1. Introduction

Phonon–polaritons are collective oscillation modes originated from coupling of electromagnetic waves with optical phonons in polar dielectrics [1], which commonly occurs at mid-IR frequencies. Near phonon resonance, permittivity of the polar materials will pass through the zero-epsilon-point or soar up to a very large value. Over epsilon-near-zero and epsilon-near-pole spectral region, these materials can behave like conventional metal strongly reflecting the light, called as Reststrahlen band. It has been recently verified that dispersion of phonon-polaritons in the van der Waals material of hexagonal boron nitride (hBN) can exhibit the natural hyperbolic dispersion over the two mid-IR Reststrahlen bands [2–9]. Its out-of-plane (axial) and in-plane (transverse) permittivities can be not only different but opposite in sign. Because of the unique dispersion, the so-called hyperbolic phonon-polaritons can exceptionally propagate with arbitrarily large transversal wavevector with a high density of states [10,11]. Moreover, the hBN HP guided mode has high volume-confinement, extremely low-loss and super subwavelength-localization, enabling a variety of applications such as thermal radiation [12–14], subwavelength imaging and focusing [6,7], wave guiding [5,15–18], perfect absorber [19–21] and amplitude modulator [22,23] in mid-IR region.

Although hBN film can support the low-loss hyperbolic polariton guiding modes that can potentially be utilized to achieve the enhanced light-matter interaction [24,25], it cannot be directly excited by the propagating waves from free-space owning to its transversal wavenumber is much larger than the photon wavenumber in free-space. Meanwhile, note that coupling of resonant modes can create the interesting hybridization, which ubiquitously exists in composite nanostructures [26,27]. Through strong mode coupling, light energy can efficiently exchange between the coupled resonant modes, producing an enhanced field in different components of hybrid system. Deep metal gratings can support a special type of localized polariton resonance, known as magnetic polaritons (MP) [28,29]. MP mode is insensitive in resonance frequency to the incident angle, which is highly appealing for a broad range of optoelectronic device applications. By simply placing the hBN thin film onto the deep metal grating, this hybrid hBN/metal grating structure can offer the in-plane wavenumber match for the efficient excitation of HP resonance. Moreover, considering hBN has hyperbolic response due to that phonon vibrations are in perpendicular crystallographic directions [3], the proposed structure provides an ideal platform to explore the basic polaritonic coupling between the hyperbolic materials and the polariton resonances in metal nanostructures. Mechanism for control of the polaritonic coupling in the proposed hBN/deep Ag-grating structure will lay the foundation for developing various practical optoelectronic devices, deserving further investigations.

In this work, we investigate the strong coupling between hyperbolic and magnetic polaritons in an hBN/deep Ag-grating hybrid system. We find that the magnetic polaritons within the grating slots can strongly interact with the hyperbolic phonon polaritons in hBN film, resulting in a large Rabi splitting with near-perfect dual band light absorption. Our results further reveal that the strong MP-HP coupling can be modified by changing geometric parameters of the structure. It is also found the two hybrid modes demonstrate exceptionally the different field patterns, which is anomalous for the strong coupling system. Moreover, resonance frequencies and near-perfect light absorption of the hybrid modes are very robust to incident angle in the proposed system. The characteristics of the strong polaritonic coupling is also theoretically studied by the strong coupling theory.

2. Numerical model and working principle

Figure 1 shows schematic of the proposed structure for strong MP-HP coupling, which comprises a silver grating covered by a thin flake of hBN. The structural parameters are initially set as: silver grating period, height and slit width of p = 4 μm, h = 1.46 μm, w = 0.16 μm and thickness of hBN thin film of d = 30 nm. Here dielectric constant of silver is described by the Drude model: ${\varepsilon _{\textrm{Ag}}}\textrm{ = }{\varepsilon _\infty }\textrm{ - }\omega _\textrm{p}^2/({{\omega^2} + i\gamma \omega } )$, where ω is the angular frequency of light, the plasma frequency, the damping constant and the permittivity at infinite frequency are ${\omega _\textrm{P}} = 13.52 \times {10^{15}}$ rad/s, $\gamma = 5.56 \times {10^{13}}$ rad/s, ${\varepsilon _\infty } = 5.0$, respectively [30]. Permittivity tensors of anisotropic hBN crystals are diagonal in the Cartesian frame and have two independent components ${\varepsilon _{xx}} = {\varepsilon _{yy}} = {\varepsilon _ \bot }$ and ${\varepsilon _{zz}} = {\varepsilon _\parallel }$. The hBN permittivity associated with in-plane and out-plane optical phonon modes is given by

 

Fig. 1. Schematic of an hBN/deep Ag-grating structure for strong interaction of magnetic polaritons and hyperbolic phonon-polaritons. (a) Top view, (b) cross-section view. Structure is illuminated by TM-polarized plane waves with magnetic field vector along y-axis.

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Fig. 2. (a) Permittivities of hBN over its Reststrahlen bands; (b) diagrams of type-I and type-II hyperbolic dispersion of hBN in lower- and upper-frequency Reststrahlen band, respectively.

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As known previously, the strong mode coupling not only enables the typical phenomenon of hybridization in composite nano-structures, but also builds a bridge for intensive energy transportation between different components of the coupled system [26,32–34]. In the investigative hBN/deep silver gating coupled system, MP mode can be resonantly excited in silver grating over a wide-angle range, featuring the enhanced magnetic fields with the induced loop current in grating grooves. Excitation frequency of this angle-insensitive polaritonic resonance can be predicted theoretically by an effective LC circuit model [29]. Due to the scattering by grating corners, light energy with the large photon wavenumber can dramatically couple into the hBN film and subsequently excite the phonon-polaritonic resonance modes therein. When the resonance frequencies of these two polaritonic modes approach one another, strong MP-HP mode interaction occurs. Dispersion of the hybrid polaritonic mode can be described by using a coupled oscillator model [35]

3. Results and discussion

3.1 Strong coupling of HP and MP modes in an hBN/ Ag-grating hybrid system

Here absorption spectra of the plain Ag gratings as a function of the grating height h is numerically computed with normal illumination of TM-polarized plane waves [inset of Fig. 3(a)]. Meanwhile, the analytical resonance wavelength of Ag gratings has been calculated by the LC circuit model for MP resonance (white solid circles) [29]. The excellent agreement of the simulated and the analytical resonance frequencies for the resonance mode herein verifies MP excitation in deep Ag gratings under study. Specifically, to realize a strong MP-HP interaction in hBN/Ag-grating structure, Ag grating was chosen to have zero-detuning of MP resonance frequency with that of HP mode. Figure 3(a) indicates the simulated absorption spectra for the bare Ag grating (black solid line) and a 30-nm-thick suspending hBN thin film (blue solid line). We see that absorption peak for the Ag grating lies at 7.33 μm, whose magnetic fields are enhanced in trenches with E-field vectors typically looped around [Fig. 3(c)], corresponding to the MP mode. Meanwhile, absorption peak for the 30nm-thick hBN nearly overlaps with that of MP mode near 7.3 μm, which is related to the hBN phonon resonance in the upper-frequency Reststrahlen band. From the absorption spectra of the hBN/Ag grating coupled structure shown in Fig. 3(b), we can clearly observe a large Rabi splitting of 40.7 meV with the dual band near-perfect light absorption. Particularly, absorptance at peaks of hybrid modes can reach 98.2% and 97.5% at 6.56 μm and 8.59 μm, respectively. Meanwhile, at the spectral valley at 7.3 μm, we see that E-fields for this coupled structure are mainly confined in hBN layer [Fig. 3(d)]. It demonstrates that the resonant phonon-polaritons in hBN essentially acts as a dark state that strongly interacts with MP mode in the proposed structure.

 

Fig. 3. Absorption spectrum for Ag grating (a) w/o and (b) with the top hBN layer, absorption curve of a bare 30-nm-thick hBN film is provided in (a), where the simulated and the analytical results are indicated by solid lines and circles. Inset displays absorption spectra of plain deep Ag grating as a function of h, where analytical MP wavelength by LC circuit model is marked with white solid circles. (c) |H| field intensity distribution at absorption peak (7.33 μm) for the plain Ag grating, with E-field vectors indicated by white arrows; |E| field intensity distributions at valley (7.3 μm) and left/right peaks (6.56, 8.59 μm) of absorption spectrum in (b) are illustrated in (d)-(f), respectively, in which boundaries of hBN layer are delimited by the white dashed lines. Modeling parameters are p = 4 μm, h = 1.46 μm, w = 0.16 μm and d = 30 nm, with normal illumination of TM-polarized plane waves from the top.

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Generally, strong mode competition and interaction are correlated to a large frequency splitting in spectral response of the hybrid system. To gain physical insights for this strong coupling phenomenon, we theoretically fit the resonant absorption spectra for MP, HP and hybrid mode in Figs. 3(a)–3(b) by Fano resonance [37], Lorentzian resonance and Alter-Towns splitting (ATS) [38,39] models, respectively, which are in good agreement with the simulated results. In common sense, the hybrid optical modes shall have very similar field distributions in most coupled systems. Intriguingly, here we find that field distributions for the hybrid modes anomaly present the entirely different patterns in the proposed coupled structure, as displayed in Figs. 3(e)–3(f). Particularly, E-fields at the left peak exhibit an exotic volume-confined Zigzag propagation in hBN film, while the other typically presents a common field-localization nearby grating corners. Mechanism of this phenomenon can be explained as the following. As can be seen in Fig. 2(a), resonance wavelength for the left peak just lies within the type-II hyperbolic Reststrahlen band of hBN, while the right one is outside this spectral region. Namely, dispersion of hBN at the left peak is hyperbolic with opposite ${\varepsilon _ \bot }$ and ${\varepsilon _\parallel }$ in sign, while that at the right peak is elliptical with ${\varepsilon _ \bot }$ and ${\varepsilon _\parallel }$ of the same sign. Owning to the completely different dispersion of hBN at resonance frequency of the two hybrid modes, light can couple into hBN layer uniquely travelling in Zigzag pattern via corner scattering at the left peak; whereas for the right one, the enhanced E fields in hBN is commonly confined near the grating corners.

3.2 Dependence of MP-HP coupling on structural parameters of silver grating

It is known that interactions between two optical modes primarily depend on their resonance frequencies. With resonance frequency of HP naturally being fixed, MP-HP coupling shall depend on the magnetic resonance frequencies, which are closely associated with the grating geometry. For the MP resonance in deep metal grating, its resonance properties were described by LC circuit model [29], whose resonance frequency reads ${\omega _{\textrm{MP}}} = {1 / {\sqrt {LC} }}$. L and C represent the effective inductance and capacitance in a unit of metal gratings.

Here Fig. 4 presents the absorption spectra dependence on height and slot width of the silver grating. From Fig. 4(a), as grating height h goes up, resonance frequency of MP red shifts to the longer wavelength. Remarkably, when h is around 1.46 μm, a pronounced spectral anti-crossing featuring a large Rabi splitting emerges near the wavelength of 7.3 μm. Rabi splitting energy is about 40.7 meV. Essentially, at the zero detuning of the resonance frequencies, strong MP-HP interaction gives rise to a Rabi splitting, which is accompanied by an intense light energy exchange between hBN and Ag grating via MP-HP hybrid mode. Over the strong-coupling spectral region, as h is reduced, the left branch bends towards the shorter wavelength, exhibiting the MP-like characteristics. While with increasing h, the hybrid mode will gradually change its nature, HP-like features appear. The eigen-frequency of the coupled polaritonic modes can be found as [40,41]

 

Fig. 4. Absorption response of the coupled structure as (a) grating height h and (b) trench width w. (c)-(d) shows the |E|-filed distributions of hybrid modes at w = 180, 360 nm, corresponding to number (1) and (2) marked in (b). Other parameters in simulation are the same as Fig. 3.

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Moreover, dependence of MP-HP coupling on slot width of grating w is shown in Fig. 4(b). Intriguingly, we can see, with w increasing, resonance frequencies of the hybrid modes (${\omega _{\textrm{HM1}}}$, ${\omega _{\textrm{HM2}}}$) are nearly unchanged, while their resonance absorption intensity gradually drops off. Note that light absorptance for each branch of hybrid modes can retain at a high lever (>95%) as w ranging from 100 nm to 280 nm in this structure. To reveal the underlying physics for the absorption reduction, Figs. 4(c)–4(d) illustrate the |E| field distributions in the left branch of hybrid modes at w = 180 nm and 360 nm. For the case with narrower slot, the volume-confined light travels in Zigzag-trajectory in hBN layer, only undergoing two times of reflection at hBN upper surface atop slot; while for the wider one, corresponding reflection times in hBN have been doubled. Considering leakage loss of the volume-confined guiding waves is proportional to the reflection times, hence increasing w will decreases the light absorptance for the hybrid mode in this strong-coupled regime.

3.3 Dependence of MP-HP interactions on hBN thickness and incident angle

Since the state density of HP resonant mode generally scales with the molecular number in hBN film, MP-HP polaritonic coupling shall rely on thickness of hBN. Accordingly, coupling between MP and HP mode as a function of hBN thickness d has been studied, as shown in Fig. 5(a). We see, as $d$ increasing, Rabi splitting gradually goes up. Particularly, as d increasing from 10 nm to 80 nm, Rabi energy gradually increases eventually reaching up to 56.4 meV, where all these cases satisfy $V > \hbar |{{\Gamma _{\textrm{MP}}} - {\Gamma _{\textrm{HP}}}} |/2$ of strong coupling condition [42]. Physically, for the structure with a larger d, a growing number of phonon-polaritons can be excited in hBN film, which will in turn strongly interact with the magnetic polaritons, rendering an increasing Rabi energy. As such, light energy can intensively exchange between hBN layer and grating trench cavity, offering an enhanced light-matter interaction in either of the hBN and the silver grooves in this coupled regime.

 

Fig. 5. (a) Absorption spectra and (b) Rabi energy of the coupled structure as a function of hBN thickness $d$; (c) absorption spectra as incident angle $\theta $, with absorption curves at $\theta $ = 0°, 10°, 20°, 30°, 40° displayed in (d). Other parameters of the structure are the same as those in Fig. 3.

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As known, realization of angle-insensitive strong light-matter interaction is highly desirable for versatile optoelectronic applications especially in practical photovoltaic and photon-detecting devices. Accordingly, the dependence of incident angle on MP-HP coupling has been given in Figs. 5(c)–5(d) under TM-polarized illumination of plane waves. As clearly observed, with $\theta $ increasing, resonance frequency of the hybrid modes nearly holds still over a wide-angle incidence from 0° to 40°. More significantly, light absorption intensity can maintain at the nearly perfect level. Physically, this angle-insensitive strong coupling should be attributed to the unique merits of magnetic polariton resonance by the designed deep metal gratings. Due to resonance frequency of MP can be naturally robust to the incident angle, zero detuning of resonance frequency for MP and HP modes (${\omega _{\textrm{MP}}} \approx {\omega _{\textrm{HP}}}$) will remain over a broader angle range, thereby enabling an angle-insensitive, dual-band intense light-matter interaction in this hybrid system.

4. Conclusions

In summary, we investigated the strong coupling of the hyperbolic-magnetic polaritons in an hBN-covered silver grating hybrid system. We find MP mode in Ag grating grooves can strongly interplay with HP mode in hBN film over its higher-frequency Reststrahlen band, rendering a large Rabi splitting with near-perfect dual band absorption. Our results reveal that MP-HP coupling can be modified by the geometric parameters of this hybrid structure. It is also found that the two hybrid modes demonstrate unusually different field patterns in this hybrid structure. Furthermore, resonance frequencies and near-perfect light absorption of the strong coupled modes are very robust to the incident angle of light. Such an angle-insensitive strong coupling of hyperbolic-magnetic polaritons within the simple structure offers new opportunities for the design of various more efficient hybrid polaritonic devices.