Abstract

A dynamic noise characterization technique for measuring narrow-linewidth frequency-sweep lasers based on phase reconstruction method is proposed. The phase and the frequency fluctuation power spectral density (PSD) of the swept optical field within a specific time window are recovered mainly by demodulating the differential phase information of the 120-degree phase difference interferometer. Then the details of the laser noise characteristics and the performance evolution law of the frequency sweep process can be observed by investigating the calculated frequency fluctuation PSD. Moreover, the integration time linewidth and Lorentzian linewidth of the swept frequency field can be obtained by introducing the integral algorithm even beyond the limit of PSD physical resolution. Meanwhile, the power of this method is verified by applying it to a kHz-linewidth frequency swept laser source based on high-order modulation-sideband injection-locking. The results show many features of the laser such as specific noise peaks and the laser characteristic evolution rules which could not be measured by other traditional methods.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, broadband and fast-tuning frequency swept laser sources with low-noise and high spectral purity have become vital tools in more and more applications including coherent optical communication [1], ground-to-satellite optical Doppler ranging [2], high-precision coherent spectroscopy [3], low-noise coherent sensing [4], and synthetic-aperture imaging laser radar [5]. For example, the ground-to-satellite high-speed coherent laser communication experiment carried out by the Micius Quantum Science Satellite in 2017, achieved a 5.12 Gbps satellite communication rate [6]. The ground station coherent receiving local oscillator laser put in use in the previous experiment is a frequency swept laser source based on high-order modulation-sideband injection-locking with a linewidth of the order of kHz and a frequency tuning speed of THz/s [7,8]. In order to satisfy the demands of fast capture and tracking, coherent locking and other performance characteristics in the coherent process, the local oscillator laser is required to maintain high linearity as well as consistent linewidth without other noise characteristics during the frequency tuning process. Therefore, an accurate measurement of the noise characteristics of the swept frequency laser during the whole swept process is very important, which may in turn become an indispensable technical guideline for optimization and improvement during development. As described in [9], the Hilbert transform approach applying for the preliminary research and development process of frequency swept laser can only obtain the average linewidth and coherence properties over at least one sweep but cannot reveal the instantaneous linewidth and noise features during the sweeping process.

In 2015, T. Butler et al. [10] proposed a method based on reconstruction of the complex electric field to measure the instantaneous optical spectrum and linewidth of the swept source system for optical coherent tomography (OCT). They eventually obtained a dynamic linewidth of 5.6 GHz in the observation time window of 250 ns in a single shot. However, it is difficult to measure the narrow-linewidth lasers directly in a relatively small observation time window because the observation time window length and the testable linewidth are ordinarily contradictory. Moreover, the detailed noise distribution cannot be obtained in the frequency domain. Despite this, they still provide a good research direction for measuring the transient characteristics of narrow-linewidth swept lasers. Assuming that the sampling points, sampling time, and the number of points used for spectrum calculation of the laser light field E(t) are N0, T0 and Nfft0 respectively, the relationship between spectral sampling time, spectral frequency range, and spectral resolution can be expressed as follows

{Spectralsamplingrate:ΔT0=T0/N0Spectralfrequencyrange:fs0=N0/T0Spectralphysicalresolution:Δfp=fs0/N0=1/T0Spectralcalculationresolution:Δfc=fs0/Nfft0=(1/T0)×(N0/Nfft0)

Therefore, once the sampling time, T0, of the field or the length of the observation time window is determined, the physical resolution of the spectrum is limited to 1/T0 (Hz), which means the minimum measurable linewidth available from the optical spectrum is 1/T0 (Hz). In this way, it is impossible to measure the value of narrow-linewidth swept lasers using a small observation time window. As for the time scale of 250 ns mentioned above, the corresponding minimum measurable linewidth is 4 MHz and to be able to measure the linewidth of 1 kHz, the observation time window should be larger than 1 ms.

In order to overcome these problems, we present a technique realizing the phase-reconstruction of the complex optical field on the basis of the 120-degree phase difference interference and then use it for the analysis of frequency fluctuation power spectral density (PSD) for further investigating the instantaneous characterizations. First, we reconstructed the laser phase according to the differential phase demodulated by the 120-degree phase difference interferometer, and then used polynomial linear fitting to investigate the frequency/phase characteristics deviating from linear sweep. By subtracting this linear component from the original laser frequency/phase, the time-domain distribution of the laser frequency/phase noise submerged in the frequency tuning feature is obtained. Then, the frequency fluctuation PSD could be estimated in certain time windows. The laser linewidth can then be calculated by integrating the frequency fluctuation PSD, which can avoid the problem that the physical resolution and the minimum measurable linewidth are limited by the scale of the observation time window as in the spectral analysis method. Besides, we can intuitively study the detailed noise features in the frequency domain for any given observation time windows. We have investigated the instantaneous characteristics of a frequency swept laser source based on high-order modulation-sideband injection-locking similar to the one used on the Micius Quantum Science Satellite. We accurately depict that the intrinsic linewidth is about 3.14 kHz in the 1.5 ms time window and fluctuating between 1.8 kHz and 4.71kHz when divided the time scale into five equal time windows of 0.3 ms. The result breaks through the limitation of the spectral physical resolution, and demonstrates the evolution law of frequency/phase noise during injection-locking and frequency tuning processes with different conditions.

2. Principle

The instantaneous phase and frequency of the laser are reconstructed using the differential phase demodulated by the interferometer, which is consistent with [10]. Then we get the quasi-linear frequency sweep function using the polynomial linear fitting method and subtract it from the original instantaneous frequency to get the residual information. The PSD is estimated directly utilizing the residual frequency fluctuation in different observation time windows. Afterwards, the linewidth and other noise features with the observation time are analyzed from the PSD.

Differential phase of the laser demodulated by a Michelson interferometer with a phase difference of 120 degrees [11] is given by the equation below

 Δφ(t)=φ(t)φ(tτ)=arctan(X2'(t)X1'(t))arctan(X2'(t)X1'(t))¯,
where φ(t) is the instantaneous phase fluctuation of the swept laser at time t, τ is the delay time difference of the two interference arms, which is 145 ns in our system (corresponding free spectral range of the Michelson interferometer is 6.9 MHz), X’1(t) and X’2(t) are expressed as:
(X1'(t)X2'(t)X3'(t))=(η1ς1ξ1η2ς2ξ2η3ς3ξ3)1(I1(t)I2(t)I3(t)),
where In(t) (n = 1, 2, 3) is the output electrical current of the interferometer with 120-degree phase difference, the parameters ηn, ζn, ξn are constants for the setup, η1 = 0.2038, η2 = −0.1181, η3 = −0.0707, ζ1 = 0.0004, ζ2 = 0.1567, ζ3 = −0.1450, ξ1 = 0.2071, ξ2 = 0.2000, ξ3 = 0.1643.

If the lasers have no dynamic frequency tuning characteristics, the PSD analysis in different observation time windows can obtain the corresponding noise characteristic parameters such as the frequency/phase PSD and the linewidth of the laser. Conversely, the frequency tuning characteristics will be dominant in the differential phase, which means the frequency fluctuation PSD mainly exhibits frequency-tuned scan variation characteristics, thereby overwhelming its noise characteristics. So, we need to characterize the effects of the frequency tuning and eliminate its influence on the noise fluctuations.

Noting that when the phase variation of the laser is slow over the entire delay time scale, the influence of the high-order terms of its Taylor expansion can be neglected [12]. Then the phase φ(t), φ(t-τ) at the points of t = t - τ and t – τ = t (t = t - τ and t – τ = t are two independent Taylor expansion points), respectively, can be further carried out Taylor expansion and mentioned as below:

{φ(t)=φ(tτ)+τφ(tτ)φ(tτ)=φ(t)τφ(t){Δφ(t)=φ(t)φ(tτ)=τφ(tτ)Δφ(t)=φ(t)φ(tτ)=τφ(t)
Substituting Eq. (4) into Eq. (2), the first derivative of the phase φ(t) can be expressed as:
φ(t)=1τΔφ(t)+12Δφ(t)=2πν(t),
The tuning frequency sweep function can be fitted from the calculated frequency by using of the fitting algorithm like least square method. On the other hand, if we know the main tuning feature, we can set the tuning function. For example, the linear tuning frequency sweep function can be expressed as follows using a linear polynomial fit
νtunable(t)=A+A1t
where A is the initial frequency of the light field. Since the initial phase may not be equal to the true value of the time when the differential phase information is acquired, A may not be the true value of the laser center frequency after a series of differential and integral operations. Hence, it is usually necessary to shift the entire spectrum to its true center frequency. The frequency noise fluctuation of the laser can then be obtained by subtracting the tuning function from the original frequency:

νniose(t)=ν(t)νtunable(t)

By estimating the PSD of phase noise fluctuations in different time windows, we obtain the corresponding phase noise PSD Sφ(t). According to the Eq. (5) in [11], the frequency fluctuation PSD, Sv(t), can be calculated. The integral linewidth at different observation time and the Lorentzian linewidth could be estimated by integrating the obtained PSD [13–15]. The linewidth will mainly depend on the value of Sv(t), while the dependence of the minimum measurable linewidth on the spectral resolution will be significantly reduced, as will be discussed in detail later.

3. Experimental setup

We use the proposed technology to investigate the instantaneous swept characteristics of a narrow-linewidth frequency swept laser, as shown in Fig. 1. The blue frame is the structure diagram of the swept laser to be tested, which is a frequency swept laser source based on high-order modulation-sideband injection-locking [7]. The master laser is a planar external cavity semiconductor laser (RIO ORIONTM, RIO) which has the same model with [7] but different wavelength of 1550.72 nm. The structure and work process of the phase lock loop (PLL) are the same with [7]. But the voltage controlled oscillator (VCO) we chose is HMC-C030 covering 8–12.5 GHz with 21 dBm output power. The pink frame is the phase-frequency noise characteristic measurement system for swept lasers, while the green one is an unbalanced Michelson interferometer with a phase difference of 120 degrees which is based on a 3 × 3 coupler and two Faraday mirrors. The entire interferometric system is packaged in a 3U chassis and filled with foam to isolate the ambient noise as much as possible, which is eventually robust against external vibrations and acoustic noise. The swept source enters into port1 of a 3-port circulator and is then divided into three beams by the 3 × 3 coupler. The amplitude of the three output ports is equivalent, while their phases are 120 degree apart. Therefore, we name the system as a 120-degree phase difference interferometer. The output signals 1, 3 of the coupler are reflected back to the coupler by the Faraday mirrors and interfere mutually with different delay times, while the output port2 is empty. Here, the Faraday mirrors can actively suppress the polarization fading as much as possible to ensure a stable interference extinction ratio [16]. Then, we used three DC-coupled detectors (Thorlabs, DET01CFC) to directly detect and a high-speed oscilloscope to capture the three interfering signals.

 figure: Fig. 1

Fig. 1 Schematic diagram of dynamic phase noise characteristics measurement system for swept lasers. RIO: Master laser; EOM: Intensity electro-optic modulator; DFB: Slave laser, distributed feedback diode laser, a single-mode DFB-type butterfly-package diode laser; RF: Radio frequency; PS: Power splitter; Ref: 100 MHz reference; VCO: Voltage controlled oscillator; LPF: Loop filter; Cir: Circulator; OC: 3 × 3 optical coupler; FRM: Faraday rotation mirror; PD: Photodiode; DAC: Data to analog card; Scope: Oscilloscope. The blue frame is a frequency swept laser source based on high-order modulation-sideband injection-locking, the pink frame is the phase noise measurement system, and the green frame is a 120-degree phase differential interferometer.

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4. Experimental results and discussions

The laser is locked in the first-order sideband of the electro-optic modulator, and the frequency synthesizer is set to sweep from 9 GHz to 10 GHz in steps of 10 kHz in the form of triangular wave with 2 ms rising edge. With such a small stepping value, the laser frequency can be seen as a quasi-continuous laser sweep. We demodulated the tuning frequency through a 120-degree phase difference interferometer in the 0.5–2 ms observation time window, shown by the green curve in Fig. 2, and obtained the linear polynomial fitting function, shown by the red curve. The time-domain distribution deviating from the linear sweep is then obtained by subtracting the fitting function from the original frequency; the grey curve represents this result. The value of fitting adjusted R-square is 0.9999. The laser frequency maintains a good linear sweep status within a sweep range of 0.75 GHz in the 1.5 ms time scale, and the standard deviation remains within 15 MHz.We further studied the variation of the noise characteristics in the whole process of injection-locking and the frequency fluctuation PSD under several conditions including the master laser, the slave laser, injection-locked without phase-locking loop (PLL), injection-locked with PLL whose fixed RF frequency is 9 GHz, and injection-locked with PLL whose sweep rate was 0.5 THz/s, as shown in Fig. 3(a). We found that the frequency fluctuation PSD of the injection-locked output is lower than that of the slave laser, while it is close to that of master laser. After applying a fixed RF frequency of 9 GHz to the VCO to achieve static phase locking, many kinds of characteristics appeared compared with those without PLL, in different frequency regions. When the frequency is less than 400 kHz, the frequency fluctuation PSD is further suppressed, but a large bulge including many ripples appears near 400 kHz, which we think is brought by the PLL circuit. As a result, the PLL bandwidth (BW) of the system is about 400 kHz. As the frequency is greater than BW, the frequency fluctuation PSD tends to be consistent with the situation of phase-unlocked state. This phenomenon can be explained by the fact that the PLL gain increases as the output power of the laser increases. So, when the frequency is lower than BW, the frequency noise component decreases. After applying the swept frequency signal to the frequency synthesizer, the relative tuning rate of the injection-locked output signal was 0.5 THz/s. Under these circumstances, we obtained the frequency fluctuation PSD in 0.5–2 ms observation time window using the phase reconstruction technique as mentioned above. Since the time scale determines the lowest frequency of the Fourier noise spectrum that can be measured with an inverse proportionality relation, the time window of 1.5 ms corresponds to minimum frequency of 667 Hz. The results show that the noise in the frequency range of 667 Hz–1 MHz is approximately white noise, and its level is approximately equal to the white noise level under static conditions. However, big spikes appear at frequencies 70 kHz, 90 kHz, and 105 kHz during the sweep process, which we think are introduced by the PLL circuit. A large bulge still exists near the frequency of 400 kHz, which is still the PLL BW. By introducing the integral algorithm, we can calculate the integration linewidth at different observation times and Lorentzian linewidth of the laser in the case of linear sweep. The calculated Lorentzian linewidth is approximately FWHM = πh0 = 3.14 kHz [13,14]. We simultaneously used the traditional self-heterodyne delay Mach–Zehnder interferometer (SDHI) linewidth measurement technology [17] to study the relationship between the static phase-locked situation at a RF frequency of 9 GHz and the swept situation, as shown in the inset of Fig. 3(a). The delay fiber length we used was 63 km and the acousto-optic modulator frequency shift was 160 MHz. Beat note of the static situation and the sweep one was approximately the same. The traces were 3 MHz, with an RBW of 100 Hz, showing the 20 dB-linewidth of the beat signal is 48 kHz, which corresponds to a Lorentzian 3 dB-linewidth of ~2.4 kHz. The results of our measurement are about 30% higher than those obtained by SDHI method because the SDHI method records the average coherence effects over at least one scanning period [10,18]. Furthermore, the SDHI method can only obtain the linewidth result under a fixed delay time, whereas our method can describe many more detailed features of the entire frequency sweeping process. In order to get a deeper understanding of the evolution law of the frequency fluctuation PSD throughout the injection-locking process, we replaced the main laser with a narrower-linewidth fiber laser (Koheras BASIK-X15, NKT-X15) with a Lorentzian linewidth of 100 Hz, as shown in Fig. 3(b). The conclusion is consistent with the case where the master laser is a RIO laser. Besides the SDHI cannot measure the Lorentzian linewidth of 100 Hz accurately. In summary, our proposed method demonstrates the noise evolution relationships between the processes of injection-locking, sweeping, and phase-locking. The results show that the injection-locking process can reduce the overall noise of the output signal, but the sweep and phase-lock process also introduce some other noise characteristics compared with the frequency PSD of master laser.

 figure: Fig. 2

Fig. 2 Frequency tuning characteristics in time domain. Green: measured tunable frequency; red: linear-fitting frequency; Grey: residual frequency error.

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 figure: Fig. 3

Fig. 3 Frequency fluctuation PSD of different processes: (a) master laser is RIO, inset shows the results of SDHI; (b) master laser is NKT-X15.

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Then the whole 1.5 ms time scale is divided into five equal time windows of 0.3 ms, for the purpose of observing the evolution law of the laser frequency fluctuation PSD during the sweeping process, as shown in Fig. 4. We found that in the case of linear sweep, the integral variation trend of the frequency fluctuation PSD for different time bands is basically the same. The characteristic peaks coincide with each other, and the PLL BW has the same value. However, the white noise level fluctuates irregularly. This indicates that the performance of the swept laser keeps changing all the time during the whole sweeping process, but will attain an optimal value at a certain moment. The Lorentzian linewidth fluctuates between 1.88 kHz and 4.71 kHz during one sweep, but the time window of 0.3 ms corresponds to the spectrum resolution of 3.33 kHz, which means the dynamic linewidth measurement technology we proposed can surpass the limit of spectrum resolution. The reason is that the linewidth measurement method we proposed obtain the linewidth mainly relying on the PSD value and avoid the directly measurement of the Fourier transform spectrum. The latter is limited by the resolution of the Fourier transform, while the PSD value does not have this limitation and is only dependent on the magnitude of the noise.

 figure: Fig. 4

Fig. 4 Evolution of frequency fluctuation PSD at five different time bands in 0.5–2 ms time period.

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Further, we set the frequency synthesizer to sweep between 9 GHz and 12 GHz, and adjusted the locking condition by changing the injection ratio so that the performance of the swept laser changed significantly during the entire rising edge. We divided the whole frequency range into five segments represented by pink, grey, purple, blue, and yellow, respectively, according to their sweeping characteristics in the time domain. At the same time, for analysis, we chose several parts with typical features expressed by the letters A–M in the observation time window of 0.5 ms. The characteristics deviating from the linear sweep in the time domain are shown in Fig. 5, where the tunable frequency is in blue, the fitting function is in red, and the residual frequency error is in green. At the same time, we calculated the corresponding frequency fluctuation PSD of these chosen areas, as shown in Fig. 6. The pink and the grey areas represent the nonlinear frequency sweeping regions. The tuning frequency of the two regions in the time domain has similar characteristics and the frequency fluctuation PSD calculated in the frequency domain is also approximately of the same order. On the whole, the value of the frequency fluctuation PSD for nonlinear frequency sweeping is larger than that in the situation of linear sweep, which is shown in Fig. 4. In addition, the variation range of frequency fluctuation PSD for nonlinear sweep is greater during the sweep process. The purple region exhibits near-linear sweep characteristics. The residual frequency error has a certain drift as a whole, but there is a large disturbance in the tuning frequency at time t = 0.02255 s. The characteristics of frequency fluctuation PSD are very close to those of the nonlinear frequency sweep region, but its white noise characteristics are flatter in the measured frequency band. In the blue region, the noise characteristics of the parts J, K, and L below the frequency of 100 kHz are approximately the same, but for the high frequency band it becomes worse as the frequency sweeps. The orange region exhibits linear sweep characteristics and relatively low value of frequency fluctuation PSD. It can be seen from the results that the better the sweep linearity, the smaller is the corresponding time domain residual frequency error, the stronger is the regularity, the lower is the frequency noise PSD, and the better is the performance.

 figure: Fig. 5

Fig. 5 Diagram of frequency with greatly varying sweep characteristics in the time domain. The blue and the red curves correspond to the left Y-axis, while the green curve corresponds to the right Y-axis.

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 figure: Fig. 6

Fig. 6 Diagram of frequency with greatly varying sweep characteristics in the frequency domain.

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5. Discussion

In 2015, the averaging method presented by Zhou Qian et al. [19] obtained the deviation of frequency from the linear sweep by subtracting the 30 times average spectrum from one specific sweep in a given time window, and thus got the frequency fluctuation PSD of the swept source. The swept sources used in this method must keep stable during multiple sweeps, and the ambient noise needs to remain at the same level at any time. As for our proposed method, the correctness greatly depends on the selected fitting function and the residual frequency error. In order to use this method correctly, we must study and understand the noise introduced by the sweep frequency nonlinearity. We define the component deviating from the linear part as the noise component, and study the characteristics of the swept source in the 0.0171 s–0.0176 s time window mentioned above. The results with different fitting functions in the time domain are as shown in Fig. 7(a)–7(d) and the corresponding adjusted R-square values are 0.9805, 0.9801, 0.9801, and 0.9075, respectively. The relative frequency fluctuation PSD is shown in Fig. 8. We found that when the adjusted R-square values are close, the residual frequency error obtained is similar, and the difference of corresponding frequency fluctuation PSD is small. However, for different fitting functions, even if the adjusted R-square values are completely equal, the frequency fluctuation PSD is still a little different. Furthermore, when the deviation of sweep nonlinearity is large, the residual frequency error in the time domain is significantly larger and the calculated frequency fluctuation PSD is increased by at least 1–2 orders of magnitude.

 figure: Fig. 7

Fig. 7 Diagram of different fitting curves in the time domain. The blue and the red curves correspond to the left Y-axis, while the green curve corresponds to the right Y-axis.

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 figure: Fig. 8

Fig. 8 Comparison of results of frequency fluctuation PSD values obtained from different fitting curves.

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6. Conclusion

Based on 120-degree phase difference interferometer and phase reconstruction technology, we proposed a phase-frequency characteristic measurement method for narrow-linewidth frequency-sweep laser. According to the relationship between the parameters of the phase-frequency characteristics [11,20], we demodulated the differential phase information of the laser field in a specific time window and then obtained the detailed phase-frequency characteristics. Besides, we calculated the integration linewidth at any observation time and the Lorentzian linewidth through integral algorithm. To our knowledge, hyperspectral resolution and high temporal resolution are always contradictory under small time window measurements. However, even if the time window is too small to calculate the integral linewidth, the Lorentzian linewidth can still be calculated. We can reconstruct the laser light field in this case based on the phase field reconstruction to obtain the linewidth of the swept laser.

We demonstrated the effectiveness of this method by investigating a frequency swept laser source based on high-order modulation-sideband injection-locking. The evolution of the noise characteristics during the injection-locking process under linear sweep was tested. In addition, we calculated the Lorentzian linewidth of the swept source in the time window of 0.5–2 ms and investigated the law of evolution in the time scale of 0.3 ms during the sweep process. The results showed that the phase-frequency characteristics of the swept laser always kept changing in real time in a small range during the whole sweeping process. It showed irregularity throughout the process, but there was an optimal value at a certain moment. Moreover, the Lorentzian linewidth obtained by the traditional SDHI linewidth measurement technique only reflects the influence of white noise components, while the detailed information of the sweep source cannot be obtained. In order to thoroughly study the advantages of the method we proposed, we changed the sweep conditions so that the swept source had five typical sweep characteristics over the entire rising edge, and re-examined the evolution of the noise characteristics at 0.5 ms time window time. We found that the better the sweep linearity, the smaller is the time domain residual frequency error, the worse is the sweep linearity, the larger is the instantaneous frequency noise PSD, and the more severely do the noise characteristics change over the entire sweep time period. At the same time, a sudden change at a certain time in the middle deteriorated the noise characteristic in the frequency domain even if the swept source was linear in the entire range of time studied.

This method can be used to measure the phase-frequency characteristics of narrow-linewidth swept lasers. It has important guidance for the characterization and selection of swept-frequency lasers in the field of high-resolution coherent optical communication, high-precision spectral measurement, and low-noise coherent sensing. At the same time, we can measure the transient characteristics continuously in a small-time window during the whole sweep process, which reflects the stability of the laser characteristics from another aspect. In turn, it can be used to monitor the characteristics of the laser employed in applications requiring long-term stable operation as well, such as space cold atomic clock [21], optical frequency comb [22], and gravitational wave detection [23].

Funding

National Natural Science Foundation of China (61875214, 61535014, and 61775225); National Key R&D Program of China (2017YFB0405501); Civil Aerospace “13th Five-Year” Preliminary Research Project (30501020107HT02) and Scientific innovation fund of Chinese Academy of Sciences (CXJJ-17S010).

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2018 (1)

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

2017 (4)

2015 (6)

2014 (1)

2013 (1)

2012 (1)

2010 (2)

2009 (1)

2007 (1)

2005 (1)

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005).
[Crossref]

2004 (1)

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

1991 (2)

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991).
[Crossref]

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarisation-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27(6), 518–520 (1991).
[Crossref]

Acef, O.

Ahn, T.-J.

Amano, H.

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

Beppu, S.

Besnard, P.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005).
[Crossref]

Biedermann, B. R.

Blin, S.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005).
[Crossref]

Boyarsky, M.

Butler, T.

Cai, H.

Chen, D.

Chen, W.-B.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Chiodo, N.

Clairon, A.

Cui, S.

Davis, M. A.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarisation-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27(6), 518–520 (1991).
[Crossref]

Di Domenico, G.

Djerroud, K.

Dong, Y.

Dong, Z.-R.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Eigenwillig, C. M.

Evans, M.

Fang, Z.

Fritschel, P.

Frolov, V.

Fromenteze, T.

Goulding, D.

Hegarty, S. P.

Henneberger, F.

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

Hou, X.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Hu, S.-J.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Hu, W.

Huber, R.

Huyet, G.

Imani, M. F.

Ji, J.-W.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Jiang, X.

Karnowski, K.

Kasai, K.

Ke, C.

Kelleher, B.

Kersey, A. D.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarisation-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27(6), 518–520 (1991).
[Crossref]

Kikuchi, K.

Kim, D. Y.

Klein, T.

Li, L.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Li, T.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Liang, Z.-G.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Liu, D.

Liu, L.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Liu, Z.

Lu, B.

Lu, H.

Lü, D.-S.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Lyu, H. C.

Ma, J.

Marrone, M. J.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarisation-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27(6), 518–520 (1991).
[Crossref]

Mercer, L. B.

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991).
[Crossref]

Mirasso, C. R.

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

Mulet, J.

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

Nakazawa, M.

O’Shaughnessy, B.

Osinski, M.

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

Pan, Z.

Pedross-Engel, A.

Peng, X.-K.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Pulido-Mancera, L.

Qin, J.

Qu, Q.-Z.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Qu, R.

Ren, W.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Reynolds, M. S.

Schilt, S.

Sciamanna, M.

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

Shi, W.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Sleasman, T.

Slepneva, S.

Smith, D. R.

Song, D.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Stéphan, G. M.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005).
[Crossref]

Sun, Y.-G.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Tam, T. T.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005).
[Crossref]

Têtu, M.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005).
[Crossref]

Thomann, P.

Tong, Y.

Vicente, R.

M. Osinski, R. Vicente, H. Amano, J. Mulet, M. Sciamanna, F. Henneberger, and C. R. Mirasso, “Simple interpretation of the dynamics of mutually coupled semiconductor lasers with detuning,” Proc. SPIE 5349, 307 (2004).
[Crossref]

Wang, B.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Wang, J.

Wang, Y.-Z.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
[Crossref] [PubMed]

Wang, Z.

Watts, C. M.

Wei, F.

Wieser, W.

Wojtkowski, M.

Wolf, P.

Xia, W.-B.

L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
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Xiang, J.-F.

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L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
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L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
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L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
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L. Liu, D.-S. Lü, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018).
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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of dynamic phase noise characteristics measurement system for swept lasers. RIO: Master laser; EOM: Intensity electro-optic modulator; DFB: Slave laser, distributed feedback diode laser, a single-mode DFB-type butterfly-package diode laser; RF: Radio frequency; PS: Power splitter; Ref: 100 MHz reference; VCO: Voltage controlled oscillator; LPF: Loop filter; Cir: Circulator; OC: 3 × 3 optical coupler; FRM: Faraday rotation mirror; PD: Photodiode; DAC: Data to analog card; Scope: Oscilloscope. The blue frame is a frequency swept laser source based on high-order modulation-sideband injection-locking, the pink frame is the phase noise measurement system, and the green frame is a 120-degree phase differential interferometer.
Fig. 2
Fig. 2 Frequency tuning characteristics in time domain. Green: measured tunable frequency; red: linear-fitting frequency; Grey: residual frequency error.
Fig. 3
Fig. 3 Frequency fluctuation PSD of different processes: (a) master laser is RIO, inset shows the results of SDHI; (b) master laser is NKT-X15.
Fig. 4
Fig. 4 Evolution of frequency fluctuation PSD at five different time bands in 0.5–2 ms time period.
Fig. 5
Fig. 5 Diagram of frequency with greatly varying sweep characteristics in the time domain. The blue and the red curves correspond to the left Y-axis, while the green curve corresponds to the right Y-axis.
Fig. 6
Fig. 6 Diagram of frequency with greatly varying sweep characteristics in the frequency domain.
Fig. 7
Fig. 7 Diagram of different fitting curves in the time domain. The blue and the red curves correspond to the left Y-axis, while the green curve corresponds to the right Y-axis.
Fig. 8
Fig. 8 Comparison of results of frequency fluctuation PSD values obtained from different fitting curves.

Equations (7)

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{ Spectral sampling rate: Δ T 0 = T 0 / N 0 Spectral frequency range: f s 0 = N 0 / T 0 Spectral physical resolution: Δ f p = f s 0 / N 0 = 1 / T 0 Spectral calculation resolution: Δ f c = f s 0 / N f f t 0 = ( 1 / T 0 ) × ( N 0 / N f f t 0 )
  Δ φ ( t ) = φ ( t ) φ ( t τ ) = arc tan ( X 2 ' ( t ) X 1 ' ( t ) ) arc tan ( X 2 ' ( t ) X 1 ' ( t ) ) ¯ ,
( X 1 ' ( t ) X 2 ' ( t ) X 3 ' ( t ) ) = ( η 1 ς 1 ξ 1 η 2 ς 2 ξ 2 η 3 ς 3 ξ 3 ) 1 ( I 1 ( t ) I 2 ( t ) I 3 ( t ) ) ,
{ φ ( t ) = φ ( t τ ) + τ φ ( t τ ) φ ( t τ ) = φ ( t ) τ φ ( t ) { Δ φ ( t ) = φ ( t ) φ ( t τ ) = τ φ ( t τ ) Δ φ ( t ) = φ ( t ) φ ( t τ ) = τ φ ( t )
φ ( t ) = 1 τ Δ φ ( t ) + 1 2 Δ φ ( t ) = 2 π ν ( t ) ,
ν tunable ( t ) = A + A 1 t
ν niose ( t ) = ν ( t ) ν tunable ( t )

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