Abstract
Nonlinear optical processes, such as harmonic generation, are of great interest for various applications, e.g., microscopy, therapy, and frequency conversion. However, highorder harmonic conversion is typically much less efficient than loworder, due to the weak intrinsic response of the higherorder nonlinear processes. Here we report ultrastrong optical nonlinearities in monolayer MoS_{2} (1LMoS_{2}): the third harmonic is 30 times stronger than the second, and the fourth is comparable to the second. The third harmonic generation efficiency for 1LMoS_{2} is approximately three times higher than that for graphene, which was reported to have a large χ ^{(3)}. We explain this by calculating the nonlinear response functions of 1LMoS_{2} with a continuummodel Hamiltonian and quantum mechanical diagrammatic perturbation theory, highlighting the role of trigonal warping. A similar effect is expected in all other transitionmetal dichalcogenides. Our results pave the way for efficient harmonic generation based on layered materials for applications such as microscopy and imaging.
Introduction
Nonlinear optical phenomena can generate highenergy photons by converting n = 2, 3, 4,… lowenergy photons into one highenergy photon. These are usually referred to as second, third, and fourthharmonic generation (SHG, THG, and FHG)^{1}. Due to different selection rules^{1, 2}, harmonic processes are distinct from optically pumped laser phenomena (e.g., optically pumped amplification^{3}), and other typical singlephoton processes (e.g., singlephoton excited photoluminescence^{1}), in which the energy of the generated photons is smaller than the pump photons. Multiphoton harmonic processes have been widely exploited for various applications (e.g., alloptical signal processing in telecommunications^{1, 4}, medicine^{5}, and data storage^{6}), as well as to study various transitions forbidden under lowenergy singlephoton excitation^{5, 6}. The physical origin of these processes is the nonlinear polarization induced by an electromagnetic field E. This gives rise to higher harmonic components, the nth harmonic component amplitude being proportional^{1} to E^{n}. Quantum mechanically, higherharmonic generation involves the annihilation of n pump photons and generation of a photon with n times the pump energy. Because an nth order nonlinear optical process requires n photons to be present simultaneously, the probability of higherorder processes is lower than that of lower order^{1}. Thus, higherorder processes are typically weaker and require higher pump intensities^{7, 8}.
Graphene and related materials are at the center of an everincreasing research effort due to their unique and complementary properties, making them appealing for a wide range of photonic and optoelectronic applications^{9,10,11}. Among these, semiconducting transitionmetal dichalcogenides (TMDs) are of particular interest due to their direct bandgap when in monolayer (1L) form^{12}, leading to an increase in luminescence by a few orders of magnitude compared with the bulk material^{12, 13}. 1LMoS_{2} has a single layer of Mo atoms sandwiched between two layers of S atoms in a trigonal prismatic lattice. Therefore, in contrast to graphene, it is noncentrosymmetric and belongs to the space group \(D_{3{\rm{h}}}^1\!\) ^{14}. The lack of spatial inversion symmetry makes 1LMoS_{2} an interesting material for nonlinear optics, since secondorder nonlinear processes are present only in noncentrosymmetric materials^{1}. However, when stacked, MoS_{2} layers are arranged mirrored with respect to one another^{14}, therefore MoS_{2} with an even number of layers (EN) is centrosymmetric and belongs to the \(D_{3{\rm{d}}}^3\) space group^{14}, producing no second harmonic (SH) signal. On the other hand, MoS_{2} with odd number of layers (ON) is noncentrosymmetric. SHG from 1LMoS_{2} was reported by several groups^{14,15,16,17,18,19,20,21}.
Here we present combined experimental and theoretical work on nonlinear harmonic generation in 1L and fewlayer (FL) MoS_{2}. We report strong THG and FHG from 1LMoS_{2}. THG is more than one order of magnitude larger than SHG, while FHG has the same magnitude as SHG. This is surprising, since one normally expects the intensity of nonlinear optical processes to decrease with n ^{1, 2}, with the SHG intensity much larger than that in THG and FHG, although evenorder processes only exist in noncentrosymmetric materials. Our results show that this expectation is wrong in the case of 1LMoS_{2}. At sufficiently low photon frequencies (in our experiments the photon energy of the pump is 0.8 eV), SHG only probes the lowenergy band structure of 1LMoS_{2}. This is nearly rotationally invariant^{22,23,24,25,26,27,28,29}, but with corrections due to trigonal warping. It is because of these corrections^{23, 26, 27}, fully compatible with the \(D_{3{\rm{h}}}^1\) space group^{1}, but reducing the full rotational symmetry of the lowenergy bands to a threefold rotational symmetry^{1}, that a finite amplitude of nonlinear harmonic processes can exist at low photon energies in ENMoS_{2}. The lack of spatial inversion symmetry is a necessary but not sufficient condition for the occurrence of SHG. A purely isotropic band structure gives a vanishing SHG signal^{30,31,32,33}, despite some terms in the Hamiltonian explicitly breaking inversion symmetry^{27, 34,35,36,37}. Terms proportional to the σ _{ z } Pauli matrix break inversion symmetry. Breaking the continuous rotational symmetry of isotropic models (e.g., by including trigonal warping) is required to obtain a nonzero secondorder response in a twoband system. In hexagonal lattices, trigonal warping is a deviation from purely isotropic bands that emerges as one moves away from the corners K and K′ of the Brillouin zone^{23, 26, 27, 36,37,38}. Since the lattice has a honeycomb structure, this distortion displays a threefold rotational symmetry^{23, 26, 27, 36,37,38}. We demonstrate that the observed THG/SHG intensity ratio can be explained by quantum mechanical calculations based on finitetemperature manybody diagrammatic perturbation theory^{39} and lowenergy continuummodel Hamiltonians that include trigonal warping^{35}. We conclude that, similar to SHG^{14,15,16,17,18}, the THG process is sensitive to the number of layers, their symmetry, relative orientation, as well as the elliptical polarization of the excitation light. Similar effects are expected for all other TMDs. This paves the way for the assembly of heterostructures with tailored nonlinear optical properties.
Results
Samples
MoS_{2} flakes are produced by micromechanical cleavage (MC) of bulk MoS_{2} ^{40} onto Si + 285 nm SiO_{2}. 1LMoS_{2} and bilayer (2LMoS_{2}) flakes are identified by a combination of optical contrast^{41, 42} and Raman spectroscopy^{43}. Raman spectra are acquired by a Renishaw microRaman spectrometer equipped with a 600 line/mm grating and coupled with an Ar^{+} ion laser at 514.5 nm. Figure 1a shows the MoS_{2} flakes studied in this work and their Raman signatures. A reference MC graphene sample is also prepared and placed on a similar substrate.
SHG and THG charcterization
Nonlinear optical measurements are carried out with the setup shown in Fig. 2 ^{44, 45}. As excitation source, we use an erbiumdoped modelocked fiber laser with a 50 MHz repetition rate, maximum average power 60 mW, and pulse duration 150 fs, which yields an estimated pulse peak power of ~8 kW^{46}. The laser beam is scanned with a galvo mirror and focused on the sample using a microscope objective. The backscattered second and third harmonic signals are split into different branches using a dichroic mirror and then detected using photomultiplier tubes (PMTs). For twochannel detection, the light is split into two PMTs using a dichroic mirror with 562 nm cutoff. After the dichroic mirror, the detected wavelength range can be further refined using bandpass filters. The light can also be directed to a spectrometer (OceanOptics QE ProFL). The average power on the sample is kept between 10 and 28 mW with a typical measurement time ~5 μs, which prevents sample damage and enables high signaltonoiseratio, even with acquisition time per pixel in the μs range.
SHG and THG images of the MoS_{2} sample are shown in Fig. 3a, b. The SH photon energy is ~1.6 eV, lower than the bandgap of 1LMoS_{2} ^{12, 13}. This is not unexpected, as harmonic generation can occur when the harmonic energy is below the bandgap^{1, 47, 48}. The SHG signal is generated in 1LMoS_{2}, while 2LMoS_{2} appears dark. As discussed above, the secondorder nonlinear response is present in 1LMoS_{2}, which is noncentrosymmetric. However, when stacked to form 2LMoS_{2}, MoS_{2} layers are mirrored^{14, 15}. Therefore, ENMoS_{2} is centrosymmetric^{14, 15}, and belongs to the \(D_{3{\rm{d}}}^3\) space group^{14, 15}, producing no SHG signal. On the other hand, ONMoS_{2} flakes are noncentrosymmetric^{14, 15}.
We note that strong THG is detected compared with SHG, even for 1LMoS_{2}, Fig. 3b. THG was previously reported for a 7LMoS_{2} flake^{18}, but here we see it down to 1LMoS_{2}. Reference ^{49} followed our work^{50} and reported THG and SHG from 1LMoS_{2}, giving effective bulklike second and thirdorder susceptibilities \(\chi _{{\rm{eff}}}^{(2)}\) and \(\chi _{{\rm{eff}}}^{(3)}\) of 2.9 × 10^{−11} mV^{−1} and 2.4 × 10^{−19} m^{2}V^{−2}, respectively. However, ref. ^{49} did not provide a detailed explanation of the large THG signal compared to the SHG. Instead it assigned the large THG/SHG ratio to a possible enhancement of THG by the edge of the B exciton. However, refs. ^{51, 52} demonstrated that SHG is enhanced only when the SHG wavelength overlaps the A or B excitons. A similar behavior is expected for THG. Thus, the explanation in ref. ^{49} may not be correct. Reference ^{53} reported highharmonic (>6thorder) generation in the nonperturbative regime with midinfrared (IR) excitation (0.3 eV), unlike our THG and FHG results with nearIR excitation (0.8 eV). We do not detect THG from the thickest areas of our flake, with N > 30, as in ref. ^{18}. The output spectrum in Fig. 3c further confirms that we observe both SHG and THG. Peaks for THG and SHG at 520 and 780 nm can be seen, as well as at 390 nm, corresponding to a fourphoton process. This is detected only in 1LMoS_{2}. Its intensity is ~5.5 times lower than SHG, and two orders of magnitude smaller than THG.
SHG signals on areas with N = 3, 5, 7 have nearly the same intensity as 1LMoS_{2}, Fig. 4a. This contrasts ref. ^{14}, where a pump laser at 810 nm was used. We attribute this difference to the fact that photons generated in the secondorder nonlinear process in our setup with a 1560 nm pump have an energy ~1.6 eV (780 nm), below the band gap of 1LMoS_{2} ^{12}, therefore are not adsorbed, unlike the SHG signal in ref. ^{14}.
Second and thirdorder nonlinear susceptibilities
Based on the measured SHG and THG intensities, we can estimate the nonlinear susceptibilities χ ^{(2)} and χ ^{(3)}. χ ^{(2)} can be calculated from the measured average powers of the fundamental and SH signals as follows^{54}:
where τ is the pulse width, P _{pump} is the average power of the incident fundamental (pump) beam, and P _{2ω } stands for the generated SH beam power, R is the repetition rate, N _{ a } = 0.5 is the numerical aperture, λ _{2} = 780 nm is the SH wavelength, τ = τ _{2} = 150 fs are the pulse durations at fundamental and SH wavelengths, \(\phi = 8\pi {\int}_0^1 { {{\rm{co}}{{\rm{s}}^{  1}}\rho  \rho {{\sqrt {1  {\rho ^2}} }^2}} \rho \,{\rm{d}}\rho = 3.56} \) from ref. ^{54}, and \({n_1} = {n_2}\sim 1.45\) are the refractive indexes of the substrate at the wavelengths of the fundamental and SHG, respectively. The effective bulklike secondorder susceptibility of MoS_{2} \(( {\chi _{{\rm{eff}}}^{(2)}} )\) can be obtained from Eq. (1) with \(\chi _{{\rm{eff}}}^{(2)} = \frac{{\chi _{\rm{s}}^{(2)}}}{{{t_{{\rm{Mo}}{{\rm{S}}_2}}}}}\), where \({t_{{\rm{Mo}}{{\rm{S}}_2}}} = 0.65\) nm is the 1LMoS_{2} thickness^{10, 24}. We obtain the effective secondorder susceptibility \(\chi _{{\rm{eff}}}^{(2)}\sim 2.2\) pmV^{−1} for 1LMoS_{2}. Reference ^{49} reported a bulklike secondorder susceptibility 29 pmV^{−1}, which is ~10 times larger than here. However, several other studies reported ~5 pmV^{−1} for 1560 nm^{20, 52, 55}. Thus, our measured \(\chi _{{\rm{eff}}}^{(2)}\) agrees well with earlier values measured with similar excitation wavelength.
The thirdorder susceptibility \(\chi _{{\rm{eff}}}^{(3)}\) can be estimated by comparing the measured THG signal from MoS_{2} to that of 1Lgraphene (SLG):
with t _{SLG} = 0.33 nm the SLG thickness, and THG_{SLG} and THG\(_{{\rm{Mo}}{{\rm{S}}_{\rm{2}}}}\) the measured signals from SLG and MoS_{2}, respectively. Our results show that THG from 1LMoS_{2} is around three times larger than THG_{SLG}, which indicates that χ ^{(3)} of 1LMoS_{2} is comparable to that of SLG, in the frequency range of our experiments. Previous reports indicate^{49, 56} that \(\chi_{{\rm{SLG}}} ^{(3)}\) is ~10^{−17}–10^{−19} m^{2} V^{−2}. Thus, based on Eq. (2), χ ^{(3)} of 1LMoS_{2} is in the same range. This is remarkable, as SLG is known to have a large χ ^{(3)} ^{56,57,58,59}. Reference ^{18} reported χ ^{(3)} of 7LMoS_{2} to be approximately three orders of magnitude smaller than \(\chi_{{\rm{SLG}}} ^{(3)}\) of ref. ^{56}. \(\chi _{{\rm{SLG}}}^{(3)}\) from ref. ^{56} is much higher than other theoretical^{58} and experimental^{49} values. We believe that our measured ratio between 1LMoS_{2} and SLG is more accurate, since we measured both materials at the same time under the same conditions.
We note that large discrepancies can be found in earlier reported effective susceptibilities for layered materials (LM). For example, there is a approximately four orders of magnitude difference in χ ^{(3)} for SLG (~10^{−15} m^{2}V^{−2} in ref. ^{57}; ~10^{−19} m^{2}V^{−2} in ref. ^{49}). There is an approximately three orders of magnitude difference in χ ^{(2)} reported for 1LMoS_{2} at 800 nm (e.g., ~10^{−7} mV^{−1} in ref. ^{15}; and ~10^{−10} mV^{−1} in ref. ^{17}). Effective susceptibilities are well defined only in threedimensional materials, since their definition involves a polarization per unit volume^{1}. Therefore, given the large discrepancies in literature, it is better to describe the nonlinear processes in LMs using the ratio between the harmonic signal power and the incident pump power (i.e., harmonic conversion efficiency). In this case, when comparing the efficiencies in our measurements with those in ref. ^{49}, our THG efficiency (~4.76 × 10^{−10}) is ~1.4 times larger than that (~3.38 × 10^{−10}) in ref. ^{49}, while our SHG efficiency (~6.47 × 10^{−11}) is twice that of ref. ^{49}. Since the effective susceptibilities are not well defined for LMs and also depend on the calculation method, we believe that the conversion efficiency is a better figure of merit for LMs.
Discussion
Our measurements show that the nonlinear response of 1LMoS_{2} and SLG are comparable in magnitude, both revealing stronger nonlinear efficiency than threedimensional nonlinear materials, such as diamond^{1} and quartz^{59}. This can be explained by considering their effective Hamiltonians^{27, 34,35,36,37, 60}. The main contribution to THG is paramagnetic. This is described by the square diagram in Supplementary Notes 1–4 (Supplementary Figs. 1–6). This paramagnetic contribution is mainly related to the strong interband coupling in the effective Hamiltonian, controlled by large velocity scales, \({v_{\rm{F}}} \approx \frac{c}{{300}}\) and \(v = \frac{{{t_0}{a_0}}}{\hbar } \approx 0.65 \times \frac{c}{{300}}\) for SLG and 1LMoS_{2}, with c the speed of light. The SLG paramagnetic thirdharmonic efficiency (PTHE) is proportional to the square of thirdorder conductivity. Since^{39} \(\sigma _{yyyy}^{(3)} \propto v_{\rm{F}}^2\), we get an overall prefactor \(v_{\rm{F}}^4\), which explains the strong nonlinear SLG response. Similarly, for 1LMoS_{2}, the square diagram contains four paramagnetic current vertices, which gives an overall prefactor v ^{4}, and an integral over the dummy momentum variables, which gives a prefactor \(\frac{1}{{{v^2}}}\) (see Supplementary Note 3). Therefore, the thirdorder response function, \(\Pi _{yyyy}^{(3)}\), is proportional to v ^{2}, which implies a scaling of PTHE as v ^{4}. Exciton physics is not considered because our experimental conditions only capture offresonance transitions.
1LMoS_{2} is transparent at this wavelength due to its ~1.9 eV gap^{12}, while SLG absorbs 2.3% of the light^{61}. Therefore, 1LMoS_{2} and other TMDs are promising for integration with waveguides or fibers for alloptical nonlinear devices, such as alloptical modulators and signal processing devices, where materials with nonlinear properties are essential^{11}.
The SHG and THG power dependence follows quadratic and cubic trends, Fig. 4b. At our power levels, THG is up to 30 times stronger than SHG. 1LTMDs have strongly bound excitons that can modify their optical properties^{62,63,64}. The exciton resonances also affect their nonlinear optical responses^{17, 65, 66}. References ^{51, 55} reported that when the SHG energy is above the A and B excitons, resonance effects are not observed. In our experiments, the energy of 3ω photons is above the A exciton but does not directly overlap with the A or B excitons. Thus, we do not assign the large THG/SHG intensity ratio to an excitonic enhancement, but to the approximate rotational invariance of the 1LMoS_{2} band structure at low energies, which is broken by trigonal warping.
SHG is weaker than expected for a noncentrosymmetric material, due to nearisotropic bands contributing to the SHG signal for our low incident photon energies (0.8 eV). Even in the presence of a weak trigonal warping, SHG and THG might be comparable above the threshold for two and threephoton absorption edges. However, this is not a resonant effect. Resonances only emerge when the laser matches a single level (like an excitonic level) rather than a continuum of states^{67}. In our analysis, SHG would be absent without trigonal warping. But, trigonal warping alone cannot explain the magnitude of the FHG signal compared to SHG and THG.
Figure 4c compares the THG/SHG ratio from experiments and calculations based on the k·p theory^{35} (see Supplementary Note 1) and finitetemperature diagrammatic perturbation theory^{39} (see Supplementary Notes 3 and 4). The calculations are a factor 2 smaller than the experiments. Considering the complexity of the nonlinear optical processes and that our calculations ignore highenergy band structure effects^{29} and manybody renormalizations^{65}, we believe this to be a satisfactory agreement, indicating the importance of trigonal warping in harmonic generation.
FHG generally derives from cascades of lowerorder nonlinear multiphoton processes^{68}. With an excitation wavelength of 1560 nm, this could be, e.g., a cascade of two SHG processes, where 780 nm photons are first generated through SHG (ω _{1560 nm} + ω _{1560 nm} ⇒ ω _{780 nm}) and then undergo another SHG process (ω _{780 nm} + ω _{780 nm} ⇒ ω _{390 nm}). To yield a FHG at 390 nm of the same intensity as SHG at 780 nm in this cascaded process, one would need a conversion efficiency (defined as P _{2ω }/P _{pump} ^{1}) for the second SHG process (i.e., ω _{780 nm} + ω _{780 nm} ⇒ ω _{390 nm}) to be close to unity. However, we observe a conversion efficiency ~10^{−10} for SHG. Therefore, we conclude that our FHG does not arise from cascaded SHGs. Another possible cascade process is based on THG (ω _{1560 nm} + ω _{1560 nm} + ω _{1560 nm} ⇒ ω _{520nm}) and sumfrequency generation (ω _{520 nm} + ω _{1560 nm} ⇒ ω _{390 nm}). We find that THG strongly increases up to N = 5, as for Fig. 4a. Therefore, we expect this cascaded process to have a similar trend with N. However, we only observe FHG in 1LMoS_{2}. Thus, we also exclude this cascade process, and conclude that this is a direct χ ^{(4)} process.
We now consider the dependence of our results on the elliptical polarization of the incident light. We consider an incident laser beam with arbitrary polarization, i.e., \({\bf{E}} = \left {\bf{E}} \right{{\hat {\varepsilon }}_ \pm }\) with \({{{\hat \varepsilon }}_ \pm } = \widehat {\bf{x}}\,{\rm{cos}}(\theta ) \pm i\widehat {\bf{y}}\,{\rm{sin}}(\theta )\). Using the crystal symmetries of 1LMoS_{2}, we derive (see Supplementary Note 2) the following expressions for the second and thirdorder polarizations P ^{(2)} and P ^{(3)}:
and
Note that θ = 0° corresponds to a linearly polarized laser along the \(\widehat {\bf{x}}\) direction, perpendicular to the \(D_{3{\rm{h}}}^1\) mirror symmetry plane, while θ = 45° corresponds to a circularly polarized laser. From Eq. (3), we expect the intensity of SHG in response to a circularly polarized pump laser to be twice that of a linearly polarized laser. Equation (4) implies vanishing THG in response to a circularly polarized pump laser.
We measure the dependence of SHG and THG on elliptical polarization using a linearly polarized laser and a rotating QWP. Depending on the angle θ between the QWP axes and the laser polarization, the excitation light is linearly (θ = 0° + m·90°) or circularly (θ = 45° + m·90°) polarized. Figure 5 shows that the experimental data are in agreement with Eqs. (3) and (4). The THG signal is maximum for a linearly polarized excitation laser, while it vanishes for circularly polarized light. SHG is always visible, but its intensity is maximum for circularly polarized light.
Given that harmonic generation is strongly dependent on the symmetry and stacking of layers and that different 1LTMDs (e.g., WSe_{2}, MoSe_{2}) all have similar nonlinear response^{11, 14, 15, 21}, one could use heterostructures (e.g., MoS_{2}/WSe_{2}) to engineer SHG and other nonlinear processes for high photonconversion efficiency for a wide range of applications requiring the generation of higher frequencies. This may lead to the use of LMs and heterostructures for applications utilizing optical nonlinearities (e.g., alloptical devices, frequency combs, highorder harmonic generation, multiphoton microscopy, and therapy etc.).
Methods
Determination of MoS_{2} thickness from SHG and THG signals
SHG and THG for FLMoS_{2} (N = 1…7) are studied on the flakes in Fig. 6a. SHG and THG images are shown in Fig. 6b, c. At 1560 nm, the contrast between 1 and 3L areas is small, as well as the contrast between 3, 5, and 7 L regions (Fig. 6b).
The THG signal increases up to N = 7, Figs. 4a and 6c. On the other hand, the SHG signal (Fig. 6b) is only generated in ON flakes, due to symmetry^{14}. Therefore, areas with intensity between the 3, 5, and 7L regions in Fig. 6c, but dark in SHG, are 4 and 6 L. The dependence of the intensities of THG and SHG on N is plotted in Fig. 4a. The combination of SHG and THG can be used to identify N at least up to 7. The THG signal develops as a function of N. Using Maxwell’s equations for a nonlinear medium with thickness t and considering the slowly varying amplitude approximation^{1, 69}, we obtain:
where I _{in} and I _{3ω } are the intensities of the incident and THG light, respectively, and χ ^{(3)}(−3ω;ω, ω, ω) is the thirdorder optical susceptibility, \({n_{j = 1,3}} = \sqrt {{\epsilon ^{(1)}}(j \omega )} \), with \(\epsilon ^{(1)}\) the TMD linear dielectric function. Δkt is the phase mismatch between the fundamental and third harmonic generated waves.
For Δkt ≈ 0, THG adds up quadratically with light propagation length (i.e., t ∝ N). The signal starts to saturate for N = 6. The possible reasons for subquadratic signal buildup can be either phase mismatch, or absorption^{13}. For THG, Δk = 3k _{in} ± k _{3ω }, where k _{in} and k _{3ω } are the wavevectors of the incident and THG signals, respectively, where the plus sign indicates THG generated in the backward direction, while minus identifies forward generated THG. Even for backward generated THG, Δkt ≈ 0 for 6LMoS_{2} (~4.3 nm^{70}). This rules out phase mismatch as the origin of the signal saturation when N ≤ 6. Therefore, we assume that the signal saturation is due to absorption of the third harmonic light.
Diagrammatic nonlinear response theory
To quantify theoretically the strength of nonlinear harmonic generation processes, we generalize the diagrammatic perturbation theory approach^{39} to the case of TMDs. We combine this technique with a lowenergy k·p model Hamiltonian \({\cal H}\left( {\bf{k}} \right)\) for 1LMoS_{2} ^{35}. In such lowenergy model, lightmatter interactions are treated by employing minimal coupling^{35, 39}, k → k + e A(t)/ħ, where A(t) is a timedependent uniform vector potential. Nonlinear response functions are calculated via the multilegged Feynman diagrams depicted in Supplementary Figs. 2 and 3.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
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Acknowledgements
We thank M.J. Huttunen and R.W. Boyd for useful discussions. We acknowledge funding from the Academy of Finland (Nos: 276376, 284548, 295777, 298297, and 304666), TEKES (NPNano, OPEC), Royal Academy of Engineering (RAEng) Research Fellowships, Fondazione Istituto Italiano di Tecnologia, the Graphene Flagship, ERC grants Hetero2D, Nokia Foundation, EPSRC Grants EP/K01711X/1, EP/K017144/1, EP/L016087/1, AFOSR COMAS MURI (FA95501010558), ONR NECom MURI, CIAN NSF ERC under Grant EEC0812072, and TRIF Photonics funding from the state of Arizona and the Micronova, Nanofabrication Centre of Aalto University.
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A.S., L.K., A.A., S.M., K.K., and Z.S. carried out the multiphoton experiments and data analysis. Samples preparation and characterization were carried out by A.L. and A.C.F. The nonlinear response function calculations were carried out by H.R. and M.P. Sample fabrication, multiphoton characterization, and analysis were coordinated by R.A.N., N.P., H.L., A.C.F., M.P., and Z.S. All authors contributed to the manuscript.
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Correspondence to Zhipei Sun.
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Säynätjoki, A., Karvonen, L., Rostami, H. et al. Ultrastrong nonlinear optical processes and trigonal warping in MoS_{2} layers. Nat Commun 8, 893 (2017). https://doi.org/10.1038/s41467017007494
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