Ultra-directional forward scattering by individual core-shell nanoparticles

We study the angular scattering properties of individual core-shell nanoparticles that support simultaneously both electric and optically-induced magnetic resonances of different orders. In contrast to the approach to suppress the backward scattering and enhance the forward scattering relying on overlapping electric and magnetic dipoles, we reveal that the directionality of the forward scattering can be further improved through the interferences of higher order electric and magnetic modes. Since the major contributing electric and magnetic responses can be tuned to close magnitudes, ultra-directional forward scattering can be achieved by single nanoparticles without compromising the feature of backward scattering suppression, which may offer new opportunities for nanoantennas, photovoltaic devices, bio-sensing and many other interdisciplinary researches.


I. INTRODUCTION
Investigations into the scattering properties of subwavelength particles are fundamentally important for researches in many directions, including sensing, optical communications, lasers, and many other interdisciplinary fields including medical and biological researches 1-3 .
Among all the demonstrations of efficient scattering shaping involving optically-induced magnetic responses, the directional forward scattering with suppressed backward scattering obtained through overlapping electric and magnetic dipoles of the same magnitude is of fundamental importance 7,9,17,[23][24][25] . This is due to the fact the lowest order dipolar modes are usually easiest to excite, and moreover those modes normally scatter dominantly and thus to some extent can decide the shape of the far-field scattering pattern. Also at the same time, this configuration of overlapping electric and magnetic dipoles exactly corresponds to the original proposal of Kerker 11 and the concept of Huygens source 12 . For some applications, such as nanoantennas 4,5 and photovoltaic devices 6 , ultra-directional forward scattering is required, which nevertheless cannot be achieved by the interferences of solely dipolar responses.
Although it has been demonstrated recently that the directionality can be enhanced through arranging the magneto-dielectric particles in arrays 9,33 , this approach sacrifices the compactness and consequently can not be used for applications such as on chip signal processing that require scattering items of small footprints. So basically ultra-directional scattering with suppressed backward scattering by individual nanoparticles would be highly preferred.
It has recently been shown 27-29 that through exciting efficiently the higher order electric and magnetic modes of dielectric particles, the scattering directionality can be significantly enhanced. Unfortunately both demonstrations to some extent rely on specific incident dipole sources 27-29 . Moreover in both configurations 27-29 , it is not directly clear how to excite the modes selectively and how to control their relative phase and magnitude, which nevertheless is vitally important for further comprehensive and systematic studies.
In this paper we investigate the angular scattering properties of individual core-shell nanoparticles with incident plane waves. The core-shell nanoparticles can be tuned to support simultaneously both electric and optically-induced magnetic modes of different orders.
Based on the mechanism of obtaining forward directional scattering and suppressed backward scattering by overlapping electric and magnetic dipoles of the same magnitude, it is demonstrated that the directionality of the forward scattering can be further improved when higher order electric and magnetic modes are efficiently excited. Since the major contributing electric and magnetic responses can be tuned to almost the same magnitudes, with single core-shell nanoparticles we achieve simultaneously ultra-directional forward scattering and suppressed backward scattering, which may shed new light to fields of nanoantennas, solar cells, bio-sensing and so on.

II. THEORETICAL MODEL WITH IDEAL ASSUMPTIONS
Without losing the generality, we confine our discussions to incident plane waves scattered by spherical nanoparticles, which could be made single-layered or multi-layered [see The scattering can be solved analytically using Mie theory 2,34 and the scattering efficiency (scattering cross section divided by the cross section of the particle) is 2 : 3 where R is the radius of the outmost layer of the spherical particle; k is the angular wave number in the background (it is vacuum in this study); a n and b n are Mie scattering coefficients, which corresponds to n-th order electric and magnetic moments respectively 2,35 . At the same time, the far-field scattering intensity (SI) is 2 : where θ and ϕ are the polar angle and azimuthal angle respectively, as shown in Fig. 1(a), and l is the distance between the observation point of SI and the center of the particle. The expressions for T 1,2 (cos θ) are 2 : n(n+1) [a n π n (cos θ) + b n τ n (cos θ)] T 2 (cos θ) = ∞ n=1 2n+1 n(n+1) [a n τ n (cos θ) + b n π n (cos θ)] Here π n (cos θ) = P 1 n (cos θ)/sinθ, τ n (cos θ) = dP 1 n (cos θ)/dθ and P 1 n (cos θ) is the associated Legendre function of the first kind 2 . It is easy to show that which indicates that π n and τ n , π n and π n+1 , τ n and τ n+1 have opposite parities with respect to cos θ. And in the forward direction θ = 0: According to Eqs. (4)- (5), it is obvious that in the backward direction (θ = 180 • ): As we will show below, those mathematical features play a key role for scattering directionality enhancement and backward scattering suppression.

A. Overlapping electric and magnetic dipoles
According to Eqs. (2)-(3), to obtain azimuthally symmetric (independent of ϕ) scattering patterns, it is required that a n = b n . This means that all the electric and magnetic moments of the same order should have the same amplitudes. If this condition is satisfied, according The main lobe angular beamwidth α is defined as the FWHM of the SI and shown in (a).
to Eqs. (3)-(6), the SI is totally suppressed [SI(π, ϕ) = 0] at the backward direction (θ = 180 • ), and enhanced at the forward direction (θ = 0). Thus we can conclude that for the configuration of plane waves scattered by spherical particles, if the scattering pattern is azimuthally symmetric, the SI at the backward direction should be suppressed. The simplest case of this is that a particle supporting only dipolar electric and magnetic modes of the same magnitude a 1 = b 1 = 0 [a n = b n = 0 (n > 1)], which is exactly what Kerker proposed 11 or the ideal Huygens source 12 . We show the corresponding two-dimensional (2D) and three-dimensional (3D) scattering patterns in Fig. 1(b) (on a scattering plane of any fixed azimuthal angle) and Fig. 1(c) respectively. We cut off part of the 3D pattern for better visibility, as is the case throughout the paper. It is clear that the SI is zero at θ = 180 • and 5 azimuthally symmetric. To characterize intuitively the directionality of the main scattering lobe, we define the full width at half maximum (FWHM) of the SI as the angular beamwidth α, as shown in Fig. 1(b). For this case, the angular beamwidth is approximately 131 • .
B. Overlapping higher order electric and magnetic modes of the same order It is natural to extend the case of overlapping electric and magnetic dipoles of the same magnitude to higher order modes, and according to Eqs. only electric dipole and electric quadrupole of the same magnitude in Fig. 2(a) [2D scattering pattern on the plane of ϕ = 0 (red curve) and ϕ = π/2 (blue curve)] and in Fig. 2(b) (3D scattering pattern). It is obvious that the backward scattering has been effectively suppressed but the scattering pattern is neither azimuthally symmetric, nor has good directionality. We note here that with the assumption of a 1 = a 2 the scattering at the backward direction is not totally suppressed [SI(π, ϕ) = 0]. However, if we assume that 3a 1 = 5a 2 , according to Eqs. (2)-(6) the scattering at the backward directional can be exactly zero. Here we show only the backward scattering suppression through overlapping electric modes of 7 different orders and this principle can certainly apply to magnetic modes of different orders.
According to Eqs. (2)-(4) and the analysis above, it is natural to proposal that through combing the interferences between not only electric and magnetic modes of the same order, but also the modes of different orders, ultra-directional forward scattering, suppression of backward scattering and side scattering lobes can be simultaneously achieved. To demonstrate this, we show in Fig. 2(c) in the red curve the 2D scattering pattern of overlapping dipoles and quadrupoles (both electric and magnetic) of the same magnitude (a 1 = a 2 = b 1 = b 2 = 0), without involving modes of higher orders [a n = b n = 0 (n > 2)].
For comparison, we show also in dashed black curve the 2D scattering pattern of overlapping quadrupoles only as already shown in Fig. 1(d). It is obvious that the side scattering lobes have been effectively suppressed without compromising much of the directionality. We also show in Fig. 2(d) the corresponding 3D scattering pattern, indicating ultra-directional forward scattering with minor side scattering lobes, which is quite different from what is shown in Fig. 1(e). Investigations have also been conducted for overlapping modes up to the third-order of hexapoles [see Fig. 2(e)- Fig. 2(f). The dashed black curve in Fig. 2(e) is the same as that shown in Fig. 1

(f)] and similar conclusions can be drawn.
We should note here that in this section we study modes up to the third order of electric and magnetic hexapoles. The directionality can be further improved with suppressed backward scattering through exciting electric and magnetic modes up to higher orders.

A. Overlapping electric and magnetic dipoles
Based on our analysis and results presented in the section above, now we turn to specific structures for realistic demonstrations. Similar to what is shown in Ref. 9 , we employ the coreshell spherical nanoparticle as shown as the inset of Fig. 3(a). The radii of the core and shell are r 1 and r 2 respectively. Firstly we demonstrate the simplest case of overlapping electric and magnetic dipoles. In Fig. 3(a) we show the scattering efficiency spectra (both total and partial efficiency spectra from electric and magnetic moments of different orders) of a coreshell nanoparticle with a silver (permittivity is taken from the experimental data 36 ) core ( r 1 = 68 nm) and dielectric shell (r 2 = 250 nm) of refractive index n = 2.5 (inset). Apparently  Fig. 3(b) and Fig. 3(c), respectively. The scattering pattern is almost identical to that ideal case shown in Fig. 1(b)-Fig. 1(c), as according to Fig. 3(a) the 9 scattering from other higher order modes is basically negligible.

B. Overlapping electric and magnetic modes up to quadrupoles
As a next step we increase the radius of the Ag core to r 1 = 93 nm and show the scattering efficiency spectra in Fig. 3(d).

C. Overlapping electric and magnetic modes up to hexapoles
At the end we investigate core-shell nanoparticles that supports modes up to the third order of hexapoles. This time we employ dielectric core-Ag shell nanoparticles which can be tuned to support overlapping electric and magnetic hexapoles of the same magnitude.  Fig. 4(b)-Fig. 4(c). For comparison, in Fig. 4(b) we also show by dashed black curve the scattering pattern of only overlapping hexapoles [the same as the curve shown in Fig. 1(f)].
In contrast to Fig. 3(e) where the side scattering lobes have only been partly suppressed, at point D, the side scattering lobes have almost been totally eliminated [ Fig. 4(b)-Fig. 4(c)], which is quite similar to that shown in Fig. 2(e)- Fig. 2(f). This is induced by the fact that besides the hexapoles, the quadrupoles have also been efficiently excited and are of comparable magnitudes [ Fig. 4(a)]. Furthermore, at point D the good directionality has also been preserved, which leads to ultra-directional forward scattering [ Fig. 4(b)-Fig. 4(c)].
Figure 4(d) shows the scattering efficiency spectra of a dielectric sphere [the same as Fig. 3(g) but at a different spectra regime] and obviously that at point E (λ = 0.947 µm) a dominant magnetic hexapole is supported. Its scattering patterns are shown in Fig. 4(e)- Fig. 4(f), which are contrastingly different from those shown in Figure 4(b)-(c), which proves again that through combining the interferences between not only electric and magnetic modes of the same order, and but also between modes of different orders, the scattering directionality can be efficiently improved with the feature of suppressed backward scattering preserved.

IV. CONCLUSIONS
To conclude, we study the scattering configuration of plane waves scattered by individual core-shell nanoparticles which are tuned to support simultaneously both electric and optically-induced magnetic modes of different orders. In contrast to the forward directional scattering and suppressed backward scattering obtained through overlapping electric and magnetic dipoles with the same magnitude, we reveal that through interferencing other higher order electric and magnetic modes which have been efficiently excited, the scattering directionality can be further improved. Since the major contributing electric and magnetic responses can be tuned to close magnitudes, by single nanoparticles we can achieve simultaneously ultra-directional forward scattering, and suppression of extra scattering lobes and backward scattering.
We note here that we confine our discussions to two-layered core-shell nanoparticles, which offers us sufficiently freedom to tune the electric and magnetic modes of a specific order to overlap with the same magnitude and central resonant wavelength [as shown in Fig. 3(a), Fig. 3(d) and Fig. 4(a)]. However for two-layered nanoparticles, we cannot tune the relative strength and phase of all the modes at the same time, which imposes some constraints on more efficient and flexible scattering shaping [e.g., as shown in Fig. 3(d), the contributions from dipoles are too small compared to those from quadrupoles and consequently the side scattering lobes cannot be totally suppressed, as shown in Fig. 3(e)]. We expect that more freedom for resonance tuning (including magnitude and phase) and thus more flexibilities for scattering shaping can be obtained by employing core-shell nanoparticles with more layers.
Moreover we should keep in mind that the ultra-directional forward scattering can be further collimated through exciting higher order modes than hexapoles or arranging the core-shell nanoparticles in arrays. It is worth noticing that the mechanism we reveal in this paper is not constrained to spherical nanoparticles and can certainly be extended to particles of other shapes, which is quite promising for various applications in the fields of nanoantennas, solar cells, bio-sensing and so on.