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Abstract
Recent studies have demonstrated that multilayer transition metal dichalcogenides can serve as promising building blocks for creating new kinds of resonant optical nanostructures due to their very high refractive indices. However, most of such studies have focused on excitonic regimes of light–material interaction, while there are few on the low-loss region below the bandgap. Here, we conceptually propose all-van der Waals photonic crystals made of electronically bulk MoS2 and h-BN, designed to operate in the telecom wavelengths. And we demonstrate that, due to extremely low absorption loss and destructive interaction between symmetry-protected and resonance-trapped bound states in the continuum, high-quality factor transmission peaks associated with electromagnetically induced transparency (EIT) are observed, thus rendering our proposed structures highly useful for applications like slow light and optical sensing. Furthermore, EIT-like effects are demonstrated in well-engineered MoS2 nanostructures with broken symmetry. We argue that this work is not only of significance for light harvesting in nanostructured van der Waals materials, but provides also a simple path of constructing classical analogues of EIT using dielectric photonic crystals.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Manipulating light at the nanoscale is of importance for numerous applications in modern optics and photonics. Since they support resonant optical modes, nanostructures, be they plasmonic, photonic, or their both, are endowed with an ability to control the light-wave dispersion at the dimensions comparable to and smaller than the light wavelength. Photonic nanostructures constituted by high-refractive-index dielectrics have recently attracted much attention thanks to their significantly reduced dissipative losses compared to plasmonic counterparts [1,2]. High-refractive-index nanophotonics have thus been a rapidly growing realm with potential for applications including lasing [3–6], holograms [7], sensing [8–11], and nonlinear harmonic wave generation [12–15]. To date, most of dielectric nanostructures have been fabricated from materials such as Si, Si3N4, and GaAs [16]. The presence of naturally layered van der Waals materials paves a new route enriching the traditional nanophotonics library based on conventional high-refractive-index materials, and provides more possibilities to miniaturize optoelectronic devices [17].
For instance, transition metal dichalcogenides (TMDCs), an emerging class of van der Waals crystals, have stimulated great research interest owing to their advanced optical and electronic functionalities [18]. Unlike conventional dielectrics aforementioned, TMDCs have weak intermolecular bonding energies, making it easy to isolate into stable, free-standing monolayer membranes. In the atomic monolayer limit, this material has a direct band gap in the visible and near-infrared regions and can thus produce a high photoluminescence, enabling it an excellent excitonic platform [19–22]. Therefore, it has been extensively investigated that TMDC monolayers are integrated onto optically resonant structures for fundamental investigations of strong light–matter interactions [23–30]. Unlike monolayers, TMDC multilayers exhibit strong anisotropy, with in-plane refractive indices higher than what is found in commonly used dielectric materials like Si and GaAs, thus making them interesting building blocks for engineering nanostructures with various functionalities. Now, bulk TMDCs have been nanostructured for designing ultrathin metasurfaces [31], resonant optical antennas [13,32–34], and one-dimensional dielectric gratings [35–37]. These studies have paid particular attention to optical properties of nanostructures near the resonant absorption peak of exciton. However, in this situation, the material’s absorption loss is large, unfavorable to creating low-loss metamaterials or photonic crystals.
In this paper, we focus on the material response in the telecom spectral range where the absorption loss can be negligible, and propose a two-dimensional dielectric photonic crystal (PhC) completely made of van der Waals materials, which can be fabricated using the traditional beam-based lithography. We demonstrate a classical analogue of electromagnetically induced transparency (EIT) using the TMDC PhC. The transparency is a result of coherent coupling of symmetry-protected bound states in the continuum (BICs) with resonance-trapped BICs, and is manifested as high-quality (Q) factor resonant peaks with nearly complete transmission. Such high-Q resonances benefit from the remarkable reduction in both radiative and non-radiative damping through the interference between BIC modes, as well as the negligible material’s loss at the telecom spectral range. Using the drastic modification in frequency dispersion related to EIT, we show the promise for the PhC in applications including slow light and optical sensing. In addition, we demonstrate a novel PhC structure with symmetry-breaking unit cell, in which EIT-like features are observed at the normal incidence of light.
2. MoS2 PhC
2.1 Materials and methods
The schematic of the suggested all-van der Waals PhC is shown in Fig. 1(A). The structure is composed of a TMDC nanodisk square relief grating made of MoS2 deposited onto a h-BN substrate. The lattice period of a = 400 nm is designed to allow our structure to work in the telecom spectral range. This dimension is considerably smaller than the wavelength of operation, inhibiting any Bragg diffraction within the structure. The MoS2 slab waveguide has a thickness of t2 = 150 nm, which supports well-defined fundamental waveguided modes in the near-infrared to the telecom wavelength range. The thickness of MoS2 disk is fixed at t1 = 50 nm, and its diameter is defined as d. The illumination source is polarized normal to the incident plane, that is, transverse-electric (TE) polarized light, as shown by the top-right-corner inset in Fig. 1(A).
Fig. 1. All-van der Waals PhCs. (A) Diagram of the MoS2 PhC on a h-BN substrate. The MoS2 PhC consists of a square nanodisk array and a thin-film dielectric waveguide, sketched by a layered medium. The inset in the top right corner shows the lattice period a = 400 nm, the thickness of nanodisk t1 = 50 nm, the thickness of slab waveguide t2 = 150 nm, the diameter of nanodisk indicated with d, and the TE-polarized incident beam with incident angle of θ. The left insets schematically show the lattice structures of the monolayer MoS2 and h-BN. (B) Dielectric functions of bulk MoS2 and (C) h-BN.
The permittivity of bulk MoS2 is taken from the experimental data by Ermolaev et al. [38], the in-plane component of which is potted in Fig. 1(B). In the wavelength range of λ > 1000 nm, the real part of the permittivity, Re(ɛ), is in excess of 16, and meanwhile its imaginary part Im(ɛ) is very small, which is significantly desirable to engineering low-loss photonic structures supporting EIT-like effects as well as BICs. Here, we would like to point out that this study can also be expanded to other TMDC materials of interest, such as MoSe2, WS2, and WSe2. The low-refractive-index van der Waals material, h-BN, is considered as a lossless dielectric at the telecom wavelengths. The dielectric function of bulk h-BN is plotted in Fig. 1(C), which is obtained according to the following dispersion formula [39]
2.2 Nonradiating BIC modes in the MoS2 PhC
We first begin the eigenmode analysis for the MoS2 PhC using the three-dimensional FEM eigenfrequency solver. Figure 2(A) shows the dispersion relation of guided modes in the structure with d = 240 nm along the M–Γ and Γ–X directions, where Γ, X, and M represent high-symmetry points of the first Brillouin zone for a square lattice. The band structure near the Γ point consists of three TE-like modes (modes 1, 2, and 4) and a transverse magnetic (TM)-like mode (mode 3) exhibiting different dispersion in the wavevector space. In the Γ–X direction, modes 1 and 4 have approximately linear dispersion; modes 2 and 3 have nearly-flat dispersion and are degenerate at the Γ point due to their same frequencies. In order to recognize these modes, we plot their magnetic-field distributions along the z axis (Hz) at the Γ point, shown by the top row in the Fig. 2(A), where the wireframe color corresponds to that of dispersion curves. As one can distinctly see, at the Γ point, mode 1 can be classified as a monopolar mode, modes 2 and 3 as a degenerate pair of dipolar modes, and mode 4 as a quadrupolar mode.
Fig. 2. Eigenmode analysis of MoS2 PhCs. (A) Band structure (inset: the first Brillouin zone) and (B) Q factor of four guided modes in the MoS2 PhC with d = 240 nm. The top row of panel A shows the Hz field distributions at the Γ point. (C) Band structure for the PhC with d = 200 nm and (D) d = 300 nm.
In order to gain a deeper insight into the physics of these modes, we calculate in Fig. 2(B) the corresponding Q-factor distribution. As can be seen, the Q factors of modes 1 and 4 are extremely sensitivity to the symmetry-breaking perturbation of the outgoing fields and drop rapidly away from the Γ point, which verifies that the monopolar and quadrupolar modes correspond to the symmetry-protected BIC mode. Without the symmetry-breaking perturbation, that is, at the Γ point, these two BIC modes can completely decouple from the radiation continuum, thus resulting in diverging Q factors. For dipolar modes 2 and 3, they are much less sensitivity to this symmetry-breaking perturbation and hold very low-Q factors near the Γ point. It has been previously demonstrated that [3], the dipolar modes are resonance-trapped BICs that rely on structural parameters and that can also have a divergent Q factor at a singular geometric parameter. Resonance-trapped BICs can preserve high-Q factors over a wide wavevector span and can operate at the off–Γ point by choosing appropriate geometric parameters in contrast to symmetry-protected ones. These versatile characters make them beneficial to practical applications.
Through modifying the diameter of MoS2 nanodisk, we can control the dispersion of these BIC modes, as shown in Figs. 2(C) and 2(D). As the disk diameter is reduced (increased) to d = 200 (300) nm, all BIC modes shift to the shorter (longer) wavelengths and meanwhile the dipolar modes are very close to the monopolar (quadrupolar) mode at the Γ point. The dipolar BIC mode is bright, which is accessible from free space, and the monopolar or quadrupolar mode is dark, which is less-accessible. For the three configurations, the bright- and dark-mode resonances are brought in close proximity in the frequency domain and can interfere giving rise to an extremely narrow EIT window [40]. The spectral feature of the EIT resonance is dominated by the coupling strength between the dark and bright modes and the detuning of resonant frequency of these two modes as well as the loss of system. In our research, the frequency detuning between modes is principally mediated by the diameter of MoS2 nanodisk.
2.3 BIC-induced EIT-like effects in the MoS2 PhC
In order to obtain the far-field information of BIC modes, we calculate the transmission spectra for these three MoS2 PhC constructions mentioned above, as shown in Fig. 3. At normal incidence, the structures only show a broad resonance that corresponds to the dipolar resonance-trapped BIC mode according to the eigenmode analysis above. And the symmetry-protected BICs exhibit entirely vanishing linewidths due to their Q factors inclined to infinity. Specifically, symmetry-protected dual BICs appear in the structure with d = 240 nm (Fig. 3(B)), which correspond to the monopolar mode at the longer wavelength and the quadrupolar mode at the shorter wavelength, respectively. As the incident angle θ increases, the TE-like dipolar mode splits into two resonances owing to its interaction with adjacent symmetry-protected BIC modes, accompanied by the appearance of a narrow EIT-like window near the normal incidence of light. The EIT resonances behave as nearly completely transmitted peaks with varying spectral linewidth. This is shown in Figs. 3(D)–3(F), where the corresponding transmission spectrum at θ = 0.2° is plotted. For the structure with d = 200 nm (Fig. 3(D)), the detuning between the monopolar mode and the BIC mode is near zero, thus resulting in a narrow transmission peak with symmetrical Lorentzian profile around λ = 1365 nm. Likewise, such resonance is formed in the structure with d = 300 nm around λ = 1375 nm due to the coupling of the dipolar mode with the quadrupolar mode (Fig. 3(F)). The BIC modes with large detuning from the dipolar modes are transformed into high-Q Fano-like resonance dips since they are embedded in a continuum of vacuum states. Accordingly, dual high-Q Fano-like resonances are observed in Fig. 3(E).
Fig. 3. Far-field spectra for the MoS2 PhCs. (A) Transmission spectra plotted as a function of incident angle θ for the PhC structure at d = 200 nm, (B) d = 240 nm, and (C) d = 300 nm. (D–E) Detailed transmission curves for structures at θ = 0.2° corresponding with panels A–C.
Also, one can note that the bandwidth of the EIT-like transmission peak is decreased as the incident angle increases and vanishes at the normal incidence. In order to quantitively visualize the EIT-like features, we only exemplify the MoS2 PhC with d = 300 nm, in which the dipolar mode resonates with the quadrupolar mode, and calculate in Fig. 4(A) the angle-dependence Q factor of the EIT-like resonance using a simple Lorentzian profile. The calculated Q factors show a trend to diverge to infinity at θ = 0°, manifesting that the EIT mode is characterized by symmetry-protected quasi-BICs.
Fig. 4. EIT-like features in the MoS2 PhC with d = 300 nm. (A) Q factors of the EIT resonance as a function of incident angle. (B) Transmission spectra at different incident angles. The open black circles correspond to the FEM calculated transmission, and the blue solid curves to the fitting results by COM. (C) Extracted fitting coefficients.
In order to shed light on the Q-factor variations in the EIT-like resonance, we carry out the quantitative description of this EIT-like system using the classical coupled oscillator model (COM). Due to the large detuning between the monopolar mode and the dipolar mode for the PhC with d = 300 nm, we can only take into account the COM including two mutual coupled oscillators subjected to a driving harmonic driving force. The bright dipolar mode is modeled as oscillator 1, driven by the harmonic force $F(t )= F{e^{ - i{\omega _s}t}}$, whereas the dark quadrupolar mode is modeled as oscillator 2. The response of the EIT-like system can be described by the following differential equations [41],
At the near-zero detuning, that is, ${\omega _1} = {\omega _2} = \omega $, the energy dissipation in the system during one period of oscillation can be calculated as follows [41]:
As shown in Fig. 4(B), we fit $1 - P(\omega )$ to the FEM calculated transmission spectra indicated by black open circles, and obtain the fitted results (blue curves) that reproduce nearly perfectly the calculated transmission. Relevant fitting coefficients, ${\gamma _1}$, ${\gamma _2}$, and $\mathrm{\varOmega }$, are extracted and plotted as a function of incident angle in Fig. 4(C). The loss of the bright mode keeps roughly constant at ${\gamma _1} \approx 0.2\; \textrm{THz}$. The dissipation rate of the dark mode, ${\gamma _2}$, is much smaller than that of the bright mode, which is an important indicator to achieve the clear regime of EIT. Typically, the loss of resonance mode, $\gamma $, is contributed by two terms, namely the radiative loss ${\gamma _R}$ and the non-radiative ${\gamma _{NR}}$. Because the absorption loss of MoS2 can be negligible in the wavelength range from 1300 to 1500 nm (see Fig. 1(B)), the non-radiative loss term ${\gamma _{NR}}$ can be substantially minimized, playing a particular active role in the increase of Q factor of EIT resonance. The coupling coefficient $\mathrm{\varOmega }$ is another equally important element in terms of achieving a high-Q EIT resonance. It is evident that the coefficient $\mathrm{\varOmega }$ of the coupling between the bright and dark modes is nearly linearly increased as the incident angle increases. This change is negatively correlated to that of the Q factor of EIT resonance (see Fig. 4(A)). Thus, the reduction of $\mathrm{\varOmega }$ value can effectively increase the Q factor of EIT resonance. We would like to stress that the above-mentioned discussion can also be done for the structure with d = 240 nm.
2.4 Slow light and sensing applications
A narrow EIT-like transparency window is highly desirable for many advanced applications such as slow light and sensing. The abrupt change in frequency dispersion occurs at the transparency window, able to slow or even halt the propagation of light. Slow light offers many possibilities in photonics. For instance, it can promote stronger light–matter interaction and provides additional control over the spectral bandwidth of this interaction. Moreover, it also allows for delaying and temporarily storing light in all-optical memories. Now, slow-light effect has found numerous applications in a large class of areas, including but not limited to optical nonlinearities [42], optical switching [43], quantum optics, and optical storage [44].
The group index can be utilizing for characterizing the performance of slow-light devices, which represents the slowdown factor of light and is given by
Fig. 5. Applications for slow light and sensing. (A) The maximum group indices at the EIT-like transparency window for the MoS2 PhC with d = 300 nm at different incident angles. (B) Principle of photonic sensing using the EIT-like transmission peak. (C) Variation in the transmission, |ΔT|, in dependence of the difference of refractive index, Δn, in the environment of the nanostructure.
Besides the slow-light effects, narrow transmission curves can be potentially applied to optical sensing due to their large sensitivity to a local change in the refractive index of the surrounding medium. The commonly used sensing scheme is tracking the shift in the resonance frequency Δω caused by the change in the refractive index of the analyte Δn. But this requires recording the entire spectrum. In addition, this method is unfriendly to the photonic structures, especially the photonic crystals. This is because the large electromagnetic-field enhancement is located inside the dielectrics, leading to an extremely small resonance shift hard to be detected. However, the shift Δω of the resonance can directly produce a detectable intensity change in the transmitted light, ΔT (Fig. 5(B)). And the more pronounced signal ΔT can be expected when the resonance becomes narrow. Thus, recording the transmission change at a fixed wavelength is a promising alternative and is highly suitable for our structure. In order to evaluate the sensing performance of the photonic system, we choose the MoS2 PhC with d = 300 nm at θ = 0.2° due to its supporting the narrower EIT-like resonance. Figure 5(C) shows the calculated transmission change ΔT as a function of the refractive index change Δn, in which the initial refractive index of surrounding medium is n = 1 and the detection wavelength close to the transmission peak position is λ = 1375 nm. As can be seen from Fig. 5(C), the structure as a sensor can detect the refractive changes as small as 10−3, accompanied by pronounced changes in the transmitted intensity of about 4%.
3. MoS2 PhC with broken symmetry
In this section, we demonstrate that EIT-like resonances can be achieved in a reduced-symmetry MoS2 PhC nanostructure at normal incidence. As shown in Fig. 6(A), the structure consists of a square array with unit cell formed by a nanodisk with a missing wedge-shaped slice. The amount of the structural symmetry-breaking is determined by the wedge angle α. The system is normally illuminated by a x-polarized light, as shown by the top-right-corner inset in this figure. We calculate the eigenfrequency as a function of α for the structures, as shown in the top panels in Fig. 6(B), revealing three photonic modes with different dispersion. Note that these modes are uncoupled mutually. It is clearly seen that mode 3 intersects with modes 1 and 2 at around the slice angle α = 120° and 20°, respectively. This can lead to coherent interaction between modes in the far field (see Fig. 6(C)). Two clear anti-crossings are observed in the transmission map, thus resulting in two narrow EIT-like transparency windows. The formation of EIT-like resonances at near α = 20° and α = 120° is respectively due to the dipole–quadrupole and dipole–monopole interactions (Fig. 6(D)), of which the physical mechanism resembles that discussed in Figs. 2(C) and 2(D). The purely real monopolar and quadrupolar BICs are highlighted by red and white dashed circles, which have Q factors diverging to infinity when the structure is symmetric (Fig. 6(B)). The two BIC modes are thus symmetry-protected. In addition, it can be seen from Fig. 6(B) that the Q factor of mode 2 is diverging at about α = 90°. This leads to a resonance with vanishing linewidth in the far-field transmission map, as marked by the black dashed circle in Fig. 6(C), which reveals the emergence of another BIC mode. The radiative loss of this BIC is almost suppressed due to the Friedrich–Wintgen scenario of destructive interference, and hence, it can be can be referred to as Friedrich–Wintgen BIC or interference-based BIC [45].
Fig. 6. EIT-like features in the MoS2 PhC with broken-symmetry unit cell. (A) The schematic illustration of the system. The parameter α is used to define the structural asymmetry. The x-polarized plane-wave beam is normally incident to the structure. The radius of the nanodisk is 300 nm, and other geometric parameters are same to those of Fig. 1(A). (B) The top panel shows the eigenfrequency of the three uncoupled modes in dependence of the asymmetric parameter α, and the other shows the Q factors of the three modes, which diverge to infinity at the two BICs. (C) Transmission spectra plotted as a function of α. BICs are highlighted by dashed circles showing two symmetry-protected BICs, namely monopolar BIC (red) and quadrupolar BIC (white), and a Friedrich–Wintgen BIC (black). The vertical solid lines indicate the two slice angles, that is, α = 20° and 120°. (D) The top and bottom panels show the detailed transmission spectrum of structures with α = 20° and 120°, respectively. The vertical shadow indicates the EIT-like transparency windows.
It should be pointed out that EIT-like effects could also be shown in other configurations with a modifying geometry of nanodisk. Some potential geometries can be employed, such as asymmetric nanorings that have been widely utilized for dielectric metamaterials [46,47], symmetry-broken cuboids [12], asymmetric one-dimensional dielectric nanowires, and other geometries involved in the literature [48].
4. Conclusion
In summary, we have investigated two-dimensional PhCs completely made of bulk van der Waals materials, working in the telecom spectral range where the systems have very low absorption losses. The structures were demonstrated to support high-Q symmetry-protected and low-Q resonance-trapped BICs near the Γ point. And we have demonstrated that through tuning the structural parameters allows an efficient coupling of the two types of BIC modes, hence resulting in the emergence of the EIT-like resonances. The potential applications of the EIT-like effect, such as slow light and sensing, were demonstrated. Under the same physical mechanism, we further gave a demonstration of the EIT-like effect in a PhC with symmetry-breaking unit cell. The physics disclosed by our findings paves a way for constructing all-optical EIT devices using PhCs. Our study does not need complicated coupled components as compared to all-dielectric metamaterial or metasurface analogues [49,50], it can also be carried out in silicon nanophotonics and it also riches the library of EIT constituting BIC-inspired metamaterials or metasurfaces [51–53]. Moreover, besides applications mentioned in this article, we believe that the systems studied are perfect candidates for BIC lasers, ultrasharp spectral filters, and the enhancement of optical nonlinearity.
Funding
National Natural Science Foundation of China (62105095, U1804261); Natural Science Foundation of Henan Province (202300410238); National Scientific Research Project Cultivation Fund of Henan Normal University (20210381, 2021PL22).
Disclosures
The authors declare no conflicts 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. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016). [CrossRef]
2. I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nature Photon 11(5), 274–284 (2017). [CrossRef]
3. A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017). [CrossRef]
4. V. Mylnikov, S. T. Ha, Z. Pan, V. Valuckas, R. Paniagua-Domínguez, H. V. Demir, and A. I. Kuznetsov, “Lasing action in single subwavelength particles supporting supercavity modes,” ACS Nano 14(6), 7338–7346 (2020). [CrossRef]
5. C. Huang, C. Zhang, S. Xiao, Y. Wang, Y. Fan, Y. Liu, N. Zhang, G. Qu, H. Ji, J. Han, L. Ge, Y. Kivshar, and Q. Song, “Ultrafast control of vortex microlasers,” Science 367(6481), 1018–1021 (2020). [CrossRef]
6. A. Tripathi, H.-R. Kim, P. Tonkaev, S.-J. Lee, S. V. Makarov, S. S. Kruk, M. V. Rybin, H.-G. Park, and Y. Kivshar, “Lasing action from anapole metasurfaces,” Nano Lett. 21(15), 6563–6568 (2021). [CrossRef]
7. X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4(1), 2807 (2013). [CrossRef]
8. N. Bontempi, K. E. Chong, H. W. Orton, I. Staude, D.-Y. Choi, I. Alessandri, Y. S. Kivshar, and D. N. Neshev, “Highly sensitive biosensors based on all-dielectric nanoresonators,” Nanoscale 9(15), 4972–4980 (2017). [CrossRef]
9. S. Romano, M. Mangini, E. Penzo, S. Cabrini, A. C. De Luca, I. Rendina, V. Mocella, and G. Zito, “Ultrasensitive surface refractive index imaging based on quasi-bound states in the continuum,” ACS Nano 14(11), 15417–15427 (2020). [CrossRef]
10. S. Romano, G. Zito, S. Torino, G. Calafiore, E. Penzo, G. Coppola, S. Cabrini, I. Rendina, and V. Mocella, “Label-free sensing of ultralow-weight molecules with all-dielectric metasurfaces supporting bound states in the continuum,” Photonics Res. 6(7), 726–733 (2018). [CrossRef]
11. M. L. Tseng, Y. Jahani, A. Leitis, and H. Altug, “Dielectric metasurfaces enabling advanced optical biosensors,” ACS Photonics 8(1), 47–60 (2021). [CrossRef]
12. Z. Liu, Y. Xu, Y. Lin, J. Xiang, T. Feng, Q. Cao, J. Li, S. Lan, and J. Liu, “High-Q quasibound states in the continuum for nonlinear metasurfaces,” Phys. Rev. Lett. 123(25), 253901 (2019). [CrossRef]
13. M. Nauman, J. Yan, D. de Ceglia, M. Rahmani, K. Z. Kamali, C. De Angelis, A. E. Miroshnichenko, Y. Lu, and D. N. Neshev, “Tunable unidirectional nonlinear emission from transition-metal-dichalcogenide metasurfaces,” Nat. Commun. 12(1), 5597 (2021). [CrossRef]
14. L. Wang, S. Kruk, K. Koshelev, I. Kravchenko, B. Luther-Davies, and Y. Kivshar, “Nonlinear wavefront control with all-dielectric metasurfaces,” Nano Lett. 18(6), 3978–3984 (2018). [CrossRef]
15. N. Bernhardt, K. Koshelev, S. J. U. White, K. W. C. Meng, J. E. Fröch, S. Kim, T. T. Tran, D.-Y. Choi, Y. Kivshar, and A. S. Solntsev, “Quasi-BIC resonant enhancement of second-harmonic generation in WS2 monolayers,” Nano Lett. 20(7), 5309–5314 (2020). [CrossRef]
16. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983). [CrossRef]
17. P. Ajayan, P. Kim, and K. Banerjee, “Two-dimensional van der Waals materials,” Physics Today 69(9), 38–44 (2016). [CrossRef]
18. S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, “2D transition metal dichalcogenides,” Nat. Rev. Mater. 2(8), 17033 (2017). [CrossRef]
19. A. Krasnok, S. Lepeshov, and A. Alú, “Nanophotonics with 2D transition metal dichalcogenides [Invited],” Opt. Express 26(12), 15972–15994 (2018). [CrossRef]
20. A. Arora, M. Koperski, K. Nogajewski, J. Marcus, C. Faugeras, and M. Potemski, “Excitonic resonances in thin films of WSe2: from monolayer to bulk material,” Nanoscale 7(23), 10421–10429 (2015). [CrossRef]
21. Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014). [CrossRef]
22. X. Li, J. Zhu, and B. Wei, “Hybrid nanostructures of metal/two-dimensional nanomaterials for plasmon-enhanced applications,” Chem. Soc. Rev. 45(11), 3145–3187 (2016). [CrossRef]
23. L. Zhang, R. Gogna, W. Burg, E. Tutuc, and H. Deng, “Photonic-crystal exciton-polaritons in monolayer semiconductors,” Nat. Commun. 9(1), 713 (2018). [CrossRef]
24. W. Liu, B. Lee, C. H. Naylor, H.-S. Ee, J. Park, A. T. C. Johnson, and R. Agarwal, “Strong Exciton−plasmon coupling in MoS2 coupled with plasmonic lattice,” Nano Lett. 16(2), 1262–1269 (2016). [CrossRef]
25. A. Bisht, J. Cuadra, M. Wersäll, A. Canales, T. J. Antosiewicz, and T. Shegai, “Collective Strong light–matter coupling in hierarchical microcavity–plasmon–exciton systems,” Nano Lett. 19(1), 189–196 (2019). [CrossRef]
26. D. Zheng, S. Zhang, Q. Deng, M. Kang, P. Nordlander, and H. Xu, “Manipulating coherent plasmon−exciton interaction in a single silver nanorod on monolayer WSe2,” Nano Lett. 17(6), 3809–3814 (2017). [CrossRef]
27. H. Wang, J. Wen, W. Wang, N. Xu, P. Liu, J. Yan, H. Chen, and S. Deng, “Resonance coupling in heterostructures composed of silicon nanosphere and monolayer WS2: a magnetic-dipole-mediated energy transfer process,” ACS Nano 13(2), 1739–1750 (2019). [CrossRef]
28. 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(4), 1900420 (2020). [CrossRef]
29. S. Lepeshov, M. Wang, A. Krasnok, O. Kotov, T. Zhang, H. Liu, T. Jiang, B. Korgel, M. Terrones, Y. Zheng, and A. Alú, “Tunable resonance coupling in single Si nanoparticle−monolayer WS2 structures,” ACS Appl. Mater. Interfaces 10(19), 16690–16697 (2018). [CrossRef]
30. X. Zong X, L. Li, K. Yu, and Y. Liu, “Strong coupling between hybrid plasmonic–photonic resonances and excitons in metallic photonic crystal–monolayer WS2 nanostructures,” Results Phys. 30, 104850 (2021). [CrossRef]
31. X. Zhang, C. De-Eknamkul, J. Gu, A. L. Boehmke, V. M. Menon, J. Khurgin, and E. Cubukcu, “Guiding of visible photons at the ångström thickness limit,” Nat. Nanotechnol. 14(9), 844–850 (2019). [CrossRef]
32. R. Verre, D. G. Baranov, B. Munkhbat, J. Cuadra, M. Käll, and T. Shegai, “Transition metal dichalcogenide nanodisks as high-index dielectric Mie nanoresonators,” Nat. Nanotechnol. 14(7), 679–683 (2019). [CrossRef]
33. T. D. Green, D. G. Baranov, B. Munkhbat, R. Verre, T. Shegai, and M. Käll, “Optical material anisotropy in high-index transition metal dichalcogenide Mie nanoresonators,” Optica 7(6), 680–686 (2020). [CrossRef]
34. N. Muhammad, Y. Chen, C. W. Qiu, and G. P. Wang, “Optical bound states in continuum in MoS2-based metasurface for directional light emission,” Nano Lett. 21(2), 967–972 (2021). [CrossRef]
35. H. Zhang, B. Abhiraman, Q. Zhang, J. Miao, K. Jo, S. Roccasecca, M. W. Knight, A. R. Davoyan, and D. Jariwala, “Hybrid exciton-plasmon-polaritons in van der Waals semiconductor gratings,” Nat. Commun. 11(1), 3552 (2020). [CrossRef]
36. Z. Chai, X. Hu, H. Yang, and Q. Gong, “Low-power all-optical tunable sharp trapped-mode resonances in asymmetrical planar WS2 exciton-polariton gratings,” Appl. Phys. Lett. 116(16), 161111 (2020). [CrossRef]
37. X. Zong, L. Li, and Y. Liu, “Photonic bound states in the continuum in nanostructured transition metal dichalcogenides for strong photon–exciton coupling,” Opt. Lett. 46(24), 6095–6098 (2021). [CrossRef]
38. G. A. Ermolaev, Y. V. Stebunov, A. A. Vyshnevyy, D. E. Tatarkin, D. I. Yakubovsky, S. M. Novikov, D. G. Baranov, T. Shegai, A. Y. Nikitin, A. V. Arsenin, and V. S. Volkov, “Broadband optical properties of monolayer and bulk MoS2,” npj 2D Mater Appl 4(1), 21 (2020). [CrossRef]
39. S. Y. Lee, T. Y. Jeong, S. Jung, and K. J. Yee, “Refractive index dispersion of hexagonal boron nitride in the visible and near-infrared,” Phys. Status Solidi B 256(6), 1800417 (2019). [CrossRef]
40. Y. C. Liu, B. B. Li, and Y. F. Xiao, “Electromagnetically induced transparency in optical microcavities,” Nanophotonics 6(5), 789–811 (2017). [CrossRef]
41. C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70(1), 37–41 (2002). [CrossRef]
42. C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010). [CrossRef]
43. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102(20), 203902 (2009). [CrossRef]
44. T. F. Krauss, “Why do we need slow light?” Nature Photon 2(8), 448–450 (2008). [CrossRef]
45. S. I. Azzam and A. V. Kildishev, “Photonic bound states in the continuum: from basics to applications,” Adv. Opt. Mater. 9(1), 2001469 (2021). [CrossRef]
46. G. Zhang, C. Lan, R. Gao, and J. Zhou, “Trapped-mode-induced giant magnetic field enhancement in all-dielectric metasurfaces,” J. Phys. Chem. C 123(47), 28887–28892 (2019). [CrossRef]
47. V. R. Tuz, V. V. Khardikov, A. S. Kupriianov, K. L. Domina, S. Xu, H. Wang, and H.-B. Sun, “High-quality trapped modes in all-dielectric metamaterials,” Opt. Express 26(3), 2905 (2018). [CrossRef]
48. K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum,” Phys. Rev. Lett. 121(19), 193903 (2018). [CrossRef]
49. Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5(1), 5753 (2014). [CrossRef]
50. Q. Wang, L. Yu, H. Gao, S. Chu, and W. Peng, “Electromagnetically induced transparency in an all-dielectric nano-metamaterial for slow light application,” Opt. Express 27(24), 35012–35026 (2019). [CrossRef]
51. R. Singh, I. A. I. Al-Naib, Y. Yang, D. R. Chowdhury, W. Cao, C. Rockstuh, T. Ozaki, R. Morandotti, and W. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99(20), 201107 (2011). [CrossRef]
52. J. F. Algorri, F. Dell’Olio, P. Roldán-Varona, L. Rodríguez-Cobo, J. M. López-Higuera, J. M. Sánchez-Pena, and D. C. Zografopoulos, “Strongly resonant silicon slot metasurfaces with symmetry-protected bound states in the continuum,” Opt. Express 29(7), 10374–10385 (2021). [CrossRef]
53. J. F. Algorri, F. Dell’Olio, P. Roldán-Varona, L. Rodríguez-Cobo, J. M. López-Higuera, J. M. Sánchez-Pena, V. Dmitriev, and D. C. Zografopoulos, “Analogue of electromagnetically induced transparency in square slotted silicon metasurfaces supporting bound states in the continuum,” Opt. Express 30(3), 4615–4630 (2022). [CrossRef]
