According to the theory in Refs. [32, 35, 46], we modeled the SPM process in the GO-coated SOI nanowires as follows:

$$\frac{\text{∂}\text{A}}{\text{∂}\text{z}}\text{= -}\frac{\text{i}{\text{β}}_{\text{2}}}{\text{2}}\frac{{\text{∂}}^{\text{2}}\text{A}}{\text{∂}{\text{t}}^{\text{2}}}\text{ + }\text{i}\text{γ}{ \left|\text{A}\right|}^{\text{2}}\text{A}\text{ - }\frac{\text{1}}{\text{2}}\text{iσ}\text{μ}{\text{N}}_{\text{c}}\text{A }\text{- }\frac{\text{1}}{\text{2}}\text{α} \text{A}$$

2

where *i* = \(\sqrt{\text{1}}\), *A*(*z*, *t*) is the slowly varying temporal pulse envelope along the propagation direction *z* of the waveguide, *β*2 is the second-order dispersion coefficient, *γ* is the waveguide nonlinear parameter, *σ*, *µ*, and *N**c* are the FCA coefficient, free carrier dispersion (FCD) coefficient, and free carrier density in silicon, respectively, and *α* is the total loss including both linear loss and nonlinear loss. The nonlinear loss includes TPA and FCA losses of bare SOI nanowires and SA loss induced by the GO films. In Eq. (2), we keep only the *β*2 item since the physical length of the waveguides is smaller than the dispersion length [50].

Based on Eq. (2), we fit the measured spectra to obtain the nonlinear parameters (*γ*’s) for the bare and hybrid waveguides. The GO-coated SOI nanowires were divided into uncoated (with silica cladding) and hybrid segments (coated with GO films) to perform numerical calculation, where the output from the previous segment was set as the input for the subsequent one. We obtain a fit *γ* of ~ 288 W− 1m− 1 for the bare SOI nanowire and fit *γ*’s for the hybrid waveguides with 1 and 2 layers of GO of ~ 675 W− 1m− 1 and ~ 998 W− 1m− 1, respectively, which are ~ 2.3 and ~ 3.5 times that of the bare SOI nanowire. These results show a good agreement with the previous work [35], indicating the high consistency and further confirming the remarkably improved Kerr nonlinearity for the hybrid waveguides.

Based on the fit *γ*‘s of the hybrid waveguides, we further extract the Kerr coefficient (*n*2) of the layered GO films using [40, 51, 52] :

$${\text{γ}}_{}\text{ =}\frac{\text{2π}}{{\text{λ}}_{\text{c}} }\frac{{\iint }_{\text{D}}^{}{{\text{n}}_{\text{0}}}^{\text{2}}\left(\text{x}\text{, }\text{y}\right){\text{n}}_{\text{2}}\left(\text{x}\text{, }\text{y}\right){{\text{S}}_{\text{z}}}^{\text{2}}\text{dxdy}}{{\left[{\iint }_{\text{D}}^{}{\text{n}}_{\text{0}}\left(\text{x}\text{, }\text{y}\right){\text{S}}_{\text{z}}\text{dxdy}\right]}^{\text{2}}}$$

3

where *λ**c* is the pulse central wavelength, *D* is the integral of the optical fields over the material regions, *S**z* is the time-averaged Poynting vector calculated using mode solving software, *n*0 (*x*, *y*) and *n*2 (*x*, *y*) are the refractive index profiles calculated over the waveguide cross section and the Kerr coefficient of the different material regions, respectively. The values of *n*2 for silica and silicon used in our calculation were 2.60 × 10− 20 m2 W− 1 [26] and 6.0 × 10− 18 m2 W− 1, respectively, with the latter obtained by fitting the experimental results for the bare SOI nanowire.

The extracted *n*2 of 1 and 2 layers of GO are ~ 1.45 × 10− 14 m2 W− 1 and ~ 1.36 × 10− 14 m2 W− 1, respectively, which are about 4 orders of magnitude higher than that of silicon and agree reasonably well with our previous measurements [35]. Note that *n*2 of 2 layers of GO is lower than that of 1 layer of GO, which we infer it may arise from the increased inhomogeneous defects within the GO layers and imperfect contact between the multiple GO layers.

Previously, we have fabricated SOI nanowires with much thicker GO films (up to 20 layers [35]), and have performed detailed theoretical analysis for the influence of GO film’s length, thickness, and coating position on the nonlinear performance of GO-coated integrated waveguides [28, 29]. In this work, the maximum GO film thickness for the fabricated device is 2 layers (i.e., ~ 4.2 nm), which is mainly used to compare the spectral broadening performance for the picosecond and femtosecond optical pulses.

Based on the SPM modeling in Eq. (2) and the fit parameters of GO, we compare the BFs of femtosecond and picosecond optical pulses after transmission through the same GO-coated SOI nanowires. Figures 5a and 5b shows the BFs of femtosecond and picosecond optical pulses versus GO film length (*L**c*) for the hybrid waveguides with 1 and 2 layers of GO, respectively. The corresponding results for the bare SOI nanowire with constant BFs are also shown for comparison. Both the picosecond and femtosecond optical pulses had the same repetition rate of ~ 60 MHz. For the picosecond pulses with a pulse duration of ~ 3.9 ps, the average input power is 3 mW, corresponding to a peak power range of 13 W ‒ the same as that used in our previous experiment [35]. For the femtosecond pulses with a pulse duration of ~ 180 fs, the average input power is 1.7 mW, corresponding to a peak power of 160 W ‒ the same as that used for the SPM measurements in Section 4.

In Fig. 5a, the BFs of femtosecond optical pulses after propagation through the hybrid waveguides first increase with *L**c* and then decrease, with the maximum values being achieved at intermediate film lengths. The optimized film length corresponding to the maximum BF for the device with 2 layers of GO is smaller than that for the device with 1 layer of GO. This reflects that the Kerr nonlinearity enhancement dominates for the devices with relatively small *L**c* and layer number *N*, and the influence of loss increase becomes more significant as *L**c* and *N* increase.

For the picosecond optical pulses in Fig. 5b, the BFs after propagation through the hybrid waveguides are lower than the BFs of the femtosecond optical pulses at the same *L**c*. This is because the femtosecond optical pulses with a much higher peak power drive more significant SPM in the hybrid waveguides. Unlike the trend in Fig. 5a, the BFs in Fig. 5b increase monotonically with *L**c*. This is mainly due to the fact that the higher peak power of femtosecond optical pulses also induces higher TPA of the SOI nanowires, resulting in a trade-off between both enhanced SPM and TPA in the hybrid waveguides.

In Table 1, we compare the nonlinear performance of different integrated waveguides incorporating GO, along with corresponding results for the bare waveguides. As can be seen, the fit *γ*, *n*2, *FOM*1 and *FOM*2 of GO-coated SOI nanowires in this work show good agreement with those in the previous work obtained by fitting the experimental results of picosecond optical pulses [35], highlighting the high consistency of our GO films. We calculated two different figure-of-merits, i.e., *FOM*1 and *FOM*2, which are widely studied and exploited in comparing nonlinear optical performance. The former one is defined from the perspective of nonlinear absorption [25, 26], whereas the latter one is defined based on the trade-off between Kerr nonlinearity and linear loss [53]. Interestingly, the two FOMs show contrary results for the different hybrid waveguides. The *FOM*1 of the GO-coated SOI nanowires is lower than the other two waveguides mainly due to the strong TPA of silicon, whereas its *FOM*2 is much higher resulting from the large *n*2 of silicon and its strong GO mode overlap.

Table 1

Comparison of nonlinear optical performance of different integrated waveguides incorporating GO. FOM: figure of merit.

Integrated waveguide | GO layer number a) | Waveguide dimension (µm) | *γ* (W− 1m− 1) b) | Fit *n*2 (×10− 14 m2/W) | *PL* (dB/cm) c) | *FOM*1 (a. u.) d) | *FOM*2 (W− 1) e) | Ref. |

SOI | *N =* 0 | 0.50 × 0.22 | 288.00 | 6.00 × 10− 4 | 4.30 | 0.74 | 0.75 | |

*N =* 1 | 668.01 | 1.42 | 24.80 | 2.07 | 0.96 | [35] |

*N =* 2 | 990.23 | 1.33 | 38.91 | 2.81 | 1.03 | |

SOI | *N =* 0 | 0.50 × 0.22 | 288.00 | 6.00 × 10− 4 | 4.30 | 0.74 | 0.75 | This work |

*N =* 1 | 675.15 | 1.45 | 24.60 | 2.08 | 0.97 |

*N =* 2 | 998.18 | 1.36 | 38.52 | 2.83 | 1.05 |

Si3N4 | *N =* 0 | 1.60 × 0.66 | 1.51 | 2.60 × 10− 5 | 3.00 | >> 1 | 0.016 | [38] |

*N =* 1 | 13.14 | 1.41 | 6.05 | 0.089 |

*N =* 2 | 28.23 | 1.35 | 12.25 | 0.099 |

Hydex | *N =* 0 | 2.00 × 1.50 | 0.28 | 1.28 × 10− 5 | 0.24 | >> 1 | 0.004 | [40] |

*N =* 1 | — | — | 1.26 | 0.007 |

*N =* 2 | 0.90 | 1.5 | 2.23 | 0.009 |

a) *N* = 0 corresponds to the results for the uncoated SOI nanowire, Si3N4, and Hydex waveguides, whereas *N* = 1 and 2 correspond to the results for the hybrid waveguides with 1 and 2 layers of GO, respectively.

b) *γ* is the nonlinear parameter. For the hybrid waveguides, *γ*’s are the effective values calculated based on Refs. [38, 40].

c) *PL* is the linear propagation loss of the GO-coated waveguides.

d) The definition of *FOM*1 = *n*2 / (*λβ**TPA*) is the same as those in Refs. [25, 26], with *n*2 and *β**TPA* denoting the effective Kerr coefficient and TPA coefficient of the waveguides, respectively, and *λ* denoting the light wavelength. The values for the Si3N4, and Hydex waveguides are > > 1 due to negligible TPA observed in these waveguides.

e) The definition of *FOM*2 = *γ × L**eff* is the same as that in Ref. [53]. Here, the GO films are uniformly integrated on the waveguides and the waveguides length for the SOI nanowire, Si3N4, and Hydex waveguides are 3 mm, 20 mm, 15 mm.

In Table 1, the *FOM*2 is a function of waveguide length *L* given by [53]

*FOM* 2 (*L*) = *γ × L**eff* (*L*) (4)

where *L**eff* (*L*) = [1 - *exp* (-*α**L* *× L*)] /*α**L* is the effective interaction length, with *γ* and *α**L* denoting the waveguide nonlinear parameter and the linear loss attenuation coefficient, respectively. Figure 6 shows *L**eff* and *FOM*2 versus waveguide length (*L*) for SOI nanowires uniformly coated with 1 and 2 GO layers, together with the result for the bare waveguide (i.e., *N* = 0). *FOM*2 first rapidly increases with *L* and then grows more progressively as *L* becomes longer. For a shorter *L*, the *FOM*2’s of the hybrid waveguides are higher than that of comparable bare waveguide, whereas the *FOM*2 of the bare waveguide gradually approaches and even exceeds those of the hybrid waveguides when *L* increases. This reflects that the negative influence induced by increased loss becomes more dominant as *L* increases.