The design of emitting nanodevices with pumping by a strong near field using plasmon and dielectric nanocavities is one of the significant and rapidly developed fields of modern nanophotonics. The authors of the first works [1, 2] combined exciton resonance of a quantum-confined chromophore and the near field of the plasmonic nanoparticle. However, large losses in metallic nanocavities strongly increased the threshold of laser generation. The use of nonlinear near-field and collective effects [3] only partially solves the problem of control of generation in such systems. New possibilities are based on the use of dielectric nanophotonics [4–6] to design low-threshold efficiently controlled micro- and nanolasers. In particular, positive feedback was reached in [7] by combining exciton resonance and high order Mie resonances in single perovskite nanoparticles at a low threshold of generation.

At the same time, to increase the output power of laser generation, such emitting nanoparticles can be assembled in a metasurface with the synchronization of their near field responses using collective subdiffraction effects. Synchronization can be achieved by exciting so-called photonic bound states in the continuum [8, 9], which lead to the generation of (quasi)trapped modes (QTMs) [10, 11] and to a strong concentration of the near field in the vicinity of nanoparticles and inside them. However, the excitation of a quasitrapped mode in the lattice of even weakly dissipative particles can initiate the collective enhancement of losses [12] due to the induction of strong fields inside each nanoparticle. Consequently, the metasurface can be used for the near field pumping, whereas an active medium can be photoluminescent two-dimensional semiconductor films with a thickness of one or a few atomic layers [13], in particular, transition metal dichalcogenides having record optical anisotropy [14] and bright exciton resonances [15], which are deposited on the metasurface.

The metasurface composed of Si disks with the radius \({{R}_{2}}\) and height \(H\), each with a circular hole with the radius \({{R}_{1}}\) shifted by \({{\Delta }_{y}}\) along the y axis (see Fig. 1a) is used for near-field pumping. Under the irradiation of this metasurface by the linearly polarized wave \({{E}_{x}}({{k}_{z}})\) of the signal field, the dipole magnetic moment component \({{m}_{z}}\) [12] perpendicular to the base plane can be excited in each disk owing to bianisotropy. The QTM regime [12] can be realized in the metasurface when its period T satisfies the condition of constructive interference of near-field responses of individual disks taking into account the effective permittivity of the metasurface (see Fig. 1b).

Fig. 1.
figure 1

(Color online) (a) Model of the metasurface of Si disks with holes. (b) Visualization of the calculated distribution of the magnetic field (indicated in color for the right disk) on the surface and the electric field (indicated by arrows) near a pair of Si disks of the metasurface in the regime of QTM excitation. (c) Reflection spectra of the metasurface placed in vacuum and on the quartz substrate, as well as at orthogonal polarizations of the signal electromagnetic wave. (d) Results of the multipole analysis for contributions from different components of the electric (\(E{{D}_{x}}\) for \({{p}_{x}}\) and \(E{{D}_{y}}\) for \({{p}_{y}}\)) and magnetic (\(M{{D}_{z}}\) for \({{m}_{z}}\)) dipoles to the total scattering cross section of a single disk of the metasurface placed on the SiO2 substrate in the regime of excitation of quasitrapped mode. The parameters of the system are R2 = 164 nm, H = 110 nm, R1 = 80 nm, Δy = 70 nm, and T = 702 nm.

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Using the strategy of search for QTMs [16], we obtain the period T = 702 nm for the metasurface consisting of Si disks with the parameters R2 = 164 nm, H = 110 nm, R1 = 80 nm, and Δy = 70 nm and placed in vacuum. In this case, QTMs are excited at a wavelength of λQTM = 980 nm (see Fig. 1c). The QTM resonance in the same metasurface placed on the SiO2 substrate is shifted to the wavelength λQTM = 1050 nm and corresponds to the reflection peak of the signal field with the full width at half maximum wFWHM = 19.7 nm (see Fig. 1с). The quality factor of this resonance is Q = λQTM/wFWHM = 54. A significant increase in the reflection coefficient of the metasurface at the QTM wavelength is due to the resonant enhancement of the emitting component \({{p}_{x}}\) of the electric dipole of each disk [17, 18] (see the results of the multipole analysis in Fig. 1d). In turn, such an enhancement is due to the bianisotropic coupling between the components \({{p}_{x}}\) and \({{m}_{z}}\); the latter component is responsible for the regime of QTM formation [11, 12]. The incident wave with the polarization \({{E}_{y}}({{k}_{z}})\) does not excite the bianisotropic component \({{m}_{z}}\) in disks (Fig. 1d) because the condition of the location of the defect (hole) with respect to the polarization of the exciting field is violated [16]. As a result, the feature of the reflection coefficient of the metasurface associated with the QTM excitation disappears (see Fig. 1c). Thus, the polarizing control of both the QTM formation in the metasurface and the intensity of the near-field response for each of its building blocks becomes possible. In addition, the electric component of the near field above the surface composed of disks in the QTM regime is oriented predominantly in the plane of the metasurface (see Fig. 1b) owing to the excitation of \({{p}_{x}}\), which is a basis for the effective control of exciton resonances with the deposition of a two-dimensional film over the metasurface.

Below, we consider the 1000 \( \times \) 1000 \( \times \) 0.7 nm flakes with a thickness of one atomic layer, which are separated from a MoTe2 crystal by, e.g., the exfoliation method [19], as an active medium of the metastructure, which completely covering the metasurface. Under these conditions, MoTe2 can be considered as a direct band gap semiconductor [20] with the photoluminescence wavelength λ0 that is determined by the width of the band gap Eg and depends on the temperature. In particular, at a temperature of 4.5 K, we obtain Eg = 1.18 eV, λ0 = 1056 nm [20], and the complex refractive index \(\bar {n}({{\lambda }_{0}}) = n({{\lambda }_{0}}) + i\alpha ({{\lambda }_{0}})\) = 4.4752 + i0.39967 [21].

For pumping, it is possible to use a cw He–Ne laser at a wavelength of λp = 633 nm [13], which normally irradiates the metastructure consisting of a MoTe2 monolayer film lying on the metasurface supporting QTM (see Fig. 2a). When the central wavelength λ0 of photoluminescence of MoTe2 is tuned to the wavelength λQTM of a quasitrapped mode, a significant enhancement of photoluminescence can be expected with the possibility of generation of coherent electromagnetic radiation by the system in the direction perpendicular to the plane of the metastructure. The initial optimization of the metasurface shown in Fig. 1a was performed so that the wavelength of the QTM in the metasurface coated with the MoTe2 monolayer film was λQTM = 1056 nm.

Fig. 2.
figure 2

(Color online) (a) Model of the metasurface in the form of thin MoTe2 film deposited on the metasurface composed of Si disks with holes. (b) Steady-state photon density \({{S}_{{\text{s}}}}\) and its derivative \(\frac{{{{d}^{2}}\log {{S}_{{\text{s}}}}}}{{{{{(d\log I)}}^{2}}}}\) versus the near-field pump intensity I. The dashed vertical straight line indicates the threshold of generation \({{I}_{{{\text{thr}}}}}\). (c, d) Visualization of the distribution of the electric field intensity in the MoTe2 film over the Si metasurface in the regime of generation of quasitrapped modes in it in the (c) absence and (d) presence of the pump field with the above-threshold intensity \(\overline I = 2.2\) kW/cm2. The near-field intensity is given in units of the incident wave intensity. The contours of disks of the metasurface located under the MoTe2 film are projected on the film by black lines. The inset in panel (d) shows the scattering pattern of one disk of the metasurface. The parameters of the Si metasurface are the same as in Fig. 1.

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Transition to the regime of generation of laser radiation can be described by rate equations for the densities of charge carriers N and photons S of the signal field [7, 22, 23] in the active MoTe2 medium of the metastructure in the form

$$\frac{{dN}}{{dt}} = \frac{{{{\alpha }_{p}}P}}{{\hbar {{\omega }_{p}}V}} - {{R}_{{{\text{nr}}}}}(N) - {{R}_{{{\text{sp}}}}}(N) - {{\upsilon }_{{\text{g}}}}g(N)S,$$
(1a)
$$\frac{{dS}}{{dt}} = - \frac{S}{{{{\tau }_{p}}}} + \Gamma {{\upsilon }_{{\text{g}}}}g(N)S + \Gamma \beta {{R}_{{{\text{sp}}}}}(N).$$
(1b)

Here, \(\hbar {{\omega }_{p}}\) is the energy of the external optical pump; αp is the imaginary part of the refractive index for MoTe2 at a pump wavelength; V is the volume of the structure; \({{\tau }_{p}} = Q{\text{/}}{{\omega }_{0}}\) and \({{\omega }_{0}} = \frac{{2\pi c}}{{{{\lambda }_{0}}}}\) are the lifetime and frequency of the lasing mode, respectively; Γ is the confinement factor of the lasing mode; β is the spontaneous emission coefficient determined by the Purcell factor; P is the pump power; Rnr = N/τnr + CN3 and Rsp = N/τsp are the nonradiative recombination and total spontaneous emission rates, respectively, where τnr and τsp are the nonradiative and spontaneous recombination times, respectively, and C is the Auger recombination coefficient; g(N) = a(N – Ntr) is the active medium gain, where a is the linear gain coefficient and Ntr is the density of electron–hole pairs necessary for the transparency regime of the medium; and \({{\upsilon }_{{\text{g}}}}\) = c/ng is the group velocity of generated radiation in the active medium, where c is the speed of light in vacuum, and we assume that ng = n(λ0). Below, following [24], we take τsp = 3 ps (cf. 4 ps in [13]), τnr = 23 ps, C = 10‒40 m6 s−1 [7], \(\beta = 0.1\), and \(\Gamma = 0.04038\) for the MoTe2 monolayer film.

The stationary solution of Eqs. (1a) and (1b) for the photon density Ss and pump power P depends on the stationary carrier density Ns and has the form [22]

$${{S}_{{\text{s}}}}({{N}_{{\text{s}}}}) = \frac{{\beta \Gamma {{\tau }_{p}}{{N}_{{\text{s}}}}}}{{{{\tau }_{{{\text{sp}}}}}(1 + \Gamma {{\upsilon }_{{\text{g}}}}a{{\tau }_{p}}({{N}_{{{\text{tr}}}}} - {{N}_{{\text{s}}}}))}},$$
(2a)
$$P({{N}_{{\text{s}}}}) = \frac{{\hbar {{\omega }_{p}}V}}{{{{\alpha }_{p}}}}\left( {CN_{{\text{s}}}^{3} + \frac{{{{N}_{{\text{s}}}}}}{{{{\tau }_{{{\text{nr}}}}}}} + (1 - \beta )\frac{{{{N}_{{\text{s}}}}}}{{{{\tau }_{{{\text{sp}}}}}}} + \frac{{{{S}_{{\text{s}}}}({{N}_{{\text{s}}}})}}{{\Gamma {{\tau }_{p}}}}} \right).$$
(2b)

Figure 2b presents the parametric gain curves for Ss versus the pump intensity I = P(Ns)/Astr, where Astr is the area of the flake. In particular, the threshold of laser generation is determined as the p-osition of the inflection point of the gain curve Ss(I) on a log–log scale, which corresponds to the condition [13, 25]

$$\frac{{{{d}^{2}}\log {{S}_{{\text{s}}}}}}{{{{{(d\log I)}}^{2}}}} = 0.$$
(3)

The complex permittivity of MoTe2 under pumping at the wavelength λp can be represented in the form [7]

$$\varepsilon (\omega ) = {{\varepsilon }_{r}}(\omega ) + \frac{{{{f}_{0}}\omega _{0}^{2}}}{{\omega _{0}^{2} - {{\omega }^{2}} - i\gamma \omega }},$$
(4)

where \({{\varepsilon }_{r}}(\omega )\) = Re\([(\bar {n}(\omega {{))}^{2}}]\) is the permittivity of MoTe2 without pumping, i.e., when \({{f}_{0}} = 0\); \({{f}_{0}}\) corresponds to the amplification amplitude at the wavelength \({{\lambda }_{0}}\) with the Lorentzian lineshape; and \(\gamma = 1{\text{/}}{{\tau }_{p}}\). As in [26], the isotropic permittivity \({{\varepsilon }_{r}}(\omega ) \approx {{\varepsilon }_{\parallel }}(\omega )\) was used for MoTe2 [21] in the calculations because the electric component of the QTM is oriented predominantly along the surface of the film (see Fig. 1b).

In the numerical simulation, we varied the imaginary part of the permittivity \(\varepsilon (\omega )\) of MoTe2 to values at which losses were completely compensated; i.e., the reflection coefficient of the signal field from the met-astructure became 1. In this case, \({{f}_{0}}\)  = Im\([(n({{\omega }_{0}}) + i{{k}_{g}}{{)}^{2}}]\gamma {\text{/}}{{\omega }_{0}}\) and the threshold gain coefficient \({{g}_{{{\text{thr}}}}}\) of the entire metastructure can be obtained using the expression \({{k}_{g}} = - \frac{{g{{\lambda }_{0}}}}{{4\pi }}\) [27, 28] for the imaginary part of the refractive index. For the case under consideration in Fig. 2a, the threshold conditions correspond to the effective permittivity εeff = 2.3207 – \(0.5012i\) (recalculated from the permittivity ε = 19.8674 − 7.1603i of the MoTe2 monolayer replacing its actual thickness by an effective one of 10 nm used in the simulation, see below) and, setting \({{N}_{{{\text{tr}}}}} = \) 1.61 × 1017 cm–3 and a = 7.08 × 10–18 m2, we obtain \({{k}_{g}} = - 0.8\), which corresponds to gthr = 95 194 cm–1 and the threshold near-field pump intensity Ithr = 23.78 kW/cm2 (with the threshold power \({{P}_{{{\text{thr}}}}} = \) 0.238 mW) and occurs at the carrier density \({{N}_{{{\text{thr}}}}} = 1.51 \times {{10}^{{18}}}\) cm–3.

The near-field response of the metastructure under consideration was numerically simulated using CO-MSOL Multiphysics software. To simulate the MoTe2 thin film with the thickness h beyond the resolution of the algorithm, we recalculated the actual permittivity \(\varepsilon (\omega )\) to the effective one εeff(ω) for a thicker film with the thickness hF similar in properties:

$${{\varepsilon }_{{{\text{eff}}}}}(\omega ) = 1 + (\varepsilon (\omega ) - 1)\frac{h}{{{{h}_{{\text{F}}}}}},$$
(5)

where h = 0.7 nm is the actual thickness of the MoTe2 film and hF = 10 nm is the effective thickness of the MoTe2 film used in the numerical simulation.

Figure 2c shows the electric field inside the MoTe2 film over the metasurface in the regime of QTM excitation at the wavelength λQTM = 1056 nm but in the absence of an additional pump field. The pump field increasing the intensity of the near-field response of disks to I = 200 kW/cm2 results in the above-threshold conditions with the effective permittivity εeff = \(2.3207 - 0.6538i\) and the corresponding gain kg = \( - 1.0435\) of the film. Since the MoTe2 film is in fact pumped by the near field at “hot points” on the surface of Si disks (see Fig. 2d), the intensity of the optical pump field Ī in the far field region necessary to implement the lasing regime decreases by a factor of about 90. In the case under consideration, this intensity is Ī = 2.2 kW/cm2 (see Fig. 2d). In fact, this corresponds to a decrease in the threshold of generation to Īthr = 264 W/cm2.

The radiation pattern presented in the inset of Fig. 2d shows that the most part of the energy of the incident wave is concentrated and scattered in the plane of the metasurface. This incoherent process is related to the excitation of nonemitting magnetic dipole \({{m}_{z}}\) in the QTM regime. In this case, only a part of the energy stored by the metasurface is reemitted owing to the bianisotropy-induced excitation of the component \({{p}_{x}}\) of the electric dipole. However, this is sufficient for the excess of threshold conditions and for the formation of a coherent signal from the entire metasurface coated with the active medium. In this case, the regime of QTM excitation and generation in the system can be controlled simply by the switching of the polarization of the signal field. A much higher energy efficiency can be expected in the implementation of QTMs on the basis of  electric bianisotropy with the excitation of the \({{p}_{z}}\) component in each disk [16]. Semiconductor quantum dots generating coherent radiation in the plane of the metasurface can serve as an active medium for such a system.

To conclude, we note that a much higher Q factor can be achieved in the presented system with pumping by QTMs [9, 16] and a decrease in the threshold of generation down to several watts per centimeter squared can be expected. Furthermore, the use of open resonator systems based on quasitrapped modes in quasi-infinite lattices makes it possible to significantly extend the dimensions of the active region and to fabricate scaled devices and metacoatings generating laser radiation. Such coatings can be placed on flexible and conducting substrates, fabricated by laser printing [29] or on the basis of liquid metamaterials [30], and controlled by an external electric field.