Main

Nonlinear optical (NLO) processes such as harmonic generation are extensively used in lasers, photonic switches1 and quantum technologies2, and as probes of material properties, including correlated quantum states3,4. Van der Waals (vdW) crystals, including transition metal dichalcogenides (TMDs) and ferroelectric materials, have emerged as promising NLO platforms, given their strong light–matter interactions5,6, nonlinear susceptibilities exceeding 100 pm V−1 (refs. 7,8,9,10), light-confinement abilities11 and facile nanostructuring for the creation of photonic resonators and metastructures12,13,14. In particular, recent studies on the nonlinear dipole-aligned rhombohedral-phase TMD 3R-MoS2 (Fig. 1a) show second-harmonic conversion efficiencies comparable to lithium niobate over 100-fold shorter propagation lengths12,15,16,17,18. The large refractive index dispersion of MoS2, however, leads to coherence lengths shorter than a micrometre at visible/near-infrared frequencies, beyond which phase-matching considerations become crucial.

Fig. 1: Imaging the propagation of FW and SH light in 3R-MoS2 waveguides.
figure 1

a, 3R-MoS2 slab waveguide on Si/SiO2 substrate. The FW is launched by focusing a pump pulse on the edge. The edge-scattered pump couples into the waveguide and outcouples at the opposite edge. A time-delayed probe pulse is used to monitor FW and SH propagation in the waveguide. b, Optical image of a 1.25-μm-thick 3R-MoS2 slab on a Si/SiO2 substrate. The left edge of 3R-MoS2 is cleaved along the armchair (a.c.) direction, marked by the black arrows. c, SH emission image of the same field of view as b, using edge excitation with a 200 fs pump pulse at EFW = 1.2 eV and detecting SH light at ESH = 2.4 eV. Light propagation wavevectors associated with angles beyond TIR are waveguided, as illustrated in the bottom panel. d, Linear reflectance R of the 3R-MoS2 slab waveguide, normalized to the Si/SiO2 substrate reflectance R0. A-Ex and B-Ex denote the energies of the A and B excitons (Ex), respectively. e, Pump–probe transient reflectance spectrum ΔR/R of the slab waveguide following EFW = 1.2 eV edge excitation, monitored midway between the two waveguide edges with a white light probe pulse. f,g, Transient signals are dominated by a dynamic Stark effect at sub-picosecond time delays (f) and by exciton generation at longer time delays (g). h,i, Spatiotemporal imaging of FW (h) and SH (i) propagation through the slab waveguide from b, following edge excitation with a 200 fs pump pulse at EFW = 1.2 eV. The colour scale refers to normalized (norm.) transient reflectance contrast. The incident pump fluence is 5.34 mJ cm2. The probe energies used to track FW and SH are 1.66 eV and 1.96 eV, respectively, indicated by arrows in e. All scale bars, 5 μm.

Recently, a periodically poled 3R-MoS2 achieved quasi-phase-matching over several micrometres, resulting in 25-fold enhancement of second-harmonic generation (SHG)19. The natural next steps would be to achieve perfect birefringent phase matching, and to integrate 3R-MoS2 in waveguides, allowing simultaneous miniaturization and optimization of NLO conversion through precise control over phase- and mode-matching conditions20,21,22,23,24. Achieving the latter, however, requires precise knowledge of the material’s linear and NLO properties, loss function and waveguide mode profile, which are difficult to extract with the required precision in realistic geometries to design ideal waveguide structures.

In this Article, we report a method that empirically accesses phase-matching angles, mode profiles and losses by imaging light propagation and SHG within waveguides with high spatiotemporal resolution, enabling systematic optimization of NLO conversion. We show that 3R-MoS2 slab waveguides support phase matching with up to few-percent SHG efficiencies under our experimental conditions, establishing their potential for on-chip integration.

Imaging of fundamental and second-harmonic wave propagation

To fabricate slab waveguides, we mechanically exfoliate multilayer flakes of 3R-MoS2 and deposit them on a Si/SiO2 substrate (Fig. 1a). Figure 1b shows an optical image of a 1.25-μm-thick slab. The as-exfoliated flake forms a multimode Fabry–Pérot cavity, resulting in periodic dips in the linear reflectance spectrum of the slab (Fig. 1d and Supplementary Note 2). Figure 1c shows the result of exciting the armchair edge of the waveguide with 1,030 nm light (EFW = 1.2 eV) and monitoring second-harmonic (SH) light at 515 nm (ESH = 2.4 eV) with conventional far-field microscopy, similar to a recent report18. The fundamental wave (FW) enters the waveguide by edge scattering. For scattered wavevectors beyond the total internal reflection (TIR) angle of ~20° at the MoS2/SiO2 interface, both the FW and SH waves are waveguided in the slab and outcouple at the opposite edge, as illustrated in the lower panel of Fig. 1c.

Instead of directly imaging waveguided FW and SH light at their respective energies, which would require high-momentum coupling such as near-field optics25,26,27,28, or upconversion29, we instead track the transient changes that these fields impart on the far-field optical properties of the material. We begin by characterizing these changes using pump–probe transient reflectance microscopy (Methods and Supplementary Note 1). Figure 1e shows the transient reflectance of the waveguide upon edge excitation with a diffraction-limited, ~200 fs pump pulse at EFW = 1.2 eV, which is below the optical gap of MoS2 (Eopt = 1.85 eV). In the first picosecond following pump excitation, all cavity modes blueshift by ~0.3 meV owing to the strong FW field in the waveguide, a characteristic signature of the dynamic Stark effect30,31,32 (Supplementary Note 3). The amplitude of the Stark effect persists only for the duration of the pump pulse.

Once the FW Stark effect signal subsides (after 1 ps pump–probe delay), Fig. 1e shows a signal lasting for a nanosecond. The transient spectrum and lifetime of this signal is the same (within experimental error) to that obtained upon above-gap photoexcitation of 3R-MoS2 (Supplementary Fig. 7), and is characteristic of photoinduced population of excitons in MoS2 (refs. 33,34). The cavity modes also shift because the dielectric function of the material is renormalized35,36 (Supplementary Fig. 8). Crucially, the amplitude of this long-lived signal increases quadratically with pump fluence when exciting below gap, but linearly with pump fluence when exciting above gap for all probe energies (Supplementary Fig. 9). The quadratic dependence confirms that below-gap excitation cannot directly populate excitons, but does so through a second-order nonlinearity. We therefore assign the long-lived signal to generation of excitons in the waveguide by SH light (ESH = 2.4 eV), which is above gap and is thus able to populate excitons in MoS2 (Fig. 1g). In centrosymmetric 2H-MoS2 waveguides of comparable thickness, in which SHG is suppressed, we observe that the long-lived signal amplitude is 2–3 orders of magnitude smaller for the same pump fluence (Supplementary Fig. 10). The latter lends strong support to our assignment and excludes possible contributions from two-photon absorption. All together, these measurements indicate that transient signals associated with the FW Stark effect and SH-generated excitons are directly proportional to the FW and SH light intensity (Supplementary Fig. 9), respectively, serving as distinct far-field reporters of FW and SH light within the waveguide (Fig. 1f,g).

To image the propagation of FW and SH fields within the waveguide, we use far-field stroboscopic scattering microscopy (stroboSCAT), a well-established approach based on transient scattering or reflectance that allows sensitively probing small pump-induced changes to a material’s dielectric function35,37. Unlike previous iterations of stroboSCAT used to track quasiparticle transport following above-gap excitation, here we excite below the material gap to track the propagation and nonlinear conversion of light. A ~200 fs, diffraction-limited pump pulse is launched into the waveguide by edge scattering. At controllable time delays, a backscattering monochromatic widefield probe spatially images the pump-induced change in material reflectance (Fig. 1a) with ~220 fs temporal resolution and sub-50 nm sensitivity to spatial motion (Supplementary Note 1). Judicious choice of the probe energies, denoted by arrows in Fig. 1e, allows isolating signals from FW (SH) fields by selecting spectral regions with a strong (weak) Stark effect response and weak (strong) SH-generated exciton response.

Key stroboSCAT results are shown in Fig. 1h,i. The striking spatiotemporal evolution is best visualized in Supplementary Videos 1 and 2. Figure 1h shows how the pump pulse at EFW = 1.2 eV, initially exciting the left edge of the slab, generates a wavepacket that is scattered within the waveguide and propagates perpendicularly to the edge, before partially bouncing back at the slanted outcoupling edge. The signal is short-lived, reporting on the spatiotemporal evolution of the FW wavepacket. By contrast, Fig. 1i tracks above-gap SH light through the latter’s generation of excitons in MoS2, which are long-lived. Therefore, Fig. 1i shows a wavefront that propagates at the same speed as the FW in Fig. 1h, but leaves a trail of long-lived excitations (FW and SH kinetics are shown in Supplementary Fig. 11). The time-resolved spatial intensity profiles of FW and SH signals in Supplementary Note 8 further highlight the distinction between FW wavepacket propagation and SH wavefront propagation.

In addition to propagating waves, Fig. 1h,i shows beating intensity fringes in both the FW and SH signals. These fringes are more evident in the spatiotemporal intensity profiles (Supplementary Fig. 12) with peaks and dips occurring at identical locations in the FW and SH traces. These fringes are due to interference between multiple waveguide modes excited by the pump pulse (Supplementary Fig. 13h), leading to spatial variations in the FW intensity. The intensity profiles are reproduced quantitatively using finite element simulations in Supplementary Note 9. The fringe intensities are amplified in the SH signal owing to the quadratic dependence on local FW intensity for SHG. Overall, by capturing light propagation using all-far-field optics, we can track the interconversion between FW and SH light even when confined within waveguides. We show below that such spatiotemporal tracking of waveguide nonlinear optics allows parameter-free determination of phase-matching conditions and waveguide losses.

Polarization-dependent SHG in 3R-MoS2 waveguides

Figure 2 details the polarization dependence of FW and SH signals in the thick waveguide of Fig. 1. The polarization axes are depicted in Fig. 2a. Figure 2b,c shows the FW and SH signals for in-plane (IP, primarily exciting TE modes) and out-of-plane (OP, primarily exciting TM modes) pump polarizations, with additional pump polarizations shown in Supplementary Fig. 14. The images allow extracting IFW and ISH for each polarization, where I corresponds to the transient reflectance intensity. Figure 2d plots the polarization dependence of IFW and of the normalized SH intensity, defined as \({I}_{\mathrm{SH}}/{I}_{{\mathrm{FW}}}^{2}\propto {{\rm{sinc}}}^{2}(\frac{\varDelta kL}{2}){L}^{2}\), where L is the propagation length, and \(\varDelta k\,=\,2{k}_{\mathrm{FW}}\,-{k}_{\mathrm{SH}}\) is the wavevector mismatch between FW and SH fields. Plotting \({I}_{\mathrm{SH}}/{I}_{{\mathrm{FW}}}^{2}\) allows normalizing for local FW intensity variations from multimode interference, which varies with pump polarization since the dispersion for TE and TM modes is different (Supplementary Figs. 15 and 16). The polar plot in Fig. 2d shows that FW waveguide coupling is highest for OP excitation, consistent with Brewster’s law. Figure 2d also shows that the normalized SH intensity is highest for IP polarization, when the FW field is aligned with the nonlinear dipole of MoS2 layers, concurring with previous SHG measurements at the outcoupling edge of the waveguide18. These results provide further confidence that the signals tracked with stroboSCAT are valid proxies for the FW and SH light intensities.

Fig. 2: Anisotropic waveguided FW and SH properties.
figure 2

a, Pump excitation geometry. Waveguided light has either IP (yellow) or OP (purple) polarizations. b,c, Snapshots of waveguided FW (at 400 fs delay (b)) and SH (at 10 ps delay (c)) in IP and OP polarizations. Scale bar, 5 μm. d, Polarization dependence (in degrees) of the FW signal and normalized SH intensity in the 3R-MoS2 waveguide, measured 10 µm away from the exciting edge, marked by dash circles in b and c. e, The ratio of SH signal for IP and OP polarizations as a function of propagation distance.

In contrast to linear measurements, our approach also tracks how the anisotropy evolves as a function of distance from the excitation edge. In Fig. 2e, we plot the anisotropy of the SH intensity defined as \(\frac{{I}_{\mathrm{SH,IP}}-{I}_{\mathrm{SH,OP}}}{{I}_{\mathrm{SH,IP}}+{I}_{\mathrm{SH,OP}}}\), where IP and OP refer to the polarization of the FW excitation. The anisotropy remains positive throughout the waveguide, indicating that preferential SHG for IP excitation is intrinsic to the system, rather than resulting from the modal structure of the waveguide. The non-monotonic trend in the anisotropy as a function of L likely results from the interplay of SH generation and SH losses, analysed below.

Birefringent phase matching in thick waveguides

The strong birefringence of MoS2 (Supplementary Fig. 15) should allow phase matching by tuning propagation angle θ (depicted in Fig. 1c) so that the refractive index n at the FW and SH frequencies is equivalent. For thick slabs where light fields are primarily confined within the waveguide, the birefringent phase-matching (BPM) angle θPM can be approximated using the geometric SHG relation (for negative uniaxial crystals)38:

$$\frac{1}{{n}_{\mathrm{IP}}{(\omega )}^{2}}\,=\frac{{\sin }^{2}\theta }{{n}_{\mathrm{OP}}{(2\omega )}^{2}}+\frac{{\cos }^{2}\theta }{{n}_{\mathrm{IP}}{(2\omega )}^{2}}$$
(1)

where ω is the angular frequency, \({n}_{\mathrm{IP}}(\omega )\) and \({n}_{\mathrm{IP}}(2\omega )\) are the ordinary refractive indices at the FW and SH frequencies, respectively, and \({n}_{\mathrm{OP}}(2\omega )\) is the extraordinary refractive index at the SH frequency. For example, using the measured dielectric functions of Supplementary Fig. 15, the calculated phase-matching angle for EFW = 1.2 eV is \({\theta }_{\mathrm{calc}}^{\mathrm{PM}}\) = 25.4° for 3R-MoS2. Figure 3a plots the measured and calculated angle-resolved reflectance spectra (with \({k}_{x}=\frac{n\omega }{c}\,\sin \theta\), where \(c\) is the speed of light) for the 1.25 μm slab waveguide of Fig. 1. In this dispersion, dips in reflectance beyond the TIR line indicate which waveguide propagation angles are allowed. Figure 3b overlays \({\theta }_{\mathrm{calc}}^{\mathrm{PM}}\) (green line) for a range of FW energies with allowed waveguide modes (yellow line) extracted from Fig. 3a. Importantly, the lines intersect at several energies, indicating that the slab should support waveguide BPM over a large frequency range.

Fig. 3: Determining phase-matching conditions through spatiotemporal imaging.
figure 3

a, Experimental (Exp., right) and calculated (Calc., left) angle-resolved reflectance spectra showing the TE mode dispersion for a 1.25 μm 3R-MoS2 slab. The yellow curve highlights the dispersion of the 11th TE mode that dominates the propagation properties of Fig. 1h. The white dashed line marks the TIR line. The full dispersion for both TE and TM polarizations is shown in Supplementary Note 12. b, Calculated phase-matching conditions (green), overlaid with TE waveguide modes (yellow) from a. The circle highlights the wavevector at which waveguide BPM is observed in our experiments. c, FW wavepacket and SH wavefront propagation extracted from stroboSCAT figures shown in Fig. 1h,i. The extracted velocity is 7.8% of light speed. Data are presented as the mean ± one standard deviation derived from the fitting error of the wavepacket arrival time. d, Group velocities of different modes calculated from the gradient of the mode dispersion. Only one mode at 1.2 eV matches the experimentally obtained group velocity of ~8% of light speed. At 1.2 eV, this mode corresponds to a momentum kx = 12.6 µm1, marked by a red circle in a. e, Normalized SH intensity \({I}_{\mathrm{SH}}/{I}_{\mathrm{FW}}^{2}\) as a function of propagation distance in the waveguide. ISH and IFW are extracted from Fig. 1h,i, respectively, through radial integration (Supplementary Fig. 12). The black line is a fit using a waveguide SHG model with explicit inclusion of SH losses (see text for details). Lossless phase-matched (PM, Δk = 0) and phase-mismatched (PMM, Δk = 25.2 µm1, corresponding to normal incidence) cases are indicated with green and blue lines, respectively. Data are presented as the mean ± one standard deviation derived from the fitting error of the integrated local FW and SH intensities.

For EFW = 1.2 eV, we identify a crossing between the 11th order waveguide mode and the phase-matching contour in Fig. 3b, highlighted with a red circle. At 1.2 eV, this mode is the first mode beyond the TIR line. To experimentally determine the dominant mode populated by edge excitation, we leverage the high spatiotemporal resolution of stroboSCAT to analyse the transport velocities of the FW wavepacket and the SH wavefront from Fig. 1h,i (Fig. 3c and Supplementary Fig. 12). The experimental FW velocity for IP excitation at EFW = 1.2 eV is vIP = 7.8% \(c\). The SH wavefront tracks the FW wavepacket at the same speed. An identical analysis for OP excitation provides a velocity of vOP = 21% \(c\) (Supplementary Fig. 16c), consistent with nIP > nOP. Figure 3d shows the expected group velocity extracted from the dispersion of Fig. 3a, \({v}_{g}=\partial \omega /\partial k\), for different modes. For EFW = 1.2 eV, only the first waveguide mode matches the experimental velocity, with an expected \({v}_{g}=\,8 \% \,c\). Thus, our spatiotemporal analysis shows that edge excitation primarily populates the lowest-k waveguide mode at 1.2 eV, and that most of the observed SH signal is generated from light propagating within this mode. The wavevector of this mode at 1.2 eV is kFW = 12.6 μm1, corresponding to θ = 24.9°. This angle matches closely with the above-calculated phase-matching angle of \({\theta }_{\mathrm{calc}}^{\mathrm{PM}}\) = 25.4° at 1.2 eV, confirming the possibility of achieving BPM.

Under perfect phase matching, the SH intensity increases quadratically with propagation distance L (ref. 38). Since the FW and SH signals in stroboSCAT are directly proportional to the FW and SH light intensities (Supplementary Fig. 9), our measurements allow empirical verification of whether BPM is achieved. In Fig. 3e, we plot the normalized SH intensity as a function of propagation distance, \({I}_{\mathrm{SH}}(L,t)/{I}_{{\mathrm{FW}}}^{2}(L,t)\), at times t corresponding to t = L/v, that is, tracking the centre of the FW wavepacket (Supplementary Fig. 12c,d). The FW signal remains approximately constant (Supplementary Fig. 18), suggesting little pump depletion, but we observe a build-up of normalized SH intensity over the ~12 μm lateral extent of the waveguide. This intensity build-up extends far beyond the coherence length of 0.94 μm for 3R-MoS2 at normal incidence18 (blue line in Fig. 3e), providing strong evidence that at least partial phase matching is achieved. Nevertheless, beyond ~7 μm, the normalized SH intensity exhibits a deviation from the L2 proportionality, owing to absorption of above-gap SH light at 2.4 eV (ref. 39). We use an established system of coupled first-order differential equations describing SHG in lossy waveguides to model our results40 (simulation details are provided in Supplementary Note 16). The black line in Fig. 3e shows the result of this model. We reach close agreement with the experimental results when assuming an SH absorption coefficient of \({\tau }_{\mathrm{waveguide}}^{-1}\) = 0.7 μm1 for 3R-MoS2. This value is much smaller than the absorption coefficient of MoS2 at normal incidence at 2.4 eV, τ−1 ≈ 30 μm−1 (ref. 41). In this BPM scheme starting with ordinary FW polarization, the SH polarization is extraordinary; since the primary optical transition dipole in MoS2 is polarized in-plane, SH light is only weakly absorbed in the waveguide. Overall, these results provide strong evidence that we achieve BPM in multimode slab waveguides of 3R-MoS2.

Polariton-assisted modal phase matching in thin waveguides

Although thick waveguides provide a convenient platform to achieve phase matching owing to a large density of waveguide modes, thin waveguides should provide stronger light confinement and larger nonlinear enhancements42,43,44,45,46. To investigate this regime, we use a single-mode 3R-MoS2 slab waveguide of 154 nm thickness (Fig. 4a). Figure 4b shows the SH emission following EFW = 1.2 eV edge excitation, confirming that waveguiding is also achieved in these thin flakes. Figure 4c overlays the calculated dispersion of the TE mode at the FW frequency (traced with a red dashed line), with the TM modes at the SH frequency (traced with green dashed lines). Modal phase matching is achieved where the lines intersect, corresponding to a matching of effective refractive indices, at EFW = 0.96 eV and 0.90 eV. Importantly, these modal crossings are facilitated by a flattening of the TM dispersion as the SH energy approaches the A- and B-exciton resonances of MoS2. This dispersion flattening is caused by hybridization between photons and excitons, resulting in the formation of part-light part-matter exciton-polaritons. Such hybridization is readily achieved in MoS2 slabs47. The two phase-matching conditions in Fig. 4c correspond to the TEFW branch crossing the lower and middle polariton branches, respectively (Supplementary Note 12). Such polariton-assisted modal phase matching in waveguide geometries remains an underexplored field and is particularly valuable for dispersion engineering in situations where phase matching is restricted by a low number of available guided modes, such as in ultrathin waveguides.

Fig. 4: Modal phase matching in thin waveguides.
figure 4

a, Optical image of a 154-nm-thick single-mode 3R-MoS2 waveguide on a glass substrate. b, Waveguided SH emission at ESH = 2.4 eV collected by a far-field objective. Scale bars in a and b, 5 μm. c, Dispersion of FW (red, TE mode) and SH (green, TM mode) in the 154-nm-thick single-mode waveguide, extracted from transfer matrix simulations. This figure overlays the FW modes onto the SH modes by rescaling the energy and momentum axes so that mode crossings correspond to matched phase velocities. d, FW signal as a function of L in the waveguide extracted from stroboSCAT (raw data in Supplementary Fig. 21b), with FW energies spanning EFW = 0.92–1.20 eV. Substantial FW depletion is observed around EFW = 0.95 eV owing to efficient phase-matched SH conversion. Incident pump fluences are kept at ~6 mJ cm2 for all FW energies. Data are presented as the mean ± one standard deviation derived from the fitting error of the integrated FW intensity. e, SH build-up extracted from stroboSCAT (raw data in Supplementary Fig. 22) for phase-mismatched and phase-matched SH generation at ESH = 2.4 eV and 1.9 eV, with incident FW pump fluences of 5.34 mJ cm2 and 6 mJ cm2, respectively. The black line is a fit using a waveguide SHG model with explicit inclusion of SH losses. Data are presented as the mean ± one standard deviation derived from the fitting error of the integrated local FW and SH intensities. f, SH emission intensity measured under waveguide (green) and normal-incidence (light green) geometries in the same slab. The pump power is 2 mW for all FW energies, using ~60 fs pulses at 1 MHz and 1.5 μm spot size. The arrows indicate the two modal phase-matching energies. The right inset illustrates the transmission and waveguide geometries.

Using stroboSCAT, we first evaluate the evolution of the FW signal propagating in the thin waveguide, for the different excitation energies highlighted with filled circles in Fig. 4c (limited by the low transmissivity of our objective below 0.92 eV). Figure 4d shows the extracted FW signal as a function of propagation distance. At frequencies that are phase-mismatched (for example, EFW = 1.2 eV), the FW is approximately constant throughout the waveguide (Supplementary Fig. 22a), indicating low losses and negligible FW intensity depletion. By contrast, we observe almost complete depletion of the FW signal over a distance of 10 μm at EFW = 0.95 eV (Fig. 4d, raw data in Supplementary Fig. 22b), suggesting efficient FW conversion. We emphasize that we do not observe FW depletion in the thick waveguide described in Fig. 3, even for identical pump energies (Supplementary Fig. 24), ruling out direct absorption losses from MoS2 at these below-gap FW energies.

To evaluate SHG efficiency in these thin waveguides, Fig. 4e plots \({I}_{\mathrm{SH}}/{I}_{\mathrm{FW}}^{2}\) extracted from stroboSCAT data for EFW = 1.2 eV (phase mismatched) and 0.95 eV (phase matched). We observe no SH build-up in the phase-mismatched case, as expected, given the short coherence length. In the phase-matched case, however, we observe an increase of the normalized SH intensity over 15 μm, although we emphasize that the absolute SH signal begins to drop after 3 μm owing to FW depletion and SH absorption (Supplementary Fig. 22b). We model this behaviour using the same system of coupled equations as that used in Fig. 3e, with the fit result shown as a black line in Fig. 4e (Supplementary Note 16). Although the data exhibit features not captured in the model, the FW depletion and SH turnover are reproduced well when using a seven fold larger nonlinear conversion efficiency compared with the multimode system of Fig. 3e, and a slightly lower SH loss of \({\tau }_{\mathrm{waveguide}}^{-1}\) = 0.48 μm1. We attribute lower losses to a larger fraction of the field extending outside the slab in thin waveguides. Supplementary Fig. 23 directly compares the simulation results for thin and thick waveguides, showing how the larger nonlinearity in thin waveguides is required to observe substantial FW depletion over few-micrometre distances. These simulations suggest that we reach a maximum possible SH conversion efficiency of ~4% in the thin waveguide, compared with 0.3% in the thick waveguide (Supplementary Fig. 23a) when using similar pump fluences of 5–6 mJ cm2. The larger SHG efficiency in the thin waveguide is likely mediated by greater field confinement and polariton-assisted enhancement48,49. These results illustrate the power of real-space spatiotemporal imaging of FW depletion within waveguides to characterize and optimize nonlinearities. We also note that a frequency-domain analysis of the periodic fringes observed in stroboSCAT of single-mode waveguides quantifies the degree of phase mismatch (Supplementary Note 19), offering an alternative approach to the real-space analysis presented here.

Finally, to ascertain that these results are consistent with traditional measurements of SH generation, we compare the outcoupled SH intensity in either waveguide or transmission geometries (Fig. 4f), using reflective optics for FW excitation to access a broad frequency range. We etch the waveguide into a 3 μm × 4 μm rectangle to outcouple SH light near its maximum absolute intensity. In transmission, SHG efficiency is measured as \({\eta }_{\mathrm{SHG}}={P}_{\mathrm{SH}}/{P}_{\mathrm{FW}}\), where PSH is the transmitted SH power and PFW is the input pump power. In the waveguide geometry, only fractions of the FW and SH light are coupled into or collected from the waveguide, limiting our comparison to relative efficiencies. The transmission measurements in Fig. 4f are consistent with previous measurements in ultrathin 3R-MoS2 flakes18, with \({\eta }_{\mathrm{SHG}}\) showing peaks at the B-exciton, and then rising rapidly at higher energies owing to nesting of electronic bands in MoS2 that drastically enhance nonlinearities50,51. In the waveguide geometry, we observe two peaks at energies near the predicted modal phase-matching conditions of Fig. 4c (ESH = 1.9 eV and 1.8 eV, highlighted with arrows in Fig. 4f). There is no measurable SHG at normal incidence at these energies, indicating large SHG enhancement in the waveguide geometry. These observations corroborate our stroboSCAT measurements, indicating that phase matching is achievable in confined slab waveguides of 3R-MoS2.

Conclusion

We have developed an approach based on nonlinear far-field microscopy to image light propagation and NLO conversion in low-loss waveguides with femtosecond and sub-50 nm spatiotemporal precision. We demonstrate the power of our approach on slab waveguides of 3R-MoS2, a material with large nonlinear susceptibility in the vanguard of current interest. We show that we can predict and realize phase matching for enhanced nonlinear conversion in both multimode and single-mode slab waveguides. Our approach allows direct extraction of phase-matching angles or degree of phase mismatch, waveguide mode profile, absorption and waveguide losses, and relative nonlinear conversion efficiencies without prior knowledge of material optical constants. Ongoing efforts to achieve better in- and out-coupling of light in these waveguides, for example, through grating couplers or tapered fibres52,53, should firmly establish 3R-MoS2 as ideal components for on-chip nonlinear optics. More generally, our imaging approach should be applicable to all semiconductor waveguides and will be valuable in the optimization of nonlinear conversion in other rhombohedral-stacked TMDs54,55 as well as ferroelectric oxyhalides9,10, a novel class of vdW materials for harmonic generation and entangled photon generation. We anticipate that such spatiotemporal imaging schemes also offer a path for understanding the fundamental properties of light in complex photonic architectures and processes, including for high-harmonic generation3,4 and time reversal transformations56,57.

Methods

Fabrication of 3R-MoS2 waveguides

3R-MoS2 crystals (HQ graphene) were tape-exfoliated onto Si/SiO2 (285 nm) substrates to create slab waveguides. The thin rectangular slab waveguide of Fig. 4 is patterned using electron beam lithography and etching. The bare flake is spin-coated with a ZEP 520A resist. Five rectangular strips are defined on the resist using electron-beam lithography (Nanobeam nB4 E-beam writing system) with subsequent resist development. The exposed 3R-MoS2 flakes are etched by reactive ion etching (Oxford Instruments Plasma Pro 100 Cobra Fluorine-based RIE/D-RIE) with SF6 plasma. The residual ZEP 520A resist is removed by N-methyl-2-pyrrolidone solvent immersion.

Far-field ultrafast optical microscopy

Spatiotemporal imaging of the waveguided FW and SH in slab waveguides is performed using ultrafast stroboscopic scattering microscopy and transient reflectance microscopy. The optical systems are detailed and illustrated in Supplementary Note 1. A Yb:KGW ultrafast regenerative amplifier (Light Conversion Carbide, 40 W, 1,030 nm fundamental, 1 MHz repetition rate, ~200 fs pulsewidth) seeds an optical parametric amplifier (OPA, Light Conversion, Orpheus-F) to generate an idler pulse tunable from 1.045 μm to 2.907 μm with ~50 fs pulsewidth. The 1,030 nm fundamental or idler serve as pump pulses, and either the signal output of the OPA or a super-continuum white light generated in a YAG window (bandpassed for stroboSCAT) is used as the probe. The pump and probe beams are directed to a home-built microscope equipped with a high numerical-aperture oil-immersion objective (Leica HC Plan Apo 63x, 1.4 NA). The pump beam focuses on the sample plane to a diffraction-limited spot. The probe beam focuses on the back focal plane of the objective by an f = 250 mm widefield lens, enabling widefield illumination of the sample. The reflected and scattered fields are imaged on a CMOS camera. Further details of the ultrafast stroboSCAT system are described in our previous reports35,58.

The linear transmittance and reflectance measurements are obtained using a different optical configuration shown in Supplementary Fig. 3. A reflective objective is used for the excitation to enable broader spectral coverage. The pump laser is focused by a reflective objective (Thorlabs LMM40X-P01, Infinity Corrected 40x, 0.5 NA). The SH light is collected by a transmissive air objective (Leica HCX PL Apo 40x, 0.85 NA, air). The light source is the same as that described above for stroboSCAT measurements. The SH light is detected using a cooled CMOS camera (Thorlabs CC505MU). The power of the SHG emission in the transmitted geometry is measured using a power meter to extract the absolute conversion efficiency.