1. Introduction

Efficient control of optical radiation at nanoscale dimensions is crucial for future light-on-chip integration. One of the interesting topics is the use of small particles to engineer specific directions of scattered light [1–3], as applied to solar energy [4], light emitting devices [5,6] and nanoantennas [7–9]. Localized surface plasmonic resonances in individual metal nanostructures have been extensively studied as they can reduce free-space light to sub-wavelength levels [10,11]. However, to achieve the aforementioned goal of controlling the directions of incident light, a specific combination of several metal nanostructures must be applied [12–14]. Furthermore, all metallic nanostructures have high metal loss, which limits their size scalability in practical applications, there by making them undesirable in signal generation and transmission [15].

One approach to avoid this limitation and still maintain similar resonant characteristics is to use high refractive index dielectric materials [16–19]. An individual high refractive index dielectric nanosphere can simultaneously exhibit electric dipole resonance (ED) and magnetic dipole resonance (MD) [20–23]. The optically induced magnetic response in a single sphere originates from a circular displacement current driven by the incident electric field, which generates a magnetic moment perpendicular to the incident electric field [24–29]. The ED interferes with the MD of the nanoparticles, which can inhibit backward scattering in a single nanosphere [23, 30–32]. Recently, the scattering properties of a single Ge nanosphere in the visible region have been reported [23]. As compared with the resonance wavelength range and peak magnitude of ED and MD of a single Si nanosphere, those of a single Ge nanosphere are more similar; thus, a single Ge nanosphere can achieve high scattering efficiency while also achieving the zero backward scattering [23]. Furthermore, Ge has a higher refractive index and extinction coefficient in the visible range as compared with Si, and the order of magnitude of its third-order nonlinear optical coefficient is also greater than that of Si. These factors make Ge more efficient when used in photothermal [33] and nonlinear optical materials [34]. However, the unidirectionality of a single nanosphere can only be achieved at a single wavelength. Several studies have proposed heterogeneous nanostructures composing metal and Si to optimize the scattering properties [35–41], such as dimers and core-shell nanoparticles. Technological advances have led to new possibilities for fabricating core-shell nanoparticles with several layers [42]. The core-shell nanaoparticles are capable to enlarge the wavelength range of the zero backward scattering and increase the scattering efficiency while achieving zero backward scattering [35–38,41]. Thus, the scattering properties of the core-shell nanostructures based on high refractive index dielectric Ge together with highly optimized features have not yet been studied.

Herein, we propose all-dielectric core-shell SiO₂-Ge-SiO₂ nanoparticles that can simultaneously achieve the broadband zero backward scattering and enhanced forward scattering. The core-shell SiO₂-Ge-SiO₂ nanoparticles satisfy the generalized Kerker condition beyond the dipole approximation. The zero value of the backward scattering efficiency can be achieved owing to the interferences involving of dipole and higher-order electromagnetic resonance modes. The zero backward scattering is efficient because the resonance wavelength ranges of ED and MD are located near the zero backward scattering. We also demonstrate that zero backward scattering with a broad wavelength range is achieved through optimization of the structural parameters. Moreover, we study the relationship between the spectral position of the broadband zero backward scattering and the thickness of each layer. Our results show promising potential for the development of nanoantennas, biosensors, and photovoltaic devices.

2. Geometry and theoretical model

Figure 1 illustrates the studied core-shell SiO₂-Ge-SiO₂ nanoparticle and the related coordinate systems where R₁ is the radius of the SiO₂ core; R₂ is the inner part of the nanoparticle comprising the SiO₂ core and Ge layer; R₃ is the radius of the entire nanoparticle, t₁=R₂−R₁ is the thickness of the Ge layer and t₂=R₃−R₂ is the thickness of the outer SiO₂ layer. The direction of the incident light and polarization is along the z-axis and x-axis, respectively. The angle between the incident and scattered light is the scattering angle θ whereas the angle between the projection of the scattered light on the x–y plane and the x-axis is the azimuth angle ϕ. Gray represents Ge, and red represents SiO₂. Light scattering of spherical particles in free space can be explained using the Mie theory; the scattering cross section is defined as the ratio of the energy of light scattered by a particle per unit of time to the intensity of the incident light:

Equation (3) indicates that, when nanoparticles are effected only by a dipole (n = 1) mode, the backward scattering efficiency can be written as follows:

 

Fig. 1 Geometry of the core-shell SiO₂-Ge-SiO₂ nanoparticle.

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3. Results and discussions

We first demonstrate the scattering properties of the core-shell SiO₂-Ge-SiO₂ nanoparticles with R₁ = 58 nm, t₁ = 73 nm and t₂ = 182 nm. The refractive index of Ge is derived from Palik [43] and the refractive index of SiO₂ is set to 1.55. Figure 2(a) shows the backward and forward scattering efficiency spectra of the core-shell SiO₂-Ge-SiO₂ nanoparticles. Zero backward scattering is clearly attained at the wavelength of λ = 720 nm, as indicated by the vertical purple dashed line. The red solid line represents the forward scattering efficiency, showing that when zero backward is achieved, highly enhanced forward scattering can also be gained. Notably, the Q_f/Q_b ratio cannot be used to characterize the scattering in our case because the ratio will be infinite as a consequence of the zero value of Q_b. To analyze the basic mechanisms behind the zero backward scattering achieved by the core-shell nanoparticles at λ = 720 nm, the contributions of the Mie multipole decomposition related term (n = 1,2,3) to the total scattering efficiency are given in Fig. 2(b). As shown by the vertical purple dashed line, the contributions of the dipole modes in addition to the quadrupole modes to the total scattering efficiency can not be ignored at λ = 720 nm. However, the primary effect is due to ED and MD, wherein, as indicated by the curves corresponding to ED (red solid line) and MD (blue solid line), the resonance range and peak magnitude of these two components are quite similar at the wavelength of λ = 720 nm. It should be noted that the contributions of the quadrupole modes are much weaker than the dipole modes, indicating that the zero backward scattering is mostly the result of the interference of ED and MD; the interference of EQ and MQ plays a secondary role. Furthermore, because the wavelength position of the zero backward scattering is located at the wavelength region of ED and MD, the total scattering efficiency reaches a high value at the wavelength of λ = 720 nm, as indicated by the black solid line in Fig. 2(b), thereby implying that this directional scattering is efficient. However, although the resonance wavelength ranges of the ED and MD overlap, the peak magnitude of ED is larger than that of MD and they do not intersect at λ = 720 nm; thus, the traditional first Kerker condition is not satisfied because of a₁ ≠ b₁. In the following section, we demonstrate that the zero backward scattering results from the generalized Kerker effects, which considers the quadrupole modes.

 

Fig. 2 Calculated scattering properties of the core-shell SiO₂-Ge-SiO₂ nanoparticles with R₁ = 58 nm, t₁ = 73 nm and t₂ = 182 nm. (a) Forward and backward scattering efficiencies spectra. (b) The contribution of the Mie multipole decomposition term to the total scattering efficiency; a₁ (ED) and b₁ (MD) for the dipole term; a₂ (EQ) and b₂ (MQ) for the quadrupole term; and a₃ and b₃ for the octupole term. (c) Far-field scattering patterns of the nanoparticles at 720 nm. (d) Two-dimensional (2D) angular distributions of the nanoparticles at 720 nm.

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To show the zero backward scattering at λ = 720 nm vividly, the far-field scattering patterns and the corresponding two-dimensional (2D) angular distributions of the core-shell SiO₂-Ge-SiO₂ nanoparticles at λ = 720 nm are shown in Figs. 2(c) and 2(d). In Fig. 2(d), the blue curve (p) represents the scenario where the scattering plane is located at ϕ = 0 and the red curve (s) represents ϕ = π/2; these scenarios correspond to the p-polarized and s-polarized components of incident light, respectively. Most of the scattered light is clearly concentrated in the forward hemisphere whereas the backward hemisphere has little scattered light, and the scattering is zero in the strict backward direction. This behavior further demonstrates that perfect zero backward scattering of the core-shell SiO₂-Ge-SiO₂ nanoparticles can be achieved at λ = 720 nm.

As mentioned previously, in Fig. 2(b), the curves representing the contributions of ED and MD to the total scattering efficiency do not intersect at λ = 720 nm, which means that the traditional first Kerker condition is not satisfied when zero backward scattering is achieved. For a clearer explanation, we need to verify the mathematical relationship of the Mie scattering coefficients for not only the dipole terms but also the dipole and quadrupole related terms. Figure 3(a) shows the real and imaginary parts of the dipole terms of the Mie scattering coefficients; the black and red solid lines represent the real part of a₁ and b₁, respectively, whereas the blue and magenta solid lines represent the imaginary part of a₁ and b₁, respectively. To deduce the wavelength position where the first Kerker condition of a₁ = b₁ is satisfied, we must find the intersects of curves for both the real part and imaginary parts of a₁ and b₁ at the same wavelength. The green line perpendicular to the x-axis at λ = 720 nm identifies the real part’s intersection, however, the curve of the imaginary part does not intersect, meaning that a₁ ≠ b₁ and the first Kerker condition is not satisfied. Therefore, the dipole approximation is not fulfilled and quadrupole modes should be considered. In Eq. (5), to achieve Q_b = 0, 5(a₂ − b₂) − 3(a₁ − b₁) should be zero, for which we have drawn the real and imaginary parts of the 3(a₁ − b₁) and 5(a₂ − b₂) related terms in Fig. 3(b). Similarly, the black and red solid lines represent the real part of the 3(a₁ − b₁) and 5(a₂ − b₂) related terms, whereas the blue and magenta solid lines represent the imaginary parts of the 3(a₁ − b₁) and 5(a₂ − b₂) related terms, respectively. The green line perpendicular to the x-axis in Fig. 3(b) simultaneously identifies the intersections of curves of the real and imaginary parts of the 3(a₁ − b₁) and 5(a₂ − b₂) terms at λ = 720 nm where zero backward scattering was achieved, which means that 3Re(a₁ − b₁) = 5Re(a₂ − b₂) and 3Im(a₁ − b₁) = 5Im(a₂ − b₂). We concluded that the zero backward scattering achieved at λ = 720 nm of the core-shell SiO₂-Ge-SiO₂ nanoparticles does not result from the interference of only ED and MD. Instead, the core-shell SiO₂-Ge-SiO₂ nanoparticles satisfy the generalized Kerker condition, where quadrupole electromagnetic modes are considered to interfere with dipole modes to achieve zero backward scattering.

 

Fig. 3 Real and imaginary parts of the multipole Mie scattering coefficients of core-shell SiO₂-Ge-SiO₂ nanoparticles with R₁ = 58 nm, t₁ = 73 nm and t₂ = 182 nm. (a) The dipole terms a₁ and b₁. (b) The dipole-related term 3(a₁ − b₁) and the quadrupole-related term 5 (a₂ − b₂).

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According to the Mie theory, the scattering properties of the core-shell SiO₂-Ge-SiO₂ nanoparticles depend on the refractive index of each layer and the geometry of the structure. Therefore, the geometric parameters R₁, t₁ and t₂ can be adjusted to shift the wavelength position of the zero backward scattering. In the following, we show that relatively broadband zero backward scattering can be obtained by optimizing the thickness of the Ge layer. Notably, perfect zero backward scattering can only be achieved where the condition of 5(a₂ − b₂) − 3(a₁ − b₁) = 0 is satisfied at certain wavelengths. In practical applications, backward scattering efficiency at very low values (Q_b < 0.02) can be approximated to the zero backward scattering, as is the case for the remaining manuscript. Figure 4(a) shows the backward scattering efficiency of the core-shell SiO₂-Ge-SiO₂ nanoparticles with R₁ fixed at 76 nm and t₂ fixed at 103 nm as the function of the wavelength and the thickness of the Ge layer t₁. For convenience, we drew the zero backward scattering region as black. Figure 4(a) shows that, when t₁ > 23 nm, broadband zero backward scattering appears in the wavelength range greater 870 nm and the spectral position of broadband zero backward scattering will be red-shifted with increasing t₁. This red-shift occurs because the resonance range and the peaks of ED and MD shift to longer wavelengths as t₁ increases. Regardless of how the spectral position of the zero backward scattering shifts, the width on the horizontal (wavelength) dimension of the broadband zero backward scattering reaches at least 23 nm. When t₁ < 23 nm, broadband zero backward scattering occurs in both the visible and near-infrared regions. The broadband zero backward scattering that appears in the visible region is the result of the interferences of the higher order electromagnetic modes. In particular, when t₁ = 8 nm the broadband zero backward scattering covers the wavelength ranges from 458 to 600 nm and from 700 to 755 nm, as shown by the horizontal white dashed line in Fig. 5(a). The zero backward scattering not only exhibits a broadband in the horizontal dimension (wavelength), but also in the vertical dimension (thickness of Ge layer). The zero backward scattering covers the range from 7 to 20 nm of the thickness of Ge layer with the incident light fixed at 451 nm, as shown by the vertical white dashed line in Fig. 4(a). Figure 4(b) shows the forward scattering efficiency of the core-shell SiO₂-Ge-SiO₂ nanoparticles with the same size as those shown in Fig. 4(a). The strong red color shows that, when zero backward scattering is achieved, highly enhanced forward scattering can be obtained, especially in the near-infrared region, corresponding to the black region in Fig. 4(a). It should be noted that when t₁ > 30 nm the broadband suppression of the forward scattering is also possible to achieve in the visible region in Fig. 4(b). However, the backward scattering is also weak in the same wavelength when t₁ > 30 nm according to Fig. 4(a), which means that the overall scattering efficiency is very low.

 

Fig. 4 The relationship between the unidirectional scattering properties of the core-shell SiO₂-Ge-SiO₂ nanoparticles and the Ge layer thickness t₁ with R₁ fixed at 76 nm and t₂ fixed at 103 nm. (a) Contour plot of the backward scattering efficiency as a function of wavelength and the Ge layer thickness t₁. (b) Contour plot of the forward scattering efficiency as a function of wavelength and the Ge layer thickness t₁. (c) Forward and backward scattering efficiency spectra of the core-shell SiO₂-Ge-SiO₂ nanoparticles with R₁ = 76 nm, t₁ = 8 nm and t₂ = 103 nm, as indicated by the horizontal white dashed line in (a). (d) Forward and backward scattering efficiency of the core-shell SiO₂-Ge-SiO₂ nanoparticles with R₁ = 76 nm and t₂ =103 nm versus t₁ when the wavelength was fixed at 451 nm, as indicated by the vertical white dashed line in (a).

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To represent the broadband zero backward scattering corresponding to the white dashed line in Fig. 4(a) more clearly, the scattering efficiency spectra of the core-shell nanoparticles with R₁ = 76 nm, t₁ = 8 nm and t₂ = 103 nm is given in Fig. 4(c), where N₁, N₂, N₃ and N₄ indicate the broadband wavelength range. As the red solid line indicates, enhanced forward scattering is achieved when the zero backward scattering is achieved, especially in the near-infrared region. Figure 4(d) shows the plot of the scattering efficiency vs the t₁ of the core-shell SiO₂-Ge-SiO₂ nanoparticles with R₁ = 76 nm, t₂ = 103 nm and the wavelength fixes at 451nm; also M₁ and M₂ identify the broadband geometry range. The far-field scattering patterns and corresponding 2D angular distributions at marked points N₁, N₂, N₃ and N₄ are given in Fig. 5. The scattering in the backward direction is almost completely suppressed at these four points, indicating a 142-nm and a 55-nm bandwidth of the zero backward scattering in the visible region and near-infrared regions for the core-shell SiO₂-Ge-SiO₂ nanoparticles. In summary, shifting the zero backward scattering to the visible and near-infrared regions through the thickness of Ge layer of the core-shell SiO₂-Ge-SiO₂ nanoparticles is effective.

 

Fig. 5 Far-field scattering patterns and corresponding 2D angular distributions of core-shell SiO₂-Ge-SiO₂ nanoparticles at different wavelengths with R₁ = 76 nm, t₁ = 8 nm and t₂ = 103 nm. (a) λ = 458 nm; (b) λ = 600 nm; (c) λ = 700 nm; and (d) λ = 755 nm.

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The relationships between the unidirectional scattering properties and the two parameters R₁ and t₂ of the core-shell SiO₂-Ge-SiO₂ nanoparticles were elucidated. Figure 6(a) shows the contour plot of backward scattering efficiency as a function of wavelength and R₁ with t₁ fixed at 16 nm and t₂ fixed at 103 nm. Figure 6(a) shows that, when R₁ is greater than 45 nm, the broadband zero backward scattering begins to appear in the near-infrared region. As R₁ increases, the zero backward scattering gradually shifts to longer wavelengths and exhibits a maximum width of 61 nm in wavelength dimension when R₁ reaches 103 nm. Figure 6(b) shows a contour plot of the forward scattering efficiency as a function of wavelength and R₁ with the same size as in Fig. 6(a). These results demonstrate that the highly enhanced forward scattering can be attained when achieving the zero backward scattering, which can be observed from the strong red region in Fig. 6(b), that corresponds to the black region in Fig. 6(a). In the visible region, a broadband zero backward scattering that moves toward longer wavelengths as R₁ increases is observed. When R₁ = 80 nm, a maximum broadband 53-nm wide is observed in the visible region. Notably, the electric and magnetic resonance modes of the core-shell SiO₂-Ge-SiO₂ nanoparticles are provided by the Ge layer. In the core-shell nanostructures with high refractive index materials shell, the ED is a result of a surface plasmon mode, whereas the MD is the result of a cavity type mode; thus, the influence of the inner SiO₂ and outer SiO₂ layer on the zero backward scattering is the result of smoothly shifting the operating resonance wavelength of electric and magnetic resonance modes; the increasing bandwidth feature of the zero backward scattering is due to the larger overlap of electric and magnetic resonance modes. The same phenomenon occurs when t₂ changes. Figure 6(c) presents a contour plot of the backward scattering efficiency as a function of wavelength and t₂, with R₁ fixed at 76 nm and t₁ fixed at 16 nm. Two broadband backward scattering whose wavelength position moves to longer wavelengths as t₂ increases appear in the visible and near-infrared region. By changing the parameter t₂, we increased the maximum wavelength width of the backward scattering as high as 54 nm in the visible region and as high as 63 nm in the near-infrared region. Similar to Fig. 6(b), strong red region exists in Fig. 6(d) corresponding to the black region of backward scattering in Fig. 6(c); thus, broadband zero backward scattering and enhanced forward scattering can be achieved simultaneously.

 

Fig. 6 (a) The contour of backward scattering efficiency as a function of wavelength and the core SiO₂ layer radius R₁ with t₁ fixed at 16 nm and t₂ fixed at 103 nm. (b) The contour of forward scattering efficiency as a function of wavelength and the core SiO₂ layer radius R₁ with t₁ fixed at 16 nm and t₂ fixed at 103 nm. (c) The contour of backward scattering efficiency contour as a function of wavelength and the outer SiO₂ layer thickness t₂ with R₁ fixed at 76 nm and t₁ fixed at 16 nm. (d) The contour of forward scattering efficiency as a function of wavelength and the outer SiO₂ layer thickness t₂ with R₁ fixed at 76 nm and t₁ fixed at 16 nm.

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4. Conclusion

In summary, we proposed the all-dielectric core-shell SiO₂-Ge-SiO₂ nanoparticles that can simultaneously achieve broadband zero backward scattering in addition to enhanced forward scattering. Zero backward scattering is efficient, especially in the near-infrared region, because the spectral position of zero backward scattering is located in the resonance wavelength range of ED and MD, which are the primary effects on zero backward scattering. Without fulfilling the dipole approximation, the interferences between dipole and higher-order electromagnetic modes must be considered to gain broadband zero backward scattering. Zero backward scattering with a broadband feature can be achieved owing to the wide wavelength range when electric and magnetic resonance modes overlap. By changing the independent geometric parameters of the core-shell SiO₂-Ge-SiO₂ nanoparticles, the wavelength range of the zero backward scattering can be as large as 140 nm in the visible region and as large as 63 nm in the near-infrared region. The unique capability of the core-shell SiO₂-Ge-SiO₂ nanoparticles may lead to their application in nano-antennas, low-loss metamaterials and photovoltaic devices.