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Abstract
Chiral plexcitonic systems exhibit a novel chiroptical phenomenon, which can provide a new route to design chiroptical devices. Reported works focused on the two-mode strong coupling between chiral molecules and nanoparticles, while multiple-mode coupling can provide richer modulation. In this paper, we proposed a three-mode coupling system consisting of a chiral Au helices array, a Fabry-Pérot cavity, and monolayer WSe2, which can provide an extra chiral channel, a more widely tunable region, and more tunable methods compared to two-mode coupled systems. The optical response of this hybrid system was investigated based on the finite element method. Mode splitting observed in the circular dichroism (CD) spectrum demonstrated that the chiroptical response successfully shifted from the resonant position of the chiral structure to three plexcitons through strong coupling, which provided a new route for chiral transfer. Furthermore, we used the coupled oscillator model to obtain the energy and Hopfield coefficients of the plexciton branches to explain the chiroptical phenomenon of the hybrid system. Moreover, the tunability of the hybrid system can be achieved by tuning the temperature and period of the helices array. Our work provides a feasible strategy for chiral sensing and modulation devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Chirality is a ubiquitous phenomenon in nature, which refers to the property that an object cannot superimpose with its mirror image by simple rotation and translation [1]. Although chirality is proposed as a geometric concept, it has been found to play an important role in biology, chemistry, and electromagnetism. For example, opposite enantiomers of the same molecule may exhibit pharmacological and toxic properties, respectively [2,3]. Furthermore, chiral objects exhibit different absorption for left (LCP) and right (RCP) circularly polarized light, which is so-called circular dichroism (CD) [4]. Benefiting from this phenomenon, CD spectroscopy is widely used to measure chirality [5].
Chiral molecules are the universal chiral objects in nature, such as DNA, amino acids, and peptides. But the resonance in the ultraviolet band and the weak chiroptical response limit their application and detection [6,7]. Many works demonstrated that combining metal nanoparticles and chiral molecules can enhance chiroptical responses and induce CD signal at resonance of nanoparticles [8–10]. This phenomenon benefits from Coulomb interactions between molecules and nanoparticles [11,12]. Chiral assembly, which uses the chiral molecules as the template to achieve the chiral arrangement of metal nanoparticles, is also a popular method [13–15]. This approach uses the strong optical response of nanoparticles to achieve chiral amplification. Meanwhile, changing the arrangement and the geometry of nanoparticles can tune the chirality of this system [16,17]. With the development of manufacturing technology, chiral plasmonic nanostructures have been found to exhibit a strongly chiroptical response, such as chemically synthesized chiral metal nanoparticles [18–20], and chiral metasurfaces [21–23]. But most of these researches are confined to the weak interaction region. New hybrid states will emerge when the light-matter interaction is at a strong coupling region, and the hybrid states will have both properties of coupled components [24–26]. So if chiral objects participate in the strong light-matter interaction,the hybrid states will perform chiral [27–29]. This mechanism offers a new route to realize chiral transfer and tuning.
This paper investigates a chiral structure-microcavity-exciton coupling system composed of a chiral Au helices array, a Fabry-P$\mathrm {\acute {e}}$rot cavity (FP-cavity), and monolayer $\mathrm {WSe_{2}}$ to explore the chiroptical responses of plexcitons. The optical response of the system is obtained by numerical simulation with COMSOL based on the finite element method. Firstly, the chiral medium is successfully modeled by modifying weak expression and boundary conditions. We confirm the three-mode coupling system can realize chiral transfer by coupling the modeled chiral medium to the FP-cavity and $\mathrm {WSe_{2}}$. Secondly, we designed an actual chiral nanostructure, the Au helices array, to replace the modeled chiral medium. Mode splitting in the circular dichroism (CD) spectrum demonstrates that the chiral response successfully shifts from the resonant position of the chiral structure to three plexcitons through strong coupling. Thirdly, we employed the coupled oscillator model to analyze the coupling behavior and obtain the energy and Hopfield coefficients of plexciton branches. And according to the calculation, we explain the chiroptical responses in the hybrid system. Finally, we explored the tunability of the system by tuning the period of the array and the temperature. Moreover, compared to the two-mode coupling system, the three-mode coupling system proposed by our work exhibits an extra chiral channel, a more widely tunable region, and more tuning methods.
2. Theory and model
As schematically presented in Fig. 1, this hybrid system is composed of a monolayer $\mathrm {WSe_{2}}$ and a layer of chiral objects positioned inside an Au Fabry-P$\mathrm {\acute {e}}$rot cavity (FP-cavity). To analyze the property of the three-mode coupling system, we first modeled the chiral object as a layer of chiral medium. And COMSOL is used to obtain the optical response of the hybrid system. Modeling method of the chiral medium we used here is by combining Maxwell’s equations with constitutive relations of chiral medium to derive the wave equation. And according to this wave equation, we modify the weak expression and boundary condition in COMSOL [30]. The constitutive relations of the chiral medium, which is non-magnetic bi-isotropic, we used here are [28]: $\left (\begin {array}{l} \mathbf {D} \\ \mathbf {B} \end {array}\right )=\left (\begin {array}{cc} \varepsilon \varepsilon _{0} & -i \kappa / c \\ i \kappa / c & \mu _{0} \end {array}\right )\left (\begin {array}{c} \mathbf {E} \\ \mathbf {H} \end {array}\right )$ , where $\varepsilon _0$ and $\mu _0$ are the vacuum permittivity and permeability, respectively, $c$ is the speed of light, $\varepsilon$ is relative permittivity, $\kappa$ is the Pasteur parameter which describes the chiral property.
Fig. 1. Schematic diagram of the coupling system illuminated by circularly polarized light. Two 30 nm thick Au mirrors form the FP-cavity, and the length between them is L. A 10 nm thick chiral layer and a 1 nm thick monolayer $\mathrm {WSe_{2}}$ are positioned at the center of the cavity. And the material of other place inside the cavity is $\mathrm {SiO_{2}}$.
Combining chiral constitutive relations with Maxwell’s equations to derive wave equation in chiral medium, and considering the electric field in COMSOL is $\mathrm {\mathbf {E}(\mathbf {r},t)=\mathbf {E}(\mathbf {r})\cdot e^{i{\omega }t}}$, we can get the chiral wave equation:
Equation (1) changes to the achiral wave equation when $\kappa =0$. Then we multiply the test function $\mathbf {\Lambda }$ in both sides of the partial differential equation and do integral [31]. And we can get the weak form :
Here $\mathbf {e_{n}}$ is the normal unit vector of the integral interface. The weak form Eq. (2) includes two parts, one is the volume integral part, and the other is the surface integral part. The surface integral part can be limited by the boundary condition, so the weak expression of Eq. (2) in COMSOL is set as follows :
As for the boundary conditions for Eq. (3), the surface integral in Eq. (2) and the continuity of the electromagnetic field are both considered. So we should set surface current density at the interface of the chiral medium as an extra boundary condition :
Here $\kappa _1$ and $\kappa _2$ refer to the chiral parameter of the medium on both sides of the interface. $n_{1,2}$ is the normal unit vector of the boundary between the first and second medium. In our system, as shown in Fig. 1, for the upper interface of the chiral medium, the media on either side of the interface are $\mathrm {SiO_{2}}$ and the chiral medium, respectively. Therefore, in the upper interface of the chiral medium, the chiral parameters are set as $\kappa _{1}=\kappa _{\mathrm {SiO_{2}}}$, $\kappa _{2}=\kappa$. As for the bottom interface, the media on either side of the interface are the chiral medium and $\mathrm {WSe_{2}}$, respectively. So, the chiral parameters are set as $\kappa _{1}=\kappa$, $\kappa _{2}=\kappa _{\mathrm {WSe_{2}}}$ for the bottom interface. The $\mathrm {SiO_{2}}$ and $\mathrm {WSe_{2}}$ are achiral, so their chiral parameters can be set as $\kappa _{\mathrm {SiO_{2}}}$=$\kappa _{\mathrm {WSe_{2}}}$=0. And the $\kappa$ is the chiral parameter of the chiral medium. Therefore, the extra boundary condition of the upper interface can be set as $J_s=\frac {i\kappa }{c\mu _0}\left (\mathbf {n}_{\mathbf {1},\mathbf {2}}\times \mathbf {E}\right )$. As for the bottom interface the extra boundary condition is set as $J_s=-\frac {i\kappa }{c\mu _0}\left (\mathbf {n}_{\mathbf {1},\mathbf {2}}\times \mathbf {E}\right )$.
The permittivities of gold mirrors are from Johnson and Christy’s data [32]. And the parameter of chiral medium is modeled by the Lorentzian permittivity [28]: $\varepsilon _{c}=\varepsilon _{B}+f_0\frac {\omega _{0}^{2}}{\omega _{0}^{2}-\omega ^{2}-i\gamma _{0} \omega }$, and the Pasteur parameter has a similar form : $\kappa =\kappa _{0}\frac {\omega _{0}^{2}}{\omega _{0}^{2}-\omega ^{2}-i\gamma _{0} \omega }$, where $\omega$ is the angular frequency of incident light. The resonant wavelength of the chiral medium is set as 800 nm whitch corresponds to the resonant angular frequency $\omega _{0}$=1.5498 eV. And $\gamma _{0}$=50 meV is the damping parameter describing the absorption resonance broadening. The permittivity of background is $\varepsilon _{B}$=2.25, $f_{0}$=0.5 is amplitude of the resonance, and $\kappa _{0}$=0.03 is the amplitude of Pasteur parameter. As for the dielectric function of the monolayer $\mathrm {WSe_{2}}$ can be described by the Lorentz models: $\varepsilon _{1}=\varepsilon _{B}+f_{1}\frac { \omega _{1}^{2}}{\omega _{1}^{2}-\omega ^{2}-i \gamma _{1}\omega }$, where $f_{1}$=0.5, $\omega _{1}$=1.659 eV($\lambda$=747.3 nm), and $\gamma _{1}$=43 meV according to the data in [33].
In this system, the thickness of gold mirrors, chiral layer, and $\mathrm {WSe_{2}}$ respectively are $\mathrm {h_{Au}=30 nm}$, $\mathrm {h_{chiral}= 10 nm}$, $\mathrm {h_{WSe_{2}}=1 nm}$. The thickness of $\mathrm {WSe_{2}}$ is set to 1 nm rather than the reported 0.7 nm [33]. Such a small variation of thickness slightly affects the absorption intensity of $\mathrm {WSe_{2}}$, but it has little impact on the strong coupling process. It is because the strong optical responses of the other coupled components would cover up the slight variation. To improve the accuracy of simulations, we choose 1 nm as the thickness of $\mathrm {WSe_{2}}$. The wavelength of incident light is 625$\sim$925 nm to cover the resonant wavelength of both the chiral medium and $\mathrm {WSe_{2}}$. For the same reason, the cavity length L is set to 180$\sim$220 nm. And their optical responses are presented in Fig. 2.
Fig. 2. (a) The transmittance spectrum of the Au FP-cavity with $\mathrm {SiO_{2}}$ inside it. The black dots correspond to the position of the resonance peak of the cavity. The white line is the linearly fits of peak position changing with L. (b) The transmittance and CD spectra of the modeled chiral medium. And the absorption of the monolayer $\mathrm {WSe_{2}}$. (c) The transmittance and CD spectra of the complex system composed of the chiral medium and $\mathrm {WSe_{2}}$. The red and blue dashed lines represent the peak position of the chiral medium and $\mathrm {WSe_{2}}$, respectively. (d) The transmittance and CD spectra of the complex system composed of the chiral medium and $\mathrm {WSe_{2}}$ when their spectra are overlapped. The red curves represent the resonances are both at 800nm. And the blue curves represent the resonances at 747.3nm. (e) The transmittance spectrum of left-handed circularly polarized light, and (f) the CD spectrum of the chiral medium-FP cavity-$\mathrm {WSe_{2}}$ coupled system, respectively. The white dashed lines are the peak position of $\mathrm {WSe_{2}}$ and the chiral medium, respectively. The background is set to $\mathrm {SiO_{2}}$.
Figure 2(a) is the transmittance spectrum of the cavity, which shows the resonant peak changes linearly with the cavity length L. Using the chiral medium modeled method we described, different optical responses illuminated by the left (LCP) and right (RCP) polarized light are obtained, and the results are shown in Fig. 2(b). Note that the CD is calculated by the difference in transmittance: $\mathrm {CD=T_{LCP}-T_{RCP}}$. And the bottom figure in Fig. 2(b) is the absorption spectrum of 1 nm thick $\mathrm {WSe_{2}}$ whose resonance is at 1.659 eV (747.3 nm) and is covered in the cavity’s tuning range. When the cavity is not involved in the interaction, the optical response of the complex system of the chiral medium and $\mathrm {WSe_{2}}$ is shown in Fig. 2(c). As we can see, two peaks appear at the position of chiral medium and $\mathrm {WSe_{2}}$, respectively. Meanwhile, the two peaks both show different intensities under LCP and RCP. However, in the CD spectrum, only one peak appears at the position of the chiral medium, which means that the chirality can’t transfer from the chiral medium to $\mathrm {WSe_{2}}$ directly. We further analyzed the complex system of the chiral medium and $\mathrm {WSe_{2}}$ when their resonance spectra are overlapped. In Fig. 2(d), we do not find mode splitting. And the CD signal of the complex system is same as the bare chiral medium. This means that whether the resonances of the chiral medium and $\mathrm {WSe_{2}}$ are overlapped, the CD signals will not transfer from the chiral medium to $\mathrm {WSe_{2}}$. It is because the chiral medium and $\mathrm {WSe_{2}}$ can’t produce a local electromagnetic field to enhance the interaction between them. When the FP-cavity is involved in the interaction between the chiral medium and $\mathrm {WSe_{2}}$, three plexciton branches are observed in the transmittance spectrum, as shown in Fig. 2(e). In addition, as shown in Fig. 2(f), three branches also can be seen in the CD spectrum instead of only one peak at the position of the chiral medium, which differs from the results in Fig. 2(c), demonstrating that the chirality transfer occurs. We noticed that the CD values in Fig. 2(f) are smaller than in Fig. 2(b). This phenomenon is caused by the handedness reversal when circularly polarized light is reflected by the cavity mirrors leading to the chirality density decreasing inside the cavity. So, utilizing the handedness-preserving mirrors to avoid handedness reversal is one way to enhance the CD values [34]. In addition, the chirality density of the superchiral field is much higher than the circularly polarized light. So, utilizing the structure which can produce a local superchiral field to replace the FP-cavity is another potential way to enhance the CD values in the coupling system [35].
To reveal the underlying mechanism behind the three-mode coupled system, we use the coupled oscillator model to analyze the hybrid states [36–38]. And this system can be modeled as follows:
3. Results and discussion
According to the above description, the chiral transfer can be realized by the three-mode coupling system. Then we designed an actual chiral object, the Au helices array, to replace the modeled chiral medium to verify the feasibility of this hybrid system.
As shown in Fig. 3(a), the helices array is placed on top of the $\mathrm {WSe_{2}}$. The geometry and position parameters are contained in the inset. The array unit isn’t a single helix but a $\mathrm {C_{4,z}}$ symmetric four helices array for avoiding circular polarization conversion [28,40]. The optical responses of the Au helices array are shown in Fig. 3(b) and (c). The resonant position of this array is at 1.5595 eV (795 nm) where the different intensity of transmittance emerges. So the CD peak appears at the resonant position as presented in Fig. 3(c). And according to the transmittance spectrum, we can get the line width of the helices array $\mathrm {\gamma _{c-}=67.05 meV}$ and $\mathrm {\gamma _{c+}=69.42 meV}$ by fitting analysis. Here, the $\mathrm {-}$ and $\mathrm +$ denote the helices array excited by LCP and RCP, respectively.
Fig. 3. (a) Schematic diagram of the coupling system composed of the helices array, $\mathrm {WSe_{2}}$, and Au FP-cavity. Two Au mirrors are both 30 nm thick. The $\mathrm {WSe_{2}}$ is 1 nm thick. The material of other places inside the cavity is $\mathrm {SiO_{2}}$. The cavity length is L which changes from 180 nm to 220 nm. The period P of the nanostructure array is 100 nm. The single helix, whose geometry parameters are set as r=4 nm, R=11 nm, and h=25 nm, is shown in the upper inset. Here, the r, R, and h represent the inside radius, the external radius, and the length of the helix, respectively. And the position parameters of the helices in a unit cell are shown in the lower inset, which exhibits a $\mathrm {C_{4,z}}$ symmetry. (b) The transmittance spectrum of the Au helices array. Orange and blue curves denote the array illuminated by LCP and RCP, respectively. (c) The CD spectrum of the helices array. The background is set to $\mathrm {SiO_{2}}$.
Figure 4(a) and (b) are the interaction between the cavity(L=206 nm) and helices array, and the interaction between cavity(L=189 nm) and $\mathrm {WSe_{2}}$, respectively. The length of the cavity is according to the zero detuning position. These two cases both exhibit clear mode splitting which is $\mathrm {\Omega _{1}=192.49 meV}$ in Fig. 4(a) and $\mathrm {\Omega _{2}=82.15 meV}$ in Fig. 4(b). So we can get the intensity of interaction: $g_{cav-c}=\frac {\Omega _{1}}{2}=96.245 \mathrm {meV}$ and $g_{cav-x}=\frac {\Omega _{2}}{2}=41.075 \mathrm {meV}$. The peaks position of the helices array and $\mathrm {WSe_{2}}$ are far apart, and the line widths are narrow enough, causing their resonant peaks to lack overlapping area, which is similar to the case shown in Fig. 2(d). So we can ignore the interaction between them: $g_{c-x}=0$. The optical responses of the three-mode coupled system are shown in Fig. 4(c) and Fig. 4(d). Note that here we only show the transmittance spectrum under LCP because the spectrum under RCP is the same as under LCP, just different in intensity. In the transmittance spectrum, there are three plexciton branches, which are the upper (UPB), middle (MPB), and lower (LPB) branches. The black dots in Fig. 4(c) are the peak position of these three branches of simulated results. Then we bring the interaction intensity ($g_{cav-c}=96.245 \mathrm {meV}$, $g_{cav-x}=41.075 \mathrm {meV}$, and $g_{c-x}=0$), resonant energy ($E_{c}$=1.5595eV, $E_{x}$=1.659eV, and $E_{cav}$), and the line width ($\gamma _{c}$=67.05meV, $\gamma _{x}$=43meV, and $\gamma _{cav}$) into the Eq. (5). Note that here, the $E_{cav}$ and $\gamma _{cav}$ are the functions of cavity length L, as shown in Fig. 2(a). By calculating, we can get the energy of three plexciton branches as the functions of L, presented by the white curves in Fig. 4(c), which exhibit anticrossing behaviors. We can see that the calculated results can fit well with the simulation. Meanwhile, we can get the Hopfield coefficient $\alpha _{1}$, $\alpha _{2}$, and $\alpha _{3}$. $\left |\alpha _{1}\right |^{2}$, $\left |\alpha _{2}\right |^{2}$, and $\left |\alpha _{3}\right |^{2}$ of the hybrid states as the functions of L are shown in Fig. 4(e). The chirality of the coupled system is described by the CD spectrum shown in Fig. 4(d). Here, we can see that the CD signal peak appears near the plexciton branches rather than at the position of the helices array, which demonstrates that the chirality transfers from the resonance of chiral nanostructure to three plexcitons through strong coupling. The proportion of the components in plexciton branches is shown in Fig. 4(e). Combining the transmittance spectrum in Fig. 4(c) and the proportion variation in Fig. 4(e), we find that the change in branches’ intensity is consistent with the variation trend of the cavity mode’s proportion. So the optical response strength of the coupled system is mainly determined by the cavity mode. For example, the intensity of UPB declines with L increasing, and this variation trend is the same as the cavity’s proportion variation in UPB. And the weak strength of MPB is caused by the low proportion of cavity mode. But for the CD spectrum, the intensity variation is different. Because the optical response of the coupled system is mainly determined by the cavity mode as our analysis of the transmittance spectrum. But the chirality is offered by the helices. This results in the CD spectrum being determined by both the cavity mode and helices. For example, the proportion of the cavity and helices declines with L increasing in UPB, and the trend of MPB is the opposite. So in the CD spectrum, we can see that the CD intensity of UPB decline with the L increasing, and the case of MPB reverses. As for the LPB, we can see that the cavity and helices change in the opposite trend. When the L is small, the optical response is mainly determined by the cavity, but its proportion is not dominant. This causes a weak optical response and leads to small CD signals. The proportion of cavity mode increases with the L increasing. This causes a strong optical response leading to the CD signals increasing. When the cavity and helices are in comparable proportions, the CD signals of the coupling system hit the maximum. As the L continues to increase, the proportion of helices declines. This means the CD signal offered by the helices declines and leads the CD signal of LPB to decrease. Therefore, the CD signal of LPB increases first and then decreases with the L increasing. To further analyze the CD variation of the three-mode coupled system, the hybrid system without $\mathrm {WSe_{2}}$ is used as contrast.
Fig. 4. (a) The interaction between the FP-cavity and helices array when cavity length is L=206 nm. (b) The interaction between the FP-cavity and $\mathrm {WSe_{2}}$ when cavity length is L=189 nm. (c) The transmittance spectrum (under LCP) and (d) the CD spectrum of the helices array-cavity-$\mathrm {WSe_{2}}$ coupled system. The horizontal and vertical axes denote the cavity length L and the resonant energy of incident light, respectively. The black dots are the peak position of the coupled system of the simulated results. The horizontal black dashed lines at 1.659eV and 1.5595eV are the peak position of $\mathrm {WSe_{2}}$ and helices array, respectively. The oblique dashed line presents the peak position of the empty cavity. The white curves are the anticrossing curves of the calculated results. (e) The proportion of the helices array, $\mathrm {WSe_{2}}$, and cavity in UPB, MPB, and LPB.
As shown in Fig. 5(a), comparing the white triangular dots to the black circular dots, we find that the LPB of the two-mode coupled system almost coincides with it of the three-mode coupled system. This means that the $\mathrm {WSe_{2}}$ has little influence on the LPB when it participates in the coupling. In other words, when three-mode coupling occurs, $\mathrm {WSe_{2}}$ mainly interacts with the UPB formed by the cavity and helices array to form new UPB and MPB. In Fig. 5(a), we further note that the CD signal is higher in the UPB than in the LPB. According to the transmittance spectrum of the cavity, the optical response of cavity mode declines with the L increasing. So the dominant cavity mode provides a high response causing the CD signal of UPB higher than LPB when L is small. As L increases, the optical response of the cavity decreases, causing the helices array to dominate in the hybrid state. So when L is large, the CD of UPB is also higher than LPB. In this way, considering the UPB and MPB of the three-mode system are from the interaction between $\mathrm {WSe_{2}}$ and UBP of the two-mode system, the CD signals of UPB and MPB are higher than LPB in the three-mode coupling system can be explained.
Fig. 5. (a) The CD spectrum of the cavity and helices array coupled system. The white triangular dots are the peak position of the two-mode coupled system, and the black circular dots denote the peak position of the three-mode system. The white dashed line presents the cavity. And the black dashed lines are $\mathrm {WSe_{2}}$ and helices array, respectively. (b) The proportion of the cavity and helices array in UPB and LPB. (c) Normalized CD spectra of the three-mode coupling system for different temperatures. The cavity length L is fixed at 200nm, and the period P is kept at 100nm. (d) Normalized CD spectra of the three-mode coupling system for different periods. The cavity length L is fixed at 200nm, and $\mathrm {WSe_{2}}$ is kept at 1.659eV.
Comparing the circular dots and triangular dots near the UPB in Fig. 5(a), we can more clearly see that the UPB splits when $\mathrm {WSe_{2}}$ participates in the coupling. This provides an extra chiral channel. In addition, as shown in Fig. 4(e), $\mathrm {WSe_{2}}$ occupies a large proportion in UPB and MPB of the three-mode coupling system. This indicates the UPB and MPB will be significantly affected by the $\mathrm {WSe_{2}}$. This point provides a route to manipulate the chirality of plexctitons by utilizing the character of the exciton. We note that the exciton used here is the monolayer $\mathrm {WSe_{2}}$, whose energy is demonstrated as temperature-dependent. So it’s natural to consider tuning the chirality of plexcitons through temperature. The relationship between monolayer $\mathrm {WSe_{2}}$ energy and temperature is given in [41]: $E_{g}(T)=E_{g}(0)-S\cdot \left \langle \hbar \omega \right \rangle \left [coth\frac {\left \langle \hbar \omega \right \rangle }{2K_{b}T}-1\right ]$, where $E_{g}(0)\approx 1.75$eV is the excitonic transition energy at 0K, $S\approx 2.33$ is a dimensionless constant presenting the strength of the electron-phonon coupling, and $\left \langle \hbar \omega \right \rangle \approx 14.1$meV is the average acoustic phonon energy involved in electron-phonon interactions, $K_{b}$ is the boltzmann constant, and T is the Kelvin temperature. The tuning results are shown in Fig. 5(c), the black dashed lines present the peak position of the three-mode coupling system. And the peaks of the two-mode coupling system are marked by the red vertical lines as the comparison. We can see that the peak of LPB remains nearly unchanged because of the low proportion of $\mathrm {WSe_{2}}$ in LPB. While UPB and MPB occur blue shift with the temperature decrease. The CD peak of MPB will shift from 1.625eV to 1.642eV and that of UPB will shift from 1.693eV to 1.724eV when temperature changes from 300K to 200K. Moreover, we note the optical response of array structures can be affected by their periods. So chiroptical responses of this coupling system can be manipulated by changing the period of the array, and the results are shown in Fig. 5(d). We can see that as the period increases, the peaks of MPB and UPB occur redshift, while that of LPB is blue-shifted. And the peak shifts of UPB and MPB are small because their peak position are mainly affected by $\mathrm {WSe_{2}}$. The CD peak of LPB shifts from 1.471eV to 1.533eV when the period P changes from 90nm to 150nm. Therefore, the CD peak of plexcitons in this coupling system can be modulated by tuning the optical response of $\mathrm {WSe_{2}}$, the helices array, and the FP-cavity.
The traditional chiral sensing system aims to enhance the CD signals and is at the weak coupling region. In our paper, the proposed chiral strong coupling system exhibits mode splitting not only in the transmittance spectra but also in CD spectra. It provides a new route to sense chirality through the perspective of frequency shift. The Rabi splitting is very sensitive to slight changes in environmental permittivity [42,43]. Therefore, frequency shift in CD spectrum can realize very sensitive chiral sensing [44]. Furthermore, the tunability of the coupling system provides a method to tune the detecting frequency without tailoring the chiral material.
4. Conclusion
In summary, we investigated the chiroptical responses in the chiral three-mode coupling system based on the finite element method. By coupling the modeled chiral medium with the FP-cavity and $\mathrm {WSe_{2}}$, we found that the chiral transfer can be realized by three-mode coupling. An actual chiral object, an Au helices array, is designed to replace the modeled chiral medium. Mode splitting in the CD spectrum, resulting in three plexciton branches, demonstrates the chiroptical responses shift from the resonance of chiral nanostructures to the position of three plexcitons. We used coupled oscillator model to obtain the energy and Hopfield coefficients of plexcitons and analyze the coupling behavior of the system. The results demonstrate the chiral structures and FP-cavity determined the intensity of chiroptical responses of the hybrid system, while $\mathrm {WSe_{2}}$ mainly affected the position of MPB and UPB. Moreover, benefiting from the temperature-dependent character of $\mathrm {WSe_{2}}$ and the period-dependent property of the helices array, the chiroptical responses of the system can be modulated by tuning the period of the helices array and the temperature. Our work proposed a chiral structure-FP cavity-$\mathrm {WSe_{2}}$ strong coupling system to realize the chiral transfer and tuning. Meanwhile, compared with the chiral two-mode coupling system, our system provides an extra chiral channel, a more widely tunable region, and more tuning methods, which exhibit application potential in chiral sensing and modulation.
Funding
National Natural Science Foundation of China (12204061, 12174037, 12204030); State Key Laboratory of Information Photonics and Optical Communications (IPOC2021ZZ02); Fundamental Research Funds for the Central Universities (2022XD-A09).
Disclosures
The authors declare no conflicts of interest.
Data availability
All data are available from the authors upon reasonable request.
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