Design of silicon waveguides integrated with 2D graphene oxide films for Kerr nonlinear optics

The Kerr nonlinear optical performance of silicon nanowire waveguides integrated with 2D layered graphene oxide (GO) films is theoretically studied and optimized based on experimentally measured linear and nonlinear optical parameters of the GO films. The strong mode overlap between the silicon nanowires and highly nonlinear GO films yields a significantly enhanced Kerr nonlinearity for the hybrid waveguides. A detailed analysis for the influence of waveguide geometry and GO film thickness on the propagation loss, nonlinear parameter, and nonlinear figure of merit (FOM) is performed. The results show that the effective nonlinear parameter and nonlinear FOM can be increased by up to ≈52 and ≈79 times relative to bare silicon nanowires, respectively. Self-phase modulation (SPM)-induced spectral broadening of optical pulses is used as a benchmark to evaluate the nonlinear performance, examining the trade-off between enhancing Kerr nonlinearity and minimizing loss. By optimizing the device parameters to balance this, a high spectral broadening factor of 27.6 can be achieved ‒ more than 6 times that achieved in previous experiments. Finally, the influence of pulse chirp, material anisotropy, and the interplay between saturable absorption and SPM is also discussed. These results provide useful guidance for optimizing the Kerr nonlinear optical performance of silicon waveguides integrated with 2D layered GO films.

Realizing nonlinear optical devices in integrated photonic chips would reap the greatest benefits in terms of device footprint, scalability, stability, and mass production. Although silicon has been a dominant platform for integrated photonic chips [14][15][16][17][18][19], its strong twophoton absorption (TPA) in the near-infrared telecom wavelength band significantly limits its nonlinear performance [1,2]. Even if the free carriers generated by TPA are swept out by p-i-n junctions [20], silicon's relatively poor intrinsic nonlinear figure of merit (FOM) in the telecom band is below what is needed to achieve superior nonlinear performance [21][22][23]. In response to this, other complementary metal-oxide-semiconductor (CMOS) compatible platforms have been explored for nonlinear optics, such as silicon nitride (SiN) [24,25] and Hydex [26 -45]. However, while these platforms have negligible TPA, they also have a comparatively low Kerr nonlinearity [2,26].
To overcome these limitations, two-dimensional (2D) materials that exhibit an ultrahigh optical nonlinearity, such as graphene [46,47], graphene oxide (GO) [48,49], black phosphorus [50,51], and transition metal dichalcogenides (TMDCs) [52,53], have been integrated onto chips to enhance the nonlinear optical performance. Amongst the different 2D materials, GO has become highly promising due to its ease of preparation as well as flexibility in tuning its material properties [54][55][56][57][58][59][60]. Previously, GO has been shown to have a giant Kerr nonlinearity -about 4 orders of magnitude higher than silicon [58,61]. Moreover, GO has a large bandgap (> 2 eV [54,62]) that yields a linear absorption that is over 2 orders of magnitude lower than graphene at infrared wavelengths [63] as well as low TPA in the telecom band [62,64]. Based on this, enhanced SPM in GO-coated silicon nanowires [48] and FWM in GO-coated SiN and Hydex devices have all been demonstrated [49,63,65]. An even more appealing advantage of GO is the ability to precisely control the film thickness, size, and position on integrated chips via large-area, transfer-free, layer-by-layer coating methods together with standard lithography and lift-off processes [62,65,66]. In contrast to the imprecise, largely unrepeatable, and unstable approach of mechanical layer transfer processes that have been widely used for other 2D materials such as graphene and TMDCs [67,68], this method enables cost-effective, large-scale, and highly precise integration of 2D layered GO films on a chip, representing a significant advance towards manufacturable integrated photonic devices incorporating 2D materials [60].
Recently [48], we demonstrated an enhanced Kerr nonlinearity in silicon-on-insulator (SOI) nanowires integrated with 2D layered GO films, verified through SPM measurements with picosecond optical pulses. We achieved a maximum spectral broadening factor (BF) of 4.34 for an SOI nanowire with a patterned GO film. In this paper, we fully analyse and optimize the Kerr nonlinear optical performance of GO-coated SOI nanowires based on experimentally measured linear and nonlinear optical parameters of the GO films. We investigate the influence of waveguide geometry and GO film thickness on the propagation loss, nonlinear parameter, and nonlinear FOM. By increasing the mode overlap with GO films, we show that the effective nonlinear parameter and nonlinear FOM of the hybrid waveguides can be increased by up to ≈52 and ≈79 times with respect to bare SOI nanowires, respectively. We find that this needs to be balanced with an accompanying increase in linear loss, and use SPM-induced spectral broadening of the optical pulses to examine the trade-off between enhancing the Kerr nonlinearity and minimizing loss. By changing the device parameters to balance this trade-off, we achieve a high spectral BF of 27.6, more than 6 times higher than what has been achieved experimentally. Finally, we discuss the influence of pulse chirp, material anisotropy, and the interplay between saturable absorption (SA) and SPM on the Kerr nonlinear optical performance. These results highlight the significant potential to improve on experimental results [48] and provide detailed solutions for optimizing the Kerr nonlinear performance of SOI nanowires integrated with 2D layered GO films.

2D GO films and device structure
Figure 1(a) shows schematics of the atomic structures and bandgaps of graphene and GO.
As compared with graphene, GO provides more flexibility to tailor its material properties by manipulation of the oxygen-containing functional groups (OFGs) in the basal plane and sheet edges, including epoxy, hydroxyl, carbonyl and carboxyl groups [54,69]. Also, in contrast to graphene that has a metallic behavior with zero bandgap, GO has a large bandgap > 2 eV [54,62] that yields both low linear light absorption and TPA in the telecom band, which are highly desirable for Kerr nonlinear processes such as FWM and SPM [60]. Figure 1(b) shows a schematic of an SOI nanowire waveguide integrated with a GO film. The fabrication of the SOI nanowire can be achieved via either deep ultraviolet photolithography or e-beam lithography followed by inductively coupled plasma etching, all of which are mature silicon device fabrication technologies [19,70]. The GO film coating, with a thickness of ~2 nm per layer [48], can be achieved using solution-based methods that yield layer-by-layer film deposition [62,63,66]. As compared with the sophisticated transfer processes for other 2D materials such as graphene and TMDCs [71,72], these coating methods enable transfer-free and conformal film coating, with high fabrication stability, repeatability, precise control of the film thickness (i.e., number of layers), and extremely good film attachment onto integrated photonic devices [60]. Precise control of the film length and coating position can be achieved by patterning the film with standard lithography and lift-off processes [65,66]. This, together with the accurate control of the film thickness, allows the optimization of the Kerr nonlinear performance by adjusting the film thickness, length, and coating position.    49,66], the GO film thickness is assumed to be proportional to N, with a thickness of 2 nm per layer. We assume that the input pulse shape has a Gaussian profile: (1) where P0 is the pulse peak power, C0 is the initial chirp, and T0 is the pulse width (the halfwidth at 1/e intensity). The corresponding pulse energy (PE) can be described as

PE C0
In the following sections, we first investigate the influence of waveguide geometry (W, H) and GO film thickness (N) on the linear and nonlinear loss of the hybrid waveguides in Section 3, followed by their effect on the effective nonlinear parameter and nonlinear FOM in Section 4. In Section 5, SPM-induced spectral broadening of optical pulses is investigated to illustrate the trade-off between optimizing the nonlinear FOM and minimizing linear loss.
Using the results of Sections 3 and 4, we optimize the spectral broadening in GO-coated SOI nanowires by adjusting the device parameters such as waveguide geometry (W, H), layer number (N), pattern length (Lc), and coating position (L0). Finally, we discuss the influence of loss, pulse chirp, material anisotropy, and the interplay between SA and SPM on the Kerr nonlinear performance in Section 6.

Linear and nonlinear loss
In this section, we investigate the linear and nonlinear loss of GO-coated SOI nanowires with different waveguide geometries (W, H) and GO film thickness (N). supports an in-plane interaction between the evanescent field and film, which is much stronger than the out-of-plane interaction due to the large optical anisotropy of 2D materials [46,66,68]. In where βTPA, Si = 5 × 10 -12 m/W and σ = 1.45 × 10 -21 m are the TPA and FCA coefficients of silicon, respectively, A(z, t) is the slowly varying temporal envelope of the optical pulse along the waveguide (i.e., z axis), and Aeff is the effective mode area. Nc is the free carrier density given by [74]: where ℏ is Planck's constant, ω is the angular frequency, and τc = ~1 ns is the effective carrier lifetime. When T0 (e.g., 3.9 ps) is much shorter than τc, the τc term in Eq. (4) can be ignored as the generated free carriers do not have enough time to recombine within the pulse duration [72,74]. The loss in Figure 3(a) decreases with waveguide height H and width W, indicating a stronger TPA and FCA in SOI nanowires that have smaller waveguide dimensions.
As noted previously, the TPA and FCA of GO is very low at near-infrared wavelengths [61,63] and so the nonlinear loss of GO is dominated by SA arising from the ground-state bleaching of the sp 2 domain with a typical energy gap of ∼0.5 eV [58,61,75]. Figure 3

(b)
shows the SA loss of the GO films for the hybrid waveguides with different waveguide geometries (H and W) but the same GO film parameters of N = 10 and Lc = 0.4 mm. The SA loss was calculated via [76,77] : where αsat is the SA coefficient, Isat is the saturation intensity, and η is the GO mode overlap.
In our calculations, the layer dependent αsat and Isat were obtained from experimental results

Nonlinear parameter and nonlinear FOM
In this section, we further investigate the influence of waveguide geometry (W and H) and GO film thickness (N) on the effective nonlinear parameter and nonlinear FOM. Figures 4(a) and (  In Figures 4(a) and (b), γeff increases with layer number N and decreases with waveguide height H and width W, showing a similar trend to the propagation loss in Figures 2(a) and   (b). This indicates that an increased mode overlap leads to both an increased Kerr nonlinearity as well as linear loss. In Figure 4(c), when N = 20, W = 400 nm, and H = 140 nm, a high γeff of 16711 W -1 m -1 is obtained, which is ≈52 times that of the associated bare SOI nanowire (with the same waveguide geometry) and ≈4 times that of a comparable hybrid waveguide (with the same GO film thickness) with W = 500 nm and H = 220 nm. These results reflect the huge improvement in Kerr nonlinearity that can be obtained by not only introducing the GO films into SOI nanowires but by properly optimizing γeff by engineering the GO mode overlap. Based on the effective nonlinear parameter of the hybrid waveguides, we further investigate the effective nonlinear FOM (FOMeff ), which is widely used to quantitively characterize the trade-off between the Kerr nonlinearity and nonlinear loss [2]. Figures 5(a) and (b) Figure 5(c). The FOMeff 's were calculated by [1,2]: where βTPA, eff is the effective TPA coefficient obtained by fitting the results in Figures 3(b) and (c) and n2, eff is the effective Kerr coefficient calculated from where γeff is the effective nonlinear parameter in Figure 4 and Aeff is the effective mode area.
The βTPA, eff 's are influenced by SA in the GO layers, which becomes more significant as the mode overlap increases. The SA decreases the overall absorption as the pulse energy PE increases, which acts oppositely to TPA and results in the effective βTPA, eff 's of the hybrid waveguides being smaller than that of comparable bare SOI nanowires having the same waveguide geometries.
As shown in Figure 5( Figures 5 (a) and (b), the effective FOM increases with layer number N and decreases with waveguide height H and width W, showing similar trends to the propagation loss in Figure 2 and effective nonlinear parameter in Figure 4. This indicates that the nonlinear FOM can be improved by increasing the GO mode overlap via reducing the waveguide geometry or increasing the GO film thickness.

SPM-induced spectral broadening of optical pulses
Although the nonlinear FOM (Eq. (7)) has been widely used to characterize the Kerr nonlinear optical performance of bulk materials [2,78], it does not represent the full picture.
The nonlinear optical performance of hybrid waveguides incorporating 2D materials (or indeed for any device), must factor in the effects of the linear propagation loss [60]. For GOcoated SOI nanowires, the increased mode overlap yields an increased nonlinear FOM, but at the expense of an increased linear loss that sometimes can be significant. In this section, we examine SPM-induced spectral broadening of optical pulses to illustrate this trade-off. We show that, in addition to the waveguide geometry and GO film thickness, other parameters such as the GO film length and coating position are also very important to optimize the Kerr nonlinear performance. where ∆ω0 and ∆ωrms are the root-mean-square (RMS) spectral width of the input and output optical spectra, respectively. The spectral broadening was calculated using a split-step Fourier method to solve the nonlinear Schrödinger equation (NLSE) as follows [72,74]: where i = √1, β2 is the second-order dispersion coefficient, µ is the free carrier dispersion (FCD) coefficient of silicon, and α is the total loss including both the linear loss (αL) in Figure 2 and the nonlinear loss in Figure 3, which can be expressed as: (11) In our calculations, the hybrid waveguides were separated into bare and GO-coated segments.
Eq. (10) was solved for each segment, with the output from the previous segment used as the input to the following segment. In Figure 6(  This could reflect the fact that the linear loss can become a limiting factor for the nonlinear performance of the hybrid waveguidesotherwise the maximum spectral broadening would have been achieved for the smallest W where the FOMeff is the greatest. We also note that the broadened spectra exhibits a slight asymmetry, which is mainly induced by the FCA and FCD in silicon [74].  gets smaller for lower propagation loss of the bare waveguides, being much lower for Hydex and SiN waveguides [49,63] versus SOI nanowires studied here.  reflecting that there is still room for improvement on the basis of the maximum BF in Figure   7(b) (i.e., 9.1) by optimizing the GO film length. Figure 8(b) depicts the corresponding results for the hybrid waveguides with W = 600 nm and H = 140 nm, which shows the best spectral broadening among 25 different waveguide geometries considered in our study. A maximum BF of 27.6 is achieved when N = 10 and Lc = 1.43 mm, which is ≈2.2 times higher than the maximum BF in Figure 8(a) and more than 6 times higher than previous experiments [48], reflecting the potential for improvement by optimizing the waveguide geometry.

Discussion
In this section, we consider the influence of loss, pulse chirp, material anisotropy, and the interplay between SA and SPM on the Kerr nonlinear performance.  [49,65]. After including αSA-GO, the overall loss decreases, enhancing the SPM and spectral broadening. In Figure 9(c), although the intrinsic TPA itself leaves the pulse spectrum symmetric, the resulting FCA makes it considerably asymmetric.
While the linear loss of the layered GO films does pose a limitation for the Kerr nonlinear performance of the hybrid waveguide, as mentioned, it is not a fundamental material property, and any reduction by optimizing the film fabrication processes would improve the performance further.   films. In Figure 10(a), we show the ratio of the power in the GO on both sidewalls to the power in entire film, which is < 3% and decreases with N. This indicates that the TE mode overlap with GO on both waveguide sidewalls is negligible compared with that for GO on the waveguide top, and so we use the in-plane n2 of GO (corresponding to TE polarization) in our calculations, neglecting any anisotropy in n2 of the 2D layered GO films.   Figure 10(c) compares the corresponding SA loss of the hybrid waveguides. The maximum difference between them is < 0.2%, reflecting that the influence of SPM on SA is negligible. This is mainly because the total length of the hybrid waveguides (3 mm) was much shorter than the dispersion length (> 1 m), and so any change in the pulse spectrum induced by SPM did not significantly affect the temporal pulse shape [79].
In Figure. 11(a), the input pulse spectra for the same absolute value of chirp (0.3) but with opposite signs overlap each other, in agreement with Eq. (12). The corresponding pulse spectra and BFs are shown in Figures 11(b) and (c), respectively. Compared with unchirped pulses (i.e., C0 = 0), the spectral broadening increases when C0 > 0 but decreases when C0 < 0, since the positive chirp induced by SPM adds to a positive C0 , while it is offset by a negative C0.

Conclusion
In summary, we theoretically investigate and optimize the Kerr nonlinear optical performance of SOI nanowires integrated with 2D layered GO films. Detailed analysis of the influence of waveguide geometry and GO film thickness on the propagation loss, nonlinear parameter, and nonlinear FOM is performed. We show that the effective nonlinear parameter and nonlinear FOM can be increased by up to ≈52 and ≈79 times relative to bare SOI nanowires, respectively. To examine the trade-off between increasing the Kerr nonlinearity and minimizing linear loss, we consider SPM-induced spectral broadening of optical pulses.
We show that a high BF of 27.6 can be achieved by properly balancing this trade-off, more than a factor of 6 higher than what has been achieved experimentally. Finally, the role of pulse chirp, material anisotropy, and the interplay between SA and SPM in SPM-induced spectral broadening is also investigated. These results highlight the significant potential of GO films to enhance the Kerr nonlinear optical performance of SOI nanowires for practical applications.