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Abstract
Delayed self-heterodyne interferometry is a commonly used technique for the measurement of laser linewidth. It typically requires the use of a very long delay fiber when measuring narrow linewidth (especially linewidths in the kHz-range) lasers. The use of long fibers can result in system losses and the introduction of 1/f noise that causes spectral line broadening. In this paper, we present a calculation method for processing the output of a delayed self-heterodyne setup using a short length of delay fiber, to determine laser linewidth. The method makes use of pairs of data points (corresponding to adjacent maxima and/or minima) in the signal generated from the self-heterodyne setup to determine the laser linewidth. Here, the power ratio or amplitude difference of the signal at these data points is of importance. One of the key benefits of this method is that it avoids 1/f noise which would otherwise be introduced into the measurement through the application of long fibers. The experimental results highlight that the method has a high calculation accuracy. Furthermore, the capacity for the method to utilize different pairs of data points in the self-heterodyne output to determine the laser linewidth, imparts a high degree of flexibility and usability to the technique when applied to real-world measurements.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Narrow-linewidth lasers with high spectral purity have low phase noise and long coherence length. As such they have found application in optical atomic clocks [1,2], nonlinear optics [3,4], gravitational wave detection [5,6], high-speed coherent optical communications [7], and lidar [8]. Laser linewidth is the width of the power spectral density of a lasers emitted electric field at a laser’s characteristic frequency or wavelength. This linewidth is typically a characteristic of the laser gain material and the laser resonator design, but it is also influenced by fluctuations in the optical phase [9] and by noise caused by mechanical vibrations and temperature fluctuations [10–11]. Therefore, the linewidth can be used as a way of characterizing the stability of the laser, and it is also an important metric by which the practicality of the laser can be judged. Linewidth affects the performance of lasers in many applications, and critically so in applications such as high-accuracy detection systems [12], sensing systems [13], and communications systems [14]. The accurate determination of laser linewidth is therefore necessary for practical application of narrow-linewidth lasers.
There exist numerous techniques for the measurement of narrow laser linewidths [15]. The technique which has found prominence, owing to its convenience and accuracy, is the delayed self-heterodyne interferometer (DSHI) proposed by Okoshi [16]. This technique can easily measure laser linewidths in the range of tens of kHz, however, to do so requires the use of very long fiber lengths, with lengths typically greater than six times the coherence length of the laser being characterized [17]. With the continuous development of laser linewidth compression technologies [18], lasers with linewidths of the order of kilohertz have become very common [19], and even lasers with linewidths of sub-hertz [20] have been reported.
Perhaps the most significant drawback of the DSHI technique is the requirement for a very long length of delay fiber. Long fibers increase the transmission loss and introduce 1/f noise [21], leading to spectral line broadening. They can also exacerbate nonlinear effects such as fiber stimulated Brillouin scattering [22], which introduce an additional level of complexity to the measurement. To counteract the effect of 1/f noise on measurement results, Chen [23] et. al. proposed a Voigt fitting scheme, which has proven more accurate than simply determining the -3 dB linewidth value directly from a spectrum; this fitting scheme however only offsets part of the 1/f noise. The application of short lengths of delay fiber in the DSHI technique prevents the onset of 1/f noise, however, the short delay now means that the spectral line of the beat signal is no longer Lorentzian. Instead, the beat signal exhibits a characteristic coherence envelope, so the Voigt fitting method and direct measurement of the laser linewidth can no longer be used.
Two methods can be used to determine the laser linewidth from the coherence envelope. One method is to treat the coherence envelope spectrum of the beat signal as an amplitude-modulated signal and use the principle of coherent demodulation to restore the original Lorentzian line shape and read the linewidth. He et. al. [24] and Xue et. al. [25] used this approach to measure narrow laser linewidths of 2.5 kHz and 151 Hz respectively. This approach is however still subject to demodulation error as it requires an initial “estimate” of the laser linewidth and repeated iterative calculations. The other method is to utilize special points within the coherence envelope spectrum to setup a system of equations with linewidth as a variable, and then solve the equations for the laser linewidth. This is the approach that Huang et. al. [26] and Wang et. al. [27] utilized in their respective “contrast difference with the second peak and the second trough” (CDSPST) method and “dual-parameter acquisition” (DPA) calculation method. Here, they were able to determine laser linewidths with accuracy as high as 100 Hz. These methods rely on the use of very select data points, which meet certain criteria. This makes these methods difficult to utilize and prone to high error when applied to results which have high noise and/or ill-defined data points.
In this paper, we propose a linewidth calculation method which utilizes the coherence envelope generated through the use of short delay fibers in the DSHI technique. The method selects the amplitude difference of any adjacent pair of extreme points (maxima and minima) on the coherence envelope and constructs an equation to solve the linewidth. We have examined the characteristics of coherence envelope generation through both simulations and experiments. From this, we have determined a set of ideal coherence envelope characteristics which yield the most accurate linewidth measurements, and which can be used to optimize the length of the short delay fiber. We demonstrate the feasibility and accuracy of the method for the determination of narrow laser linewidths by comparing measurements with values provided by the lasers manufacturer, and with the laser linewidth as determined using a traditional DSHI measurement with a long fiber delay line.
2. Computational principle and simulation
The delayed self-heterodyne structure is shown in Fig. 1. The laser beam passes through an isolator and is then split into two beams using coupler 1. One beam passes through the delay fiber, and the other beam enters an acousto-optic modulator (AOM) which is used to generate a frequency shift. The delayed and frequency-shifted beams are then coupled via coupler 2 and are then detected using a high-speed photodetector (PD). An electrical spectrum analyzer (ESA) displays the detected signal and is also used for data acquisition.
The power spectral density (PSD) function of the beat signal can be expressed as [17,26]:
In the PSD equation, Δf, ${\tau _d}$, and f are the independent variables and $S(f )$ is the dependent variable. Using the equations, the spectral lines obtained from the simulation are shown in Fig. 2. Figure 2(a) shows the simulation results of $S(f )$ for different delay fiber lengths and a fixed laser linewidth of 1 kHz. Figure 2(b) shows the simulation results of $S(f )$ for different laser linewidths and a fixed fiber length of 3 km. From to the simulation results shown in Fig. 2(a), it can be seen that when the laser linewidth is fixed, the shorter the delay fiber length, the more the shape of the envelope of the PSD changes; the longer the fiber length, the closer the PSD spectral line shape is to a Lorentzian. Among the three functions composing $S(f )$, ${S_1}$ is a Lorentzian function whose line shape is not affected by the fiber length. ${S_3}$ is an impulse function, and when $f - {f_1} \ne 0$, ${S_3} = 0$. Therefore, the source of the difference in spectral envelopes lies in the ${S_2}$ function. As shown in Fig. 2(c), ${S_2}$ is a periodic function whose amplitude and period are determined by the delay fiber length. The longer the delay fiber length, the smaller the amplitude and period of the function (the closer the first extreme point (a minima) is to the center frequency). When the fiber length increases to a certain point, ${S_2} \approx 1$, and this function no longer influences $S(f )$; in this case, and the PSD exhibits a Lorentzian line shape. In the case of a fixed fiber length, the period of ${S_2}$ does not change, and the envelope period of $S(f )$ is also fixed. Therefore, in Fig. 2(b), though the linewidth value changes, it will not affect the frequency-location of the extreme points (maxima and minima) of the envelope.
Fig. 2. (a) Simulation results showing the power spectral density $S(f )$ as a function of frequency for different delay fiber lengths and a laser linewidth of 1 kHz. (b) Simulation results showing the power spectral density $S(f )$ as a function of frequency with different laser linewidth and a fixed fiber delay line length of 3 km. (c) Simulation results showing the function ${S_2}$ as a function of frequency for different fiber delay lengths.
In the case of practical measurements, the linewidth is fixed and of a finite value, so the period of ${S_2}$ will have a certain value. The change in delay fiber length affects the period of the fine-structure of the PSD and hence the frequency-locations of the peaks and troughs. Since the values of $\varDelta f$ and ${\tau _d}$ are known in these simulations, the plot of $S(f )$ changing with f can be determined. Therefore, on the premise that ${\tau _d}$ and $f$ are known, the curve of $S(f )$ changing with $\varDelta f$ can be obtained.
In practical measurements, $S(f )$ is typically displayed on a spectrum analyzer using dBm as the ordinate value unit (the abscissa is the frequency), a value which is based on the power received by the PD. Measuring the optical power reaching the detector adds experimental and computational complexity. Based on Eqs. (1–4), the PSD value is affected by the laser power. It means different power manifests as an up and down amplitude shift of the whole spectrogram. To facilitate the calculation, we take the difference between the values of the vertical axis at pairs of extreme points (magnitude difference) for the calculation. Since the magnitude difference measured in dB is a relative value, the influence of laser power on linewidth measurement does not need to be considered. On the premise that ${\tau _d}$ and f are known, the relationship between $\varDelta S$ and $\varDelta f$ can then be constructed. ${\tau _d}$ is controlled by the fiber length, $f$ can take the frequency value at the extreme point of the envelope, and the calculation relationship can be expressed as:
where ${S_H}$ and ${S_L}$ are the pairs of extreme points on the envelope. $\Delta S$ is not only the magnitude difference between the two extreme points in the simulated spectral line, but also the magnitude difference between the corresponding pair of extreme points in the data collected by the spectrum analyzer.The period of ${S_2}$ is ${\tau _d}$, and ${\tau _d}$ can be expressed in terms of fiber length as
where n is the fiber’s refractive index, and $L$ is the fiber length. Although ${\tau _d}$ has a certain impact on the frequency of the envelope extreme according to the Eq. (3), a slight error of n and L can be ignored in the measurement because ${\tau _d}$ is the exponential power of e [see Eqs. (5) and (6)]. It is consistent with the error analysis in Ref. 27 by Wang et. al. According to the period ${\tau _d}$, it can be deduced that the difference between the frequency value at the extreme point and the center frequency $\varDelta {f_m}$ is: $m$ is a natural number, as shown in Fig. 3, $m = 0$ at the nearest extreme point (a minima) from the center frequency, $m = 1$ at the second extreme point (a maxima) closest to the center frequency, and so on. Therefore, although a narrower linewidth corresponds to a more ambiguous local maximum/minimum for a fixed fiber length (as shown in Fig. 2), the local extreme value can be found accurately by inferring the frequency position of the extreme point via Eq. (7).Fig. 3. Positions of the extreme points (maxima and minima) corresponding to different m values in a plot of a laser power spectral density function.
By incorporating Eq. (7) with Eq. (5), the relationship between $\varDelta f$ and $\varDelta S$ can be express as:
In the experiment, both the delay fiber length and the delay fiber refractive index are known, so $\varDelta f$ and $\varDelta S$ have a linear relationship. Taking Eq. (9) as an example, the smaller the envelope difference, the larger the laser linewidth, as shown in Fig. 4. The method therefore follows the two key steps of: selecting the pair of adjacent extreme points in the experimental data and calculating their amplitude difference; and then incorporating the amplitude difference data into the calculation formula to solve for the laser linewidth.
Fig. 4. The relationship between $\varDelta f$ and $\varDelta S$ as per Eq. (9). Here the length of the delay fiber was $L = 5000$ m.
3. Experiment and analysis
The measurement setup was built according to Fig. 1, and an optical fiber isolator (ISO) was used to ensure the unidirectional transmission of the laser. Optical couplers were used to achieve splitting and coupling of laser beams. An acousto-optic modulator (AOM: 100 MHz, G-1064-100-L-D-T-AA-A-T-H) was used to achieve frequency shifting of the optical signal. The polarization controller was used to match the polarization states of the upper and lower arms of the interferometer. A photodetector (PD, PDA1GC10KFC) was used to detect the beat signal, and an electrical spectrum analyzer (ESA, RIGOL, RSA5030B) was used to collect experimental data and visualize the signal. The operating mode of the ESA was set to the real-time spectrum analysis (RTSA) mode, and the RTSA mode was used for seamless capture of the PSD coherence envelope.
The tested laser was a narrow linewidth semiconductor laser manufactured by RIO (RIO0175-5-07-1) with a fixed output power of 20.8 mW. The manufacturer’s laser test report shows a linewidth value of 5.1 kHz, a value which was determined by measuring and integrating the frequency noise of the laser. In this experiment, a 105 m long delay fiber was connected to the test device. The modulator centers the beat notes at 100 MHz, which avoids the interference of the environmental disturbance to the experiment. Although environmental noise such as mechanical vibration will cause the PSD to fluctuate in the low-frequency band around 0 Hz, the signal in our test frequency area (around 100 MHz) remains stable for a long time as shown in Fig. 5. In the frequency sweep range of 10 MHz, a total of three orders of extreme points can be found. The numerical results obtained by taking two different pairs of extreme points are shown in Table 1. It is worthwhile to mention that as the extreme points searching is affected by the resolution of the ESA, hence the sweep bandwidth can be appropriately reduced when the extreme points amount meets the requirements, and therefore to obtain a higher resolution at the same sampling points.
Table 1. Measured data and resultant, calculated linewidth using two different pairs of extreme pointsa
In analyzing the results summarized in Table 1, it can be seen that the result obtained by taking the pair of first and second- order extreme points ($m = 1$, $k = 0$) is 7.68 kHz. This value is close to the nominal value of 5.1 kHz, but still has a level of error. When the pair of second- and third-order extreme points ($m = 1$, $k = 2$) are used, the result is 16.88 kHz, and this value is significantly different. In examining the data in Fig. 5 and the simulations shown in Fig. 2, we speculate that the extreme points of the envelope are located far from the center frequency because the delay fiber is too short. Although ΔS is not sensitive to the laser power changes, some data may be submerged in the noise floor of the measurement system at low power (especially when the fiber is too short). Under the influence of the noise floor, ΔS is smaller than the ideal value, and the calculated result is larger than the actual value. The presence of a prominent noise floor emphasizes the need for strong input laser signals when using this measurement configuration. It means that increasing the input power is of help to improve the linewidth measurements accuracy. However, lasers which feature narrow linewidth and high power do not typically go hand-in-hand. A shorter fiber can produce a more obvious envelope, but the error introduced when the fiber is too short will also significantly affect the measurement accuracy of the test system. In this configuration, the presence of the modulator introduces frequency noise with a 1/f characteristic, which may lead to the broadening of spectral lines. However, the data submersion phenomenon of the noise floor becomes more serious when removing the AOM, and the measurement linewidth becomes much larger. Therefore, compared with the weak 1/f noise effect of the AOM, the noise floor is dominant. In addition, the stability of the signal is not affected by environmental noise, which gives rise to the increase in noise floor and measurement errors.
We also examined the performance of this linewidth measurement technique when characterizing another narrow-linewidth fiber laser (CONNET, CoSF-D-YB-M). We changed the length of the delay to 5 km to investigate the effect of the noise floor. As shown in Fig. 6, the farther that pairs of extreme points were chosen from the center frequency, the greater the difference between the measured and simulated data. When the frequency range exceeds the yellow area, the envelope of the experimental data is almost completely submerged in the noise floor. In order to get a clear envelope, we changed the 5 km fiber to 1 km, as shown in Fig. 6(b). By selecting the data above the noise floor, we get amplitude difference between the first valley and the second peak as 7.63 dB, and then obtained the linewidth of 5.04 kHz. This result is within the linewidth range (1-10 kHz) given by the manufacturer’s report. The effect of noise clearly limits the capacity of the technique to make accurate linewidth measurements.
Fig. 6. Fiber laser (CONNET) measurement. (a) Difference between experimental data and simulation result under the noise floor. (b) Experimental data using a delay fiber length of 1000 m.
To make the results obtained from this technique more accurate, the length of delay fiber should be selected according to the noise characteristics of the measurement system and that of the laser being characterized. When the fiber length is too short, the extreme points are far from the center frequency, and the noise floor significantly impacts the results. When the fiber length is too long, the generation of 1/f noise will also affect the results. However, when the fiber length is ideal, pairs of adjacent extreme points will be clearly distinguishable, and close to the center frequency of the laser being characterized. We investigated the performance of this measurement technique with the application of delay fiber lengths of 105 m, 1000 m, 3000 m, and 5000 m. Figure 7(a) shows the experimental results obtained with different delay fiber lengths. The data at 105 m-delay is disturbed by the noise floor, resulting in a smaller amplitude difference than the ideal level and hence a larger linewidth. 1/f frequency noise is introduced at delays of 3000 m and above, resulting in spectral broadening. In addition, the amplitude and period of the envelope become smaller when the fiber is longer than 3000 m, and the spectral is in the transition state between the coherent envelope and the Lorentzian [28]. The envelope calculation method shows a certain error in this state. For the RIO laser used in our study, the PSD at 1000 m delay is not affected by the noise floor and has no 1/f frequency noise. Therefore, it shows a clear envelope with a large amplitude difference. The experimental data (6.10 kHz) at 1000 m delay is shown in Fig. 7(b), which is the closest to the actual linewidth.
Fig. 7. (a) Plot of the experimentally determined linewidth of the RIO (RIO0175-5-07-1) laser as a function of delay fiber length. The values were determined using our developed measurement and analysis technique. (b) Plots of the experimental data for fiber length of 1000 m.
In the section detailing the theory behind our method, we explained that a long delay fiber makes the influence of the ${S_2}$ function on $S(f )$ disappear so that the envelope of the PSD measurement becomes Lorentzian. To compare the results obtained using our method to that obtained using a traditional DSHI approach, we significantly increased the length of the delay fiber to 50 km. A Lorentzian line shape was detected on the ESA, and this is plotted in Fig. 8. Since the 1/f noise has a more prominent effect on broadening the spectral line near the center frequency [21], the result of directly taking the -3 dB bandwidth is often larger than the true linewidth. The -20 dB bandwidth value was instead taken and the laser linewidth is calculated to be 14.05 kHz by the relationship shown in Table 2. The results highlight how the laser linewidth can be directly read from the Lorentzian-shaped spectrum when the delay fiber is long, however, 1/f broadening greatly impacts the measurement accuracy.
Fig. 8. The captured power spectrum of our test laser using the DSHI method with a long delay fiber length of 50 km.
Table 2. Relation between N dB bandwidth and laser linewidth [29]
It is clear that there are discrepancies between the linewidth values that we calculate using our method and that stated by the manufacturer. We believe that this discrepancy may be due to different sources of error inherent in the two different methods used to determine the laser linewidth. In the case of the manufactures applied method of measuring the laser linewidth, the laser frequency noise is used. Here, the frequency noise is converted into linewidth through β line theory [30], but there are certain errors in the conversion process. As a result, it is difficult to judge the true accuracy of the measurement.
In our method, errors are predominantly a result of noise within the experimental measurement. In the case where the length of the delay fiber is short, the extreme points of the coherence envelope are far from the center value. They may hence become obscured by the noise, leading to amplitude differences which are smaller than the actual value, which in turn yield larger linewidth values. We have demonstrated that the influence of noise can be minimized through the use of delay fibers of the optimal length. Here, the effect is to reduce the apparent noise level within the measurement and improve the measurement of the amplitude difference between pairs of adjacent extreme points. By tracking the change in measured linewidth as a function of delay fiber length, we clearly show that a minimum value (narrowest linewidth) can be determined. We have also compared our method to the traditional DSHI method wherein a very long length of delay fiber is used. Here, the influence of 1/f noise is clearly apparent and the linewidth value directly measured from the Lorentzian PSD is significantly higher than the values obtained using our method, and that quoted by the manufacturer. So while the linewidth values measured using our method do not perfectly match that quoted by the manufacturer, they are very close and it cannot be said with certainty which method (ours or the manufacturers) is the most accurate. What should be highlighted however is that our method yields results which are far closer to the manufacturers value than that obtained using a traditional DSHI approach. As such, we believe that our method is highly suitable for the measurement of the linewidth of narrow linewidth lasers.
4. Conclusion
In summary, we have demonstrated a method which can be used to determine the linewidth of narrow-linewidth lasers and makes use of short lengths of delay fiber in a delayed self-heterodyne setup. We deduce and verify the feasibility of this method through both simulations and experiments. In the experiment, we successfully applied our method to the measurement of the linewidth of a commercial semiconductor laser, achieving results close to the manufacturers stated value and far better than that achieved using traditional delayed self-heterodyne interferometer techniques.
Funding
National Natural Science Foundation of China (61905061, 61927815).
Acknowledgments
Z. Z. thanks support of Hebei Provincial Department of Education's Postgraduate Student Innovation Capacity Development Grant Program (CXZZSS2022057). R.P.M. acknowledges support from the Asian Office of Aerospace Research and Development (AOARD).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L Campbell, M. Bishof, X. Zhang, and J. Ye, “An optical lattice clock with accuracy and stability at the 10∓18 level,” Nature 506(7486), 71–75 (2014). [CrossRef]
2. N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, and A. D Ludlow, “An atomic clock with 10–18 instability,” Science 341(6151), 1215–1218 (2013). [CrossRef]
3. H. Chen, Z. Bai, X. Yang, J. Ding, Y. Qi, B. Yan, Y. Wang, Z. Lu, and R. P. Mildren, “Enhanced stimulated Brillouin scattering utilizing Raman conversion in diamond,” Appl. Phys. Lett. 120(18), 181103 (2022). [CrossRef]
4. W. Loh, A. A Green, F. N. Baynes, D. C. Cole, F. J. Quinlan, H. Lee, and S. A Diddams, “Dual-microcavity narrow-linewidth Brillouin laser,” Optica 2(3), 225–232 (2015). [CrossRef]
5. B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, and K. D. Giardina, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72(7), 076901 (2009). [CrossRef]
6. A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, and M. E. Zucker, “LIGO: The laser interferometer gravitational-wave observatory,” Science 256(5055), 325–333 (1992). [CrossRef]
7. Z. Bai, R. J. Williams, O. Kitzler, S. Sarang, D. J. Spence, Y. Wang, Z. Lu, and R. P. Mildren, “Diamond Brillouin laser in the visible,” APL Photonics 5(3), 031301 (2020). [CrossRef]
8. C. A. Hostetler, M. J. Behrenfeld, Y. Hu, J. W. Hair, and J. A. Schulien, “Spaceborne lidar in the study of marine systems,” Annu. Rev. Mater. Sci. 10(1), 121–147 (2018). [CrossRef]
9. A. Schawlow and C. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958). [CrossRef]
10. K. F. Renk, Basics of Laser Physics (Springer Cham, 2012).
11. A. E. Siegman, Lasers (University Science Books, 1986), pp. 647–648.
12. L. Shi, T. Zhu, Q. He, and S. Huang, “Effect of laser linewidth on phase-OTDR based distributed vibration sensing regime,” Proc. SPIE 9157, 91576H (2014). [CrossRef]
13. C. Q. Li, T. F. Wang, H. Y. Zhang, J. J. Xie, L. S. Liu, and J. Guo, “Effect of laser linewidth on the performance of heterodyne detection,” Acta Phys. Sin. 65(8), 084206 (2016). [CrossRef]
14. S. Savory and A. Hadjifotiou, “Laser Linewidth Requirements for Optical DQPSK Systems,” IEEE Photonics Technol. Lett. 16(3), 930–932 (2004). [CrossRef]
15. Z. Bai, Z. Zhao, Y. Qi, J. Ding, S. Li, X. Yan, and Z. Lu, “Narrow-linewidth laser linewidth measurement technology,” Front. Phys. 9, 768165 (2021). [CrossRef]
16. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980). [CrossRef]
17. L. E. Richter, H. I. Mandelburg, M. S. Kruger, and P. A. McGrath, “Linewidth determination from self-heterodyne measurements with subcoherence delay times,” IEEE J. Quantum Electron. 22(11), 2070–2074 (1986). [CrossRef]
18. S. Huang, T. Zhu, G. Yin, T. Lan, F. Li, L. Huang, and M. Liu, “Dual-cavity feedback assisted DFB narrow linewidth laser,” Sci. Rep. 7(1), 1185 (2017). [CrossRef]
19. L. Huang, P. Chang, X. Li, X. Li, W. Zhang, Y. Li, F. Gao, F. Bo, L. Gao, W. Huang, M. Liu, and T. Zhu, “All-fiber narrow-linewidth ring laser with continuous and large tuning range based on microsphere resonator and fiber Bragg grating,” Opt. Express 26(25), 32652–32661 (2018). [CrossRef]
20. S. Gundavarapu, G. M. Brodnik, M. Puckett, et al., “Sub-hertz fundamental linewidth photonic integrated Brillouin laser,” Nat. Photonics 13(1), 60–67 (2019). [CrossRef]
21. L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991). [CrossRef]
22. E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539–541 (1972). [CrossRef]
23. M. Chen, Z. Meng, J. Wang, and W. Chen, “Ultra-narrow-linewidth measurement based on Voigt profile fitting,” Opt. Express 23(5), 6803–6808 (2015). [CrossRef]
24. Y. He, S. Hu, S. Liang, and Y. Li, “High-precision narrow laser linewidth measurement based on coherent envelope demodulation,” Opt. Commun. 50, 200–205 (2019). [CrossRef]
25. M. Xue and J. Zhao, “Laser linewidth measurement based on long and short delay fiber combination,” Opt. Express 29(17), 27118–27126 (2021). [CrossRef]
26. S. Huang, T. Zhu, Z. Cao, M. Liu, M. Deng, J. Liu, and X. Li, “Laser linewidth measurement based on amplitude difference comparison of coherent envelope,” IEEE Photonics Technol. Lett. 28(7), 759–762 (2016). [CrossRef]
27. Z. Wang, C. Ke, Y. Zhong, C. Xing, H. Wang, K. Yang, and D. Liu, “Ultra-narrow-linewidth measurement utilizing dual-parameter acquisition through a partially coherent light interference,” Opt. Express 28(6), 8484–8493 (2020). [CrossRef]
28. S. Huang, M. Wan, J. Wu, D. Lu, B. Zhang, Y. Zheng, and X. Fang, “Precise laser linewidth measurement by feature extraction with short-delay self-homodyne,” Appl. Opt. 61(7), 1791–1796 (2022). [CrossRef]
29. D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).
30. G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt. 49(25), 4801–4807 (2010). [CrossRef]
