1. Introduction

Scattering of light by small particles has many applications in several areas of physics, such as astrophysics, spectroscopy, nuclear physics, meteorology, optical communications, sensing, and biophysics [1,2]. Moreover, plasmonic nanoparticles are able to confine light in subwavelength volumes and, consequently, to achieve huge local field enhancement at specific (resonance) frequencies. These frequencies can be tuned over a relatively broad spectrum by changing the materials and geometrical parameters involved [3]. These properties render these so-called particle plasmon polaritons suitable for enhancing light-matter interaction.

For many applications, control of the scattering directionality represents an important aspect. Kerker et al. predicted theoretically that is possible to obtain both, zero-backward and near-zero-forward scattering (Kerker conditions) via hypothetical spheres with identical electric permittivity and magnetic permeability [4–6]. More precisely, the Kerker conditions rely on the interference of electric and magnetic dipole resonances [7,8] and can thus also be satisfied by small dielectric particles of sufficiently high refractive index [9,10]. Similarly, a generalized Kerker condition based on electric dipole and electric quadrupole resonances has been reported [11]. In addition, directional light scattering can be realized via magneto-optical materials [12–14], all-dielectric [15] and magneto-electric [16] core-shell spheres. While in these systems, zero-backward-scattering and near-zero-forward scattering occur for a narrow spectral region only, broadband zero-backward and near-zero-forward scattering has recently been demonstrated for metallo-dielectric core-shell particles that operate beyond the dipole limit [17]. Moreover, broadband suppression of backscattering at optical frequencies using wavelength-sized dielectric spheres with low permittivity has been reported [18].

However, for many applications based on multiple light scattering, preferential backscattering is generally desired. In the case of backscattering, the average cosine of the scattering angle, the so-called asymmetry parameter $g$ is negative. In fact, when the radius of an ordinary scatterer is much smaller than the incoming wavelength, the scattering is essentially dipole-like and hence $g\sim 0$. For ordinary Mie scatterers of radius equal or longer than the incident wavelength, the scattering is mostly in the forward direction so that $g\sim 1$. Therefore, negative $g$ hardly occurs in ordinary scatterers. Nonetheless, it has been shown that magnetic particles with large values of the permeability [19] and lossless dielectric nanospheres with large values of the dielectric constant (silicon or germanium) [20] exhibit negative $g$ in the infrared region. In addition, the application of an external magnetic field breaks the scattering isotropy of magneto-optical cylinders in the dipole regime leading to preferential backward scattering [21]. Finally, specific clusters composed of short-range correlated scatterers exhibit negative values of the asymmetry parameter [22] and these characteristics have been associated with Bragg-like scattering [23]. Specifically, negative values of the asymmetry parameter lead to an unusual multiple-light scattering regime in which the scattering mean free path exceeds the transport mean free path [20].

In this work, we present an alternative strategy to achieve preferential backscattering by using composite plasmonic nanoparticles with core-shell morphology [24]. We demonstrate that attaining negative $g$-values relies on the interference of the hybridized surface plasmon polariton modes that form at the respective metal-dielectric interfaces. Specifically, we propose simple design rules for achieving preferential backscattering, combined with strong light-matter interaction.

2. Theory

The analytical expression for the backscattering efficiency of a spherical scatterer is [1]

The normalised total light power scattered in all directions is expressed by the scattering efficiency [1]:

For low concentrations of identical scatterers, multiple scattering theory [26] relates the transport mean free path $l_{\mathrm {t}}$ and the extinction mean free path $l_{\mathrm {ext}}$ to the single-particle scattering and extinction efficiencies, $Q_{\mathrm {sca}}$ and $Q_{\mathrm {ext}}$, and the asymmetry parameter $g$ according to

In essence, such strong reductions of the transport mean free path via negative asymmetry parameters lead to more pronounced multiple scattering processes relative to extinction-related processes and this is essential for many applications. For instance, the fabrication-tolerant broad-band operation of random spectrometers [27] requires a well-developed diffusive regime of light propagation in the spectrometers’ active area. This is tantamount to stating that the size of the spectrometers’ scattering area must exceed the transport mean free path. In the ordinary scattering regime where $l_{\mathrm {t}} > l_{\mathrm {ext}}$, this also means that considerable extinction is incurred and, with a view on the dynamic ranges of many available (and affordable!) detectors, this may seriously limit the device operation. In the anomalous scattering regime with negative asymmetry parameter where $l_{\mathrm {t}} < l_{\mathrm {ext}}$, extinction is a lesser concern. As a consequence, considerable design freedom is obtained which can be utilized in several ways, ranging from the use of cheaper detectors via smaller device footprints all the way to spectrometers with enhanced sensitivity.

3. Results and discussion

To start with, we consider a homogeneous sphere of radius $S= 100$ nm with relatively low permittivity $\epsilon =4$ and permeability $\mu =1$, in air. The scattering efficiency of this particle in the visible region is characterized by two pronounced peaks, at the blue end of the spectrum, stemming from the dipolar Mie resonances which are a combination of magnetic and electric type (Fig. 1(b)). Due to the interference between the magnetic and electric dipole moment, the asymmetry parameter is positive corresponding to scattering in the forward direction. Moreover, the magnetic and electric dipole moment overlap at 350 nm and 459 nm, where the first Kerker condition is fulfilled $(a_{1} = b_{1})$. Especially at the long wavelength side (459 nm), where the interference is maximum, a peak ($g=0.55$) in the asymmetry parameter spectrum is observed (Fig. 1(c)). However, moving to shorter wavelengths, second-order multipole contributions become significant. It is worth noting that third- and even higher-order multipoles ($n=3,\ldots$) are negligible due to the size of the scatterer relative to the incident wavelength.

 

Fig. 1. Panel (a): Scattering efficiency $Q_{\mathrm {sc}}$ of a dielectric sphere of radius 100 nm with permittivity $\epsilon =4$, in air, and the relative (with respect to the incident plane wave) electric field amplitude distribution in the plane of polarization, at the resonance wavelength. Panel (b): Electric (red lines) and magnetic type (blue lines) Mie coefficients for n=1 (solid lines) and n=2 (dashed lines). Panel (c): Asymmetry parameter $g$ and the contribution of the corresponding terms (Eq. (4)). Inset: Backward (black line) and forward (red line) scattering efficiency.

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We now consider a composite particle composed of a spherical core of radius 90 nm and $\epsilon =4$, $\mu =1$, coated with a concentric spherical silver shell, 10 nm thick. In this case, since there are two dielectric-metal interfaces, particle- and cavity-like surface plasmon polariton modes are formed at the outer and inner surfaces of the shell, respectively. These modes interact with each other and give rise to hybridized surface plasmon polariton modes, by analogy to the formation of bonding and antibonding electron states of a molecule by symmetric and antisymmetric linear combination of the atomic orbitals, respectively [28–30]. The short wavelength antibonding hybrid mode is mostly localized around the shell (particle-like), while the long wavelength bonding hybrid mode extends into the core region (cavity-like). These particle- and cavity-like dipoles as well as the quadrupole surface plasmon polariton modes manifest themselves as peaks in the scattering cross section (Fig. 2(a)). In our calculations, we employ the actual optical constants of silver, deduced from the experiment [31]. As shown in Fig. 2(b), the long wavelength surface plasmon polariton modes are mainly of electric type (red lines), while the short wavelength particle-like surface plasmon polariton mode is a combination of electric and magnetic type. The interference between the electric- and magnetic-type Mie coefficient for $n=1$ at 674 nm gives rise to negative asymmetry parameter as we observe from the corresponding term $\Re ( a_{n}b^{*}_{n})$ (Fig. 2(c)). Moreover, another mechanism takes place at the same wavelength region. The overlap of the electric dipole and electric quadruple moment enhances the backscattering and the asymmetry parameter exhibits a minimum $g=-0.51$ at 640 nm, which implies $l_{\mathrm {t}}=0.81l_{\mathrm {ext}}$, for a low density dispersion of such scatterers for which the independent scatterer approximation of multiple scattering theory [26] is applicable.

 

Fig. 2. Panel (a): Scattering efficiency $Q_{\mathrm {sc}}$ of a composite nanoparticle, composed of a dielectric core of radius 90 nm and permittivity $\epsilon =4$, coated with a concentric silver shell, 10 nm thick, in air, along with the relative (with respect to the incident plane wave) electric field amplitude distributions in the plane of polarization, at the three resonances. Panel (b): Electric (red lines) and magnetic type (blue lines) Mie coefficients for n=1 (solid lines) and n=2 (dashed lines). Panel (c): Asymmetry parameter $g$ and the contribution of the corresponding terms (Eq. (4)). Inset: Backward (black line) and forward (red line) scattering efficiency.

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More generally, it is worth noting that in the wavelength range where negative $g$ occurs, the backward scattering efficiency is higher than the forward scattering efficiency $Q_{\mathrm {b}}> Q_{\mathrm {f}}$ (Fig. 2(c)). Moreover, the backscattering occurs at wavelengths where the scattering efficiency is relatively low. By adjusting the geometrical parameters, like the thickness of the shell, we can achieve backscattering combined with strong light-matter interaction. In Fig. 3, we display a systematic variation of the surface plasmon polariton modes and the corresponding asymmetry parameter with respect to the shell thickness. By increasing the shell thickness, the resonances come closer and the antibonding surface plasmon polariton modes at lower wavelengths are more pronounced due to the weaker plasmon hybridization. In this way, we can achieve stronger scattering at those wavelengths where the backscattering occurs. It is worth noting that for thicker shells, the asymmetry parameter takes on smaller negative values due to the destructive interference between $\Re ( a_{n}b^{*}_{n})$ and $\Re ( a_{n}a^{*}_{n+1})$. Therefore, the choice of the shell thickness depends on the desired characteristics. Thick shells lead to a weaker backscattering combined with relatively high scaterring efficiency, while thinner shells exhibit stronger backscattering and weaker light-matter interaction.

 

Fig. 3. Panels (a), (b) and (c): Scattering efficiency, Mie coefficients and asymmetry parameter of a composite nanoparticle, composed of a dielectric core of radius 80 nm and permittivity $\epsilon =4$, coated with a concentric silver shell, 20 nm thick, in air, respectively. Panels (d), (e) and (f): Scattering efficiency, Mie coefficients and asymmetry parameter of a composite nanoparticle, composed of a dielectric core of radius 70 nm and permittivity $\epsilon =4$, coated with a concentric silver shell, 30 nm thick, in air, respectively. In both cases the total radius of the scatterer is $S = 100$ nm. Insets: Backward (black line) and forward (red line) scattering efficiency.

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A similar analysis can be carried out for 2D cylindrical scatterers and delivers similar results. We assume an infinite cylinder of radius $S = 100$ nm, composed of a dielectric core ($\epsilon = 4$), coated with a concentric cylindrical silver shell, in air. In Fig. 4, we display the results of a systematic variation of the scattering efficiency and the asymmetry parameter with respect to the shell thickness. In all cases, we observe that due to the interference of the surface plasmon polariton modes, the asymmetry parameter takes on high positive and negative values. Consequently, by adjusting the thickness of the shell, we can obtain high directional scattering also in 2D cylindrical scatterers.

 

Fig. 4. Panels (a), (b) and (c): Wavelength dependence of the scattering efficiency (black solid lines, left ordinate) and asymmetry parameter (red dotted lines, right ordinate) of infinite dielectric cylinders, coated with a concentric silver shells of thickness $D=30$ nm, $D=20$ nm, and $D=10$ nm, respectively. In all cases, the radius of the composite scatterer is $S = 100$ nm.

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We are thus in a position to formulate the aforementioned design rules: In the visibly frequency range, simple lossless dielectric nanoparticles ($R \sim 100$ nm) with relatively low values of the permittivity $(\epsilon \le 4)$ such as glass or polymer spheres or cylinders provide only forward scattering. By coating them with a metallic shell, we obtain scattering in the backward direction. In order to achieve significant backscattering, the nanoparticles should be relatively small (with respect to the incident wavelength) so that the Kerker conditions are fulfilled for $n=1$ and $n=2$ and the higher-order contributions are washed out. Moreover, the thickness of the metallic shell should be tuned properly so that the interference of cavity- and particle-like hybrid plasmon polariton modes produces negative values for the asymmetry parameter $g$. Finally, we can further tune the shell thickness in order to adjust the relative importance of asymmetry and scattering strength.

4. Conclusions

In summary, using Mie theory, we have demonstrated the existence of preferential backscattering by a subwavelength composite nanoparticle with a dielectric-metal core-shell morphology. Such composite nanoparticles offer a versatile platform for engineering hybrid surface plasmon polariton modes. Moreover, the interaction of the particle- and cavity-like modes gives rise to enhanced backward scattering. By properly adjusting the thickness of the shell, we can achieve high backward directional scattering combined with strong overall scattering (i.e., strong light-matter interaction). The negative asymmetry parameter leads to an anomalous scattering regime where the transport mean free path is reduced below the extinction mean free path for a dilute suspension of scatterers. Our results will be useful for many applications such as sensing and spectroscopy.