1. Introduction

Four-wave mixing (FWM) is an important nonlinear process that can be used to control and manipulate optical signals within integrated photonics platforms. Owing to the large third-order nonlinearity and high optical confinement, silicon waveguide systems are particularly well suited to the observation of FWM processes using continuous-wave (CW) pump sources with optical powers typically employed in communication systems [1]. Capitalizing on the compact and integrated geometries, silicon-based FWM has been exploited for high-speed and high-capacity data conversion [2] and regeneration [3]. However, without resorting to pulsed pump sources [4] or complex resonator geometries [5,6], the conversion efficiencies in these chip-based systems are typically very low ($\sim -30\,$dB), with the maximum value of $\sim -10\,$dB obtained for a waveguide with very precise dispersion engineering [7].

A convenient approach to enhancing the conversion in FWM interactions is to couple the parametric nonlinear process with the resonant Raman response of the material. Raman-enhanced FWM (RE-FWM) has been explored in silicon systems for down-conversion of data signals using coherent anti-Stokes Raman scattering (CARS) processes [8], which converts Stokes photons into anti-Stokes photons under quasi-phase-matching conditions. However, as of to date the conversion efficiencies obtained via this process when using CW pump sources have been limited and, for example, the largest reported conversion of $\sim -50\,$ dB required a pump of $700\,$mW due to the strong phase mismatch [9]. Significantly, as the gain efficiency of Raman scattering is higher at the Stokes wavelength compared to the anti-Stokes, higher gains should be possible using a coherent Stokes Raman scattering (CSRS) process, where the signal is positioned at the anti-Stokes wavelength for up-conversion [10]. Indeed, RE-FWM of the Stokes signal has recently been demonstrated using a silicon core fiber structure, where it was shown that a conversion efficiency of $\sim -44\,$dB was possible for a low power pump ($28\,$mW), representing an enhancement of $15\,$dB, even for a non-phase-matched process [11]. With further optimization of the waveguide designs to achieve phase-matching, it is anticipated that large RE-FWM conversion efficiencies could be obtained with traditional CW pump sources.

In this paper, we investigate design parameters for integrated silicon waveguides to optimize the RE-FWM conversion of an anti-Stokes signal at $1430\,$nm, to the Stokes wave at $1680\,$nm, using a CW telecom band pump source at $1545\,$nm. Our work starts by considering the dispersion properties of different waveguide structures etched into silicon-on-insulator (SOI) platforms with silicon thicknesses of $220\,$nm and $500\,$nm. Comparison of the FWM bandwidths reveals that larger rib structures etched into the $500\,$nm silicon are preferable for maximizing the interaction with the Raman gain. Following selection of the optimal structure parameters to achieve efficient nonlinear conversion, the tolerances for the waveguide parameters such as width and etch depth are studied to validate the fabrication precision and the experimental feasibility. The simulation results indicate that a maximum conversion efficiency of about $45\,$dB is possible for a pump power of only $130\,$mW, which is more than $50\,$dB higher than the best results obtained for FWM conversion in optimized planar silicon waveguides.

2. Description of the nonlinear propagation

Figure 1(a) shows a schematic of the proposed setup for the observation of RE-FWM, where a CW pump source at $\lambda _p=1545\,$nm is coupled into a silicon waveguide together with a tunable signal $\lambda _s$. The nonlinear processes that contribute to RE-FWM are schematically illustrated in Fig. 1(b) and (c), where (1) is degenerate FWM, (2) is stimulated Raman scattering (SRS), and (3) is inverse Raman scattering (IRS). Here Fig. 1(b) corresponds to CSRS conversion and Fig. 1(c) is CARS. For CSRS, the Stokes light is originally generated by FWM, which is enhanced due to the existence of SRS. The compensation of the pump power as a result of IRS further enhances the Stokes light, which is what leads to the higher gains for this process. For CARS, the anti-Stokes light is generated by FWM with the assistance of phonons generated by SRS with the right energy and momentum.

 

Fig. 1. (a) Schematic of the design of the experimental setup. (b) Schematic of the CSRS process. (c) Schematic of the CARS processes. In these images the arrows represent a transfer of energy via (1) FWM, (2) SRS and (3) IRS.

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To quantitatively evaluate the nonlinear performance, the spectral evolution resulting from the nonlinear propagation is simulated using the nonlinear Schrödinger equation (NLSE) [12]:

For the terms on the left-hand side of the equation, $A$ is the pulse amplitude, $\alpha$ is the linear propagation loss, $\beta _n$ is the $n$th order dispersive term, and $\sigma$ ($\sigma = 1.45 \times 10^{-21}\,$m$^2$ [13]) is the free-carrier parameter, with $\mu$ and $N_c$ representing the free-carrier dispersion and density, respectively. We note that dispersion terms up to the fourth-order ($\beta _4$) are included in our simulations as these are most important for determining the FWM phase matching. The terms on the right-hand side of Eq. (1) describe the nonlinear propagation, where $\gamma$ is the nonlinearity parameter, $\gamma _1 = \mathrm {d}\gamma /\mathrm {d}\omega \approx \gamma /\omega _0$, and $R(t)$ is the nonlinear response function, including the delayed Raman contribution as given in [14]. Owing to the nonlinear absorption in silicon, $\gamma$ is expressed in the form: $\gamma =k_0n_2/A_{\rm {eff}}+i\beta _{\rm {TPA}}/(2A_{\rm {eff}})$, in which $k_0$ is the propagation constant, $n_2$ ($n_2 = 5.5 \times 10^{-18}\,\rm {m^{2}/W}$ at $1545\,\rm {nm}$ [14]) is the nonlinear refractive index, $\beta _{\rm TPA}$ ($\beta _{TPA} = 10\times 10^{-12}\,$m/W at $1545\,$nm) is the two-photon absorption (TPA) coefficient, and $A_{{\rm eff}}$ is the effective mode area.

To accurately model the effects of the nonlinear absorption, Eq. (1) is coupled to the equation for the free-carrier density $N_c$, which is given by:

Finally, a useful metric from which to evaluate the performance of the FWM process is to calculate the conversion efficiency (CE), defined as the ratio between the signal power $P_{s}$ and the power of the generated idler $P_{i}$ as:

3. Structural design

Our starting point for designing the waveguides was to consider two different structures: strip waveguides formed from $220\,$nm SOI and rib waveguides formed from a thicker $500\,$nm SOI platform. Both structures can be defined using standard photolithography and etching techniques to produce waveguides with cross-sections, as illustrated in Fig. 2(a). In these figures, H is the thickness of the silicon on top of the insulator layer, h is the etch depth, and W is the width. The longitudinal waveguide design is the same in both cases, as depicted in the bottom image of Fig. 2(a), where a straight section is sandwiched between two coupling regions (a down and up taper) to ensure good coupling into the few hundred nanometer-sized waveguides. We note that although several of our waveguide designs support multiple guided modes, by optimizing coupling into the tapered regions it is possible to ensure that the fundamental mode is predominantly excited.

 

Fig. 2. (a) Structural diagram of the planar waveguide designs. The top two images depict cross-sections of a rib waveguide with the top silicon layer of $500\,$nm and strip waveguide with a top silicon layer of $220\,$nm. The bottom image shows the longitudinal variation including input and output coupling regions. Second- and fourth-order dispersion parameters (b) and FWM bandwidth (c) for the two optimized structures illustrated in (a).

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Of the two cross-section designs, the thicker waveguides are typically associated with lower propagation loss values, with strip waveguide losses often reported to be $\sim 2-4\,$dB/cm and rib waveguides having loss values $\sim 1\,$dB/cm [15]. However, there is another advantage of the rib structure in that by having access to two optimization parameters (width and etch depth) it is possible to have greater control of the dispersion properties. To illustrate this, Fig. 2(b) shows the second- and fourth-order dispersion parameters ($\beta _2$ and $\beta _4$, respectively), as calculated via modal analysis using COMSOL, for two waveguide designs with the dimensions of h$\,=350\,$nm and W$\,=790\,$nm for the rib, and W$\,=790\,$nm for the strip. These waveguide designs were chosen specifically so that the zero-dispersion wavelength (ZDW: $\beta _2=0$) is close to the desired pump wavelength of $1545\,$nm, and thus represent an optimum configuration for efficient FWM. Figure 2(c) then plots the FWM conversion efficiency for the two structures as a function of wavelength, as calculated via the NLSE using Matlab for a pump power of $100\,$mW, a signal power of $5\,$mW, and assuming the same loss value of $1\,$dB/cm. The propagation length was set to $1\,$cm to avoid the generation of higher-order Stokes and anti-Stokes light due to the accumulation of Raman gain with increasing propagation length. The sharp peaks seen on the conversion curves are due to RE-FWM, where the short-wavelength peak is associated with CSRS and the long-wavelength peak is CARS. It is clear from these results that there is a notable difference in the achievable FWM bandwidth, with the thicker $500\,$nm rib waveguide providing the better overlap with the Raman peaks to achieve a higher maximum gain ($\sim -6\,$dB). Thus, we fixed on the design of the rib structure in the $500\,$nm thick silicon, but continued to investigate the role of the width and etch depth to ensure the optimal design had been achieved.

Figure 3(a) plots the dispersion profile for various rib waveguide dimensions (h and W values) for the chosen pump wavelength of $1545\,$nm. As already mentioned, it is preferable to operate near the ZDW to obtain a broad and efficient FWM bandwidth, which is depicted by the black dash line. From this figure it is clear that the zero-dispersion condition can be achieved for different combinations of h and W, with larger widths simply corresponding to larger etch depths. Moreover, $\beta _2$ changes slowly along the width, but dramatically along the etch depth, which suggests that the dispersion profile is more sensitive to the critical dimension of waveguides. To illustrate the influence of the different waveguide dimensions, Fig. 3(b) shows conversion efficiencies for three different combinations of h and W as a function of the signal wavelength. As for Fig. 2(c), the simulations were conducted with a pump power of $100\,$mW, a signal power of $5\,$mW, a length of $1\,$cm, and loss value of $1\,$dB/cm. Again, the strong RE-FWM peaks are observed on the spectra, but this time they are all well positioned in the middle of the flat section of the bandwidth. Moreover, in all cases the CSRS is stronger than the CARS by a conversion efficiency of $\sim 4\,$dB, which originates from the inherent higher energy transfer efficiency at the Stokes wavelength compared to the anti-Stokes wave. Therefore, CSRS is the focus in the following discussion about the performance of RE-FWM.

 

Fig. 3. (a) Second-order dispersion parameter ($\beta _2$) varying with etch depth (h) and width (W). The black dash line represents $\beta _2=0$. (b) Plot of conversion efficiency as a function of $\lambda _s$ for waveguide designs with different combinations of h and W that have $\beta _2=0$ at the $1545\,$nm pump wavelength (i.e., the dimensions are selected from the black dash line in (a)). The simulations are conducted with a pump power of $100\,$mW, a signal power of $5\,$mW and a propagation length of $1\,$cm.

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As seen in Fig. 3(b), the conversion efficiency for CSRS remains approximately stable for the different combinations of h and W. The bandwidth, defined as the signal wavelength range where the conversion efficiency drops by $3\,$dB from the maximum value at the centre of the curve, is largest for the etch depth of $350\,$nm. This is due to the difference in the values of $\beta _2$ and the corresponding values of $\beta _4$, which combine to minimise the total dispersion. Additionally, there is a trade-off between the effective mode area and the propagation loss caused by the interaction between the guided light and the sidewalls. As a result, we have settled on a target structure with h$=350\,$nm and W$=790\,$nm as a compromise between a small mode area to achieve a high mode intensity, but with the potential for low losses through a modest etch depth. This structure has dispersion values of $\beta _2= -2.49\times$10$^{-3}$ ps$^2$/m and $\beta _4=3.53\times$10$^{-^7}$ ps$^4$/m. In the following simulations, the linear loss is assumed to be $1\,$dB/cm, based on typical values from our fabrication of waveguides from this platform [15].

4. Performance of CSRS

Following identification of the optimal structure, the influence of the structural parameters on the RE-FWM conversion efficiency was investigated to validate the design for the existing fabrication techniques. In these simulations, the system parameters are the same as those used for Fig. 3. Figure 4(a) displays the conversion efficiency as a function of $\lambda _s$ as the etch depth h is changed from $280\,$nm to $420\,$nm. As expected, the bandwidth is at a maximum when h is equal to the optimum value of $350\,$nm, but decreases quite sharply as h is increased or decreased away from this. Figure 4(b) then shows the efficiency as the waveguide width W is changed from $670\,$nm to $900\,$nm. Again, the bandwidth is maximum for the designed width of $790\,$nm, but the rate of decrease of the bandwidth as the width is adjusted is less sharp than for the changing etch depth.

 

Fig. 4. (a) Conversion efficiency as functions of h and $\lambda _s$. (b) Conversion efficiency as functions of W and $\lambda _s$. (c) and (d) show the conversion efficiencies of the CSRS processes, marked with black dash boxes in (a) and (b), respectively, where signal is tuned to the anti-Stokes wavelength of $1430\,$nm. In all cases the pump wavelength is $1545\,$nm with a power of $100\,$mW, the signal power is $5\,$mW and the propagation length is $1\,$cm.

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In both cases it is clear that CSRS, marked by the black dash boxes, is strongly excited when the bandwidth of FWM overlaps with the Raman gain spectrum. Figure 4(c) and Fig. 4(d) display the conversion efficiency as a function of h and W, which are extracted from Fig. 4(a) and 4(b), respectively. The reference conversion efficiency is set as $-10\,$dB, as denoted by the dash lines in Fig. 4(c) and (d), which is around $4\,$dB less than the maximum conversion for the $100\,$mW pump. It is noteworthy that, even with a lower pump power and a larger effective mode area compared to the system previously used to report the highest FWM conversion efficiency [7], higher conversion efficiencies can be easily achieved via our geometry by coupling the nonlinear processes of FWM, SRS and IRS. Based on this reference value, we can determine the fabrication tolerance of the etch depth h to be $20\,$nm and width W to be $47\,$nm, which are well within the attainable resolutions for existing photolithography technologies [16]. The smaller structural tolerance of h compared with that of W is consistent with the dispersion analysis, which further confirms that the nonlinear performance is more sensitive to the smallest dimension of planar waveguides.

Having established the fabrication tolerances for this design, we subsequently investigated the performance of the RE-FWM by comparing the conversion efficiency and Stokes output power for the CSRS process. Figure 5(a) plots a colour map of the conversion efficiency as functions of pump power and wavelength length, with the remaining parameters for the pump, signal and waveguide the same as used previously for Fig. 4. From this result we see that as the pump power increases above $\sim 200\,$mW, the conversion efficiency saturates at a propagation length of $\sim 3\,$cm. We attribute this saturation to the fact that for high pump powers the Stokes light is sufficiently enhanced to transfer energy to the anti-Stokes wave, as well as to higher order Stokes waves. As a result, the optimal pump power to achieve a maximum conversion efficiency of approximately $45\,$dB is only $130\,$mW, with a propagation length of $4.6\,$cm. We note that for such efficient conversion, the deviation between CSRS and CARS is increased to $>50\,$dB, which reinforces the advantages of this pumping scheme. Although this pump power is higher than used in other studies of FWM [17,18], such powers are readily available within telecom systems and the advantage of our approach is that it only relies on a simple waveguide design, without the need for extremely low linear loss or ring resonator designs, to achieve an enhancement in the conversion efficiency of $\sim 50\,$dB. In fact, even in the case of higher linear losses $\sim 4\,$dB/cm, although the maximum conversion efficiency decreases to $\sim -0.5\,$dB, the enhancement of CSRS compared with pure FWM still reaches up to $\sim 21\,$dB.

 

Fig. 5. (a) Conversion efficiency and (b) Stokes output power as functions of the pump power and waveguide length when pumping at $1545\,$nm with a $5\,$mW signal beam positioned at $1430\,$nm.

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Figure 5(b) then shows a colour map of the achievable Stokes output power as functions of pump power and waveguide length. It shows that the output idler power saturates for both increasing power and length, reaching a maximum of $560\,\mu$W for a $130\,$mW pump at a length of $3.2\,$cm. We attribute the saturation associated with increased power both to the conversion of the first order Stokes wave to higher order waves as well as nonlinear absorption associated with TPA and free-carrier effects. Thus, an effective way to improve the Stokes output power would be to shift the pump wavelength beyond $2\,\mu$m, where the nonlinear absorption diminishes. In contrast, the saturation for increasing propagation length is primarily due to the linear loss, so that the Stokes output power could also be increased through further reduction of the transmission losses. Although the maximum Stokes power does not occur at the same length as the maximum conversion efficiency, it is worth noting the conversion efficiency is still as high as $14\,$dB in this instance. Therefore, a single waveguide could be used to achieve the maximum Stokes output power and considerable conversion efficiency. The inconsistency in the optimized propagation length for the maximum conversion efficiency and Stokes output power is attributed to the presence of IRS, which leads to a decrease in anti-Stokes wave power that appears in the efficiency calculation. This results in a slightly slower saturation of conversion efficiency as a function of propagation length. Comparing the maximum achieved Stokes power with previous reports for pure SRS, it is quite remarkable that the power generated using the RE-FWM scheme is $7\,$dB larger than what has been obtained in a waveguide of comparable length but with a p-i-n structure built in to reduce the free-carrier absorption [19]. Significantly, if we reduce the losses to $0.3\,$dB/cm in our simulations, which corresponds to the lowest linear loss for planar waveguides in the telecom band at this time [20], we can achieve idler powers up to $1.47\,$mW for a pump power of only $80\,$mW and a length of $10\,$cm.

5. Conclusion

In this work, we have explored optimum designs for planar waveguide structures to exploit RE-FWM processes in the telecom band. For a pump at $1545\,$nm, a rib structure formed from a $500\,$nm thick SOI platform with an etch depth of $350\,$nm and a width of $790\,$nm was found to provide a good compromise between achieving a good overlap between the gain bandwidths of the nonlinear processes, but with potential for low propagation losses. Moreover, the design was shown to have good fabrication tolerance to achieve strong conversion efficiencies even with variations of $20\,$nm or more in the waveguide parameters. A maximum conversion efficiency of $\sim 45\,$dB could be obtained with a pump power of $130\,$mW over a propagation length of $4.6\,$cm, which is over $50\,$dB higher than typical reports of FWM in silicon planar waveguides. Applying the same pump power, a maximum Stokes power of over $560\,\mu$W could be achieved within a slightly shorter length of $3.2\,$cm, and additional simulations have shown that the output power could be increased further by reducing the losses to less than $1\,$dB/cm. The design principle illustrated in this paper allows for the application potential of RE-FWM to be explored at any desired wavelength by optimizing the structure to achieve the maximum required output. Thus, we expect the high conversion efficiencies and large Stokes output powers that can be generated through RE-FWM processes could be exploited in wide ranging areas extending well beyond traditional signal processing and communications applications.