Design analysis of 2D graphene oxide films integrated with microring resonators for FWM

—We theoretically investigate and optimize the performance of four-wave mixing (FWM) in microring resonators (MRRs) integrated with two-dimensional (2D) layered graphene oxide (GO) films. Owing to the interaction between the MRRs and the highly nonlinear GO films as well as to the resonant enhancement effect, the FWM efficiency in GO-coated MRRs can be significantly improved. Based on previous experiments, we perform detailed analysis for the influence of the GO film parameters and MRR coupling strength on the FWM conversion efficiency (CE) of the hybrid MRRs. By optimizing the device parameters to balance the trade-off between the Kerr nonlinearity and loss, we achieve a high CE enhancement of ~18.6 dB relative to the uncoated MRR, which is ~8.3 dB higher than previous experimental results. The influence of photo-thermal changes in the GO films as well as variations in the MRR parameters such as the ring radius and waveguide dispersion on the FWM performance is also discussed. These results highlight the significantly improved FWM performance that can be achieved in MRRs incorporating GO films and provide a guide for optimizing their FWM performance. uncoated MRR is achieved, which is ~8.3 dB higher than what has been achieved experimentally. We also discuss the influence of photo-thermal changes in the GO films as well as the variation of other MRR parameters such as ring radius and waveguide dispersion on the FWM performance. These results highlight the significant potential to improve on previous experimental results [18] and provide a guide for optimizing FWM performance of MRRs integrated with GO films.

I.  INTRODUCTION raphene oxide (GO) has become a rising star in the family of two-dimensional (2D) materials owing to its potential for mass production as well as the flexibility in tuning its material properties [1][2][3][4]. Recently, the excellent nonlinear optical properties of GO have attracted significant interest [5][6][7][8][9]. It has been reported that GO has an ultrahigh Kerr nonlinearity (n2) that is about 4 orders of magnitude higher than nonlinear bulk materials such as silicon and chalcogenide glasses [5,6,10]. In addition, GO has a large optical bandgap (typically > 2 eV [1,11]), which yields a material absorption that is over 2 orders of magnitude lower than graphene as well as negligible two-photon absorption (TPA) in the telecom band [12,13]. Another important advantage of GO is that it can be mass produced from natural graphite [3]. This, together with facile solution-based fabrication processes [14], is attractive for large-scale manufacturing of integrated devices that incorporate GO films [2,15,16].
Based on these advantages, many high performance nonlinear photonic devices that incorporate GO films [13, 17- 21] have been demonstrated -especially those based on complementary metal-oxide-semiconductor (CMOS) compatible integrated platforms [13,[17][18][19]. Enhanced fourwave mixing (FWM) in GO-coated doped silica and silicon nitride (SiN) waveguides has been reported [13,17], where conversion efficiency (CE) enhancements of up to 6.9 dB and 9.1 dB relative to the uncoated waveguides were achieved. Significant spectral broadening of optical pulses in GO-coated silicon waveguides induced by self-phase modulation (SPM) has also been observed [19], achieving a high spectral broadening factor of 4.34 for a device with a patterned film including 10 layers of GO. A significant enhancement in the nonlinear figure of merit (FOM) for silicon nanowires by a factor of 20 was also achieved, resulting in a FOM > 5.
In our previous work [18], we experimentally demonstrated enhanced FWM in CMOS compatible doped silica microring resonators (MRRs) integrated with 2D layered GO films. Due to the resonant enhancement effect [22,23], an increase of up to ~10.3 dB in the FWM CE was achieved. In this paper, we fully investigate and optimize the FWM performance of GOcoated MRRs based on previous experimental measurements of the GO film properties such as loss and Kerr nonlinearity, which are distinct from bulk materials and show a strong dependence on the film thickness and optical power. We perform a detailed analysis of the influence of the GO film parameters and MRR coupling strength on the FWM CE of the hybrid MRRs. By properly balancing the trade-off between the Kerr nonlinearity and loss, a high CE enhancement of ~18.6 dB relative to the uncoated MRR is achieved, which is ~8.3 dB higher than what has been achieved experimentally. We also discuss the influence of photo-thermal changes in the GO films as well as the variation of other MRR parameters such as ring radius and waveguide dispersion on the FWM performance. These results highlight the significant potential to improve on previous experimental results [18] and provide a guide for optimizing FWM performance of MRRs integrated with GO films. 2 II. DEVICE STRUCTURE Fig. 1(a) shows a schematic of an integrated MRR made from doped silica, with 1 layer of patterned GO film being coated on the planarized waveguide top surface. Inset shows a schematic illustration for the atomic structure of GO, including different oxygen-containing functional groups (OFGs) such as hydroxyl, epoxide, and carboxylic decorated on a graphene-like carbon lattice. In contrast to graphene, which has a metallic behavior with a zero bandgap [24], pristine GO is a dielectric material with a bandgap > 2 eV [1,12]. This is larger than both the single photon (~0.8 eV) and two-photon (~1.6 eV) energies around 1550 nm, resulting in negligible linear light absorption or TPA in the telecom band. We consider MRRs that are fabricated on a high index doped silica glass (Hydex) platform [25] via CMOS compatible processes. More details about the Hydex device fabrication can be found in Refs. [22,26,27]. As compared with GOcoated waveguides, GO-coated MRRs can dramatically improve the FWM efficiency by virtue of the resonant enhancement of the optical intensity within the resonant cavities [22,23], thus significantly reducing the required power. The upper cladding of the doped silica MRR is removed by chemical mechanical polishing (CMP) to obtain a planarized waveguide top surface for GO film coating. The GO film coating can be achieved using a solution-based method that yields layer-by-layer film deposition and precise control of the film thickness with an ultrahigh resolution of ~2 nm [12,28]. Unlike graphene or other 2D materials that are typically prepared via non-solution-based deposition followed by cumbersome layer transfer processes [29][30][31][32], our coating method enables large-area, transfer-free, and high-quality GO film coating with high fabrication stability, mass producibility, and excellent film attachment onto integrated waveguides [2,19]. Patterning of the films can be achieved via standard lithography and lift-off processes [18,28]. This, together with the layer-by-layer deposition of GO films, forms the basis for the optimization of the FWM performance of the hybrid MRRs with different GO film thicknesses and pattern lengths. Fig. 1(b) shows a schematic of the waveguide cross section of the hybrid MRR in Fig. 1(a). The corresponding transverse electric (TE) mode profile is shown in Fig. 1(c). We chose the TE polarization in our following analysis because it supports an in-plane interaction between the film and the evanescent field leaking from the MRR, which is much stronger than the   Table. b t 2 + κ 2 = 1 for lossless coupling is assumed for the directional couplers. c The circumference of the MRR is L = 2πR. d Here we show the extinction coefficient and Kerr coefficient at PCW = 25 dBm for N = 1 -50 based on the measured results in Ref. [18]. e Following our previous experimental measurements [18], the GO film thickness is assumed to be proportional to N, with a thickness of 2 nm per layer.
3 out-of-plane interaction due to the significant optical anisotropy of 2D films [28,31,32]. Table І summarizes the parameters of the doped silica MRR, the GO film, and the continuous-wave (CW) laser used in our following analysis, with the former two being further classified into material and physical parameters. Four-port MRRs with two identical directional couplers are used in our following analysis, which is consistent with that used in Ref. [18]. The GO-coated MRRs are designed based on, but not limited to, the Hydex platform.
In the following sections, we first investigate the powerdependent propagation loss and nonlinear parameters of the hybrid MRRs induced by photo-thermal changes in the GO films. Next, by properly balancing the trade-off between loss and the Kerr nonlinearity, we optimize the FWM CE in the hybrid MRRs by regulating the GO film parameters (N, Lc) and the MRR coupling strength (t). Finally, we discuss the influence of photo-thermal changes in the GO films as well as the effect of varying other MRR parameters such as the ring radius and waveguide dispersion on the FWM performance of the hybrid MRRs.

III. POWER-DEPENDENT PROPAGATION LOSS AND NONLINEAR PARAMETERS
As reported in previous work [18], the linear loss (k) and Kerr nonlinearity (n2) of GO films coated on integrated waveguides change with input CW power, particularly at high powers. This can be attributed to photo-thermal changes in the GO films, which is a combined result of power sensitive photo-thermal reduction as well as self-heating and thermal dissipation in the multilayered film structure [17,18,33]. Such changes are not permanent and can revert back after the power is turned off. In addition, these changes have a slow time response on the order of millisecond, which is different to FWM and TPA that have ultrafast response times on the order of femtoseconds [17]. Photo-thermal changes in the GO films lead to power-dependent propagation loss and nonlinear parameters for GO-coated waveguides, and this is further amplified in GO-coated MRRs due to resonant enhancement. In this section, we investigate the power-dependent propagation loss and nonlinear parameters of the hybrid MRRs induced by the photo-thermal changes in GO films.
We first calculate the resonant build-up factor () of a MRR as a function of its coupling strength (t) and round-trip field transmission factor (A). The  reflects the relationship between the input CW power (PCW) and the intracavity power (Pintra) in a MRR, which will be used for calculating Pintra directly related to the propagation loss and nonlinear linear parameters of the hybrid MRRs in our following analysis. Fig.   2(a) shows  versus t and A. The  was calculated at resonant wavelengths based on [34,35]: In Eq. (1), A can be further expressed as: where αc, u and Lc, u are the loss factors and lengths of the GO coated and uncoated waveguide segments, respectively. In Fig.  2(a), the maximum A is 0.989, which corresponds to the uncoated MRR (unless otherwise specified, the MRR radius used is 592 μmthe same as in Ref. [18]). The maximum  is achieved at t = 0.994 and A = 0.989, which is determined by the balance between t and A, as reflected by Eq. (1). Fig. 2(b) shows the MRR's extinction ratio (ER) versus t and A. The ER increases with A but decreases with t, mainly due to the change in the difference between intracavity loss and external coupling loss of a four-port MRR with two identical directional couplers [36][37][38][39][40][41][42]. In previous work [18], we measured the propagation loss and nonlinear parameters versus input CW power (PCW) for hybrid MRRs with 1 − 5 layers of uniformly coated and 10 − 50 layers of patterned (50-µm-long) GO, respectively. The coupling strength of the uncoated MRR was 0.912. In Figs. 3(a) and (b), we fit the measured power-dependent propagation loss and nonlinear parameters as functions of the intracavity power Pintra, which will be used for calculating FWM CE in next section. The input CW power PCW in Ref. [18] (from 15 dBm to 25 dBm) is converted to corresponding intracavity power Pintra based on the calculated  in Fig. 2(a).
In Fig. 3(a), the propagation loss increases with GO layer number N. This is mainly due to an increase in mode overlap for the hybrid MRRs with thicker GO films. A small contribution is from an increase in the material absorption arising from inhomogeneous defects and imperfect contact between the multiple GO layers [13,28]. As Pintra increases, the hybrid MRRs show an increased propagation loss, in contrast to the uncoated MRR that manifests a constant propagation loss. This further confirms the power sensitive photo-thermal changes in GO films. Following the same trends with the propagation loss, the nonlinear parameter γ in Fig. 3(b) increases with both N and Pintra. This reflects the trade-off between the Kerr nonlinearity and linear loss, which  Fit γ(Pintra), c(Pintra) based on the experimental results in Ref. [18] Matlab Measured γ and c in Ref. [18] Step is critical for optimizing the FWM performance. Note that in our calculation we neglect the influence of power-dependent loss on the round-trip field transmission factor A, since accounting for it would only lead to a maximum difference in For a fixed input power PCW, varying the GO film parameters such as layer number N and coating length Lc changes the intracavity loss and hence intracavity power Pintra. Therefore, the power dependent propagation loss and nonlinear parameters of the hybrid MRRs are also affected by N and Lc. Fig. 4(a) shows Pintra versus Lc for the hybrid MRRs with films including (i) 1 − 5 and (ii) 10 − 50 layers of GO. The other parameters are kept constant: t = 0.912 and PCW = 25 dBm -taken from our previous experiments [18]. To clearly show the difference, we choose different ranges for Lc in Figs. 4(a-i) and (a-ii) -with a smaller range for thicker films (N ≥ 10). As can be seen, Pintra decreases with Lc and N, resulting from an increased intracavity loss in the hybrid MRRs. Figs. 4(b) and (c) show the corresponding propagation loss and nonlinear parameters γ versus Lc, respectively. Both the propagation loss and nonlinear parameters γ decrease with Lc, showing a trend similar to that of Pintra in Fig. 4(a) and reflecting that the power dependent propagation loss and nonlinear parameters of the hybrid MRRs is strongly dependent on Pintra.

IV. OPTIMIZING FWM PERFORMANCE
In this section, we investigate the influence of the GO film parameters (N, Lc) and MRR coupling strength (t) on the FWM performance of the GO-coated MRRs, taking into account the power-dependent propagation loss and nonlinear parameter discussed in Section III.
The FWM CE of the GO-coated MRRs (CEMRR) is calculated by [23,43]    where Pidler, out and Psignal, in are the output power of the idler and input power of the signal, respectively. CEWG is the CE of an equivalent waveguide with the same length as the circumference of the MRR. The calculation of CEWG is based on the theory in Refs. [13,17]. For the MRRs with patterned GO films, CEWG is calculated by dividing the equivalent waveguides into coated and uncoated segments that have different propagation loss and nonlinear parameters. FEp,s,i in Eq. (3) are resonant field enhancement factors for the pump, signal, and idler, respectively, which can be expressed as [18,22,44]: where t and κ are the field transmission and coupling coefficients defined in Table I, respectively. ϕ p, s, i are the round-trip phase shift of the pump, signal, and idler, respectively, which can be given by: kpc, sc, ic and kpu, su, iu are the wavenumbers of the pump, signal, and idler for the GO coated and uncoated segments, respectively. Table II  This reflects the fact that the Kerr nonlinearity enhancement dominates for the devices with relatively small Lc and N, and the influence of loss increase becomes more significant as Lc and N increase.
In Table III, we compare the calculated CE of the hybrid MRRs with optimized GO film lengths and the measured CE in our previous experiment where we fabricated devices with fixed film coating lengths of ~3.67 mm (i.e., the circumference of the MRR) for N = 1 -5 and 50 μm for N = 10 -50 [18]. For the devices with optimized GO film lengths, there is an improvement in the CE for all the considered GO layer numbers. Particularly, a maximum CE of -34.7 dB is achieved for N = 50 and Lc = 17 µm, which corresponds to a CE enhancement of 13.7 dB compared to the uncoated MRR and 3.4 dB further improvement relative to previous experimental result.
In addition to GO film parameters (N, Lc), the MRR coupling strength (t) also significantly affects the FWM performance of the hybrid MRRs. Based on the process flow in Table II, we further calculate the FWM CE of hybrid MRRs with different coupling strength (t). In our calculations, we chose 20 different values of t ranging from 0.812 to 0.997. For each of them, the calculation processes for Fig. 5 with a fixed t were repeated to obtain the optimized film length Lc' and the corresponding maximum CE for different numbers of GO layers N. Fig. 6(a) shows the calculated Lc' versus t, (i) for N = 1 -5 and (ii) for N = 10 -50. The other device parameters are kept the same, i.e., R = 592 µm and Pp = Ps = 22 dBm. As can be seen, Lc' decreases with t. This reflects that the positive impact of the GO films in improving the FWM CE decreases with t. Fig. 6(b) shows the maximum CE of the hybrid MRRs corresponding to the calculated Lc' in Fig. 6(a). The results for the uncoated MRRs (N = 0) are also shown for comparison. The CE enhancement compared to the uncoated MRR is further extracted from Fig. 6(b) and plot in Fig. 6(c). A maximum CE enhancement of 18.6 dB is achieved at t = 0.812, Lc = 42 µm, and N = 50, which is 4.9 dB higher than the maximum CE enhancement when t = 0.912. This reflects the fact that reducing t further yields a better CE enhancement. The difference in CE between the hybrid and uncoated MRRs becomes smaller as t increases, which is consistent with the trend for Lc' in Fig. 6(a). When t is close to 1, the CE enhancement approaches zero, indicating that incorporating GO films would not bring any benefits in improving the FWM performance in this case. In Fig. 6(d), we plot the insertion loss (at the drop port) of the hybrid MRRs with optimized film lengths Lc' in Fig. 6(a). It can be seen that the insertion loss increases with t and becomes > 8 dB when t is close to 1, which is mainly induced by the four-port MRRs with two identical directional couplers. This indicates that despite the MRR with a weak coupling strength (i.e., high t) has a high CE, it suffers from a high insertion loss that limits their use in practical applications.

V. DISCUSSION
In this section, we discuss the influence of photo-thermal changes in the GO films as well as the effect of varying some of the other MRR paramteres such as ring radius and waveguide dispersion on the FWM performance. This, together with the analysis in Section IV, provides a systematic approach for designing GO-coated MRRs in order to optmize the FWM performance.
As discussed in Section III, photo-thermal changes in the GO films lead to power-dependent propagation loss and  Fig. 7(a), after including photo-thermal changes, the CE decreases, with a more notable difference occurring at higher powers. This reflects the fact that the influence of an increase in loss is more significant than the increase of γ for the device with a thin GO film. In Fig. 7(b), the CE obtained when including photo-thermal effects is lower at low pump powers, while as the pump power increases, it gradually overtakes the CE obtained without including photo-thermal effects. This reflects a more complex influence of the photo-thermal changes on the FWM performance for the hybrid MRRs with thick GO films, which can be attributed to an increase of defects and imperfect contact as well as more obvious thermal dissipation issue in the thick GO films.
Due to the resonant enhancement effect in the MRRs, the FWM CE can be significantly improved in GO-coated MRRs as compared with GO-coated waveguides. In Fig. 8, we compare the FWM CE of GO-coated MRRs and comparable GO-coated waveguides, (i) for the devices with 1 layer of GO and (ii) for the devices with 50 layers of GO. Similar to the case of Fig. 7, optimized film lengths were chosen for the hybrid MRRs and the other device parameters are kept the same as those in Fig. 7. The hybrid waveguides have the same length as the circumference of the MRRs, and both the MRR and the waveguides have the same GO film length. For the hybrid waveguides, we neglect the slight variation induced by photo-thermal changes in the GO films. As can be seen, the CEs of the hybrid MRRs are much higher than those of the hybrid waveguides for both N = 1 and N = 50, clearly reflecting the huge CE improvement enabled by the resonant structure.
For practical device fabrication, hybrid MRRs with uniformly coated GO films are easier to be fabricated since they do not need lithography or lift-off processes for film patterning. In Fig. 9, we further investigate the FWM performance of these hybrid MRRs. Fig. 9(a-i) shows the   MRR's ER versus its radius R and coupling strength t when N = 1 and Pp = Ps = 22 dBm. The ER decreases with both R and t the former results from the increase of the intracavity loss with R, while the latter is consistent with the trend in Fig. 2(b). Fig. 9(a-ii) shows the CE versus R and t. The CE enhancement relative to the uncoated MRR is further extracted from Fig.  9(a-ii) and shown in Fig. 9(a-iii). In our calculation, we neglect the slight difference in the MRR coupling strength t between the GO coated and uncoated MRRs, since including it would result in a difference of only < 0.3%. In Fig. 9(a-ii), the CE (-40.8 dB) at R = 592 µm and t = 0.912 is marked, which corresponds to a CE enhancement of 7.6 dB in Fig. 9(a-iii), showing good agreement with the experimental result in Ref. [18]. The maximum CE (-24.9 dB) at R = 135 µm and t = 0.992 is also marked, which is 15.9 dB higher than the CE at R = 592 µm and t = 0.912 and corresponds to a CE enhancement of -1.8 dB. In Fig. 9(a-iii), a maximum CE enhancement of 14.6 dB is achieved at R = 135 µm and t = 0.812, which is different to the point corresponding to the maximum CE. This reflects the trade-off between achieving the maximum CE versus the maximum relative CE enhancement for the device design, which is consistent with the results in Figs. 6(b) and (c). Fig. 9(b) shows the corresponding results for N = 2. The maximum CE enhancement is improved further by ~4.3 dB as compared with that for N = 1, while both the maximum ER and CE decrease due to the increase in loss with film thickness. This, on one hand, indicates that a high CE enhancement can be achieved for the hybrid MRRs with small radii even without the use of film patterning, while on the other hand, it reflects the fact that the CE significantly decreases with GO film thickness for the uniformly coated MRRs.
Finally, we investigate the influence of waveguide dispersion on the FWM performance of hybrid MRRs. Fig.  10(a) shows the group-velocity dispersion β2 for the hybrid MRRs with (i) N = 1 and (ii) N = 50 layers of GO, together with the β2 of the uncoated MRR. The material dispersion of GO and doped silica was taken from Refs. [12,27]. The β2 of the hybrid MRRs is slightly lower as compared with the uncoated MRR, with the difference becoming more significant for the device with thicker films. The reduced β2 induced by the GO films yields an enhanced anomalous dispersion and consequently better phase matching for FWM [45]. Fig. 10(b) shows the CE versus Δλ (defined as wavelength spacing between pump and signal) for the hybrid MRRs, (i) for N = 1, Lc = 3.4 mm and (ii) for N = 50, Lc = 17 µm. The corresponding result for the uncoated MRR is also shown for comparison. The other parameters are kept the same as t = 0.912 and R = 592 µm. The CE slightly decreases with Δλ, with a difference < 2 dB for Δλ / FSR = 30 when N = 50, Lc = 17 µm. This reflects the fact that both the doped silica and the GO film have a low material dispersion, which allows highly effective phase matching for broadband FWM.

VI. CONCLUSION
In summary, the FWM performance of MRRs integrated with 2D layered GO films is theoretically studied and optimized based on material and device parameters from previous experiments. A detailed analysis for the influence of GO film parameters and MRR coupling strength on the FWM CE of the GO-coated MRRs is performed. By redesigning the device parameters to properly balance the trade-off between the Kerr nonlinearity and loss, up to ~18.6 dB enhancement in the FWM CE is achieved, corresponding to ~8.3 dB further improvement over what was achieved experimentally. The influence of photo-thermal changes in the GO films as well as the variation of some other MRR parameters such as ring radius and waveguide dispersion is also investigated. These results confirm the effectiveness of introducing GO films to improve the MRR's FWM performance and serve as a roadmap for optimizing the FWM performance of GO-coated MRRs.