Scalable functionalization of optical fibers using atomically thin semiconductors

Atomically thin transition metal dichalcogenides are highly promising for integrated optoelectronic and photonic systems due to their exciton-driven linear and nonlinear interaction with light. Integrating them into optical fibers yields novel opportunities in optical communication, remote sensing, and all-fiber optoelectronics. However, scalable and reproducible deposition of high quality monolayers on optical fibers is a challenge. Here, we report the chemical vapor deposition of monolayer MoS2 and WS2 crystals on the core of microstructured exposed core optical fibers and their interaction with the fibers' guided modes. We demonstrate two distinct application possibilities of 2D-functionalized waveguides to exemplify their potential. First, we simultaneously excite and collect excitonic 2D material photoluminescence with the fiber modes, opening a novel route to remote sensing. Then we show that third harmonic generation is modified by the highly localized nonlinear polarization of the monolayers, yielding a new avenue to tailor nonlinear optical processes in fibers. We anticipate that our results may lead to significant advances in optical fiber based technologies.

of optical resonators is a widely used method to enhance the light matter interaction 3-8 . It is, however, naturally limited to narrowband resonances, while broadband, ultrafast operation cannot easily be implemented. On the other hand, by integrating TMDs on waveguides or optical fibers the interaction length can be enhanced greatly in a broadband, non-resonant manner 9 . The resulting 2Dfunctionalized waveguides (2DFWG) can utilize the optical properties of TMDs via interaction with parts of the evanescent fields of the guided modes. 2DFWGs can show remarkable features, leveraging from, e.g., the nonlinear or excitonic properties of the TMDs. Previous attempts to fabricate 2DFWG rely on mechanical transfer of exfoliated TMDs onto the waveguides or optical fibers [10][11][12][13][14] . However, this approach is prone to induce uncontrollable stress fields and lacks reproducibility and scalability. Thus, such methods are hardly suitable for future large-scale integration. A process to grow high quality monolayer TMDs directly on optical fibers or waveguides is therefore required to take 2DFWGs to the next level. We tackle this challenge by directly growing monolayer TMDs on the guiding core of all-silica microstructured exposed core optical fibers (ECFs) 15 , turning them into 2DFWGs, in a scalable process. Specifically, we show the growth of monolayer MoS 2 and WS 2 crystals on the guiding core of all-silica ECF and investigate their interaction with the evanescent fields of highly confinement guided modes. The growth is facilitated by a modified chemical vapour deposition (CVD) process [18], yielding interspersed monolayer TMD crystals of high quality with a typical length of 30 on ECFs with a length of a few centimetres. Our scalable technique paves the way for 2DFWGs as a new tool for integrated optical architectures, active fiber networks, distributed sensing, and photonic chips.
We highlight the possible functionalities of our 2DFWGs in two case studies. The first demonstrates infiber excitation and collection of exciton-driven photoluminescence (PL) which may pave ways towards future experiments in excitonics and remote fiber-based sensing schemes. The second is focussed on the way the highly nonlinear TMD coating modifies the nonlinear wave dynamics in ECFs, by investigating enhanced third harmonic generation (THG). In general, this shows that 2DFWGs can be used to enhance and tailor the nonlinear response of integrated wave systems, without any modification to the guided modes themselves, leading to new applications in nonlinear light conversion and optical signals processing. The overall concept of both of our experiments is displayed in Fig. 1 (a). The ECFs have been coated with MoS 2 and WS 2 crystals on the entire grooved surface, which also forms the upper surface of the ECF's core. A laser is coupled into the fundamental mode (FM) of the ECF, which interacts with the TMDs via the evanescent part of the mode. The resulting polarization; e.g., PL or third harmonic (TH) light, is coupled back into the fiber modes or into free space, where it can be collected for further analysis. An optical microscopy image of the coated exposed side of the ECF is given in Fig. 1 (b), showing high quality MoS 2 crystals. The focal plane of the image is chosen to coincide with bottom of the groove running along the entire 60 mm length of the coated ECF, which is also the top of the exposed core.  Fig. 1 (c) displays a cross sectional scanning electron microscopy (SEM) image of the ECF's core area (an SEM image of the entire ECF cross section is provided in Fig. Supp. 2 (a)). The core is suspended by three struts of pure silica to a cladding structure. The upper boundary of the core forms the bottom of a groove, which is running down the entire length of the ECF. When placed in the CVD reactor 16 the upper boundary of the core is thus completely exposed to the CVD reactants. Thus, as monolayer TMD crystals are grown on the entire surface of the ECF, they are also grown on the exposed surface of the core. Their lateral size, distribution and thickness can be tuned in the growth process. After careful optimization, monolayers are almost grown exclusively, as can be seen from the inset of Fig. 1 (b), which displays a typical Raman spectrum of the MoS2 crystals on the ECF showing characteristic spacing between the Raman modes for monolayers 17 . Examples of alternative growth modes together with their Raman spectra are displayed in Fig. Supp. 2 (c-e).
The ECF's core has a diameter of ~2 µm and supports two nondegenerate FMs, which are mostly polarized along the x and y direction. However, the x-polarized FM, were the polarization is aligned parallel to the coated surface, has a better field overlap with the TMD layer (see Tab. Supp. 1) and its polarization is aligned with the large (3) -components of the TMDs nonlinear tensor 18 . Hence, all experiments and simulations are carried out with the x-polarized FM. Its field distribution was calculated numerically (more details in Supplement 5) and is shown for a wavelength 0 = 1570 nm in Fig. 1 (d). Because of the small size of the core, the FM is well-confined with an effective area of approximately 3.5 μm 2 at 1570 nm. A fraction of 1.6 % of the electromagnetic energy is flowing in the air region above the ECF's core, which can thus interact effectively with the TMD crystals.

Guided-wave Photoluminescence
Next, we verify the location and PL activity of the TMD crystals, grown on the curved ECF core by performing a cross sectional PL emission mapping. The PL map displayed in Fig. 1 (e) is superimposed on a cross sectional SEM image of the ECF for easier understanding. Illumination and collection of PL light were sideways through the groove of the uncut ECF. We note that the PL follows the outline of the ECF grove, which indicates that the TMD crystals have grown in direct contact with the entire surface of the ECF. Note that the part of PL light extending downwards from the center of the image is caused by the diffraction at the ECF core and its interaction with the confocal setup and does not indicate the presence of TMDs within the core. A PL map along the propagation direction and a PL spectrum can be found in Fig. Supp. 3.
We now focus on guided wave excitation of PL in 2DFWGs, which is mediated by excitation and decay of excitons in the TMD coating. Excitons in TMDs 19 are particularly appealing because they exhibit spin valley coupling 20-23 and are important for the emission of single photons 24-28 . We couple an uncontrolled polarization green laser ( = 532 nm) into the ECF, which excites excitons via the evanescent field of the FM (for experimental setup is see Fig. Supp. 1.). PL from the TMD monolayers is either emitted into free space or coupled back into the ECF's mode. Remission into guided modes has been observed by imaging the end facet of the fiber and by measuring the spectrum. The results are displayed in Fig. 2 (a) and (b). The image of the PL at the end facet of the ECF, is long pass filtered to remove residual laser radiation and then displayed in Fig. 2 (a). For better orientation, a microscope image of the ECF itself is superimposed. The PL light is clearly emitted from guided modes at the core of the ECF. This light is then analysed spectroscopically, with exciton peaks at 678 nm (MoS2) and 622 nm (WS2) being recorded, respectively 16 . This means that the evanescent field of guided modes can be used to both excite and collect PL 29-31 , making 2DFWGs, such as TMD coated ECFs, highly interesting for integrated excitonics and remote sensing applications.
Lateral emission into free space was observed with a camera mounted sideways, imaging the bottom of the ECFs groove. We attain compound images of the distribution of PL over a substantial section of the ECF and thus an image of PL active TMD crystals. One image such of an MoS2 coated fiber is displayed in Fig. Supp. 4 (a). From this, we can extract the distribution and cumulative length of monolayer crystals on the ECF. For this specific sample we observe 39 distinct MoS2 crystals with an average length of 25 µ per crystal and a filling factor of 5.4%, although the coverage and crystal size have been significantly increased in later batches after further optimization of our growth procedure.
A transmission spectrum through the ECF is displayed in Fig. 2

Enhancement of Third Harmonic Generation
2D TMDs are also highly interesting because of their strong nonlinear optical response per unit thickness 32 . For third order processes this is quantified by the nonlinear refractive index 2 with a reported value of 2 MoS 2~2 .7 ⋅ 10 −16 m 2 /W for TMDs transferred on waveguides 14 . At roughly four orders of magnitude more than that of silica it is extremely large, although lower values have been reported on planar substrates 18 . Thus, a TMD coating may have a substantial contribution to nonlinear effects in ECFs, although less than 10 −4 of the energy flow of the FM is localized in the TMD at any wavelength (see Fig. Supp. 5 (e)). The influence on the nonlinearity can be quantified by calculating the respective contributions of the MoS2 coating and the SiO2 core to the overall self phase modulation coefficient = MoS 2 + SiO 2 33 . Indeed, we find MoS 2 > SiO 2 for wavelength in excess of 1470 nm (see Supplement 6 and Fig. Supp. 5(g)), i.e. the nonlinear contribution of the TMD coating dominates for long wavelengths. Note, that even larger MoS 2 may be obtained in the future by optimizing the field overlap of the FM with the TMD coating, opening e.g. the path for TMD enhanced supercontinuum generation experiments.
While many third order nonlinear processes are observed in fibers, THG is particularly fascinating, because it relies on the simultaneous interplay of nonlinearity, mode matching, and phase matching (PM). We found that there is no appreciable modification of the PM by the TMD coating, because all linear mode properties, except loss, are unaffected by the TMD coating (see Fig. Supp. 5). The ECFs of the design used here, exhibit PM only for higher-order TH modes (HOMs) at a TH wavelength of roughly 550 nm, corresponding to a fundamental wavelength of 1650 nm 34-36 .
To show that THG is indeed enhanced we excited TH with a pulsed laser operating at 0 = 1570 nm and a pulse duration of 32 fs (see Fig. Supp. 6 for pulse characterization 37 ). Fig. 3 (a) displays the TH spectrum for three different input energies for a bare and an MoS2 coated ECF. We consistently observe more THG in the MoS2 coated ECF, which signifies that the TMD coating does enhance the THG process. This is particularly noteworthy, as the MoS2 coated ECFs experience roughly 60% linear loss over the length of the ECF and the comparison was made for equal input energy. As dictated by PM we observe TH not at exactly a third of 0 but in a spectral band ranging from 540 to 560 , marked in Fig. 3 (a). The fundamental wave (FW) spectrum must thus first nonlinearly broaden (more details in Fig. Supp. 7) into a THG-relevant subband between 1620 nm to 1680 nm before TH is generated. This explains the somewhat stronger-than-cubic scaling in the inset Fig. 3 (a). The similarity of TH spectrum for both ECF types reassures us that the phase matching (PM) between FW and TH is indeed unaffected by the MoS2 coating.
Because a PM modification is thus ruled out, the observed TH enhancement must be related to the process of nonlinear light generation itself. This process is driven by the nonlinear polarization field THG ( 0 3 ) = 3 ( , ; 0 ) ⋅ (3) ( , ) . Here, ( , ) is the x-component of the electric field of the FW mode an (3) ( , ) is the dominating element of the nonlinear tensor of the THG process for the TMD coating and the silica core, respectively. The shape of THG is displayed in Fig. 3 (b) in logarithmic scaling. There are two major contributions: the spatially smooth nonlinear polarization from the SiO2 core and the strong but highly localized contribution of the TMD coating, better visible in the inset.
The so-generated TH radiation is distributed onto the TH modes, the magnitude of which is described by an overlap coefficient (THG) for every TH mode (see supplement). Thus, the addition of the TMD coating does not enhance the nonlinear interaction for all HOMs equally but it boosts those, which are localized close to the surface and with predominant x-polarization. This mode-selective nonlinear enhancement is quantified in Fig. 3 (c), displaying (THG) values for all TH HOMs close to the PM point.
The PM region contains 11 HOMs and is marked by the shading (see supplement for determination of PM bandwidth Δ ). The contribution of SiO2 and MoS2 are marked in different colours. While (THG) grows for all HOMs, it does so very differently from mode, i.e. the enhancement is mode-selective.
To confirm this model and to compare with the experiment, we only discuss the three HOMs with the largest (THG) marked with 1 to 2 in Fig. 3 (c). The most dominant HOM is the 1 mode at 1 eff = 1.379. 1 (THG) is approximately doubled due to the MoS2 coating. The second strongest contribution for the bare ECF comes from the 2 mode at 2 eff = 1.366, 2 (THG) increases by ~2.5 upon application of the TMD coating. Upon coating; however, it is superseded by the 3 mode at 2 eff = 1.383, which almost quadruples its 3 (THG) . The modes profiles of 1 , 2 , and 3 can be found in Fig. Supp. 8.
The mode-selective enhancement of the overlap coefficients is reflected in spatial distribution of the TH light as seen in images recorded by a camera, focussed to output plane of the ECFs, displayed in Fig. 3 (d) and (e), for the bare and the MoS2 coated ECF, respectively. The single peak at the top of bare the ECF is replaced by a broader and weaker triple peaked distribution for the MoS2 coated ECF. Moreover, the field is less localized and extends further into the bottom strut for the MoS2 coated ECF. Qualitatively, both distributions can be reproduced by a simple superposition of the 1 , 2 and 3 modes, displayed in Fig

Summary
In summary, we have shown that high quality crystalline monolayer TMDs, e.g. MoS2 and WS2, can be grown directly on the core of microstructured exposed core fibers in a scalable CVD process. This process functionalizes the optical fibers, creating a new platform to investigate and utilize the electrooptic properties of 2D TMDs. Excitonic and nonlinear functionalization is demonstrated in two case studies. First, we excite and collect excitonic photoluminescence from monolayers in the optical fiber, which may give access to, e.g. remote sensing schemes and provide a new platform to investigate excitonic effects. We have also demonstrated that 2D materials modify nonlinear optical processes intricately by investigating mode-selective enhancement of third harmonic generation. This may will enhance the design freedom for highly nonlinear guided wave systems and may be utilized in nonlinear fiber devices. Altogether the direct growth of 2D materials on waveguides is opening a novel path towards the scalable and reproducible functionalization of waveguides, fibers, and other integrated optical systems.

ECF fabrication:
The fused silica ECF was fabricated using an ultrasonic drilled silica preform that was then cut open on one side to expose one part of the central section of the fiber. The preform was then caned and inserted into a jacket tube, which is then drawn into an ECF. Active pressurization was used during the draw on the central cane piece, to stabilize the structure 38 . The fiber has an outer diameter of 220 μm [ Fig. 1(a)] and an effective core diameter just above 2 μm.

CVD growth of TMDs on fibers:
MoS2 and WS2 crystals were grown on the ECF (fixed on a quartz holder) by a modified CVD growth method in which a Knudsen-type effusion cell is used for the delivery of sulfur precursor 16 . The grown TMDs on the ECFs were initially characterized using optical microscopy (Zeiss Axio Imager Z1.m) and Raman spectroscopy (Bruker Senterra spectrometer operated in backscattering mode using 532 nm wavelength obtained with a frequency doubled Nd:YAG Laser, a 100x objective and a thermoelectrically cooled CCD detector.) Microscopic PL mapping and spectroscopy: Photoluminescence mapping was carried out with a commercial confocal PL lifetime microscope (Picoquant Microtime 200), with an excitation laser operating at 530 nm. The maps, where created by moving the sample along the focus of the microscope's objective, which had a magnification of 64x. The resulting spatial resolution is estimated be in the range of 500 nm. Detection of the PL signal was carried out with an avalanche photodiode. Alternatively, the PL microscope was connected to grating spectrometer (Horiba Jobin Yvon Triax) equipped with a cooled CCD detector to measure PL spectra.
In-fiber and transverse PL mapping and spectroscopy: To record PL and transmission spectra, the incoming light was focused into the fiber core. For PL spectroscopy the outcoming light was passed through two of 550 nm long-pass filters and then imaged into onto a spectrometer (Horiba Jobin Yvon Triax), with a cooled Si-CCD-detector. PL was excited with a 532 nm laser (lighthouse Photonics Sprout), whereas transmission spectra have been excited with a white light diode. Alternatively the light was imaged onto a CMOS camera (Zyla 4.2 sCMOS), to image the PL at the output facet. The camera was alternatively mounted laterally together with a 10x objective imaging the sideways emission of PL from these crystals. Again, a set of 550 nm long-pass filters was used to reject scattered light from the excitation laser. The camera and the objective had been mounted on a motion stage to map larger sections of the fiber side.

THG measurements:
Nonlinear experiments were carried out with a femtosecond laser emitting pulsed with a duration of 32 fs at a wavelength of 1570 nm at a repetition rate of 80 MHz (Toptica FemtoFiber pro IRS-II) and focussed into the ECF with an aspheric lens. Light leaving the fiber was collimated with a microscope objective and coupled into two optical spectrum analysers, one for the measurement of the FW and the other for the measurement of the TH spectra.

Supplementary Information Experimental Setup
A sketch of the experimental setup is depicted in Fig. Supp. 4. To record PL, and transmission spectra, the incoming light was focused into the fiber core using a 40X microscope objective. An identical objective was used to collimate the light, leaving the other end of the fiber. For PL the outcoming light was passed through two consecutive of 550 nm long-pass filters, to remove residual laser radiation. It was then refocused into the entrance of a multimode fiber with 200 µm diameter connected to a HORIBA spectrometer, with a cooled Si-CCD-detector. The spectrometer was used to measure both PL and transmission spectra. PL was excited with a 532 nm laser, whereas transmission spectra have been using excitation with a fiber coupled white light diode.
The fiber and all objectives have been mounted on 3-axis translation stages, to optimize light coupling and imaging. A CCD camera was placed behind the setup with a flip mirror, with a tube lens. This way the output facet could be observed, which was used to obtain a good coupling into the fundamental mode for PL and transmission and further used to obtain the modal images of the THG field. It was replaced by an InGaAs-camera to optimize the incoupling for the THG experiments.
A highly sensitive Zyla 4.2 sCMOS camera and a 10x objective was utilized to determine the distribution and size of MoS2 crystals on the fiber, by imaging the sideways emission of PL from these crystals. A set of 550 nm long-pass filters was used in this case to reject scattered light from the excitation laser. The camera and the objective had been mounted on a motion stage to map larger sections of the fiber side. To demonstrate the THG in the ECF, the pump beam was focused into the ECF using an aspheric lens (Thorlabs C230TMD-C), instead of the microscope objective for better IR focussing. The incident power and polarization were adjusted by a combination of half wave plate and polarizer. The output was imaged using a 40X microscope objective and an IR camera to confirm the coupling into the fiber core. The output was coupled into a graded index multimode fiber (1mm of diameter) connected to an optical spectrum analyser (ANDO 6315A). The device has a noise level of -80 dBm for the range 350-1750 nm. All signals should be above this detection limit.

Fiber Geometry and Raman Spectra of TMD on ECFs
ECFs are microstructured optical fibers of the suspended core type, i.e. they consist exclusively of glass with air insertions that run along the length of the fibers. The air insertions form a silica core, which is suspended by three (or more) thin glass struts. ECFs 15 are a special subclass of such suspended core fibers in that one of the air insertions is massively enlarged to point where it entire fact of the fiber is exposed to the outside world, i.e. it forms a trench which runs along one end of the fiber. SEM images of the fiber cross section and a zoom into the core area are depicted in Fig. Supp. 2 (a) and (b), respectively.
The ECFs have been placed in a CVD chamber with the trench being exposed to the gas flow of the reactor 16 . As a result, TMD crystals are grown along the entire surface of the fiber, and also in the trench. Some of the crystals grow on the exposed side of the core and can thus interact with the fibers modes. Depending on the specific growth conditions, both single layer, as well as multi layer crystals have been observed after completion of the growth process. Images of such crystals are displayed in Fig. Supp. 2 (c) and Fig. Supp. 2 (e), together with Raman spectra, which underline the respective nature of the two types of crystals in Fig. Supp. 2 (d). Note, that by optimization of the growth parameters, we eventually were able to fabricate fibers, which are almost exclusively coated with single layer crystals.

Fig. Supp. 2: (b) SEM cross section of the entire an ECF. The groove running along the length of the fiber is visible at the top. (b) SEM cross section of the core area of the ECF located at the orange box in (a). (b) Microscope image of a monolayer MoS2 crystal CVD grown on the exposed surface of the core of an ECF. (d) Raman spectra of the monolayer crystals displayed in subfigures (c) and (e). (e) Microscope image of a multilayer MoS2 crystal CVD grown on the exposed surface of the core of an ECF.
External PL Mapping and Spectra of MoS 2 on an ECF PL was initially tested with a commercial confocal PL microscope using a 530 nm laser for excitation. The microscope has a spatial resolution in order of 500 nm. The fiber was placed flat on the specimen table and thus perpendicular to the optical path. An x-y-map (parallel to the specimen table and along the fiber) was first measured to determine the location of a PL active monolayer crystal. It is displayed in Fig. Supp. 3 (a).
Due to the confocality of the setup and the curved nature of the ECF's core surface, we only map PL from material close to the ECF core itself. Spectra were then measured from various spots along the length of the fiber, with a typical result recorded below. A double gaussian was then fitted for the MoS2 spectrum, as done in Fig. Supp. 3 (b) to judge the relative contribution of the fundamental exciton and the trion to the overall PL. Overall, it was found that both the of the peaks, their width and their relative strength all coincide with the values for high quality CVD grown TMDs on planar surfaces. We then measured a cross section PL map, by scanning perpendicular to the fiber. The results are reported in the main text of the paper. We note that the TMDs have indeed grown confocally to the curved inside surface of the ECF, with no appreciably loss of PL quality confirmed by the measurement of further PL spectra.

Determination of the Distribution and Size of MoS 2 crystals on a Fiber Section
Using the technique described in the experimental setup we could image PL light emanating from the ECF sideways with a resolution of approximately 10 µm. Translation of the objective and camera with respect to the fiber allowed us to ascertain a set of images for a substantial part of the fiber. Mechanical limitations however prevented us from imaging the entire length of the fiber, particuarily the first and the last part of the fiber are difficult to access, simply due to the mechanical space required by the incoupling and outcoupling objectives.
After recording of the images a compound PL image, such as the one displayed in Fig. Supp. 4 (a). was generated and a statistical analysis of the acquired dataset was carried out. A threshold just above the noise level of the camera was set and each connected spot with values above this threshold was considered to be a PL active monolayer crystal. Using the data, we could map both total light emitted by each crystal together with their size (e.g. length), as displayed in Fig. Supp. 4 (b). The linear connection between the two quantities suggest that each crystal has a comparable brightness per unit length and thus we conclude that their material quality does not change from crystal to crystal.

Fundamental Mode Properties of the ECF
Dispersive and nonlinear properties of the coated sections of the ECF were calculated from the geometry measured by the SEM displayed in Fig. Supp. 2 (a) and (b) with a high-index layer of a thickness of 0.65 nm superimposed onto the upper surface of the ECF core. Modal properties have been obtained Finite using element simulations (COMSOL v5.4). Scattering boundary conditions were applied to analyse FM and HOMs. As a result of not fully symmetric geometry, the first two modes are non-degenerate and almost mutually orthogonally polarized. Power flow distributions, i.e. the zcomponent of the Poynting vector, of the fundamental mode is displayed at three different wavelengths in Fig. Supp. 5 (a-c).
The refractive index of SiO2 was calculated directly by COMSOL as a function of frequency using the Sellmeier equation. The refractive index of MoS2 was taken from 39 ,as displayed in displayed Fig. Supp. 5 (f). while the refractive index of air was set to be 1.0. In this simulation, the ECF core was established as a lossless material, so the damping of the travelling light, as displayed in Fig. Supp. 5 (d)) is caused by the high extinction coefficient of MoS2 monolayer.
Propagation loss was also confirmed experimentally. For the bare ECFs, we achieve a coupling efficiency of 29% at = 1570 nm, with no appreciable propagation loss (less than 1 dB/m) 38 . For the MoS2 coated ECF, we assume the same coupling efficiency. However, the transmitted power is roughly 60% lower at otherwise identical settings. The total fiber length was 60 mm, so that we measure an overall net loss of 0.1 dB/mm. Using the filling factor of 5.4% established in Supplement 4, this means that the MoS2 coated sections of the ECF are responsible for a loss of ~1.8 dB/mm. This value is fairly consistent with the simulated values in Fig. Supp. 5(d) 2, which predict a value of 1.2 dB/mm and an independent cut back measurement, which yielded 1.3 dB/mm. . 5 (g) were then calculated according to the procedure outlined below. It can be seen that for longer wavelengths at > 1450 nm, the TMD's contribution to the SPM coefficients is larger than that of the silica core.  .7 ⋅ 10 −16 m 2 /W have been reported on TMDs transferred on waveguides, indicating that the specific value of 2 may depend on the geometry and substrate material 14 , particularly in 2DFWGs. In our work we have used the latter, higher value, because of the similarity of the geometry in 14 with our ECFs and the consistency with our experimental findings.

Cross-Sectional part of the ECF
Nonlinear effects in guided mode systems, however, do not only depend on the material nonlinearity but in the specific shape of modes and the distribution of electromagnetic energy over the nonlinear materials in the is mode. Coefficients for Self Phase Modulation and THG have been calculated from overlap integrals, which take the highly vecorial nature of the mode into account.
For SPM we have resorted to the method developed in 33 , which also accounts for the propagation losses induced by the TMDs. There are two distinct contributions to the overall SPM coefficient , with are related to the two sections of the geometry filled with SiO2 and TMD, respectively. The coefficient is calculated from their sum, i.e. = SiO 2 + MoS 2 and the contributions are computed according to: Where is the material index (SiO2 or MoS2), is the cross sectional area occupied by this material indicated by the material index (i.e. either SiO2 or MoS2), ( ) and 2 are material specific linear and nonlinear refractive indicies, where the dispersion of the linear index is taken into account.
( , ; ) and ( , ; ) are the electric and magnetic field of a particular mode propagating inside the fiber at a given wavelength , and 0 and 0 are the vacuum permittivity and permeability, respectively. ̂ is the unit vector along the propagation direction.
We use the SPM coefficients to judge if the overall contribution of the TMD coating to the nonlinear effects shown in the fiber are substantial at all. Indeed we find that the TMD coating has a substantial impact on the ECF's nonlinear properties. At 0 = 1570 nm, we find that, the overall contribution of the TMD to the self phase modulation coefficient = MoS 2 + SiO 2 with MoS 2 ( 0 ) = 0.046 (Wm) −1 is slightly higher than that of SiO 2 ( 0 ) = 0.037 (Wm) −1 33 . The evolution of both contributions as a function of the wavelength is given in Fig. Supp. 5 (g). Here we find that for longer wavelegths > 1450 nm, the TMD contribution is stronger than that of the silica core. For shorter wavelengths this is note the case. We attribute this to the contraction of the fundamental mode, as the wavelength is decreased and the consecutive reduction of the energy propagating in the TMD coating, as displayed in Fig. Supp. 5 €. This means that the TMD coating may be used as a highly selective method to enhance the nonlinearities and tailor nonlinear 2DFWGs, while leaving linear properties of the 2DFWGs virtually unchanged.
The overlap coefficient of the THG is likewise a sum of the material specific contributions (2) Where the subscripts denote the polarization direction of the electric field and , is the electric field of the third harmonic higher order mode with mode index and , , and denote the fundemntal mode at the fundamental wavelength. Because the fields are mostly polarized in anddirection and there is no appreciable value for the out-of-plane nonlinear tensor of the TMD, the equation simplifies to: (3)

Laser Pulse Characteristics for the Excitation of Third Harmonic
The pulse duration was estimated by measuring the spectrum and the non-collinear intensity autocorrelation (AC) of the pulse incident to the fiber (see Fig. Supp. 6 (a) and (b)). The measured AC was compared to the AC of the transform-limited pulse displayed in Fig. Supp. 6 (c) as calculated from the measured spectrum. Both are found in excellent agreement in the central peak region as well as in good agreement in the wings. We conclude that the FWHM pulse duration is close to the transformlimit of 30 fs. The final estimate for the pulse duration of 34fs at the fiber facet was obtained by further numerical propagation through the used aspheric lens.

HOM Third Harmonic Generation
Third harmonic generation (THG) was excited by a femtosecond laser emitting at 0 = 1570 nm. It was observed with a spectrometer according to the procedure described above. Spectra for three different input power levels, for a bare and an MoS2 coated ECF are displayed in Fig. Supp. 7 (a) and in the main text. Note that for the MoS2 coated ECF we have inferred the input power from the output power and the previously inferred loss per unit length of the fiber. This means that in fact for the MoS2 coated ECF there is less pulse energy available to drive the THG process as it continuously drops along the length of the fiber. Nevertheless, we observe a 47% increase in THG for an input energy of 0.59 nJ, 41% for 0.67 nJ and 18 % for 0.70 nJ input energy, as indicated by the ratio THG = MoS 2 THG / Bare THG , displayed in the first row of Fig. Supp. 7 (c).
The TH light does not correspond with a third of the wavelength of the FW. In fact, most of the output spectrum is centered at wavelengths between 540 and 560 nm, which corresponds to a FW wavelength of 1520 to 1580 nm. Both ranges are marked with orange bands in Fig. Supp. 7 (a) and (b). In this spectral range some higher order modes of the TH wavelength are phase matched with the FM, i.e. if the effective refractive index of the FW mode FW and of the TH mode TH , fulfils the inequality Δ = | FW − TH | < 0 /( c ). Here is the number of the higher order mode (HOM). c is the characteristic length of the system, i.e. the typical length of an MoS2 crystal, which is 34.7 ( Fig.  Supp. 4), yielding Δ = 0.012. For our ECFs only HOMs with ≫ 1 are phase matched.
Because of this mismatch of laser wavelength and PM band, any THG must be precipitated by a spectral broadening of the FW into this band, via nonlinear processes, which explains the higher than cubic power scaling of the THG process displayed in the inset of MoS2 coated Fig. Supp. 7 (a).
The broadening process of the FW into the THG relevant sub-band differs somewhat for the MoS2 coated and bare ECFs. While a more systematic investigation of spectral broadening in MoS2 coated ECFs is subject to further investigation, we here resort to estimate its impact on the THG efficiency only. We therefore measure the energy in the THG generation sub band of the FW and compare these values between MoS2 coated and bare ECFs. The corresponding ratio FW is calculated as Using these two ratios and the fact that the THG efficiency scales with the cube of the FW energy and linearly with the TH energy, we can estimate the relative efficiency of the THG process of the MoS2 coated ECFs vs. the bare ECFs as THG ⋅ FW 3 . The results are displayed in the bottom row of Fig. Supp. 7(c). They indicate a corrected enhancement factor of 30% for = 0.59 nJ, 53% for = 0.67 nJ, and 67% for = 0.70 nJ, respectively. This corrected enhancement factor is consistently higher than the uncorrected one and underlines that the THG process is fundamentally enhanced by the TMD coating. The value of the overlap coefficient for the THG process depends critically on the shape of the specific HOM mode, or more specifically their x-polarized components. In the phase matched regions we have identified three modes which have a large overlap coefficient and which contribution to the experimentally recorded picture of the TH modes.
The modes have been designated as 1 , 2 and 3 in the main text. Their shape, determined from the FEM simulations discussed above, is displayed below in Fig. Supp. 8 (a-c). It can be seen that each of the modes does have at least one field maximum close to the center of the top surface of the ECF. It is thus obvisous that each of these three mode's overlap coefficients does growth considerably upon the application of the TMD coating.