Unveiling delay-time-resolved phase noise dynamics of narrow-linewidth laser via coherent optical time domain reflectometry

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 dynamics and its intrinsic linewidth of a highly coherent laser remains ambiguous and challenging. Here, we present an approach capable of revealing delay-time-resolved phase noise dynamics of a coherent laser based on coherent optical time domain reflectometry (COTDR), in which distributed Rayleigh scattering along a delay fibre 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 a precise measurement of ultra-narrow laser linewidth by exploiting its delay-time-resolved phase jitter statistics, confirmed with the analytical modelling and numerical simulations.

Coherent laser sources with highly purity spectrum are at the heart of highprecision 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 currently enables a coherent laser radiation with an extremely narrow linewidth down to mHz [8][9][10][11][12][13] , however, the characterization of such 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 intrinsic quantum-limit linewidth generally reflects the shortterm 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 2 nd identical laser 16 , is proposed and intensively employed to characterize the linewidth as well as the phase noise [17][18][19][20] . However, in spite of its simplicity, the DSHI approach turns out to be physically incompetent for an ultranarrow linewidth (e.g. < 1 kHz) due to a remarkable attenuation (>40 dB) over hundreds of kilometres delay fibre 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][22][23] or the strong coherent envelope analysis associated to a relative short delay [24][25][26] . However, its robustness remains elusive because of a definite delay-time-dependent behaviour 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][29][30] , leading to a dilemma for an intrinsic linewidth characterization. On the other hand, interferometric recovery of phase or frequency noise of the laser by means of a short-delay-time interferometer are theoretically 31 and experimentally [32][33][34][35][36] demonstrated to estimate the intrinsic laser linewidth albeit with a non-straightforward manner, however, these rather sophisticated measurement systems place additional stringent constrictions, resulting in either an over-optimistic estimation of the intrinsic linewidth 37 or hindering the flexibility for practical implementation. Consequently, a precise characterization of the fundamental laser linewidth subject to a quantum-limit phase noise dynamics is highly demanded not only being oriented by practical applications but also in a more fundamental prospective of laser physics.
In this communication, we demonstrate a novel technique to reveal the phase jitter dynamics 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 fibre 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 dynamics, 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.

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: (1) 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 38  (2) has zero mean and its variance 2 is linearly proportional to the time delay .
Theoretical results reveal that the relation between the intrinsic linewidth and the phase jitter variance 39,40 , (3) 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.
Given a laser with a lorentzian linewidth of 1 kHz, a random walk of the laser phase jitter can be simulated by Wiener process with computational random numbers, as shown in Fig. 1(a). The electrical field of the laser can be recalculated by adding the random phase jitter into its original pure sinusoidal waves. Then, the laser linewidth can be determined by the 3-dB bandwidth of the power spectral density of the laser spectrum. As shown in Fig. 1(d), the linewidth of the laser with quantum noise-induced random phase jitter was broadened to a Lorentz line shape with a 3-dB linewidth of 1-kHz.  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 σ 2 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 , which is in agreement with the theoretical analysis.

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 fibre occasionally deteriorates the heterodyne beating notes, leading to a notorious signal fading 45 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.
Here, is the calculated PCC between two beating note traces at the central delay time of 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 2π × 10 −4 rad 2 . 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 45 , can be alleviated to a large extent since the occasional ultra-weak signal fading occurs at certain points within single pulse width cell 44 . 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 | | at the delay time of τ could be statistically calculated as, It is worth mentioning that, the time interval Δt between two calculated beating note traces should be longer than the coherent time of the laser in order to mitigate any correlation between them for independent statistical analysis of the phase jitter variance.
In this way, the delay-time resolved phase jitter variance can be mathematically 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 42 .  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)

Delay-time-resolved laser phase noise dynamics and statistics
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( ~10 ms). Ultimately, the delay-time resolved phase jitter variances of a narrowlinewidth laser were retrieved with the removal of any fading problems, in accordance with numerical and theoretical simulations, as shown in Fig. 6 (g). Note that, the phase jitter variance with finer delay time step can be quasi-continuously retained, albeit with the expense of computational budget.

Intrinsic linewidth characterization of a highly coherent laser
The above-mentioned technique evidently reveals the phase noise dynamics of a coherent laser by statistically retrieving its delay-time resolved phase jitter variance.
One of its potential applications can be explicitly utilized to characterize the quantumnoise-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.

Discussions
We demonstrate a technique relying on COTDR system to retrieve the delay-timeresolved phase noise dynamics of a coherent laser, yielding a high-precision determination of its intrinsic linewidth subject to the quantum limit. The delay-timeresolved phase jitter variance was deployed with up to dozens of microsecond delay time in the aid of Rayleigh scattering along few-kilometre-long fibres, which discriminately mitigates the dominated 1/f frequency noise over a long delay time (e.g., >1ms). Revealing of π-beyond phase jitter over a longer delay time, which currently is restricted by the PCC-based retrieved phase confinement within [0 π], however, can be computationally solved by the phase unwrapping algorithm 49 . In addition, alternative possibilities in terms of fading-diminishing phase retrieval approaches, including I/Q demodulation 50,51 and 3×3 coupler-based interferometer 52 merged with polarization diversity detection 53 , would definitely diversify the options for a high-fidelity characterization of the intrinsic linewidth.
Considering the fact that slow phase fluctuations can be technically compensated 8,54 , 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; for instance, the intrinsic linewidth governed by diverse phase noise dynamics were intensively studied in a plenty of lasers involving various gain mechanisms such as gas laser 55 , fibre laser 56 , semiconductor laser 10,39 , quantum cascade laser 9,32 , photonic integrated laser 13 and micro-resonator laser 57 . To demonstrate the different phase noise dynamics of lasers with various gain mechanism, we have experimentally measured the delay-time-resolved phase jitter variance of a fiber laser compared to that of an external cavity semiconductor laser, which are presented in Supplementary Figure S1. It is believed that the proposed intrinsic linewidth determination via a statistical analysis of laser phase noise dynamics offers salient robustness in terms of reliability and flexibility, opening new windows for discovering the next frontiers of laser physics and measurement science.

Coherent optical time domain reflectometry. Coherent optical time domain reflectometry
(COTDR) is a ubiquitous technique that typically captures the beating notes of a cw light source and its pulse-modulated signal from Rayleigh scattering along an optical fibre for a time-offlight measurement of position-resolved response from external disturbance. Either intensity 58 or phase 43,50 interrogation based on COTDR could be implemented to achieve a distributed acoustic sensing.
Delayed self-heterodyne interferometer approach. In comparison, a conventional delayed self-heterodyne interferometer (DSHI) was utilized to evaluate the laser linewidth. The laser beam was launched into a fibre-based Mach-Zehnder interferometer which was deployed by utilizing an acousto-optic modulator with a 40-MHz carrier frequency shift in one arm and a long delay fibre of SMF in the another arm. Sufficiently long delays (i.e., larger than laser coherent time) are basically required to decorrelate two beams, and the detected power spectral density of the beating signal turns to a self-convolution of the laser spectrum and then the laser linewidth can be determined. In our experiments, the heterodyne beating signal was converted by an AC-coupled photodetector (PDB450CAC, Thorlabs) and its power spectral density (PSD) is displayed by a signal analyzer (FSW50, R&S). To eliminate the 1/f frequency noise induced Gaussian linewidth broadening, the lorentzian linewidth of a coherent laser based on DSHI method is practically estimated by the width at 20-dB lower than the maximum of the measured PSD instead of the conventional 3-dB full-width-half-maximum linewidth.