Abstract
Optical bound states in the continuum (BICs) provide a way to engineer very narrow resonances in photonic crystals. The extended interaction time in these systems is particularly promising for the enhancement of nonlinear optical processes and the development of the next generation of active optical devices. However, the achievable interaction strength is limited by the purely photonic character of optical BICs. Here, we mix the optical BIC in a photonic crystal slab with excitons in the atomically thin semiconductor MoSe_{2} to form nonlinear excitonpolaritons with a Rabi splitting of 27 meV, exhibiting large interactioninduced spectral blueshifts. The asymptotic BIClike suppression of polariton radiation into the far field toward the BIC wavevector, in combination with effective reduction of the excitonic disorder through motional narrowing, results in small polariton linewidths below 3 meV. Together with a strongly wavevectordependent Qfactor, this provides for the enhancement and control of polariton–polariton interactions and the resulting nonlinear optical effects, paving the way toward tuneable BICbased polaritonic devices for sensing, lasing, and nonlinear optics.
Introduction
Optical bound states in the continuum (BICs), supported by photonic crystal structures of certain geometries, have received much attention recently as a novel approach to generating extremely spectrally narrow resonant responses^{1,2}. Since BICs are uncoupled from the radiation continuum through symmetry protection^{3} or resonance trapping^{4}, they can be robust to perturbations of photonic crystal geometric parameters^{5}. This robustness enables a broad range of practical applications, including recently demonstrated spectral filtering^{6}, chemical and biological sensing^{7,8}, and lasing^{4}.
Providing an efficient lighttrapping mechanism, optical BICs are particularly attractive for enhancing nonlinear optical effects^{9,10}, with recent theoretical proposals discussing enhanced bistability^{11} and Kerrtype focusing nonlinearity^{12}. However, for the practical realization of these proposals, a significantly stronger material nonlinear susceptibility than that generally available in dielectricbased photonic crystals is needed.
An attractive approach to the enhancement of the effective nonlinearity is to use excitonpolaritons—hybrid quasiparticles that inherit both the coherent properties of the photonic modes and the interaction strength of the excitons^{13,14,15}. Hybrid nanophotonic systems incorporating atomically thin transition metal dichalcogenides (TMDs) have emerged as a particularly promising platform owing to their ease of fabrication and the possibility of roomtemperature operation^{16,17,18}. In addition to conventional microcavitybased designs, TMD excitonpolaritons have been observed in plasmonic lattices^{19}, photonic crystal slabs (PCSs)^{20,21}, and other nanophotonic structures^{22}.
Coupling TMD excitons to optical BICs in photonic crystals will not only boost the potentially achievable nonlinearities but also provide control on the resonant BIC properties through the excitonic fraction in the polariton, as has been proposed theoretically^{23}.
Here, we experimentally demonstrate and investigate nonlinear polaritons formed via the strong coupling of excitons in monolayer (1 L) MoSe_{2} and optical BICs in a 1D PCS, with Rabi splitting of >27 meV and BIClike radiation suppression in the surfacenormal direction. Despite the large ∼9 meV inhomogeneous broadening of the MoSe_{2} excitonic line, we achieve a small polariton linewidth below 3 meV, corresponding to a very well resolved splittingtolinewidth ratio of ∼9 and Qfactors up to 900. Using the strongly wavevectordependent Qfactor of the photonic crystal dispersion, we show a controllable reduction in the polariton linewidth by a factor of 5–10 when approaching the BIC. The narrow polariton lines allow us to accurately measure the polariton–polariton interaction strength through powerdependent spectral blueshifts in the resonant reflectance experiment, corresponding to an exciton–exciton interaction strength of g_{X} ∼ 1.0 μeV μm^{2}. This polariton nonlinearity is comparable with the values measured in III–V materials^{24,25} and significantly larger than those previously observed in TMD monolayerbased systems^{22}, paving the way toward quantum applications of excitonpolaritons in atomically thin semiconductors.
In the experiment, we fabricate a 1D PCS sample consisting of 90 nm thick Ta_{2}O_{5} bars on a SiO_{2}/Si (1 μm/500 μm) substrate, as schematically shown in Fig. 1a, with a scanning electron microscopy image shown in Fig. 1b. The PCS geometry (see “Methods”) is designed for large refractive index modulation to open a photonic band gap and support an optical BIC close to the exciton resonance in monolayer MoSe_{2}. As illustrated in the photonic band structure shown in Fig. 1d, the BIC is expected to form on the lowerenergy m = 2 TE mode (red) at the Γ point in the crystal momentum space^{23}, with characteristic confinement and antisymmetric spatial distribution of the optical field (c) with respect to the mirror symmetry plane of the PCS cell (see Supplementary Fig. S6).
We measure the PCS band structure via angleresolved reflectance spectroscopy (see “Methods” and Supplementary Fig. S1). Figure 1e shows the experimental differential reflectance spectra for varying angle θ, where the signal from the unpatterned Ta_{2}O_{5}/SiO_{2}/Si substrate is subtracted for clarity: δR (θ, ω) = R_{PCS} (θ, ω) − R_{Sub} (θ, ω). Three modes, a broad symmetric one (m = 3) and two narrower antisymmetric ones (m = 2), are clearly observed in the figure, in agreement with the theory (d, red dashed box). We fit the lowerenergy antisymmetric mode peak in the reflectance spectra using a Fanolike line shape F(ω) ∝ (qγ/2 + ω − ω_{0})^{2}/(γ^{2}/4 + (ω − ω_{0})^{2}) with resonance frequency ω_{0}, linewidth γ, and asymmetry parameter q, which arises due to interference with the broad symmetric mode and an even broader Fabry–Perot response of the layered substrate (see Supplementary Fig. S2).
The extracted Fano fit parameters are plotted in Fig. 1f, g as functions of the ratio k_{x}/k = sinθ, where k_{x} is the inplane wavevector component, k is the freespace wavevector magnitude, and θ is the angle with respect to the surface normal. Toward the Γ point (k_{x} → 0), the reflectivity associated with the mode sharply decreases (f, red), while the spectral line narrows (g, black circles), resulting in a sharply increasing Qfactor (g, red open circles), defined as Q = ω_{0}/γ. This behavior is a characteristic of an atΓ optical BIC^{26}, where the interference of optical waves outgoing in opposite directions leads to effective light trapping in the near field and vanishing farfield radiation. This is in contrast to the case studied recently^{20,21}, where only radiating PCS modes with smaller and largely angleindependent Qfactors were considered for strong coupling to excitons in 2D semiconductors.
The theoretically predicted radiative Qfactor of the BIC diverges toward infinity at k_{x} = 0. In practice, the measured Qfactor is limited predominantly by nonradiative losses^{5,26}. The two major contributions in our case are (1) intrinsic absorption in Ta_{2}O_{5} and (2) surfaceroughnessinduced symmetry breaking and scattering^{27}, limiting Q to ~10^{3}. Additional resonance broadening comes from leaky losses in the Si due to nearfield penetration through the SiO_{2} layer^{27} and finite sample size effects^{3,28}.
We simulate the PCS dispersion and associated Qfactors with the Fourier modal method, taking into account these four loss mechanisms (see Supplementary Note 1 and Fig. S3). As shown in Fig. 1g (dashed lines), good agreement with the experiment can be achieved by considering the scattering losses through an additional imaginary part of the Ta_{2}O_{5} refractive index δn ∼ 0.002i (Fig. 1g, dashed lines).
We then create polaritons by coupling the observed BIC to excitons in the monolayer MoSe_{2} in a vertically stacked structure consisting of 1 L MoSe_{2}, multilayer hexagonal boron nitride (hBN), and a PCS, as illustrated in Fig. 2a. To maximize the Qfactor of the resulting polariton modes, we use largearea multilayer hBN and monolayer MoSe_{2} flakes of ∼100 μm in size, covering ∼200 periods of the PCS, as demonstrated in the optical microscope image in (b). The hBN spacer plays a threefold role: it avoids MoSe_{2} flakes “sagging” into the PCS grooves, reduces the influence of the Ta_{2}O_{5} surface roughness, and provides tunability of the BIC frequency through the hBN thickness. In our case, a 9 nm thick hBN spacer shifts the PCS mode and spectral position of the BIC by ∼30 meV to bring it close to resonance with the neutral exciton in the 1 L MoSe_{2} at 7 K, with ħω_{X} = 1.65 eV (Fig. 1f, blue squares).
We study the polaritons experimentally via angleresolved reflectivity and photoluminescence (PL) measurements at 7 K (see “Methods”), with the results of TEpolarized detection shown in Fig. 2c–f. In comparison with Fig. 1e, the lowerenergy antisymmetric PCS mode observed in reflectivity (Fig. 2c, d) is now redshifted by ∼30 meV owing to the presence of hBN/MoSe_{2} and split into upper and lower polariton branches (UPB and LPB, respectively) owing to strong coupling with the neutral exciton in the 1 L MoSe_{2} centered at ħω_{X} = 1.65 eV (see also Supplementary Fig. S4). Both the LPB and UPB retain BIClike behavior near the Γ point, exhibiting several distinctive properties.
First, at the Γ point, both the LPB and UPB are “dark,” as radiation into the far field becomes symmetry forbidden, effectively extending the interaction time for potential enhancement of the optical nonlinearities. Second, close to the Γ point, polaritons possess a negative effective mass and associated negative group velocity inherited from the PCS mode, providing a potential platform for studying TMDbased polariton selffocusing and soliton formation, similar to what has been discussed theoretically and studied experimentally in other polariton systems^{14,29}. Third, the strong variation in the PCS mode linewidth in the vicinity of the BIC results in a wavevectordependent Qfactor of both the LPB and UPB, enabling control of the polariton linewidth with the angle. These strongly modulated Qfactors, when combined with polariton–polariton interactions, can lead to novel phenomena such as the emergence of the socalled “weak lasing” state of matter^{30} or the spontaneous formation of superfluid polariton currents^{31}.
Further details of the optical response are revealed by TEpolarized angleresolved PL spectra (e), showing emission from both polariton branches and the uncoupled neutral exciton (X^{0}). The latter is increasingly enhanced toward small wavevectors and exhibits a slight redshift of ∼1 meV, which we attribute to weak coupling to the higher frequency and broader m = 3 symmetric mode (see Supplementary Note 2 and Fig. S5). Charged exciton (trion, X^{−}) emission is also observed at ħω_{T} = 1.62 meV independent of k_{x}, implying a weak coupling.
We analyze the wavevectordependent behavior of the LPB, UPB, uncoupled neutral exciton, and trion by fitting the PL spectra for each k_{x} with four Lorentzian functions L_{i} (ω) ∝ ((ω − ω_{i})^{2} + (γ_{i}/2)^{2})^{−1}, as shown in Fig. 2f, and extracting the spectral position ω_{i} and linewidth γ_{i} as the full width at half maximum (FWHM) for each peak. The extracted parameters are plotted as functions of the inplane wavevector in Fig. 3, with spectral positions (a) for the UPB (red symbols), LPB (blue symbols), and uncoupled neutral exciton (orange symbols), corresponding values of the FWHM (b), and calculated Qfactors (c). The parameters for the uncoupled excitons extracted from the TMpolarized PL are plotted as green dots.
We then fit the extracted spectral positions of the UPB (ω_{+}) and LPB (ω_{−}) with a coupled oscillator model^{32}, using the spectral position and homogeneous linewidth for the uncoupled neutral exciton \(\tilde \omega _{X} = {\omega}_{X} + i\gamma _{X}\) and for the PCS/hBN mode \(\tilde \omega _{C}\left( {k_x} \right) = \delta \omega _{C} + \omega _{C}\left( {k_x} \right) + i\gamma _{C}\left( {k_x} \right)\):
Here, Ω_{R} is the Rabi splitting between the UPB and LPB, and the two fit parameters are the coupling strength κ and additional spectral shift δω_{C} of the PCS mode^{20} due to the presence of 1 L MoSe_{2}. The fit curves for the spectral positions of the UPB and LPB are shown in Fig. 3a by the red and blue solid lines, respectively. The uncoupled PCS/hBN mode, indicated by the black squares, comes into resonance with the uncoupled neutral exciton at \(k_x^{{\mathrm{res}}}/k \simeq \pm 0.014\), corresponding to an angle of \({\theta^{\mathrm{res}}} \simeq \pm 0.8^\circ\). From the fits in Fig. 3a, we extract a coupling strength of κ = 13.9 meV, which corresponds to a Rabi splitting of Ω_{R} = 27.4 meV and splittingtolinewidth ratio of ∼9, exceeding the values recently reported for a WSe_{2}/PCS system^{20} and theoretical estimates for strong coupling to an optical BIC^{23}. Because Ω_{R} is larger than the sum of the exciton (∼9 meV) and PCS mode linewidth (∼3−11 meV), the hybrid MoSe_{2}/hBN/PCS system is unambiguously in the strong coupling regime.
Quantitatively, the polariton linewidth γ_{±} is expected to vary between that of the exciton (γ_{X}) and PCS modes (γ_{C}) depending on the excitonic fraction in the polariton. However, our experimentally observed values of the polariton linewidth (Fig. 3b, open symbols) close to resonance \(k_x = k_x^{{\mathrm{res}}}\) are significantly smaller than both γ_{X} (green) and γ_{C} (black). We attribute this to polariton motional narrowing, similar to the effects studied previously for quantum wells in microcavities^{33,34,35,36,37}.
Here, the large polariton mode size (tens of μm) together with the large Rabi splitting lead to effective averaging over excitonic disorder^{38} in the 1 L MoSe_{2} over a broad (nm–μm) range of length scales. As a result, the excitonic contribution to the polariton FWHM close to resonance is given by only the homogeneous exciton linewidth \(\gamma _{\mathrm{X}}^h\)^{39}, while away from resonance, where the polariton frequency overlaps with the exciton peak, it changes toward the inhomogeneous linewidth \(\gamma _{\mathrm{X}}^{inh}\) due to an increasing interaction with disorder and associated scattering with higher momenta excitonic states as well as absorption^{36,40}. We use a phenomenological model that accounts for homogeneous and inhomogeneous contributions to the polariton linewidth (see Supplementary Note 3). The model shows good agreement with the experimental data (Fig. 3b, blue curve) for a homogeneous linewidth of \(\gamma _{\mathrm{X}}^h\sim 1\) meV (b, black dashed line), which is within the range of recently reported values^{41,42,43,44} for the lowtemperature radiative decay rate of excitons in monolayer MoSe_{2}.
As a result of excitonic and photonic disorder averaging, the Qfactors achieved for polaritons around the Γ point in our structure are ∼2 times higher than those for the bare PCS mode (Fig. 3c), reaching Q ∼ 900. These higher Qfactors offer the potential for further improvement of the polariton linewidth through the fabrication of macroscopic photonic crystal samples^{3} with improved surface quality and the use of large TMD flakes grown by chemical vapor deposition. In addition, the strongly kdependent Qfactor of the PCS mode in the vicinity of the BIC enables precise control of the polariton linewidth and corresponding Qfactors via angle or temperature tuning (see Supplementary Note 4 and Figs. S7 and S8).
Mixing photonic modes with excitons in our hybrid MoSe_{2}/hBN/PCS structures leads to a dramatic enhancement of the associated optical nonlinearities. We probe the underlying polariton–polariton interaction due to the excitonic contribution by measuring the pumpdependent frequency shifts of the polariton peaks in the reflectivity spectra. The polariton modes are excited resonantly in both the frequency and wavevector domains by ∼130 fs laser pulses (Fig. 4a, inset), with the incident fluence varying from 0.1 μJ/cm^{2} to 3.0 μJ/cm^{2} (see “Methods”).
Figures 4a, b show the measured pumpdependent reflectivity spectra of the LPB resonance (solid black curves) for selected values of the incident fluence, increasing from bottom to top, and for two different xcomponents of the wavevector: \(k_x^{\left( 1 \right)}/k = 0.078\) (a) and \(k_x^{\left( 2 \right)}/k = 0.024\) (b). Due to the exciton–exciton interaction, which increases with the density of the created quasiparticles^{45}, the lowerenergy polariton resonance in the reflectivity spectra is continuously shifted with the fluence toward higher energies, as clearly seen from the Lorentzian curves L(E) based on the resonance frequency and linewidth extracted from the fits (Fig. 4a, b, bottom panels). We observe larger blueshifts (b) for wavevectors closer to the anticrossing condition \(k_x^{{\mathrm{res}}}/k = 0.014\), as expected for stronger polariton–polariton interactions associated with the increasing exciton fraction in the polariton.
The top panel in Fig. 4c shows the blueshift values for \(k_x^{\left( 1 \right)}/k = 0.078\) (black squares) and \(k_x^{\left( 2 \right)}/k = 0.024\) (red circles) extracted from the Fano line shape fitting (a, b, red dashed curves) for varying fluence, together with the corresponding linear fits (black and red lines). Calculating the polariton density n_{P} for each fluence (see Supplementary Note 5), we obtain the polariton–polariton interaction strength g_{P} = dE_{P}/dn_{P} of \(g_P( {k_x^{\left( 1 \right)}} ) \sim 0.04\) μeV μm^{2} and \(g_P( {k_x^{\left( 2 \right)}} ) \sim 0.16\) μeV μm^{2}. Furthermore, from the \(g_P\left( {k_x} \right) \propto g_X\left {X\left( {k_x} \right)} \right^4\) dependence on the Hopfield coefficient X(k_{x}), which describes the exciton fraction in the polariton, we estimate the exciton–exciton interaction strength in our measurement to be g_{X} = 1.0 ± 0.4 μeV μm^{2}. This value is on the same order as the theoretical estimate g_{X} ~ 1.6 μeV μm^{2}, as well as the value g_{X} ~ 1.4 μeV μm^{2} we extract from a direct measurement of the pumpdependent excitonic blueshifts in TM polarization (see Supplementary Note 6 and Fig. S9).
While it is difficult to directly compare our g_{X} values with those observed in III–V materials due to the few orders of magnitude difference in the reported numbers, our interaction strength is not much lower than the values of g_{X} ≃ 10 μeV μm^{2} recently extracted from careful measurements in GaAsbased polaritonic systems^{25}. On the other hand, our nonlinearities are considerably larger than those reported previously for WS_{2}based polaritons^{22}, where the estimation of the exciton–exciton interaction strength could possibly be uncertain due to efficient local heating at 300 K and the use of high excitation densities leading to higherorder interaction effects and associated redshifts.
The exciton densities in our experiment (≤10^{12} cm^{−2}) are far below the Mott transition density (~10^{14} cm^{−2}), and the observed polariton nonlinearity is mostly due to the exciton–exciton interaction, with phase space filling effects likely playing only a minor role. As seen from the fluencedependent linewidth plots in the bottom panel of Fig. 4c, the increased interaction at higher densities also leads to a faster polariton decay, manifested as powerdependent broadening. In addition, we note that the observed nonlinearities are fast at least on a 100 fs scale, providing future opportunities for developing polaritonbased ultrafast modulators and switches.
In summary, we present the first experimental demonstration and investigation of optical BICbased polaritonic excitations. The formation of BIClike polaritons in a hybrid system of a monolayer semiconductor interfaced with a PCS, with suppressed radiation into the far field and line narrowing due to effective disorder averaging, extends the polariton–polariton interaction time, which enhances the nonlinear optical response. In the future, these “dark” states can be accessed through near fields using guided modes excited by grating coupling or by nonlinear frequency conversion. With the strength of the underlying exciton–exciton interaction g_{X} ~ 1.0 μeV μm^{2}, our polaritons exhibit strong excitonfractiondependent optical nonlinearities that are fast on a 100 fs time scale. In addition, the planar geometry of our structure allows straightforward fabrication of the electrical contacts for electrostatic control of the polaritons and associated interactions, while the use of atomically thin semiconductors in principle allows roomtemperature operation. Thus, the formation of BICbased polaritons can enable not only significantly enhanced but also controllable and fast nonlinear optical responses in photonic crystal systems due to the strong excitonic interaction in monolayer semiconductors and can open a new way to develop active and nonlinear alloptical onchip devices.
Methods
Sample fabrication
Ta_{2}O_{5} layers of 90 nm thickness were deposited on commercial SiO_{2}/Si substrates via ebeam assisted ionbeam sputtering. PCSs were fabricated by patterning the Ta_{2}O_{5} layers via a combination of electronbeam lithography and plasma etching to yield the following geometric parameters: pitch p = 500 nm, groove width w = 220 nm, and depth d = 90 nm, as characterized by scanning electron and atomic force microscopy measurements. Largearea highquality flakes of multilayer hBN and monolayer MoSe_{2} were mechanically exfoliated from commercial bulk crystals (HQ Graphene) and stacked vertically onto the photonic crystal sample surface via dry transfer to form a hybrid 1 L MoSe_{2}/hBN/PCS structure.
Optical measurements
Angleresolved reflectance spectroscopy was performed in a backfocalplane setup with a slit spectrometer coupled to a liquidnitrogencooled imaging CCD camera (Princeton Instruments SP2500+PyLoN), using white light from a halogen lamp for illumination (see Supplementary Fig. S1). For pumpdependent reflectivity measurements, the sample was excited by 130 fs pulses from a wavelengthtuneable Ti:sapphire oscillator (SpectraPhysics, Tsunami, 80 MHz repetition rate) with wavevector control via laser beam positioning within the back focal plane of the objective. A singleslit optical chopper with a duty cycle of 0.001 was used in the laser beam to avoid sample heating. Angleresolved PL measurements were performed in the same setup with offresonant excitation by monochromatic light from a HeNe laser with a wavelength λ_{exc} = 632.8 nm. The sample was mounted in an ultralowvibration closedcycle helium cryostat (Advanced Research Systems) and maintained at a controllable temperature in the range of 7−300 K. The cryostat was mounted onto a precise xyz stage for sample positioning. Spatial filtering in the detection channel was used to selectively measure signals from the 1 L MoSe_{2}/hBN/PCS sample area.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors acknowledge funding from the Ministry of Education and Science of the Russian Federation through Megagrant No. 14.Y26.31.0015. A.I.T. and D.N.K. acknowledge the UK EPSRC grant EP/P026850/1. I.A.S. acknowledges the project “Hybrid polaritonics” of Icelandic Science Foundation. Numerical calculations of the angleresolved reflectivity maps were funded by RFBR according to the research project № 183200527. Sample fabrication was funded by RFBR, project No 193290269. Timeresolved measurements were partly funded by the Russian Science Foundation (Grant No. 197230003). V.K. acknowledges support from the Government of the Russian Federation through the ITMO Fellowship and Professorship Program. This work was in part carried out using equipment of the SPbU Resource Centers “Nanophotonics” and “Nanotechnology”. We thank M. Zhukov, A. Bukatin, and A. Chezhegov for their assistance with the sample characterization and A. Bogdanov for the helpful discussion.
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Kravtsov, V., Khestanova, E., Benimetskiy, F.A. et al. Nonlinear polaritons in a monolayer semiconductor coupled to optical bound states in the continuum. Light Sci Appl 9, 56 (2020). https://doi.org/10.1038/s413770200286z
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DOI: https://doi.org/10.1038/s413770200286z
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