The structure of ML-TMDs is formed by two hexagonal lattices of chalcogen atoms embedding a plane of metal atoms arranged at trigonal prismatic sites located between chalcogen neighbors32. Figure 1c shows the lattice structure for MX2 ML-TMDs (M = Mo, W, and X = S, Se), and Fig. 2a, b show the valence and conduction bands for MoS2 as obtained from tight-binding calculations39. The electronic band structure of other MX2 materials is considered to be qualitatively similar. The direct bandgap is ~1.5 eV, which implies optical transparency for infrared radiation; the linear surface conductivity has a very small real part (corresponding to absorption) and a higher imaginary part at infrared wavelengths. Figure 2c shows the wavelength dependence of the linear surface conductivities of MX2. In the presence of an external pump field with angular frequency ω3, the ML-TMD second-order nonlinear processes lead to the generation of down-converted signal and idler waves with angular frequencies ω1 and ω2, such that ω3 = ω1 + ω2. Figure 2e illustrates the PDC mixing surface conductivities for MoS2. Both linear and nonlinear conductivities are calculated by a perturbative expansion of the tight-binding Hamiltonian for MX2 (see Methods and Supplementary Material). For infrared photons with energy smaller than the bandgap, extrinsic doping by an externally applied gate voltage (see Fig. 1c) modifies the optical properties, leading to increased absorption due to free-carrier collisions and to smaller PDC mixing conductivities. Figure 2d, f show the dependence of the linear and nonlinear surface conductivities on the Fermi level EF. As detailed below, extrinsic doping generally leads to a decrease in PDC efficiency.
Figure 1b shows the parametric oscillator design incorporating ML-TMDs. The cavity consists of a dielectric slab (thickness L) surrounded by two Bragg grating mirrors (BGs); the ML-TMD is placed on the left BG inside the cavity. The cavity is illuminated from the left by an incident (i) pump field (frequency ω3), and the oscillator produces both reflected (r) and transmitted (t) signal and idler fields with frequencies ω1 = (ω3 + Δω)/2 and ω3 = (ω3 − Δω)/2, where Δω is the beat-note frequency of the parametric oscillation (PO).
As detailed in the Materials and methods, the cavity equations for the fields do not contain the wavevector mismatch Δk. Indeed, due to their atomic thickness, ML-TMDs are not optically characterized by a refractive index but rather by a surface conductivity. Hence, the parametric coupling produced by the quadratic surface current in ML-TMDs is not hampered by dispersion; thus, no PM condition is required. To observe signal and idler generation, only the PO condition must be satisfied along with the signal resonance (SR) and idler resonance (IR) conditions, leading to a significant reduction in the intensity threshold (see Methods). Since there is no PM requirement, such conditions can be met by adjusting either the cavity length L or the pump incidence angle θ as tuning parameters. For SR and IR, one needs highly reflective mirrors for both signal and idler (see Materials and methods), as realized by locating the stop band of the micron-sized BGs at half of the pump frequency ω3/2. Figure 3 shows the PO analysis for a cavity composed of two BGs with polymethyl methacrylate (PMMA) and MoS2 deposited on the left mirror. The infrared pump has a wavelength of λ3 = 780 nm, which lies in the same spectral region showing very pronounced nonlinear properties for MoS2 (see Fig. 2e). The BGs are tuned with their stop bands centered at 1560 nm (= 2λ3). In Fig. 3a–i, we consider the case of normal incidence θ = 0 and plot the PO (black), SR (red), and IR (green) curves in the (L/λ3, Δω/ω3) plane. Doubly resonant POs (DRPOs) corresponding to the intersection points of these three curves are labeled by dashed circles. Therefore, for normal incidence of the pump, degenerate (Δω = 0) and non-degenerate (Δω ≠ 0) DRPOs exist at specific cavity lengths. Note that such oscillations also occur for sub-wavelength cavity lengths (L < λ3). Each oscillation starts when the incident pump intensity ${\it{I}}_3^{\left(i \right)}$ is increased above a threshold ${\it{I}}_{3Th}^{\left(i \right)}$ (see Materials and methods). Figure 3b–i shows the threshold for two specific degenerate and non-degenerate DRPOs. Figure 3b, f shows the thresholds (black curves on the shadowed vertical planes) corresponding to the PO (black) curves; one can observe that the minimum thresholds occur at SR and IR (identified by the intersection between the red and green curves). The minimum intensity thresholds are on the order of GW cm−2, with the non-degenerate DRPO threshold greater than the degenerate DRPO threshold because the reflectivity of the Bragg mirror is maximum at Δω = 0 (i.e., at half the pump frequency, as discussed above). Figure 3c–e (and, analogously, Fig. 3g–i) shows the basic DRPO features by plotting the intensities ${\it{I}}_1^{\left(t \right)}, {\it{I}}_2^{\left(t \right)}, {\it{I}}_3^{\left(t \right)}$ of the transmitted signal, idler, and pump fields as functions of the scaled cavity length L/λ3 and the incident pump intensity. Note that, in the considered example, the range of L/λ3 where the oscillation actually occurs is rather narrow due to the high reflectivity of the adopted BG.
We emphasize that tuning of the PO may be realized by adjusting the pump incidence angle θ, with negligible effect on the oscillation thresholds. In Fig. 3j–n, we analyze the DRPOs by using θ as a tuning parameter for a given cavity length. In particular, in Fig. 3j, k, we consider a cavity with a fixed length, as in Fig. 3b–e. The PO, SR, and IR curves of Fig. 3j intersect at a degenerate DRPO point at $\theta \simeq 6^{\circ}$. In Fig. 3k, we plot the transmitted signal intensity ${\it{I}}_1^{\left(t \right)}$ as a function of the pump incidence angle and intensity ${\it{I}}_3^{\left(i \right)}$; one can observe that the intensity threshold is comparable to the case shown in Fig. 3b–e, with PO occurring for a range of angles θ on the order of a hundredth of a degree, which is experimentally feasible. We show similar results in Fig. 3m, n, where the non-degenerate DRPO of Fig. 3f–i is investigated for a cavity with a slightly different length and is shown to exist at a finite incident angle with unchanged note-beat frequency Δω. A more accurate analysis of Fig. 3l also reveals that, for a given L, the cavity sustains multiple DRPOs (both degenerate and non-degenerate) at different incidence angles θ. In Fig. 3n, we plot the transmitted intensity of a degenerate DRPO that grows with pump intensity above the ignition threshold.
Until now, our analysis has been based on the basic oscillator geometry sketched in Fig. 1b, where the ML-TMD is placed on top of the right mirror. It is, however, also instructive to investigate the dependence of the PO phenomenology on the location of the ML-TMD inside the cavity. Consequently, we consider a different parametric oscillator design whose geometry is sketched in Fig. 4a, with the same Bragg mirrors and cavity dielectric (of thickness L = 3.05λ3) as above but with the ML-TMD placed at a distance 0 < d < L from the left mirror. For simplicity, we focus here on degenerate DRPOs (Δω = 0), triggered by the same pump as above (λ3 = 780 nm), as in this case, due to the physical coincidence of the signal and idler fields, the SR and IR conditions coincide and the PO condition is automatically satisfied (see Materials and methods). In Fig. 4b, we plot the SR = IR curve identifying the incidence angle θ at which the DRPO occurs as a function of the normalized distance d/L. Note that the PO angle periodically depends on d/L and is always close to $\theta \left(0 \right) = 38.57\; {^{\circ}}$ (compare with Fig. 3n) as a consequence of the slight modification of the free cavity modes produced by the presence of the ML-TMD. In Fig. 4c, we plot the pump intensity threshold ${\it{I}}_{3Th}^{\left(i \right)}$ of the POs shown in Fig. 4b as a function of d/L. The marked periodic dependence of the intensity threshold on the location of the ML-TMD is particularly evident, together with the existence of minima and very large maxima. Such features can be easily understood by noting that at different locations inside the cavity, the ML-TMD experiences a spatially periodic cavity modal field (which is observed, as detailed above, to be slightly dependent on the location of the ML-TMD) and therefore shows minima and maxima for the intensity threshold at the anti-node and node positions (where the modal field strength is maximal and zero, respectively).
It is also worth stressing that such features are strictly a consequence of the two-dimensional character of the ML-TMD, which can additionally be exploited to tune and control the parametric oscillator behavior.
The novel PO utilizing ML-TMDs as nonlinear media are PM free because of the atomic size of the ML-TMDs. Several examples of POs with MoS2 can also be designed using other families of ML-TMDs, leading to qualitatively similar results. In the Supplementary Material, we compare the calculated dependence of the pump intensity threshold as a function of wavelength λ3 for parametric oscillators built using MoS2, WS2, and MoSe2, WSe2; we find that the chosen material affects the minimal threshold intensity in a given spectral range. One can optimize the choice of the material for a desired spectral content and threshold level. In this respect, we emphasize that these functionalities are enabled by the inherently large nonlinear surface conductivities of ML-TMDs. A heuristic comparison with standard photonic media may be accomplished by introducing an effective second-order nonlinear mixing susceptibility $\chi _{ {eff}}^{\left(2 \right)}\left({\omega _1, \omega _2} \right)$ for the ML-TMDs, which is found to be of the order $\chi _{ {eff}}^{\left(2 \right)}\left({\omega _1, \omega _2} \right) \approx 10^{ - 10}\; { {mV}}^{ - 1}$ (≈2 orders of magnitude higher than that of LiNbO3, which is one of the most widespread and efficient materials used for second-order nonlinear optical functionalities42). Therefore, by using standard photonic media instead of ML-TMDs (in the envisaged micro-cavity), parametric oscillations would require a pump threshold that is at least 4 orders of magnitude higher (the threshold intensity depends inversely on the product $\left| {\tilde \sigma _{23}\tilde \sigma _{13}^ \ast } \right|$, see Materials and methods), and second-order nonlinear effects due to other photonic components of the proposed device are expected to be irrelevant.
A further degree of freedom offered by ML-TMDs lies in the electrical tunability afforded by the application of an external gate voltage, as depicted in Fig. 1c. The gate voltage increases the Fermi level and hence affects the nonlinearity and absorption because of electron–electron collisions in the conduction band (see Fig. 2d, f). Although electrical tunability of MX2 has not been hitherto experimentally demonstrated, to the best of our knowledge, we emphasize that such an additional degree of freedom is absent in traditional parametric oscillators. In the Supplementary Material, we calculate the pump intensity threshold as a function of the Fermi level of MoS2, and we show that the threshold may increase by one order of magnitude. Consequently, an external gate voltage can be used to switch-off PO at a fixed optical pump intensity, with potential for realization of rapid electrical modulation of the output signal and idler fields.
Finally, we emphasize that experimental realization of the discussed micron-sized phase-matching-free parametric oscillators is heavily facilitated by the inherent flexibility offered by these devices. Indeed, in contrast to traditional parametric oscillators, the key tunability (by means of the external pump incidence angle) unlocks the cavity size, which remains arbitrary. While the narrow angular selectivity found in our calculations can be easily overcome by using focused pump beams with finite size, the reflectivity of the Bragg mirrors heavily affects the parametric oscillation threshold. Thus, high-reflectivity Bragg mirrors with leakage ≈10−4 are desirable for reaching thresholds on the order of GW cm−2, which are achievable using pulsed infrared lasers with picosecond-like single pulse duration. Accurate control of the TMD layer number remains the only experimentally critical limiting factor: since TMDs with even layer numbers are centrosymmetric, it is imperative for the oscillator design to embed TMDs with an odd layer number. In addition, increasing the layer number hampers relaxation of the phase-matching condition; therefore, TMD monolayers are considered to be the best materials in terms of design optimization.