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All-optical polarization and amplitude modulation of second-harmonic generation in atomically thin semiconductors

All-optical polarization and amplitude modulation of second-harmonic generation in atomically... Articles https://doi.org/10.1038/s41566-021-00859-y All-optical polarization and amplitude modulation of second-harmonic generation in atomically thin semiconductors 1 1 2 2,3 2,3 Sebastian Klimmer    , Omid Ghaebi , Ziyang Gan , Antony George    , Andrey Turchanin , 4 1,3  ✉ Giulio Cerullo and Giancarlo Soavi    Second-harmonic generation is of paramount importance in several fields of science and technology, including frequency con- version, self-referencing 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 all-optical triggers. Electrical control of the harmonic generation efficiency offers large modulation depth at the cost of low switching speed, by contrast to all-optical nonlinear devices, which provide high speed and low modulation depth. Here we demonstrate all-optical modulation of second-harmonic generation in MoS with a modulation depth of close to 100% and speed limited only by the fun- damental pulse duration. This result arises from a combination of D crystal symmetry and the deep subwavelength thickness 3h 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 high-speed integrated frequency converters, broadband autocorrela- tors for ultrashort pulse characterization, and tunable nanoscale holograms. temming from the first demonstration of optical harmonic However, all of the electrical and all-optical schemes that have generation , nonlinear optics has been in the spotlight of sci- been proposed so far for SHG modulation in two-dimensional Sence and technology for more than half a century. In particu- materials have considerable downsides. On one hand, electrical lar, second-harmonic generation (SHG) is a second-order nonlinear modulation has been demonstrated in tungsten diselenide (WSe ) process widely used for frequency conversion, self-referencing of monolayers by tuning the oscillator strength of neutral and charged 2 3,4 frequency combs , crystal symmetry and Rashba effect studies , exciton resonances through electrostatic doping, and also in molyb- 5 6 27 sensing , interface spectroscopy and ultrashort pulse characteriza- denum disulfide (MoS ) homobilayers by breaking the naturally tion . Aside from free-space applications, there is increasing interest occurring inversion symmetry through electrical gating, in the lat- towards the realization of microscale integrated nonlinear devices. ter case with a large modulation depth of up to a factor of 60; how- Here, a major challenge comes from the centrosymmetric nature of ever, electronics is intrinsically slower than optics and photonics. On silicon (Si) and silicon nitride (Si N ), which forbids second-order the other hand, all-optical SHG modulation has been achieved by 3 4 nonlinearities. Large efforts have been devoted to the integration of quenching of the exciton oscillator strength following ultrafast opti- 8,9 29,30 nonlinear crystals such as lithium niobate , or to symmetry break- cal excitation in MoS (refs. ). This approach offers high modula- 10 11 ing in Si and Si N , for instance, via strain , electric fields or the tion speed and is limited in principle only by the excited state/exciton 3 4 photogalvanic effect . lifetime (approximately tens of picoseconds); however, the largest Two-dimensional materials such as graphene and transition depth in all-optical SHG modulation reported so far in TMDs is metal dichalcogenides (TMDs) hold great promise for nonlinear 55%, with a strong dependence on the excitation wavelength and flu- optical applications. They have a strong and broadband optical ence. Furthermore, this scheme for all-optical SHG modulation is 13,14 response , combined with the possibility of harmonic genera- only effective for excitation and frequency conversion above-gap or tion enhancement at excitonic resonances in TMDs and at mul- at excitonic resonances and it is not applicable for below-gap excita- 16,17 tiphoton resonances in graphene’s Dirac cones . Furthermore, tion, thus leading to a naturally limited spectral bandwidth. thanks to their flexibility and mechanical strength , they can be Here we demonstrate a novel approach for the all-optical control easily integrated into photonic platforms. Various functionalized of the second-harmonic (SH) polarization in MoS and show that this devices for sensing and frequency conversion have been demon- can be used for all-optical modulation of the SH efficiency with modu- 19 20 21 strated on fibres , waveguides and microrings , while direct lation depth close to 100% and speed limited only by the fundamental patterning of TMDs has been used to realize atomically thin frequency (FF) pulse duration. Our method relies solely on symmetry 22,23 24,25 meta-lenses and nonlinear holograms . Furthermore, har- considerations in combination with the deep subwavelength thick- monic generation in two-dimensional materials can be efficiently ness of the sample and thus does not require resonant enhancement 16,26–28 29,30 tuned by external electrical or all-optical excitation , or above-gap excitation for its implementation. Moreover, the same offering an extra degree of freedom for the design of advanced approach can be extended to any two-dimensional material belong- nanoscale devices. ing to the D symmetry group, thus for instance to any material of 3h 1 2 Institute of Solid State Physics, Friedrich Schiller University Jena, Jena, Germany. Institute of Physical Chemistry, Friedrich Schiller University Jena, Jena, 3 4 Germany. Abbe Center of Photonics, Friedrich Schiller University Jena, Jena, Germany. Dipartimento di Fisica, Politecnico di Milano, Milan, Italy. e-mail: giancarlo.soavi@uni-jena.de NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics 837 AC NATurE PHoToNicS Articles a c d Mo Probe AC E (ω) E (ω) P(2ω) 1 2 Control Fig. 1 | MoS symmetry properties, optical selection rules and an all-optical ShG modulation scheme. a, Top view of a MoS crystal. The arrows inside 2 2 the hexagon highlight the D three-fold rotational symmetry. b, A schematic of the resulting SH polarization for different combinations of FFs along the AC 3h (horizontal arrows) and ZZ (vertical arrows) directions. c,d, A sketch of the all-optical SH polarization modulation. The control pulse is polarized along the ZZ direction whereas the probe pulse is polarized along the AC direction of the MoS sample. c, When the delay between the control and probe pulses is larger than the FF pulse duration, both will generate a SH signal polarized along the AC direction. d, For zero delay between the probe and control pulses the SH signal will be emitted along the ZZ direction. the TMD family. Our findings provide a new strategy for the tuning where E and E correspond to the FF fields with polarization AC ZZ 39,41–43 of the fundamental properties of light (polarization and amplitude) along the AC and ZZ directions, respectively . The SHG from in the nonlinear regime and in the two-dimensional thickness limit, two electric fields with the same polarization (either along AC or and thus pave the way to the design of novel advanced functionalities ZZ) will thus always result in an emitted SH intensity with polariza- in high-speed frequency converters, nonlinear all-optical modulators tion along the AC direction, as depicted in Fig. 1b. This is indeed 31,32 and transistors , interferometric autocorrelators for ultrashort pulse the case for all of the SHG experiments on two-dimensional materi- 24 15,36,39,43 characterization and tunable atomically thin holograms . als performed so far . On the other hand, two ultrashort FFs with perpendicular polarization (along the AC and ZZ directions) Nonlinear optical characterization and with the same amplitude will generate a SH signal along the AC For the experiments, we used high-quality monolayer MoS flakes direction if they do not overlap in time (Fig. 1c), whereas they will 33,34 fabricated by a modified chemical vapour deposition method generate a SH signal along the ZZ direction at zero delay (Fig. 1d), on thermally oxidized Si/SiO substrates. By contrast to thin films thus leading to an ultrafast 90° polarization switch within the FFs where surface SHG is allowed due to an out-of-plane component of pulse duration. Finally, for circularly polarized FFs, the emitted SH 6,35 the second-order susceptibility , TMDs belong to the D symme- has opposite circular polarization due to valley-dependent selection 3h 44–46 try group and thus have only one non-vanishing in-plane compo- rules (see the analysis of equation (4)) . 35–40 nent of the nonlinear optical susceptibility (Methods) The SHG measurements were performed using the set-up shown in Fig. 2a and described in Methods. To realize all-optical polarization switching and SH modulation, it is crucial to first (2) (2) (2) (2) (2) χ ≡ χ = −χ = −χ = −χ , (1) yyy yxx xyx xxy characterize the relative orientation between the FFs and the MoS where x and y refer to the in-plane Cartesian coordinates of the SH sample. To do so, we first performed SHG experiments using a tun- polarization and of the two FFs. A sketch of the hexagonal lattice able near-infrared optical parametric oscillator (OPO) as the FF. for MoS is shown in Fig. 1a, in which the Cartesian coordinates The emitted SHG power for the FF wavelengths used in our exper- are defined with respect to the two main lattice orientations: the iments (between 1,360 nm and 1,560 nm) is shown in Fig. 2b. The armchair (AC) and zigzag (ZZ) directions. In this framework, the slope of 2 in the double logarithmic plot is further proof of a genu- 2ω SH intensity I as a function of the FF for any TMD along the AC ine SHG process. The crystal orientation of the MoS sample was and ZZ directions can be written as determined for each FF wavelength by SH polarization-dependent experiments, with an extra polarizer in front of the detector to 2ω 2 2 2 I ∝|E −E | , (2) 36,39,43,47 AC AC ZZ measure the SH parallel to the excitation polarization : the SH intensity is proportional to cos [3(θ − ϕ )], where θ is the 2ω 2 FF polarization angle and ϕ is the rotation of the AC direction I ∝|2E E | , (3) ZZ AC ZZ 0 NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics ZZ ZZ NATurE PHoToNicS Articles WL CMOS Sample DM ×40 HWP Filter SPAD BS HWP Pump b c 90° 120° 60° 150° 30° 180° 1,560 nm 1,480 nm 210° 330° 1,400 nm 20 μm 1,360 nm 240° 300° 0.1 1 10 270° FF power (mW) Fig. 2 | All-optical modulation set-up and ShG characterization. a, A sketch of the set-up used for the experiments. For the FF we used an OPO tunable between ~1.3 μm and 2.0 μm. The FF beams are separated into two perpendicular replicas inside of a Mach–Zehnder interferometer and subsequently focused onto the MoS sample with a ×40 microscope objective. The backscattered SH signal is spectrally filtered and detected with a SPAD. CMOS, complementary metal-oxide semiconductor camera; bS, beam splitter; P, polarizer; HWP, half-wave plate; WL, white light. DM, dichroic mirror. b, The power dependence of the SHG signal for all wavelengths used in our experiments. The grey dashed line is a guide to the eye to indicate a slope of 2, which is typical of the SHG process. c, A polar plot of the normalized SH intensity as a function of the excitation polarization angle θ with SH polarization detection always parallel to the FF. blue circles show experimental data and the solid blue line indicates the cos [3(θ − ϕ )] fit. relative to the p-polarization in the laboratory frame. Figure 2c is Nonlinear all-optical modulation an example of the SH polar plot for a FF wavelength of 1,400 nm Having defined the AC and ZZ directions of our sample, we now and shows that the AC direction is tilted by ϕ = 13.6° + n × 60° demonstrate all-optical SH polarization and amplitude modula- (where n is an integer) with respect to p-polarization in the labora- tion. We separate the FF beam into two perpendicular replicas, tory coordinates. Furthermore, Fig. 1c confirms the absence of any align them along the AC and ZZ directions of the sample using a detectable strain in our sample, as uniaxial strain would result in a half-waveplate, and control their relative delay with a motorized 38,39 symmetric attenuation along its direction of action . The small mechanical stage (see Methods for details). For large delays (that is, asymmetry of the lobes along the two AC directions (~15°/70° and longer than the FF pulse duration) between the two perpendicular ~190°/250°) is attributed to polarization-scrambling effects due to FFs, SH will be emitted by each individual FF along the AC direc- the use of a dichroic mirror in reflection geometry . Finally, on tion following equation (2). At zero delay (when the two FFs overlap the basis of the results in Fig. 2b, we determine the modulus of perfectly in time) the SH intensity along the AC direction will go the complex second-order nonlinear susceptibility at the FF wave- to zero and the SH signal will be emitted only along the ZZ direc- lengths used in our experiments. To do so, we estimate the optical tion. Figure 3a shows the SH average power emitted along the ZZ losses of the set-up from the SH emission at the sample position direction as a function of the delay between the two perpendicularly to the detection on the single-photon avalanche diode (SPAD) polarized FFs and for a FF wavelength of 1,480 nm. The Gaussian (2) and calculate the SH tensor element χ of MoS , as described in fit (the blue curve in Fig. 3a) has a full-width at half-maximum Methods. We thus obtained effective second-order susceptibility (FWHM) of ~250 fs, which corresponds to the autocorrelation values for our FF wavelengths at 1,360 nm, 1,400 nm, 1,480 nm function of our OPO pulse with a duration of ~175 fs. Moreover, the –1 –1 –1 and 1,560 nm of ~282.1 pm V , ~153.7 pm V , ~44.7 pm V and SH signal along the ZZ direction is now ideally background free, –1 ~24.2 pm V , respectively. The highest value obtained at 1,360 nm demonstrating the potential of ultrafast SH polarization switching 15,49 FF wavelength is due to exciton resonant SH enhancement . All and the close-to-100% amplitude modulation of our approach. values are in good agreement with those previously reported by We further note that this result is solely based on symmetry 36,42,49–51 experiments performed at similar FF wavelengths and pre- considerations and thus provides an ultrabroadband response that 52,53 dicted by theory . It is worth noting that for single-layer TMDs is not limited to above-gap or resonant exciton pumping. We where interlayer interference is absent, SHG is insensitive to the obtained the same result for all of the FF wavelengths used in our phase of the nonlinear optical response. experiments, as shown in Fig. 3b. The possibility to emit SH along NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics SHG power (fW) NATurE PHoToNicS Articles a b c 6.0 ZZ AC 1,560 nm Delay 4.5 1,480 nm 3.0 1,400 nm 1.5 1,560 nm AC ZZ ZZ AC 1,360 nm 1,480 nm 1,400 nm Delay Delay 1,360 nm –500 0 500 500 0 500 0.1 1 10 Delay (fs) Delay (fs) FF power (mW) Fig. 3 | ultrafast polarization and amplitude all-optical switching. a, Second-harmonic intensity for an FF wavelength of 1,480 nm (black line) measured along the ZZ direction as a function of the delay between the two perpendicularly polarized FFs (illustrated in the inset). The blue line corresponds to a Gaussian fit of the autocorrelation curve, whereas the grey shaded area represents the noise level at ~3.5 fW of our experiments. b, The normalized SH intensities and noise level for all of the FF wavelengths used in our experiments. c, A double logarithmic plot of the emitted SH power as a function of the incident FF power along the AC direction. The grey dashed line is a guide to the eye, representing a slope of 1. perpendicular directions (AC and ZZ) with the same efficiency (at τ = ± delay, where τ is the FF optical cycle), to +45° with is a unique feature that arises from the combination of symmetry respect to the AC–ZZ directions (at delay). This result is τ = ± (2) and deep subwavelength thickness of TMDs, which relaxes the consistent with the theoretical SH polarization P generated by phase-matching constraints typical of harmonic generation. This an arbitrary elliptically polarized FF after a simple basis transfor- result could have an immediate application in background-free mation to account for the rotation by −45° with respect to AC/ZZ ultrabroadband and ultrashort pulse characterization. For instance, directions: in the most advanced commercial systems for ultrashort pulse characterization, one has to switch between collinear and non- (2) (2) 2 P = ϵ χ |E| ZZ ±i sin (2ϑ)AC , (4) collinear geometries to collect either the interferometric or the background-free intensity autocorrelation signals, respectively. By contrast, in our approach both signals are accessible using the same Here ϑ = 0° denotes a linearly polarized FF at 45° with respect to geometry and by simply switching the SH detection from AC to ZZ. the AC/ZZ direction, whereas ϑ = 45° corresponds to a circularly Further, following equation (3), the power scaling of the emitted polarized FF. This clearly shows that the SH component emitted SH along the ZZ direction is linear with respect to each of the FF along the AC direction oscillates with a period of as a function of intensities. This is confirmed by the power-dependent SHG mea- the FF polarization, by contrast to the SH emitted along the ZZ surements reported in Fig. 3c, where we show the emitted SH power direction. This underpins the interferometric precision required to along the ZZ direction at zero delay between the two FFs and as a fully capture the modulation along the AC direction. The exper- function of the AC-polarized FF power. imental results show a weak modulation also for the SH emitted To gain more insight into the temporal evolution of the emit- along the ZZ direction, although this is not expected from theory. ted SH polarization and amplitude, we scan the delay between the This could arise from weak strain (that is, below the limit detectable 39,56 two perpendicularly polarized FFs with interferometric precision by our SHG polarization measurements ), small deviations in and measure the emitted SH along both the AC and ZZ directions the alignment of the detection polarizer with respect to the AC/ZZ (2) (Fig. 4a). To control the delay between two perpendicular pulses with directions or from extra terms in the χ arising from the valley 48,57 the desired sub-optical-cycle precision, we used the common-path degree of freedom . Looking at Fig. 4c, one can indeed appreciate delay generator sketched in Fig. 4b and described in Methods. As that the SH is emitted only along the ZZ direction at zero delay expected, the SH power is emitted only along the AC direction for (FF at −45° with respect to AC/ZZ directions), whereas the emit- delays longer than our pulse duration, and no signal is detected ted SH components along AC and ZZ are identical at delay, as along the ZZ direction (Fig. 4a). Instead, for delays close to zero expected for circular polarization. we observe a strong ultrafast modulation of the SH power emitted along the AC direction. This can be better appreciated by looking Discussion at Fig. 4c, which shows the emitted SH power along the AC and ZZ In conclusion, we have demonstrated all-optical polarization switch- directions at 1,480 nm for delays between −10 fs and +10 fs. ing and amplitude modulation of SHG in MoS . Our approach sur- It is useful to note that, for delays much shorter than the pulse passes all previously reported electrical and all-optical attempts of duration, our interferometric measurement is the analogue of SH tuning in terms of modulation depth and speed, providing a 90° tuning the polarization of one FF pulse along the orthodrome of polarization switch, modulation depth close to 100%, and speed the Poincaré sphere (see the inset in Fig. 4a). This corresponds limited only by the FF pulse duration. Moreover, our method is to a rotation of the FF polarization from −45° with respect to the intrinsically broadband as it only relies on the crystal symmetry of AC–ZZ directions (at zero delay), to left/right circular polarization TMDs. We thus foresee a direct impact of our results on a variety NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics SHG power (fW) SHG power (normalized) SHG power (fW) NATurE PHoToNicS Articles a b ZZ e nn e o o AC –50 0 50 100 150 200 –10 –5 0 5 10 Delay (fs) Delay (fs) Fig. 4 | Phase-locked all-optical Sh modulation along the AC and ZZ directions. a, Second-harmonic power for an FF wavelength of 1,480 nm along the AC (blue) and ZZ (red) directions as a function of the relative delay between the two perpendicularly polarized FFs. The inset shows the Poincaré sphere with polarization directions on its orthodromes. The red line indicates the change in polarization when tuning the delay between the two FFs with a birefringent delay line. b, A schematic of the modified common-path delay generator. 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Ultra-strong nonlinear optical processes and trigonal unless indicated otherwise in a credit line to the material. If material is not included in warping in MoS layers. Nat. Commun. 8, 1–8 (2017). the article’s Creative Commons license and your intended use is not permitted by statu- 47. Li, Y. et al. Probing symmetry properties of few-layer MoS and h-bn by tory regulation or exceeds the permitted use, you will need to obtain permission directly optical second-harmonic generation. Nano Lett. 13, 3329–3333 (2013). from the copyright holder. To view a copy of this license, visit http://creativecommons. 48. Mouchliadis, L. et al. Probing valley population imbalance in transition metal org/licenses/by/4.0/. dichalcogenides via temperature-dependent second harmonic generation imaging. npj 2D Mater. Appl. 5, 1–9 (2021). © The Author(s) 2021 NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics 842 NATurE PHoToNicS Articles Calculation of the second-order nonlinear susceptibility. The sheet SH tensor Methods (2) element χ can be calculated from the FF and SH average powers using the Polarization-dependent SH intensity. The vectorial components of the S (2) equation : second-order polarization P (2ω) for a material with D symmetry (such as 3h TMDs) are given by 3 2 c ϵ fπr t (1 +n ) P (2ω) (2) 0 FWHM 2 SHG   2 χ = √ , (5) E (ω) 2 2 16 2Sω P (ω) FF     E (ω)     y where c is the speed of light, ϵ is the permittivity of free-space, f = 76 MHz is the   (2) (2) 0 P (2ω) 0 0 000 χ   x xxy pump laser repetition rate, r ≈ 1.85 μm is the focal spot radius, t ≈ 200 fs is the   2 FWHM     E (ω)   (2)    (2) (2) FWHM of the pulse, n ≈ 1.45 is the substrate refractive index, S = 0.94 is a shape   2 P (2ω)   = ϵ  χ χ 0 000  y 0 yxx yyy       factor for Gaussian pulses, ω is the FF angular frequency, and P (2ω) – P (ω) are   SHG FF 2E (ω)E (ω) y z   (2) the SH and FF average powers, respectively. The effective bulk-like second-order P (2ω) 0 0 0000   (2)   2E (ω)E (ω) x z susceptibility χ can be subsequently calculated from equation (5) as   eff (2) (2) χ 18,46 χ = , where the thickness of MoS d is 0.65 nm (refs. ). 2 MoS 2E (ω)E (ω) d 2 x y eff MoS Pulse duration of the FFs. To prove that our method is solely limited by the (2) (2) (2) (2) where χ = −χ = −χ = χ . If we now consider a TMD oriented in pulse duration of the FFs, we performed a standard characterization of a temporal yyy xxy yxx profile of our OPO source at different wavelengths (Extended Data Fig. 1). For such way that the ZZ and AC directions lie along the x and y Cartesian coordinates, the measurements, we used a home-built autocorrelator based on a Michelson respectively, and we neglect the z (out-of-plane) direction, we obtain the following interferometer equipped with a motorized and computer-controlled translation expression: stage (HPS60-20X-M5, Optosigma). Two identical and temporally delayed replicas   ( ) (2) P (2ω) 2E (ω)E (ω) of the OPO pulse were then focused onto a 1-mm-thick beta barium borate crystal ZZ AC ZZ (2)   = ϵ χ (2ω,ω,ω) 0 (BBO-652H, Eksma Optics) in non-collinear geometry and the background-free (2) 2 2 E (ω) −E (ω) P (2ω) ZZ AC SHG autocorrelation intensity was detected on a silicon photodetector (DET10A2, AC 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 Finally, as the SH intensity is proportional to the absolute square value of the 220 fs, respectively. second-order polarization, we retrieve equations (2) and (3) shown in the main text: Data availability (2) 2ω 2 2 2 2 The data that support the plots within this paper and other findings of this study I = |P | ∝|E −E | AC AC ZZ AC are available from the corresponding author on reasonable request. Source data are (2) 2ω 2 2 I = |P | ∝|2E E | provided with this paper. AC ZZ ZZ ZZ r eferences 58. Preda, F. et al. Linear and nonlinear spectroscopy by a common-path SHG set-up. For the FF we used an OPO (Levante IR from APE) pumped by an ytterbium-doped mode locked laser (FLINT12, Light Conversion) with a birefringent interferometer. IEEE J. Sel. Top. Quant. Electron. 23, 88–96 (2016). 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 Acknowledgements replicas whose relative delay is tuned with two different approaches: a We acknowledge H. Rostami and F. Preda for helpful discussions. This work was computer-controlled motorized stage (M-414.2PD, PI) in a Mach–Zehnder supported by the European Union’s Horizon 2020 Research and Innovation programme interferometer configuration and a commercial common-path birefringent under Grant Agreement GrapheneCore3 881603 (G.S. and G.C.). This publication is part interferometer (GEMINI, NIREOS) . Compared with standard home-built of the METAFAST project that received funding from the European Union’s Horizon interferometers, the GEMINI provides subwavelength interferometric stability 2020 Research and Innovation programme under Grant Agreement No. 899673 (G.S. and with precise control on the delay between the two replicas with attosecond G.C.). We acknowledge the German Research Foundation DFG (CRC 1375 NOA project precision. The polarization of the FFs was tuned using a half-waveplate numbers B2 (A.T.) and B5 (G.S.)) and the Daimler und Benz foundation for financial (AHWP05M-1600, Thorlabs) and the power on the sample was controlled by support (G.S.). Open-access funding was provided by Friedrich-Schiller-Universität Jena. two polarizers (LPNIR050 and WP12L-UB, both Thorlabs). Finally, the two collinear and perpendicularly polarized FFs were focused on the sample using Author contributions a custom built microscope equipped with a ×40 reflective objective (LMM- S.K. and G.S. conceived the experiments. S.K. and O.G. performed the all-optical 40X-P01, Thorlabs). The backscattered SH signal is spectrally separated using modulation measurements. Z.G., A.G. and A.T. fabricated and provided the high-quality a dichroic mirror (DMSP950, Thorlabs), further spectrally purified by filters MoS sample. S.K., G.C and G.S. wrote the manuscript, with contributions from all (FELH0650, FESH850, FESH0950, all Thorlabs) and detected with a SPAD authors. All authors participated in the discussion and commented on the manuscript. (C11202-050, Hamamatsu). The SH polarization was measured using a wire grid polarizer (WP12L-UB, Thorlabs). Competing interests The authors declare no competing interests. Estimate of the optical losses of the set-up. To quantify the SH signal generated directly at the sample position, optical losses of the different components of the set-up must be considered. Although the transmission coefficients for the Additional information investigated SH wavelengths of the filters and the dichroic mirror (all >96%) Extended data is available for this paper at https://doi.org/10.1038/s41566-021-00859-y. were taken from the manufacturer’s website, the values for polarizers and Supplementary information The online version contains supplementary material the microscope objective were determined experimentally. A transmission of available at https://doi.org/10.1038/s41566-021-00859-y. ~79% was determined for the wire grid polarizer, whereas we determined a transmission of 50% for the reflective objective. Last, the responsivity of the Correspondence and requests for materials should be addressed to Giancarlo Soavi. SPAD was taken into account, which ranges depending on the investigated Peer review information Nature Photonics thanks Goki Eda and the other, anonymous, SH wavelength between ~17% and ~31%. In total, we estimated our optical reviewer(s) for their contribution to the peer review of this work. losses from the SH emission to the detector to be ~86–92%, depending on the Reprints and permissions information is available at www.nature.com/reprints. wavelength. NATure PhOTONiCS | www.nature.com/naturephotonics NATurE PHoToNicS Articles 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. NATure PhOTONiCS | www.nature.com/naturephotonics http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nature Photonics Springer Journals

All-optical polarization and amplitude modulation of second-harmonic generation in atomically thin semiconductors

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Articles https://doi.org/10.1038/s41566-021-00859-y All-optical polarization and amplitude modulation of second-harmonic generation in atomically thin semiconductors 1 1 2 2,3 2,3 Sebastian Klimmer    , Omid Ghaebi , Ziyang Gan , Antony George    , Andrey Turchanin , 4 1,3  ✉ Giulio Cerullo and Giancarlo Soavi    Second-harmonic generation is of paramount importance in several fields of science and technology, including frequency con- version, self-referencing 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 all-optical triggers. Electrical control of the harmonic generation efficiency offers large modulation depth at the cost of low switching speed, by contrast to all-optical nonlinear devices, which provide high speed and low modulation depth. Here we demonstrate all-optical modulation of second-harmonic generation in MoS with a modulation depth of close to 100% and speed limited only by the fun- damental pulse duration. This result arises from a combination of D crystal symmetry and the deep subwavelength thickness 3h 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 high-speed integrated frequency converters, broadband autocorrela- tors for ultrashort pulse characterization, and tunable nanoscale holograms. temming from the first demonstration of optical harmonic However, all of the electrical and all-optical schemes that have generation , nonlinear optics has been in the spotlight of sci- been proposed so far for SHG modulation in two-dimensional Sence and technology for more than half a century. In particu- materials have considerable downsides. On one hand, electrical lar, second-harmonic generation (SHG) is a second-order nonlinear modulation has been demonstrated in tungsten diselenide (WSe ) process widely used for frequency conversion, self-referencing of monolayers by tuning the oscillator strength of neutral and charged 2 3,4 frequency combs , crystal symmetry and Rashba effect studies , exciton resonances through electrostatic doping, and also in molyb- 5 6 27 sensing , interface spectroscopy and ultrashort pulse characteriza- denum disulfide (MoS ) homobilayers by breaking the naturally tion . Aside from free-space applications, there is increasing interest occurring inversion symmetry through electrical gating, in the lat- towards the realization of microscale integrated nonlinear devices. ter case with a large modulation depth of up to a factor of 60; how- Here, a major challenge comes from the centrosymmetric nature of ever, electronics is intrinsically slower than optics and photonics. On silicon (Si) and silicon nitride (Si N ), which forbids second-order the other hand, all-optical SHG modulation has been achieved by 3 4 nonlinearities. Large efforts have been devoted to the integration of quenching of the exciton oscillator strength following ultrafast opti- 8,9 29,30 nonlinear crystals such as lithium niobate , or to symmetry break- cal excitation in MoS (refs. ). This approach offers high modula- 10 11 ing in Si and Si N , for instance, via strain , electric fields or the tion speed and is limited in principle only by the excited state/exciton 3 4 photogalvanic effect . lifetime (approximately tens of picoseconds); however, the largest Two-dimensional materials such as graphene and transition depth in all-optical SHG modulation reported so far in TMDs is metal dichalcogenides (TMDs) hold great promise for nonlinear 55%, with a strong dependence on the excitation wavelength and flu- optical applications. They have a strong and broadband optical ence. Furthermore, this scheme for all-optical SHG modulation is 13,14 response , combined with the possibility of harmonic genera- only effective for excitation and frequency conversion above-gap or tion enhancement at excitonic resonances in TMDs and at mul- at excitonic resonances and it is not applicable for below-gap excita- 16,17 tiphoton resonances in graphene’s Dirac cones . Furthermore, tion, thus leading to a naturally limited spectral bandwidth. thanks to their flexibility and mechanical strength , they can be Here we demonstrate a novel approach for the all-optical control easily integrated into photonic platforms. Various functionalized of the second-harmonic (SH) polarization in MoS and show that this devices for sensing and frequency conversion have been demon- can be used for all-optical modulation of the SH efficiency with modu- 19 20 21 strated on fibres , waveguides and microrings , while direct lation depth close to 100% and speed limited only by the fundamental patterning of TMDs has been used to realize atomically thin frequency (FF) pulse duration. Our method relies solely on symmetry 22,23 24,25 meta-lenses and nonlinear holograms . Furthermore, har- considerations in combination with the deep subwavelength thick- monic generation in two-dimensional materials can be efficiently ness of the sample and thus does not require resonant enhancement 16,26–28 29,30 tuned by external electrical or all-optical excitation , or above-gap excitation for its implementation. Moreover, the same offering an extra degree of freedom for the design of advanced approach can be extended to any two-dimensional material belong- nanoscale devices. ing to the D symmetry group, thus for instance to any material of 3h 1 2 Institute of Solid State Physics, Friedrich Schiller University Jena, Jena, Germany. Institute of Physical Chemistry, Friedrich Schiller University Jena, Jena, 3 4 Germany. Abbe Center of Photonics, Friedrich Schiller University Jena, Jena, Germany. Dipartimento di Fisica, Politecnico di Milano, Milan, Italy. e-mail: giancarlo.soavi@uni-jena.de NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics 837 AC NATurE PHoToNicS Articles a c d Mo Probe AC E (ω) E (ω) P(2ω) 1 2 Control Fig. 1 | MoS symmetry properties, optical selection rules and an all-optical ShG modulation scheme. a, Top view of a MoS crystal. The arrows inside 2 2 the hexagon highlight the D three-fold rotational symmetry. b, A schematic of the resulting SH polarization for different combinations of FFs along the AC 3h (horizontal arrows) and ZZ (vertical arrows) directions. c,d, A sketch of the all-optical SH polarization modulation. The control pulse is polarized along the ZZ direction whereas the probe pulse is polarized along the AC direction of the MoS sample. c, When the delay between the control and probe pulses is larger than the FF pulse duration, both will generate a SH signal polarized along the AC direction. d, For zero delay between the probe and control pulses the SH signal will be emitted along the ZZ direction. the TMD family. Our findings provide a new strategy for the tuning where E and E correspond to the FF fields with polarization AC ZZ 39,41–43 of the fundamental properties of light (polarization and amplitude) along the AC and ZZ directions, respectively . The SHG from in the nonlinear regime and in the two-dimensional thickness limit, two electric fields with the same polarization (either along AC or and thus pave the way to the design of novel advanced functionalities ZZ) will thus always result in an emitted SH intensity with polariza- in high-speed frequency converters, nonlinear all-optical modulators tion along the AC direction, as depicted in Fig. 1b. This is indeed 31,32 and transistors , interferometric autocorrelators for ultrashort pulse the case for all of the SHG experiments on two-dimensional materi- 24 15,36,39,43 characterization and tunable atomically thin holograms . als performed so far . On the other hand, two ultrashort FFs with perpendicular polarization (along the AC and ZZ directions) Nonlinear optical characterization and with the same amplitude will generate a SH signal along the AC For the experiments, we used high-quality monolayer MoS flakes direction if they do not overlap in time (Fig. 1c), whereas they will 33,34 fabricated by a modified chemical vapour deposition method generate a SH signal along the ZZ direction at zero delay (Fig. 1d), on thermally oxidized Si/SiO substrates. By contrast to thin films thus leading to an ultrafast 90° polarization switch within the FFs where surface SHG is allowed due to an out-of-plane component of pulse duration. Finally, for circularly polarized FFs, the emitted SH 6,35 the second-order susceptibility , TMDs belong to the D symme- has opposite circular polarization due to valley-dependent selection 3h 44–46 try group and thus have only one non-vanishing in-plane compo- rules (see the analysis of equation (4)) . 35–40 nent of the nonlinear optical susceptibility (Methods) The SHG measurements were performed using the set-up shown in Fig. 2a and described in Methods. To realize all-optical polarization switching and SH modulation, it is crucial to first (2) (2) (2) (2) (2) χ ≡ χ = −χ = −χ = −χ , (1) yyy yxx xyx xxy characterize the relative orientation between the FFs and the MoS where x and y refer to the in-plane Cartesian coordinates of the SH sample. To do so, we first performed SHG experiments using a tun- polarization and of the two FFs. A sketch of the hexagonal lattice able near-infrared optical parametric oscillator (OPO) as the FF. for MoS is shown in Fig. 1a, in which the Cartesian coordinates The emitted SHG power for the FF wavelengths used in our exper- are defined with respect to the two main lattice orientations: the iments (between 1,360 nm and 1,560 nm) is shown in Fig. 2b. The armchair (AC) and zigzag (ZZ) directions. In this framework, the slope of 2 in the double logarithmic plot is further proof of a genu- 2ω SH intensity I as a function of the FF for any TMD along the AC ine SHG process. The crystal orientation of the MoS sample was and ZZ directions can be written as determined for each FF wavelength by SH polarization-dependent experiments, with an extra polarizer in front of the detector to 2ω 2 2 2 I ∝|E −E | , (2) 36,39,43,47 AC AC ZZ measure the SH parallel to the excitation polarization : the SH intensity is proportional to cos [3(θ − ϕ )], where θ is the 2ω 2 FF polarization angle and ϕ is the rotation of the AC direction I ∝|2E E | , (3) ZZ AC ZZ 0 NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics ZZ ZZ NATurE PHoToNicS Articles WL CMOS Sample DM ×40 HWP Filter SPAD BS HWP Pump b c 90° 120° 60° 150° 30° 180° 1,560 nm 1,480 nm 210° 330° 1,400 nm 20 μm 1,360 nm 240° 300° 0.1 1 10 270° FF power (mW) Fig. 2 | All-optical modulation set-up and ShG characterization. a, A sketch of the set-up used for the experiments. For the FF we used an OPO tunable between ~1.3 μm and 2.0 μm. The FF beams are separated into two perpendicular replicas inside of a Mach–Zehnder interferometer and subsequently focused onto the MoS sample with a ×40 microscope objective. The backscattered SH signal is spectrally filtered and detected with a SPAD. CMOS, complementary metal-oxide semiconductor camera; bS, beam splitter; P, polarizer; HWP, half-wave plate; WL, white light. DM, dichroic mirror. b, The power dependence of the SHG signal for all wavelengths used in our experiments. The grey dashed line is a guide to the eye to indicate a slope of 2, which is typical of the SHG process. c, A polar plot of the normalized SH intensity as a function of the excitation polarization angle θ with SH polarization detection always parallel to the FF. blue circles show experimental data and the solid blue line indicates the cos [3(θ − ϕ )] fit. relative to the p-polarization in the laboratory frame. Figure 2c is Nonlinear all-optical modulation an example of the SH polar plot for a FF wavelength of 1,400 nm Having defined the AC and ZZ directions of our sample, we now and shows that the AC direction is tilted by ϕ = 13.6° + n × 60° demonstrate all-optical SH polarization and amplitude modula- (where n is an integer) with respect to p-polarization in the labora- tion. We separate the FF beam into two perpendicular replicas, tory coordinates. Furthermore, Fig. 1c confirms the absence of any align them along the AC and ZZ directions of the sample using a detectable strain in our sample, as uniaxial strain would result in a half-waveplate, and control their relative delay with a motorized 38,39 symmetric attenuation along its direction of action . The small mechanical stage (see Methods for details). For large delays (that is, asymmetry of the lobes along the two AC directions (~15°/70° and longer than the FF pulse duration) between the two perpendicular ~190°/250°) is attributed to polarization-scrambling effects due to FFs, SH will be emitted by each individual FF along the AC direc- the use of a dichroic mirror in reflection geometry . Finally, on tion following equation (2). At zero delay (when the two FFs overlap the basis of the results in Fig. 2b, we determine the modulus of perfectly in time) the SH intensity along the AC direction will go the complex second-order nonlinear susceptibility at the FF wave- to zero and the SH signal will be emitted only along the ZZ direc- lengths used in our experiments. To do so, we estimate the optical tion. Figure 3a shows the SH average power emitted along the ZZ losses of the set-up from the SH emission at the sample position direction as a function of the delay between the two perpendicularly to the detection on the single-photon avalanche diode (SPAD) polarized FFs and for a FF wavelength of 1,480 nm. The Gaussian (2) and calculate the SH tensor element χ of MoS , as described in fit (the blue curve in Fig. 3a) has a full-width at half-maximum Methods. We thus obtained effective second-order susceptibility (FWHM) of ~250 fs, which corresponds to the autocorrelation values for our FF wavelengths at 1,360 nm, 1,400 nm, 1,480 nm function of our OPO pulse with a duration of ~175 fs. Moreover, the –1 –1 –1 and 1,560 nm of ~282.1 pm V , ~153.7 pm V , ~44.7 pm V and SH signal along the ZZ direction is now ideally background free, –1 ~24.2 pm V , respectively. The highest value obtained at 1,360 nm demonstrating the potential of ultrafast SH polarization switching 15,49 FF wavelength is due to exciton resonant SH enhancement . All and the close-to-100% amplitude modulation of our approach. values are in good agreement with those previously reported by We further note that this result is solely based on symmetry 36,42,49–51 experiments performed at similar FF wavelengths and pre- considerations and thus provides an ultrabroadband response that 52,53 dicted by theory . It is worth noting that for single-layer TMDs is not limited to above-gap or resonant exciton pumping. We where interlayer interference is absent, SHG is insensitive to the obtained the same result for all of the FF wavelengths used in our phase of the nonlinear optical response. experiments, as shown in Fig. 3b. The possibility to emit SH along NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics SHG power (fW) NATurE PHoToNicS Articles a b c 6.0 ZZ AC 1,560 nm Delay 4.5 1,480 nm 3.0 1,400 nm 1.5 1,560 nm AC ZZ ZZ AC 1,360 nm 1,480 nm 1,400 nm Delay Delay 1,360 nm –500 0 500 500 0 500 0.1 1 10 Delay (fs) Delay (fs) FF power (mW) Fig. 3 | ultrafast polarization and amplitude all-optical switching. a, Second-harmonic intensity for an FF wavelength of 1,480 nm (black line) measured along the ZZ direction as a function of the delay between the two perpendicularly polarized FFs (illustrated in the inset). The blue line corresponds to a Gaussian fit of the autocorrelation curve, whereas the grey shaded area represents the noise level at ~3.5 fW of our experiments. b, The normalized SH intensities and noise level for all of the FF wavelengths used in our experiments. c, A double logarithmic plot of the emitted SH power as a function of the incident FF power along the AC direction. The grey dashed line is a guide to the eye, representing a slope of 1. perpendicular directions (AC and ZZ) with the same efficiency (at τ = ± delay, where τ is the FF optical cycle), to +45° with is a unique feature that arises from the combination of symmetry respect to the AC–ZZ directions (at delay). This result is τ = ± (2) and deep subwavelength thickness of TMDs, which relaxes the consistent with the theoretical SH polarization P generated by phase-matching constraints typical of harmonic generation. This an arbitrary elliptically polarized FF after a simple basis transfor- result could have an immediate application in background-free mation to account for the rotation by −45° with respect to AC/ZZ ultrabroadband and ultrashort pulse characterization. For instance, directions: in the most advanced commercial systems for ultrashort pulse characterization, one has to switch between collinear and non- (2) (2) 2 P = ϵ χ |E| ZZ ±i sin (2ϑ)AC , (4) collinear geometries to collect either the interferometric or the background-free intensity autocorrelation signals, respectively. By contrast, in our approach both signals are accessible using the same Here ϑ = 0° denotes a linearly polarized FF at 45° with respect to geometry and by simply switching the SH detection from AC to ZZ. the AC/ZZ direction, whereas ϑ = 45° corresponds to a circularly Further, following equation (3), the power scaling of the emitted polarized FF. This clearly shows that the SH component emitted SH along the ZZ direction is linear with respect to each of the FF along the AC direction oscillates with a period of as a function of intensities. This is confirmed by the power-dependent SHG mea- the FF polarization, by contrast to the SH emitted along the ZZ surements reported in Fig. 3c, where we show the emitted SH power direction. This underpins the interferometric precision required to along the ZZ direction at zero delay between the two FFs and as a fully capture the modulation along the AC direction. The exper- function of the AC-polarized FF power. imental results show a weak modulation also for the SH emitted To gain more insight into the temporal evolution of the emit- along the ZZ direction, although this is not expected from theory. ted SH polarization and amplitude, we scan the delay between the This could arise from weak strain (that is, below the limit detectable 39,56 two perpendicularly polarized FFs with interferometric precision by our SHG polarization measurements ), small deviations in and measure the emitted SH along both the AC and ZZ directions the alignment of the detection polarizer with respect to the AC/ZZ (2) (Fig. 4a). To control the delay between two perpendicular pulses with directions or from extra terms in the χ arising from the valley 48,57 the desired sub-optical-cycle precision, we used the common-path degree of freedom . Looking at Fig. 4c, one can indeed appreciate delay generator sketched in Fig. 4b and described in Methods. As that the SH is emitted only along the ZZ direction at zero delay expected, the SH power is emitted only along the AC direction for (FF at −45° with respect to AC/ZZ directions), whereas the emit- delays longer than our pulse duration, and no signal is detected ted SH components along AC and ZZ are identical at delay, as along the ZZ direction (Fig. 4a). Instead, for delays close to zero expected for circular polarization. we observe a strong ultrafast modulation of the SH power emitted along the AC direction. This can be better appreciated by looking Discussion at Fig. 4c, which shows the emitted SH power along the AC and ZZ In conclusion, we have demonstrated all-optical polarization switch- directions at 1,480 nm for delays between −10 fs and +10 fs. ing and amplitude modulation of SHG in MoS . Our approach sur- It is useful to note that, for delays much shorter than the pulse passes all previously reported electrical and all-optical attempts of duration, our interferometric measurement is the analogue of SH tuning in terms of modulation depth and speed, providing a 90° tuning the polarization of one FF pulse along the orthodrome of polarization switch, modulation depth close to 100%, and speed the Poincaré sphere (see the inset in Fig. 4a). This corresponds limited only by the FF pulse duration. Moreover, our method is to a rotation of the FF polarization from −45° with respect to the intrinsically broadband as it only relies on the crystal symmetry of AC–ZZ directions (at zero delay), to left/right circular polarization TMDs. We thus foresee a direct impact of our results on a variety NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics SHG power (fW) SHG power (normalized) SHG power (fW) NATurE PHoToNicS Articles a b ZZ e nn e o o AC –50 0 50 100 150 200 –10 –5 0 5 10 Delay (fs) Delay (fs) Fig. 4 | Phase-locked all-optical Sh modulation along the AC and ZZ directions. a, Second-harmonic power for an FF wavelength of 1,480 nm along the AC (blue) and ZZ (red) directions as a function of the relative delay between the two perpendicularly polarized FFs. The inset shows the Poincaré sphere with polarization directions on its orthodromes. The red line indicates the change in polarization when tuning the delay between the two FFs with a birefringent delay line. b, A schematic of the modified common-path delay generator. 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Appl. 5, 1–9 (2021). © The Author(s) 2021 NATure PhOTONiCS | VOL 15 | NOVeMber 2021 | 837–842 | www.nature.com/naturephotonics 842 NATurE PHoToNicS Articles Calculation of the second-order nonlinear susceptibility. The sheet SH tensor Methods (2) element χ can be calculated from the FF and SH average powers using the Polarization-dependent SH intensity. The vectorial components of the S (2) equation : second-order polarization P (2ω) for a material with D symmetry (such as 3h TMDs) are given by 3 2 c ϵ fπr t (1 +n ) P (2ω) (2) 0 FWHM 2 SHG   2 χ = √ , (5) E (ω) 2 2 16 2Sω P (ω) FF     E (ω)     y where c is the speed of light, ϵ is the permittivity of free-space, f = 76 MHz is the   (2) (2) 0 P (2ω) 0 0 000 χ   x xxy pump laser repetition rate, r ≈ 1.85 μm is the focal spot radius, t ≈ 200 fs is the   2 FWHM     E (ω)   (2)    (2) (2) FWHM of the pulse, n ≈ 1.45 is the substrate refractive index, S = 0.94 is a shape   2 P (2ω)   = ϵ  χ χ 0 000  y 0 yxx yyy       factor for Gaussian pulses, ω is the FF angular frequency, and P (2ω) – P (ω) are   SHG FF 2E (ω)E (ω) y z   (2) the SH and FF average powers, respectively. The effective bulk-like second-order P (2ω) 0 0 0000   (2)   2E (ω)E (ω) x z susceptibility χ can be subsequently calculated from equation (5) as   eff (2) (2) χ 18,46 χ = , where the thickness of MoS d is 0.65 nm (refs. ). 2 MoS 2E (ω)E (ω) d 2 x y eff MoS Pulse duration of the FFs. To prove that our method is solely limited by the (2) (2) (2) (2) where χ = −χ = −χ = χ . If we now consider a TMD oriented in pulse duration of the FFs, we performed a standard characterization of a temporal yyy xxy yxx profile of our OPO source at different wavelengths (Extended Data Fig. 1). For such way that the ZZ and AC directions lie along the x and y Cartesian coordinates, the measurements, we used a home-built autocorrelator based on a Michelson respectively, and we neglect the z (out-of-plane) direction, we obtain the following interferometer equipped with a motorized and computer-controlled translation expression: stage (HPS60-20X-M5, Optosigma). Two identical and temporally delayed replicas   ( ) (2) P (2ω) 2E (ω)E (ω) of the OPO pulse were then focused onto a 1-mm-thick beta barium borate crystal ZZ AC ZZ (2)   = ϵ χ (2ω,ω,ω) 0 (BBO-652H, Eksma Optics) in non-collinear geometry and the background-free (2) 2 2 E (ω) −E (ω) P (2ω) ZZ AC SHG autocorrelation intensity was detected on a silicon photodetector (DET10A2, AC 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 Finally, as the SH intensity is proportional to the absolute square value of the 220 fs, respectively. second-order polarization, we retrieve equations (2) and (3) shown in the main text: Data availability (2) 2ω 2 2 2 2 The data that support the plots within this paper and other findings of this study I = |P | ∝|E −E | AC AC ZZ AC are available from the corresponding author on reasonable request. Source data are (2) 2ω 2 2 I = |P | ∝|2E E | provided with this paper. AC ZZ ZZ ZZ r eferences 58. Preda, F. et al. Linear and nonlinear spectroscopy by a common-path SHG set-up. For the FF we used an OPO (Levante IR from APE) pumped by an ytterbium-doped mode locked laser (FLINT12, Light Conversion) with a birefringent interferometer. IEEE J. Sel. Top. Quant. Electron. 23, 88–96 (2016). 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 Acknowledgements replicas whose relative delay is tuned with two different approaches: a We acknowledge H. Rostami and F. Preda for helpful discussions. This work was computer-controlled motorized stage (M-414.2PD, PI) in a Mach–Zehnder supported by the European Union’s Horizon 2020 Research and Innovation programme interferometer configuration and a commercial common-path birefringent under Grant Agreement GrapheneCore3 881603 (G.S. and G.C.). This publication is part interferometer (GEMINI, NIREOS) . Compared with standard home-built of the METAFAST project that received funding from the European Union’s Horizon interferometers, the GEMINI provides subwavelength interferometric stability 2020 Research and Innovation programme under Grant Agreement No. 899673 (G.S. and with precise control on the delay between the two replicas with attosecond G.C.). We acknowledge the German Research Foundation DFG (CRC 1375 NOA project precision. The polarization of the FFs was tuned using a half-waveplate numbers B2 (A.T.) and B5 (G.S.)) and the Daimler und Benz foundation for financial (AHWP05M-1600, Thorlabs) and the power on the sample was controlled by support (G.S.). Open-access funding was provided by Friedrich-Schiller-Universität Jena. two polarizers (LPNIR050 and WP12L-UB, both Thorlabs). Finally, the two collinear and perpendicularly polarized FFs were focused on the sample using Author contributions a custom built microscope equipped with a ×40 reflective objective (LMM- S.K. and G.S. conceived the experiments. S.K. and O.G. performed the all-optical 40X-P01, Thorlabs). The backscattered SH signal is spectrally separated using modulation measurements. Z.G., A.G. and A.T. fabricated and provided the high-quality a dichroic mirror (DMSP950, Thorlabs), further spectrally purified by filters MoS sample. S.K., G.C and G.S. wrote the manuscript, with contributions from all (FELH0650, FESH850, FESH0950, all Thorlabs) and detected with a SPAD authors. All authors participated in the discussion and commented on the manuscript. (C11202-050, Hamamatsu). The SH polarization was measured using a wire grid polarizer (WP12L-UB, Thorlabs). Competing interests The authors declare no competing interests. Estimate of the optical losses of the set-up. To quantify the SH signal generated directly at the sample position, optical losses of the different components of the set-up must be considered. Although the transmission coefficients for the Additional information investigated SH wavelengths of the filters and the dichroic mirror (all >96%) Extended data is available for this paper at https://doi.org/10.1038/s41566-021-00859-y. were taken from the manufacturer’s website, the values for polarizers and Supplementary information The online version contains supplementary material the microscope objective were determined experimentally. A transmission of available at https://doi.org/10.1038/s41566-021-00859-y. ~79% was determined for the wire grid polarizer, whereas we determined a transmission of 50% for the reflective objective. Last, the responsivity of the Correspondence and requests for materials should be addressed to Giancarlo Soavi. SPAD was taken into account, which ranges depending on the investigated Peer review information Nature Photonics thanks Goki Eda and the other, anonymous, SH wavelength between ~17% and ~31%. In total, we estimated our optical reviewer(s) for their contribution to the peer review of this work. losses from the SH emission to the detector to be ~86–92%, depending on the Reprints and permissions information is available at www.nature.com/reprints. wavelength. NATure PhOTONiCS | www.nature.com/naturephotonics NATurE PHoToNicS Articles 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. NATure PhOTONiCS | www.nature.com/naturephotonics

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