by the Drude model with plasma frequency ωp = 2π×2.175×10 15 rad/s and damping frequency ωc = 2π×6.5×10 12 rad/s [36]. We first use the source-representation, in which case the response of the nanostructure can be described by the irreducible dipoles and the toroidal dipole. The dipole moments can be calculated (Eqs. (2) and (3)) using the current density obtained with a finite-element method package COMSOL [37]. The calculated scattering cross sections (see Supplemental material) for these induced dipole moments are shown in Fig. 1 respectively. The current density distribution for the resonance at ~71 THz is illustrated in Fig. 1(b), where the threading of the magnetic field lines (black lines) around the nanostructure confirms the excitation of a toroidal dipole resonance [12].
We show the z-component of the optical forces (Eq. (4)) in Fig. 1(d), with two peaks at the frequencies coinciding with those of the scattering cross sections in Fig. 1(c). We now use the field-representation. We first calculate the partial scattering cross sections (see Supplemental material) from each renormalized dipoles and the results are shown in Fig. 2 Figure 2(b) shows that the strong enhancement of force at the resonance of ~71 THz is contributed by z F p , coming from the interaction between the incident field and the renormalized electric dipole moment, while the same resonance force in the source-representation has been interpreted as a summation of the contributions from the irreducible electric dipole and the toroidal dipole, with major contribution coming from the toroidal dipole in Fig. 1(d). The resonance at ~64 THz in Fig. 2(b) can be attributed to the renormalized magnetic dipole, in agreement with the interpretation in Fig. 1(d). By comparing the force contributions between the two representations (see  , with n denoting the radial direction [30]. At the far-field, one cannot tell whether the radiation field comes from an irreducible electric dipole or a toroidal dipole, but only find a renormalized electric dipole instead. We also note that the similarity of the far field radiation pattern of the irreducible electric dipole and the toroidal dipole was very recently exploited to produce 'anapoles' [38].

Figs. 1(d) and 2(b)), we can infer that 
In general, toroidal resonances are weak and their contributions to the scattering power are usually obscured by the much stronger irreducible dipoles. In this case, optical force can serve to detect the toroidal response. To illustrate this point, we plot in Fig. 3 the sum of the scattering cross sections due to all dipolar components (i.e. x, y, z) of the nanostructure shown in Fig. 1(a). The results based on the source-representation agree well with those based on the field-representation. While the effect of the toroidal dipole on the total scattering cross section is barely noticeable (the small peak of sca C  pt or sca C p at ~71 THz), it gives the dominant contribution to the optical force at ~71 THz (see Fig. 1(d)). The strong sca We note that our theory can also be applied to other toroidal structures [12,13].

Supplementary material for "Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole"
Xu-Lin Zhang, 1,2 S. B. Wang, 1 Zhifang Lin, 3,4 Hong-Bo Sun, 2 and C. T. Chan 1,* where ω is the angular frequency. The scattering cross section can be determined by  (2) For renormalized dipoles: To determine the scattering cross section for the renormalized dipoles, we consider the vector spherical harmonics defined by Bohren and Huffman [3], where even and odd modes were distinguished. The corresponding scattering coefficients can be worked out using the relations: where the subscript "e" and "o" denote even and odd, respectively, and       With the help of these scattering coefficients, the scattering cross section can be evaluated by [3]    

II Calculation of optical forces based on field-representation
In this part, we give a detailed description for calculating the optical force using the multipole expansion based on vector spherical functions.