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Abstract
Effective manipulation of the interactions between light and matter is crucial for the advancement of various high-performance optoelectronic devices. It is noted that the toroidal dipole resonance refers to an electromagnetic excitation that exists beyond the conventional understanding of electric and magnetic multipoles, which shows great potential for enhancing light-matter interactions. In this work, we investigate the strong coupling properties of electric toroidal dipole (ETD) and magnetic toroidal dipole (MTD) with excitons in (PEA)2PbI4 perovskite metasurfaces. The nanostructure consists of two identical nanobars on a SiO2 substrate, which support ETD and MTD responses. The strong coupling between ETD/MTD modes and perovskite excitons is achieved when adjusting oscillator strength f0, which can be charactered by the clearly anti-crossing behavior appeared in the transmission spectra. The Rabi splitting can be readily tuned by controlling f0. When f0 increases to 1.0, their Rabi splitting values reach as high as 371 meV and 300 meV, respectively. The proposed strong coupling between excitons and ETD/MTDs paves the way for large-scale, low-cost integrated polaritonic devices operating at room temperature.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Strong light-matter interactions at the nanoscale has sparked significant interest in both fundamental research and practical applications within nanophotonics, such as nanolasing, all-optical switching, and quantum information processing, etc [1]. When the decay rate of the optical cavities and excitons are lower than the energy-exchanged rate between these subsystems, exciton-polaritons are formed with half-light and half-matter components [2], resulting in changes in the corresponding emission (or absorption) line shape and emission (or absorption ) frequency [3,4], at this time, this light-matter interactions belong to the strong coupling regime. In recent years, diverse nanostructures have been employed to enhance strong coupling, such as metallic nanoantennas [5], metallic nanoparticles [6], hybrid two-dimensional materials-based nanostructures [7] etc. Metal-based nanostructures support localized surface plasmon resonance, which provides strong optical confinement at the deep subwavelength scale. Metallic nanostructures often suffer from their unavoidable intrinsic ohmic losses in the optical region [8]. Another important candidate is the dielectric metasurfaces [9–11]. All-dielectric nanostructure composed of high-refractive-index and low-loss materials support electric and magnetic Mie resonances [12–15]. The intensity and position of the resonant mode with high Q-factor can be adjusted by thermal, optical and electrical means, thus realizing a series of high-performance optoelectronic devices [16–21]. At present, many studies have focused on strong coupling based on all-dielectric metasurfaces. Such as, Wang et al. explored the Mie surface lattice resonance in a silicon nanoparticle array to research the strong coupling between 1L WS$_2$ and Mie modes [22]. Xie et al. and Al-Ani et al. realized strong coupling in transition metal dichalcogenide monolayer by employing different metastructures supporting bound states in the continuum [23,24].
In addition, through ingenious structural design, the dielectric nanostructures can support an elusive resonance called a toroidal dipole (TD) resonance [25–27], which enables a distinctly highly confined optical field. The TD resonance is a special resonance mode distinct from the electric dipole (ED) and magnetic dipole (MD) responses, which can achieve strong field confinement in nanostructured systems due to the reduction of radiation losses [28]. The TD is first introduced by Zel’dovich in 1957 [29], can be categorized into electric toroidal dipole (ETD) and magnetic toroidal dipole (MTD) [30,31]. The ETD arises due to the circulation of polar current along the meridian of the torus. It can be characterized by the MD moment formed by the head-to-tail connection of the current. Correspondingly, the MTD is created by the circular polarization current flowing along the torus and can be described by the ED moment formed by the head-to-tail connection of the current [31,32]. TD moments have long been elusive because their toroidal moment-related responses in natural materials are feeble and often masked by electromagnetic multipoles [33]. Fortunately, one can design subwavelength-scale artificial micro-nanostructures to excite strong ETD resonance [28]. However, less attention has been paid to the MTD response because optimizing or achieving ED loop closure in nanostructure is more challenging. To the best of our knowledge, there are only a few studies on the strong coupling achieved by ETD resonance [34], and in particular, the strong coupling realized by the MTD resonance has not been reported.
In this paper, the strong coupling of ETD/MTD modes and excitons based on (PEA)$_{2}$PbI$_{2}$ perovskite metasurfaces is investigated. Firstly, the ETD/MTD resonances of perovskite metasurfaces are designed, which are demonstrated by near-field analysis and multipole decomposition. Secondly, the strong interactions between ETD/MTD modes and excitons is invesitigetd by adjusting the oscillator strength. Finally, the Rabi splitting values of the strong coupling between ETD/MTD modes and excitons for different exciton oscillation strengths and quality (Q) factors are calculated separately. The Rabi splitting values can reach to 371 meV and 300 meV, respectively.
2. Results and discussion
It is well known that the choice of active material can strongly influence the exciton-polariton coupling strength and critical temperature via the oscillator strength $f_0$ and the binding energy E$_b$ of excitons [35]. In recent years, appealing organic-inorganic lead halide perovskite hybrid materials, which inherit the significant advantages of organic-inorganic semiconductors, such as high light absorption efficiency, wide resonance tunability, large excitonic binding energy and cheap fabrication techniques, etc[35]. These advantages make it an excellent platform for achieving superior performance in photonics and optoelectronic devices. In particular, two-dimensional (2D) layered perovskite materials possess ultralarge exciton binding energies (up to hundreds of meV) and excellent oscillation strength, which allow strong coupling to be observed at room temperature [36]. Although there have been strong coupling studies based on (C$_6$H$_5$C$_2$H$_4$NH$_3$)$_2$PbI$_4$ (abbreviated as (PEA)$_2$PbI$_4$) perovskite metasurfaces [14]. However, achieving strong coupling between the ETD/MTD resonances and (PEA)$_2$PbI$_4$ perovskite excitons remains an open problem.
2.1 Design of ETD and MTD resonant metasurfaces
The design of (PEA)$_2$PbI$_4$ perovskite metasurfaces is illustrated in Fig. 1(a). It consists of a periodic square array of two identical nanobars, placed on a low-refractive-index substrate (SiO$_2$). The specific parameter layout of unit cell is shown in Fig. 1(b), characterized by its length ($L$), width ($W$) and thickness ($H$), airgap ($G$). While $P_x$ and $P_y$ are the periods of the array in the $x$ and $y$ directions, respectively. The crystal structure of (PEA)$_2$PbI$_4$ is shown in Fig. 1(c). We perform simulations of this structure using the Finite-Difference-Time-Domain (FDTD) method. During the simulations, the planar wave is incident perpendicular to the (PEA)$_2$PbI$_4$ metasurfaces, as shown Fig. 1(a). Periodic boundary conditions are set in the $x$- and $y$-directions, and perfectly matched layer conditions are used in the $z$-direction. In our calculations, the artificial dielectric constant of (PEA)$_2$PbI$_4$ can be described by the Lorentz oscillator as Eq. (1) [37],
where $\varepsilon _0$=5.76 is the background permittivity. $f_0$ is the oscillator strength, $\omega _{ex}$=2.4 eV and $\Gamma _{ex}$=2$\gamma _{ex}$=30 meV are the exciton’s transition energy and full linewidth. It is important to note that when we consider it solely as a background index material, $f_0$=0. As shown in Appendix A, the real and imaginary parts of the permittivity of (PEA)$_2$PbI$_4$ perovskite also change accordingly with the change of $f_0$, and its peaks also increases regularly, and the position of (PEA)$_2$PbI$_4$ perovskite excitons is around $\lambda$=516.6 nm.
Fig. 1. (a) A schematic diagram of ETD/MTD-based (PEA)$_2$PbI$_4$ perovskite metasurfaces. (b) Detailed parameter layout diagram of the unit cell. (c) Crystal structure diagram of (PEA)$_2$PbI$_4$.
Firstly, the (PEA)$_2$PbI$_4$ perovskite metasurfaces supporting ETD and MTD resonances are designed, respectively. Prior to this, we explore the (PEA)$_2$PbI$_4$ perovskite film structure and find that the transmission spectra did not splitting with the change of $f_0$, that is, no strong coupling occurred obviously, as shown in Appendix B. Therefore, we design the (PEA)$_2$PbI$_4$ perovskite as a specific metasurfaces for better strong coupling. In this simulations, the $L$ and $W$ of the nanobars are fixed at 283 nm and 116 nm, and the array period is fixed at $P_x=P_y$=300 nm. To unveil the characteristics of the ETD/MTD resonances, the excitonic oscillator strength is set to 0. We calculate the resonant transmission spectra and the near-field distribution and far-field radiation. Figures 2(a) and 2(e) show the transmission spectra of ETD and MTD metasurfaces. It can be seen that the resonance wavelength is located at 516.6 nm, which ensures overlap with material excitons. This lays a good foundation for the next study on strong coupling between ETD/MTD and excitons in (PEA)$_2$PbI$_4$ perovskite metasurfaces. It should be noted that the thickness $H$ and the airgap $G$ of the ETD resonator are 234 nm and 13 nm, while for the MTD resonator, they are 165 nm and 44 nm, respectively. Furthermore, the normal incidence of plane waves polarized along the $y$-direction and $x$-direction are used for ETD and MTD resonances, respectively. The electric field and displacement current distributions of $x$-$y$ and $x$-$z$ sections at resonant wavelengths are calculated for ETD resonance. As can be noticed from Figs. 2(b) and 2(c), there is a noticeable local field enhancement in the airgap, exhibiting electric and magnetic fields’ enhancement by 20 and 42 folds, respectively. More specifically, by forming two circular electric field vector distributions in opposite directions in the $x$-$y$ plane, the two MDs along the $z$-axis can be excited. In combination with the incident light, an ETD response along the $y$-axis is then produced by the circular MD in the $x$-$z$ plane. Remarkably, two opposing MDs work together in their intersection region, enhancing the field significantly. Similarly, for MTD resonance, combining Figs. 2(f) and 2(g), two circular magnetic fields with opposite directions are formed to excite an ED along the $z$-axis, which in turn excites an MTD response along the $y$-axis. It is worth emphasizing that at this time, as seen in Fig. 2(f), the electric field is well localized inside the device.
Fig. 2. The properties of ETD/MTD resonances. (a), (e) The transmission spectra when $f_0$=0. (b), (g) The electric field and displacement current distributions at the resonance wavelength. (c), (f) The magnetic field and magnetic vector distributions. (d), (h) The total scattered power and the contribution of different multipoles.
To gain a deeper insight into the physical properties of ETD and MTD resonances, electromagnetic multipole decomposition in Cartesian coordinates is employed to assess the contribution of various multipole components on the far-field radiation [30,31]. The displacement current is represented by $\vec {j}=-i\omega \epsilon _{0} (n^2-1) \vec {E}$, where $\vec {E}$ is the electric field, $\omega$ is the angular frequency, $n$ is the refractive index of the structure, and $\varepsilon _0$ is the permittivity of the vacuum. And then the scattered powers of the multipole moments are calculated from [30–32]: $I_P = \frac {2{\omega ^4}}{3{c^3}}{\left | \vec P \right |^2}, {I_M}= \frac {2{\omega ^4}}{3{c^3}}{\left | \vec M \right |^2}, {I_T} = \frac {2{\omega ^6}}{3{c^5}}{\left | \vec T \right |^2}, I_{G} =\frac {2\omega ^{6} }{3c^{5} } \left | \vec {G} \right | ^{2}, {I_{Q^{(e)}} = \frac {\omega ^6}{5{c^5}}\sum {\left | {\vec Q _{\alpha \beta }^{(e)}} \right |}^2}, {I_{Q^{(m)}} = \frac {\omega ^6}{40{c^5}}\sum {\left | {\vec Q _{\alpha \beta }^{(m)}} \right |}^2},$ where $c$ and $\omega$ are the speed and angular frequency of light, respectively. The $\vec {P}$, $\vec {M}$, $\vec {T}$, $\vec {G}$, $\vec {Q}^{\left (e\right )}$ and $\vec {Q}^{\left (m\right )}$ are the electric dipole (ED) moment, magnetic dipole (MD) moment, electric toroidal dipole (ETD) moment, magnetic toroidal dipole (MTD) moment, electric quadrupole (EQ) moment and magnetic quadrupole (MQ) moment, respectively. As shown in Fig. 2(d), the contribution of ED, MD, MTD, EQ and MQ are very weak, the contribution of ETD is dominant at the resonance wavelength. Similarly, for MTD response, a similar analysis is shown in Fig. 2(h). It can be seen that ED, ETD, MQ, MD and EQ are significantly suppressed, and the far-field radiation of MTD is dominant. All the far-field and near-field analysis results mutually prove that the devices we designed support ETD and MTD resonances.
2.2 Strong coupling of ETD/MTD modes-excitons
The appearance of ETD/MTD resonances implies an enhanced electromagnetic field inside the device structure, suggesting that they can be used to achieve strong light-matter interactions. Next, we explore the strong coupling between ETD/MTD and excitons in perovskite metasurfaces. As shown in Fig. 3(a), the transmission spectra of both ETD and MTD resonances with excitons are splitting into three dips when $f_0$=0.4, labeled as LP, D and UP, respectively. This can be explained by the fact that when the coupling is "turned on" by using a dielectric constant that includes exciton resonances and a high background index, the two spectral features of the ETD/MTD modes and the (PEA)$_2$PbI$_4$ perovskite excitons are hybridized, resulting in two transmission spectra valleys in the 480 nm-560 nm range. LP and UP are low and high energy branches, in addition to these two dips, we also observe a broader intermediate dip (D), which originates from the absorption of excitons not coupled to the ETD/MTD modes, which appears slightly redshift compared to that without excitons ($f_0$=0). All these results are an obvious feature of the strong interactions between photons and excitons. Moreover, the electromagnetic field distribution at LP, D and UP in $x$-$y$ section are shown in Figs. 3(b) and 3(c). Compared with Figs. 2(b) and 2(f), it can be found that the fields enhancement at the resonance wavelength is significantly reduced, which is attributed to the occurrence of the strong interactions, and most of the energy has been transferred to the splitting dips LP and UP.
Fig. 3. (a) The transmission spectra for $f_0$=0 (dashed lines) and $f_0$=0.4 (solid lines), where the red/blue correspond to ETD/MTD resonance. LP/D/UP are the three splitting dips, where the subscript E/M represents ETD/MTD. (b), (c) The electric/magnetic field distribution of the ETD/MTD resonances coupled with excitons at three splitting dips.
The interactions of resonant mode and excitons is viewed as a coupling between harmonic oscillators. The coupled mode theory is expressed as Eq. (2) [4],
In short, only when Eqs. (5) and (6) are satisfied simultaneously can the system be strong coupling.
In this case, the transmission spectra of the uncoupled ($f_0$=0) and coupled ($f_0$=0.4) are further studied with the parameter $W$ increased from 108 nm to 121 nm. It can be seen from Figs. 4(a) and 4(c) that with the increase of $W$, the ETD/MTD resonance positions appear red-shifted when uncoupled. However, after coupling, as shown in Figs. 4(b) and 4(d), the position of the excitons remains unchanged (red dotted line). Still, the ETD/MTD resonance has obvious dispersion and anti-crossing behavior. The eigenvalues $E_{LP,UP}$ for ETD/MTD resonances are fitted as shown in Figs. 4(b) and 4(d). The difference between the upper polariton branch (UP) and lower polariton branch (LP) at the anticrossing point in the fitted data are the Rabi splitting of ETD/MTD-excitons strong coupling 231 meV and 184 meV, respectively. The excitons damping rate is known to be 15 meV, while the ETD/MTD resonance damping rate is easily extracted from Figs. 2(a) and 2(e), as shown in Appendix C, the half-line width of the transmission spectrum is obtained by fitting with Fano formula, which are 8 meV and 7.8 meV, respectively. According to Eq. (4), the coupling strength $g$ of the ETD/MTD resonances with excitons ($f_0$=0.4) can be calculated as 115 meV and 92 meV, respectively. Excitingly, all results are satisfied the strong coupling criterion. The anti-crossing behavior combined with the splitting of dips appearing in the transmission spectra, further demonstrates that the ETD/MTD resonances and excitons indeed realize strong coupling.
Fig. 4. Transmission maps with different nanobar width ranging from 108 to 121 nm for ETD/MTD modes-excitons (a), (c) uncoupled and (b), (d) strong coupling.
Furthermore, we also demonstrate that the Rabi splitting of the ETD/MTD modes-excitons strong coupling can be readily tuned by controlling the excitonic oscillator strength of the bulk (PEA)$_{2}$PbI$_{4}$ perovskite. Figure 5(a) shows the change of the Rabi splitting value of ETD/MTD modes-excitons strong coupling as $f_0$ increases from 0.2 to 1.0. Its transmission spectra are shown in Fig. 9 in Appendix D. It can be clearly noticed that the greater the exciton oscillation strength, the more pronounced the Rabi splitting value. The Rabi splitting value of the ETD-excitons strong coupling is 371 meV at an oscillation strength of 1.0. It is worth mentioning that under the same circumstance, the Rabi splitting value of MTD-excitons strong coupling is also considerable (300 meV). In addition, the $Q$-factors of an optical resonator determines its lifetime and the degree of overlap of resonant transmission spectra between the cavity mode and the material excitons, which directly affects the coupling between the cavity and the excitons. Here, by adjusting the geometric structure parameters of metasurfaces unit cell, resonant cavity with different $Q$-factors can be easily obtained, see in Appendix C for $Q$-factors calculation, and the simulated transmission spectra of the (PEA)$_{2}$PbI$_{4}$ perovskite metasurfaces agrees well with the fitted result. Four ETD/MTD resonators with different $Q$-factors are modulated, and their Rabi splitting is found to change slightly, as shown in Fig. 5(b).
Fig. 5. Rabi splitting values of ETD/MTD modes-excitons strong coupling for (a) different exciton oscillation strengths and (b) different $Q$-factors.
3. Conclusion
To summarize, the strong interactions between the toroidal dipole modes and (PEA)$_{2}$PbI$_{4}$ perovskite excitons is studied in the metasurfaces system. Firstly, the resoannt (PEA)$_{2}$PbI$_{4}$ perovskite metasurfaces composed of two identical nanobars placed on a SiO$_{2}$ substrate are proposed, the ETD and MTD resonances are demonstrated through the analysis of the electromagnetic field distribution and the contribution of multipole far-field radiation. Three distinct splitting dips in the transmission spectra of ETD/MTD-excitons resonances are obseved by changing the oscillator strength. Further, the obvious anti-crossing behavior in transmission spectra maps also confirme the occurrence of strong coupling by adjusting the width of the nanobars. The Rabi splitting value reach 371 meV and 300 meV when $f_0$=1.0, respectively. In addition, it is found that the Rabi splitting value of this system changes very slightly when adjusting the $Q$-factor of metasurfaces. Our results are beneficial for regulating the light-matter interactions at the nanoscale to achieve high-performance optoelectronic devices.
Appendix
A. Permittivity of (PEA)$_{2}$PbI$_{4}$ perovskite
It is noted that the $f_0$ can be controlled by changing the transition frequency (energy) of electrons, such as carried out an external electromagnetic field. The real and imaginary parts of the permittivity of PEA)$_{2}$PbI$_{4}$ perovskite are shown in Fig. 6.
Fig. 6. Permittivity of (PEA)$_{2}$PbI$_{4}$ perovskite with $f_0$=0-1.0: (a) real part; (b) imaginary part.
B. The transmission spectra of the (PEA)$_{2}$PbI$_{4}$ perovskite thin film structure
From the transmission spectra shown in Fig. 7, it can be clearly seen that with the increase of $f_0$, the (PEA)$_{2}$PbI$_{4}$ perovskite with a film structure has no splitting, and it can be considered that there is no strong coupling phenomenon.
C. Transmission spectra of the designed ETD/MTD resonance at different $Q$-factors
The transmission spectra of the designed ETD/MTD resonance with different $Q$-factors is fitted by the classical Fano formula of Eq. (7).
Fig. 7. The transmission spectra of the perovskite thin film structure with a thickness of 165 nm at different $f_0$.
Fig. 8. The transmission spectra of ETD/MTD resonator with a $Q$-factor of 300: (a) Simulated (blue solid line) and Fano fitted (green dots) for ETD, (b) Simulated (orange solid line) and Fano fitted (light blue dots) for MTD. (c) (d) are the transmission spectra of ETD/MTD resonators when $Q$ is 300, 500, 800 and 1000, respectively.
D. Transmission maps for ETD/MTD modes-excitons at different $f_0$
Funding
National Natural Science Foundation of China (12004084, 12164008); Guizhou Provincial Science and Technology Department (ZK[2021]030); Science and Technology Innovation Team Project of Guizhou Colleges and Universities ([2023]060); Science and Technology Talent Support Project of the Department of Education in the Guizhou Province (KY[2018]043); The Growth Foundation for Young Scientists of Education Department of Guizhou Province (QJHKY [2017]214); Industry and Education Combination Innovation Platform of Intelligent Manufacturing and Graduate Joint Training Base at Guizhou University (2020-520000-83-01-324061); Research Fund for the Doctoral Program of Guizhou Education University (2021BS013).
Disclosures
The authors declare that they have no conflict of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. Törmä and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Reports on Pro. Phys. 78, 013901 (2014). [CrossRef]
2. F. Deng, H. Liu, L. Xu, S. Lan, and A. E. Miroshnichenko, “Strong exciton–plasmon coupling in a ws2 monolayer on au film hybrid structures mediated by liquid ga nanoparticles,” Laser Photonics Rev. 14, 1900420 (2020). [CrossRef]
3. K. As’ham, I. Al-Ani, L. Huang, A. E. Miroshnichenko, and H. T. Hattori, “Boosting strong coupling in a hybrid wse2 monolayer–anapole–plasmon system,” ACS Photonics 8, 489–496 (2021). [CrossRef]
4. L. huang, A. Krasnok, A. Alú, Y. Yu, D. Neshev, and A. E. Miroshnichenko, “Enhanced light–matter interaction in two-dimensional transition metal dichalcogenides,” Reports on Prog. Phys. 85, 046401 (2022). [CrossRef]
5. E. Eizner, O. Avayu, R. Ditcovski, and T. Ellenbogen, “Aluminum nanoantenna complexes for strong coupling between excitons and localized surface plasmons,” Nano Lett. 15, 6215–6221 (2015). [CrossRef]
6. K. Okamoto, K. Okura, P. Wang, S. Ryuzaki, and K. Tamada, “Flexibly tunable surface plasmon resonance by strong mode coupling using a random metal nanohemisphere on mirror,” Nanophotonics 9, 3409–3418 (2020). [CrossRef]
7. K. As’ham, I. Al-Ani, M. Alaloul, S. Abdo, A. Abdulghani, W. Lei, H. T. Hattori, L. huang, and A. E. Miroshnichenko, “Enhanced strong coupling in the hybrid dielectric-metallic nanoresonator and ws2 monolayer,” Phys. Rev. Appl. 19, 054049 (2023). [CrossRef]
8. J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotechnol. 10, 2–6 (2015). [CrossRef]
9. G. W. Castellanos, S. Murai, T. Raziman, S. Wang, M. Ramezani, A. G. Curto, and J. Gómez Rivas, “Exciton-polaritons with magnetic and electric character in all-dielectric metasurfaces,” ACS Photonics 7, 1226–1234 (2020). [CrossRef]
10. R. Paniagua-Domínguez, B. Luk’yanchuk, A. Miroshnichenko, and J. A. Sánchez-Gil, “Dielectric nanoresonators and metamaterials,” J. Appl. Phys. 126, 150401 (2019). [CrossRef]
11. S. Murai, D. R. Abujetas, L. Liu, G. W. Castellanos, V. Giannini, J. A. Sánchez-Gil, K. Tanaka, and J. Gómez Rivas, “Engineering bound states in the continuum at telecom wavelengths with non-bravais lattices,” Laser Photonics Rev. 16, 2100661 (2022). [CrossRef]
12. L. Huang, R. Jin, C. Zhou, G. Li, L. Xu, A. Overvig, F. Deng, X. Chen, W. Lu, A. Alù, and A. E. Miroshnichenko, “Ultrahigh-q guided mode resonances in an all-dielectric metasurface,” Nat. Commun. 14, 3433 (2023). [CrossRef]
13. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016). [CrossRef]
14. K. As’ham, I. Al-Ani, W. Lei, H. T. Hattori, L. Huang, and A. Miroshnichenko, “Mie exciton-polariton in a perovskite metasurface,” Phys. Rev. Appl. 18, 014079 (2022). [CrossRef]
15. D. Zeng, S. Zong, G. Liu, W. Yuan, X. Liu, J. Chen, C. Tang, and Z. Liu, “Near-perfect quantitatively tunable q factors of quasi-bound states in the continuum via material-based thermal-optic perturbations,” Opt. Lett. 48, 3981–3984 (2023). [CrossRef]
16. H. Yu and F. Qiu, “Compact thermo-optic modulator based on a titanium dioxide micro-ring resonator,” Opt. Lett. 47, 2093–2096 (2022). [CrossRef]
17. Z. Liu, W. Tan, G. Fu, X. Liu, G. Liu, J. Chen, and C. Tang, “Multipolar silicon-based resonant meta-surface for electro-optical modulation and sensing,” Opt. Lett. 48, 2969–2972 (2023). [CrossRef]
18. G. Liu, M. Liu, G. Fu, X. Liu, and Z. Liu, “Kerr nonlinear medium assisted double-face absorbers for differential manipulation via an all-optical operation,” Opt. Express 30, 26597–26608 (2022). [CrossRef]
19. S.-X. Xia, X. Zhai, L.-L. Wang, and S.-C. Wen, “Polarization-independent plasmonic absorption in stacked anisotropic 2d material nanostructures,” Opt. Lett. 45, 93–96 (2020). [CrossRef]
20. S. Xia, X. Zhai, L. Wang, Y. Xiang, and S. Wen, “Plasmonically induced transparency in phase-coupled graphene nanoribbons,” Phys. Rev. B 106, 075401 (2022). [CrossRef]
21. J. Wang, J. Kühne, T. Karamanos, C. Rockstuhl, S. A. Maier, and A. Tittl, “All-dielectric crescent metasurface sensor driven by bound states in the continuum,” Adv. Funct. Mater. 31, 2104652 (2021). [CrossRef]
22. S. Wang, T. Raziman, S. Murai, G. W. Castellanos, P. Bai, A. M. Berghuis, R. H. Godiksen, A. G. Curto, and J. Gómez Rivas, “Collective mie exciton-polaritons in an atomically thin semiconductor,” The J. Phys. Chem. C 124, 19196–19203 (2020). [CrossRef]
23. P. Xie, Z. Liang, T. Jia, D. Li, Y. Chen, P. Chang, H. Zhang, and W. Wang, “Strong coupling between excitons in a two-dimensional atomic crystal and quasibound states in the continuum in a two-dimensional all-dielectric asymmetric metasurface,” Phys. Rev. B 104, 125446 (2021). [CrossRef]
24. I. A. Al-Ani, K. As’Ham, L. huang, A. E. Miroshnichenko, and H. T. Hattori, “Enhanced strong coupling of tmdc monolayers by bound state in the continuum,” Laser Photonics Rev. 15, 2100240 (2021). [CrossRef]
25. A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric metamaterials with toroidal dipolar response,” Phys. Rev. X 5, 011036 (2015). [CrossRef]
26. C. Zhou, L. Huang, R. Jin, L. Xu, G. Li, M. Rahmani, X. Chen, W. Lu, and A. E. Miroshnichenko, “Bound states in the continuum in asymmetric dielectric metasurfaces,” Laser Photonics Rev. 17, 2200564 (2023). [CrossRef]
27. S. You, M. Zhou, L. Xu, D. Chen, M. Fan, J. huang, W. Ma, S. Luo, M. Rahmani, C. Zhou, E. A. Miroshnichenko, and L. Huang, “Quasi-bound states in the continuum with a stable resonance wavelength in dimer dielectric metasurfaces,” Nanophotonics 12, 2051–2060 (2023). [CrossRef]
28. T. Kaelberer, V. Fedotov, N. Papasimakis, D. Tsai, and N. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010). [CrossRef]
29. I. B. Zel’Dovich, “Electromagnetic interaction with parity violation,” Sov. Phys. JETP 6, 1184–1186 (1958).
30. Z.-J. Yang, Y.-H. Deng, Y. Yu, and J. He, “Magnetic toroidal dipole response in individual all-dielectric nanodisk clusters,” Nanoscale 12, 10639–10646 (2020). [CrossRef]
31. X. Wang, D. Chen, W. Ma, S. you, M. Zhou, L. Wang, Q. Zhang, S. Luo, and C. Zhou, “Tuning the magnetic toroidal dipole response in dielectric metasurfaces,” JOSA B 40, 560–566 (2023). [CrossRef]
32. X. Liu, J. Li, Q. Zhang, and Y. Wang, “Dual-toroidal dipole excitation on permittivity-asymmetric dielectric metasurfaces,” Opt. Lett. 45, 2826–2829 (2020). [CrossRef]
33. N. Papasimakis, V. Fedotov, V. Savinov, T. Raybould, and N. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15, 263–271 (2016). [CrossRef]
34. S. You, Y. Zhang, M. Fan, S. Luo, and C. Zhou, “Strong light–matter interactions of exciton in bulk ws 2 and a toroidal dipole resonance,” Opt. Lett. 48, 1530–1533 (2023). [CrossRef]
35. W. Du, S. Zhang, Q. Zhang, and X. Liu, “Recent progress of strong exciton–photon coupling in lead halide perovskites,” Adv. Mater. 31, 1804894 (2019). [CrossRef]
36. N. H. M. Dang, D. Gerace, E. Drouard, G. Trippé-Allard, F. Lédée, R. Mazurczyk, E. Deleporte, C. Seassal, and H. S. Nguyen, “Tailoring dispersion of room-temperature exciton-polaritons with perovskite-based subwavelength metasurfaces,” Nano Lett. 20, 2113–2119 (2020). [CrossRef]
37. L. Lu, Q. Le-Van, L. Ferrier, E. Drouard, C. Seassal, and H. S. Nguyen, “Engineering a light–matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points,” Photonics Res. 8, A91–A100 (2020). [CrossRef]
