1. Introduction

Enhancement of electromagnetic scattering from subwavelength objects is of critical importance for the study of fundamental research and potential applications in optical physics, including sensing, bio-medical imaging, energy harvesting, antenna [1–4]. Based on the Mie scattering theory, the scattering field of subwavelength particles is intrinsically attributed to electromagnetic multipole resonances [5]. However, for a subwavelength atom in a three-dimensional (3D) vacuum environment, there exists a constraint for the maximum scattering cross section with $3\lambda ^{2}/2\pi$ [6], known as a single-channel limit, where $\lambda$ is the wavelength of light in vacuum. Similarly, for a 2D scattering configuration, this upper limit can approach to be $2\lambda /\pi$ [7].

Over the past decade, many theoretical studies on the scattering enhancement of subwavelength particles have been demonstrated to overcome the aforementioned single-channel limit and realize the "superscattering" phenomenon [8–20]. Initially, a pioneering work of 2010 by Ruan and Fan [9], firstly proposed that the superscattering can be achieved from a subwavelength multilayer nanorod structure composed of plasmonic and dielectric materials, through overlapping the different angular momentum channels at a single frequency. Subsequently, various subwavelength structures with combinations of different materials are employed to generate the superscattering, such as alternating graphene/dielectric [11–13], hBN/dielectric [14], or dielectric/dielectric [15] layered structure. Recently, Qian et al. [21] have reported the first experimental investigation for the superscattering from metasurface-based multilayered structures, by utilizing the degenerate resonances of confined surface waves. Nevertheless, the desirable performance of superscattering effect is subject to realistic material loss, structural complexity, and the lack of broad tunability.

On the other hand, great interest has recently been devoted to layered Van der Waals crystals consisting of individual atomic planes [22–31], i.e. single layer (or a few layers) of graphene, hBN or their combinations, due to their unusual electronic and optical properties. For example, in comparison with metal, graphene is 2D semi-metallic material with low loss and highly electrical tunability, supporting highly confined surface plasmon polaritons (SPPs). Also, hBN is known as a kind of natural hyperbolic material, and a thin slab of hBN allows for several multimode resonances - hyperbolic phonon polaritons (HPPs) within its two Reststrahlen (RS) bands. This fact may offer the opportunity to generate simultaneous multimode excitations at a single frequency or even multiple frequencies, and thus observe the superscattering from hBN-based structure [14]. What’s more, since both SPPs in graphene and the HPPs in hBN can occur in the mid-infrared frequency range, the combined graphene/hBN heterostructure may marry their respective attributes, electrical tunability in the former and multimode resonances in the latter, through the excitation of the hybrid plasmon-phonon eigenmodes [27–29].

Based on a simple combination structure composed of graphene and hBN materials, in this work we aim to explore the possibility to achieve tunable multifrequency superscatterer with high performance. A transfer matrix method (TMM) is employed to obtain the Mie scattering coefficients and also the exact solutions for the discrete multimode resonances. In accordance with the general way to generate the superscattering, we then seek the explicit conditions of resonantly overlapping different eigenmode resonances by scanning external tunable parameters, i.e. the chemical potential of graphene via the gate voltage. The performance of multifrequency superscattering effect is further validated from the scattering spectra, near-field and far-field distributions. In addition, the tolerance of multifrequency superscattering to structure vibrations and realistic material loss is also investigated in the numerical calculations.

2. Model and methods

Let us consider a subwavelength layered cylindrical rod structure consisting of a dielectric core, monolayer graphene and a hBN shell. The 2D cross-sectional view in the $xy$ plane is schematically depicted in Fig. 1(a), and the rod structure is homogenous along the $z$ direction. The outer radius, the outer surface conductivity, and dielectric function of core and shell layers are set with ($\rho _{l},\sigma _{l},\bar {\epsilon }_{l}$), $l=1,2$. As for the possible anisotropy, the dielectric function $\bar {\epsilon }_{l}$ in the cylindrical system is set by a diagonal matrix with components ($\epsilon _{l\rho }, \epsilon _{l\phi }, \epsilon _{lz}$). The entire structure is embedded in another dielectric medium with permittivity $\bar {\epsilon }_{3}$. All the materials are non-magnetic.

 

Fig. 1. (a) Schematic diagram of a multilayer cylindrical rod structure at a cross-sectional view in the $xy$ plane. The regions in green/yellow colors indicate core materials and shell layer, respectively, while the interface layer, i.e. graphene, is shown with a thin blue region. For convenience, the outer radius, the outer surface conductivity, and dielectric function of individual layers are set with ($\rho _{l},\sigma _{l},\bar {\epsilon }_{l}$), $l=1,2$. The entire structure is embedded in another dielectric medium with permittivity $\bar {\epsilon }_{3}$. (b) The discrete bands of ED ($m=\pm 1$) and QD ($m=\pm 2$) resonances around the first RS band of hBN for the dielectric/graphene/hBN multilayer structure, where the ideal lossless configuration is considered. The notations $\mathbf {D}_{m,n}$ ($\mathbf {Q}_{m,n}$) are used to identify the different ED (QD) resonances, where $m$ and $n$ denote the angular momentum channel and the mode number along the radial direction, respectively. The field profiles of several eigenmodes are also shown in (b). The first RS band edge of hBN are indicated by the dotted lines, with the critical frequencies being $f_{TO_{1}}$ and $f_{LO_{1}}$. Other parameters are set with the radius of core layer $R_{1}=0.04$ $\mu$m, the thickness of hBN $d=R_{2}-R_{1}=4$ $\mu$m and the chemical potential of graphene $\mu _{c}=0.8$ eV.

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The monolayer graphene can be modeled with a complex-valued surface conductivity $\sigma _{g}$, including the contributions from the intraband and interband transitions [32–35], i.e., $\sigma _{g}=\sigma _{intra}+\sigma _{inter}$, where

The optical properties of hBN can be characterized by the permittivity $\epsilon _{j}=\epsilon _{j}(\infty )+s_{v,j}\omega _{v,j}^{2}/(\omega _{v,j}^{2}-i\gamma _{v,j}\omega -\omega ^{2}), j=\perp, \parallel$, with $\epsilon _{\perp }$ and $\epsilon _{\parallel }$ being the components of the relative permittivity perpendicular and parallel to hBN plane [14]. For the out-of-plane direction, i.e., $j=\perp$, $\epsilon _{j}(\infty )=2.95$, the dimensionless coupling factor $s_{v,j}=0.61$, the normal frequency of vibration $\hbar \omega _{v,j}=92.5$ meV, and the amplitude decay rate $\hbar \gamma _{v,j}=0.25$ meV [28,36–38]. And for the in-plane direction, i.e., $j=\parallel$, $\epsilon _{j}(\infty )=4.87$, $s_{v,j}=1.83$, $\hbar \omega _{v,j}=170.1$ meV, and $\hbar \gamma _{v,j}=0.87$ meV [28,36–38]. Notice that there exist two distinct RS bands for hBN crystals, where the first RS band over the frequency range of ($f_{TO_{1}}, f_{LO_{1}}$) corresponds to type-I hyperbolicity ($\epsilon _{\perp }>0$, $\epsilon _{\parallel }<0$), and the second RS band ($f_{TO_{2}}, f_{LO_{2}}$) is associated with type-II hyperbolicity ($\epsilon _{\perp }<0$, $\epsilon _{\parallel }>0$). Here, the components of the permittivity of hBN shell in the cylindrical system are $\epsilon _{2\rho }=\epsilon _{\perp }$ and $\epsilon _{2\phi }=\epsilon _{2z}=\epsilon _{\parallel }$.

We start with a 2D scattering problem of the proposed rod structure in Fig. 1(a), under the normal illumination of an external TM-polarized plane wave with the magnetic field $\mathbf {H}=H_{0}e^{ik_{0}x}\hat {z}$. The $\exp (-i\omega t)$ time-dependent convention for the field is used, and $k_0=\omega \sqrt {\mu _{0}\epsilon _{0}}$ is the wavenumber in vacuum. For convenience, we solve this problem by using a system of local cylindrical coordinates ($\hat {\rho },\hat {\phi },\hat {z}$) with an origin located at the center of the rod structure. According to Mie scattering theory, the magnetic field in the $l$th layer can be written by $H_{l,z}(\rho,\phi )=H_{0}\sum _{m=-\infty }^{\infty }i^{m}[A_{l,m}J_{lm}(k_{l}\rho )+B_{l,m}H_{lm}^{(1)}(k_{l}\rho )]e^{im\phi }$, where $J_{lm}$ and $H_{lm}^{(1)}$ respectively denote the $(lm)$th-order Bessel function and Hankel function of the first kind with the arguments of $k_{l}\rho$, $k_{l}=\sqrt {\epsilon _{l\phi }}k_{0}$ and $lm=m\sqrt {\epsilon _{l\phi }/\epsilon _{l\rho }}$. Note that $lm$ reduces to be integer $m$ for the isotropic case $\epsilon _{l\rho }=\epsilon _{l\phi }$.

The coefficients ($A_{l,m}$, $B_{l,m}$) of adjacent layers are linked by a transfer matrix of the interface with [39,40]

By considering the singularity of the Hankel function at the origin, we can set $A_{1,m}=1$ and $B_{1,m}=0$, and then get the scattering coefficients $S_{m}$ for the incident plane wave in the surrounding medium, $S_{m}=\mathcal {M}_{21}/\mathcal {M}_{11}$, from the transfer matrix components $\mathcal {M}_{11}$ and $\mathcal {M}_{21}$. The pole locations of $S_{m}$ at the angular momentum channel $m$ can show explicitly the multipole resnonances for this layered structure, i.e., electric dipoles (ED;$~\mathbf {D}$) with $m=\pm 1$, and electric quadrupoles (QD;$~\mathbf {Q}$) with $m=\pm 2$.

3. Results

As a concrete example, we show in Fig. 1(b) the discrete bands of ED and QD resonances around the first RS band of hBN for the dielectric/graphene/hBN multilayer structure depicted in Fig. 1(a). The isotropic dielectric material in core layer is used, i.e., $SiO_{2}$ with permittivity $\bar {\epsilon }_{1}=2.1$, and the surrounding medium is set as vacuum. A specific ideal case of lossless graphene and hBN is firstly assumed, and other parameters are set with $R_{1}=0.04$ $\mu$m, the thickness of hBN $d=R_{2}-R_{1}=4$ $\mu$m and $\mu _{c}=0.8$ eV. In addition to the angular momentum channel $m$, we also introduce the mode number $n$ (the number of the field maximum points) along the radial direction and use the notations $\mathbf {D}_{m,n}$ ($\mathbf {Q}_{m,n}$) to identify the different ED (QD) resonances [see the corresponding field distribution in Fig. 1(b)]. Apparently, besides the hybrid surface plasmon-phonon resonance ($\mathbf {D}_{11}$ and $\mathbf {Q}_{21}$) near the graphene layer below a frequency $f_{{TO}_{1}}$, there exist multiple branches of hyperbolic plasmon-phonon polaritons ($\mathbf {D}_{12}, \mathbf {D}_{13},\ldots$ and $\mathbf {Q}_{22}, \mathbf {Q}_{23},\ldots$) within the first RS band of hBN between $f_{{TO}_{1}}$ and $f_{LO_{1}}$, as shown in Fig. 1(b). This fact provides the possibility of the simultaneous excitations of multimode resonances at a single frequency or even multiple frequencies. In contrast to several previous studies [11,12,41] that the mode solutions in cylindrical structure are evaluated approximately by those in the equivalence case of 1D planar waveguide model based on the Bohr condition, here we adopt the rigorous Mie scattering theory to predict the resonance locations and study the scattering properties.

To further explore the possibility of the superscattering, we next investigate in Fig. 2 the dependence of multimode resonances on the chemical potential $\mu _{c}$. Remarkably, it is found that several branches of mode solutions can intersect within hBN’s first RS band. For example, $\mathbf {D}_{12}\&\mathbf {Q}_{23}$ coincide spectrally at point A ($\mu _{c}=0.89$ eV, $f=22.85$ THz). In addition, $\mathbf {D}_{13}\&\mathbf {Q}_{25}$ and $\mathbf {D}_{14}\&\mathbf {Q}_{27}$ can also be seen to overlap each other at point B ($\mu _{c}=0.36$ eV, $f=23.47$ THz) or C ($\mu _{c}=0.22$ eV, $f=23.85$ THz), respectively. Thereby, the hybrid hyperbolic plasmon-phonon modes in this proposed rod structure can be finely manipulated to obtain the coincidence at multiple frequencies by tuning the chemical potential of graphene via some external parameters, i.e., gate voltage or chemical doping. Previously, Qian et al. [14] reported the superscattering can be constructed at dual-frequency regimes in a hBN-based structure, but the superscattering points occur at the two specified frequencies close to the hBN’s RS band edges, suffering from the lack of flexible tunability. Here, by combining the advantages between graphene and hBN in our proposed structure, it is expected to achieve and further manipulate the superscattering flexibly and accurately at a multiple-frequency regime.

 

Fig. 2. The dependence of multimode bands on the chemical potential of graphene $\mu _c$. Three superscattering points A, B, and C are found within hBN’s first RS band, due to the respective spectral overlapping $\mathbf {D}_{12}\&\mathbf {Q}_{23}$, $\mathbf {D}_{13}\&\mathbf {Q}_{25}$, and $\mathbf {D}_{14}\&\mathbf {Q}_{27}$. The geometrical parameters are the same as those used in Fig. 1(b).

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We now turn to directly see the scattering properties at these predicted superscattering points. In order to highlight the underlying physics, we examine the total scattering cross-section by considering the dominant contributions from ED and QD resonances supported in subwavelength objects. Figure 3 plots the spectra of total normalized scattering cross-section (NSCS$=\sum _{m=-\infty }^{\infty }|S_{m}|^{2}$) as well as those spectra contributed only by the individual ED ($m=\pm 1$) or QD ($m=\pm 2$) resonances with the chemical potential $\mu _c=0.89$ eV, $0.36$ eV and $0.22$ eV. Under the ideal lossless assumption [see Fig. 3(a), 3(c), 3(e)], the superscattering beyond single-channel scattering limit (NSCS>1) can appear at multiple frequency regions. Particularly, total NSCS approaches to be the maximum (NSCS$\sim 4$) at the frequencies of points A, B and C, where the contributions from individual scattering channels ($m=\pm 1, \pm 2$) are completely close to the single-channel limit. In contrast, when the realistic loss in graphene and hBN layers is considered [see Fig. 3(b), 3(d), 3(f)], the superscattering points A, B, and C experience a slight shift, and the corresponding chemical potentials are respectively changed with $\mu _c=0.84$ eV, $0.34$ eV, and $0.19$ eV, but the resonant frequencies remains unchanged. For these cases, the scattering power degrade dramatically, and the performances of superscattering deteriorate or even disappear. Nevertheless, the NSCS still exceeds the single-channel scattering limit near points A, B, or C, and thereby the superscattering phenomenon shows a high tolerance to material loss for this simple layered system. As a consequence, we here attain the goal of a multifrequency superscatterer with high tunability based on the graphene-hBN cylindrical rod structure.

 

Fig. 3. Total scattering cross-section (normalized by $2\lambda /\pi$) and the dominant contributions from individual scattering channels $m=\pm 1$, $\pm 2$, when the realistic loss in materials is absent (a)(c)(e) or present (b)(d)(f). The particular case is considered in each figure, where the chemical potentials of graphene are associated with those at supercattering points A, B, and C. (a) $\mu _{c}=0.89$ eV (b) $\mu _{c}=0.84$ eV (c) $\mu _{c}=0.36$ eV (d) $\mu _{c}=0.34$ eV (e) $\mu _{c}=0.22$ eV (f) $\mu _{c}=0.19$ eV. The black dashed line represents the single channel limit of scattering cross-section. The geometrical parameters are the same as those used in Fig. 1(b).

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To confirm the desired results, we show the real part of total magnetic field in the far field [Fig. 4(a)-4(b)] as well as the intensity of scattered waves in the near-field regimes [Fig. 4(c)-4(d)], when an incident plane wave with unity amplitude normally illuminates the rod structure from the left side. As an example, for the lossless case at point A [Fig. 4(a) and 4(c)], the incident wave is largely disturbed by the presence of the rod structure, and a significant shadow forms behind it with the size much larger than the rod diameter. Meanwhile, the near-field distributions of scattered waves at the superscattering point clearly display the mixed pattern among multipole resonances, including ED, QD and other higher-order resonances as well. In comparison, when the realistic loss in materials is present [Fig. 4(b) and 4(d)], the contributions to the scattering come basically from ED and QD resonances, and the scattering is greatly influenced. Nevertheless, the superscattering is still observable in the presence of realistic loss, which is in agreement with our analytical prediction in Fig. 3.

 

Fig. 4. (a)(b) The real part of total magnetic field Re$[H_z]$ in the far-field regime (c)(d) The scattering field intensity $|H_{z}|^{2}$ in the near-field regime. All the cases are in a particular configuration of Fig. 3(a) or 3(b), associated with superscattering point A, when an incident TM-polarized plane wave with unity amplitude normally illuminates the rod structure from the left side. The realistic loss in materials is absent (a)(c) or present (b)(d). For clarity, the size of rod structure is indicated by the gray region. The units used in x or y axis are normalized by the light wavelength $\lambda _{A}$ at superscattering point A.

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We finally investigate in Fig. 5 the impact of structure variation on the multifrequency superscattering. Here we fix the radius of the shell layer with $R_{2}=3.8~\mu$m and consider the realistic material loss in the structure. Figure 5(a) shows the evolution of the aforementioned superscattering points A, B, and C, when the radius $R_1$ of the core layer varies. Clearly, it is seen that, three superscattering points can still be readily attained in various structures by tuning appropriately the chemical potential of graphene. This fact then tells us that a high tolerance of multifrequency superscattering to the structure variation should be expected. In Fig. 5(b), we further plot the multifrequency superscattering regions within which NSCS>1 for different structural configurations, when a fixed chemical potential $\mu _c=0.3$ eV is taken. The different regions filled with blue/dark yellow/purple colors indicate the achievements of superscattering around points A, B, and C, respectively. Therefore, the proposed simple structure may provide a flexible and controllable platform in practical implementation to realize multifrequency superscattering over a broad range of structural combinations and configurations.

 

Fig. 5. (a) The evolution of superscattering points A, B, and C as a function of the radius $R_1$ of the core layer. (b) The multifrequency superscattering regions within which NSCS>1 for different structural configurations, when a fixed chemical potential $\mu _{c}=0.3$ eV is taken. The different regions filled with blue/dark yellow/purple colors indicate the achievements of superscattering around points A, B, and C, respectively. The radius of the shell layer is fixed with $R_{2}=3.8~\mu$m in (a)(b).

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4. Conclusion

In conclusion, we explicitly construct the superscattering beyond single-channel scattering limit at a multiple-frequency regime with a simple graphene-hBN cylindrical structure. Based on a rigorous multimode engineering approach, two different resonance channels between ED and QD can be tuned delicately to spectrally overlap each other through external parameters, i.e. the chemical potential of graphene. We remark that such a method can be applied to other multilayered systems and also seek the possibility of additional superscattering points created by the spectral overlapping among three or more multipoles resonances. In addition, more studies on a off-to-on transition from superscattering to invisibility may be carried out through other controllable parameters, such as the use of external static magnetic field [19] or phase-change materials [20]. Here our results may suggest an accurate and efficient method to actively tune and modulate optical signals at the subwavelength scale.