Abstract

We report the stabilization of a soliton’s carrier frequency from a silicon nitride microresonator using a spatial interferometer to close the loop around thermal tuning. The spectral offset of the soliton carrier frequency was derived from spatial fringe pattern generated by 160 GHz repetition rate soliton pulses. Results were compared to real-time measurements by an Optical Spectrum Analyzer (OSA). The spatial interferometer and the OSA control results were in agreement and the resulting stabilization level is presented.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Kerr Frequency Combs (KFC) in microresonators are one method to generate high repetition rate, ultrashort pulses to form Optical Frequency Combs (OFC) [1]. Kerr combs are achieved by optically coupling a continuous wave laser into a nonlinear microresonator that use parametric effects to generate equidistant comb lines [2,3]. However, these unique devices require stabilization in order for them to function as part of a system for precision timing or spectroscopy. Two key metrics have been identified that define the unique characteristics of a KFC, they are Self Frequency Shift (SFS) produced by a Raman frequency shift and Spectral Recoil (SR) created by crossing modes that enhances a comb frequency. The SFS and SR have been extensively reported by [47], additional SR studies by [8,9] have also been reported.

Stone’s work on stabilizing a 22 GHz Silica microresonator consisted of utilizing RF beat signals from a stabilized cavity laser and a master oscillator to control pump power and frequency [10]. Lucas imprinted a weak-phase-modulation sideband imprinted into the pump laser to detune a 14 GHz MgF2 whispering gallery resonator to stabilize comb power and bandwidth [8]. Zhang stabilized the carrier-envelope offset frequency of a 56MHz femtosecond laser using an RF signal derived from a heterodyne interferometer by cross-correlation of pulses [11]. A hybrid SiN waveguide and a 100MHz fiber comb were used to control the offset frequency to within 30MHz over a 7.5 hr period [12]. This paper, an extension of our earlier publication on a spatial interferometer used to track the soliton carrier frequency from a 160 GHz SiN microresonator [13], presents results from stabilization of the soliton’s carrier frequency based on a spatial fringe pattern generated by interfering two soliton pulses onto a camera. Our method allows for simple capturing of GHz level repetition rate soliton pulses and is simple relative to an RF system that requires other complexities from which to generate an RF signal and could be used as a method to maintain and stabilize a Kerr comb’s spectrum variability to within 19 GHz.

We followed the same process set forth by Joshi to achieve soliton mode lock using thermal tuning. [14]. Our revised approach used the additional fringe pattern from the pump frequency as a reference spatial frequency. The spatial frequency difference, $\Omega$, between the two fringe patterns identifies the soliton’s carrier frequency as a spectral offset due to SFS ($\Omega _{SFS}$) and SR ($\Omega _{SR}$). The interpretation of the spatial frequency difference was used to generate an error signal and when combined with a Frequency Transfer Function (FTF) altered temperature of the microresonator via the voltage from a WFG. Our work sought to maintain the spectral offset between pump and soliton carrier frequency. We used the overall comb power and bandwidth measured from an OSA as our independent verification. As in our earlier report the SiN ring resonator was provided by Columbia University’s Lipson Nanophotonics Group and Gaeta Quantum and Nonlinear photonics group. Figure 1 is a block diagram of the system setup depicting the two feedback controls to the resonator heater and the AOM through the WFG based on fringe data captured from our spatial inteferometer [13].

 figure: Fig. 1.

Fig. 1. Block diagram of control setup for thermal ring resonator tuning with heater strips and pump power tuning through the EDFA. The OSA’s spectral centroid, comb power and bandwidth of the soliton’s frequency spectrum is used as independent verification.

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2. Thermal and pump power tuning control setup

The soliton can exist within a red detuned range defined by the limits of the detuning parameter, $\delta \omega$, shown as

$$\frac{\sqrt(3)}{2}\kappa < \delta \omega < \delta \omega_{max} = \kappa \frac{\pi^2 P_{in}}{16 P_{th}},$$
where $P_{th} = \frac {\kappa ^2 \hbar \omega _o}{8 \eta g_o}$ and $P_{in}$ is the optical input power [8]. Once the geometry and materials for the SiN resonator are set the formation of a soliton is controlled by input power, $P_{in}$, and the detuning parameter, $\delta \omega$. In our case, thermal detuning is employed as at fixed optical pump frequency. The soliton metrics that will be used as control variables are the pulse duration, $\tau _s$, as shown in Eq. (2) [8]
$$\tau_s = \frac{1}{D_1}\sqrt{\frac{D_2}{2\delta \omega}}$$
and spectral offset, which is a combination of SFS and SR, first term and second term, respectively, in Eq. (3) [6]:
$$\Omega = \frac{-8 \tau_R D_2}{15 \kappa_A D_1^2 \tau_s^4} - \frac{r \kappa_B}{\kappa_A E(1-\Gamma^2)} |h_{r-}|^2.$$

Our focus will be on the first term, SFS, where four terms are shown that influence the spectral shift. Two of those terms are related to the material and geometric dispersion of the microresonator, $D_1$ and $D_2$. Both of these terms are also a product of the mode solution for a given optical pump frequency which produces the effective index, $n_{eff}$. The rate of loss for different modes, $\kappa _A$ and $\kappa _B$, are a function of optical pump power and waveguide construction impacting the soltion’s peak power. The Raman time constant is $\tau _R$ and $\Gamma$ is the soliton pump efficiency, $\Gamma = P_{sol}/P_{min}$. Lastly, the soliton pulse width, $\tau _s$, has an inverse fourth power impact to the SFS, as shown relative to the pump frequency in Fig. 2(a). We also observed changes in the soliton pulse width from detuning the heater voltages on the microresonator. The results are shown in Fig. 2(b), where the detuning of the microresonator has an inverse relation with the pulse width as expected from Eq. (2). The linear fit from the increasing heater voltage data has a slope and intercept of $-3.521e^{-7} + 2.286$, with units of $Vdc*ps^{-4}$. Combining the detuning versus pulse width fit with the slope and intercept shown in Fig. 2(a) of $1.839e^{-5} + 0.2705$, with units of $THz*ps^{-4}$ we obtain a slope of 52.2 GHz/mv. This value is within $4.2\%$ of our OSA to spatial interferometer calibration slope, 50 GHz/mv, which will be discussed in the following section. Based on the separate observations between detuning via the heater voltage, the spectral offset and pulse width from the OSA data it is a clear indication that the detuning is the primary control of the SFS in the soliton.

 figure: Fig. 2.

Fig. 2. (a), Plot of the soliton spectrum’s centroid and pulse width taken from OSA data during calibration confirms the relationship with Eq. (3) [15]. (b), Plot of soliton pulse calculated from the OSA spectrum versus the heater voltage. Note the pulse width hysteresis is speculated to be from a longer cooling time constant in the microresonator after increasing the temperature via the heater voltage.

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3. Spatial interferometer stabilization metrics

3.1 Thermal and pump power tuning impacts

Figure 3 is a photograph of the ring resonator with Pt heaters deposited directly onto the SiN and attached to voltage probes that lead to a WFG. Research has shown that changes in index of refraction can effectively control repetition rate and ultimately the carrier frequency of the soliton [16]. Since our SiN ring resonator is thermally tuned into resonance and then into soliton mode lock, any additional thermal energy can change the index of refraction and thus the group velocity of the propagating pulse. To thermally tune a SiN resonator into soliton mode lock a ramp function with a voltage peak-peak and an offset is applied via the voltage probes and attached to channel one output of an WFG. The offset voltage is adjusted until a resonance is obtained. After resonance, a chaotic regime is entered where, due to SPM, frequencies begin to form without any mode lock. Voltage adjustment continues until Four Wave Mixing (FWM) begins, followed by frequency cascade [17]. At that point the WFG is setup to operate in burst mode and triggered to create a mode locked soliton [14], where an un-stabilized mode locked soliton forms. Previous spatial interferometer work had shown how the spectral shift, $\Omega$, represents the shift of the soliton’s carrier frequency away from the pump frequency and can be tracked by the spatial frequency of a spatial interferometer. The instrument presented in [13] is based on a fringe pattern generated by Young’s double slit experiment. We project two fringe patterns, one for the known pump and the other from the KFC, onto a pixelated camera, process the fringe patterns and extract the carrier frequency through a series of Matlab routines. Here, we show how we stabilize $\Omega$ by adjusting the resonator temperature, monitoring the spatial frequency of the fringe pattern from our spatial interferometer. The SiN microresonator setup is shown in [13] with the exception that the RIO pump laser is replaced with a ultralow expansion glass Fabry-Perot stabilized laser at 1565nm. After the ring resonator had been thermally tuned into a soliton mode, the system was allowed to operate in Open Loop (OL), meaning no outside intervention, for a period of time depending on the experiment. Prior to transitioning to Closed Loop (CL) the Matlab routine calculates a CL setpoint, $\Omega _{sp}$. The CL uses a simple PID routine to minimize the error signal between $\Omega _{sp}$ and $\Omega _{rt}$ by adjusting the offset voltage on a two channel Agilent 33520B Wave-Form Generator (WFG) to change the temperature which alters the group index of the microresonator. An AA opto-electronics fixed frequency Acousto-Optical Modulator (AOM) controlled through the second channel of the WFG was used to control optical power coupled into the SiN microresonator.

 figure: Fig. 3.

Fig. 3. SiN micro-resonator ring created using CMOS methods shown with Platinum heater strips.

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An AOM driver and a fiber pig tailed modulator was used to adjust optical pump power into SiN chip. The second channel of the WFG was used to output a DC voltage to the AOM driver and control the pump input power into the chip. The AOM driver input voltage was set to 4.5 Vdc to correspond to approximately 30-34 mw of chip pump power. Varying the WFG output voltage between 2.74 to 4.74 Vdc and measuring the output power with a power meter produced a slope of 7.96 mw/Vdc. To determine the impact on soltion carrier frequency, the AOM driver voltage was varied by $\pm 50$ mv about the setpoint. Figure 4(b) illustrates the results of a $\pm 0.4$ mw pump power, netting a change in soliton carrier frequency of 314 KHz/mw. Clearly the heater voltage control has a larger impact on the carrier frequency of the soliton than the AOM as expected [8,10].

 figure: Fig. 4.

Fig. 4. (a), Illustration of voltage profile applied to the SiN ring resonator after soliton mode lock (red) and the resulting changes in the soliton’s carrier frequency as determined from a spatial interferometer(blue) and the OSA (blue squares). (b), AOM voltage profile (red) versus soliton carrier frequency (blue) and OSA (blue squares). (c), Heater voltage profile (red) versus soliton’s integrated amplitude (blue) and comb power from OSA (blue squares). (d), Illustration of AOM voltage profile control pump input power (red) with changes in the soliton’s integrated amplitude from the spatial interferometer (blue) and comb power from OSA (blue squares).

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Use of the AOM to control pump input power had two goals. The first was to compensate for a opto-mechanical drift that occurs with the free space optical coupling of the lensed pump fiber. Any drop in optical power, $P_{in},$ affects the detuning range where the soliton can exist as shown in Eq (1). Secondly, the AOM control was to maintain a relatively constant comb power during detuning. To asses impact of the pump power on our tuning metrics via AOM control, the Soliton’s Integrated Amplitude (SIA) was treated as an independent metric from the heater control. The SIA was merely the summation of spatial fringe power [13]. Likewise, the heater voltage’s impact on the SIA was measured and shown in Fig. 4(c). Figure 4(d) shows that as the pump power increases the SIA decreases. Previous work has shown a small correlation between the SIA and the soliton’s spectral bandwidth, indicating increased bandwidth increased the total amount of comb power thus the spatial fringe power as well [13]. Both the AOM and heater control have inverse relationship with the SIA, indicating an inverse relation with the spectral bandwidth of the soliton. The AOM and heater control impacted the soliton’s normalized comb power by $16\%$ and $7\%$, respectively.

4. Results

Our control method will employ the technique of controlling the spectral offset, $\Omega$ (Eq. (3)), with respect to the pump frequency by modulating the heater voltage at a constant pump power. Actual implementation begins with collecting 2D fringes and converting them to 1D data for both pump and soliton. The spatial frequency for both are converted to optical frequencies by calibrating the spatial interferometer to the OSA. The method is to tune the heater voltage above and below the soliton voltage set point while simultaneously collecting OSA spectrum and spatial frequency data from the resulting fringes. Post processing of the soliton spectrum by curve fitting a $\textrm{sech}^2$ envelope using $A\textrm{sech} [\frac {(\omega - \Omega )\pi \tau _s}{2}]$, where A is the amplitude of the spectrum, $\omega$ the angular frequency, $\Omega$ is the spectral offset and $\tau _s$ as the pulse width. A linear fit from the plot of fringe spatial frequency, from the interfereometer, against the OSA calculated spectrum centroid was used to calibrate the spatial interferometer. Figure 4(a) illustrates a 22 mv increase in heater voltage between time intervals of 150 to 250 seconds resulting in a slope of 50 GHz/mv to the soliton carrier frequency as measured by the OSA and closely tracked by our interferometer. The calibration slope of 50 GHz/mv agrees well with the 52 GHz/mv slope determined by the data shown in Figs. 2(a) and 2(b). A time delay between voltage shifts was applied to allow the system to settle and collect OSA samples, thus producing the stair step affect in the plots. Further details of the calibration method is discussed in [13].

The start of a soliton stabilization control begins a length of time defined by the user as OL before a transition to CL, followed by another OL segment identical in length to the initially OL. During the OL segment a setpoint, $\Omega _{sp}$, is determined based on the difference between the last N=10 average pump and soliton carrier frequencies prior to transitioning to CL, shown in Eq. (4). The variables $f_p^o$ and $f_s^o$ refer to pump and soliton optical frequencies derived from spatial fringes, respectively:

$$\Omega_{sp} = \frac{1}{N}\sum_{\textrm{n=1}}^{N}f_{pn}^o - \frac{1}{N}\sum_{\textrm{n=1}}^{N}f_{sn}^o$$
$$\epsilon = \Omega_{sp} - \sum_{\textrm{f=1}}^{frames}\left(\frac{1}{N}\sum_{\textrm{n=f}}^{N+f}f_{pn}^o - \frac{1}{N}\sum_{\textrm{n=f}}^{N+f}f_{sn}^o\right).$$

The initial WFG voltage offset, $V_{init}$, used to create the soliton mode lock, is read by the software and stored during the OL segment. Once the spectral offset set point is established and the loop transitions to CL an error signal, ($\epsilon$), described by Eq. (5) is calculated. The error signal is based on the difference between the calculated pump and soliton carrier frequency for each frame and the setpoint. The error signal combined with a Frequency Transfer Function (FTF) in units of mv/GHz determines the change in WFG voltage to maintain the setpoint, $\Omega _{sp}$. Initially the FTF was based on the calibration slope of 50 GHz/mv, however, further changes were required to empirically tune the control loop to sustain a viable soliton over the course of the measurement cycle and minimize standard deviation of the soliton carrier frequency. The FTF can often vary depending on the room temperature and optical power coupled into the resonator. Two separate FTF values were used in our data collection, either 1mv/250 GHz or 1mv/1000 GHz. The updated offset voltage, $V_n$, is determined from the previous voltage, $V_{n-1}$ value plus the multiplication of the error signal with a FTF variable as shown in Eq. (6). The frequency transfer function is FTF = 1/X (mv/GHz) , where $X = (250,1000)$. The loop rate of our processing demonstrated a mean = 54ms with a standard deviation = 23ms while our camera operates at a 30ms frame rate.

$$V_{n} = V_{n-1} + \epsilon*FTF$$

Figure 5(a) is a graph of the pump frequency calculated from the pump fringe data (black) over a 2 hour period compared to the pump frequency obtained from the OSA (red-stars). Fluctuations in the optical pump frequency calculated from pump spatial fringes are a result of pump fringe amplitude variations during fringe processing algorithm. The fringe amplitude variations manifest themselves as a change in spatial frequency and will have to be further investigated. Figure 5(b) plots the Pump Integrated Amplitude (PIA) (red) calculated during the data collection. Note the peaks and valleys of the PIA correlates to the pump frequency variations. We calculated the Root Mean Squared (RMS) error between the OSA and the measured pump frequency from the spatial interferometer to be 14GHz.

Figure 5(c) illustrates the soliton carrier frequency during the OL and CL segments, where the heater control began ~168 seconds into the data collection as shown by the text arrows denoted as "Open Loop" at the start and end of the data set. Note that the slow drift in the calculated carrier frequency(blue) tracks the pump frequency drift shown in the previous plot. Figure 5(d) shows the soliton carrier frequency within the first 1000 seconds of data collection. As seen, the OL carrier frequency has an upward drift along with a larger amount of jitter. Once the set point, $\Omega _{sp}$, is established and the loop is closed the processing begins control of the soliton carrier frequency with the offset voltage from the WFG. The carrier frequency removes the upward drift by slowly shifting down to the set point which is maintained for the 2 hour duration. The carrier frequency maintains a reduction of jitter by roughly $70\%$ for over 2 hours. Longer data collection sets are possible, but tend to suffer from opto-mechanical drift in the free space optics, namely the coupling pump fiber and output capture lens which suffered from lab temperature swings. We calculated the RMS error between the OSA and the measured soliton’s spectral centroid frequency from the spatial interferometer to be 19GHz.

 figure: Fig. 5.

Fig. 5. (a), Plot of the pump frequency from the spatial interferometer(black) and OSA (red) with an RMS error of 14GHz. Drifts in the pump frequency can be related back to the variations in the pump fringe amplitude. The calculated pump and soliton carrier frequency (c) are plotted on the same frequency scale to maintain context of the variation. (b), Plot of the pump (red) and soliton fringe (blue) integrated amplitudes. The metrics indicates changes in both amplitude and width of the fringe profile. Typically changes can be tracked down to power drifts in pump coupling. (c), Plot of the carrier frequency as determined by the spatial interferometer (blue) and the OSA (red squares) with an RMS error of 19GHz. The OSA data and interferometer data match well with one another during the 2 hour data collection. (d), A time zoomed look at the soliton carrier frequency drifts upwards until the transition from OL to CL around 168 seconds. The data quickly settles down and shifts to the setpoint established during OL.

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The actual control variable, soliton spectral offset calculated as the difference between the pump and carrier, is shown in Fig. 6(a) and centered about the setpoint, $\Omega _{sp}$ (black dash). Figure 6(b) graphs the WFG voltage control, where the voltage decreased by 14 mv, from the initial value, during the data collection period to counteract the carrier frequency upward trend noted earlier. A typical open loop maximum voltage deviation is $\pm 5mv$ before the soliton ceases to exist, however in this data set the voltage exceeded that maximum by almost three times its open loop limit. Unfortunately, a combination of control with the AOM and the heater voltage were unsuccessful leading to premature failure of the soliton each time. It is suspected that since the fringe amplitude exhibited changes induced by opto-mechanical drift coupled with heater changes the AOM tended to overcompensate the optical pump power into the microresonator leading to the soliton failure. Further, work to stabilize the opto-mechanical coupling in/out of the microresonator is required before a meaningful AOM/heater control loop can be processed.

 figure: Fig. 6.

Fig. 6. (a), Difference frequency, $\Omega$, plot (orange) and the setpoint(black dash). The difference between the pump and soliton frequency were used as the control metric for the CL segment. (b), Plot of the WFG dc voltage as a function of time to control the microresonator heaters. The OL and OSA paused sections are noted where data was not collected.

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As an independent measurement of our CL control, OSA data of the soliton’s optical spectrum is taken simultaneously. Figure 7(a) illustrates the data calculated from the captured OSA spectrum during the complete data collection cycle. The total normalized comb power, calculated by summing up the amplitude of each comb tooth in the spectrum then normalizing, is shown by blue circles. The comb power drops as the control switches from OL to CL, then stabilizes where the comb power only varies by 1.5$\%$. Likewise, the comb 3 dB bandwidth (red stars) drops and stabilizes after the control loop closes to maintain a bandwidth with a standard deviation of 100 GHz. The pulse width from the OSA data is calculated to range between 77-78 fs with a standard deviation of 0.58 fs during the CL segment. Figure 7(b) is a 2D plot of several variables calculated from the OSA spectrum and displayed as a function of time in the y-axis and wavelength in the x-axis. Visualization of the OSA spectrum responses to the OL and CL segments are clearly evident over time for each frequency. The plot shows the comb frequencies that are higher/lower than the expected $\textrm{sech}^2$ curve fit for each spectrum captured shown as yellow/blue. The pump frequency (black), the spectral centroid or soliton carrier (blue) and the 3dB bandwidth lines (red) are all plotted on top of the OSA spectrum power deviations.

 figure: Fig. 7.

Fig. 7. (a), OSA spectrum derived metrics of comb power(blue circles) and 3 dB bandwidth (red stars) during the complete 2 hour data collection cycle. Both metrics went through a settling down period before stabilizing out when transitioning from OL to CL. The OSA paused between 500 - 1500 seconds. (b), OSA power deviation, as a function of time and wavelength, from the sech2 curve fit. The combs above the fit shade to yellow while those below the fit shade to blue. The pump frequency (black) is shown next to the spectrum carrier frequency (blue) and the 3 dB bandwidth boundaries are identified (red). Enhanced combs at higher wavelengths (yellow) coincided with shifts in the carrier frequency and bandwidth of the spectrum.

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Another indication of the improvement between OL and CL is shown in Fig. 8(a) and Fig. 8(b). The histograms with 100 frequency bins are visual indicators of the frequency spread from collected data and derived from the spatial interferometer. The pump histogram with a fringe based optical frequency standard deviation of 13 GHz has a spread wider than would be expected from a Fabry-Perot stabilized cavity, however, recall the bulk of the perceived spread is believed to be induced by amplitude variations of the pump fringes. The derived pump frequency variations also imprint themselves onto the soliton carrier frequency since we use the pump as a reference. The soliton carrier frequency spread is very evident in the OL mode with a standard deviation of 63GHz. However, once the CL segment is reached the frequency distribution narrows with a standard deviation of 19GHz. The true indication of how well our approach stabilized the soliton is by viewing the frequency difference, $\Omega _{sp}$ between pump and soliton carrier. Figure 9(a) illustrates the frequency spread of $\Omega$ with a standard deviation of 12.3 GHz (100 pm), well within the expected values of our system performance. The data illustrates a real improvement in controlling the resonator drift with active changes to the heater offset voltage using the changes in soliton carrier frequency relative to the pump from a spatial interferometer. In this case the more aggressive FTF of 1/250 provided the better control metrics. A comparison of SFS versus pulse width, Fig. 2(a), from the calibration data to the actual OL/CL data collected off the OSA is shown in Fig. 9(b). The OL OSA data shows a larger distribution in both axis of the plot, where as the CL OSA data distribution is reduced by roughly 28$\%$ in both axis. The CL data distribution also has a shift in both axis that coincides with the frequency set point relative to the optical pump frequency and the resulting pulse shift in the soliton’s spectrum once the system settles down, as shown inFig. 7(a).

 figure: Fig. 8.

Fig. 8. (a), Pump histogram during the complete data collection cycle. Standard deviation of 13 GHz was higher than expected due to fringe amplitude impacts. (b), Soliton carrier frequency for both OL and CL segments showing a standard deviation of 63 and 13 GHZ, respectively. The CL distribution tends towards a Gaussian distribution.

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 figure: Fig. 9.

Fig. 9. (a), Histogram of the frequency difference between pump and soliton carrier frequency used in the error signal for closed loop control. The system maintained a standard deviation of 12.3 GHz over a 2 hour period. (b), Graph comparing the OL (green) data and the CL (orange) data to the calibration data. The CL data shows a tighter grouping versus the OL data, both sets of data points follow the same linear trend as the calibration data. Offset between calibration and control data are from different environment influences.

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5. Conclusion

We were able to show the control of the spectral offset of a SiN microresonator using a spatial interferometer as a feedback system. The spatial frequency and integrated amplitude of the resulting fringe were related to the soliton’s carrier frequency and spectral bandwidth. Through the use of integrated heater elements deposited on the microresonator the voltage was used to change the cavity temperature to alter the spectral offset. An AOM was used to maintain consistent soliton cavity power due to compensate for thermal drifts in the launch fiber and in the resonator cavity. The results were a control of spectral shift to standard deviation of 12.3 GHz(100 pm), comb power maintained to within 1.5$\%$ and bandwidth of standard deviation of 100 GHz.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

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3. K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express 21(1), 1335–1343 (2013). [CrossRef]  

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16. X. Xue, Y. Xuan, C. Wang, P.-H. Wang, Y. Liu, B. Niu, D. E. Leaird, M. Qi, and A. M. Weiner, “Thermal tuning of kerr frequency combs in silicon nitride microring resonators,” Opt. Express 24(1), 687–698 (2016). [CrossRef]  

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References

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  1. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
    [Crossref]
  2. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
    [Crossref]
  3. K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express 21(1), 1335–1343 (2013).
    [Crossref]
  4. M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
    [Crossref]
  5. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-q silica microresonator,” Optica 2(12), 1078–1085 (2015).
    [Crossref]
  6. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Theory and measurement of the soliton self-frequency shift and efficiency in optical microcavities,” Opt. Lett. 41(15), 3419–3422 (2016).
    [Crossref]
  7. X. Yi, Q.-F. Yang, X. Zhang, K. Y. Yang, and K. Vahala, “Single-mode dispersive waves and soliton microcomb dynamics,” Nat. Commun. 8(1), 14869 (2017).
    [Crossref]
  8. E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
    [Crossref]
  9. Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Spatial-mode-interaction-induced dispersive waves and their active tuning in microresonators,” Optica 3(10), 1132–1135 (2016).
    [Crossref]
  10. J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
    [Crossref]
  11. X. Zhang, G. Wu, M. Hu, and S. Xiong, “Stabilizing carrier-envelope offset frequency of a femtosecond laser using heterodyne interferometry,” Opt. Lett. 41(18), 4277–4280 (2016).
    [Crossref]
  12. D. R. Carlson, D. D. Hickstein, A. Lind, S. Droste, D. Westly, N. Nader, I. Coddington, N. R. Newbury, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Self-referenced frequency combs using high-efficiency silicon-nitride waveguides,” Opt. Lett. 42(12), 2314–2317 (2017).
    [Crossref]
  13. D. K. Mefford and P. J. Reardon, “Towards a stabilized kerr optical frequency comb with spatial interference,” Appl. Opt. 59(26), 7930–7937 (2020).
    [Crossref]
  14. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Thermally controlled comb generation and soliton modelocking in microresonators,” Opt. Lett. 41(11), 2565–2568 (2016).
    [Crossref]
  15. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996).
    [Crossref]
  16. X. Xue, Y. Xuan, C. Wang, P.-H. Wang, Y. Liu, B. Niu, D. E. Leaird, M. Qi, and A. M. Weiner, “Thermal tuning of kerr frequency combs in silicon nitride microring resonators,” Opt. Express 24(1), 687–698 (2016).
    [Crossref]
  17. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. 38(18), 3478–3481 (2013).
    [Crossref]

2020 (1)

2018 (1)

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

2017 (3)

D. R. Carlson, D. D. Hickstein, A. Lind, S. Droste, D. Westly, N. Nader, I. Coddington, N. R. Newbury, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Self-referenced frequency combs using high-efficiency silicon-nitride waveguides,” Opt. Lett. 42(12), 2314–2317 (2017).
[Crossref]

X. Yi, Q.-F. Yang, X. Zhang, K. Y. Yang, and K. Vahala, “Single-mode dispersive waves and soliton microcomb dynamics,” Nat. Commun. 8(1), 14869 (2017).
[Crossref]

E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
[Crossref]

2016 (6)

2015 (1)

2014 (1)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

2013 (2)

2011 (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref]

1996 (1)

Brabec, T.

Brasch, V.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Briles, T. C.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

Carlson, D. R.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

D. R. Carlson, D. D. Hickstein, A. Lind, S. Droste, D. Westly, N. Nader, I. Coddington, N. R. Newbury, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Self-referenced frequency combs using high-efficiency silicon-nitride waveguides,” Opt. Lett. 42(12), 2314–2317 (2017).
[Crossref]

Coddington, I.

Diddams, S. A.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

D. R. Carlson, D. D. Hickstein, A. Lind, S. Droste, D. Westly, N. Nader, I. Coddington, N. R. Newbury, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Self-referenced frequency combs using high-efficiency silicon-nitride waveguides,” Opt. Lett. 42(12), 2314–2317 (2017).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref]

Drake, T. E.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

Droste, S.

Foster, M. A.

Gaeta, A. L.

Geiselmann, M.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

Gorodetsky, M. L.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Guo, H.

E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
[Crossref]

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

Hänsch, T. W.

Herr, T.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Hickstein, D. D.

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref]

Hu, M.

Jang, J. K.

Ji, X.

Johnson, A. R.

Joshi, C.

Jost, J. D.

E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Karpov, M.

E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
[Crossref]

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

Kippenberg, T. J.

E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
[Crossref]

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref]

Klenner, A.

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Kordts, A.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

Krausz, F.

Lamont, M. R. E.

Leaird, D. E.

Levy, J. S.

Lind, A.

Lipson, M.

Liu, Y.

Lucas, E.

E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
[Crossref]

Luke, K.

Mefford, D. K.

Miller, S. A.

Nader, N.

Newbury, N. R.

Niu, B.

Okawachi, Y.

Papp, S. B.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

D. R. Carlson, D. D. Hickstein, A. Lind, S. Droste, D. Westly, N. Nader, I. Coddington, N. R. Newbury, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Self-referenced frequency combs using high-efficiency silicon-nitride waveguides,” Opt. Lett. 42(12), 2314–2317 (2017).
[Crossref]

Pfeiffer, M. H. P.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

Poppe, A.

Qi, M.

Reardon, P. J.

Saha, K.

Salem, R.

Shim, B.

Spencer, D. T.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

Spielmann, C.

Srinivasan, K.

Stone, J. R.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

Suh, M.-G.

Vahala, K.

Wang, C.

Wang, C. Y.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Wang, P.-H.

Weiner, A. M.

Westly, D.

Wu, G.

Xiong, S.

Xu, L.

Xuan, Y.

Xue, X.

Yang, K. Y.

Yang, Q.-F.

Yi, X.

Zervas, M.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

Zhang, X.

X. Yi, Q.-F. Yang, X. Zhang, K. Y. Yang, and K. Vahala, “Single-mode dispersive waves and soliton microcomb dynamics,” Nat. Commun. 8(1), 14869 (2017).
[Crossref]

X. Zhang, G. Wu, M. Hu, and S. Xiong, “Stabilizing carrier-envelope offset frequency of a femtosecond laser using heterodyne interferometry,” Opt. Lett. 41(18), 4277–4280 (2016).
[Crossref]

Appl. Opt. (1)

Nat. Commun. (1)

X. Yi, Q.-F. Yang, X. Zhang, K. Y. Yang, and K. Vahala, “Single-mode dispersive waves and soliton microcomb dynamics,” Nat. Commun. 8(1), 14869 (2017).
[Crossref]

Nat. Photonics (1)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Optica (2)

Phys. Rev. A (1)

E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, “Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative kerr solitons in optical microresonators,” Phys. Rev. A 95(4), 043822 (2017).
[Crossref]

Phys. Rev. Lett. (2)

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in kerr-microresonator frequency combs,” Phys. Rev. Lett. 121(6), 063902 (2018).
[Crossref]

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116(10), 103902 (2016).
[Crossref]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Block diagram of control setup for thermal ring resonator tuning with heater strips and pump power tuning through the EDFA. The OSA’s spectral centroid, comb power and bandwidth of the soliton’s frequency spectrum is used as independent verification.
Fig. 2.
Fig. 2. (a), Plot of the soliton spectrum’s centroid and pulse width taken from OSA data during calibration confirms the relationship with Eq. (3) [15]. (b), Plot of soliton pulse calculated from the OSA spectrum versus the heater voltage. Note the pulse width hysteresis is speculated to be from a longer cooling time constant in the microresonator after increasing the temperature via the heater voltage.
Fig. 3.
Fig. 3. SiN micro-resonator ring created using CMOS methods shown with Platinum heater strips.
Fig. 4.
Fig. 4. (a), Illustration of voltage profile applied to the SiN ring resonator after soliton mode lock (red) and the resulting changes in the soliton’s carrier frequency as determined from a spatial interferometer(blue) and the OSA (blue squares). (b), AOM voltage profile (red) versus soliton carrier frequency (blue) and OSA (blue squares). (c), Heater voltage profile (red) versus soliton’s integrated amplitude (blue) and comb power from OSA (blue squares). (d), Illustration of AOM voltage profile control pump input power (red) with changes in the soliton’s integrated amplitude from the spatial interferometer (blue) and comb power from OSA (blue squares).
Fig. 5.
Fig. 5. (a), Plot of the pump frequency from the spatial interferometer(black) and OSA (red) with an RMS error of 14GHz. Drifts in the pump frequency can be related back to the variations in the pump fringe amplitude. The calculated pump and soliton carrier frequency (c) are plotted on the same frequency scale to maintain context of the variation. (b), Plot of the pump (red) and soliton fringe (blue) integrated amplitudes. The metrics indicates changes in both amplitude and width of the fringe profile. Typically changes can be tracked down to power drifts in pump coupling. (c), Plot of the carrier frequency as determined by the spatial interferometer (blue) and the OSA (red squares) with an RMS error of 19GHz. The OSA data and interferometer data match well with one another during the 2 hour data collection. (d), A time zoomed look at the soliton carrier frequency drifts upwards until the transition from OL to CL around 168 seconds. The data quickly settles down and shifts to the setpoint established during OL.
Fig. 6.
Fig. 6. (a), Difference frequency, $\Omega$, plot (orange) and the setpoint(black dash). The difference between the pump and soliton frequency were used as the control metric for the CL segment. (b), Plot of the WFG dc voltage as a function of time to control the microresonator heaters. The OL and OSA paused sections are noted where data was not collected.
Fig. 7.
Fig. 7. (a), OSA spectrum derived metrics of comb power(blue circles) and 3 dB bandwidth (red stars) during the complete 2 hour data collection cycle. Both metrics went through a settling down period before stabilizing out when transitioning from OL to CL. The OSA paused between 500 - 1500 seconds. (b), OSA power deviation, as a function of time and wavelength, from the sech2 curve fit. The combs above the fit shade to yellow while those below the fit shade to blue. The pump frequency (black) is shown next to the spectrum carrier frequency (blue) and the 3 dB bandwidth boundaries are identified (red). Enhanced combs at higher wavelengths (yellow) coincided with shifts in the carrier frequency and bandwidth of the spectrum.
Fig. 8.
Fig. 8. (a), Pump histogram during the complete data collection cycle. Standard deviation of 13 GHz was higher than expected due to fringe amplitude impacts. (b), Soliton carrier frequency for both OL and CL segments showing a standard deviation of 63 and 13 GHZ, respectively. The CL distribution tends towards a Gaussian distribution.
Fig. 9.
Fig. 9. (a), Histogram of the frequency difference between pump and soliton carrier frequency used in the error signal for closed loop control. The system maintained a standard deviation of 12.3 GHz over a 2 hour period. (b), Graph comparing the OL (green) data and the CL (orange) data to the calibration data. The CL data shows a tighter grouping versus the OL data, both sets of data points follow the same linear trend as the calibration data. Offset between calibration and control data are from different environment influences.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

( 3 ) 2 κ < δ ω < δ ω m a x = κ π 2 P i n 16 P t h ,
τ s = 1 D 1 D 2 2 δ ω
Ω = 8 τ R D 2 15 κ A D 1 2 τ s 4 r κ B κ A E ( 1 Γ 2 ) | h r | 2 .
Ω s p = 1 N n=1 N f p n o 1 N n=1 N f s n o
ϵ = Ω s p f=1 f r a m e s ( 1 N n=f N + f f p n o 1 N n=f N + f f s n o ) .
V n = V n 1 + ϵ F T F