## Abstract

The interference between conventional multipoles (e.g., electric and magnetic dipole, electric quadrupole, etc.) is known as the cause of unidirectional backward and forward scattering of nanoparticles. However, an unconventional multipole moment, toroidal dipole moment is generally overlooked in the unidirectional scattering. In this work, we systematically investigate the unidirectional scattering in the system of plasmonic nanoparticles. It is found that the toroidal dipole moment can play a significant role in the unidirectional backward scattering. The structural tunability of the unidirectional scattering is also demonstrated. Our results can find applications in the design of nanoantennas.

© 2017 Optical Society of America

## 1. Introduction

Control of the directionality of scattered electromagnetic (EM) waves is of significance to many fields such as nanoantennas [1], photovoltaic devices [2] and sensors [3], etc. The directionality of scattered EM waves is dependent on the excitation of the multipole moments of the scatterer. As is known, for the Rayleigh scattering of a dielectric particle, the scattering pattern is symmetric for the forward scattering (FS) and backward scattering (BS) since only the electric dipole is excited [4]. However, by considering the magnetic dipole or other higher order multipoles involved in the scattering process, the unidirectional scattering can be achieved, based on the destructive or constructive interference of the scattered waves generated from different multipole moments (i.e., electric and magnetic dipole, electric quadrupole, etc.) [5–12]. To be specific, for a single particle made by magnetic materials [5], high-index semi-conducting materials [6–8], topological insulators [9] and spoof localized surface plasmon resonators [10], unidirectional scattering can be realized by utilizing the interference of electric and magnetic dipoles. The unidirectional scattering has also been demonstrated in core-shell particles [11-12] with quadrupoles taken into account.

On the other hand, toroidal moments have received considerable interest in recent years [13]. It has been demonstrated that the toroidal moments are significant but generally neglected in the conventional multipole expansion theory. The omission can be attributed to the fact that the toroidal moment is a high-order term [14] and the corresponding radiation is relatively small for scatterers in the subwavelength scale. Recently, it is shown that the radiation from the toroidal dipole moment can be greatly enhanced in the structures of split-ring resonators [13,15], high index meta-materials [16] and other artificial structures [17] by mimicking a closed loop of magnetic dipoles. Resonant toroidal dipole modes can also be supported by an assembly of plasmonic nanoparticles as long as the electric dipole resonance exists for these particles [18]. The radiation pattern from the toroidal dipole is indistinguishable from that of the conventional electric dipole since the toroidal dipole can be incorporated in a more general definition of the electric dipole as a high-order correction [19]. However, other characteristics of toroidal moments, such as resonance frequency, quality factor and phase of radiation can be much different from those of the conventional electric dipole. Many interesting optical responses induced by the toroidal dipole moment are thus revealed, including, non-trivial optical transparency [20], anapole [21], lasing [22], all-optical Hall effect [23] and nanoantennas [24], etc.

In this paper, we show that the toroidal dipolar excitation can indeed play an important role in the unidirectional scattering of an assembly of nanoparticles. The unidirectional backward scattering can be interpreted by the interference between the toroidal dipole and other conventional multipoles. Furthermore, the enhanced BS and FS of this assembly can be tuned by geometric parameters of the constituent nanoparticles.

## 2. Multipole expansion method

Here, we consider the scattering properties of a system composed of three electric dipoles, which has been proved to be a platform for toroidal moments in a wide range of frequencies [18]. The system under study is shown in Fig. 1. Three plasmonic nanospheres are aligned along the *y*-axis. The center-to-center distance between the neighboring particles is *d*. The radius of the central sphere is *R*_{2}, different from the radii of the side ones (*R*_{1} = *R*_{3}). The background is the free space in our study. The permittivity of plasmonic nanospheres is described according to the Drude model with $\epsilon (\omega )=1-{\omega}_{p}^{2}/({\omega}^{2}+i\gamma \omega )$, where *ω*_{p} is the plasma frequency and γ is the electron scattering rate. We consider an incident plane wave propagating along the positive *y* axis with ${E}^{\text{ext}}={\widehat{e}}_{z}{E}_{0}{e}^{i{k}_{i}y}$, where *E*_{0} is the amplitude, *k _{i}* = ω/

*c*is the wave vector,

*c*is the speed of light in vacuum. The factor of time harmonics

*e*

^{-}

^{i}^{ω}

*is omitted here. The nanospheres can be treated as point dipoles since the radii of nanospheres we consider are much smaller than the incident wavelength. To justify the dipole approximation, the scattering cross section*

^{t}*C*

_{sca}for single plasmonic nanospheres calculated by the Mie theory [4] is shown in Fig. 2(a) for different radii. It is found that the dipolar term

*n*= 1 is dominant in the spectral range we are interested in. The induced dipole moment

**p**

*(*

_{i}*i*= 1, 2, 3) located at

**r**

*can be determined by the following coupled dipole equation [25]:*

_{i}*i*-th nanosphere,

*a*

_{1}(ω) is the electric dipolar term of the Mie coefficients [4], ${E}_{i}^{\text{ext}}$ represents the external electric field on the

*i*-th nanosphere, $\overleftrightarrow{g}({r}_{i}-{r}_{j})$ is the dyadic Green function. The dipole moment for each nanoparticle can be obtained through the above equation. As a result, the total multipole moments for the tri-particle assembly, such as the electric dipole

**P**, magnetic dipole

**M**, toroidal dipole

**T**, electric quadrupole

**Q**

_{e}, and magnetic quadrupole

**Q**

_{m}can also be obtained through the above-calculated point dipoles

**p**

*as follows [18]:electric dipole moment:magnetic dipole moment:toriodal dipole moment:*

_{i}*R*(

*r*) =

*k*

_{0}

^{2}

*e*

^{ik}^{0}

*/*

^{r}*r*is the radial factor of the scattering electric fields,

*r*is the radial distance of the far-field detector,

**F**(θ, ϕ) is the polar (θ) and azimuthal (ϕ) factor of the scattered electric field. The subscripts of

**F**(θ,ϕ) stand for different multipole moments with

**F**

_{p}=**n**× (

**p**×

**n**),

**F**

_{T}=**n**× (

*ik*

_{0}

**T**×

**n**),

**F**

_{m}=**M**×

**n**, ${F}_{{Q}_{e}}=i{k}_{0}n\times \left(n\times {\overleftrightarrow{Q}}_{}^{(e)}n\right)/2$ and ${F}_{{Q}_{m}}=i{k}_{0}\left(n\times {\overleftrightarrow{Q}}_{}^{(m)}n\right)/2,$ Here

**n**=

**r**/

*r*is the unit vector in the radial direction. For the FS or BS, the unit vector

**n**is parallel or opposite to the direction of incident wave vector

**k**

_{i}, respectively. Noted that

**F**

*(θ, ϕ) and*

_{m}**F**

_{Q}_{e}(θ, ϕ) are odd functions of

**n**and their signs will be flipped from FS to BS, whereas

**F**

_{p,T}_{,}

_{Q}_{m}(θ, ϕ) are even function of

**n**and the scattered electric fields generated by these three multipole moments are symmetric for BS and FS.

## 3. Results and discussions

We can investigate the scattered fields in the far field by summing up the contributions from all the multipoles in Eq. (8). The scattering intensity *I* = *E _{s}**

*E*

_{s}versus the incident frequency for the FS and BS are shown in Fig. 2(b). Here, we set

*R*

_{1}=

*R*

_{3}= 20 nm,

*R*

_{2}= 25 nm,

*d*= 60 nm, ω

_{p}= 6.18 eV, and the nanospheres are assumed to be lossless with

*γ =*0. The spectra of scattering intensity for BS and FS are quite different. The maximal ratio of BS to FS intensity appears at the frequency of 0.532ω

_{p}, at which the scattering intensity of FS has a dip. The peak ratio of BS to FS intensity is about 20, which is much larger than the typical maximal ratio of BS to FS for a PEC sphere with 9:1 [29]. The peak ratio of BS to FS is denoted by BS/FS

_{max}(dashed gray line). The results simulated by COMSOL Multiphysics are also shown for comparison. It is noted that there only exists a small deviation of frequency for BS/FS

_{max}between the analytical and numerical results. This small deviation comes from the finite-size effect of nanospheres. The angular distributions of the scattering intensity in the far field are shown in the insets of Fig. 2(c) for three different frequencies indicated by the red dots.

To analyze the underlying mechanism of the unidirectional scattering, the radiation power from different channels of multipole moments are given in Fig. 3(a). The parameters are kept the same as those in Fig. 2(b). Two types of resonant modes in our system are efficiently excited. One mode with resonance frequency ω~0.528ω_{p} possesses a resonant peak of toroidal and electric dipole moment. This has been demonstrated to be a special kind of symmetric mode where the side dipoles are out of phase with the central one [18]. The other is a resonant magnetic dipole mode (resonance frequency ω~0.547ω_{p}). This is an asymmetric mode with the two side dipoles oscillating out of phase while the dipole moment of the central one vanishing. The frequency of BS/FS_{max} locates between the resonances frequencies of these two modes, which is represented by the dashed gray line in Fig. 3(a). It is noted that the radiation power from the total electric dipole moment **P** almost reduces to zero at the frequency of BS/FS_{max}. The corresponding near field distribution of the **E** and **H** field at the frequency of BS/FS_{max} is shown in Fig. 3(b). The electric field has been enhanced remarkably near the surface of spheres. Meanwhile, a “toroidal-like” distribution of the magnetic field is observed clearly, manifesting the significant role that the toroidal moment plays in this unidirectional scattering. For the FS, the scattered electric field **E**_{s} are decomposed into different channels of multipole moments as shown in Figs. 3(c) and 3(d). In Fig. 3(c), the amplitude of **F*** _{T}* +

**F**

_{Q}_{m}is almost the same as that of

**F**

*+*

_{m}**F**

_{Q}_{e}at the frequency of BS/FS

_{max}(|

**F**

*+*

_{T}**F**

_{Q}_{m}|/|

**F**

*+*

_{m}**F**

_{Q}_{e}|~0.9). However, the phase difference between these two pairs, namely, (

**F**

*,*

_{T}**F**

_{Q}_{m}) and (

**F**

*,*

_{m}**F**

_{Q}_{e}), is nearly π at the frequency of BS/FS

_{max}as shown in Fig. 3(d). Note that the scattered electric fields generated by

**F**

*and*

_{T}**F**

_{Q}_{m},

**F**

*and*

_{m}**F**

_{Q}_{e}are in phase for the FS. Thus, they destructively interfere in the far field, leading to the significant suppression of FS. For the BS (not shown), the scattered electric fields generated by these two pairs, (

**T**,

**Q**

_{m}) and (

**M**,

**Q**

_{e}), constructively interfere in the backward direction since the radiation phase of (

**M**,

**Q**

_{e}) are reversed. Thus the BS can be dramatically enhanced in the far field.

Tunable directionality is highly desirable in the design of nanoantennas. In Fig. 4(a), the ratio of BS intensity to FS intensity versus the radius of central nanosphere *R*_{2} and the incident frequency is given. The equal intensity with BS/FS = 1 is marked by the black curves. The directionality of scattered waves of the tri-particle assembly changes as we tune the radius of *R*_{2}. For instance, the angular distribution of scattered light can be switched from primarily FS to BS at the frequency of ω~0.53ω_{p} when *R*_{2} varies from 21 nm to 26 nm. For *R*_{2}>20 nm, the frequency regime for the backward directionality broadens as *R*_{2} increases. Meanwhile, the maximal ratio of BS/FS becomes larger. For *R*_{2}<20 nm, however, the FS is dominant over the whole spectrum. Interestingly, the ratio of BS/FS can be extremely small (less than 0.02) as *R*_{2}~21 nm, represented by the black region in Fig. 4(a), manifesting the significant suppression of BS. In Fig. 4(b), we plot the angular distribution of the scattering intensity in the far field (*y-z* plane) for the points A and B marked in Fig. 4(a). The angular distributions are completely different at these two frequencies. Strong suppression of BS is found at the point A, while significant enhancement of BS is observed at the point B. The directionality here can also be interpreted by the interference of multipoles. The amplitude and phase of scattered electric fields radiated from each multipole moments are shown in Figs. 4(c) and 4(d). At the frequency of point A (orange dashed line in Fig. 4(c)), the amplitude of the scattered electric fields contributed from **P** and **M** (**Q**_{e}) are much larger than those from **T** and **Q**_{m}, therefore we can only consider **P** and **M** (**Q**_{e}) at point A. The phase difference of scattered electric fields generated by **P** and **M** (**Q**_{e}) are very small. Hence, they will constructively interfere in forward direction, yielding strong FS at point A. At point B (gray dashed line in Fig. 4(c)), the scattered electric field induced by **T** is the dominant one. However, the phase of scattered fields generated by **T** (**Q**_{m}) differs by almost π compared to those of **P** and **M** (**Q**_{e}) as shown in Fig. 4(d), leading to the destructive interference in the forward direction. As a result, the strongly enhanced BS can be observed.

The BS/FS ratio as a function of separation distance *d* and incident angle θ_{i} are shown in Fig. 5(a) and Fig. 5(b) respectively. It is found that the unidirectional backward scattering can exist as the separation distance d varies. The maximal ratio of BS/FS increases as d decreases, and the frequency for BS/FS_{max} undergoes a red shift. For different incident angle θ_{i}, we calculated the ratio of far-field scattering intensity in the –*y* direction to that in the *y* direction. The unidirectional scattering also exists in a wide range of incident angles as illustrated in Fig. 5(b). The scattering pattern in the far field for different incident angles θ_{i} are shown in Figs. 5 (c) and 5(d) for the frequency ω = 0.531ω_{p} and ω = 0.538 ω_{p}, respectively. It is found that small incident angle is preferred for the unidirectional scattering.

The plasmonic nanospheres can be embedded in dielectric background of glass, or TiO_{2} etc. In Fig. 6(a), we show BS/FS for the system embedded in different background medium. The frequency for unidirectional backward scattering has a red shift as ε_{b} increases. To achieve large toroidal moments, the optimization of *R*_{1} and *R*_{3} is needed. In the quasi-static limit, the pure toroidal resonance occurs at *R*_{2} = (15/8)^{1/3}*R*_{1}≈1.23 *R*_{1} which can be derived analytically [18]. However, deviation exists as the retardation effect is considered. In Fig. 6(b), BS/FS for different radii (*R*_{1}, *R*_{3}) of side nanospheres are shown, where we fix *R*_{2} = 25 nm and *d* = 60 nm. It is found that the maximum of BS/FS increase as the radii of side nanospheres decrease, meanwhile, a blue shift of unidirectional backward scattering takes place. The plasmonic dissipation can reduce the performance of toroidal dipole in the assembly of plasmonic nanoparticles. Compared with the lossless case with *γ =* 0, the maximum of BS/FS decreases as we take *γ =* 0.003 ω_{p}. For silver with realistic permittivity [30], the ratio of BS/FS are given in Fig. 6(c), where we set the refractive index of the background medium *n*_{b} = 2.4 (e.g., TiO_{2} in the visible frequency), *R*_{1} = *R*_{3} = 15 nm, *R*_{3} = 25 nm and *d* = 45 nm. The maximal BF/FS is about 3. The corresponding radiation spectra for different multipole moments are shown in Fig. 6(d). The line shape is quite similar to those in Fig. 3(a). Again, the toroidal dipolar excitation plays a key role in the unidirectional scattering. The assembly of nanospheres can be realized experimentally by capillary-driven self-assembly method [31] or other techniques [32-33]. Since we use point dipole model in our study, the nanospheres can also be replaced by nano-disks [18,34] or other kind of nanoparticles.

Note that the unidirectional backward scattering has also been revealed in many systems [7, 11, 35-36] at visible frequencies. In experiments, the maximal BS/FS is about 2 for single silicon nanospheres [7] and gold-silicon nanosphere dimers [35]. Theoretically, it can reach about 10 for silicon sphere dimers [36] and 12 for metallic-dielectric core-shell spheres [11]. In the system of plasmonic nanospheres, the maximal ratio of BS/FS can reach ~50 if no loss is considered. The maximal BS/FS can remain in the order of 10 for *γ~*0.003ω_{p} with loss taken into account. The ratio of BS/FS_{max} can be further enhanced by optimizing parameters (such as the shape of nanoparticles, separation distance etc.). Meanwhile, the loss may be compensated by gain media in order to further enhance the ratio of BS/FS [37].

## 4. Conclusion

In summary, we study the unidirectional scattering for the system of three plasmonic nanoparticles. The asymmetric scattering in the forward and backward direction is analyzed and illustrated by the estimation of the radiation from all the multipole moments. We find that in the unidirectional scattering, the toroidal dipole moment can be significant and work together with other conventional multipole moments. The maximal ratio of BS to FS can be interpreted by the fact that the scattered fields of **T**, **Q**_{m} are out of phase with those of **M**, **Q**_{e} (in some case, **P** is not negligible and should be taken into account) in the forward direction, while they are in phase in the backward direction. Theoretically, the unidirectional scattering for three electric dipoles can also be interpreted by analyzing the phase and magnitude of each electric dipole. However, estimation of total multipole moments of the whole system serves as a simple way to achieve the unidirectional scattering, especially for the case of large number of nanoparticles or other complicated structures. The unidirectional scattering may also be found in other artificial structures which support toroidal dipole moments, providing another mechanism for the unidirectional scattering. Our study of the unidirectional scattering may find not only potential applications in the design of nanoantennas, but also theoretical interest in the investigation of EM properties of the elusive toroidal moment.

## Funding

National Natural Science Foundation of China (NSFC) (Grant No. 11574037); Fundamental Research Funds for the Central Universities (Grant No. CQDXWL-2014-Z005 and 106112016CDJCR301205); Key Laboratory of Micro- and Nano-Photonic Structures (Ministry of Education).

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