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Abstract
Laser with high spectral purity plays a crucial role in high-precision optical metrology and coherent communication. Thanks to the rapid development of laser frequency stabilization, the laser phase noise can be remarkably compensated, allowing its ultra-narrow linewidth subject to mostly quantum limit. Nevertheless, the accurate characterization of phase noise statistics and its linewidth of a highly coherent laser remains ambiguous and challenging. Here, we present an approach capable of revealing delay-time-resolved phase noise statistics of a coherent laser based on coherent optical time domain reflectometry (COTDR), in which distributed Rayleigh scattering along a delay fiber essentially allows a time-of-flight mapping of a heterodyne beating signal associated with delay-time-dependent phase information from a single laser source. Ultimately, this novel technique facilitates precise measurement of ultra-narrow laser linewidth by exploiting its delay-time-resolved phase jitter statistics of random fiber laser with pump lasers of various linewidths, confirmed with the analytical modeling and numerical simulations.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Coherent laser sources with highly purity spectrum are at the heart of high-precision measuring science, including laser interferometer gravitational-wave observatory (LIGO) [1], optical atomic clock [2] and high-resolution spectroscopy [3]. Versatile applications such as coherent communication [4,5], narrow-linewidth microwave/terahertz photonic generation [6,7] also demand an ultra-narrow linewidth laser involving squeezed phase noise to upgrade their performance towards a fundamental limit. A rapid development of frequency stabilization technique based on optical or electrical feedback currently enables a coherent laser radiation with linewidth as narrow as mHz over certain time frame which is even smaller than the intrinsic linewidth [8–13], however, the characterization of the narrow linewidth remains challenging.
The linewidth of a laser is fundamentally governed by the laser radiation coupled with a quantum process of spontaneous emission (i.e., pure white noise), namely “Shawlow-Townes linewidth” [14], which theoretically exhibit a lorentzian lineshape [15]. As a crucial parameter, this quantum noise-limit linewidth generally reflects the short-term stability as well as the temporal coherence of a laser. It can be evaluated by the heterodyne detection of beating notes superimposed with another identical laser, which is however not usually available in a breeze, particularly for an elaborately stabilized laser system. Alternatively, a delayed self-heterodyne interferometer (DSHI) approach, in which one portion of de-correlated laser beam after a long delay time is obligatorily required to replace the 2nd identical laser [16], is proposed and intensively employed to characterize the linewidth as well as the phase noise [17–20]. However, in spite of its simplicity, the DSHI approach turns out to be physically incompetent for an ultra-narrow linewidth (e.g. < 1 kHz) due to a remarkable attenuation (>40 dB) over hundreds of kilometers delay fiber even though at a minimum loss spectral window around 1.55 µm. This challenge was overcome in part by a loss-compensated recirculating technique with an extended long delay [21–23] or the strong coherent envelope analysis associated to a relative short delay [24–26]. However, its robustness remains elusive because of a definite delay-time-dependent behavior in all above-mentioned approaches [27]. More importantly, the inevitable 1/f frequency noise over a long delay beyond the laser coherence time imposes a deviation of the intrinsic linewidth from a lorentzian shape to a Gaussian broadening one [28–30], leading to a dilemma for an intrinsic linewidth characterization. Alternatively, the linewidth could be derived from suitably weighted integral of the laser frequency noise spectrum, which can be readily recovered by means of intensity fluctuations at the output of an unbalanced interferometer [31–37]. The precision of the linewidth estimation would rely on the frequency noise measurement and, more importantly, be crucially determined by approximations of the integral weight in the frequency domain, resulting in either an over-optimistic estimation of the intrinsic linewidth [38] or hindering the flexibility for practical implementation. Consequently, a precise characterization of the fundamental laser linewidth subject to quantum-limit phase noise statistics is highly demanded not only being oriented by practical applications but also in a more fundamental prospective of laser physics.
In this paper, we demonstrate a novel technique to reveal the phase noise statistics of a coherent laser by employing a coherent optical time domain reflectometry (COTDR), for the first time to the best of our knowledge, in which distributed Rayleigh scattering along a segment of delay fiber technically introducing a time-of-flight for the heterodyne beating note carrying a delay-time-resolved phase information of the laser under test. A fading-free phase jitter demodulation based on Pearson correlation coefficient (PCC) is developed, which allows a statistical analysis of phase noise, agreeing well with theoretical analysis and numerical simulation. In this scenario, the intrinsic linewidth is ultimately characterized by means of the statistics of the delay-time resolved phase jitter. The proposed approach breaks through the dilemma of narrow linewidth characterization in a conventional DSHI approach, highlighting promising potentials in fundamental laser physics and coherent communication.
2. Principle
2.1 Laser intrinsic linewidth modelling
Spontaneous emission as a quantum process is fundamentally ubiquitous during a laser generation, which primarily introduces the phase fluctuation of the laser field in a random fashion and deviates its output electrical field from an ideal sinusoidal wave. A quasi-monochromatic laser field can be modelled as a sinusoidal wave with random fluctuations of the phase:
where E0 is a stationary value for the amplitude, f0 is the nominal central optical frequency, respectively. The white frequency noise-induced phase fluctuation φ(t) is a continuous random walk, which can be modelled by the Wiener process [39]. Thus, phase jitter after a positive time delay τ can be defined as ${\varDelta _\tau }\varphi (t )= \varphi ({t + \tau } )- \varphi (t )$, which is a random variable with a probability density function of a Gaussian distribution,
Fig. 1. Simulation of laser phase noise and intrinsic linewidth. (a) Simulated random-walk phase noise by Wiener process, in which the phase jitter variance is given by ${\boldsymbol \sigma }_{\boldsymbol \varphi }^2 = 2{\boldsymbol \pi \varDelta }{{\boldsymbol \nu }_{\boldsymbol c}}{\boldsymbol \tau }$ with ${\boldsymbol \varDelta }{{\boldsymbol \nu }_{\boldsymbol c}}$ of 1 kHz. (b) and (c) are zooming in results of random phase noise and corresponding electrical field of beating signals with (blue) and without (red) random phase jitter. The beating signal originates from a self-heterodyne of the laser field with a frequency shift of 80 MHz; (d) Power spectral density (PSD) of the beating signals with (blue) and without (red) random phase jitter. The peak is normalized to a maximum of 0 dB, and the center frequency is normalized to 80 MHz, which is in agreement with theoretical lorentzian laser line shape (magenta dash line) with a full-width-half-maximum of 1 kHz subject to quantum-limit phase noise.
2.2 Laser phase noise in COTDR system
Coherent optical time domain reflectometry (COTDR) is an intriguing technique for characterization and fault location in optical fiber transmission systems [42,43] as well as distributed fiber-optic dynamic strain sensing aiming to structural health monitoring [44,45]. In COTDR system, the lightwave from a narrow-linewidth laser source is divided into two paths: the lightwave in the first path is temporally modulated into a pulse with a central frequency shift fs and then launched into one-end of long fiber while the lightwave in the second path remains cw as the local reference. As the light pulse injects into the optical fiber with numerous randomly localised Rayleigh scatters originating from inhomogeneities of the refractive index, a backscattered coherent speckle signal can be formed as the vector sum of these back reflections. Then, the Rayleigh backscattered light beams are recaptured and superimposed with the local reference. By ignoring the polarization mismatch, the electrical field of the beating note can be expressed as,
By removing the DC component and down-converting the frequency into the baseband of a balanced photodetector (BPD), the detected heterodyne beat-note can be described as,
According to Eq. (3), the intrinsic linewidth is theoretically determined by the delay-time-dependent phase jitter variance. In Fig. 2, we simulate the COTDR-based heterodyne beating notes of a laser with 1-kHz intrinsic linewidth at different delay times of 0.1, 1, 10 µs, following random phase noises by repeating 1000 times Wiener process computation. It can be seen that, the pattern of beating notes at 0.1-µs delay time exhibits a minor variation over 1000 repeating cycles while larger ripples are significantly introduced as the delay time increases to 10 µs, indicating a worse phase jitter variance. Correspondingly, the delay-time-dependent phase jitter variance σ2 can be statistically obtained at each delay time. In Fig. 2(g), numerical results show that the phase jitter variance is proportional to the delay time with a linear slope of $2\pi \varDelta {\upsilon _c}$, which is in agreement with the theoretical analysis.
Fig. 2. Simulation of the phase noise statistics of a coherent laser with a quantum-limit linewidth of 1-kHz. (a)-(c) show numerically simulated evolutionary trajectories of 1000 heterodyne beating notes in COTDR at delay time of 0.1 µs, 1 µs and 10 µs. Random phase noise of laser with 1-kHz intrinsic linewidth was numerically generated by a Wiener process from random number generation. Note that, Rayleigh scattering-induced ${\phi _R}(z )$ in each fiber location z was setting as null for simplicity without loss of generality. (d)-(f) represent corresponding probability histograms (blue bar) of the phase jitter at delay time of 0.1 µs, 1 µs and 10 µs, respectively; (g) Numerical phase jitter variance as a function of delay time (blue star dots), corresponding well with the theoretical curve (red dash line) given by ${\boldsymbol \sigma }_{\boldsymbol \varphi }^2 = 2{\boldsymbol \pi \varDelta }{{\boldsymbol \nu }_{\boldsymbol c}}{\boldsymbol \tau }$.
2.3 Fading-free retrieval of delay-time-resolved phase jitter
In typical COTDR, the random nature of Rayleigh backscattering as well as the random state of polarization of the backscattered signals along the fiber occasionally deteriorates the heterodyne beating notes, leading to a notorious signal fading [46] which is detrimental to the phase recovery. To address it, we develop a fading-free approach based on Pearson Correlation coefficient (PCC) for the phase jitter retrieval in COTDR. Figure 3 shows the schematic diagram to demodulate the delay-time resolved phase jitter variance by statistical analysis with assistant of COTDR. In COTDR system, optical pulses modulated from the laser under test are consecutively reflected by distributed Rayleigh scattering along optical fiber, allowing a time-of-flight mapping of the heterodyne beating signal associated to delay-time-dependent laser phase noise. By launching optical pulses with a time interval larger than the laser coherent time (Δt = tj – tk >tc), a pair of coherent COTDR traces (Ij, Ik) carrying the delay-time-dependent phase from the laser under test can be acquired. Taking into account of cosine function type of beating note traces, the absolute value of the phase jitter at delay time $\tau $ can be deduced by (See Appendix A),
Fig. 3. Schematic diagram of delay-time-resolved phase jitter retrieval and statistical analysis relying on COTDR.
Here, $\rho $ is the calculated PCC between two beating note traces at the central delay time of ${\tau _z}$ within a small time window width of tw,
It is important to mention that, the PCC-based phase jitter retrieval with a moderately small time window (∼100 ns) basically guarantees a negligible phase variation within this small time window, particularly for a highly coherent laser source. For instance, for a 1-kHz laser, the corresponding phase jitter variance within a timescale of 100 ns is around $2{\pi } \times {10^{ - 4}}$ rad2. In this scenario, the fading problems in COTDR, caused by the destructive interference of random Rayleigh scattering as well as the polarization mismatch between the backscattered signal and the local reference [46], can be alleviated to a large extent since the occasional ultra-weak signal fading occurs at certain points within single pulse width cell [45]. Eventually, the proposed PCC-based approach turns out to be effective for the fading-free phase jitter retrieval.
By launching a number of optical pulses and repeating Ns times beating note traces acquisition, the variance of the absolute phase jitter $|{{\varDelta _\tau }\varphi } |$ at the delay time of ${\tau _z}$ could be statistically calculated as,
3. Experimental setup
In Fig. 4, the lightwave beam from the laser under test is split into two branches through an optical coupler. The upper branch was modulated into an optical pulse through an acoustic-optic modulator (AOM) with a frequency-shift (fAOM), which is driven by a pulse generator (PG) with a pulse width of 100 ns and a pulse repetition rate of 10 kHz. The generated optical pulses are amplified by an EDFA and then pass through a narrowband filter to suppress undesired noises from amplified spontaneous emission. By launching optical pulses into the delay fiber through an optical circulator (CIR), the lightwave is backscattered by numerous distributed Rayleigh scatters along the delay fiber by introducing the time-of-flight for variable delay times. The 10% lower branch is used as the local oscillator (LO) and to superimpose with backscattered signals through another 50/50 coupler. The delay fiber is 1-km standard single mode fiber (SMF) which was placed in a sound-proof box for ambient acoustic isolation. The symbols of fs and fAOM represent the center frequency of the laser source and the frequency shift induced by the AOM, respectively. A polarization controller (PC) is inserted in order to optimize the polarization matching between the LO and scattered signals. Afterwards, the beating signals was converted into electrical signals by a balanced photo-detector (BPD) (PDB130C, Thorlabs) and then digitized by a high-speed oscilloscope (MSOS804A, Keysight). Here, the balanced heterodyne detection is commonly adopted to reduce the dynamic range requirement of the detector as well as improve the sensitivity [43].
Fig. 4. Experimental setup for the delay-time resolved measurement of laser phase jitter. (Component label abbreviations: AOM—Acoustic-optic modulator, PG—Pulse generator, EDFA—Erbium doped fiber amplifier, CIR—optical circulator, PC—polarization controller, BPD—balanced photodetector.)
4. Results and discussions
4.1 Delay-time-resolved laser phase noise statistics
A 1550 nm External Cavity Laser (ECL) (ORION Module, RIO) with a narrow linewidth of 2 kHz was utilized as the laser under test. Figures 5(a)–5(c) display the recorded beating note trace evolutions within a time window of 100 ns by consecutively launching 1000 optical pulses with a repetition rate of 10 kHz in COTDR system. It can be seen that the beating notes pattern show a small trace-by-trace variation at 1-µs delay time while the increased random phase noise apparently degenerates the beating note pattern as the delay time increasingly reaches 10 µs, which is coincident as the theoretical prospection. With the PCC-based phase jitter retrieval method, the absolute phase jitter $|{{\varDelta _\tau }\varphi } |$ at delay time τ can be obtained through data processing. Considering the estimated coherent time of the 2-kHz laser (i.e., ${t_c}{\; }\sim {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\varDelta {\upsilon_c}}}} \right.}\!\lower0.7ex\hbox{${\varDelta {\upsilon _c}}$}} = 0.5{\; }ms)$, two beating note traces with a time interval Δt = 0.6 ms (e.g., the trace pairs of the 1st/7th, the 2nd/8th, …, the 994th/1000th) are consecutively selected to recover the phase jitters for statistical analysis. As illustrated in Figs. 5(d)–5(f), the probability of the retrieved absolute value of the phase jitter $|{{\varDelta _\tau }\varphi } |$ at delay times of 1, 5, 10 µs are statistically obtained, showing consistent matching with numerical and theoretical simulations. It is worth mentioning that the proposed PCC-based phase retrieval would not only highlight a fading-overwhelmed Rayleigh scattering-based distributed sensor, but also provide a theoretical framework based on delay-time-resolved phase dynamic and statistics to determine the phase stability of Rayleigh based distributed sensors and its dependence on laser frequency stability.
Fig. 5. Phase noise dynamic and statistics retrieval of a 2-kHz External Cavity Laser. (a)-(c) Evolutionary trajectory of 1000 beating notes at different delay times of: (i) 1 µs, (ii) 5 µs, (iii) 10 µs. (The time window tw is 100 ns for each delay time case). (d)-(f) show the experimental probability histograms (Blue) of the corresponding resolved absolute phase jitter $|{{\varDelta _\tau }\varphi } |$, compared to numerical (Orange) and theoretical (Red dash line) simulation results.
The accuracy of statistically demodulated phase jitter variances would be affected by several factors in data acquisition and processing, such as the sampling rate (Fs) of the data acquisition, total beating note trace sample number (Ns), the time window (tw) for absolute phase jitter recovery and the time interval (Δt) between two beating note trace samples. As shown in Fig. 6(a), the measured phase jitter variance at different delay times decrease as we increase the sampling rate of the data acquisition and remain stable at a sampling rate larger than 1 GSa/s. In statistical analysis, the calculated phase jitter variances were convergent as the sample number increasingly reached 1000, as depicted in Fig. 6(b). To go around of fading problems in COTDR system, a time window tw was properly chosen to demodulate the absolute phase jitter. In Fig. 6(c), the fading-induced error turns to be significant at a time window of less than 50 ns while the recovered phase jitter remained stable as tw exceeds 100 ns, implying an excellent performance in signal fading error suppression. In statistical analysis of the phase jitter variance, two consecutive trace samples should be independent which can only be guaranteed by an adequate time interval (Δt > tc) between two optical pulses modulated from the same laser source. As can be seen in Fig. 6(d), smaller phase jitter variances can be found as the trace time interval is less than the coherent time of the 2-kHz laser (${t_c}\sim {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\varDelta {\upsilon_c}}}} \right.}\!\lower0.7ex\hbox{${\varDelta {\upsilon _c}}$}} = 0.5{\; }ms$). The recovered phase jitter variances saturated at constant levels for different delay time cases at Δt >0.5 ms. On the other hand, the phase jitter variance increased divergently as the trace time interval is longer than several milliseconds (∼3 ms, defined as te) in our experiments, particularly at a larger delay time. It mainly attributes to additional 1/f frequency noise caused by the external disturbance imposed from the thermal and acoustic noise of surrounding environment [47], which primarily dominates in a low frequency domain of <100 Hz (corresponding to a time interval of ∼10 ms). Ultimately, the delay-time resolved phase jitter variances of a narrow-linewidth laser were retrieved with the removal of any fading problems, in accordance with numerical and theoretical simulations, as shown in Fig. 6(e). Thermal and acoustic noise from the environment would result in Rayleigh scattering-induced extra phase noise. In our experiment, the phase jitter variance errors at each discrete delay time can be obtained by the difference between the measured and the theoretical value. Then, the measurement uncertainty can be estimated by the absolute mean value, i.e., 0.005 rad2 in this case. It is worth noting that the measurement error exhibits no dependence with the delay time whilst the measured phase jitter variance is linearly proportional to the delay time. Consequently, the Rayleigh scattering-induced phase noise can be technically suppressed by the linear fitting for the linewidth estimation. Note that, the phase jitter variance with finer delay time step can be quasi-continuously retained, albeit with the expense of computational budget.
Fig. 6. Delay-time resolved phase jitter variance. Experimental investigation of data acquisition parameters: (a) Sampling rate Fs, (b) Trace sampling number Ns, (c) Time window tw, (d) Trace time interval Δt. Discrete delay times were selected as 2 µs (black), 4 µs (red), 6 µs (blue), 8 µs (megenta), 10 µs (green). (e) Delay-time-resolved phase jitter variance. (Parameters setting as Fs = 2 GSa/s, Ns = 1000, tw = 100 ns and Δt = 0.6 ms.)
4.2 Intrinsic linewidth characterization of a highly coherent laser
The above-mentioned technique evidently reveals the phase noise statistics of a coherent laser by retrieving its delay-time resolved phase jitter variance. One of its potential applications can be explicitly utilized to characterize the quantum-noise-induced linewidth of a laser source, particularly for ultra-narrow linewidth of <1 kHz. According to Eq. (3), the white-noise-induced laser intrinsic linewidth can be readily deduced by the slope efficiency of the phase jitter variance with respect to the delay time in a straightforward manner.
For the proof-of-concept demonstration, a sub-kHz Brillouin random fiber laser (BRFL) [48] was investigated by our proposed approach. As a new breed of fiber lasers, BRFLs have shown unique spectral dynamics and noise properties, arousing immense potentials in underlying fundamental research as well as practical applications [49]. Providing that the linewidth of the BRFL is unknown, the acquisition of the delay-time resolved phase jitter variance was first deployed with variable trace time intervals from 0.1 ms to 10 ms. According to Eq. (3), the linewidth can be mathematically estimated as the slope efficiency by a linear curve fitting of the phase jitter versus the modified delay time, as shown in Fig. 7(a). By plotting the estimated linewidth as a function of the trace time interval, two areas can be divided in terms of the relation between the trace time interval Δt and the laser coherent time (${t_c}{\; } \cong {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\varDelta {\upsilon_c}}}} \right.}\!\lower0.7ex\hbox{${\varDelta {\upsilon _c}}$}}$), as depicted in Fig. 7(b). Taking account of the requirement of the trace time interval ($\varDelta t{\; } \ge {t_c})$ in the statistical retrieval of the phase jitter, the ultimate linewidth can be reasonably determined at the value close to the crossing point with a virtual curve of $t = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\varDelta {\upsilon_c}}}} \right.}\!\lower0.7ex\hbox{${\varDelta {\upsilon _c}}$}}$. As a result, the linewidth of the BRFL was given by the least square method-based linear fitting slope (i.e. $554.0 \pm 10.0$ Hz) of the delay-time resolved phase jitter variance at the trace interval of 1.8 ms. In Fig. 7(c), a normalized lorentzian PSD of the laser spectrum was re-established by substituting $\varDelta {\upsilon _c}$ with the measured linewidth of 554 Hz in our proposal, compared to the measured results (675${\pm} $100 Hz) via the DSHI with 200-km delay fiber which is estimated with the 20-dB linewidth (13.5 kHz) divided by 20. Evidently, the conventional DSHI with hundreds of kilometer fiber delay inevitably imposed a deviation of the intrinsic lorentzian linewidth with a Gaussian-like broadening in the central of the laser shape, even though corrected by the 20-dB linewidth rather than a direct 3-dB linewidth.
Fig. 7. Phase noise statistics and intrinsic linewidth characterization of a sub-kHz Brillouin random fiber laser (BRFL). (a) Phase jitter variance as a function of the delay time with the beating note trace interval of (i) 0.5 ms, (ii) 1.8 ms, (iii) 3.0 ms; the delay time is modified by multiplying a factor of 2π to facilitate the linewidth fitting. (b) Linewidth estimation as a function of trace time interval ranging from 0.1 ms to 10 ms. (c) Recovered laser spectrum PSD with a lorentzian linewidth estimated by our proposal in comparison to the DSHI results with 200-km delay fiber. (d) Investigation of BRFL pump laser impact on phase noise statistics and intrinsic linewidth. The BRFL was pumped by three different pump lasers: (1) AOI laser with 50 kHz (DFB, AOI Inc.), (2) RIO laser with 30 kHz (RIO009x, RIO Inc.) and (3) NP laser with 3.4 kHz (Rock Module, NP Photonics Inc.). Accordingly, the intrinsic linewidth of the BRFLs with three pump lasers were mathematically obtained as 554.0(±10.0) Hz, 998.1(±23.0) Hz, 1600.0(±30.2) Hz, respectively.
Furthermore, the dependence of the pump laser on the BRFL linewidth was explicitly investigated by exploiting the delay-time-resolved phase jitter variance. As shown in Fig. 7(d), a 3.4-kHz narrow-linewidth pump laser delivers the random laser emission with a lowest phase jitter variance as well as a narrowest sub-kHz intrinsic linewidth while the pump lasers with larger linewidth would also remarkably deteriorate the phase noise of the random laser radiation and readily broaden its intrinsic linewidth beyond 1 kHz. It indicates the feasibility and the robustness of our proposal as an effective approach for exploiting underlying laser physics. Although both electrical and optical components such as the photodetector, the AOM diver and the EDFA would impose extra noise, the proposed technique is based on the statistics analysis over thousands of measurements rather than one-shot in all existing methods, and hence it compensates the extra noise to a large extent. Furthermore, the additive noise on the phase variance would not affect its slope in delay time, as well as the intrinsic linewidth measurement, indicating remarkable advantages in terms of the robustness for the linewidth characterization.
4.3 Discussions
In our experiments, the delay-time-resolved phase jitter variance was deployed with approximately 10 microsecond delay time in the aid of Rayleigh scattering along 1-kilometre-long fiber, which discriminately mitigates the dominated 1/f frequency noise over a long delay time (e.g., >1 ms for at least 200-km delay fiber in DSHI-based 1kHz linewidth characterization). The PCC-based retrieved phase confinement technique is restricted to phase excursions less than π, however phase unwrapping algorithms can be implemented to overcome this limitation [50]. In addition, alternative possibilities in terms of fading-diminishing phase retrieval approaches, including I/Q demodulation [51–53] and 3${\times} $3 coupler-based interferometer [54] merged with polarization diversity detection [55], would definitely diversify the options for a high-fidelity characterization of the intrinsic linewidth.
Since the phase variance at each delay time τz is resolved by correlation-based demodulation within a moderately small time window, the uncertainty of the measured phase variance is mainly determined by the intrinsic linewidth $\varDelta {\nu _c}$ of the laser and the time window tw, i.e., Error($\sigma _\varphi ^2$) ${\approx} 2\pi \varDelta {\nu _c}{t_w}$. In our experiment, the time window tw is 100 ns (determined by the pulse width) and the total Rayleigh fiber length is 1 km. It means that the measured phase jitter variances at the delay times of above 1 µs (i.e., Rayleigh fiber length longer than 100 m) exhibits at least one order of magnitude higher than the measurement uncertainty, which guarantees the precision of this method. Note that, this restriction could be released by utilizing a narrower time window/pulse width and a longer length of Rayleigh fiber to reveal its delay-time-resolved phase statistics. For example, the phase noise statistics of a laser even with 10-mHz intrinsic linewidth could be measured by launching 10-ns optical pulse sequence into over-10 km Rayleigh fiber, albeit with the expense of higher speed data acquisition and processing. Instead of Rayleigh scattering, other randomly distributed reflection scheme such as the arrays of weak FBGs [56] or random fiber gratings [57] could alternatively replace the Rayleigh scattering for a significant signal loss reduction and the alleviation of high pulse power requirement. Furthermore, the proposed measurement system would be definitely limited by the thermal and acoustic noise from the external environment, which hence can be improved by further adiabatic and vibration isolation of the delay fiber (e.g., an adiabatic vacuum shield on a vibration-isolated mounting). Alternatively, this could be solved by taking differential phase variation among two equal section length at two closed fiber locations (say some centimeters), as the environmental temperature and acoustic noise will be similar, the differential phase change measurement will compensate the temperature and acoustic signal change impact on Rayleigh scattering.
5. Conclusions
In this paper, the phase noise statistics of a coherent laser was retrieved with the aid of a COTDR system, in which distributed Rayleigh scattering introduced a time-of-flight mapping of the heterodyne beating note carrying the delay-time-resolved phase information of the laser. With a PCC-based phase jitter demodulation, the phase noise statistics can be analyzed for a high-precision determination of its intrinsic linewidth subject to the quantum limit, agreeing well with theoretical analysis and numerical simulation. Considering the fact that slow phase fluctuations can be technically compensated, noise spectra at low frequencies are not always of concern while the intrinsic linewidth characteristic associated with fundamental limits is non-trivially paid extensive attentions. It is believed that the proposed intrinsic linewidth determination via a statistical analysis of laser phase noise offers salient robustness in terms of reliability and flexibility, opening new windows for discovering the next frontiers of laser physics and measurement science.
Appendix A: Pearson correlation coefficient-based phase jitter retrieval
Given a pair of zero-mean variables (X, Y), the formula for Pearson correlation coefficient $\rho $ can be written as,
Appendix B: Derivative of the phase jitter variance
Given the random variables x, y each satisfying the following probability density function,
Funding
Canada Research Chairs (950231352); Natural Sciences and Engineering Research Council of Canada (RGPIN-2015-06071; STPGP-506628-17).
Acknowledgment
We thank Chengxian Zhang for loaning the 2-kHz External Cavity Laser and the helpful assistance and fruitful discussions during the experiments.
Disclosures
The authors declare no conflicts of interest.
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