Switchable strong coupling between dual hyperbolic phonon polaritons and photons in hybrid structure of metasurfaces and h-BN slab

We propose an all-dielectric hybrid structure combined with hexagonal boron nitride slab and strontium titanate (STO) metasurfaces to excite dual hyperbolic phonon polaritons (HPhPs) and an additional optical (TO) phonon, and achieve their strong coupling with photons. The metasurfaces, supporting tunable guided-mode resonance via adjusting the external temperature, consists of STO two-dimensional grating and STO layer. Thus, the strong coupling can be switched and tuned actively between the dual HPhPs and TO phonon via adjusting the external temperature of metasurfaces. This work has numerous potential applications on multi-channel biosensors, filters and tunable source and detectors.


Introduction
Metasurfaces, an artificial composite material consists of sub-wavelength building blocks with small thickness, are able to support three-dimensional manipulation of light. Utilization of arrangement and interaction of basic building blocks allows for modifying the features of electromagnetic waves such as direction of propagation, and phase [1,2]. Particularly, all-dielectric metasurfaces recently drawn numerous attentions due to low thermal conductivity and dissipation comparing metallic metasurfaces [3,4]. The guided-mode resonance can be excited in all-dielectric metasurfaces, and shows a strong field enhancement [5]. Therefore, all-dielectric metasurfaces could provide an ideal platform for implementing enhanced light-matter interaction.
In couple system, strong coupling, as an unique regime of light-matter interaction, occurs when the rates of exchange energy between matter and photon becomes faster than the dissipation rates of other loss mechanisms. Strong coupling regime shows coherent energy oscillations (Rabi oscillations) between matter and photon, i.e., the electric field of radiation manifest a periodic beating mode at Rabi frequency in the time domain. In spectra, it results the generation of two new mixed eigenstates separated by normal mode splitting, called Rabi splitting [6,7]. Strong coupling has huge potential applications in the fields of quantum information [8], all-optical logics [9,10] and controlling chemical reactions and interactions. Recently, many emerging material platforms have been demonstrated to obtain strong coupling, such as topological [11,12] and perovskite [13,14] and semiconductor heterostructures [15,16]. And researchers have successfully realized tunable strong coupling by using transition metal dichalcogenides materials [17,18], phase changes materials [19] and anionic fluorescent dye [20]. However, most studies of tunable strong coupling concentrate on single-mode, hence, tunable multi-modes strong coupling still remains to be realized.
In recent years, hexagonal boron nitride (h-BN) has received extensive attentions, owing to it has potential to alternate metals to become new building blocks of nanophotonic devices in mid-infrared to THz frequencies [21,22]. h-BN is a prototypical hyperbolic material, which exhibits anisotropic phonons and supports hyperbolic phonon polariton (HPhP) modes [21,23,24]. Especially, HPhP in h-BN has extremely long lifetimes [25,26], thus, it can be applied to strong coupling [27]. To date, the HPhP in h-BN can be excited at far-field by its manufacturing special structure, such as grating [28], ribbons [29], cones [21,30,31], thin flake with rectangular hole array [32] and heterostructures [33][34][35] or far-field Raman scattering process [36]. In addition, scanning near-field optical microscopy is used for exciting and detecting the HPhP in h-BN at near-field [25,37]. However, until now excitation of the HPhP in h-BN slab at far-field, has remained unexplored.
Strontium titanate (STO), as an excellent phase-change materials [38,39], its permittivity can be tuned by controlling external temperatures, which can be used for designing tunable metasurfaces [40][41][42]. In this work, we propose an all-dielectric hybrid structure combined with h-BN slab and all-dielectric metasurfaces to excite dual HPhPs and an additional optical (TO) phonon in h-BN slab, and achieved their switchable strong coupling with photons. For the first time to the best of our knowledge, utilizing h-BN slab at far field successfully excite HPhPs. The all-dielectric metasurfaces consist of STO rectangular bars and STO layer, therefore support the tunable guided-mode resonance due to the temperature dependence of the phase of STO. By changing the temperature, the guided-mode resonance wavelength can be tuned. Moreover, the electric field distributions E z are analyzed to demonstrate the excitation of dual HPhPs in hybrid system. Importantly, the hybrid system can implement strong coupling at dual HPhPs modes and TO phonon by analyzing the dispersion curves and the dynamic properties of reflectivity electric field. As a result, the strong coupling can be switched and tuned actively between dual HPhPs and TO phonon by tuning temperature of metasurfaces. h-BN as an anisotropic material can support two type HPhPs in the mid-infrared frequency range. Type I (ε < 0, ε ⊥ > 0) and type II (ε > 0, ε ⊥ < 0) HPhPs can accounts for out-of-plane (ω TO, = 780 cm −1 , ω LO, = 830 cm −1 ) and in-plane (ω TO,⊥ = 1370 cm −1 , ω LO,⊥ = 1610 cm −1 ) phonon mode, respectively. The real and imaginary part of the relative permittivity of h-BN (ε = ε x,y and ε ⊥ = ε z ) as shown in figure 2(a), and its dielectric function is given by [43]:

Hybrid structure of STO metasurfaces and h-BN slab
The relative permittivity of STO can be expressed as [44,45]: where ε ∞ ≈ 9.6 represent the high-frequency bulk permittivity, f is the temperature-dependant oscillator strength and the value is 2.6 × 10 6 cm −2 , ω 0 represent the soft mode frequency. ω and γ are the frequency and soft mode damping parameter, respectively. ω 0 and γ can be calculated as: The real and imaginary part of permittivity of the STO can be calculated by equations (2)-(4), and the results are illustrated in figure 2(b). The left and right axis represent the real and imaginary part of permittivity, respectively. By adjusting temperature, the relative dielectric constant of STO can be controlled due to temperature-dependant of ω 0 and γ. Thus, the STO can be applied to tunable metasurfaces. All-dielectric metasurfaces in this work consists of STO two-dimensional grating and STO layer, which can excite a guided-mode resonance mode. Obviously, as the temperature of STO increases from 225 K to 325 K, the real and imaginary part of permittivity both decrease in figure 2  of wavelength, the real part of permittivity has not significantly change, but the imaginary part of permittivity is decreasing. These result shows temperature-dependant of the relative dielectric constant of STO, hence, the guided-mode resonance of metasurfaces can be modulated via controlling the temperature (T ) of STO two-dimensional grating and STO layer.
The COMSOL Multiphysics is employed to investigate the proposed hybrid system. The unit cell periodical boundary conditions are imposed in x-axis and y-axis directions, and the perfectly matched layers are applied in z-axis directions. A x-polarized plane wave is a normal incident.

Analysis
In order to illustrate the underlying mechanism of the hyperbolic photon-phonon polaritons and photon-phonon polaritons in h-BN slab and metasurfaces, the dispersion distributions of the uniform h-BN structure and the metasurfaces with h-BN slab can be derived from the imaginary part of the Fresnel reflection coefficient r p . Utilizing the transfer matrix method, the complex reflectivity r p can be calculated  [46]. For n layers structure, yielding the following analytical expression for M: Here, R i, j and T j are the boundary condition matrices and propagation matrix, respectively r i, j and t i, j are the Fresnel reflection, transmission coefficient for the interface between layers i and j, respectively. k i,z is the out-of-plane wave-vector in layer i, and d i is the thickness of layer i. r i, j and t i, j can be given by: where ε i is the relative permittivity of layer i. Finally, the complex reflectivity r p for the n layers structure is given by a ratio of M ba and M aa The uniform h-BN slab structure are three layers, consisting of air/h-BN/air. Thus, the parameters used where q designates the magnitude of the wavevector in the x-y plane. For the metasurfaces with h-BN slab, there are five layers, consisting of air/STO two-dimensional grating/STO/h-BN/air. The effective permittivity of two-dimensional grating for p-polarization can be calculated as [47,48]: where f x = W P x and f y = L P y represent the fill factors in the x-axis and y-axis, ε 1 = 1 and ε 2 = ε STO are the relative permittivity of air and STO, respectively. The other parameters are ε 3 = ε STO , ε 4 = ε ⊥ , ε 5 = 1, The dispersion maps of uniform h-BN slab with the thickness d 2 = 250 nm, and the dispersion maps of the metasurfaces with h-BN slab at three different STO temperatures (300 K, 273 K and 250 K) can be obtained via the above calculations as depicted in figure 3. In the three the metasurfaces with h-BN slab cases, the thicknesses of STO two-dimensional grating and STO layer are set as d 2 = d 3 = 215 nm and the thickness of h-BN slab is d 4 = 250 nm. As illustrated by figure 3(a), HPhP modes are bounded inside the Reststrahlen band (RSB), and the numbers of the HPhP mode and the distributions in h-BN greatly influenced by its thickness. As shown in figures 3(b)-(d), for the metasurfaces with h-BN slab, the hyperbolic photon-phonon polaritons inside the RSB arise from the strong coupling between HPhP and photons. The photon-phonon polaritons out of the RSB is formed result of the strong coupling between photons and TO phonon polaritons. Moreover, the dispersion branch outside and inside the RSB move to higher wavelength with the STO temperatures decrease. The dispersion branch differences in the hybrid system at three different STO temperatures can be explained by temperature-dependant of the relative permittivity of STO. As the STO temperatures change, the real and imaginary part of permittivity will change so that the hybrid system is change in the energy loss.
The reflectivity spectra of metasurfaces at different temperature are illustrated in figure 4(a), where h 1 = 215 nm. It is shown that the resonance wavelength gradually red-shift with temperature of STO decrease. In order to investigate the metasurfaces resonance mechanism, the electric field distribution |E| are calculated and presented in figure 4(b). It can be clearly observed that guided-mode resonance occurs at resonance wavelength λ = 6.446 μm with the typical standing wave profile. While the incident wave diffracts in the two-dimensional grating, the wave vector can be modulated to match waveguide mode which result in the radiation of waveguide mode and guided-mode resonance can be successfully excited [5,[49][50][51]. In addition, the reflectivity spectrums of uniform h-BN layer are shown in figure 4(c) together with the spectrum got for the metasurfaces with h-BN, where h 1 = 180 nm, h 2 = 250 nm and T = 300 K. The corresponding metasurfaces resonance at same parameters is shown for comparison. For uniform h-BN layer, a sharp resonance dip can be observed at λ h−BN = 7.333 μm, which is transverse TO phonon of h-BN27. In the spectrum for the hybrid system, the dips of guided-mode resonance and h-BN transverse TO phonon and the other two dips are observed around λ M = 5.734 μm, λ TO = 7.299 μm, λ I = 6.484 μm and λ II = 6.853 μm, respectively. The guided-mode resonance and TO phonon in system hybrid both occurred frequency shift comparing the uniform h-BN and metasurfaces, due to the dielectric features of the h-BN. In order to investigate the origin of the two dips (λ I and λ II ), the electric field distributions E z at λ I and λ II are simulated in figures 4(d) and (e). Notably, the fields exhibit a 'zigzag' polaritonic rays pattern inside the h-BN layer (x-z plane), which is a feature of HPhP and demonstrate the excitation of dual HPhPs in hybrid system [25,32,37]. For the convenience of analysis, dual HPhPs mode are defined as HPhP I (λ I ) and HPhP II (λ II ), respectively. When metasurfaces resonance λ M is tuned at λ I , λ II or λ TO , the hybrid system implement strong coupling and energy oscillates between the dual HPhPs or TO phonon and the electromagnetic filed, respectively. Meanwhile, the periodic energy transfer (Rabi oscillations) happens at Rabi frequencies.
In order to verify strong coupling between the three modes (dual HPhPs and TO phonon) and the photons, the reflectivity spectra of hybrid system with different resonance wavelength of metasurfaces are simulated, where temperature of STO is T = 300 K. In figures 5(a)-(c), the metasurfaces resonance is tuned from 5.974 μm to 7.901 μm by varying the thickness of metasurfaces h 1 from 194 nm to 312 nm, where the x-axis and y-axis represent metasurfaces resonance and the spectra, respectively. The anti-crossing behavior as hallmark of strong coupling can be observed from the two polaritonic branches around λ I , λ II and λ TO in figures 5(a)-(c). Moreover, the two polaritonic branches converge nicely toward the λ I , λ II and λ TO , which mean the anti-crossing wavelengths approach the λ I , λ II and λ TO . With the resonance wavelength of metasurfaces increase, the photons mode successively coupled with the HPhP I, HPhP II and TO phonon modes, and two hybrid modes can be noticed from all reflectivity spectra in figures 5(d)-(f). The spectral positions and strength of two hybrid modes are modified by the detuning between the three modes (dual HPhPs and TO phonon) and the photons. At zero-detuning, the two hybrid modes exhibit distinct anti-crossing behavior and the Rabi splitting can be observed.
Furthermore, the dynamic properties of Rabi oscillations are investigated to further verify the hybrid system implement strong coupling. In this part, the T is fixed as 300 K. Three mid-infrared ultrafast Gaussian pulse (0.5 ps, 1.50 ps and 1.30 ps pulse width), respectively tuned to anti-crossing wavelengths λ I = 6.484 μm (h 1 = 215 nm), λ II = 6.853 μm (h 1 = 237 nm) and λ TO = 7.299 μm (h 1 = 260 nm), are used for simultaneously exciting the one mode of the HPhPs and TO phonon of h-BN layer and photons mode of metasurfaces. In this case, the rates of exchange energy between the three modes (dual HPhPs and TO phonon) and the photons becomes faster than the dissipation rates of other loss mechanisms, which resulting in periodic energy transfer between the h-BN layer and the metasurfaces. Therefore, the electric field of radiation manifest a periodic beating mode at Rabi frequency in the time domain. As depicted in figures 6(a)-(c), the time evolution of reflectivity pulses of coupled system exhibit oscillations behavior with periods of 1.00 ps, 3.75 ps and 4.50 ps, respectively. These oscillations are Rabi oscillations and directly reflect the periodic energy transfer between the three modes (dual HPhPs and TO phonon) and the photons, which demonstrate the features of strong coupling in time-domain [6,7,15].
On the other hand, the switchability of strong coupling modes can be realized, due to metasurfaces resonance wavelength can be tuned via controlling the temperature of STO bars and layer. Figure 6(d) shows the reflectivity spectra of the proposed hybrid system at different temperatures of STO bars and layer. With the temperature decrease from 300 K to 250 K, the metasurfaces resonance wavelength red-shifts, and the hybrid system achieved the strong coupling at λ I (T = 300 K), λ II (T = 273 K) and λ TO (T = 250 K) in turn. Therefore, strong coupling can be switched and tuned actively between dual HPhPs and TO phonon by selecting the corresponding temperature.

Conclusions
In conclusion, an all-dielectric hybrid structure with STO two-dimensional grating, STO layer and h-BN slab is proposed and demonstrated to implement the switchability of strong coupling between dual HPhPs (and an additional TO phonon) and the photons. The all-dielectric metasurfaces supporting guided-mode resonance consists of STO two-dimensional grating and STO layer. The dispersion relations are analyzed to demonstrated the all-dielectric metasurfaces with h-BN slab at different STO temperatures can support hyperbolic photon-phonon polaritons inside RSB region and photon-phonon polaritons outside of the RSB region. The guided-mode resonance wavelength can be tuned via controlling the temperature of STO two-dimensional grating and layer. In addition, the TO phonon is obtained in uniform h-BN slab. The result of electric field distribution E z demonstrates the excitation of dual HPhPs in hybrid system. Moreover, the dispersion curves and the dynamic properties of reflectivity electric field both confirm the all-dielectric hybrid system can implement strong coupling at dual HPhPs and TO phonon mode. Consequently, owing to the tunability of guided-mode resonance, the strong coupling can be switched and tuned actively between the three modes (HPhP I, HPhP II and TO phonon) and photons via selecting the corresponding temperature. The proposed all-dielectric hybrid system has numerous potential applications on multi-channel biosensors, filters and tunable source and detectors.

Funding
National Natural Science Foundation of China (11874019); Natural Science Foundation of Guangdong Province under Grant (2019A1515011172).