Optical Parametric Amplification in Photonic Chips Integrated with 2D Materials

1. Introduction

Nonlinear integrated photonic devices, which enable all-optical signal generation and processing without inefficient optical-electrical-optical (OEO) conversion, offer far superior processing speed and energy efficiency compared to electronic devices, along with added benefits such as compact device footprint, low power consumption, high stability, and mass production [1,2,3]. Silicon platform that leverages the well-developed complementary metal-oxide-semiconductor (CMOS) fabrication technologies from integrated circuit industry has been a leading platform for integrated photonic devices [2]. Nevertheless, the absence of χ⁽²⁾ optical nonlinearity and the strong two-photon absorption (TPA) at near-infrared telecommunication wavelengths significantly limit the use of silicon devices for many nonlinear optical applications [4,5]. Although other CMOS compatible platforms, such as silicon nitride (SiN) and doped silica, exhibit much weaker TPA [6,7], they also suffer from limited nonlinear conversion efficiency due to their much lower χ⁽³⁾ optical nonlinearity compared with silicon.

The era of big data has witnessed an unprecedented surge in data generation and processing, leading to an explosive demand for higher processing speed and energy efficiency in data centers that bear the brunt of this growth. This has driven the pursuit for high-performance nonlinear integrated photonic devices, motivating the hybrid integration of advanced nonlinear optical materials to overcome the limitations of existing integrated platforms [8,9,10]. Since the ground-breaking discovery of graphene in 2004 [11], 2D materials with atomic thickness and extraordinary properties have emerged at the forefront of global research, providing vast opportunities for both fundamental science and industrial applications [12,13,14,15]. In recent years, significant progress has been made in on-chip integration of 2D materials to implement hybrid devices that surpass the performance limits of conventional counterparts and unlock new capabilities [16,17,18]. A substantial body of research has demonstrated the enhancement of fundamental nonlinear optical processes, such as four-wave mixing (FWM) and self-phase modulation (SPM), in hybrid integrated photonic devices incorporating 2D materials [19,20]. Although the ultrahigh optical nonlinearity of 2D materials has been widely recognized, their considerably higher light absorption compared with conventional bulk integrated materials limits their use in many advanced nonlinear optical applications, particularly optical parametric amplification (OPA) [21,22,23] and optical micro-comb generation based on OPA [24,25,26]. As illustrated in Figure 1, in these applications critical questions arise as to whether the high parametric gain (PG) provided by 2D materials with high optical nonlinearity can compensate for their high loss to achieve net gain (NG), and how close current technologies are to realizing this goal. These issues are fundamental and of broad interest to the related research community, yet remain unaddressed.

Recently [1], we experimentally demonstrated NG of up to ~24 dB for femtosecond optical pulses propagating through SiN waveguides integrated with 2D graphene oxide (GO) films, representing the first demonstration of achieving NG via OPA by integrating 2D materials onto photonic chips. In this perspective, based on practical material parameters obtained from this work, we systematically analyze the feasibility of achieving NG via OPA in silicon waveguides incorporating GO and using continuous-wave (CW) light. This is highly challenging due to the strong TPA of silicon at telecommunication wavelengths and the much lower peak power of CW light compared with optical pulses. Nevertheless, it is of great significance given silicon’s dominant role in integrated photonic devices and the more widely use of CW pumping for nonlinear optical applications. Our simulation results reveal that although a gap remains toward achieving this goal, it can be bridged through coordinated advances in optimizing waveguide structure, reducing loss of GO, and improving GO’s thermal stability. We also investigate and analyze the existing gap for different types of silicon waveguides including nanowire waveguides, rib waveguides, etchless waveguides, and slot waveguides, as well as for other 2D materials such as graphene and molybdenum disulfide (MoS₂). This work presents a clear roadmap and practical insights to guide future efforts towards achieving NG via OPA in integrated photonic devices incorporating 2D materials.

2. Limitations for Achieving NG via OPA in Bulk Integrated Photonic Devices

OPA underpins many modern optical applications, such as optical communications, broadband optical sources, and optical micro-comb generation [21,23]. Compared with optical amplifiers based on rare-earth-doped fibers and III-V semiconductors, OPA-based amplifiers can offer gain over much broader wavelength ranges and enable optical amplification using passive devices [22,23]. In this section, we discuss the limitations of conventional SiN and silicon waveguides in achieving NG via OPA at telecommunication wavelengths (centered at 1550 nm). Based on the theory detailed in Note 1 of Supplementary information (SI), we performed simulation using typical material parameters and waveguide loss factors aligned with state-of-the-art fabrication capabilities. Our discussion of OPA in this paper focuses on that based on χ⁽³⁾ optical nonlinearity, although we note that OPA can also be induced by χ⁽²⁾ optical nonlinearity [27,28,29].

In Figure 2, we analyze the PG for SiN waveguides. Note that the PG values here and in our following discussion refer to net parametric gain over and above the intrinsic loss of the waveguides (i.e., achieving NG requires PG > 0). This differs from the “on/off parametric gain” quoted in Refs. [30,31], where the waveguide loss is excluded, resulting in higher PG values.

To simplify the discussion, our PG simulation here and in the following analysis assume that the signal and pump wavelengths are closely spaced, so that the phase mismatch can be neglected.

Figure 2(a) shows the waveguide schematic and transverse-electric (TE) mode profile (at 1550 nm), where the SiN waveguide has a typical cross-section of 1600 nm × 720 nm [1,32]. In our following discussion, we focus on investigating the PG for TE modes as they are more widely used than transverse-magnetic (TM) modes for nonlinear integrated photonic devices, particularly those incorporating 2D materials [9,33,34].

Figure 2(b) shows the simulated PG for the SiN waveguide in Figure 2(a) excited by using a CW pump. In our simulation, we used typical waveguide nonlinear parameter (γ = ~1.1 W^-1m^-1) and dispersion reported in Refs. [1,32]. The typical optical bandgap of SiN is ~5.0 eV, far exceeding twice the photon energy at 1550 nm (i.e., ~1.6 eV) and resulting in negligible TPA at telecommunication wavelengths. The waveguide propagation loss (PL) is a critical factor affecting the PG. For comparison, we used two different values for the PL of the SiN waveguides in our simulation. The first one is ~0.5 dB/cm, which is typical for SiN waveguides fabricated with current average fabrication capabilities [1,35]. The second one is ~0.02 dB/cm, corresponding to state-of-the-art fabrication capabilities according to Refs. [22,36]. In Figure 2(b), the white dashed lines denote the condition of PG = 0, and the black points and asterisks mark the results simulated using the parameters of practical devices reported in Refs. [1] and [22], respectively. In Figure 2(b-i), the simulated PG values show good agreement with the experimental results in Refs. [1,22], confirming the accuracy of our simulation. At a pump power of P_CW = 2 W, the PG remains negative for waveguide length L ≤ 200 cm when PL = 0.5 dB/cm. In contrast, NG is achievable for 20 cm ≤ L ≤ 200 cm when PL = 0.02 dB/cm. This highlights the significance of reducing the waveguide PL in achieving NG, which is also reflected by the results in Figure 2(b-ii) and 2(b-iii).

Figure 2(c) shows the corresponding results under pulsed pumping, which is based on FWM between a CW light as the signal and optical pulses as the pump ‒ consistent with our experimental demonstration in Ref. [1]. Compared with CW pumping, pulsed pumping enables high peak powers while maintaining low average powers, which helps suppress the material property changes induced by thermo-optic effects. This is particularly important for 2D materials such as GO, which can be reduced at high temperatures [37]. In our simulation, we used optical pulses with a pulse width of 180 fs and a maximum peak power P_pulse of 180 W ‒ consistent with the pulse parameters employed in our previous experiments [1,38]. As can be seen, NG can be easily achieved even for PL = 0.5 dB/cm, and it becomes more significant for PL = 0.02 dB/cm. This is mainly because the high peak power of the pumping pulses significantly enhances the efficiency of the OPA process. It should be noted that the PG values shown inFigure 2(c) and in the following pulsed pumping analysis correspond to the maximum values achieved at the peak powers, which cannot be sustained over long times due to the short pulse duration.

Following the analysis performed for SiN waveguides, the corresponding results for silicon nanowire waveguides are provided in Figure 3. In our simulation, the silicon nanowire waveguides have a typical cross-section of 450 nm × 220 nm, and we used typical waveguide nonlinear parameter (γ = ~288 W^-1m^-1) and dispersion reported in Ref. [20]. Compared with SiN waveguides, the nonlinear parameter of silicon nanowire waveguide is two orders of magnitude larger ‒ owing to silicon’s much higher Kerr coefficient (~6.0 × 10^-18 m²/W [20,39]) relative to SiN (~2.6 × 10^-19 m²/W [32]) and the much smaller effective mode area. Unlike SiN with a large bandgap that yields negligible TPA at telecommunication wavelengths, the much smaller optical bandgap of silicon (~1.12 eV [2,40]) leads to significant TPA in this range. The nonlinear optical absorption induced by TPA, as well as parasitic free-carrier effects (i.e., free carrier dispersion and free carrier absorption [2]) were considered in our simulation for silicon devices (see details in Note 2 of SI).

Similar to that in Figure 2, we used two different values for the PL of the silicon nanowire waveguides in our simulation: 4.3 dB/cm representing average fabrication capabilities [20] and 1.0 dB/cm corresponding to state-of-the-art fabrication capabilities [41]. We performed simulation for both CW and pulsed pumping, and in both cases, the PG remains negative within the investigated power and waveguide length ranges. This is mainly due to the strong TPA that induces significant nonlinear optical loss that increases with input power, highlighting the challenges of achieving NG in silicon waveguides. We note that Ref. [30] reported achieving NG in silicon nanowire waveguides with a cross-section of 600 nm × 300 nm, so we also performed simulation for this waveguide geometry. The simulation results show that achieving NG remains challenging in this case when TPA is taken into account (see details in Note 3 of SI). In fact, the results in Ref. [30] were achieved based on FWM between two synchronized picosecond optical pulses, for which the TPA-induced free carrier effects can be neglected since the generated free carriers with long life times (typically on the order of ~1 ns [42,43]) do not have sufficient time to recombine within the pulse duration. This differs from FWM between two CW lights or between a CW light and a pulsed pump, as discussed in Figure 2(c) and 3(c), where TPA induced free carrier effects are no longer negligible.

In Figure 4 and Figure 5, we analyze the PG for other types of silicon waveguides including rib, etchless, slab, and slot waveguides. Each of these waveguides has its own characteristics and has been widely studied and employed in diverse applications [44,45,46].

Figure 4 shows the results for silicon rib, etchless, and slab waveguides that exhibit lower PL than nanowire waveguides. In Figure 4(a), the rib waveguides have a cross-section of 1000 nm × 380 nm, and the cap has an etched depth of 70 nm. The rib waveguides with a shallow-etched structure exhibit weaker optical confinement, which reduce mode overlap with scattering sidewalls and consequently lower the PL [47]. In Figure 4(b), the etchless waveguides have a cross-section of 1000 nm× 70 nm. These waveguides can be fabricated by first growing and patterning a thermal oxide mask, followed by defining the waveguide core through thermal oxidation of silicon, where the silicon layer remains completely free from the etching process, resulting in extremely smooth silicon/silica interface and significantly reduced PL [48]. In Figure 4(c), the slab waveguides have a cross-section of 900 nm × 60 nm. The low PL is enabled by the ultra-thin waveguide structure that exhibits lower sensitivity to sidewall roughness [49]. In our simulation, the PL values of the silicon rib, etchless, and slab waveguides were 0.4, 0.3, and 0.6 dB/cm, respectively. These values were consistent with those reported in Refs. [47,48,49]. The corresponding nonlinear parameters of these waveguides were ~30, ~89, and ~69 W^-1m^-1, respectively, calculated using the same n₂ of silicon for the nanowire waveguide but with effective mode areas specific to each waveguide configuration. As can be seen, although these waveguides offer lower PL values compared with nanowire waveguides, this comes at the expense of lower optical nonlinearities. Similar to the results in Figure 3, for both CW and pulsed pumping, the PG remains negative within the power and waveguide length ranges investigated in Figure 4. This further highlights the challenges of overcoming the TPA of silicon and achieving NG in silicon waveguides at telecommunication wavelengths.

Figure 5 shows the simulation results for silicon slot waveguides. Owing to their ability to confine light tightly within nanoscale slot regions, slot waveguides offer exceptionally strong field confinement and greatly enhanced light-matter interaction, making them widely used for optical sensing, nonlinear photonics, and electro-optic modulation [50,51]. We analyze slot waveguides with different slot widths of 100, 150, and 200 nm inFigure 5. Each silicon waveguide on both sides of the slot region has a cross-section of 225 nm × 220 nm. The slot region and the upper cladding are made of silica, and the fabrication of such slot waveguides can be achieved using techniques detailed in Ref. [41]. In our simulation, the PL values of the slot waveguides with 100-nm, 150-nm, and 200-nm slot widths were 4.1, 5.0, and 6.3 dB/cm, respectively ‒ consistent with those reported in Ref. [41]. The calculated nonlinear parameters of these waveguides were ~13, ~8.8, and ~6.4 W^-1m^-1, respectively. Similar to the results in Figure 3 and Figure 4, the PG remains negative for both CW and pulsed pumping. A key reason is that the material filling the slot region is silica, which has a very low Kerr coefficient (~2.6 × 10^-20 m²/W [52]). By filling the slot region with highly nonlinear materials such as nonlinear polymers [53] and 2D materials [54], the waveguide optical nonlinearity can be significantly improved, offering a promising route toward achieving NG. Detailed analysis for achieving NG in slot waveguides with highly nonlinear 2D materials is provided in Section 3.

3. Challenges and Opportunities for Achieving NG via OPA in Hybrid Integrated Photonic Devices Incorporating 2D Materials

The limitations for achieving NG via OPA in bulk nonlinear integrated photonic devices motivate on-chip integration of advanced materials to address them. In this section, we systematically analyze the challenges and opportunities for achieving NG via OPA in integrated photonic devices incorporating 2D materials. Our discussion focuses on devices with 2D GO films based on our experimental work in Ref. [1], and comparisons between GO and other 2D materials such as graphene and MoS₂ will also be provided.

In Figure 6, we analyze the PG of SiN waveguides coated with 2D monolayer GO films. GO is an oxidized derivative of graphene, featuring carbon networks decorated with diverse oxygen-containing functional groups (OCFGs) such as hydroxyl, carboxyl, and carbonyl groups [14,55]. Among various 2D materials, GO provides several key advantages for OPA applications, including high optical nonlinearity (~1.5 × 10^-14 m²/W, which is about 5 orders of magnitude higher than that of SiN [32,38]) and a large optical bandgap (2.1 – 3.6 eV [56]) that yields low linear optical absorption and nonlinear TPA at near-infrared wavelengths. In addition, GO has facile solution-based synthesis processes [57] and transfer-free film coating with precise control over the thickness [58], showing strong capability for large-scale manufacturing and high compatibility with integrated device platforms [10].

Figure 6(a) shows the waveguide schematic and TE mode profile (at 1550 nm). The SiN waveguide structure is the same as that in Figure 2(a), except that the silica upper cladding is removed to enable coating of a GO film on the top surface. InFigure 6(b-i), we investigate the PG versus waveguide length L and CW pump power P_CW under current fabrication capabilities [1]. In our simulation, the material property parameters of GO, including refractive index (n = ~1.97 at 1550 nm), extinction coefficient (k =~ 0.012 at 1550 nm), film thickness (~2 nm), and Kerr coefficient (n₂ = ~1.5 × 10^-14 m²/W) were consistent with those obtained from our experiments in Refs. [1,32]. The calculated nonlinear parameter of the hybrid waveguide and GO-induced excess propagation loss (EPL_GO, after excluding PL of the uncoated SiN waveguide) were ~14.5 W^-1m^-1 and ~3 dB/cm, respectively ‒ both of which are typical of current GO film fabrication capabilities [1,32]. The PL of the uncoated SiN waveguide (~0.2 dB/cm) was estimated according to the state-of-art PL of clad SiN waveguides (~0.02 dB/cm) and the current etching techniques. The asterisk marks the simulated PG that agrees well with the experimental result in Ref. [1]. As can be seen, the PG remains negative within the investigated ranges of waveguide length and pump power. This indicates achieving NG under CW pumping in the GO-SiN hybrid waveguides remains challenging under current fabrication capabilities. Our simulation also shows that the hybrid waveguides exhibit higher PG than comparable SiN waveguides without GO (see details in Note 4 of SI). This reflects the fact that, although incorporating 2D GO does not enable NG, it is still beneficial for improving the OPA performance.

The reduction of GO at high temperatures or under high optical powers, which breaks the chemical bonds between the OCFGs and the carbon network, is an intrinsic feature of GO that can limit its use for high-power OPA applications. As the degree of reduction increases, reduced GO (rGO) exhibits decreased optical bandgaps and increased optical absorption, which can result in decreased PG. InFigure 6(b-i), the dashed-dot line marks the CW pump power threshold (P_thres = ~0.2 W) above which significant GO reduction occurs, according to the experimental results in Ref. [37]. It should be noted that the P_thres is not a fundamental limit and can be further increased (as indicated by the arrow) by improving the thermal stability of GO. For instance, our previous work showed that modifying the degree of oxidation of GO using an electrochemical method allows GO to preserve a high Kerr nonlinearity while significantly enhancing its thermal stability (>100 times) under high-power laser illumination [59].

In addition to thermal stability, there remains room to further reduce the linear loss of 2D GO films and optimize their performance for OPA applications. In principle, GO with a bandgap > 2 eV exhibits negligible linear optical absorption at near infrared wavelengths, and the linear loss of practical GO films is dominated by defect-induced absorption and scattering loss arising from imperfect layer contact and film unevenness [60,61]. As these loss sources are fabrication-dependent and difficult to quantify, estimating the minimum linear loss achievable in practical GO films is challenging. What is clear, however, is that there remains significant room to reduce the EPL_GO arising from these sources by optimizing the film fabrication processes. InFigure 6(b-ii), we plot PG of GO-SiN waveguides versus L and EPL_GO. In our simulation, the input CW pump power is 0.2 W. For comparison, all the parameters of the hybrid waveguide are the same as those used in Figure 6(b–i), except for the varying EPL_GO. The dashed line marks the current loss level of EPL_GO = 3 dB/cm (i.e., k = ~ 0.012) [1], and the arrow indicates the desired direction for future optimization. As can be seen, reducing the GO-induced loss alone is still insufficient for achieving NG in the GO-SiN hybrid waveguides.

InFigure 6(b-iii), we further investigate the feasibility of achieving NG through a combined increase in P_thres and decrease in EPL_GO, where the PG versus EPL_GO and P_CW is plotted. The waveguide length in our simulation was L = 2 cm. It can be seen that NG can be achieved with a maximum EPL_GO of ~0.5 dB/cm (i.e., k < 0.002) and a minimum P_CW of 1 W. This result reveals that maintaining the current level of optical nonlinearity of GO while reducing its extinction coefficient k (by a factor of ~6) and optimizing its thermal stability (> ~5 times), can yield NG under CW pumping in the GO-SiN hybrid waveguides. At P_CW = 2 W and EPL_GO = 0.1 dB/cm, only a small PG of ~0.2 dB is achieved. We also performed simulation for a longer hybrid waveguide with L = 10 cm (see details in Note 5 of SI), which shows a higher PG of ~4.3 dB at P_CW = 2 W and EPL_GO = 0.1 dB/cm.

Figure 6(c) shows the corresponding results under pulsed pumping. For comparison, the pulse parameters are consistent with those in Section 2. For pulsed pumping, a low average power can still provide a high peak power (at a low repetition rate) without inducing significant thermal reduction of GO. Therefore, we assume that there are no pump power thresholds in Figure 6(c) and the following analysis on pulsed pumping. Under pulsed pumping, NG can be easily achieved even under the current loss level of EPL_GO = 3 dB/cm, together with significant PG enhancement compared to the uncoated SiN waveguides (see details in Note 4 of SI). These results agree with our experimental results in Ref. [1].

Extending the analysis for GO-SiN hybrid waveguides in Figure 6, Figure 7 provides the corresponding results for GO-coated silicon nanowire waveguides. The waveguide structure is the same as that in Figure 3. The self-assembly coating method [58] enables conformal coating of the GO film around the silicon nanowire waveguide. In our simulation, the nonlinear parameter for the GO-silicon hybrid waveguide was 668 W^-1m^-1 [20], and the PL of the uncoated silicon waveguide was 1.0 dB/cm [41]. Compared GO-SiN waveguides, the stronger GO mode overlap in silicon nanowire waveguides yields a higher nonlinear parameter, and this comes at the expense of a higher EPL_GO of ~20 dB/cm) and a lower P_thres of ~0.02 W [39]. As can be seen, even with decreased EPL_GO and increased P_thres achievable through fabrication and material optimization, the GO-coated silicon nanowire waveguides remain unable to achieve NG over the investigated CW and pulse pump power ranges. The main limitation for this is still the significant TPA of silicon, and the hybrid waveguides also exhibit higher PG than comparable waveguides without GO (see details in Note 6 of SI).

In Figure 8 and Figure 9, we analyze the PG for other types of silicon waveguides coated with monolayer GO films, including rib, etchless, slab, and slot waveguides as discussed in Figure 4 and Figure 5.Figure 8shows the PG versus EPL_GO and pump power for the hybrid rib, etchless, and slab waveguides. For each type of waveguide, the results for both CW and pulsed pumping are provided. To clearly show the evolution of PG, we chose small ranges for the CW (< 1 W) and pulsed (< 20 W) pump powers. We also selected L = 1 mm as an optimized length for achieving NG, and the corresponding results for a longer length of L = 2 cm is provided in Note 7 of SI. Compared to the uncoated waveguides in Figure 4, the incorporation of GO significantly enhances the optical nonlinearity, yielding calculated γ values of ~38, ~1626, and ~1669 W^-1m^-1, respectively. This enhancement is accompanied by increased excess propagation loss, with calculated EPL_GO of ~1.9, ~19.8, and ~20.4 dB/cm based on current GO loss of k = ~0.012. We also calculated P_thres values (only for CW pumping) of the three hybrid waveguides based on the GO mode overlap, which are ~0.05 W, ~0.01 W, and ~0.01 W, respectively.

In Figure 8, the hybrid rib waveguides do not show NG for both CW and pulsed pumping, even with potentially decreased EPL_GO and increased P_thres. In contrast, the hybrid etchless and slab waveguides exhibit NG for both CW and pulsed pumping. For CW pumping, NG can be achieved by either reducing EPL_GO or increasing P_CW. At P_CW = 1 W and EPL_GO = 5 dB/cm, the PG values for the hybrid etchless and slab waveguides are ~2.0 dB and ~2.7 dB, respectively. For pulsed pumping, NG becomes much easier to achieve and can be realized even with the current GO loss level. In addition, the hybrid slab waveguide exhibits better performance, providing a higher maximum PG value (~4.3 dB) and a broader pulse power range for achieving NG (P_pulse < 13 W) compared with the hybrid etchless waveguide (with a maximum PG of ~3.0 dB and P_pulse < 7 W for achieving NG). The superior performance of the hybrid slab waveguides compared to other types of GO-silicon waveguides highlights the importance of increasing GO mode overlap (to enhance optical nonlinearity) and suppressing silicon mode overlap (to minimize TPA) when optimizing the waveguide structure for achieving high PG.

Figure 9 shows the results for GO-silicon slot waveguides, which have the same waveguide structures as those in Figure 5 except that the slot region is conformally coated with monolayer GO films. The optical nonlinearity of the slot waveguides can be significantly improved by incorporating 2D GO films, showing calculated γ values of ~2517, ~1774, and ~1335 W^-1m^-1 for the waveguides with 100-nm, 150-nm, and 200-nm slot widths, respectively. The calculated EPL_GO were ~59.3, ~51.0, and ~44.9 dB/cm for these waveguides, with corresponding P_thres values of ~0.006 W, ~0.007 W, and ~0.008 W, respectively. Similar to Figure 8, we analyze the PG versus EPL_GO and pump power for both CW and pulsed pumping. For CW pumping, the 100-nm and 150-nm slot waveguides can achieve NG at the current GO loss level by increasing P_thres. At P_CW = 2 W and with current GO loss of k = ~0.012, a PG of ~0.6 dB can be obtained for the 100-nm slot waveguides. On this basis, a much higher PG of ~6.7 dB could be achieved by reducing the GO loss to k = ~0.003. For the 200-nm slot waveguides with weaker mode confinement, the PG is lower at the same P_CW and k, and achieving NG requires a combined decrease in EPL_GO and increase in P_thres. For pulsed pumping,

NG can be readily achieved at the current GO loss level when P_pulse is less than ~20 W, ~30 W, and ~40 W for the 100-nm, 150-nm, and 200-nm slot waveguides, respectively. At P_pulse = 10 W and with current GO loss level, the PG values for the three waveguides are ~1.8 dB, ~4.1 dB, and ~5.2 dB, respectively. The 200-nm slot waveguide achieves a higher PG than the 100-nm and 150-nm slot waveguides, showing a trend opposite to the results for CW pumping. This is mainly due to the fact that the much stronger mode confinement in the 100-nm and 150-nm slot waveguides also significantly enhances TPA under pulsed pumping with high peak powers. It is also worth noting that these PG values of the hybrid slot waveguides are higher than comparable values of ~-0.7 dB for the hybrid slab waveguide in Figure 8(c-iii) and ~-15.8 dB for the hybrid nanowire waveguide in Figure 7(c-iii), reflecting that the slot waveguide structure exhibits the best performance for achieving high PG via OPA among all the waveguide structures we investigated.

As mentioned previously, a phase-matched condition is assumed for the simulation in Figure 2Figures –Figure 9. Therefore, the simulated PG values represent the maximum achievable values. Phase mismatch over a large wavelength range degrades the OPA performance, and we have also performed simulations to investigate the influence of phase mismatch on the PG. Detailed results are provided in Note 8 of SI.

In Figure 10, we compare the PG of SiN and silicon waveguides incorporating different 2D materials, including GO discussed previously, as well as graphene and MoS₂ as representative examples. For comparison, we performed simulation for the devices incorporating monolayer GO, graphene, MoS₂ films with typical thicknesses of ~2 nm [1], ~0.3 nm [62], and ~0.7 nm [63], respectively. Among these three 2D materials, GO with a bandgap of ~2.0 eV ‒3.5 eV exhibits the lowest third-order optical nonlinearity (n₂ = ~1.5 × 10^-14 m²/W [64]) and the lowest linear optical absorption (k = ~0.012 at ~1550 nm [1]) at the near-infrared wavelengths we investigated. Graphene with zero bandgap exhibits the highest absolute value of third-order optical nonlinearity (n₂ = ~-1.0 × 10^-13 m²/W [65]) and the highest linear optical absorption (with k = ~1.35 at ~1550 nm [66]). MoS₂ is a representative transition metal dichalcogenide (TMDC) with a direct bandgap of ~1.8 eV ‒ 1.9 eV in the monolayer form, exhibiting moderate third-order optical nonlinearity (n₂ = ~2.3 × 10^-14 m²/W [67]) and linear optical absorption (k = ~0.1 at ~1550 nm [67]). In our simulation, the nonlinear optical absorption of graphene was taken into account using the nonlinear absorption coefficient of β = 1.0 × 10^-7 m/W reported in Ref. [65]. For GO and MoS₂, their nonlinear optical absorption at near infrared wavelengths was not considered given their large optical bandgaps, although we notice that doping or defect states can induce strong nonlinear optical absorption in these materials [34,67].

Figure 10(a-i) shows PG versus pump power for SiN waveguides incorporating the three 2D materials under CW pumping. Except for the coated 2D materials, the SiN waveguides discussed here are the same as those in Figure 6. In our simulation, a relatively short waveguide length of L = 1 mm was chosen to mitigate performance degradation induced by the loss of 2D materials. The simulation of GO was performed based on its current level of linear loss and thermal stability. The calculated nonlinear parameter and graphene-induced excess propagation loss for the graphene-coated SiN waveguide were γ = ~-50 W^-1m^-1 and EPL_grahene = ~123 dB/cm, respectively. For the MoS₂-coated SiN waveguide, these parameters were γ = ~32 W^-1m^-1 and EPL_MoS2 = ~24 dB/cm. As can be seen, none of the three waveguides achieves NG under CW pumping, and the GO hybrid waveguide exhibits the highest PG among them. The latter indicates a better trade-off between enhanced optical nonlinearity and increased loss for the GO hybrid waveguide. Figure 10(a-ii) shows the corresponding results of Figure 10(a-i) under pulsed pumping. As can be seen, all the three waveguides can achieve NG when P_pulse is sufficiently high. The graphene-coated waveguide still exhibits the lowest PG among the three, but the MoS₂-coated waveguide achieves higher PG than the GO-coated waveguide when P_pulse > 50 W, reflecting that at high power levels MoS₂ exhibits a better trade-off between optical nonlinearity and loss.

In Figure 10(b), we compare the PG of silicon nanowire waveguides incorporating the three 2D materials under both CW and pulsed pumping. Except for the coated 2D materials, the silicon nanowire waveguides are the same as those inFigure 7. The calculated γ and EPL_graphene for the graphene-coated silicon waveguide were ~1146 W^-1m^-1 and ~516 dB/cm, respectively. For the MoS₂-coated silicon waveguide, these parameters were γ = ~1716 W^-1m^-1 and EPL_MoS2 = ~193 dB/cm. These values are much higher than those for the graphene-coated and MoS₂-coated SiN waveguides due to the much stronger mode overlap with 2D materials in these waveguides. For CW pumping in Figure 10(b-i), NG is still not achieved in any of the three hybrid waveguides, and the GO hybrid waveguide achieves the highest PG ‒ similar to those observed in Figure 10(a-i). For pulsed pumping in Figure 10(b-ii), none of the three waveguides achieves NG, and the PG decreases as P_pulse increases, exhibiting a trend different from that observed in Figure 10(a-ii). This reflects the fact that the strong TPA of silicon remains dominant in these hybrid waveguides. The GO hybrid waveguide exhibits much higher PG than the other two waveguides under both CW and pulsed pumping, highlighting the smaller gap for GO towards achieving NG and motivating the focus on GO optimization in this work.

Finally, we note that some 2D materials, such as doped graphene and GO with defect states [68,69], exhibit saturable absorption (SA) behavior, featured by a decrease in loss with increasing input power ‒ opposite to that of TPA [10,70]. The decrease in loss enabled by the SA of 2D materials is beneficial for boosting the OPA processes and can be utilized to improve the PG, but it predominantly manifests under pulsed excitation with high peak powers, rather than under CW pumping. Although in this paper we mainly discuss the OPA arising from χ⁽³⁾ optical nonlinearity, it can also be realized by leveraging χ⁽²⁾ optical nonlinearity of non-centrosymmetric 2D materials. Recently [29], experimental demonstration of OPA in 2D TMDCs based on their high χ⁽²⁾ optical nonlinearity has been reported. Owing to the atomic thickness of the TMDCs, the PG was achieved without phase-matching constraints, and the efficiency was independent of the in-plane polarization of the signal and pump lights. Similar to TMDCs, GO exhibits a non-centrosymmetric atomic structure that yields a large χ⁽²⁾ optical nonlinearity that is absent in undoped graphene with a centrosymmetric atomic structure [71,72]. This property provides opportunities for future OPA applications. It is also worth noting that the family of 2D materials is rapidly growing, and many newly discovered 2D materials with ultrahigh optical nonlinearity are emerging, such as rhenium sulfur (ReS₂, a member of TMDCs with a large χ⁽²⁾ = ~1.0× 10^-9 m/V [73]), gallium selenide (GaSe, a member of IIIA-VIA[76–106 compounds with a large χ⁽³⁾ = ~1.4 × 10^-16 m²/V² [74]), and bismuth selenide (Bi₂Se₃, a typical topological insulator with ultra-broad response bandwidth spanning from the visible to the terahertz regions [75]). This work has broad implications for microcombs [76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106] and their applications to microwave photonics, neuromorphic processors and communications. [107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157] The addition and use of 2D materials [158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203] will add extra functionality to microcomb chips for potential applications to quantum photonics [204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219] and other areas. [220,221,222,223,224,225,226,227,228,229] As new materials continue to be discovered, this field is expected to see further exciting progress in the future.

4. Conclusions

In this perspective, we discuss the current challenges and future opportunities towards achieving NG via OPA in photonic chips integrated with 2D materials. Based on material parameters obtained from our previous experiments, we systematically analyze the feasibility of achieving NG via OPA in silicon and SiN waveguides incorporating 2D GO films under both CW and pulsed pumping. Our analysis reveals that, although a gap remains in achieving NG with current fabrication capabilities, it is not fundamental and can be bridged through combined efforts in optimizing waveguide structure, reducing loss of GO, and improving GO’s thermal stability. We also compare the performance for different waveguide structures and 2D materials, and analyze the existing gap in each case. Our work provides a roadmap for engineering OPA-induced PG in integrated photonic devices incorporating 2D materials. With continued improvements in fabrication capabilities and optimization of material properties, nonlinear integrated photonic devices incorporating advanced 2D materials are expected to progressively deliver on their high promise in the future.