In the last years, transition metal dichalcogenide (TMD) crystals have been a heavily investigated material class. Their composition of stoichiometric monolayers, bound in the stack only by van der Waals forces, enables to realize very thin crystals down to few- or single monolayers providing promising optical and electronic properties [1–5]. Especially in the monolayer (ML) form, TMDs have been shown to be remarkably efficient light sources. Most interest in this regard has been paid to the strong photoluminescence (PL), which is due to a transition from an indirect band gap in bulk TMDs to a direct band gap in a ML. On the other hand, MLs of TMDs (as well as odd-numbered few-layer crystals) have a broken inversion symmetry and thus feature a second-order optical nonlinearity [6,7]. Second-order nonlinearities enable the coherent generation of light by nonlinar three-wave mixing. These nonlinear frequency-conversion processes are parametric, i.e. they maintain the quantum state of the material during the interaction. The most notable parametric three-wave mixing process is second-harmonic generation (SHG), where a long-wavelength fundamental wave (FW) is converted into a second-harmonic (SH) wave at half the FW wavelength, as already shown in MLs of the TMDs MoS$_2$ [6,7], WS$_2$[8], MoSe$_2$ [9], and WSe$_2$[10,11]. In TMD MLs, for specific wavelengths nonlinear processes are enhanced due to excitonic resonances or band-nesting effects [6,7,12–14], giving rise to considerable nonlinear efficiencies despite the nanoscopic thicknesses and small volumes of the TMD MLs. Since parametric nonlinear processes strongly depend on the properties of the nonlinear crystal, e.g. the crystal symmetry and the refractive index, in TMD MLs they so far have been mostly used as a probe of the crystal structure, e.g. to determine the crystal orientation [15–17], unveil the existence of edge states [15,18], or find domain boundaries in grown TMD ML flakes [16,19].

From an application point of view, TMD MLs can also be used as versatile sources for parametrically generated light. Due to their nanometer thickness, TMD MLs can be used to functionalize surfaces or integrated optical structures to enable parametric nonlinear interactions for classical [6,7,20] or quantum [21,22] light generation. Furthermore, SHG from TMD MLs can be modulated electrically [23,24], enabling dynamically controlled light sources. However, spatial control of the nonlinearly generated light has not been demonstrated yet. For PL emission, which is incoherent across the TMD ML surface, such control can be achieved only by coupling to photonic structures that scatter light into specific directions and thus enable a direction dependent emission enhancement [25,26].

Since they preserve the coherence of the pump light field, parametric nonlinear processes also enable spatial control by shaping the interference of light generated at different locations of a TMD ML flake. For SHG in TMD MLs, it was recently shown that a doughnut-like FW vortex beam will create a SH beam with the same shape [27]. Whereas this approach requires additional optical elements to shape the FW beam, similar control of the SH spatial shape can also be obtained by influencing the local geometry of the nonlinear medium. This has been successfully demonstrated in bulk nonlinear media, where the local orientation of the nonlinear susceptibility can be influenced by electric-field poling. This is mostly used for achieving quasi-phase matching [28]. Furthermore, electric-field poling can also be used to create so-called nonlinear photonic crystals [29] that can be used for nonlinear beam shaping [30,31]. Similar effects can also be achieved in thin metasurfaces, where arrays of nonlinear nanoresonators with spatially varying properties enable nonlinear beam control resulting in effects like nonlinear diffraction [32,33], beam steering [34], nonlinear lensing [35–37], and nonlinear holograms [38,39]. TMD MLs as atomically thin materials with below 1 nm thickness can enable such nonlinear beam control close to the ultimate limit of miniaturization.

Here we experimentally demonstrate direct nonlinear beam control of SHG from structured MLs of molybdenum disulfide (MoS$_2$). By partially removing the MoS$_2$ in a specific pattern according to the targeted SH beam shape, we spatially modulate the source of nonlinearly generated radiation as schematically shown in Fig. 1(a), thus enabling nonlinear diffraction and the generation of an optical vortex beam. Our results show that TMD MLs, specifically MoS$_2$, can be versatile and controllable sources for complex radiation patterns generated by parametric nonlinear interactions.

 

Fig. 1. a) Schematic representation showing the excitation of the patterned MoS$_2$ ML flake and the radiation of the generated second harmonic (SH). b) Fabrication scheme: (i) CVD growth of MoS$_2$ ML on SiO$_2$/Si substrate and (ii) focused ion beam milling. c) Atomic force microscope measurement of an inscribed grating. The blue dotted line indicates the line where the height profile plotted below with the blue line is taken.

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To control the spatial intensity distribution of the SH beam generated in a TMD ML for excitation with a simple Gaussian beam, one has to spatially modulate the nonlinear susceptibility tensor $\chi ^{\left (2\right )}$ giving rise to SHG. The highest nonlinear efficiency could be achieved by locally controlling the orientation of the crystal and hence the nonlinear tensor, thus imprinting a spatially varying phase on the SH beam. This approach has been realized in nonlinear crystals by electric field poling [29] or in nonlinear metasurfaces by reorientation of the constituting nanoresonators [32]. However, although vertical stacking of TMD MLs with controlled orientation is possible [40], laterally controlling the crystal orientation with high spatial resolution could not be achieved yet. Alternatively, the magnitude of the nonlinear susceptibility can be modulated. In TMD MLs, the simplest way to achieve such modulation is by locally removing the ML. This approach can make use of well-developed micro- and nanostructuring technologies, thus enabling a very precise and variable structure definition. Nano- and microstructuring of MoS$_2$ MLs was achieved by electron-beam lithography [41], by block copolymer lithography [42], and in few and single layers of MoS$_2$ by focused ion beam milling (FIB) [43,44].

Here, we spatially modulate the magnitude of the nonlinear susceptibility of MoS$_2$ MLs by realizing one-dimensional gratings and demonstrate nonlinear diffraction of the SH generated in these structures. We use MoS$_2$ ML flakes grown on a silicon substrate with 300 nm SiO$_2$ thermal oxide layer by chemical vapour deposition using MoO$_3$ and sulfur as precursors. The sulfur evaporation was controlled using a Knudsen-type effusion cell [45]. The flakes, which have a thickness of only 0.7 nm, are patterned by FIB milling (FEI Helios NanoLab G3 UC), schematically shown in Fig. 1(b). Here, a beam of Ga$^+$ ions is focused on the ML and removes the material locally. The ions are accelerated to an energy of 30 keV. The ion current is reduced to 1.1 pA to achieve a very small spot of the focused beam and thereby a high resolution. The pattern is realized by scanning the beam across the flake. To improve the quality of the final pattern we increase the dose by scanning 20 times over the whole design, resulting in a dose of $\sim 11$ ions/nm$^2$.

By this process, complex spatial patterns can be realized. Several gratings of different period (625 nm, 675 nm and 725 nm) were fabricated. All periods are chosen to lie well below the FW wavelength, so the FW does not get diffracted. However, these periods are large enough to collect the first diffraction order of the SH with a microscope objective of numerical aperture NA=0.85. An AFM scan of a realized one-dimensional grating designed with a period $p=725$ nm is shown in Fig. 1(c). The period is measured by the AFM scan to be $p=713\,\textrm {nm}\pm 13\,\textrm {nm}$. The blue plotted line shows the height profile of the dotted line in the AFM image. The processed region exhibits a height increase which can be associated with an amorphization of the solid substrate [46]. The height difference between unexposed substrate and unexposed MoS$_2$ monolayer is around 1 nm, indicated by two dotted lines in the height profile, which confirms the atomically thin monolayer thickness of the CVD grown flake.

From the AFM scan we can estimate the duty-cycle $dc=a/p$ as the ratio of the unstructured stripe width $a$ divided by the period $p$ as depicted in Fig. 1(c). The result is $0.40\pm 0.02$ although designed to be $dc_{\textrm {designed}}=0.5$. The origin for this mismatch is the scattering of the Ga$^+$-ions in the substrate, which increases the damaged surface area compared to the area exposed by the beam.

Next we investigate the SH signal from the structured MoS$_2$ ML flakes using a home-built SH microscope. A Ti:Sapphire laser (80 MHz, $\sim$100 fs pulse length) is used to excite the sample at a FW wavelength of 840 nm. This coincides with the C-absorption peak (or C exciton) [6,47,48] at the corresponding SH wavelength of 420 nm, where the nonlinear susceptibility is resonantly enhanced and a larger SHG efficiency is expected [6,13]. The sample is excited by focusing through a microscope objective with NA=0.85, where the excitation spot size can be controlled using an aperture in the FW beam path. The generated SH is collected by the same microscope objective in reflection and imaged onto an EM-CCD camera for analysis. A short-pass filter with a cut-off wavelength of 500 nm is placed in front of the camera to block the FW beam as well as potential PL, which may be excited by two-photon absorption. With the help of an additional lens in the SH beam path, we can also measure the angular spectrum of the SH, i.e. the angular distribution of the SH intensity, by imaging the back-focal plane of the collection objective.

Figure 2(a) shows a microscope image of the whole patterned grating with period $p=725$ nm already presented by the AFM scan in Fig. 1(c). The stripes of unexposed MoS$_2$ are visible due to the considerable absorption of the atomically thin crystal. In Fig. 2(b), we provide a map of the emitted SH for the grating shown in Fig. 2(a), taken by scanning the FW beam across the sample and integrating the SH power incident on the camera for every position. Clearly, we can identify the unstructured portion of the MoS$_2$ ML flake by its strong SH amplitude, whereas the patterned area emits a weaker signal. Due to the FW excitation spot size of $\sim 1\,\mu$m full-width at half-maximum (FWHM), the patterned stripes of the grating are not resolved. For these excitation conditions, an average FW input power of 2 mW is used and the maximum of the measured SH power corresponds to 800 fW. The conversion efficiency is thus around $4\cdot 10^{-10}$, comparable to similar experiments on MoS$_2$ flakes [49].

 

Fig. 2. a) Contrast-enhanced optical microscope image of a patterned flake with a one-dimensional grating. b) Map of SH signal scanned over the patterned flake. c) SH signal at 0$^\circ$ polarization from the structured and unstructured areas of the monolayer flake dependent on the input polarization. d) Dependence of the SH power measured at one position on the unstructured MoS$_2$ on the FW power $P_{\mathrm {FW}}$.

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The ratio of the SH power emitted from the structured and the unstructured area is $0.23\pm 0.03$. This is much less than would be expected from the duty cycle of $dc = 0.4$ measured by the AFM scan, as the emitted SH power should linearly depend on the area of the nonlinear material. However, as the scanning FW beam excites only one of the remaining MoS$_2$ stripes at once, the source of SH radiation has only a width of 290 nm, much smaller than the SH wavelength. Hence, the SH in this case is radiated into a large range of angles transverse to the grating and cannot be completely collected by the microscope objective, leading to a reduced measured SH power.

Due to its crystal structure and internal symmetries, the second-order nonlinear tensor of MoS$_2$ only has in-plane components, all of which have an equal magnitude of $\chi ^{\left (2\right )}_{yyy}=-\chi ^{\left (2\right )}_{yxx}=-\chi ^{\left (2\right )}_{xxy}=-\chi ^{\left (2\right )}_{xyx}=\chi ^{\left (2\right )}\approx 130 \textrm {pm} V^{-1}$ for SH at the C-exciton [50,51]. Here, $x$ and $y$ are orthogonal directions along the armchair and zig-zag directions of the hexagonal lattice of MoS$_2$, respectively. The connection between SH and the crystal structure can be used to analyze the crystal in the patterned area and compare it with the unstructured area. To this end, we measure the polarization dependence of the generated SH by rotating the linear FW input polarization and measuring the SH power after a linear polarizer. The normalized SH powers of structured and unstructured areas are plotted in Fig. 2(c) in dependence on the FW polarization angle. Both curves show the characteristic four-fold symmetry and the same orientation. Hence, the crystal structure of the remaining MoS$_2$-stripes with its particular symmetries is not modified by the ion beam exposure.

Figure 2(d) shows the SH power dependent on the FW power in a double-logarithmic plot. Comparing with a fitted function of FW power squared, the data points clearly follow the expected quadratic power dependence.

To investigate the nonlinear diffraction of the SH signal from the MoS$_2$ ML gratings, we excite with a larger spot of $\sim 7\,\mu$m FWHM, obtained by reducing the FW beam width by using only a small part of the aperture of the focusing objective. To be able to collect large diffraction angles, we use again the microscope objective with a numerical aperture of NA=0.85. In Fig. 3(a), the spatial SH distribution generated in the grating depicted in Fig. 3(b) is shown. Several stripes were excited, which are clearly visible and whose spacing corresponds to the designed grating period $p=675\,$nm. The period is chosen such that linear diffraction of the exciting FW beam does not take place, however, the first diffraction orders of the SH are propagating. This is evident from the SH angular spectrum in the back-focal plane of the microscope objective, which is shown in Fig. 3(c). The measured intensity is multiplied by the factor $\cos (\theta )$, where $\theta$ is the emission angle with respect to the optical axis, to correct the apodization error from the microscope objective [52]. We clearly observe three maxima corresponding to the zero and first diffraction orders of the SH.

 

Fig. 3. a) Spatial distribution of the second-harmonic (SH) signal from the patterned flake using an excitation spot of 7 $\mu$m diameter. b) Contrast enhanced optical microscope image of the investigated monolayer flake. c) Back-focal plane image (angular spectrum) of the SH signal from the patterned flake showing the zero and the first diffraction orders. d) Angle of the first diffraction order for different periods of the grating.

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The emission angle of the first SH diffraction orders $\theta _{\pm 1} = 39^\circ$ is consistent with the theoretical prediction for this grating period. We confirmed this further by measuring gratings of different periods. The observed angle of the first diffraction order for several gratings is plotted in Fig. 3(d). It agrees well with the theoretical prediction by the diffraction law $p \sin (\theta _m)=m\lambda$, where $p$ is the period, $m$ is the diffraction order, and $\lambda$ is the wavelength of the diffracted light, i.e. the SH wavelength of $\lambda =420$ nm.

The ratio between the SH powers in the zero and first diffraction orders is $R=1.72\pm 0.12$. $R$ depends on the duty cycle $dc$, as the latter influences the diffraction efficiency and thus the amount of SH in the first diffraction order. We theoretically model the SH diffraction from the MoS$_2$-grating by calculating the linear diffraction pattern of the modulated SH field emitted from the multiple-slit grating. Using this model, the experimentally found ratio $R$ can only be explained if we assume a duty cycle of $dc=0.30\pm 0.02$ in the experiment. This is again lower than measured by the AFM scan shown in Fig. 1(c). This discrepancy is most likely due to a damage of the MoS$_2$ crystal structure at the sides of the ridges forming the grating. Here, the ion dose was not high enough to remove the MoS$_2$ but still induced a large number of defects in the crystalline lattice that reduce the nonlinearity. Thus, the width of the MoS$_2$ ridge contributing to the SHG is narrower than indicated by the AFM measurement. This effect could potentially be reduced by using further optimized writing strategies with even lower ion doses but more passes of the ion beam, ion beams of optimized shapes, or different ions than gallium.

Next, we demonstrate that also more complicated diffraction effects can be realized. To this end, we aim to nonlinearly create a vortex beam, i.e. an optical beam carrying nonzero optical angular momentum or topological charge, featuring the characteristic doughnut beam shape with zero intensity at the beam center. Such beams can be generated using fork-like dislocations in otherwise periodic gratings as shown in Fig. 4(a). In principle this is a one-dimensional grating deranged by an additional semi-infinite stripe, which includes a topological dislocation and therefore acts as the center for a special type of interference generating a vortex beam in the first diffraction order of the grating. This grating structure is realized by a spatially varying transmittance $t(x,\;y)=\frac {1}{2}(1+\textrm {sign}(\cos (2\pi x/p+l\arctan (x/y))))$ [53,54] with $p$ being the period and $l$ being the topological charge. Similar structures were used in the linear regime [55]. In the nonlinear regime, SH vortex beams from such a dislocation structure were shown from poled Lithium tantalate [56] and third-harmonic vortex beams from a gold grating [54]. To generate such a vortex beam using a MoS$_2$ ML, we inscribe a dislocated grating using the technique described before. We designed the structure according to the equation given above with $p=725\,$nm and $l=1$ (see Fig. 4(a)). In Fig. 4(b), we show the patterned MoS$_2$ ML flake, the fork-like defect of the grating is visible in the contrast-enhanced zoomed-in section.

 

Fig. 4. a) Design of the inscribed pattern with an introduced defect providing a fork-like structure. b) Optical microscope image of the structure with contrast-enhanced zoomed-in inset of the defect singularity. c) Back-focal plane image (angular spectrum) of the SH signal from the structure showing the zero and the first diffraction orders, where the latter show a zero-intensity center (ring shape).

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To investigate the spatial characteristics of the SHG from this structure, we used the same setup as described previously with a Ti:Sapphire laser providing 200 fs pulses centered at 800 nm.

The observed SH in the back-focal plane image (see Fig. 4(c), again correcting the apodization by multiplying $\cos (\theta )$) shows a very strong zero order and comparably weak first diffraction orders. To measure the shape of the latter, we needed to use a very long exposure time which saturated the camera at the position of the zero order. Nevertheless, the shape of the SH signal at the first diffraction orders in the angular spectrum showed a ring with zero intensity in the center. This is the confirmation of generating a vortex beam in the first diffraction order by a patterned MoS$_2$ ML.

To summarize, by patterning MoS$_2$ ML flakes, we were able to shape the spatial characteristics of their second-harmonic response. This we used to control the emission direction and intensity distribution of SH radiation generated in the patterned MoS$_2$. In particular, we experimentally showed that MoS$_2$ grating structures lead to diffraction of the generated SH in accordance with theoretical predictions. This basic result can be used to implement ultra-thin nonlinear diffractive optical elements generating more complicated spatial patterns of the SH. As a first example, we created a vortex beam in the first diffraction order by inscribing a grating featuring a topological dislocation into the atomically thin MoS$_2$. Our results show, that patterned MoS$_2$, as well as other TMDs with second-order nonlinearity, is a suitable platform for nonlinear beam shaping. Due to their small thickness, they can be used for customized functionalization of a large number of surfaces, thus enabling to use spatially controlled nonlinear effects in conjunction with many other optical elements. In our work, we focused on spatially modulating the magnitude of the emitted SH to achieve control of the emission direction. Modulating the phase may be possible if two suitably structured flakes with opposite crystal directions are combined, e.g. by transferring them on each other. In this case, SH emission with 0 or $\pi$ phase from neighboring domains may be achieved, further enhancing the potential of this approach. After acceptance of our manuscript, we became aware of a related work investigating nonlinear diffraction and holography for microstructured WS₂ sheets [57].