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Abstract
The radio frequency (RF) spectrum of microcombs can be used to evaluate its phase noise features and coherence between microcomb teeth. Since microcombs possess characteristics such as high repetition rate, narrow linewidth and ultrafast dynamical evolution, there exists strict requirement on the bandwidth, resolution and frame rate of RF measurement system. In this work, a scheme with 1.8-THz bandwidth, 7.5-MHz spectral resolution, and 100-Hz frame rate is presented for RF spectrum measurement of microcombs by using an all-optical RF spectrum analyzer based on cross-phase modulation and Fabry Perot (FP) spectrometer, namely FP-assisted light intensity spectrum analyzer (FP-assisted LISA). However, extra dispersion introduced by amplifying the microcombs will deteriorate the bandwidth performance of measured RF spectrum. After compensating the extra dispersion through monitoring the dispersion curves measured by FP-assisted LISA, the more precise RF spectra of microcombs are measured. Then, the system is used to measure the noise sidebands and line shape evolution of microcombs within 2s temporal window, in which dynamic RF combs variation at different harmonic frequencies up to 1.96 THz in modulation instability (MI) state and soliton state are recorded firstly. Therefore, the improved bandwidth and resolution of FP-assisted LISA enable more precise measurement of RF spectrum, paving a reliable way for researches on physical mechanism of microcombs.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Like rulers of light, optical frequency combs allow for precise measurement and have brought revolutionary improvement to the metrology field [1]. Recently, Kerr frequency combs based on high-Q micro-resonator, namely microcombs, have characteristics such as high repetition rate, compact footprint and low power consumption [2]. Especially, the emergence of dissipative Kerr solitons (DKSs) could improve the performance of conventional frequency combs in applications where high coherence is desired, such as spectroscopy, astronomical measurement and optical coherence tomography (OCT) [3,4,5], etc. Moreover, soliton microcombs also offer abundant dynamic nonlinear phenomena, such as soliton state switching [6], breathing soliton [7], soliton crystals [8,9]. Such physical dynamics can be characterized in three domains, including optical, temporal and radio frequency (RF) domains. Generally, spectral and temporal information of microcombs can be directly obtained by optical spectrum analyzer (OSA) and autocorrelator, respectively [10,11]. In RF domain, the high repetition rate of microcombs raises high demand on the bandwidth of RF measurement system. The fundamental frequency of microcombs has been obtained [12], while the higher RF harmonic frequency is rarely studied.
When it comes to the scenario of generating the DKSs, microcombs intrinsically possess characteristics such as high repetition rate from GHz to THz, narrow linewidth under 1 GHz (owing to the high Q above 105 [12]) and fast scanning frequency process (generally under 1 s [13]). Accordingly, its RF measurement system should satisfy the following requirements: large bandwidth (at least THz), high resolution (under 1 GHz) and high frame rate. The traditional measuring method is using an electrical spectrum analyzer (ESA) with a high-speed photodetector (PD) [10]. Although it can provide high resolution of several Hz, the bandwidth is typically below 100 GHz limited by PD and ESA [14]. Another modified approach is based on frequency down conversion by adding extra electro-optic modulation [15] or pulse modulation [16], and the bandwidth can be extended from 100 GHz to 200 GHz. All the schemes described above enable the measurement of fundamental frequency while fail to record the higher order RF harmonic frequency information. Besides, the dynamic evolution process of broadband RF spectrum has been rarely observed. Thus, it is still challenging to achieve the RF spectrum measurement with large bandwidth, high resolution, and high frame rate.
In this paper, the RF spectrum of microcombs is experimentally measured and analyzed using an improved all-optical RF spectrum analyzer. In this scheme, the RF spectrum of microcombs is transformed to optical spectrum of the probe via cross-phase modulation (XPM) and then observed by a Fabry-Perot (FP) spectrometer. This all-optical light intensity spectrum analyzer (LISA) assisted with FP spectrometer, named as FP-assisted LISA, has achieved 1.8-THz bandwidth, 7.5-MHz resolution and 100-Hz frame rate. It satisfies the requirements of microcombs’ RF spectrum characterization very well. In order to get better measurement results, the microcombs need to be amplified before entering the FP-assisted LISA. As a result, the dispersion introduced by erbium-doped fiber amplifier (EDFA) would distort the pulse and decrease the peak intensity of microcombs [17]. In experiment, the dispersion is accurately compensated with a programmable filter through monitoring the dispersion curves obtained from RF spectrum firstly. Moreover, the dynamic noise sidebands and coherence of higher order RF harmonic frequencies (up to 40th harmonic frequencies) are successfully characterized and analyzed at modulation instability (MI) state and soliton state within 2 s (200 consecutive frames). These RF spectrum measurement results verify that the proposed method has the advantage of large bandwidth, high resolution, and high frame rate. It may reveal the rich dynamic information of RF spectrum, leading to a deeper understanding of microcombs. Compared to the conventional RF analyzers, the FP-assisted LISA can observe higher order RF harmonic frequency, which can reflect the coherence between comb teeth with larger spacing. In other words, it can reveal the second-order spectral coherence in optical domain [18], which can provide the insight into the spectral coherence dynamics of microcombs. Such a level of measurement capability is prohibitively unattainable in conventional scheme using ESA with limited spectral range typically ten of GHz.
2. Operating principle and experiment setup
Firstly, the theoretical expression of microcombs’ RF spectrum is derived for a more intuitive understanding. For simplicity, the Gaussian pulse form is used here which is reasonable owing to the similar trend with hyperbolic secant type [19]. Optical spectrum field of microcombs can be written as:
The Δw, FSR represent the line width and free spectral range of the microcombs, respectively. RF spectrum Scomb(ω) of microcombs represents the power spectrum of its temporal intensity. According to the convolution theorem [20], Scomb(ω) can be written as:
It is obvious that the RF spectrum of microcombs is also the combs with equal FSR, named RF combs. The j-th tooth in RF spectrum is the sum of the beat notes of all comb teeth with iFSR spacing in the optical spectrum. In addition, the higher order RF comb teeth can reflect the coherence between comb teeth with larger spacing in optical spectrum, which has been reported in mode-locked fiber lasers [21].
The traditional method based on ESA with PD has limited bandwidth of 100 GHz and can only observe little information due to the large FSR of microcombs. All-optical RF spectrum analyzer shown in Fig. 1(a) effectively solves the problem of insufficient bandwidth [22]. Firstly, supposing that the probe light is continuous wave (CW) with frequency ωprobe as: E(t) = exp(-iωprobet). Because of XPM in highly nonlinear fiber (HNLF), the temporal intensity of microcombs is transformed to the phase of CW probe as: E(t) = exp(-iωprobet)·exp(-imIcomb(t)), where XPM-induced nonlinear phase shift is mIcomb(t), m is equal to 4πLn2/λ (neglecting the transmission loss), n2 is the nonlinear index coefficient, L is the length of HNLF, λ is the center wavelength of microcombs, and Icomb(t)=|Ecomb(t)|2 represents the temporal intensity of the microcombs.
Fig. 1. (a) Schematic principle of microcomb’s RF spectrum measured by all-optical RF spectrum analyzer, using the HNLF as nonlinear medium, E(t) is the electric field of microcombs. (b) The simulation results of RF spectrum Scomb(ω) and the RF spectrum of microcombs modulated in the optical spectrum Io(ω).
Assuming that the nonlinear phase shift satisfies small-signal approximation, i.e., |mIcomb(t)| << 1 at all times, the temporal electric field can be also written as:
And the intensity of output optical spectrum can be obtained from the temporal electric field Eq. (4) [22] and combined with Eq. (3), the RF spectrum of microcombs around the probe is:
The detailed experimental setup for microcombs’ RF spectrum measurement is illustrated in Fig. 2(a), and can be divided into three parts: microcombs generation, dispersion compensation and FP-assisted LISA. In the first part, the kernel component is a packaged high-index doped silica glass micro-ring resonator which has 49-GHz FSR with high Q up to ∼2.05 × 106 [23]. The auxiliary and pump are amplified to 3.5 W and coupled into the microcavity from two opposite direction to deterministically generate Kerr solitons. The optical spectrum of typical single soliton at point B is depicted in Fig. 2(b). In dispersion part, the EDFA is utilized to amplify the power to occur the XPM in HNLF. But the additional dispersion induced by the EDFA and connection fiber will distort the pulse and the RF spectrum. A L-band programmable filter (Finisar waveshaper 4000s) is leveraged to compensate the dispersion.
Fig. 2. (a) Experimental setup of microcombs’ RF spectrum measurement, including three parts: comb generation, dispersion compensation and FP-assisted LISA. (b) Optical spectrum of microcombs at point B. (c) RF spectrum after XPM at point C. (d) Amplified RF spectrum observed from OSA and two shots measurement result from the FP spectrometer with the FSR of 1.5 GHz.
In FP-assisted LISA part, the power of amplified microcombs is around 15 dBm after EDFA and reduced to 12 dBm after a polarization controller (PC) and an optical coupler. Then the microcombs and CW probe at 1550 nm (NKT Basik E15) are coupled into the HNLF with L = 100 m, γ = 10 w-1km-1 and dispersion slope S of 0.0228 ps/nm2/km. It is noted that the power of the soliton should be carefully chosen to achieve high signal-to-noise ration and avoid the spectral overlap. The RF spectrum of microcombs can be obtained from the probe’s optical spectrum as shown in Fig. 2(c), which has similar trend with the simulation results in Fig. 1(b). In addition, the observed RF spectrum bandwidth can be as large as THz owing to the femtosecond response time of XPM. The harmonic frequencies of RF spectrum have shown up to 3 THz (from 1525 nm to 1550 nm) with 49-GHz FSR. The arbitrary harmonic radio frequency can be filtered out 0.4 nm by a C-band programmable filter, and then analyzed by the FP spectrometer consisting of a FP interferometer (SA200-12B) and a real-time oscilloscope. The FP spectrometer can measure 1.5 GHz in one frame, which is limited by the FSR of FP interferometer. As Fig. 2(d) shows, the RF harmonic frequency at 1.96 THz is firstly observed by Grating-based OSA (blue line in left) with 1.25-GHz resolution and then observed by the FP spectrometer with 7.5-MHz resolution.
3. Results
3.1 Parameter measurement
Bandwidth, resolution, and frame rate are three critical parameters of the RF spectrum analyzer. To evaluate these parameters of FP-assisted LISA, two co-polarized CW sources are applied to generate the high beat-note Ω and the temporal intensity is I(t) = cos(2πΩt). In order to test the resolution and frame rate, one of the CW sources is fixed at 1568 nm and the other is set at 1572 nm to generate around 500-GHz radio frequency. Two consecutive RF spectra are measured by the FP-assisted LISA in Fig. 3(a), and it displays the frame rate of 100 Hz and the temporal interval of 10 ms. The pulse width is 0.05 ms corresponding to 7.5 MHz resolution, according to the mapping factor of FP interferometer. Moreover, in order to test the bandwidth, one laser is still fixed at 1568 nm while the other is scanned from 1570 nm to 1587 nm with a step of 0.8 nm (around 100 GHz). The scanning range of the generated RF signal is from 0.2 THz to 2.4 THz, and the peak intensity of measured pulse is recorded by the FP-assisted LISA. As is depicted in Fig. 3(b), the measured results are plotted with blue dots while the fitting curve is red line. In this experiment, the HNLF with low dispersion slope is used, and the wavelengths of pump and probe are also located symmetrically with zero dispersion point (1561 nm) to achieve large measured bandwidth [22]. As a result, the 3-dB bandwidth is extended as large as 1.8 THz. the small intensity fluctuation is probably caused by the uneven gain of optical amplifier. It is noted that the 1.8-bandwidth is achieved by multi-step test. To sum up, the FP-assisted LISA has 7.5-MHz resolution, 100-Hz frame rate and bandwidth up to 1.8 THz, which can be applied to measure the RF spectrum of microcombs.
Fig. 3. (a) The frame rate (100 Hz) and resolution (7.5 MHz) measured from the FP interferometer. (b) The multi-step measured bandwidth of FP-assisted LISA.
3.2 Dispersion compensation
The Fourier transform of RF spectrum is the autocorrelation of temporal intensity, so the temporal change of microcombs can be reflected in its RF spectrum, as the Eq. (2) shows. In other words, an all-optical RF spectrum analyzer with large bandwidth can be applied to monitor dispersion compensation of the microcombs. In the dispersion compensation part as shown in Fig. 2(a), the programmable filter not only filters out L-band microcombs but also adds appropriate spectral phase to compensate the normal dispersion in L-band EDFA. In addition, the filtering shape is hyperbolic secant with the 3-dB bandwidth of 10 nm and the center wavelength of 1580 nm. Then corresponding radio frequency of microcombs is monitored by FP-assisted LISA to evaluate the matching degree of dispersion.
Firstly, the monitoring scheme is verified in simulation. The simulation parameters are consistent with the experimental system parameters. And the peak intensities of different radio frequencies are recorded as a function of compensated dispersion. Simulated dispersion curves (normalized) at different radio frequencies of 0.049, 0.245, 0.49, 0.735, 0.98, 1.225, 1.47 THz are plotted in Fig. 4(a). The peak of dispersion curves at different harmonic frequencies are all at the dispersion value of -0.1552 ps2, which represents the best dispersion-compensation point. Moreover, the steepness of dispersion curve represents the dispersion sensitivity. The dispersion sensitivity is higher with the increasing radio frequencies, which means that the dispersion degrades the coherence between the comb teeth with a large frequency spacing, according to the Eq. (5).
Fig. 4. (a) - (b) Simulation and experiment results (normalized) of dispersion carves at different harmonic radio frequencies, and both have the same legend. (c) The well-compensated (blue line) and uncompensated (red line) power spectra measured by FP-assisted LISA within 0.15 GHz and RF fundamental frequency transformed from temporal waveform (green line). (d) The autocorrelation trace of microcombs reconstructed by numerical processing of broadband RF spectrum.
Subsequently, the experimental dispersion curves measured by FP-assisted LISA confirmed the simulation results as shown in Fig. 4(b). The dispersion measurement is discretized due to the limitation of the minimum dispersion step size. When the RF is lower than 0.735 THz, the experimental dispersion curves fits well with the simulated results. Beyond 0.735 THz, the steepness of different dispersion curves is slightly deviated from the simulation ones, which may be caused by the lower signal-noise ratio (SNR) at higher radio frequency. At the same time, the increasing dispersion sensitivity with frequency also shows that the FP-assisted LISA with large bandwidth is helpful to compensate the dispersion. Considering the dispersion curve at 0.735 THz, the best dispersion-compensation value is -0.1056 ps2, and changing the corresponding phase through programmable filter can effectively eliminate the excess dispersion in the system. And comparing to the theoretical value of -0.1552 ps2, the deviation may be caused by the experimental parameter deviations and adjustment accuracy of the programmable filter.
To highlight the adverse impact of dispersion on RF spectrum measurement, two dispersion-compensation values of -0.1056 ps2 (well-compensated) and -0.2556 ps2 (uncompensated), like A and B shown in Fig. 4(b), are select to obtain the power spectra measured by FP-assisted LISA as shown in Fig. 4(c). It can be seen clearly that the compensated dispersion can effectively improve the SNR and observing range of microcombs. Besides, the RF fundamental frequency of 49 GHz has been converted from 100-μs temporal waveform measured by 59-GHz real time oscilloscope (KEYSIGHT DSAZ594A), as shown in Fig. 4(c) (green lines). It is noted that the electrical method with 10-kHz resolution consumes several minutes to get the results, and cannot measure the higher-order harmonic frequency.
In order to verify the dispersion compensation, the autocorrelation trace of single soliton has been transformed from RF spectrum according to the Wiener-Khintchine theorem as shown in Fig. 4(d) [11]. It shows the autocorrelation pulse width of 556 fs that corresponds to the sech2 pulse width of 380 fs. There is slight difference to the theoretical calculation result of 263 fs (10nm 3-dB bandwidth sech2 filtering) owing to imperfect dispersion compensation.
3.3 Dynamic RF spectrum of microcombs
After compensating the dispersion introduced by EDFA, the microcombs with original pulse width have been amplified to 15 dBm and measured by FP-assisted LISA. The dynamic phenomenon of microcombs is measured at different harmonic frequencies as illustrated in Fig. 5. Firstly, the MI state is obtained by detuning the laser and then measured by FP-assisted LISA. The harmonic frequencies of 0.49 THz, 0.98 THz, 1.47 THz and 1.96 THz are selected for microcombs’ dynamic analysis, which is displayed in Fig. 5(a). During the formation process of the subcombs, the multiple lines may exist in a single resonance, so the sidebands around the harmonic frequency in RF spectrum are generated in the MI state [8]. The multiple peaks, narrow single or noise sidebands’ features can be observed from different RF spectrum of MI state [8], which depends on the detuning between pump and resonance. So higher resolution is necessary to observe the narrower linewidth and more precise peak intensity. And the FP-assisted LISA with 7.5 MHz resolution is sufficient to meet the requirements of microcombs RF measurement.
Fig. 5. (a)-(b): Radio frequency dynamic evolution of MI state and single soliton state at 0.49 THz, 0.98 THz, 1.47 THz, 1.96 THz within two seconds, and corresponding spectra of two states.
Firstly, the MI state above the Fig. 5(a) is selected to observe the RF beat notes, and the line shapes of beat notes at corresponding harmonic frequencies can be seen from the left pictures. In the generating process of microcombs, multiple lines will exist in a single resonance, so the noise sidebands will occur around the RF comb teeth [12]. Therefore, the linewidths of the harmonic frequencies at MI state are much wider than the 7.5-MHz resolution, which reflects the large phase noise of the MI state. Moreover, significant different line shapes of harmonic frequencies are owing to the randomness of phase noise in the MI state. The coherence between the comb lines is gradually getting worse as the optical frequency spacing increases. Therefore, the noise sidebands of RF comb in the left of Fig. 5(b) broaden from the original 0.234 GHz at 0.49 THz to 0.405 GHz at 1.96 THz, indicating the broadening characteristic with the increasing harmonic frequencies. It may be caused by the worse coherence of the comb teeth with larger spacing. In other words, the large observation bandwidth of the RF spectrum can reveal the coherence strength of different harmonic frequencies. The two-dimensional graphs also show the dynamic evolution of radio frequency intensity within 2 s (including 200 successive frames). As the picture shows, the slight evolution at four harmonic frequencies in a 2-second observation window indicates the considerable temporal stability of the MI state. On the contrary, the single soliton has the advantage of high coherence as Fig. 5(b) shows. And the average Pearson correlation coefficient of single soliton is 0.97, which indicates the better temporal stability in 2 s compare to the 0.7 correlation of MI state [24]. The peak intensity of soliton state is also higher than the MI state. And the linewidths of different RF combs are around 7.5 MHz limited by the resolution of LISA. In a word, large-bandwidth, high-speed, high-resolution FP-assisted LISA has an outstanding advantage in analyzing the noise and coherence characteristics of microcombs.
4. Conclusion
In conclusion, we have experimentally demonstrated dispersion compensating and dynamic characterizing through microcombs’ RF spectrum by an improved all-optical RF spectrum analyzer. The proposed analyzer achieves 7.5-MHz resolution, 1.8-THz bandwidth and 100-Hz frame rate, which are suitable for RF spectrum characterization of microcombs. Then extra dispersion introduced by EDFA is compensated precisely according to the dispersion-sensitive curve at 735-GHz frequency, which effectively improves the microcombs’ RF measurement accuracy. Subsequently, the FP-assisted LISA successfully measured the noise sidebands characteristics and dynamics of four harmonic frequencies in microcombs’ RF spectrum including 0.49 THz, 0.98 THz, 1.47 THz and 1.96 THz in 2s (200 frames). The results verified large noise and poor coherence at MI state, and the high coherence of single soliton. From the results, this scheme is suitable for the wideband RF characterization of the microcombs, especially the microcombs with the FSR larger than 200 GHz. Moreover, the method can monitor larger harmonic frequency of RF spectrum and may pave a new way to investigate different MI states or soliton states.
Funding
National Natural Science Foundation of China (61505060, 61631166003, 61675081, 61735006, 61927817); National Key Research and Development Program of China (2019YFB2203102); China Postdoctoral Science Foundation (2018M640692).
Disclosures
The authors declare no conflicts of interest.
References
1. T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361(6402), eaan8083–162 (2018). [CrossRef]
2. N. G. Pavlov, S. Koptyaev, G. V. Lihachev, A. S. Voloshin, A. S. Gorodnitskiy, M. V. Ryabko, S. V. Polonsky, and M. L. Gorodetsky, “Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes,” Nat. Photonics 12(11), 694–698 (2018). [CrossRef]
3. M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354(6312), 600–603 (2016). [CrossRef]
4. E. Obrzud, M. Rainer, A. Harutyunyan, M. H. Anderson, J. Liu, M. Geiselmann, B. Chazelas, S. Kundermann, S. Lecomte, M. Cecconi, A. Ghedina, E. Molinari, F. Pepe, F. Wildi, F. Bouchy, T. J. Kippenberg, and T. Herr, “A microphotonic astrocomb,” Nat. Photonics 13(1), 31–35 (2019). [CrossRef]
5. W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6(1), 1–8 (2015). [CrossRef]
6. 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]
7. M. Yu, J. K. Jang, Y. Okawachi, A. G. Griffith, K. Luke, S. A. Miller, X. Ji, M. Lipson, and A. L. Gaeta, “Breather soliton dynamics in microresonators,” Nat. Commun. 8(1), 1–7 (2017). [CrossRef]
8. D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11(10), 671–676 (2017). [CrossRef]
9. P. Ryczkowski, M. Närhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018). [CrossRef]
10. P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546(7657), 274–279 (2017). [CrossRef]
11. Y. Zhao, L. Chen, W. Wang, R. Wang, H. Hu, X. Wang, C. Zhang, W. Zhang, and X. Zhang, “Repetition rate multiplication control of micro-combs assisted by perfect temporal Talbot effect,” APL Photonics 5(4), 046102 (2020). [CrossRef]
12. T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6(7), 480–487 (2012). [CrossRef]
13. Y. He, Q.-F. Yang, J. Ling, R. Luo, H. Liang, M. Li, B. Shen, H. Wang, K. Vahala, and Q. Lin, “Self-starting bi-chromatic LiNbO3 soliton microcomb,” Optica 6(9), 1138–1144 (2019). [CrossRef]
14. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107(6), 063901 (2011). [CrossRef]
15. P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid Electro-Optically Modulated Microcombs,” Phys. Rev. Lett. 109(26), 263901 (2012). [CrossRef]
16. Q. Li, T. C. Briles, D. A. Westly, T. E. Drake, J. R. Stone, B. R. Ilic, S. A. Diddams, S. B. Papp, and K. Srinivasan, “Stably accessing octave-spanning microresonator frequency combs in the soliton regime,” Optica 4(2), 193–203 (2017). [CrossRef]
17. Y. Geng, X. Huang, W. Cui, Y. Ling, B. Xu, J. Zhang, X. Yi, B. Wu, S. Huang, K. Qiu, C. W. Wong, and H. Zhou, “Terabit optical OFDM superchannel transmission via coherent carriers of a hybrid chip-scale soliton frequency comb,” Opt. Lett. 43(10), 2406–2409 (2018). [CrossRef]
18. S. M. Kobtsev and S. V. Smirnov, “Coherent properties of super-continuum containing clearly defined solitons,” Opt. Express 14(9), 3968–3980 (2006). [CrossRef]
19. Y. Li, S.-W. Huang, B. Li, H. Liu, J. Yang, A. K. Vinod, K. Wang, M. Yu, D.-L. Kwong, H.-T. Wang, K. K.-Y. Wong, and C. W. Wong, “Real-time transition dynamics and stability of chip-scale dispersion-managed frequency microcombs,” Light: Sci. Appl. 9(1), 1–10 (2020). [CrossRef]
20. J. W. Goodman, Introduction to Fourier Optics3rd ed. (Roberts & Company Publishers, 2004).
21. B. Li, Y. Yu, X. Wei, Y. Xu, K. K. Tsia, and K. K. Y. Wong, “Real-time observation of round-trip resolved spectral dynamics in a stabilized fs fiber laser,” Opt. Express 25(8), 8751–8759 (2017). [CrossRef]
22. C. Dorrer and D. N. Maywar, “RF Spectrum Analysis of Optical Signals Using Nonlinear Optics,” J. Lightwave Technol. 22(1), 266–274 (2004). [CrossRef]
23. Z. Lu, W. Wang, W. Zhang, S. T. Chu, B. E. Little, M. Liu, L. Wang, C.-L. Zou, C.-H. Dong, B. Zhao, and W. Zhao, “Deterministic generation and switching of dissipative Kerr soliton in a thermally controlled micro-resonator,” AIP Advances 9(2), 025314 (2019). [CrossRef]
24. J. Benesty, J. Chen, and Y. Huang, “On the Importance of the Pearson Correlation Coefficient in Noise Reduction,” IEEE Transactions on Audio Speech, and Language Processing. 16(4), 757–765 (2008). [CrossRef]
