1 INTRODUCTION

Dielectric spheres exhibit a wide range of modes (numerous electromagnetic resonances referred to as Mie modes) with different spectral characteristics, suggesting them as a perfect platform for a wide range of applications. For example, in the last decade, analogs of Fano resonances have been observed in various plasmonic nanostructures [1, 2] with the Mie size parameter \(q = \pi D{\text{/}}\lambda < 1\) (D is the particle diameter and λ is the wavelength of the illuminating radiation). Light scattering by spherical nanospheres consisting of metal/insulator composites was considered in [3]. It was shown that light in a dielectric material is stronger amplified under optimal conditions. In this case, the behavior of the Fano resonance depends on a gap between the core and shell of a sphere [4].

The resonance characteristics of a bilayer nanosphere 180 nm in diameter in the wavelength range of 320–450 nm were reported in [5]. It is shown that the resonances of the core and shell of the layered spherical particle can overlap with each other in the energy spectrum, which gives rise to Fano resonances in forward or backscattering spectra. Unexpected color effects in hollow silica nanoparticles about 400 nm in diameter, comparable to the wavelength of visible light, with thin shells (<15 nm) were studied in [6]. The resonance characteristics of whispering gallery modes in a liquid-filled hollow glass sphere with a diameter of 20 to 115 µm were analyzed in [7], whereas the resonance characteristics of subwavelength dielectric spheres in the optical and terahertz ranges were studied in [8]. Such dielectric structures allow an extremely intense manipulation of light, similar to that achieved with plasmonic nanostructures [1–4].

Interest in dielectric spherical particles with a diameter on the order of a wavelength arose more than a century ago in studies of unusual optical effects of light scattering by finely dispersed sulfur suspensions [9]. However, a number of new unusual optical phenomena have been discovered for such particles (with the Mie size parameter q ~ 10) in the last decade, including magnetic light, photonic nanojets, optical nanovortices, Fano resonances of a high (l > 5) order, etc., which are reviewed in [10–12].

Whispering gallery modes in bilayer dielectric particles about 8 μm in diameter were studied in [13], and those in a hollow 90-µm sphere with a 892-nm-thick shell were reported in [14]. Resonance tuned by changing the plasma Fano frequency in a multilayer particle consisting of a spherical plasmon core and three-layer dielectric shells was studied in [15]. The features of subwavelength localization of light in mesoscale dielectric spheres with a cavity were considered in [16, 17].

In [10, 18], it was demonstrated that high-order Fano resonances related to internal Mie modes appear at certain values of the size parameter q and the refractive index of the particle and can lead to the enhancement of both magnetic and electric fields by a factor of 105−107. Note that resonant excitation of internal essentially subwavelength localized magnetic fields in a dielectric particle is a nontrivial problem in optics, since strong magnetism in the optical range cannot be achieved in natural dielectric materials [19]. High-order Fano resonances, for which the gain in the field can achieve the aforementioned values, are quite sensitive to both the particle size parameter and dissipative losses in its material [10–12], as well as to the characteristics of the environment [20, 21].

Superresonance conditions [10, 18] at a fixed refractive index of the solid spherical particle material were achieved in the cited works by the accurate choice of the radius of the particle. At the same time, the effect of superresonance in a hollow dielectric spherical particle with the predetermined refractive index and outer diameter has not yet been demonstrated to the best of our knowledge.

In order to expand the ranges of existence and mechanisms of this effect for particles with a given refractive index and outer size, in this work, we excite high-order Fano resonances in a mesoscale hollow dielectric spherical particle using our earlier approach [10, 18]. For the particle with a refractive index of 1.5, which is characteristic of dielectrics in the optical range, we show that a giant local enhancement of the magnetic and electric fields can also be achieved near the poles of the hollow dielectric sphere with the optimally chosen spherical cavity size. As before, we use the exact Mie theory [22], determine the contribution of an individual mode, and confirm that a high-order Fano resonance (52nd mode) with a significant enhancement (up to a factor of 108) of the magnetic and electric fields can be excited in the hollow spherical particle with the size parameter q ~ 10.

2 MODEL

The wave structure within the hollow spherical particle and near its surface can be accurately determined using the Mie scattering theory in terms of particle material, environmental characteristics, particle size parameter, and cavity size. The scattered field outside the spherical particle with the cavity is also represented as a multipole expansion, where the expressions relating various partial cross sections with the scattering coefficients formally coincide with those for a single-layer particle [22–24]. However, the scattering coefficients themselves are determined through recurrence relations [24]. At the same time, an additional degree of freedom appears in this configuration: by changing the spatial size of the cavity, one can control the position of various resonances. Without reducing the generality of the problem, in accordance with the Mie theory [22], we consider the scattering of a linearly polarized plane electromagnetic wave by the hollow spherical particle with a refractive index of 1.5, which is characteristic of most dielectric materials in the optical range [10], and the size parameter q ~ 10 (with a fixed outer diameter of 5, 6, and 8 μm) at the wavelength of a helium−neon laser (λ = 632.8 nm [24]). It is assumed that the cavity and the sphere are located in vacuum (with the refractive index n = 1). The effects of energy dissipation in the sphere are assumed to be negligibly small [10, 21]. The numerical simulation technique is described in more detail in [10, 18, 21]. The algorithm described in [24] was used to simulate scattering by the sphere with the cavity. The geometry of the problem is illustrated in the inset of Fig. 1a.

Fig. 1.
figure 1

(Color online) Spectra of scattering by a hollow sphere with an outer diameter of (a) 5, (b) 6, and (c) 8 μm versus the relative cavity radius δ.

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3 SIMULATION RESULTS AND THEIR DISCUSSION

In Fig. 1, we show the spectra of scattering by hollow dielectric spheres with the outer diameter of 5, 6, and 8 μm versus the ratio δ = Rcav/R of the radius of the cavity to the outer diameter of the sphere (Fig. 1a). The relative resonant cavity radius δ in this case is ch-osen to reach the maximum relative intensity of the r-esonance peaks at the shadow pole of the sphere (point A in the inset of Fig. 1a).

The distribution of electric and magnetic fields for spheres of various outer diameters with an optimal cavity size is shown in Figs. 2 and 3 on the linear and logarithmic scales, respectively. As seen in Figs. 2 and 3, the situation is similar to the case of a solid spherical particle [10, 18, 21]; i.e., the distribution of the field in the spherical particle under the superresonance conditions has the shape characteristic of that for a single resonant eigenmode. Such a field structure in the Mie theory [22] corresponds to a significant predominance of only one term in the series describing internal fields (the Mie coefficients cn or dn [22] depending on the type of resonance, electric or magnetic), which is responsible for the excited resonant mode [10, 26].

Fig. 2.
figure 2

(Color online) Distribution of the relative intensities of the (left panels) electric and (right panels) magnetic fields in hollow spheres with an outer diameter of (a) 5, (b) 6, and (c) 8 µm on a linear scale.

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Fig. 3.
figure 3

(Color online) Same as in Fig. 2, but on a logarithmic scale.

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Superresonance [10, 18, 21] is characterized by the existence of two “hot spots” at the poles of the sphere along the direction of radiation propagation (in the illuminated and shadow parts, see Fig. 2). The amplitudes of the internal magnetic and electric fields at these points increase sharply due to the constructive interference of one resonant mode with numerous modes in the particle [10, 18]. In addition, the configuration of the fields corresponding to whispering gallery modes [27, 28] (Fig. 3) is preserved, but their intensity is 4–5 orders of magnitude lower than that at hot spots, which is typical of Fano resonances [10, 18], but is not observed in the structure of the field characteristic of the whispering gallery resonance [27] and of cylindrical particles. As seen in Figs. 2c and 3c, the magnetic field under the superresonance conditions in the case under study is more than an order of magnitude higher than the electric field.

Following [10, 18, 21, 26], the analysis of the internal Mie coefficients for hollow spheres demonstrates that the amplitudes of all modes are quite small, except for the only resonant mode. The number of these modes depends on the outer size of the sphere and the resonant diameter of the cavity. The dominant effect of the high-order internal resonance mode on superresonance is illustrated in Figs. 4a–4c for a nonabsorbing spherical particle with a refractive index of n = 1.5 and an outer diameter of 5, 6, and 8 µm, respectively. Table 1 presents the resonance characteristics of hollow dielectric spheres (here, |eAn| and |mAn| are the amplitudes of the coefficients of the electromagnetic field components under the superresonance conditions, which are proportional to the Mie coefficients cn and dn, respectively [21, 24]).

Fig. 4.
figure 4

(Color online) Amplitudes of lth modes in a hollow dielectric sphere with an outer diameter of: (a) 5, (b) 6, and (c) 8 µm. The arrow marks the resonant mode.

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Table 1. Resonance characteristics of mesoscale hollow spheres
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With an increase in the outer diameter of the sphere from 5 to 8 μm, both the resonant inner diameter of the cavity and the number of the resonant mode increase, similarly to the solid sphere [18]. The amplitude of the 32nd resonant mode in the hollow spherical particle with a size parameter of q ~ 24 is about a factor of 5 higher than that for the other modes, whereas the amplitude of the 52nd resonant mode in the particle with q ~ 39 is about a factor of 60 higher than for the other modes. Such a high Mie coefficient is due to the constructive interference of a single partial wave in the hollow sphere. Moreover, the magnetic field in the particle with q = 24.8 exceeds the electric one only by a factor of about 2, this excess in the particle with q = 39.75 already exceeds a factor of 15. Thus, intermode coupling can be controlled by varying the size of the internal cavity.

Note that high-Q resonances are quite sensitive to the accuracy of manufacturing the surface of a spherical particle. Both bulk inhomogeneities and the surface roughness can be described by fluctuations in the permittivity of the sphere material [29]. In the optical range, the thickness of the scattering surface layer is usually several nanometers [30] or even less [31], which is much smaller than both the thickness of the layer supporting whispering gallery modes and the characteristic sizes of hot spots at the poles of the sphere in superresonance. In this case, the scattering Q-factor is inversely proportional to the squared roughness of the sphere surface [29–31]. Under the superresonance conditions, the width of the resonance line is about λ/Q, where Q ~ 107–1011 is the Q‑factor [12, 21]. Therefore, the width of the resonance line is proportional to the surface roughness squared. In this case, any inhomogeneities about the width of the resonance line will likely shift and break the resonance. However, the effect of the symmetry of the sphere and requirements for its surface and material on the morphology of magnetic resonances and the related essentially three-dimensional singularities requires a separate complex study, which is beyond the scope of this work and will be reported elsewhere. The above results for the “perfect” spherical particle can be considered as a theoretical upper limit on the level of generated fields.

4 CONCLUSIONS

A seemingly simple object, such as a dielectric mesosphere with a cavity, is actually a complex system for modeling and understanding. A spherical particle with a size parameter q > 10 supports several spectrally sharp Mie scattering modes, and the accurate simulation of the effect of an air-filled cavity on them is a difficult task, since simple morphological perturbations (the presence of a cavity) lead to significant changes in the mode distribution.

The excitation of high-order Fano resonances requires precise tuning of the parameters of the sphere and cavity size at a predetermined refractive index of the dielectric material and the outer diameter of the sphere. We have shown that the low-loss scattering of light by a mesoscale hollow dielectric sphere allows one to observe optical effects related to high-order Fano resonances previously discovered for monolithic spheres [10, 18, 21]. In a monolithic dielectric sphere, superresonance is ensured by the accurate choice of the outer diameter of the particle, whereas in a hollow sphere with a predetermined outer size, superresonance is achieved by the accurate choice of the cavity radius. In the mesotronics of spherical dielectric particles, the Fano resonance provides a sharp transition from relatively weak localized fields to giant magnetic fields. So, for a particle with a refractive index of 1.5 and a size parameter of q ~ 40, the resonant mode has the number l = 52. In this case, the generation of electric and magnetic fields with relative magnitudes at the sphere poles about 106−107 is possible if the magnetic field exceeds the electric field by more than a factor of 15. The effects under study expand the list of possible methods for generating extremely high magnetic fields by mesoscale dielectric spheres. Note that one of the possible technologies for manufacturing a hollow sphere using borosilicate glass [21] is described in [32].