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Abstract
Broadband continuous-wave parametric gain and efficient wavelength conversion is an important functionality to bring on-chip. Recently, meter-long silicon nitride waveguides have been utilized to obtain continuous-traveling-wave parametric gain, establishing the great potential of photonic-integrated-circuit-based parametric amplifiers. However, the effect of spiral structure on the performance and achievable bandwidth of such devices have not yet been studied. In this work, we investigate the efficiency-bandwidth performance in up to 2 meter-long waveguides engineered for broadband operation. Moreover, we analyze the conversion efficiency fluctuations that have been observed in meter-long Si3N4 waveguides and study the use of temperature control to limit the fluctuations.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The ever-increasing need for bandwidth calls for innovative technological solutions that can scale up. All-optical integrated signal processing is key to overcome the limitations of the existing electronic technology, as nonlinear optical interactions can be leveraged to implement the required processing elements in optical communication systems. Four-wave mixing (FWM) is a well-established nonlinear process that has been studied to amplify, copy and multicast high speed data signals as the process maintains the data format and coherence while providing broadband operation [1]. Yet, the limited conversion efficiency (CE) of integrated continuous-wave (CW) FWM has hindered its implementation [2].
Various material platforms have been widely investigated for the integration of FWM functionality on chip, such as Hydex [3], silicon (Si) [4], silicon carbide (SiC) [5], chalcogenide [6], silicon nitride ($\rm {Si}_3\rm{N}_4$) and III-V semiconductors [7]. Si$_3$N$_4$ is one of the strong candidate materials for monolithic integration of the nonlinear medium owing to its CMOS fabrication compatibility, absence of two-photon absorption (TPA) in the telecom band and wide transparency window. Numerous nonlinear interactions have been demonstrated in Si$_3$N$_4$ such as the generation of broadband and efficient optical frequency combs [8,9], supercontinuum generation expanding from the visible to the mid-infrared [10–13], 3rd order optical parametric oscillation [14], or even second harmonic generation and other three-wave mixing processes [15–19]. Furthermore, broadband FWM and distant wavelength conversion have also been demonstrated in Si$_3$N$_4$ waveguides [20] [21]. However, the relatively small nonlinearity of Si$_3$N$_4$ compared to that of lower bandgap alternatives, such as for AlGaAs [7], for a long time limited the maximum CE that could be obtained in CW pumping regime. In their work, Pu et al. managed to obtain a CE around -4 dB in mm-long AlGaAs waveguides while maintaining more than 800 nm bandwidth owing to the high nonlinearity of AlGaAs ($\gamma =630\;\rm{W}^{-1}{\rm m}^{-1}$). This limitation in CE was surpassed recently in Si$_3$N$_4$ by increasing the effective length and using spiral waveguides. Although nonlinear spiral waveguides have been already studied in various platforms [22–24], reaching order of meters has only been feasible recently in Si$_3$N$_4$ thanks to high-quality fabrication and ultra-low optical loss [25,26]. Consequently, up to 9.5 dB of phase-sensitive parametric gain [27] and 12 dB of phase-insensitive parametric gain [28] in a traveling wave configuration were demonstrated in Si$_3$N$_4$ waveguides by coupling pump powers in the order of a few watts. However, in their work, Ye et al. observed fluctuations in the measured CE values and deviations from the expected values in Archimedean spiral Si$_3$N$_4$ waveguides. Similarly, we had observed strong fluctuations in our previous work utilizing constant bent segments of 230 µm radii of curvature [21].
In this work, we further investigate the potential of meter-long $\rm {Si}_3\rm{N}_4$ waveguides for efficient and broadband FWM, and the source of CE fluctuations. In addition, we address the wavelength and optical power dependent loss in the spirals through mode-mixing and we show that temperature-tuning can reduce the fluctuations.
2. Design and experimental procedure
The $\rm {Si}_3\rm{N}_4$ waveguides used in this study are fabricated by the Damascene process [25] in the form of long spirals having straight segments and bent segments with a 230 µm radius of curvature (ROC). Inverse tapers are utilized in order to excite the fundamental mode in the waveguide. We aim at engineering the dispersion of the waveguides, similar to previous works [29], to guarantee broadband conversion from a telecom pump near 1550 nm in the TE mode of operation, which allows for the lowest propagation loss and highest coupled pump power in the device. Previously, we had shown geometries that allowed such CW operation at around 2 µm of pump wavelength. We achieved maximum CE of -30 dB in a 50 cm long waveguide and a two-sided 3 dB bandwidth of 120 nm. For TM polarization, the dispersion was optimized for 1600 nm pump reaching maximum CE of -22 dB and a two-side 3 dB bandwidth of 150 nm [21]. We show in Fig. 1(a) the simulated group velocity dispersion ($\beta _2$) at 1550 nm for varying waveguide height and width, where the black line indicates the zero dispersion value. Based on the available nominal height around 0.67 µm, we can see that waveguide widths near 2.2 µm would result in a slight anomalous dispersion, very close to the zero dispersion value as needed for broadband conversion. However, we can also see that while the dispersion is less sensitive to width, height variations in the order of 10’s of nm can significantly shift the boundary between normal and anomalous dispersion. This sensitivity is illustrated in Fig. 1(b). With the targeted waveguide dimension of 2.2 µm x 0.67 µm, the dispersion is expected to be slightly anomalous at 1550 nm (red arrow) for TE polarization. We also plot the expected theoretical dispersion given the fabrication tolerance of $\pm$ 10 nm. Owing to the very specific targeted dispersion, such a small deviation can significantly detune the device from its optimal operation, shifting it close to 1500 nm or 1640 nm, for the 0.66 µm and 0.68 µm height, respectively.
Fig. 1. (a) Simulated contour graph of group-velocity dispersion (GVD) of $\rm {Si}_3\rm{N}_4$ waveguides for various heights and widths for TE polarization. Black line indicates the zero dispersion value. (b) GVD vs the wavelength for 2.2 µm width and the aimed 0.67 µm height (solid line) together with a possible $\pm$ 10 nm deviation (dashed lines). The arrow represents the aimed zero-dispersion wavelength (ZDW) at 1550 nm. (c) Intensity mode overlap factor between fundamental mode in the straight segment and fundamental mode (left axis) and TE$_{10}$ and TE$_{20}$ modes (right axis) in the circular segment
The 2.2 µm x 0.67 µm waveguide supports 3 TE and 3 TM modes at 1550 nm. Although inverse tapers are designed for fundamental mode coupling, the light can scatter into the higher order modes due to the presence of many straight-to-bend transitions. The bends cause a change in the shape of the wavefront such that some portion of the fundamental mode light overlaps with higher-order modes and excites them proportional to the bend curvature. The simulated intensity overlap integral of the fundamental TE mode (TE$_{00}$) with the first two higher order modes (TE$_{10}$ and TE$_{20}$) as a function of ROC is plotted in Fig. 1(c). Although the intensity overlap integral between a straight and bent section of the waveguide is only 0.15% for the 230 µm ROC between TE$_{00}$ and TE$_{10}$, excitation of higher order modes can build up since meter-long waveguides can have more than 400 bent sections. We also see that as the ROC decreases below 100 µm, the intensity overlap integral increases above 1% while it also becomes possible to excite a remarkable amount of TE$_{20}$ mode.
The experimental setup to study FWM includes a pump-probe architecture utilizing S/C or L band sources covering 1440 nm to 1660 nm. All sources are set to TE polarization and combined using various wavelength-division multiplexers (WDM) to minimize losses. The power of the pump and of the signal can be independently controlled before being coupled to the waveguide under test with a lensed fiber. The output light is then collected with a lens and collimated to the optical spectrum analyzer (OSA) Yokogawa AQ6375B or to power meters in order to calibrate the powers. The total coupling loss is estimated around 5 dB. The polarization states of the pump, signal and idler beams at the output are monitored with the help of a polarization beam splitter (PBS) to ensure stable operation.
3. Efficiency and bandwidth in m-long ${{\mathrm{Si}}_{\mathrm{3}}{\mathrm N}_{\mathrm 4}}$ waveguides
Broadband parametric conversion requires that the phase matching condition be ensured for large detuning between the fixed pump and signal wavelengths. The CE between the signal and idler power, $P_s$ and $P_i$, respectively, is given by:
where $P_p$ is the pump peak power, $g=\sqrt {(\gamma P_p)^2 - {(\kappa /2)^2}}$, $L_{eff}=\left \{\frac {1-{\rm exp}(-\alpha L)}{\alpha }\right \}$, $\alpha$ is the linear loss, $\gamma = \frac {2\pi n_2}{\lambda A_{eff}}$ is the nonlinear coefficient, with $n_2$ the nonlinear index and $A_{eff}$ the effective mode area. $\kappa$ is the total phase mismatch which depends on the signal, idler and pump propagation constants $\beta _s$, $\beta _i$, and $\beta _P$, respectively, and the nonlinear contribution $2\gamma P_p$. Under low-gain approximation, the CE can be approximated toGiven $\Omega$, the angular frequency detuning between pump and signal (identical to pump and idler), and considering up to the fourth order dispersion coefficients ($\beta _2$ and $\beta _4$), we have that:
In the case of long waveguides, the gain bandwidth $\Delta \Omega$ is set by the waveguide length. Using the estimation that the bandwidth is reached when $|\kappa L|=\pi$, it is given by Eq. (5) and (6) for the small and large gain limit, respectively defined for $2\gamma P_pL<<\pi$ and $\Delta \Omega < \Omega _p$ with $\Omega _p$ the peak of the parametric gain when $\kappa = 0$:
In the small gain limit around the ZDW, where $\beta _4$ dominates over $\beta _2$ we obtain
In all cases, the bandwidth scales with an inverse trend of the length, particularly limiting in the large gain limit, while the CE increases with the square of the length, introducing the well-known bandwidth trade-off, which is particularly crippling when the pump power is limited as is the case for CW pumping, and in low $\gamma$ platforms.
We plot in Fig. 2(a)-(c) the simulated normalized CE evolution as a function of waveguide length for 1550 nm pumping, for the three cross sections shown in Fig. 1(b). We see that for short waveguide lengths, the bandwidth of the process can be very broad in all cases as the phase matching requirements are relaxed. In particular, for a 2.2 µm x 0.68 µm waveguide the main phase-matched region around the pump merges with the higher-order far detuned phase-matched bands. However, for propagation beyond a few millimeters, the bandwidth is significantly decreased in the case of non-optical dispersion (Fig. 2(a) and (c)). While the propagation loss has been experimentally measured between 2 and 4.7 dB/m in our samples, in agreement with the previously reported loss values between 1.7 to 5 dB/m [25], the theoretical calculations are carried with a 4 dB/m loss to illustrate the CE bandwidth trend which is consistent with all our possible loss values.
Fig. 2. (a), (b) and (c) Theoretical normalized CE contour graph for a 1 W of coupled pump power at 1550 nm for various signal wavelengths for various lengths of $\rm {Si}_3\rm{N}_4$ waveguides having the width of 2.2 µm and the heights of 0.66, 0.67 and 0.68 µm, respectively. (d), (e) and (f) Normalized experimental CE (dots) and theoretical CE (lines) for pumping at 1550 nm in the waveguide lengths of 0.5, 1 and 1.5 m.
We experimentally tested the behavior of waveguides with nominal cross-section 2.2 µm x 0.67 µm and with lengths of 0.5 m, 1 m and 1.5 m. The pump is positioned at 1550 nm and the coupled power is estimated to be 1.7 W, while the signal is swept on the short wavelength side of the pump with an estimated coupled power of 20 mW. The normalized CEs as a function of signal detuning are shown in Fig. 2(d)-(f) together with the theoretical curves obtained from simulations for the waveguide height of 0.67 µm $\pm$ 10 nm. We observe that as expected, the bandwidth indicated at the -3 dB values by the red dashed arrow, decreases with length from 48 nm (one-sided) for the 0.5 m long waveguide to 15 nm for the 1.5 m waveguide. We also observe that the experimental data does not perfectly fit the theoretical expectation, being slightly narrower than for the optimal case (0.67 µm height). Moreover, possible fluctuations in waveguide cross-section can become more impactful as the length of the device increases from typical millimeter to now meter scale. In addition, during the experiment, although the signal power was kept constant during the sweep, we observed significant signal power fluctuations at the output of the waveguides. It should be pointed out that such behavior makes the estimation of the exact input and output losses, and hence coupled power, quite challenging.
The maximum values of CE for these three waveguides, and for a coupled pump power of estimated around 1.7 W, were measured to be -10 dB, -3.7 dB and -4.6 dB, for the 0.5 m, 1 m and 1.5 m waveguides respectively. While as expected the CE increases when the length is doubled to 1 m, we do not see a further increase with the 1.5 m length waveguide. This behavior could be the result of the 1.5 m long waveguide having higher propagation losses than the 1 m long one, and/or due to more power being lost to higher-order modes.
In order to better characterize CE trends as a function of lengths, waveguides with 2.5 µm x 0.67 µm cross-section are then tested as 4 different lengths (0.5 m, 1 m, 1.5 m and 2 m) were available for this given geometry even though the dispersion was slightly detuned from optimal for broadband conversion. The results for a fixed pump-signal detuning of 1.5 nm are presented in Fig. 3(a). Although the CEs follow the trend of the square of the pump power (shown by the guide for the eye dashed lines), strong fluctuations in the data can be observed. At 32 dBm estimated coupled pump power and in the 2 m long waveguide, the CE of around 3 dB is obtained. The nonlinear parameter, $\gamma$, is found to be around 0.6-0.8 $\rm {W}^{-1} \rm{m}^{-1}$, from the linear fit obtained for 0.5, 1, 1.5, and 2 m long waveguides and assuming propagation loss of 4 dB/m in the waveguides as an average loss. This range of values for $\gamma$ is in agreement with the one obtained from the simulated $A_{\rm eff}$ and $n_2 = 2.5 \times 10^{-19} \rm {m}^{2} \rm{W}^{-1}$, which is 0.75 $\rm {W}^{-1} \rm{m}^{-1}$. The variations in the measured data could be due to fluctuations of the propagation loss and coupling loss between different waveguides and to the difficulty in knowing the exact effective pump power that could be lower than the coupled power owing to the excitation of higher order modes. To determine the possible boundaries of operation, the CE as a function of length for high (1.7 W) and low (0.25 W) estimated coupled pump powers are plotted in Fig. 3 (b). We also plot the theoretical trends in the expected CE as a function of length for the two extreme values of extracted propagation loss, namely 2 and 4.7 dB/m. First, as expected the change in the propagation loss has a more profound effect as the waveguide gets longer. Second, the range of effective loss is within expected values (2 to 4.7 dB/m), with the 1 and 1.5 m waveguides showing slightly lower performance.
Fig. 3. (a) Experimental CE (dots) and guides for the eye (dashed lines) as a function of estimated coupled pump power for the waveguide length of 0.5, 1 and 1.5 m. Inset shows the spectra of FWM at around 1536.5 nm showing parametric gain in 2 m long waveguides at 32 dBm of estimated pump power (b) The experimental and theoretical CE for 0.25 and 1.7 W of estimated coupled pump power together with simulated CE vs length curves for 2 dB/m and 4.7 dB/m propagation loss. (c) Optical spectra of cascaded idler generation as a result of 31 dBm and 29 dBm coupled pump power at 1537 and 1542 nm.
The 2 m long waveguide, which showed the best performance in terms of CE, is also characterized by cascaded idler generation. In this case, 1536.5 and 1541 nm light is coupled into the waveguide at estimated coupled power of 31 and 29 dBm, respectively. Such configuration act as a two-pump parametric amplifier and leads to the generation of first and higher-order idlers owing to the efficient FWM with constant separation of 5 nm, as shown in Fig. 3 (c). We see that first order idlers experience parametric gain. The formed idlers cover more than a 100 nm wavelength span. Such cascaded idler formation with coherent sources can be intriguing for tunable optical frequency comb generation in Si$_3$N$_4$ waveguides.
4. Mode-mixing in meter-long spiral waveguides
As previously mentioned, the waveguides under test support 3 TE and 3 TM modes at 1550 nm, and the bends can cause excitation of higher order mode even if only the fundamental mode is launched in the waveguide. The excitation of the modes depends on the interference between the fundamental mode and the higher order at the straight-to-bend transitions which can interfere constructively or destructively with the fundamental mode of the straight section. The simulation of such interference is quite intricate as the length of the segments is much larger than the wavelength of operation and is very sensitive to the local temperature. Therefore, the exact simulation is beyond the scope of this work.
For FWM process in m-long waveguides, the pump, signal, and idler can all excite higher order modes. However, for combinations where all three are not in the same mode, a strong phase mismatch would significantly impact the CE. Moreover, when the pump is in a higher order mode, the CE is also reduced owing to the larger effective area and hence lower $\gamma$. So the dominant mechanism for idler generation is when the pump, signal, and idler are all in fundamental mode, and excitation to higher order modes will yield lower CE due to strong phase mismatch and larger effective areas. In addition, mode mixing also affects the idler and signal coupled to the OSA causing more fluctuations in the measured CEs.
In order to investigate this aspect, we run the CE vs wavelength detuning experiments of $\rm { Si}_3\rm{N}_4 $ spiral waveguides having a cross-section of 2.0 µm x 0.8 µm. While again this cross section is not intended to give the largest bandwidth, these waveguides provide us with an important testbed in terms of ROC (75 µm and 230 µm) and length (7.4, 16, 22, and 32 cm), as summarized in Table 1. The CE vs wavelength detuning is experimentally tested in these waveguides and presented in Fig. 4, together with theoretical fit according to Eqs. (2) and (3). The square root of the reduced chi-square of the fit is extracted as a figure of merit for the fluctuations and presented in Table 1 as well. We can clearly observe that the experimental measurements fit well in the case of the 230 µm bend radius and the 22 and 32 cm waveguides (Fig. 4(a),(b)), compared to the m-long ones, due to lower excitation to the higher order modes in shorter waveguides. We also obtain a lower square root of chi-square of 1.12 dB in the 22 cm long waveguide compared to the 32 cm long waveguide with 1.42 dB. To confirm the effect of ROC, the data for 75 µm and with lengths of 7.4, 16, and 32 cm, is shown in Fig. 4(c)-(e), respectively. The fluctuations in CE are more pronounced in these waveguides due to lower ROC. The square root of the chi-square rises from 2.18 dB to 3.05 and 4.78 dB as the length of the waveguide increases from 7.4 to 16 and 32 cm. In the 32 cm long waveguide, the fit looks arbitrary as the fluctuations wash out the dispersion characteristics. We can also note that for all cases, the decrease in bandwidth with the length is visible.
Fig. 4. (a), (b), (c), (d) and (e) Experimental CE (dots) and Fitted CE (dashed lines) in the one-side tapered waveguides I, II, III, IV and V, respectively. The shape of the corresponding spiral waveguide has been depicted in the lower left corner of each graph.
Table 1. Tabulation of length and radius of curvature (ROC) in the bend segments for one-side tapered $\rm {Si}_3\rm{N}_4$ waveguides used in this work together with the reduced-chi square of theoretical fit according to Eq. (2).
4.1 FWM experiment with temperature control
As the main source of the CE fluctuation is the interference of the different modes, a slight change in the refractive indices of the modes can cause a substantial effect in CE in meter-long spirals having hundreds of straight-to-bend transitions. One method to tune the refractive indices, and hence the phase mismatch between the modes, is to alter the temperature profile. Tuning the temperature profile slightly changes the refractive index difference between the different modes causing a change in their interference. It is also important to note that the coupling loss of the waveguide serves as a strong heat source that can alter the temperature profile of the waveguide.
The FWM experiment is repeated with the chip positioned on a stage integrated with a heatsink and a Peltier controller. The power scaling experiment in 2.5 µm x 0.67 µm cross-section is repeated, similar to Fig. 3(a), but this time for a fixed temperature set at 35$^{\circ }$ C. The measured CE obtained in 0.5 m, 1 m and 2 m long waveguides are presented in Fig. 5(a). The CE fluctuations are observed to be more limited with the pump power and more stable over time. The CE shows good agreement with the square of the pump power in the 0.5 m long waveguide with a minimal amount of fluctuations. In the 1 m long waveguide, the CEs undershoot the theoretical fit at low pump powers and overshoot at the high pump powers. The reason behind the deviation is expected to be the temperature gradient in the chip with the coupling losses serving as a heat source. The intermodal interference in the straight-to-bend transitions can lead to more constructive or destructive interference depending on the temperature of the transitions. In the 2 m long waveguide, we observe two transitions between more efficient and less efficient than the theoretical fit. More portion of the light is scattered to the higher-order modes due to more straight-to-bend transitions in longer waveguides. In addition, a slight change in the $\Delta \beta$ of different modes with temperature can have a more profound change in the phase. Hence, the fluctuations are more pronounced in the 2 m long waveguide compared to the 1 m and 0.5 m long waveguides.
Fig. 5. (a) Experimental CE (dots) and guide to eye (dashed lines) as a function of estimated coupled pump power for the waveguide length of 0.5, 1 and 2.0 m the cross-section of 2.5 µm x 0.67 µm after the integration of a heat sink and Peltier controller (b) Contour graph of experimental CE for various pump power and set temperature of temperature controller in the $\rm {Si}_3\rm{N}_4$ waveguide with the cross-section of 2.2 µm x 0.67 µm and length of 1 m at 1550 nm pump wavelength and 5 nm wavelength detuning. (c) The idler power and for an optimized set temperature at every signal wavelength. The dashed line shows the simulated CE for 2.2 µm x 0.67 µm and 1 meter of length.
In order to measure the effect of the temperature on wavelength detuning for broadband FWM, the waveguide with the 2.2 µm x 0.67 µm cross-section and 1 m length is chosen as an example. We start by monitoring the CE as a function of temperature and pump power at a fixed 5 nm pump-signal detuning to observe the optimal CE formation. We extract the CE for the coupled pump powers of 25, 27, 29, and 30.8 dBm and for every set temperature between 15$^{\circ }$ C to 50$^{\circ }$ C with 1$^{\circ }$ C steps. The results of this measurement are presented in Fig. 5(b). The CE fluctuates more than 15 dB with the change in the set temperature even at the same input pump power. We also observe an optimal set temperature, around 35$^{\circ }$ C and 40 $^{\circ }$ C, where the highest CE is obtained. The optimal set temperature tends to shift to the higher values as the pump power lowers, which can be a result of less heating of the waveguide from optical loss. The CEs at the optimal temperature agree better with the theoretical expectation of scaling with the square of the pump power.
The optimal set temperature can be the temperature for which the interference between the different waveguide modes is mostly constructive at the transitions between the straight and bend segments, such that most of the light stays in the fundamental mode. Thus, optical powers are expected to show fewer fluctuations with wavelength detuning provided that the FWM is carried at the optimal set temperature at each wavelength detuning. To test if the fluctuations can indeed be reduced, the idler power is optimized with the set temperature at each wavelength while keeping the input pump and signal powers constant. The results are shown in Fig. 5(c). It can be seen that optimizing the set temperature can reduce fluctuations and improve CE by up to 10 dB at some detunings. The experimentally measured trend gets closer to the theoretical one, although a discrepancy attributed to the exact waveguide cross-section is still noticeable.
5. Discussion and conclusion
Recent studies have demonstrated that the relatively lower nonlinearity of $\rm {Si}_3\rm{N}_4$ compared to other nonlinear optical platforms can be compensated by increasing the interaction length thanks to the ultra-low optical loss, by increasing the pump power thanks to the high power handling and through precise dispersion engineering of $\rm {Si}_3\rm{N}_4$ waveguides. However, the use of longer waveguides inevitably gives rise to a reduction in the bandwidth for a limited pump power, as well as possible CE fluctuations due to mode-mixing. The mode mixing in the waveguides introduces a trade-off between compactness and fluctuations. Having more bends enables dense packing of the waveguide in a small area whereas more portion of the light excites into the higher-order modes. The other structures offering a smoother change in the curvature of the waveguide might also reduce the excitation of the higher order modes [30–32]. Another method for overcoming the mode mixing problem is utilizing waveguides supporting only one mode by shrinking the dimensions. However, shrinking dimensions can prevent obtaining a ZDW in C-band, limit the necessary dispersion engineering for broadband operation, and increase the propagation loss. Such optimization of the structure of the spirals or the dimensions of the waveguide will be an important step forward but is beyond the scope of this study. Pushing further the performance and maturity of materials with higher nonlinearity might be an alternative path for reaching broadband parametric amplification without the drawbacks observed in this work as the length and power requirements could be lowered.
Introducing a temperature controller into the system is a method of addressing the mode-mixing and bandwidth reduction without the need to change the dimensions of the waveguide or the spiral structure. Operating at the optimized set temperature at a given wavelength detuning of the waveguide allows us to reduce the fluctuations remarkably. Coupling with an optimized grating coupler [33] or an inverse taper [34] design can reduce the heating at the input coupling leading to a higher degree of pump power independence in the CE.
The parametric gain formation in $\rm { Si}_3\rm{N}_4 $ waveguides has also been confirmed in this work. Such gain can be utilized in the two-pump scheme to obtain cascaded idler generation. This work enables further studies controlling the frequency combs and stable wavelength conversion.
Funding
Swiss National Science Foundation (182414).
Acknowledgments
The authors would like to thank Dr. Marco Clementi for their assistance in the integration of the temperature controller unit into the stage.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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