# Nonlinear optics in SiN waveguides with integrated 2D MoS 2 films

David Moss

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David Moss. Nonlinear optics in SiN waveguides with integrated 2D MoS 2 films. 2025. hal-05356106

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# Nonlinear optics in SiN waveguides with integrated 2D MoS₂ films

David J. Moss

¹Optical Sciences Centre, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
E-mail: dmoss@swin.edu.au

**Keywords:** Integrated photonics, 2D materials, nonlinear optics.

## Abstract

Significantly enhanced spectral broadening induced by self-phase modulation (SPM) is experimentally demonstrated in silicon nitride (Si₃N₄) waveguides integrated with 2D monolayer molybdenum disulphide (MoS₂) films. Monolayer MoS₂ films with ultrahigh optical nonlinearity are synthesized via low-pressure chemical vapor deposition (LPCVD) and subsequently transferred onto Si₃N₄ waveguides, with precise control of the film length and placement achieved by selectively opening windows on the chip upper cladding. Detailed SPM measurements are performed using fabricated devices with various MoS₂ film lengths. Compared to devices without MoS₂, improved spectral broadening of optical pulses is observed for the hybrid devices, achieving a broadening factor of up to ~2.4 for a device with a 1.4-mm-long MoS₂ film. Theoretical fitting of the experimental results further reveals an increase of up to ~27 fold in the nonlinear parameter (γ) for the hybrid waveguides and a Kerr coefficient (n₂) of MoS₂ nearly 5 orders of magnitude higher than Si₃N₄. These results confirm the effectiveness of on-chip integration of 2D MoS₂ films to enhance the nonlinear optical performance of integrated photonic devices.
# 1. Introduction

As a fundamental third-order ($\chi^{(3)}$) nonlinear optical process, self-phase modulation (SPM) occurs when high-peak-power optical pulses propagate through a nonlinear medium, where the Kerr optical effect induces an intensity-dependent refractive index change that results in phase modulation and pulse spectral broadening [1-3]. As a key all-optical modulation mechanism, SPM has found wide applications in realizing broadband optical sources [4, 5], optical spectroscopy [6, 7], optical logic operations [8, 9], pulse compression [10, 11], optical diodes [12, 13], optical modulators [14, 15], and optical coherence tomography [16, 17].

Realizing SPM based on photonic integrated circuits (PICs) offers attractive advantages such as compact device footprint, high stability and scalability, and cost-effective mass production [18-21]. Benefiting from excellent compatibility with well-developed complementary metal-oxide-semiconductor (CMOS) fabrication technology, silicon has been a dominant material platform for PICs [22, 23]. Despite having a Kerr coefficient $n_2$ over an order of magnitude higher than silicon nitride ($Si_3N_4$) [24], the strong two-photon absorption (TPA) of silicon at near-infrared wavelengths poses a fundamental limitation for its nonlinear optical performance in the telecom band [25]. As alternatives, other CMOS-compatible material platforms such as $Si_3N_4$ and high-index doped silica glass (Hydex), which exhibit negligible TPA in the telecom band, have been widely studied [26, 27]. Nevertheless, their relatively low Kerr nonlinearities ($n_2 \sim 2.6 \times 10^{-19} \text{ m}^2/\text{W}$ for $Si_3N_4$ and $n_2 \sim 1.3 \times 10^{-19} \text{ m}^2/\text{W}$ for Hydex, both over an order of magnitude lower than silicon [28, 29]) remain a limiting factor for the efficiency of nonlinear optical processes in these devices [30, 31].

To address the limitations of existing photonic integrated platforms, the incorporation of two-dimensional (2D) materials with ultrahigh optical nonlinearity has emerged as a promising
strategy. The remarkable Kerr nonlinearity of 2D materials such as graphene [31-34], graphene oxide (GO) [35-39], molybdenum disulphide (MoS₂) [40-44], and tungsten disulfide (WS₂) [45] has been extensively studied and leveraged to implement nonlinear optical devices that overcome performance barriers and offer new capabilities.

Recently, we reported the synthesis of high-quality 2D MoS₂ films with precise control over defect formation [46, 47]. In this work, we integrated as-prepared 2D MoS₂ films onto Si₃N₄ waveguides and experimentally demonstrate significantly improved SPM performance for the hybrid devices. Due to strong mode overlap between the Si₃N₄ waveguides and the MoS₂ films with ultrahigh Kerr nonlinearity, the hybrid waveguides exhibit substantially enhanced spectral broadening of optical pulses after propagating through them. High-quality monolayer MoS₂ films are synthesized via low-pressure chemical vapor deposition (LPCVD) and integrated onto Si₃N₄ waveguides through a polymer-assisted transfer process. Window opening in silica upper cladding enables precise control over the film length and placement on photonic integrated chips. We performed detailed SPM measurements for fabricated devices with various MoS₂ film lengths and at various pulse peak powers. Results show that the hybrid waveguides exhibit improved spectral broadening of optical pulses compared with bare Si₃N₄ waveguides, achieving a maximum broadening factor (BF) of ~2.4. By fitting the experimental results with theory, we obtain an improvement of up to ~27 times in the nonlinear parameter (γ) for the hybrid waveguides, as well as a Kerr coefficient (n₂) of MoS₂ that is about 5 orders of magnitude higher than Si₃N₄. These results highlight the strong potential of on-chip integration of 2D MoS₂ films for enhancing the nonlinear optical performance of Si₃N₄ devices.
## 2. Device design and fabrication

**Figure 1(a)** illustrates the atomic structure of a MoS₂ monolayer, where a hexagonal sheet of molybdenum (Mo) atoms is sandwiched between two hexagonal sheets of sulfur (S) atoms, with adjacent Mo and S atoms linked by strong covalent bonds. As a representative transition metal dichalcogenide [38, 39], MoS₂ is a semiconducting material with a direct bandgap of ~1.8–1.9 eV in its monolayer form, along with moderate carrier mobility, atomic-scale thickness (~0.7 nm), and high stability. These properties have enabled a wide range of photonic, electronic, and optoelectronic applications, such as nonlinear optical devices [40, 41, 48], field-effect transistors [49-51], optical modulators [52-54], and broadband photodetectors [55-57]. For nonlinear optical applications, the bandgap of monolayer MoS₂ exceeds twice the photon energy at 1550 nm (i.e., ~1.6 eV), yielding relatively low linear light absorption and TPA at near infrared wavelengths. It is also worth noting that the quality of 2D MoS₂ films, which is strongly affected by intrinsic structural defects such as sulfur vacancies [41, 42], plays a critical role in determining their optical properties such as optical bandgap, refractive index, light absorption, and optical nonlinearity.

**Figure 1(b)** illustrates a Si₃N₄ waveguide integrated with a monolayer MoS₂ film. The cross-section view and top view of this hybrid waveguide is shown in **Figure 1(c)** and **Figure 1(d)**, respectively. In this work, we choose Si₃N₄ integrated platform with a large bandgap of ~5.0 eV [58] that yields negligible TPA at near infrared wavelengths. A window is opened on the silica upper cladding to expose the Si₃N₄ waveguide to air, enabling MoS₂ film coating and its interaction with the waveguide evanescent field.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** (a) Schematic atomic structure of monolayer molybdenum disulfide (MoS<sub>2</sub>). (b) Schematic of a Si<sub>3</sub>N<sub>4</sub> waveguide integrated with a monolayer MoS<sub>2</sub> film. The pulse schematics on the left and right sides illustrate the optical pulse spectra before and after propagation through the waveguide, respectively. (c) – (d) Cross-section view and top view of the hybrid waveguide in (b), respectively. (e) TE mode profile of the hybrid waveguide in (b). (f) Microscopic image of a window-opened Si<sub>3</sub>N<sub>4</sub> chip coated with monolayer MoS<sub>2</sub>. (g) Microscopic image of a monolayer MoS<sub>2</sub> film grown on a sapphire substrate.</figcaption></figure>

**Figure 1(e)** shows the transverse electric (TE) mode profile (at 1550 nm) for the hybrid waveguide, which was simulated using commercial mode-solving software (COMSOL Multiphysics). In our simulation, the width and height of the Si<sub>3</sub>N<sub>4</sub> waveguide were *W* = 1.60 µm and *H* = 0.66 µm, respectively. The thickness of the monolayer MoS<sub>2</sub> film was ~0.7 nm. The refractive index (*n*) and extinction coefficient (*k*) of MoS<sub>2</sub> were *n*<sub>TE</sub> = ~3.8 and *k*<sub>TE</sub> = ~0.107, respectively. These values were obtained from our previous measurements reported in Refs. [46,
59]. The light-matter interaction between the highly nonlinear MoS₂ film and the waveguide evanescent field can boost the nonlinear optical processes in the hybrid waveguide, forming the basis for enhancing the SPM performance. It is also worth mentioning that, due to the polymer-assisted transfer method employed in our device fabrication (which will be introduced later), the monolayer MoS₂ film does not conformally cover the sidewalls of the Si₃N₄ waveguide, leaving air gaps between the sidewalls and the film. This, however, has minimal influence on the MoS₂ mode overlap in the hybrid waveguide, as it mainly depends on the interaction between the evanescent field and the MoS₂ film on the waveguide top surface. Mode simulations further indicate that the mode overlap with the MoS₂ film near the sidewalls is more than an order of magnitude lower than that on the top surface.

**Figure 1(f)** shows a microscopic image of the fabricated Si₃N₄ chip coated with a monolayer MoS₂ film, showing high transmittance and good surface morphology. In our fabrication, we first fabricated the bare Si₃N₄ waveguides using a CMOS-compatible, annealing-free, and crack-free method reported in Refs. [60, 61]. The process began with a two-step deposition of Si₃N₄ film (330 nm-thick layer in each step), achieved via LPCVD for strain management and crack prevention. Subsequently, 248-nm deep ultraviolet lithography and CF₄/CHF₂/O₂ fluorine-based dry etching were employed to pattern the waveguide layout, followed by deposition of the silica upper cladding via high-density plasma enhanced chemical vapor deposition (HDP-PECVD). Finally, lithography and RIE were used to open windows in the silica cladding and etch down to the BOX layer, which allows the coating of MoS₂ films. All the fabricated Si₃N₄ waveguides had the same length of ~2.0 cm. The lengths of the opened windows varied between ~0.2 mm and ~1.4 mm, enabling precise control of the MoS₂ film length that interacts with Si₃N₄ waveguide.
<figure><img src="image_3.png" /><figcaption>**Figure 2.** (a) Schematic illustration showing the chemical vapor deposition (CVD) process flow used for synthesizing monolayer MoS₂ films. (b) X-ray diffraction (XRD) spectrum for the synthesized MoS₂ film. (c) Measured Raman spectra of a Si₃N₄ chip before (bottom) and after coating a monolayer MoS₂ film (top).</figcaption></figure>

Following fabrication of the Si₃N₄ waveguides, a LPCVD method we developed in Ref. [46] were used to synthesize high-quality MoS₂ films shown in **Figure 1(g)**. This approach enables precise control of intrinsic atomic defects arising from sulfur vacancies, and facilitates the direct growth of large-area MoS₂ films with excellent uniformity. As illustrated in **Figure 2(a)**, MoS₂ monolayers were synthesized by a chemical vapor deposition (CVD) process carried out in a two-temperature-zone furnace. The process began by drop-casting the Mo precursor onto an ultrasonically cleaned sapphire substrate placed in Zone 2, which was maintained at 750 °C. Subsequently, Sulfur powder heated to 180 °C in Zone 1 was vaporized and transported downstream into Zone 2 by a 70-sccm argon gas flow. Finally, the Mo and S species reacted on the substrate surface to yield MoS₂ single crystals. During the entire process, a chamber
pressure of ~1 torr was sustained, thereby preserving the low-pressure CVD environment.

**Figure 2(b)** shows the X-ray diffraction (XRD) spectrum for the synthesized MoS₂ film. The diffraction rings can be indexed to the (100), (103), and (105) reflections of hexagonal MoS₂ crystal structure, showing good agreement with the XRD spectra for MoS₂ reported in Refs. [54, 55].

The as-synthesized monolayer MoS₂ film was transferred onto the fabricated Si₃N₄ chip via a polymer-assisted transfer process [62]. The as-grown film was first spin-coated with a polystyrene (PS) support layer, after which the PS/MoS₂ stack was exfoliated from the growth substrate via a water-assisted delamination process driven by interfacial surface energy differences [63]. The stack was then stamped onto the Si₃N₄ chip through van der Waals interactions, and finally, the PS layer was removed by dissolution in toluene.

**Figure 2(b)** shows the Raman spectra of the same Si₃N₄ chip before and after coating a monolayer MoS₂ film, which were characterized using a ~514 nm excitation laser. Except for a peak at ~517 cm⁻¹ arising from the substrate, two additional peaks at ~384 cm⁻¹ and ~404 cm⁻¹ were observed in the Raman spectrum for the MoS₂-coated chip, which corresponds to the in-plane (E¹₂g) TE and out-of-plane (A₁g) vibrational modes of MoS₂, respectively. Moreover, the frequency separation of ~20 cm⁻¹ confirms the monolayer nature of the MoS₂ film. These features are consistent with previously reported CVD-grown monolayer MoS₂ [41, 52, 53], providing evidence for successful integration of high-quality 2D MoS₂ film onto the Si₃N₄ chip.

### 3. Loss measurements

The integration of MoS₂ films onto Si₃N₄ waveguides introduces additional propagation loss.

**Figure 3** shows a schematic of the experimental setup used for both loss measurements in this
section and SPM measurements in **Section 4**. In our measurements, lensed fibers were employed to butt couple light into and out of the fabricated devices with specially designed inverse-taper couplers at both ends. The fiber-to-chip coupling loss was ~7.5 dB / facet. For both the loss and SPM measurements, TE-polarized input light was selected, as it enables in-plane interaction between the waveguide evanescent field and the $MoS_2$ film, in contrast to the much weaker out-of-plane interaction due to the significant anisotropy of 2D materials [31, 35, 60].

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Experimental setup used for measuring loss and SPM for $MoS_2$-coated $Si_3N_4$ waveguides. CW laser: continuous-wave laser. FPL: fiber pulsed laser. PC: polarization controller. VOA: variable optical attenuator. OPM: optical power meter. DUT: device under test. CCD: charge-coupled device. OSA: optical spectrum analyzer.</figcaption></figure>

We first measured the linear loss of the fabricated devices using a continuous-wave (CW) light. **Figure 4(a)** shows the measured TE-polarized insertion loss (IL) versus $MoS_2$ film
coating length $L_c$ for the hybrid waveguides. For comparison, the corresponding results for the uncoated $Si_3N_4$ waveguides (i.e., $L_c$ = 0 mm) are also shown. For different devices, the power and wavelength of the input CW light were kept the same as $P_{CW}$ = ~10 dBm and $\lambda$ = ~1550 nm, respectively. Unless otherwise specified, the input power of CW light or optical pulses in our following discussion indicates the power coupled into the waveguide after excluding the fiber-to-chip coupling loss. The data points depict the average values of measurements on three duplicate devices, and the error bars reflect the variations among them. As can be seen, the IL increases almost linearly with $L_c$. By using the cut-back method [64], we obtained a propagation loss of ~0.5 dB/cm for the uncoated $Si_3N_4$ waveguides. Based on this value and the measured IL values in **Figure 4(a)**, the excess propagation loss induced by monolayer $MoS_2$ film was extracted to be ~24 dB/cm. Compared with our earlier reports on $Si_3N_4$ waveguides coated with monolayer GO films [65, 66], the loss induced by $MoS_2$ is about 10 times higher, yet remains about five times lower than graphene-induced excess propagation loss in graphene-coated $Si_3N_4$ waveguides [60, 67].

**Figure 4(b)** shows the measured TE-polarized IL versus input CW power for an uncoated $Si_3N_4$ waveguide and a hybrid waveguide with a 1.4-mm-long $MoS_2$ film. For both waveguides, the IL does not show any obvious variation with the input CW power. This indicates that the $MoS_2$ film exhibited negligible power-dependent loss in the measured power range. At an input CW power of ~400 mW (i.e., ~2 W prior to waveguide coupling), we had already reached the maximum output of our erbium-doped fiber amplifier (EDFA). Yet, no observable changes were observed in the measured IL for the hybrid waveguide. This indicates the excellent thermal stability of $MoS_2$, a key advantage for nonlinear optical applications that require high powers to boost nonlinear interactions.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** (a) Measured insertion loss (IL<sub>CW</sub>) of hybrid MoS<sub>2</sub>-Si<sub>3</sub>N<sub>4</sub> waveguides versus MoS<sub>2</sub> coating length (L<sub>c</sub>). The corresponding results for uncoated Si<sub>3</sub>N<sub>4</sub> waveguide (i.e., L<sub>c</sub> = 0 mm) are also shown for comparison. (b) Measured insertion loss (IL<sub>CW</sub>) versus input CW power (P<sub>CW</sub>) for an uncoated Si<sub>3</sub>N<sub>4</sub> waveguide and a hybrid waveguide with a 1.4-mm-long MoS<sub>2</sub> film. (c) Measured insertion loss (IL<sub>CW</sub>) versus input CW wavelength for the two waveguides in (b). (d) Measured insertion loss (IL<sub>pulse</sub>) versus input power of optical pulses for the two waveguides in (b). In (a) and (b), the input CW wavelength is ~1550 nm. In (a) and (c), the input CW power is ~10 dBm. In (a), (b), and (d), the data points depict the average of measurements on three samples, and the error bars illustrate the variations among the different samples.</figcaption></figure>

**Figure 4(c)** shows the measured TE-polarized IL versus input CW wavelength for the two waveguides measured in **Figure 4(b)**. The input CW power was maintained at ~10 dBm. Due to the limited wavelength tuning range for our CW laser, we could only measure the IL between 1500 and 1600 nm. Within this range, the flat spectral response of both uncoated and MoS<sub>2</sub>-coated Si<sub>3</sub>N<sub>4</sub> waveguides indicates negligible wavelength-dependent material absorption or coupling loss.

In addition to linear loss measurements performed using CW light, we employed optical pulses generated by a fiber pulsed laser (FPL, as illustrated in **Figure 3**) to characterize the
nonlinear loss of the fabricated devices. **Figure 4(d)** shows the measured IL versus input power of optical pulses for the two waveguides measured in **Figures 4(b)** and **4(c)**. The FPLs generated nearly Fourier-transform limited optical pulses centered at ~1557 nm. The pulse duration and repetition rate were ~580 fs and ~60 MHz, respectively. The average input powers ranged between ~1.9 mW and ~3.2 mW, which corresponds to peak powers of ~57 W – ~91 W. In contrast to CW light that has a peak power equaling to its average power, optical pulses have peak powers that are significantly higher than their average powers, making them suitable for characterizing nonlinear optical loss without introducing significant thermo-optic effects. Similar to those observed in **Figure 4(b)**, the measured IL of both uncoated and hybrid Si₃N₄ waveguides showed no significant variation with increasing input power of optical pulses. This indicates the absence of significant nonlinear optical absorption such as TPA and saturable absorption effects for both Si₃N₄ and MoS₂ – a behavior consistent with their relatively large optical bandgaps as previously discussed.

**4. SPM measurements**

Following the loss measurements, we performed SPM measurements to characterize the spectral broadening of optical pulses after propagating through the fabricated devices. In our SPM measurements, we employed the same FPL and fabricated devices as those used for loss measurements in **Section 3**. As illustrated in **Figure 3**, optical pulses generated from the FPL were coupled into the fabricated devices, with a variable optical attenuator (VOA) and a polarization controller (PC) employed to adjust the power and polarization of the input light, respectively. After passing through the fabricated devices, the output signal was collected by an optical spectrum analyzer (OSA) to monitor spectral broadening. We did not employ any fiber
couplers in our SPM measurements to avoid spectral filtering distortions, given that their
operational bandwidth was insufficient to cover the full spectrum of the optical pulses generated
by the FPL.

**Figure 5(a)** shows the normalized spectra of optical pulses before and after propagating through the uncoated Si₃N₄ waveguide and the hybrid waveguide with a 1.4-mm-long MoS₂ film. The peak power of the input femtosecond pulse was maintained at ~91 W. Compared with the input pulse spectrum, slight spectral broadening was observed in the output spectrum from the uncoated Si₃N₄ waveguide, which was mainly caused by SPM in the Si₃N₄ waveguide. The MoS₂-coated Si₃N₄ waveguide exhibited much greater spectral broadening than the uncoated Si₃N₄ waveguide, indicating the significantly enhanced SPM in the hybrid waveguide.

**Figure 5(b)** shows the normalized spectra of optical pulses after propagating through hybrid waveguides with various MoS₂ coating lengths ($L_c$). Here we measured five hybrid devices, with $L_c$ varying between ~0.2 mm and ~1.4 mm. The corresponding result for an uncoated Si₃N₄ waveguide is also shown for comparison. It can be seen that more significant spectral broadening was observed for an increased MoS₂ coating length, further confirming that the enhanced SPM in the hybrid waveguides was induced by the coated MoS₂ films.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** SPM experimental results. (a) Normalized spectra of optical pulses before and after propagating through an uncoated Si₃N₄ waveguide and a hybrid waveguide with a 1.4-mm-long MoS₂ film. (b) Normalized spectra of optical pulses after propagating through hybrid waveguides with various MoS₂ coating lengths ($L_c$) ranging between ~0.2 mm and ~1.4 mm. The output spectrum from an uncoated Si₃N₄ waveguide is also shown for comparison. (c) Optical spectra measured at different input pulse peak powers ($P_{peak}$) for the hybrid waveguide with a 1.4-mm-long MoS₂ film. In (a) and (b), the $P_{peak}$ is ~91 W.</figcaption></figure>

**Figure 5(c)** shows the optical spectra measured at different input pulse peak powers ($P_{peak}$) for the hybrid waveguide with $L_c = 1.4$ mm. Here we show the results for five different $P_{peak}$ values ranging from ~57 W to ~91 W – identical to those used for nonlinear loss measurements
in **Figure 4(d)**. As the input peak power was raised, the output spectra exhibited increasing broadening, in agreement with the expected behavior of SPM-induced spectral broadening.

## 5. Theoretical analysis and discussion

In this section, we analyze the SPM experimental results in **Section 4** by fitting them with theory to extract nonlinear parameter $\gamma$ of the hybrid waveguide and Kerr coefficient $n_2$ of MoS$_2$.

To quantitatively compare spectral broadening across different waveguides, broadening factors ($BF$) were calculated from the measured output spectra in **Figure 5**. The $BF$ is defined as [33, 37, 60]:

$$BF = \frac{\Delta\omega_{rms}}{\Delta\omega_0} \qquad (1)$$

where $\Delta\omega_{rms}$ and $\Delta\omega_0$ are the root-mean-square (RMS) spectral widths of the signals before and after propagation through the waveguides (either uncoated or coated with MoS$_2$), respectively.

**Figure 6(a)** shows the calculated $BFs$ for the measured optical spectra in **Figure 5(b)**.

Increasing the MoS$_2$ coating length leads to a rise in the $BF$, showing a trend consistent with that in **Figure 5(b)**. At $L_c = \sim 1.4$ mm, a maximum $BF$ of $\sim 2.4$ is achieved for the hybrid waveguide. In comparison, the calculated $BF$ for the uncoated $Si_3N_4$ waveguide is only $\sim 1.2$.

**Figure 6(b)** shows the calculated $BFs$ for the measured optical spectra in **Figure 5(c)**. With the input pulse peak power increasing from $\sim 57$ W to $\sim 91$ W, the $BF$ increases from $\sim 1.4$ to $\sim 2.4$, agreeing well with the trend in **Figure 5(c)**.
<figure><img src="image_9.png" /><figcaption>**Figure 6.** (a) Calculated broadening factor (BF) versus MoS₂ coating length ($L_c$) for the measured optical spectra in Figure 5(b). (b) Calculated BF versus input pulse peak power ($P_{peak}$) for the measured optical spectra in Figure 5(c). (c) – (d) Nonlinear parameters for the MoS₂-coated segments ($\gamma_{hybrid}$) in the hybrid waveguides obtained by fitting the results in Figures 5(b) and (c), respectively. (e) – (f) Kerr coefficient ($n_2$) of MoS₂ extracted from the results in (c) and (d), respectively.</figcaption></figure>

To fit the experimental results with theory, we simulated the SPM-induced spectral broadening of optical pulses by numerically solving the nonlinear Schrödinger equation as follows [21, 37]:

$$ \frac{\partial A}{\partial z} = -\frac{i\beta_2}{2} \frac{\partial^2 A}{\partial t^2} + i\gamma |A|^2 A - \frac{1}{2} \alpha A \quad (2) $$
where $i = \sqrt{-1}$, $A(z, t)$ is the slowly varying temporal pulse envelope along the propagation direction $z$, $\beta_2$ is the second-order dispersion coefficient, $\gamma$ is the waveguide nonlinear parameter and $\alpha$ is the loss factor. In Eq. (2), only the second-order dispersion term $\beta_2$ is considered, since the physical length of the waveguides (i.e., ~2 cm) is much shorter than the dispersion length [68]. According to our measurements in Figure 4(d), both $Si_3N_4$ and $MoS_2$ exhibited negligible nonlinear optical absorption, so only linear loss was considered for $\alpha$. In our simulation, we divided the $MoS_2$-coated $Si_3N_4$ waveguides into uncoated (with silica cladding) and hybrid segments (coated with $MoS_2$ films). For each segment, Equation (2) was numerically solved, with the output of one segment serving as the input for the next. The output from the final segment was used to fit the measured spectra in Figure 5 and extract the waveguide nonlinear parameter $\gamma$.

We first extracted the $\gamma$ for the uncoated $Si_3N_4$ waveguide, which is ~1.2 $W^{-1}m^{-1}$ and consistent with previously reported values in Refs. [58, 59, 61]. Subsequently, we proceeded to extract the $\gamma$ values for the $MoS_2$-coated segments in the hybrid waveguides. Figures 6(c) and (d) show the $\gamma$ values for the hybrid segments obtained by fitting the experimental results in Figures 5(b) and (c), respectively. Despite variations in $MoS_2$ coating length and input pulse peak power, the extracted $\gamma$ values remain nearly unchanged, reflecting the consistency of our SPM measurements. The average $\gamma$ for the hybrid segments is ~32 $W^{-1}m^{-1}$ – about 27 times higher than that of the uncoated $Si_3N_4$ waveguide. This highlights the substantial improvement in the waveguide’s optical nonlinearity after incorporating 2D $MoS_2$.

The Kerr coefficient ($n_2$) of 2D $MoS_2$ are further extracted based on the fit $\gamma$ values of the hybrid waveguides using [37, 60, 70-75]:
$$ \gamma = \frac{2\pi \iint_D n_0^2(x, y)n_2(x, y)|E(x, y)|^4 dxdy}{\lambda_c \left[ \iint_D n_0(x, y)|E(x, y)|^2 dxdy \right]^2} \quad (3) $$

where $\lambda_c$ is the light wavelength, $D$ is the integral of the optical fields over the material regions, $E(x, y)$ is the electric field distribution calculated by mode solving software, and $n_0(x, y)$ and $n_2(x, y)$ are the refractive index and Kerr coefficient profiles across the waveguide cross section, respectively. The values of $n_2$ for silica and $Si_3N_4$ used in our calculation were ~2.6 × 10⁻²⁰ m²/W [24] and ~2.5 × 10⁻¹⁹ m²/W, respectively, with the latter obtained by fitting the $\gamma$ for the uncoated $Si_3N_4$ waveguide. **Figures 6(e) and (f)** show the Kerr coefficient ($n_2$) of $MoS_2$ extracted from the fitted $\gamma$ values in **Figures 6(c) and (d)**, respectively. Consistent with the results in **Figures 6(c) and (d)**, the extracted $n_2$ does not show any significant variations. The average $n_2$ of $MoS_2$ is ~2.3 × 10⁻¹⁴ m²/W – about 5 orders of magnitude higher than $Si_3N_4$. This highlights the ultrahigh third-order optical nonlinearity of monolayer $MoS_2$ as compared to bulk materials. According to **Eq. (3)**, the nonlinear parameter $\gamma$ is determined by the weighted contributions of the Kerr coefficients ($n_2$) from different material regions in the waveguide cross section. Although the 2D $MoS_2$ film occupies only a small fraction, its exceptionally high Kerr coefficient enables a substantial enhancement of $\gamma$ for the hybrid waveguides. We note that the $n_2$ value of $MoS_2$ obtained here is much higher than those reported in Refs. [42, 44], which can probably be attributed to our film being strictly monolayer (in contrast to multi-layered films used in Refs. [42]), as well as its high quality with minimal defects enabled by our developed synthesis method [46]. These results highlight the promise $MoS_2$-coated waveguides have for enhancing nonlinear optics, on par with both GO [76–82] and other 2D materials [83–89] with potential applications to microcomb based photonics. [90–160]
<figure><img src="image_10.png" /><figcaption>**Figure 7.** (a) Simulated spectral evolution of optical pulses propagating along an uncoated Si₃N₄ waveguide. (b) Zoom-in view of (a) around the window opening area ranging from 0.7 mm to 2.1 mm. (c) Simulated spectral evolution of optical pulses propagating along a hybrid waveguide coated with a 1.4-mm-long MoS₂ film. (d) Zoom-in view of (c) around the window opening area ranging from 0.7 mm to 2.1 mm. In (a) – (d), the input pulse peak power is ~91 W.</figcaption></figure>

**Figures 7(a) and (c)** show the simulated spectral evolution along the entire length (i.e., ~2 cm) of the uncoated and hybrid Si₃N₄ waveguides, respectively. They were simulated by using the fitted $\gamma$ values for the uncoated and MoS₂-coated waveguides obtained in **Figure 6**. Zoom-in views around the window opening area (between 0.7 mm and 2.1 mm, near the input ports) are shown in **Figures 7(b) and (d)**. Unlike the gradual spectral broadening along the entire length of the uncoated Si₃N₄ waveguides, dramatic spectral broadening occurs within the MoS₂-coated segment in the hybrid waveguide, followed by gradual broadening in the subsequent uncoated segment. This further reflects enhanced SPM in the MoS₂-coated segment enabled by integrating MoS₂ with ultrahigh optical nonlinearity.
## 6. Conclusion

In this work, we experimentally demonstrate significantly enhanced SPM-induced spectral broadening of optical pulses in Si₃N₄ waveguides integrated with 2D MoS₂ films. High-quality monolayer MoS₂ films are synthesized via LPCVD, followed by a polymer-assisted transfer process that enables their on-chip integration. Selective window openings in the upper cladding allow accurate control of the film length and position on the integrated chips. Detailed SPM measurements are carried out using fabricated devices with various MoS₂ film lengths and at various pulse peak powers. Compared with uncoated Si₃N₄ waveguides, the hybrid waveguides exhibit enhanced spectral broadening for optical pulses propagation through, achieving a maximum *BF* of ~2.4. Analysis of the experimental results through theoretical fitting further reveals up to ~27-fold increase in the nonlinear parameter for the hybrid waveguides, together with an extracted *n*₂ of MoS₂ that is nearly five orders of magnitude higher than Si₃N₄. These results validate the integration of 2D MoS₂ films onto Si₃N₄ devices as a powerful approach to enhance their Kerr nonlinear optical performance.

## Conflict of Interest

The authors declare no competing financial interest.

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