Abstract
Secondharmonic generation is of paramount importance in several fields of science and technology, including frequency conversion, selfreferencing of frequency combs, nonlinear spectroscopy and pulse characterization. Advanced functionalities are enabled by modulation of the harmonic generation efficiency, which can be achieved with electrical or alloptical triggers. Electrical control of the harmonic generation efficiency offers large modulation depth at the cost of low switching speed, by contrast to alloptical nonlinear devices, which provide high speed and low modulation depth. Here we demonstrate alloptical modulation of secondharmonic generation in MoS_{2} with a modulation depth of close to 100% and speed limited only by the fundamental pulse duration. This result arises from a combination of D_{3h} crystal symmetry and the deep subwavelength thickness of the sample, it can therefore be extended to the whole family of transition metal dichalcogenides to provide great flexibility in the design of advanced nonlinear optical devices such as highspeed integrated frequency converters, broadband autocorrelators for ultrashort pulse characterization, and tunable nanoscale holograms.
Main
Stemming from the first demonstration of optical harmonic generation^{1}, nonlinear optics has been in the spotlight of science and technology for more than half a century. In particular, secondharmonic generation (SHG) is a secondorder nonlinear process widely used for frequency conversion, selfreferencing of frequency combs^{2}, crystal symmetry and Rashba effect studies^{3,4}, sensing^{5}, interface spectroscopy^{6} and ultrashort pulse characterization^{7}. Aside from freespace applications, there is increasing interest towards the realization of microscale integrated nonlinear devices. Here, a major challenge comes from the centrosymmetric nature of silicon (Si) and silicon nitride (Si_{3}N_{4}), which forbids secondorder nonlinearities. Large efforts have been devoted to the integration of nonlinear crystals such as lithium niobate^{8,9}, or to symmetry breaking in Si and Si_{3}N_{4}, for instance, via strain^{10}, electric fields^{11} or the photogalvanic effect^{12}.
Twodimensional materials such as graphene and transition metal dichalcogenides (TMDs) hold great promise for nonlinear optical applications. They have a strong and broadband optical response^{13,14}, combined with the possibility of harmonic generation enhancement at excitonic resonances in TMDs^{15} and at multiphoton resonances in graphene’s Dirac cones^{16,17}. Furthermore, thanks to their flexibility and mechanical strength^{18}, they can be easily integrated into photonic platforms. Various functionalized devices for sensing and frequency conversion have been demonstrated on fibres^{19}, waveguides^{20} and microrings^{21}, while direct patterning of TMDs has been used to realize atomically thin metalenses^{22,23} and nonlinear holograms^{24,25}. Furthermore, harmonic generation in twodimensional materials can be efficiently tuned by external electrical^{16,26,27,28} or alloptical excitation^{29,30}, offering an extra degree of freedom for the design of advanced nanoscale devices.
However, all of the electrical and alloptical schemes that have been proposed so far for SHG modulation in twodimensional materials have considerable downsides. On one hand, electrical modulation has been demonstrated in tungsten diselenide (WSe_{2}) monolayers^{26} by tuning the oscillator strength of neutral and charged exciton resonances through electrostatic doping, and also in molybdenum disulfide (MoS_{2}) homobilayers^{27} by breaking the naturally occurring inversion symmetry through electrical gating, in the latter case with a large modulation depth of up to a factor of 60; however, electronics is intrinsically slower than optics and photonics. On the other hand, alloptical SHG modulation has been achieved by quenching of the exciton oscillator strength following ultrafast optical excitation in MoS_{2} (refs. ^{29,30}). This approach offers high modulation speed and is limited in principle only by the excited state/exciton lifetime (approximately tens of picoseconds); however, the largest depth in alloptical SHG modulation reported^{29} so far in TMDs is 55%, with a strong dependence on the excitation wavelength and fluence. Furthermore, this scheme for alloptical SHG modulation is only effective for excitation and frequency conversion abovegap or at excitonic resonances and it is not applicable for belowgap excitation, thus leading to a naturally limited spectral bandwidth.
Here we demonstrate a novel approach for the alloptical control of the secondharmonic (SH) polarization in MoS_{2} and show that this can be used for alloptical modulation of the SH efficiency with modulation depth close to 100% and speed limited only by the fundamental frequency (FF) pulse duration. Our method relies solely on symmetry considerations in combination with the deep subwavelength thickness of the sample and thus does not require resonant enhancement or abovegap excitation for its implementation. Moreover, the same approach can be extended to any twodimensional material belonging to the D_{3h} symmetry group, thus for instance to any material of the TMD family. Our findings provide a new strategy for the tuning of the fundamental properties of light (polarization and amplitude) in the nonlinear regime and in the twodimensional thickness limit, and thus pave the way to the design of novel advanced functionalities in highspeed frequency converters, nonlinear alloptical modulators and transistors^{31,32}, interferometric autocorrelators for ultrashort pulse characterization and tunable atomically thin holograms^{24}.
Nonlinear optical characterization
For the experiments, we used highquality monolayer MoS_{2} flakes fabricated by a modified chemical vapour deposition method^{33,34} on thermally oxidized Si/SiO_{2} substrates. By contrast to thin films where surface SHG is allowed due to an outofplane component of the secondorder susceptibility^{6,35}, TMDs belong to the D_{3h} symmetry group and thus have only one nonvanishing inplane component of the nonlinear optical susceptibility (Methods)^{35,36,37,38,39,40}
where x and y refer to the inplane Cartesian coordinates of the SH polarization and of the two FFs. A sketch of the hexagonal lattice for MoS_{2} is shown in Fig. 1a, in which the Cartesian coordinates are defined with respect to the two main lattice orientations: the armchair (AC) and zigzag (ZZ) directions. In this framework, the SH intensity I^{2ω} as a function of the FF for any TMD along the AC and ZZ directions can be written as
where E_{AC} and E_{ZZ} correspond to the FF fields with polarization along the AC and ZZ directions, respectively^{39,41,42,43}. The SHG from two electric fields with the same polarization (either along AC or ZZ) will thus always result in an emitted SH intensity with polarization along the AC direction, as depicted in Fig. 1b. This is indeed the case for all of the SHG experiments on twodimensional materials performed so far^{15,36,39,43}. On the other hand, two ultrashort FFs with perpendicular polarization (along the AC and ZZ directions) and with the same amplitude will generate a SH signal along the AC direction if they do not overlap in time (Fig. 1c), whereas they will generate a SH signal along the ZZ direction at zero delay (Fig. 1d), thus leading to an ultrafast 90° polarization switch within the FFs pulse duration. Finally, for circularly polarized FFs, the emitted SH has opposite circular polarization due to valleydependent selection rules (see the analysis of equation (4))^{44,45,46}.
The SHG measurements were performed using the setup shown in Fig. 2a and described in Methods. To realize alloptical polarization switching and SH modulation, it is crucial to first characterize the relative orientation between the FFs and the MoS_{2} sample. To do so, we first performed SHG experiments using a tunable nearinfrared optical parametric oscillator (OPO) as the FF. The emitted SHG power for the FF wavelengths used in our experiments (between 1,360 nm and 1,560 nm) is shown in Fig. 2b. The slope of 2 in the double logarithmic plot is further proof of a genuine SHG process. The crystal orientation of the MoS_{2} sample was determined for each FF wavelength by SH polarizationdependent experiments, with an extra polarizer in front of the detector to measure the SH parallel to the excitation polarization^{36,39,43,47}: the SH intensity is proportional to cos^{2}[3(θ − ϕ_{0})], where θ is the FF polarization angle and ϕ_{0} is the rotation of the AC direction relative to the ppolarization in the laboratory frame. Figure 2c is an example of the SH polar plot for a FF wavelength of 1,400 nm and shows that the AC direction is tilted by ϕ_{0} = 13.6° + n × 60° (where n is an integer) with respect to ppolarization in the laboratory coordinates. Furthermore, Fig. 1c confirms the absence of any detectable strain in our sample, as uniaxial strain would result in a symmetric attenuation along its direction of action^{38,39}. The small asymmetry of the lobes along the two AC directions (~15°/70° and ~190°/250°) is attributed to polarizationscrambling effects due to the use of a dichroic mirror in reflection geometry^{48}. Finally, on the basis of the results in Fig. 2b, we determine the modulus of the complex secondorder nonlinear susceptibility at the FF wavelengths used in our experiments. To do so, we estimate the optical losses of the setup from the SH emission at the sample position to the detection on the singlephoton avalanche diode (SPAD) and calculate the SH tensor element χ^{(2)} of MoS_{2}, as described in Methods. We thus obtained effective secondorder susceptibility values for our FF wavelengths at 1,360 nm, 1,400 nm, 1,480 nm and 1,560 nm of ~282.1 pm V^{–1}, ~153.7 pm V^{–1}, ~44.7 pm V^{–1} and ~24.2 pm V^{–1}, respectively. The highest value obtained at 1,360 nm FF wavelength is due to exciton resonant SH enhancement^{15,49}. All values are in good agreement with those previously reported by experiments performed at similar FF wavelengths^{36,42,49,50,51} and predicted by theory^{52,53}. It is worth noting that for singlelayer TMDs where interlayer interference^{54} is absent, SHG is insensitive to the phase of the nonlinear optical response.
Nonlinear alloptical modulation
Having defined the AC and ZZ directions of our sample, we now demonstrate alloptical SH polarization and amplitude modulation. We separate the FF beam into two perpendicular replicas, align them along the AC and ZZ directions of the sample using a halfwaveplate, and control their relative delay with a motorized mechanical stage (see Methods for details). For large delays (that is, longer than the FF pulse duration) between the two perpendicular FFs, SH will be emitted by each individual FF along the AC direction following equation (2). At zero delay (when the two FFs overlap perfectly in time) the SH intensity along the AC direction will go to zero and the SH signal will be emitted only along the ZZ direction. Figure 3a shows the SH average power emitted along the ZZ direction as a function of the delay between the two perpendicularly polarized FFs and for a FF wavelength of 1,480 nm. The Gaussian fit (the blue curve in Fig. 3a) has a fullwidth at halfmaximum (FWHM) of ~250 fs, which corresponds to the autocorrelation function of our OPO pulse with a duration of ~175 fs. Moreover, the SH signal along the ZZ direction is now ideally background free, demonstrating the potential of ultrafast SH polarization switching and the closeto100% amplitude modulation of our approach.
We further note that this result is solely based on symmetry considerations and thus provides an ultrabroadband response that is not limited to abovegap or resonant exciton pumping. We obtained the same result for all of the FF wavelengths used in our experiments, as shown in Fig. 3b. The possibility to emit SH along perpendicular directions (AC and ZZ) with the same efficiency is a unique feature that arises from the combination of symmetry and deep subwavelength thickness of TMDs, which relaxes the phasematching constraints typical of harmonic generation. This result could have an immediate application in backgroundfree ultrabroadband and ultrashort pulse characterization. For instance, in the most advanced commercial systems^{55} for ultrashort pulse characterization, one has to switch between collinear and noncollinear geometries to collect either the interferometric or the backgroundfree intensity autocorrelation signals, respectively. By contrast, in our approach both signals are accessible using the same geometry and by simply switching the SH detection from AC to ZZ. Further, following equation (3), the power scaling of the emitted SH along the ZZ direction is linear with respect to each of the FF intensities. This is confirmed by the powerdependent SHG measurements reported in Fig. 3c, where we show the emitted SH power along the ZZ direction at zero delay between the two FFs and as a function of the ACpolarized FF power.
To gain more insight into the temporal evolution of the emitted SH polarization and amplitude, we scan the delay between the two perpendicularly polarized FFs with interferometric precision and measure the emitted SH along both the AC and ZZ directions (Fig. 4a). To control the delay between two perpendicular pulses with the desired subopticalcycle precision, we used the commonpath delay generator sketched in Fig. 4b and described in Methods. As expected, the SH power is emitted only along the AC direction for delays longer than our pulse duration, and no signal is detected along the ZZ direction (Fig. 4a). Instead, for delays close to zero we observe a strong ultrafast modulation of the SH power emitted along the AC direction. This can be better appreciated by looking at Fig. 4c, which shows the emitted SH power along the AC and ZZ directions at 1,480 nm for delays between −10 fs and +10 fs.
It is useful to note that, for delays much shorter than the pulse duration, our interferometric measurement is the analogue of tuning the polarization of one FF pulse along the orthodrome of the Poincaré sphere (see the inset in Fig. 4a). This corresponds to a rotation of the FF polarization from −45° with respect to the AC–ZZ directions (at zero delay), to left/right circular polarization (at \(\tau =\pm \frac{T}{4}\) delay, where τ is the FF optical cycle), to +45° with respect to the AC–ZZ directions (at \(\tau =\pm \frac{T}{2}\) delay). This result is consistent with the theoretical SH polarization P^{(2)} generated by an arbitrary elliptically polarized FF^{46} after a simple basis transformation to account for the rotation by −45° with respect to AC/ZZ directions:
Here ϑ = 0° denotes a linearly polarized FF at 45° with respect to the AC/ZZ direction, whereas ϑ = 45° corresponds to a circularly polarized FF. This clearly shows that the SH component emitted along the AC direction oscillates with a period of \(\frac{T}{2}\) as a function of the FF polarization, by contrast to the SH emitted along the ZZ direction. This underpins the interferometric precision required to fully capture the modulation along the AC direction. The experimental results show a weak modulation also for the SH emitted along the ZZ direction, although this is not expected from theory. This could arise from weak strain (that is, below the limit detectable by our SHG polarization measurements^{39,56}), small deviations in the alignment of the detection polarizer with respect to the AC/ZZ directions or from extra terms in the χ^{(2)} arising from the valley degree of freedom^{48,57}. Looking at Fig. 4c, one can indeed appreciate that the SH is emitted only along the ZZ direction at zero delay (FF at −45° with respect to AC/ZZ directions), whereas the emitted SH components along AC and ZZ are identical at \(\frac{T}{4}\) delay, as expected for circular polarization.
Discussion
In conclusion, we have demonstrated alloptical polarization switching and amplitude modulation of SHG in MoS_{2}. Our approach surpasses all previously reported electrical and alloptical attempts of SH tuning in terms of modulation depth and speed, providing a 90° polarization switch, modulation depth close to 100%, and speed limited only by the FF pulse duration. Moreover, our method is intrinsically broadband as it only relies on the crystal symmetry of TMDs. We thus foresee a direct impact of our results on a variety of photonic devices, such as highspeed frequency converters, nonlinear alloptical modulators and transistors^{31,32}, autocorrelators for ultrashort pulse characterization, and atomically thin optically tunable nonlinear holograms^{24}.
Methods
Polarizationdependent SH intensity
The vectorial components of the secondorder polarization P^{(2)}(2ω) for a material with D_{3h} symmetry (such as TMDs) are given by
where \({\chi }_{yyy}^{(2)}={\chi }_{xxy}^{(2)}={\chi }_{yxx}^{(2)}={\chi }^{(2)}\). If we now consider a TMD oriented in such way that the ZZ and AC directions lie along the x and y Cartesian coordinates, respectively, and we neglect the z (outofplane) direction, we obtain the following expression:
Finally, as the SH intensity is proportional to the absolute square value of the secondorder polarization, we retrieve equations (2) and (3) shown in the main text:
SHG setup
For the FF we used an OPO (Levante IR from APE) pumped by an ytterbiumdoped mode locked laser (FLINT12, Light Conversion) with a repetition rate of 76 MHz, pulse duration of 100 fs, and generating pulses tunable between ~1.3 μm and 2.0 μm. The FF is then separated into two perpendicular replicas whose relative delay is tuned with two different approaches: a computercontrolled motorized stage (M414.2PD, PI) in a Mach–Zehnder interferometer configuration and a commercial commonpath birefringent interferometer (GEMINI, NIREOS)^{58}. Compared with standard homebuilt interferometers, the GEMINI provides subwavelength interferometric stability with precise control on the delay between the two replicas with attosecond precision. The polarization of the FFs was tuned using a halfwaveplate (AHWP05M1600, Thorlabs) and the power on the sample was controlled by two polarizers (LPNIR050 and WP12LUB, both Thorlabs). Finally, the two collinear and perpendicularly polarized FFs were focused on the sample using a custom built microscope equipped with a ×40 reflective objective (LMM40XP01, Thorlabs). The backscattered SH signal is spectrally separated using a dichroic mirror (DMSP950, Thorlabs), further spectrally purified by filters (FELH0650, FESH850, FESH0950, all Thorlabs) and detected with a SPAD (C11202050, Hamamatsu). The SH polarization was measured using a wire grid polarizer (WP12LUB, Thorlabs).
Estimate of the optical losses of the setup
To quantify the SH signal generated directly at the sample position, optical losses of the different components of the setup must be considered. Although the transmission coefficients for the investigated SH wavelengths of the filters and the dichroic mirror (all >96%) were taken from the manufacturer’s website, the values for polarizers and the microscope objective were determined experimentally. A transmission of ~79% was determined for the wire grid polarizer, whereas we determined a transmission of 50% for the reflective objective. Last, the responsivity of the SPAD was taken into account, which ranges depending on the investigated SH wavelength between ~17% and ~31%. In total, we estimated our optical losses from the SH emission to the detector to be ~86–92%, depending on the wavelength.
Calculation of the secondorder nonlinear susceptibility
The sheet SH tensor element \({\chi }_{{\mathrm{S}}}^{(2)}\) can be calculated from the FF and SH average powers using the equation^{42}:
where c is the speed of light, ϵ_{0} is the permittivity of freespace, f = 76 MHz is the pump laser repetition rate, r ≈ 1.85 μm is the focal spot radius, t_{FWHM} ≈ 200 fs is the FWHM of the pulse, n_{2} ≈ 1.45 is the substrate refractive index, S = 0.94 is a shape factor for Gaussian pulses, ω is the FF angular frequency, and P_{SHG}(2ω) – P_{FF}(ω) are the SH and FF average powers, respectively. The effective bulklike secondorder susceptibility \({\chi }_{\mathrm{eff}}^{(2)}\) can be subsequently calculated from equation (5) as \({\chi }_{\mathrm{eff}}^{(2)}=\frac{{\chi }_{S}^{(2)}}{{d}_{\mathrm{Mo{S}_{2}}}}\), where the thickness of MoS_{2} \({d}_{\mathrm{Mo{S}_{2}}}\) is 0.65 nm (refs. ^{18,46}).
Pulse duration of the FFs
To prove that our method is solely limited by the pulse duration of the FFs, we performed a standard characterization of a temporal profile of our OPO source at different wavelengths (Extended Data Fig. 1). For the measurements, we used a homebuilt autocorrelator based on a Michelson interferometer equipped with a motorized and computercontrolled translation stage (HPS6020XM5, Optosigma). Two identical and temporally delayed replicas of the OPO pulse were then focused onto a 1mmthick beta barium borate crystal (BBO652H, Eksma Optics) in noncollinear geometry and the backgroundfree SHG autocorrelation intensity was detected on a silicon photodetector (DET10A2, Thorlabs). From this we obtained values for the autocorrelation in the range of 217 fs to 310 fs with Gaussian fits, corresponding to pulse durations between 150 fs and 220 fs, respectively.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request. Source data are provided with this paper.
Change history
21 February 2022
In the version of this Article initially published, the following metadata was omitted and has now been included: Open access funding provided by FriedrichSchillerUniversität Jena (1010).
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Acknowledgements
We acknowledge H. Rostami and F. Preda for helpful discussions. This work was supported by the European Union’s Horizon 2020 Research and Innovation programme under Grant Agreement GrapheneCore3 881603 (G.S. and G.C.). This publication is part of the METAFAST project that received funding from the European Union’s Horizon 2020 Research and Innovation programme under Grant Agreement No. 899673 (G.S. and G.C.). We acknowledge the German Research Foundation DFG (CRC 1375 NOA project numbers B2 (A.T.) and B5 (G.S.)) and the Daimler und Benz foundation for financial support (G.S.). Openaccess funding was provided by FriedrichSchillerUniversität Jena.
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S.K. and G.S. conceived the experiments. S.K. and O.G. performed the alloptical modulation measurements. Z.G., A.G. and A.T. fabricated and provided the highquality MoS_{2} sample. S.K., G.C and G.S. wrote the manuscript, with contributions from all authors. All authors participated in the discussion and commented on the manuscript.
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Extended data
Extended Data Fig. 1 Normalized SHG intensity autocorrelations of various OPO wavelengths in the investigated interval.
Colored dots are for experimental data and solid lines are for gaussian fits. The plots are vertically shifted for clarity.
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Source Data Fig. 2
Raw data from Fig. 2b,c.
Source Data Fig. 3
Raw data from Fig. 3a–c.
Source Data Fig. 4
Raw data from Fig. 4a,c.
Source Data Extended Data Fig. 1
Raw data from Extended Data Fig. 1.
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Klimmer, S., Ghaebi, O., Gan, Z. et al. Alloptical polarization and amplitude modulation of secondharmonic generation in atomically thin semiconductors. Nat. Photon. 15, 837–842 (2021). https://doi.org/10.1038/s4156602100859y
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DOI: https://doi.org/10.1038/s4156602100859y
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