1. Introduction

Strong coupling between light and matter is an ongoing topic in nanophotonics as it promises various applications including but not limited to quantum manipulation [1], ultra-fast optical switching [2], low-threshold semiconductor lasers [3], Bose-Einstein condensation [4]. To reach strong coupling, the coherent energy exchange rate between light and matter needs to surpass their average dissipation rate. A considerably large Rabi splitting on the spectral response of a photonic system is the hallmark of strong coupling along with generating new hybrid modes known as light-matter hybrid polaritons [5,6]. In the past decades, a variety of hybrid photonic systems have been exploited to realize strong coupling such as coupled plasmonic and Fabry-Pérot (F-P) cavities [7–10], plasmonic nanoantennas loaded with quantum dots, Si nanodisks covered with uniform molecular J-aggregates [11], etc. Among these hybrid systems, optical microcavities coupled with transition metal dichalcogenide (TMDC) monolayers have attracted particular interests [12–15] owing to the intriguing excitonic features of TMDC monolayers including large binding energies and sharp resonances. In this respect, the most extensive studies are strong coupling between plasmonic resonances and excitons, namely plasmon-exciton strong coupling, which has been realized in both individual and periodic metallic nanostructures integrated with TMDC monolayers [16–19]. In a similar configuration, strong coupling between Mie-type resonances and excitons has been observed in individual Si nanoparticles covered with WS₂ monolayers [20,21]. In addition, it has also been demonstrated that photon-exciton strong coupling can be achieved in TMDC monolayers inserted in F-P cavities with metallic or DBR mirrors [22–24].

Besides the intriguing excitonic features, TMDC layers inherently possess a high refractive index (n > 4) in the visible regime [25,26], which offers the possibility to generate transverse electric (TE) polarized waveguide modes in a TMDC layer with sub-nanometric thickness. However, theoretically speaking such TE polarized waveguide mode in an unpatterned TMDC membrane is a bonded mode which is challenging to be probed by far-field measurements. Very recently, this issue has been settled by pattering TMDC membrane into photonic crystal (PhC) slabs with ångström thickness, and then the TE waveguide mode dispersion is folded to generate guided-mode resonances (GMRs) that can be probed via far-field measurements [28]. Compared to an unpatterned TMDC membrane, it is then expected that the coupling between a TMDC PhC slab and an optical microcavity will be more subtle because GMRs must be involved in addition to exciton resonances. It is also worth noting that the spectral responses of the GMR of a TMDC PhC membrane is readily to be tuned by adjusting the period, which is quite a desirable feature in terms of studying its coupling effect with other modes.

Considering the study of coupling effects between a TMDC PhC slab and an optical microcavity is still lack, here in this work we investigate a hybrid WS₂ PhC−F-P cavity system aiming to study the interactions between F-P cavity modes, exciton emissions, and GMRs. We first show that the strong coupling between exciton resonance and F-P cavity modes can be achieved with a Rabi splitting of 51 meV at room temperature. We further found that GMRs of the WS₂ PhC can also interact with F-P cavity modes and their coupling strength intimately depends on the geometry parameters of the hybrid system. At an optimal condition, a strong coupling between GMRs and F-P modes is reached with a Rabi splitting as large as 63 meV. Finally, we deliberately blue shifts the GMR to be spectrally overlapped with the exciton resonance by decreasing the period of the WS₂ PhC. Under this condition, a triple mode anti-crossing behavior is observed near the degenerate energy when changing the cavity length. By fitting the spectral response with a three-coupled oscillator model, the Rabi splitting between the F-P mode, the exciton, and the GMR is obtained as 120 meV which we believe reaches a strong coupling between three modes. Our results unambiguously demonstrate that GMRs of a TMDC PhC provide an additional degree of freedom for the manipulation of collective optical responses of a TMDC–optical cavity hybrid system.

2. Two mode strong coupling between F-P mode and GMR (exciton)

Figure 1(a) schematically shows the hybrid structure investigated in this study, which consists of an F-P cavity constructed by two 40 nm thick Ag mirrors and a WS₂ PhC inserted in the cavity. The PhC is a patterned five-layer (5 L) WS₂ membrane with total thicknesses (denoted as t) of 3.09 nm and air holes in a 2D square array on x-y plane. The radius of all the air holes is kept as 250 nm and the period (Λ) of the square array is assumed to be 700 nm. In the following theoretical and numerical studies, the complex permittivity of Ag is fitted based on the experimentally measured data given in [27], and that of the WS₂ membranes with layer numbers from 1 L to 5 L is fitted based on the experimentally measured data given in [28]. The hybrid structure is excited by an x-polarized plane wave propagating along the -z direction. Without loss of generality, the surrounding medium of the whole structure and that between the WS₂ and Ag mirrors is supposed to be air. To study the coupling effects between different modes in such hybrid structure, we first inspect the resonances of the corresponding uncoupled components, i.e. the bare WS₂ PhC and the empty F-P cavity. Figure 1(b) shows the numerically calculated transmission spectra of the bare WS₂ PhC (blue lines) and reflection spectra of the F-P cavity modes (green dotted and dashed lines) with two selected cavity lengths by using a commercial simulation package FDTD solution. It is seen that the transmission spectrum of the bare WS₂ PhC has two dips that correspond to the exciton resonance near 620 nm and the GMR near 710 nm, respectively. The upper right inset in Fig. 1(b) shows the profile of the electric field amplitude in a single unit cell of the WS₂ PhC at the transmission dip of the GMR. It is seen that the electric field amplitude exhibits an interference pattern in y-direction, which indicates such GMR corresponds to the resonance of guided waves with diffraction orders of $({0, \pm 1} )$. As to the empty F-P cavity, when the cavity length is 870 nm the F-P mode contributes to a reflection dip around the same spectral range as that of the exciton of the bare WS₂ PhC. When increasing the cavity length to 970 nm, the F-P mode red shifts to about 675 nm which is slightly shorter than the resonant wavelength (∼710 nm) of the GMR of the bare WS₂ PhC.

 

Fig. 1. (a) Schematic view of the hybrid structure composed of a WS₂ PhC and an F-P cavity with two 40 nm thick Ag films as mirrors. The WS₂ PhC is a 5 L WS₂ membrane drilled with air holes in a 2D square array with period of Λ = 700 nm and the radius of air holes are r = 250 nm. The length of the F-P cavity is H and the WS₂ PhC is placed at a distance of Z from the bottom mirror. The whole structure is excited by a normally incident x-polarized plane wave. (b) Numerically calculated transmission spectra of a bare 5 L WS₂ PhC (blue lines) and reflection spectra of the empty F-P cavities of two selected cavity lengths (green dotted and dashed lines). The upper right inset shows the profile of electric field amplitude of the GMR in a single unit cell of the 5 L WS₂ PhC membrane where the black dashed circle represents the boundary of the air hole. The lower right inset shows the profile of electric field amplitude of the 3^th F-P mode where the white dashed lines represent the boundaries of the Ag mirrors towards the cavity. (c) Reflection spectra of the hybrid structures with the selected cavity lengths. (d) Reflection spectra for hybrid structures when the cavity length H changes from 700 to 2500 nm. The vertical white-dash-dotted lines mark the resonant wavelength of the exciton resonance and the GMR. The horizontal green-dotted lines mark the cavity lengths of 870 nm and 970 nm.

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We then calculated the reflection spectra of the hybrid structures by placing the WS₂ PhC at the center ($Z = {H / 2}$) of the F-P cavities with the above selected cavity lengths, as shown in Fig. 1(c). For the hybrid structure with cavity length of 870 nm (dashed line), the coupling between the exciton resonance and the F-P mode results in new hybrid polariton states manifested as two reflection dips that are spectrally separated about 51 meV. Interestingly, for the hybrid structure with cavity length of 970 nm the reflection spectrum (solid line) also shows two sharp dips separated about 63 meV with almost the same depth. This implies that significantly coupling between the GMR of the WS₂ PhC and the F-P mode happens though their resonance wavelengths obtained from the uncoupled structures are not exactly the same.

To gain further insight of the coupling effects between different modes in the hybrid structure, Fig. 1(d) shows the calculated contour plot of the reflection as a function of the wavelength (x-axis) and the cavity length (y-axis). First, it is seen that F-P modes of different orders contribute to the predominant bands of reflection dips of which the resonant wavelengths vary as the cavity length changes. Second, multiple spectral anti-crossing can be observed at several different wavelengths. Specifically, at the wavelength of the exciton resonance, i.e. 620 nm as marked by the vertical white-dotted line, anti-crossing appears but only on the bands of the odd F-P modes, indicating that only F-P modes of odd orders can effectively couple to the exciton resonance. Meanwhile, as the cavity length increases the coupling strength between F-P modes and the exciton resonance reduces, which results in smaller Rabi splitting. Similarly, the coupling between the GMR and F-P modes of odd orders leads to more distinct anti-crossing features on the bands of reflection dips around 710 nm. However, a different feature must be noted that the center of the Rabi splitting between the GMR and the F-P modes blue shifts as the cavity length decreases. This observation implies that the resonant wavelength of the GMR is also affected by the cavity length. For a bare WS₂ PhC under normal incidence, one can expect that the resonant wavelength of the GMR is mainly determined by the period (Λ) of the lattice. When put the WS₂ PhC into an F-P cavity, the influence of image charges induced in the cavity mirrors on the resonant wavelength of the GMR must be considered. This can be verified by noticing that the center of Rabi splitting eventually red shifts to 710 nm as the cavity length increases over 2400 nm. For shorter cavity length, for instance 970 nm, the center of Rabi splitting significantly blue shifts to about 675 nm because of the strong interaction with image charges. And for even shorter cavity length, the affection of image charges will be strong enough to completely kill the GMR. The anti-crossing appears at other regions as marked by the white boxes in Fig. 1(d) can be attributed to the coupling between high order GMRs and F-P modes, which we will not pay much attention. Here we stress that it is possible to observe the coupling (spectral anti-crossing) between exciton or GMR and the F-P modes of even orders only if the WS₂ PhC is placed offset respect to the center of the cavity. This is because the center of the cavity is exactly a node of the standing wave of an even order F-P mode. On the contrary, the center of the cavity is an antinode of the standing wave of an odd order F-P mode. To illustrate this point, the profile of the electric field amplitude of the 3^th order F-P mode is given in the lower right inset in Fig. 1(b), where the electric field amplitude maximizes at the center of the cavity. Details about how the coupling strength between the F-P mode and the GMR is modulated by the position of the WS₂ PhC will be discussed later.

To unravel the underlying physics of the above observations, we employ the coupled oscillator model (COM) to analyze the spectral responses of the hybrid structure [29,30]. For the coupling between the exciton and the F-P cavity modes, the Hamiltonian of the COM is given in Eq. (1) when ${e^{i\omega t}}$ is applied for the time dependence of a harmonic wave with angular frequency $\omega $.

Rabi splitting (Ω) can be obtained from Eq. (2) by implementing E_FP = E_Exciton, i.e.:

In Eq. (2), E_FP as a function of the cavity length is readily known based on the F-P resonance condition, and E_Exciton is fixed as 1.98 eV as it is the inherent excitonic feature of the 5 L WS₂. After fitting the reflection dips of the hybrid modes by Eq. (2), coupling strength g is obtained to further calculate the Rabi splitting (Ω) based on Eq. (3). In Fig. 2(a), we append the results of the COM on the zoomed reflection contour plots (Fig. 1(d)) around the anti-crossing between the exciton resonance (blue lines) and the F-P modes (green lines) of 3^th, 5^th and 7^th order. It is seen that curves of eigenvalues (black-dashed lines) calculated based on Eq. (2) as the function of the cavity length match well with the reflection dips of the hybrid modes. Taking the coupling between the 3^th order F-P cavity modes and exciton resonance as an example (left panel of Fig. 2(a)), the Rabi splitting is calculated as 51 meV based on Eq. (3). It is well known that the criteria of two mode strong coupling is the Rabi splitting must exceed their average dissipation rate, i.e. Ω > (γ_FP + γ_Exciton)/2. Here γ_FP (γ_Exciton) can be obtained by fitting the reflection (transmission) spectra of the empty F-P cavity (bare WS₂ PhC) using a single resonance Lorentz formula. However, we notice that the spectra in Fig. 1(b) near the resonances of both the F-P modes and the GMR exhibit a slightly asymmetric profile which is the sign of a Fano resonance. On the basis of this observation, we also fit the spectra in Fig. 1(b) near the resonances of F-P modes and the GMR using a single resonance Fano formula given in [31]. For comparison, the fitted damping rates of different modes by the Lorentz formula (${\gamma _{\textrm{Lorentz}}}$) and the Fano formula (${\gamma _{\textrm{Fano}}}$) are given in Table 1. It is seen that ${\gamma _{\textrm{Fano}}}$ is only slightly smaller than ${\gamma _{\textrm{Lorentz}}}$ for the F-P modes of lower orders and the GMR. Therefore, in the following discussions we only need to compare the Rabi splitting with the larger average dissipation rate calculated based on ${\gamma _{\textrm{Lorentz}}}$. For the 3^th order F-P mode and the exciton resonance, the average dissipation rate is calculated as 28.75 meV (γ_FP = 10 meV and γ_Exciton= 47.5 meV), confirming that the coupling between them reaches strong coupling regime. The modulus squares of eigen coefficients, i.e. |α|² and |β|², are known as Hopfield coefficients that represent the proportion of uncoupled states in the mixed state. The insets in the left panel of Fig. 2(a) show the Hopfield coefficients of the upper (denoted as U_p) and lower (denoted as Lp) polariton states as the function of the cavity length. It shows that the Hopfield coefficients of the exciton state and F-P state have opposite trends as the cavity length changes for the Up and Lp polariton states, and both equal to 0.5 at the center of the Rabi splitting. As a matter of fact, the coupling strength between two resonances can be straight forwardly calculated by the overlap integral of their electromagnetic fields based on Eq. (4) [32]:

 

Fig. 2. (a) Zoomed reflection contour plots around the anti-crossing between the exciton resonance (blue lines) and the F-P modes (green lines) of 3^th, 5^th and 7^th order from left to the right panel. (b) Zoomed reflection contour plots around the anti-crossing between the GMRs (yellow lines) and the F-P modes (green lines) of 3^th, 5^th and 7^th order from left to the right panel. The black dashed lines are the results of the COM and the insets give the Hopfield coefficients for the U_p (top-left) and L_p hybrid polariton states as a function of H.

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Table 1. Comparison of the fitted damping rates by Fano and Lorentz formulas (cavity lengths are given in the brackets below the name of the F-P mode)

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To distinguish with g in Eq. (1), here the couple strength is denoted as $\kappa $, H_i (E_i) represents the electric field of the i-th mode, ɛ = max (ɛ₁, ɛ₂) where ɛ_i is the dielectric profiles of the i-th mode, and ɛ_bg = ɛ₁ + ɛ₂ – ɛ is the background dielectric profile. The coupling strength κ between the 3^th order F-P cavity mode and the exciton resonance calculated based on Eq. (4) is 30 meV that is close to g (25.5 meV) obtained from the COM, i.e. Equation (2), which confirms the validity and accuracy of the above analyzing method. Similarly, parameters related to the coupling between the exciton resonance and F-P cavity modes of other orders (5^th and 7^th) are obtained based on Eqs. (2–4) and listed in Table 2. It is seen that the coupling between the exciton resonance and the F-P cavity modes of 3^th, 5^th and 7^th order all reaches strong coupling regime, though the coupling strength is declining as the cavity length increases.

Table 2. Parameters of exciton – F-P coupling and GMR – F-P coupling (${E}_\textbf{{FP}}^0$ represents the theoretical resonant energy of the F-P modes while E_FP is the actual value used in the COM, A. D. R is short for the average dissipation rate)

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When the F-P cavity mode is around 700 nm, the GMRs are coupled to the F-P modes of odd orders as shown by the zoomed reflection contour plots near the anti-crossing points in Fig. 2(b). It is seen that results of the COM (black-dashed lines) is consistent with the reflection dips of the hybrid modes as well. More details about the coupling between the GMRs and F-P cavity modes can be found from the parameters listed in Table 2. First of all, it is apparently that the coupling between the GMRs with the F-P modes of all the considered odd orders reaches strong coupling regime. More importantly, their strong coupling is more prominent than that between the exciton resonances and F-P modes in terms of the magnitude of Rabi splitting. For example, the Rabi splitting between the GMR and the 3^th order F-P cavity is 63 meV which is much larger than the average dissipation rate (γ_FP + γ_GMR)/2, where γ_GMR ≈ 3 meV and γ_FP ≈ 8 meV. This can be understood by the fact that as a quasi-bonded mode the GMR has much smaller radiative loss and therefore quite large quality factor (Q) as verified by the extremely narrow transmission peak shown in Fig. 1(b) for the spectrum of a bare WS₂ PhC. Moreover, fields of the GMR of the WS₂ PhC are confined around the membrane, resulting in a small model volume which is also beneficial to a large Rabi splitting. Results of Fig. 2(b) are one of the most important contributions of our work, which clearly shows that the GMR in a WS₂ PhC is quite suitable for realizing strong coupling in TMDC photonic hybrid resonant structures.

Next we study how the strong coupling between the GMR and the F-P modes can be modulated by the geometry of the structure. Figure 3(a) shows the reflection contour plot of the hybrid structure as a function of the wavelength and the position of the WS₂ PhC (Z) when the cavity length is kept as 970 nm. It is seen that there is a variation of Rabi splitting as Z changes. In detail, when the WS₂ PhC resides at the antinodes of the standing wave of the F-P mode of odd orders, for example at the center of the cavity (Z = 485 nm), Rabi splitting reaches the maximum (marked by the triangle). Correspondingly, when the WS₂ PhC resides at the nodes of the standing wave (Z = 660 nm or Z = 300 nm), the GMR cannot even be excited (marked by the circles) since the amplitudes of the fields vanish at these positions. In between the nodes and antinodes, Rabi splitting varies following the intensity profile of the standing wave. The coupling coefficient κ as the function of the position Z shown in Fig. 3(b) further confirms that the variation of the Rabi splitting is a result of the modulation of coupling strength. Another geometry parameter that may have impacts on the Rabi spitting is the radius of air holes since it is one of the factors determining the complex effective mode index of the guided-mode. Figure 3(c) shows reflection contour plot of the hybrid structure as a function of the wavelength and the radius of air holes. It is seen that the center of the Rabi splitting blue shifts as the radius of air holes increases from 100 nm to 350 nm, which can be explained by the modulation of the resonance wavelength of the GMR by the radius of air holes. More interestingly, the Rabi splitting shows a non-monotonic variation as the radius of air holes changes but has an optimum at radius of 250 nm that corresponds to the radius where the coupling coefficient $\kappa $ reaches the maximum as shown in Fig. 3(d).

 

Fig. 3. Modulating the strong coupling between GMRs and F-P modes by the geometry of the hybrid structure. (a) Reflection spectra for the hybrid structure of H = 970 nm when the position of the WS₂ PhC Z is varied from 120 to 840 nm. (b) Coupling coefficient ($|\kappa |$) as a function of Z. (c) Reflection spectra for the hybrid structure of H = 970 nm when the radius of air holes is varied from 100 to 350 nm. (d) Coupling coefficient ($|\kappa |$) as a function of the radius of air holes.

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The last parameter that we want to investigate is the layer number of the WS₂ PhC because it affects the refractive index of the membrane. Figure 4(a) shows the transmission spectra of WS₂ PhC with varying layer numbers from 1 L to 5 L. It is seen that as the same as [28], the transmission dips associated with the GMRs red shift slightly and broaden as the layer number increases, which means a more efficient coupling of the GMR to the excitation field. As a result, hybrid states of the hybrid structure red shifts and the Rabi splitting becomes larger as seen in Fig. 4(b). Correspondingly, the coupling coefficient κ shows a similar uptrend as the layer number increases as shown in Fig. 4(c).

 

Fig. 4. (a) Transmission spectra of the WS₂ PhC with varying thickness from 1 L to 5 L. Inset: The enlarged plots of the 1 L and 2 L WS₂ PhCs at GMRs. (b) Reflection spectra of the hybrid structure with different layer numbers of the WS₂ PhC. (c) Coupling coefficient ($|\mathrm{\kappa } |$) between GMRs and F-P modes as a function of the number of layers.

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3. Triple mode strong coupling between F-P mode, GMR and exciton

Finally, we show that it is possible to realize a triple mode strong coupling through tuning the resonance wavelength of the GMR to spectrally overlap with that of the exciton resonance by adjusting the period Λ of the WS₂ PhC. Figure 5(a) shows the transmission of a bare 5 L WS₂ PhC as a function of the wavelength and the period Λ. Apparently, the resonance wavelength of the guided-mode is readily to be tuned by changing Λ. More importantly, it is found that the resonance wavelength of the guided-mode can spectrally overlap with that of the exciton resonance at Λ = 625 nm while keeping a very weak coupling between them. This is further confirmed by the transmission spectrum (blue line) of the bare 5 L WS₂ PhC with Λ = 625 nm shown in the upper panel of Fig. 5(b), where only a single transmission dip is present. When put this 5 L WS₂ PhC into the F-P cavity with H = 860 nm, a triple mode coupling involving exciton resonance, the GMR and the F-P cavity mode happens, resulting in three hybrid modes that correspond to three spectrally separated transmission dips as show in the lower panel of Fig. 5(b). To further study such triple mode coupling, Fig. 5(c) shows the reflection spectra of the hybrid structure against the cavity length H when Λ = 625 nm. It is seen that the triple mode coupling manifested as an interesting anti-crossing and produces three hybrid polariton states which we name them as upper (Up), middle (Mp) and lower (Lp) polariton state in turn from the short to the long wavelength. To analyze this triple mode Rabi splitting, we use the COM with three oscillators, where the Hamiltonian reads [33,34]:

 

Fig. 5. (a) Transmittance spectra for 5 L WS₂ PhC with different array periods. (b) Upper panel: transmission of the bare 5 L WS₂ PhC with Λ = 625 nm (blue line) and reflection spectra of the empty F-P cavity (green dashed line) with H = 860 nm; Lower panel: reflection spectra of the corresponding hybrid structure. (c) Reflection spectra of the hybrid structure with Λ = 625 nm when the cavity length H changes from 700 to 950 nm. The yellow, blue and green lines represent the resonance wavelengths of GMRs, exciton resonances, and F-P modes, respectively. Black dotted lines correspond to theoretical fits based on the COM. (d) Hopfield coefficients as a function of the wavelength for F-P modes, GMRs and exciton resonances of the upper (Up), middle (Mp) and lower (Lp) polaritons.

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Table 3. Parameters of Exciton − GMR – F-P triple mode coupling.

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To determine whether the interaction between these three modes reaches the strong coupling regime, a criterion of the triple mode strong coupling must be applied [33,34]:

Using the Hopfield coefficients in Fig. 5(d), the damping rate of each hybrid state is obtained as:

By substituting the fitted damping rates of the F-P mode (γ_FP ≈ 10 meV), the GMR (γ_GMR ≈ 38.6 meV), and the exciton (γ_Exciton ≈ 38.6 meV) into Eq. (8), the damping rate of each hybrid state can be calculated. Then, the criterion of a triple mode strong coupling is finally obtained as Ω> 29.3 meV. According to the Rabi splitting listed in Table 3, i.e. Ω = 120 meV, we can declare that a tripled mode strong coupling between the F-P mode, the GMR and the exciton resonance is reached in the hybrid structure. We also emphasis that the Rabi splitting of the triple mode coupling is significantly enhanced compared with the coupling of the two modes, which may find applications in such as biosensing and multi-model nanolasing.

4. Conclusion

In conclusion, optical responses of a hybrid structure consisting of a WS₂ PhC and an F-P cavity are investigated in this work, which show interesting strong coupling effects between exciton emissions, GMRs and F-P cavity modes. In addition to the coupling between F-P modes and exciton emissions, GMRs of WS₂ PhCs can also interact with other modes, which provides extra freedoms to manipulate the collective optical responses of the hybrid structure. We have used the COM and overlap integral to analyze the strong coupling between different modes and obtain the optimal Rabi splitting of 51 meV, 63 meV and 120 meV for the F-P—exciton coupling, the F-P—GMR coupling, and the triple mode coupling, respectively. Compared with the exciton resonances, we show that the GMRs have much higher Q factor and smaller modal volume that both can boost the strong coupling effect. The detailed numerical results of optical responses of the hybrid structure as a function of geometry parameters can be a guidance for relevant experimental designs and studies. It is reasonable to expect that similar strong coupling effects can be observed in other optical cavities embedded with TMDC PhCs, thus our results open a new avenue for studying multi-mode strong coupling based on TMDC materials.