1. Introduction

Scattering and absorption from a small particle under resonance is particularly interesting. For a sufficiently small isotropic sphere compared with the wavelength, resonance occurs when the permittivity of a sphere is negative and its amplitude is two times larger than the permittivity of the surroundings, known as the Fröhlich condition [1]. A classic example in optics is a metallic particle, which resonates because of the excitation of localized surface plasmons typically in the visible region. Strong scattering from such plasmons finds applications in sensing, imaging, and guiding [2]. A small particle made of an inorganic material that has strong photonic vibrational modes also shows resonance, typically in the mid-infrared [1].

Another class of resonating small particles is one comprising the strongly anisotropic particles characterized by hyperbolic dispersion [3–6]. Because of their strong anisotropy, waves having large wavenumbers (high-k waves) are supported inside the particles. Thus, the size of the hyperbolic metamaterial (HMM) particle can be sufficiently small compared with the incident wavelength.

Whereas the resonance modes and the scaling law of HMM particles have been studied [4], the scattering and absorption properties, which are critical in applications, of such particles have not been investigated yet. In this work, we present our systematic study on scattering and absorption from an HMM nanoparticle. By comparing with a plasmonic nanoparticle, we unveil its unique scattering and absorption features.

2. Model and methodology

Hyperbolic metamaterials are extreme instances of uniaxial materials whose principal components of the permittivity tensor have opposite signs. Because of its hyperbolic dispersion, an HMM can ideally support infinitely large wavenumbers, leading to sub-diffraction imaging [7–10], sub-wavelength interference [11–13], and large photonic density of states [14–16], to name a few. The capability to support high-k waves is beneficial in the making of resonators that are small compared with the wavelength. Because the refractive indexes of conventional dielectrics in the optical range are on the order of unity, the size of a particle required to support whispering gallery modes is comparable to the incident wavelength. With HMMs, the size of a particle to resonate becomes much smaller than the incident wavelength.

In the following, the permittivity of the HMM particle considered in the current work is the effective permittivity of silver-silicon multilayers. The effective permittivity (ε = diag(ε_x, ε_x, ε_z)) is calculated from the effectively medium theory (EMT) [17], ${\epsilon }_{\text{x}}=r{\epsilon }_{\text{m}}+(1-r){\epsilon }_{\text{d}},{\epsilon }_z^{-1}=(1-r){\epsilon }_{\text{d}}^{-1}+r{\epsilon }_{\text{m}}^{-1}$, where r is the volume fraction of the metal and ε_m and ε_d are the permittivties of metal (silver) and dielectric (silicon), respectively. Here, the optical axis as well as the z-axis is perpendicular to the multilayers. Note that although non-locality is not included in this second-order EMT, it serves as a reasonable estimate under the condition, |ε_m(ω)| >> ε_d. To simplify the situation, the form of the permittivity of silver used in our work derives from the Drude-Lorentz model [18] whose parameters were obtained by a fit to the experimental data [19]. The permittivity of silicon is assumed to be non-dispersive and set to 11.56 [18]. As a comparison, the scattering properties of a silver nanosphere are considered as well.

The scattering properties of the HMM particles are computed by finite element method in scattered field formulation. The boundary is truncated by a perfectly matched layer. In post processing, the scattering and absorption cross-sections of a particle are evaluated from the far field and the consumed energy, respectively [20]. The scattering and absorption of an isotropic sphere is calculated from Mie theory [21]. Following convention, the scattering and absorption of a particle are discussed in terms of efficiency [1]. The scattering efficiency (Q_scat) and absorption efficiency (Q_abs) are calculated by dividing the scattering and absorption cross-sections by the cross-section of the particle, respectively.

3. Results and discussion

The effective permittivity of the silver-silicon multilayers, which are used in the following computation, is plotted in Fig. 1(a). The silver filling ratio (r) is two third. Within the wavelength range of the plot, ε_x< 0 and ε_z >0, hence dispersion is hyperbolic. Next, an HMM nanosphere of 60 nm radius is placed in air where the optical axis is aligned with the z-axis. The incident electromagnetic wave is linearly polarized along the x-axis and propagates in the z-axis.

 

Fig. 1 (a) Effective permittivity of silver-silicon multilayer where the optical axis is parallel to the z-axis. (b) Scattering efficiency (Q_scat) and absorption efficiency (Q_abs) of the HMM sphere. The white arrow in the sphere of the inset indicates the optical axis, which is parallel to the z-axis. (c) The magnetic and electric field amplitudes of the HMM sphere at 1564 nm in the x-z plane and the y-z plane. (d) Scattering and absorption efficiencies of the silver sphere in x-z plane and y-z plane. (e) The magnetic and electric field amplitudes of the silver sphere at 360 nm. The white dashed lines on (c) and (e) indicate the sizes of the HMM and Ag spheres, respectively.

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The scattering and absorption efficiencies of the HMM sphere are plotted in Fig. 1(b); a strong resonance occurs at 1564 nm wavelength. At this condition, the sphere radius (r) is sufficiently small with respect to the resonance wavelength in free space (λ₀), which gives r/ λ₀ = 0.038. To see the modes inside the HMM sphere, the magnetic as well as electric field amplitudes at 1564 nm have been plotted in Fig. 1(c). The magnetic field clearly shows the lowest resonant mode and is similar to the magnetic resonance of a high-refractive index nanoparticle [22]. Similar resonance modes are observed in coupled metallic nanostrips [23]. The detail analysis in terms of magnetic resonances will be discussed elsewhere.

As a comparison, a silver sphere placed in air is considered. To have the same radius-to-wavelength ratio to the HMM sphere, the radius of the silver sphere is set to 13.7 nm. Figure 1(d) shows the scattering and absorption efficiencies for the silver sphere. There is a strong resonance peak at 360 nm, where r/ λ₀ = 0.038. The magnetic field amplitudes as well as electric field amplitudes of the silver sphere at 360 nm are plotted in Fig. 1(e) to visualize the difference between the HMM sphere. By comparing Fig. 1(b) and (d), we see that the scattering and absorption efficiencies for both the HMM sphere and the silver sphere having the identical radius-to-wavelength ratio are of the same order. These results show that the HMM spheres can perform as good scatterers as well as local heaters [24–26] in the near infrared.

The unique resonant modes of an HMM sphere can be seen when the host refractive index dependence as well as size dependence are studied. Figure 2(a) plots the absorption efficiencies of the same HMM sphere as in Fig. 1(b) but placed in different hosts. These hosts are assumed to be isotropic and homogenous. From Fig. 2(a), even if the host refractive index changes from unity to two, the resonance wavelength barely shifts. Insensitivity to the host refractive index is due to the effectively large refractive index of the HMM sphere. The situation contrasts the plasmonic particles whose resonance wavelengths red-shift with increasing host refractive index. For the same silver sphere as illustrated in Fig. 1(d), the resonance wavelength shifts from 360 nm to 368 nm when the host refractive index changes from unity to two (figure not shown).

 

Fig. 2 (a) Normalized absorption efficiencies of the HMM sphere in different hosts. (b) Normalized absorption efficiencies of the HMM sphere having different radii. Each plot in (a) and (b) is normalized to its peak value.

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The scattering and absorption efficiencies of three HMM spheres having different radii (r = 48, 60, and 72 nm) are plotted in Fig. 2(b). Except for the radii, all conditions are the same as for Fig. 1(b). From Fig. 2(b), the resonance wavelength shifts by more than 100 nm when the radius is increased or decreased by 20%. The resonance mode which is similar to a dielectric cavity mode makes the HMM spheres sensitive to the radii. As a comparison, the radius dependence is also calculated for the silver sphere. Even though the radius is increased or decreased by 20%, the shift of the resonance wavelength is around 1 nm (figure not shown). It is well known that the plasmonic resonance becomes independent of the sphere radius in the limit of sufficiently small radius; this is known as the Rayleigh scattering condition [1].

The strong anisotropy of the HMM sphere becomes obvious when the incident angle is oblique to the optical axis of the HMM sphere. Except for the incident angle, an identical situation is considered in Fig. 1(b). The two different sets of angular dependences are considered. The angle θ is defined in the x-z plane and the magnetic field is always parallel to the y-axis. The other angle ϕ is defined in the y-z plane and the electric field is always parallel to the y-axis (see Fig. 3(a)). The θ- and ϕ-dependencies of the absorption efficiencies are plotted in Fig. 3(b). The absorption efficiency, while hardly depending on θ, largely depends on ϕ. The far-field amplitudes of the electric fields are plotted with respect to θ and φ, in Fig. 3(c)–(f). Note that there are far-fields patterns parallel to the incident plane (|E_Π|) and perpendicular to the incident plane (|E_⊥|) for each θ and ϕ plot. It is important to mention that the maximum angle of the far-field patterns is always 180 degrees (forward direction) and does not depend on θ or ϕ. Such angular-independent scattering properties could be beneficial for unidirectional light extraction and sensing. Obviously, the parallel components of the scattered far-field patterns of the silver sphere rotate as the incident angle becomes larger (see Fig. 3(g) and (j)). Because the radius of the silver sphere is sufficiently small compared with the wavelength, the perpendicular components of the scattered far-field patterns are nearly circular and show little angular dependence (see Fig. 3(h) and (i)).

 

Fig. 3 (a) Definition of θ and ϕ. The optical axis indicated by the white arrow for each HMM sphere is parallel to the z-axis. (b) Incident angle dependences (θ and ϕ) of the normalized absorption efficiencies for the HMM sphere at 1564 nm. (c)–(j) Polar plots of the electric far-field amplitudes for θ and ϕ dependencies for the HMM sphere at 1564 nm ((c)–(f)) and for the silver sphere at 360 nm ((g)–(j)). In (h) and (i), all four curves overlap.

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Additionally, we note the shape of the far-field patterns. Regarding patterns with θ = 0 for the HMM sphere (electric field parallel to the x-axis), the shapes of the field patterns in the x–z plane (parallel plane) and in the y–z plane (perpendicular plane) are circular and figure-of-eight (8) in form, respectively (see Fig. 3(c) and (d)). It is well known that when the size of the isotropic sphere is sufficiently small, its parallel and perpendicular components exhibit patterns similar to a figure-of-eight and a circle, respectively (see Fig. 3(g) and (h)) [1]. Thus, the far-field patterns of an HMM sphere and an isotropic sphere are 90 degrees rotated with respect to the incident direction (z-axis in Fig. 3).

So far, we have worked using effective parameters when studying the properties of the HMM sphere. To finish, we consider a practical HMM-particle structure. Take a HMM particle consisting of silver-silicon multilayers whose total dimensions are 120 nm (x) by 120 nm (y) by 120 nm (z). The layer thicknesses of silver and silicon in z-axis are 13.3 nm and 6.7 nm, respectively, thus the HMM particle is consists of twelve layers. The permittivities of silver and silicon are the same as before. Figure 4(a) plots the scattering and absorption efficiencies of the multilayer cubic particle placed in air. The incident fields are linearly polarized in the x-axis and propagating in the z-axis. At 1690 nm, there is a strong resonance; the field patterns are shown in Fig. 4(b), the resonance modes being identical with those in Fig. 1(c). As a comparison, the scattering and absorption efficiencies of a homogenous cubic is also plotted in Fig. 4(a). The dimension is identical to the multilayer particle and the effective permittivity shown in Fig. 1(a) is used. Although the resonance wavelength differ by 33 nm between the two structures, EMT serve as a good estimate. From Fig. 4, we could expect that the unique scattering and absorption properties that we examined for the homogenous HMM sphere hold true for an HMM particle consisting of multilayers.

 

Fig. 4 (a) Scattering and absorption efficiencies of the silver-silicon multilayer cubic particle as well as the homogenous HMM cubic particle. The multilayer cubic as well as the homogenous cubic are indicated as layers and EMT, respectively. The inset shows the schematic of the multilayer cubic. (b) Magnetic field amplitude of the multilayer cubic at 1690 nm in the x-z plane and the y-z plane.

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4. Summary

In summary, the scattering and absorption of HMM particles which are sufficiently smaller than the wavelengths have been studied. Whereas the resonance wavelength is insensitive to the host refractive index, it is sensitive to particle size. The maximum angles of the far-field scattering patterns do not depend on the incident angle. Because the strength of the scattering and absorption are comparable to a plasmonic particle of the same radius-to-wavelength ratio, it is expected that HMM particles are potentially useful scatterers as well as local heaters in the near infrared where noble metal particles do not resonate.