Characterization of Laser Frequency Stability by Using Phase-Sensitive Optical Time-Domain Reﬂectometry

: A new method to measure laser phase noise and frequency stability based on the phase-sensitive optical time-domain reﬂectometry is proposed. In this method, the laser under test is utilized in a phase-sensitive optical time-domain reﬂectometer, which employs phase-modulated dual pulses and acts as an optical frequency discriminator: laser frequency ﬂuctuations are deduced from the analysis of the reﬂectometer data corresponded to phase ﬂuctuations along the vibration-damped and thermally insulated ﬁber spool. The measurement results were validated by comparison with direct optical heterodyning of the tested and more coherent reference lasers. The use of dual pulses generated by an acousto-optic modulator makes it easy to adjust the time delay during measurements, which distinguishes favorably the proposed method from standard optical frequency discriminators. The method is suitable for testing highly coherent lasers and qualifying their parameters such as linear drift rate, random frequency walk rate, white frequency noise (which is directly related to laser instantaneous linewidth), and ﬂicker noise level.

Complex E-field of the laser output can be written as where ϕ(t) is the phase fluctuations, ν 0 is the average carrier optical frequency, P 0 is the average optical power, and δP(t) is the optical power fluctuations, which for modern, highly coherent lasers can be mostly neglected in many applications [10].In many cases, it is more convenient to use frequency fluctuations, which are related to ϕ(t) by the formula The laser frequency fluctuations are usually characterized either by their power spectral density (PSD) or by their Allan variance [11].In the case of stationary laser operation, when long-term frequency drift can be neglected, the laser frequency PSD often can be expanded into a power series [11]: where S 0 characterizes white frequency noise, which determines the instantaneous (Lorentzian) laser linewidth ∆ν L = πS 0 , S −1 is flicker noise level, which determines the Gaussian laser linewidth, and S −2 defines a rate of frequency random walk.The Lorentzian linewidth of modern, highly coherent lasers is less than 1 kHz, and the flicker noise level is ~10 5 -10 6 Hz 2 [12].
In the case when the laser carrier frequency is not stabilized, there is long-term frequency drift caused by factors such as ambient temperature fluctuations or due to the laser warming up.The laser frequency variation, in this case, is almost linear ∆ν(t) = at where a is the frequency drift rate.If such linear drift is present, the calculation of laser frequency power spectral density and estimations of S 0 , S −1 , S −2 require subtraction of linear frequency drift term (a numerical procedure also known as detrending) [13] and/or use of the Bartlett/Welch methods for estimating the frequency noise PSD [14].
An alternative method for characterizing the laser frequency fluctuations is to measure the Allan variance for different time intervals τ [11]: where ν k is mean value of frequency ν(t) over time interval [t k , t k + τ].The Allan variance characterizes both long-and short-term frequency fluctuations.In the case when linear drift or random walk dominates over white and flicker noise, we will assume that this is the long-term frequency fluctuations, and for the short-term, vice versa.
As mentioned above, a typical Lorentzian linewidth of currently available highly coherent lasers is less than 1 kHz and cannot be resolved by standard techniques based on diffraction gratings or scanning Fabry-Perot interferometry.Still, slow wavelength/carrier frequency drifts can be measured by high-precision wavelength meters based on Fizeau interferometers [15].The resolution of such devices (typically ~0.5 MHz) is still insufficient for carrier frequency measurements with higher precision.For this reason, optical homo- [16], hetero- [17], and self-heterodyning [12,13,18] approaches are used.
In optical heterodyning, the output of the laser under test (LUT) at the carrier frequency ν c is combined with the output of a reference laser at the carrier frequency ν lo , resulting in a beat note at a difference frequency |ν lo − ν c |, which can be detected by a fast enough photodetector (PD) and acquired using a high-speed analog-to-digital converter (ADC) built in an electrical spectrum analyzer (ESA) or a digital storage oscilloscope (DSO).Optical heterodyning is suitable for both short-and long-term frequency fluctuations measurements.In this method, the carrier frequencies of the reference laser and the LUT must be very close to each other, so the beat note |ν lo − ν c | is within the electrical bandwidth of a PD and digitizer.The reference laser is not required if homo-and self-heterodyne techniques are used.Instead, the output of the LUT is split into two arms of an unbalanced Mach-Zehnder or Michelson interferometer, and the beat note is formed due to the interference of the laser output with itself delayed by time τ d , where τ d = l d n c , l d is the length of the delay line, n is the group refractive index, and c is a speed of light in vacuum.In the delayed self-heterodyne interferometer (DSHI), an acousto-optic modulator (AOM) is installed in one of the interferometer arms to shift the frequency of the beat note from zero central frequency, thus the flicker noise of the electronics is insignificant [16].The DSHI allows one to estimate white noise level S 0 , and in case τ d ≥ 3S 0 /S −1 , also flicker noise level S −1 [12].The length of delay line l d in the fiber interferometer is limited by fiber attenuation and usually does not exceed 100 km that corresponds to τ d ≤ 0.5 ms.
In contrast to the DSHI, the homodyne method with a short fiber delay line l d = 5-10 m (τ d = 25-50 ns) is suggested to measure the slow laser frequency drifts [19].In that case, the unbalanced interferometer is used as a frequency discriminator where the frequency drift ν(t) is proportional to phase difference and time delay.However, it was shown in [19] that in the case of using a single photodetector for beat note registration, only the absolute value of ν(t) could be measured, but not the sign.
In this paper, we propose a new method of laser phase noise and frequency instability characterization based on dual-pulse phase-sensitive optical time-domain reflectometry (DP-ϕOTDR).In the method, the LUT output is used to form sequences of periodically phase-shifted dual pulses to probe the fiber spool placed in a vibroacoustically isolated box.Analysis of sequences of backscattered optical signals allows one to calculate optical phase changes along the fiber and thus permits evaluation of instantaneous (Lorentzian) linewidth, and frequency changes slow in comparison with the pulse repetition rate.
The proposed method does not require a reference laser and allows an easily adjustable delay time τ d between the paired pulses, which is essentially equivalent to adjusting the delay time in the unbalanced interferometer used for delayed homodyning and selfheterodyning techniques.

Theory
The operation principle of DP-ϕOTDR is described in detail in [20][21][22][23][24][25][26][27][28].In DP-ϕOTDR, the fiber spool is probed by a sequence of four pairs of pulses (dual pulses) with a time delay τ d between them and periodically changing phase difference ∆φ ∈ 0, π 2 , π, 3π 2 .While propagating along the fiber, these pulses are scattered by fiber inhomogeneities (so-called scattering centers), and two Rayleigh backscattering lightwaves 2n is the spatial shift of the backscattered lightwaves.These two lightwaves result in interferences and are acquired by the photodetector of DP-ϕOTDR.The resulting signal is called DP-ϕOTDR trace.Analysis of a sequence of four DP-ϕOTDR traces allows evaluation of the phase difference accumulated over the length l d along the fiber: ( This phase difference varies along the fiber length and allows one to track in time variations in fiber temperature δT, strain δε, and probing laser frequency [22][23][24][25][26][27][28][29].The overall variation in phase difference is given by the formula Here, δΦ Rayleigh are random phase jumps caused by the interference fading [10,[23][24][25][26][27][28][29], α T , α ε , α ν are thermal, strain, and frequency drift coefficients, and δν is a phase shift due to laser frequency change between the probing pulses.In the case of laser white frequency noise, δϕ las is a random quantity with zero mean and standard deviation of √ γ2πτ d ∆ν L determined by laser linewidth and delay time between the pulses in probing pair; the γ factor depends on the DP-ϕOTDR parameters.In order to reduce the impact of random jumps and receiver noise, averaging should be performed along the fiber length much greater than l d [25].The result of the measurement is equivalent to the phase measurement using an unbalanced interferometer with the optical path difference equal to l d = τ d c 2n .The thermal coefficient equals to α T = 2π λ cτ d β, where β = 1 ∂T includes a thermooptic and a thermal expansion coefficient and approximately equals to 0.915 × 10 −6 •K −1 for germanium-doped fused silica fiber at room temperature [30].The strain coefficient equals to α ε = 2π λ cτ d ξ, where ξ = 0.78 is a coefficient, which accounts for photoelastic effect contribution [31].The frequency drift coefficient is proportional to time delay between the pulses α ν = 2πτ d .It was shown in [29] that probe pulses carrier frequency shifts lead to the uniform phase difference changes over the entire fiber length.Taking τ d = 300 ns and λ = 1550 nm, we obtain α T = 3300 rad•K −1 , α ε = 0.25 rad•nanostrain −1 , and α ν = 1.9 rad•MHz −1 .Thus, easily measurable optical phase variation of 1 radian cor-responds to either a 0.3 mK temperature change, fiber stretch of 4 nanostrains, or laser frequency shift of 530 kHz.It follows from these estimates that in order to measure the laser frequency drift at the level of 50 kHz/s, the fiber spool temperature change rate should be smaller than 0.1 K/h.This requirement was achieved by placing the fiber spool inside a thermally insulated box located on a vibrationally isolated table in a regular air-conditioned lab so that strain variations caused by external vibrations could also be neglected.Therefore, laser frequency changes can be calculated as where angular brackets denote averaging along the fiber spool. Unwrapping of phase difference data δΦ(z, t) z along time is highly recommended to be applied to increase dynamic range of the measurements [32].In order to separate white noise from relatively slow (which time scales are much larger than the pulse repletion period) frequency deviations, additional digital low-pass filtering can be performed.In the same time, using (4) Allan variance of frequency measured by DP-ϕOTDR allows for characterizing laser frequency instability.It can be shown that the Allan variance for DP-ϕOTDR at small time scales is determined by the Lorentzian linewidth and equals to where τ 0 = t p N p is phase difference trace sampling time, which equals to the pulse repetition period t p multiplied to a number of DP-ϕOTDR traces N p required to obtain a single phase difference trace Φ(z).Depends of DP-ϕOTDR design N p can equal 3 [20] or 4 [21].In our case, N p = 4.

Experimental Setup
A scheme of experimental setup is shown in Figure 1 and consists of DP-ϕOTDR and an additional optical heterodyning setup in order to compare experimental results obtained by both approaches.The output of the LUT is divided into two parts using 10/90 splitter.The larger part of the output is directed to an acousto-optic modulator (AOM) to form phase-modulated dual pulses.Then, dual pulse energy is boosted using an erbium-doped fiber amplifier (EDFA 1) and launched into the fiber spool located inside a thermal insulated box placed on the vibration-insulating table.The temperature inside the box is measured using a thermocouple connected to a 6 1 /2 -digit multimeter.The Rayleigh backscattered signal from the fiber spool propagates through the optical circulator and an erbium-doped fiber preamplifier (EDFA 2), where it is amplified.An optical filter with a 100 GHz bandwidth (OF) is installed at the output of the preamplifier to reduce influence of amplified spontaneous emission.The amplified Rayleigh backscattered signal is detected by the photodetector (PD 1), connected to the analog-to-digital converter (ADC), and is processed by a field-programmable gate array (FPGA).The repetition rate of the probing pulses f p are varied up to 8 kHz.The data are taken over a 1 km long fiber section.The smaller portion of the LUT output is used for optical heterodyning.For this purpose, the LUT output is combined with the output of the reference laser using a 50/50 coupler.The beat note of two laser outputs is detected by photodetector Alphalas UPD-15-IR2 (PD2) and acquired by a spectrum and signal analyzer Rohde&Schwarz FSW8 (ESA) operated in spectrogram mode, which allows carrying out long-term observations of changes in beat note frequency.
probing pulses fp are varied up to 8 kHz.The data are taken over a 1 km long fiber sect The smaller portion of the LUT output is used for optical heterodyning.For this purp the LUT output is combined with the output of the reference laser using a 50/50 coup The beat note of two laser outputs is detected by photodetector Alphalas UPD-15-(PD2) and acquired by a spectrum and signal analyzer Rohde&Schwarz FSW8 (ESA) erated in spectrogram mode, which allows carrying out long-term observations changes in beat note frequency.The description of the tested lasers is given in Table 1.The description of the tested lasers is given in Table 1.

Determination of γ for Different Parameters of Dual Pulse
The waterfall plots of the phase difference sequence obtained by DP-ϕOTDR with pulse repetition rate f p = 1 kHz for different pulse duration τ p and time delays τ d using an RIO Orion as laser source are shown in Figure 2a-c.For each waterfall time, the dependence of phase difference was averaged over the length L = 1 km, and all are shown below in Figure 2d-f.The left Y-axes correspond to the phase difference in rad, and the right Y-axes correspond to the frequency in MHz calculated using (7).Notice that in Figure 2d-f, the left Y-axes are fixed; meanwhile, the right Y-axes are different because of the division by τ d in conversion Formula (7).In order to separate the contribution of white phase noise and laser frequency drift, Butterworth high-and low-pass filters with 10 Hz cut-off frequency were applied to the phase difference time dependencies.The laser frequency drift can vary over time due to ambient temperature variations or self-warming during the laser operation.The obtained frequency drifts lie in a range of 60-90 kHz/s for the RIO.
To determine the coefficient γ, we measured the root mean square (RMS) values of white phase noise s Φ of DP-ϕOTDR signals for different combinations of pulse duration τ p and time delay τ d .The experimental dependencies s Φ = s Φ (τ p , τ d ) for different combinations of pulse duration and time delay are shown in Figure 3a dependencies are in good agreement with the results of the numerical simulation, which considers the Lorentzian linewidth ∆ν L = 2 kHz for RIO [12].
below in Figure 2d-f.The left Y-axes correspond to the phase difference in rad, and the right Y-axes correspond to the frequency in MHz calculated using (7).Notice that in Fig- ure 2d-f, the left Y-axes are fixed; meanwhile, the right Y-axes are different because of the division by d τ in conversion Formula (7).In order to separate the contribution of white phase noise and laser frequency drift, Butterworth high-and low-pass filters with 10 Hz cut-off frequency were applied to the phase difference time dependencies.The laser frequency drift can vary over time due to ambient temperature variations or self-warming during the laser operation.The obtained frequency drifts lie in a range of 60-90 kHz/s for the RIO.

Determination of Laser Phase and Frequency Instability Noise Parameters
The DP-ϕOTDR Allan deviations σ DP−ϕOTDR (τ) = σ DP−ϕOTDR 2 (τ) are calculated from ν(t) using (4).The set of data for the RIO for time delays τ d = 300 and 600 ns at pulse repetition rate f p = 1000 Hz, downsampled to 250 Hz, was experimentally measured, and σ DP−ϕOTDR (τ) are evaluated and shown in Figure 4. Earlier in our previous works [12,17], using both heterodyning and DSHI, the Lorentzian linewidth and flicker noise level of tested RIO were measured and evaluated as ∆ν L = 2 kHz and S −1 = 2 × 10 6 Hz 2 , respectively.Therefore, these parameters were used in numerical simulations of DP-ϕOTDR [22,23,33] at the same f p , τ d values with RIO as a laser source.In order to clarify how flicker noise affects the DP-ϕOTDR Allan deviation, some simulations take into account only white frequency noise, while others take both white and flicker frequency noise into account [12].The linear frequency drift rate in simulations is selected so that it coincides with the experimental data.The dots in Figure 4 correspond to the experimental data, and the solid and dashed lines represent numerical simulation results.The dotted lines for different combinations f p and τ d match Formula (9) with γ = τ d / τ d + τ p /2 and fit well both experimental and simulated data for time intervals ≤0.03 s for f p = 250 Hz and ≤0.01 s for f p = 1 kHz.It is shown that for time intervals 0.1-0.5 s, frequency instability of RIO can be characterized as linear drift and fitted with σ DP−ϕOTDR (τ) = a √ 2 τ (the black dashed line in Figure 4) with the rate a = ~70 kHz/s.The combination of the DP-ϕOTDR Allan variances for white noise and linear drift fits the experimental and simulations data f p = 250 Hz well, but for f p = 1 kHz, it does not.There are discrepancies at τ = 10 −2 -10 −1 s for simulations, which do not consider the flicker noise (solid lines).If simulations do take into account flicker noise, their data (corresponded blue and red dashed lines in Figure 4) show good agreement with experimental results and can be fit well with the expression Photonics

Determination of Laser Phase and Frequency Instability Noise Parameters
The DP-φOTDR Allan deviations tively.Therefore, these parameters were used in numerical simulations of DP-φOTDR [22,23,33] at the same fp, d τ values with RIO as a laser source.In order to clarify how flicker noise affects the DP-φOTDR Allan deviation, some simulations take into account only white frequency noise, while others take both white and flicker frequency noise into account [12].The linear frequency drift rate in simulations is selected so that it coincides with the experimental data.The dots in Figure 4   It was shown that when using the DP-ϕOTDR Allan variance, the white and the flicker frequency noise levels could be evaluated as well as linear frequency drifts.The higher probe pulse rates and longer time delays are preferred to estimate S −1 .With the exception of the white noise of the frequency, the obtained DP-ϕOTDR Allan variance dependences coincide with the standard formulas from [10].

Comparison with Optical Heterodyning
To compare our technique with heterodyning, we used ITLA PurePhotonics PPCL200, of which its output was split for simultaneous use in DP-ϕOTDR and heterodyning with RIO as a reference of a more stable laser.The example of the waterfall plot of the phase difference sequence obtained with DP-ϕOTDR (f p = 1 kHz and τ d = 300 ns) using an ITLA as a laser source is shown in Figure 5a.The laser carrier frequency drifts obtained using DP-ϕOTDR and optical heterodyning approaches are shown in Figure 5b.One can see that the obtained dependences are in good agreement.The calculated Allan deviations for the laser carrier frequency drifts measured by each approach are shown in Figure 5c.The obtained Allan deviations are fit well on time intervals of 0.1-10 s.Slight differences at times up to 0.1 s in the results can be explained by the ESA resolution operated in the spectrogram mode.The laser carrier frequency drifts obtained by both approaches at time intervals of 0.1-3 s are well approximated by a random walk [11] with the level S −2 = 1.2•10 12 Hz 3 .The linear frequency drifts are not observed for ITLA.
Allan variances for white noise and linear drift fits the experimental and simulations data fp = 250 Hz well, but for fp = 1 kHz, it does not.There are discrepancies at τ = 10 −2 -10 −1 s for simulations, which do not consider the flicker noise (solid lines).If simulations do take into account flicker noise, their data (corresponded blue and red dashed lines in Figure 4) show good agreement with experimental results and can be fit well with the expression It was shown that when using the DP-φOTDR Allan variance, the white and the flicker frequency noise levels could be evaluated as well as linear frequency drifts.The higher probe pulse rates and longer time delays are preferred to estimate S−1.With the exception of the white noise of the frequency, the obtained DP-φOTDR Allan variance dependences coincide with the standard formulas from [10].

Comparison with Optical Heterodyning
To compare our technique with heterodyning, we used ITLA PurePhotonics PPCL200, of which its output was split for simultaneous use in DP-φOTDR and heterodyning with RIO as a reference of a more stable laser.The example of the waterfall plot of the phase difference sequence obtained with DP-φOTDR (fp = 1 kHz and d τ = 300 ns) using an ITLA as a laser source is shown in Figure 5a.The laser carrier frequency drifts obtained using DP-φOTDR and optical heterodyning approaches are shown in Figure 5b.One can see that the obtained dependences are in good agreement.The calculated Allan deviations for the laser carrier frequency drifts measured by each approach are shown in Figure 5c.The obtained Allan deviations are fit well on time intervals of 0.1-10 s.Slight differences at times up to 0.1 s in the results can be explained by the ESA resolution operated in the spectrogram mode.The laser carrier frequency drifts obtained by both approaches at time intervals of 0.1-3 s are well approximated by a random walk [11] with the level S−2 = 1.2•10 12 Hz 3 .The linear frequency drifts are not observed for ITLA.
Thus, it was shown that DP-φOTDR can be used to directly measure laser frequency drifts at times of more than 0.1 s.To estimate laser Lorentzian linewidth and flicker noise level, the use of a high probe pulse rate is recommended.However, in the case of limited data memory resources to measure long-term frequency stability (>1 s), the high probe pulse rate is not necessary.Therefore, in order to meet both requirements to characterize each laser under the DP-φOTDR test, data were recorded twice at a probe pulse rate of 1 and 8 kHz with total measuring times ≤10 −1 and ≤10 s, respectively.The DP-φOTDR Allan deviations of all lasers under test for fp = 1 kHz are shown on the right side of Figure 6.RIO, OEW, and INV are characterized by linear drifts with a rate of a = ~70, ~300, and ~500 kHz/s, respectively.In contrast, ITLA and DFB have random frequency walks with the levels S−2 = 10 12 and 10 11 Thus, it was shown that DP-ϕOTDR can be used to directly measure laser frequency drifts at times of more than 0.1 s.
To estimate laser Lorentzian linewidth and flicker noise level, the use of a high probe pulse rate is recommended.However, in the case of limited data memory resources to measure long-term frequency stability (>1 s), the high probe pulse rate is not nec-essary.Therefore, in order to meet both requirements to characterize each laser under the DP-ϕOTDR test, data were recorded twice at a probe pulse rate of 1 and 8 kHz with total measuring times ≤10 −1 and ≤10 s, respectively.The DP-ϕOTDR Allan deviations of all lasers under test for f p = 1 kHz are shown on the right side of Figure 6.RIO, OEW, and INV are characterized by linear drifts with a rate of a = ~70, ~300, and ~500 kHz/s, respectively.In contrast, ITLA and DFB have random frequency walks with the levels S −2 = 10 12 and 10 11 Hz 3 , respectively.The DP-ϕOTDR Allan deviations of all lasers for f p = 8 kHz are shown on the left of Figure 6.The experimental data are fit by σ DP−ϕOTDR (τ) = 4∆ν L / π 2τ d + τ p τ f p + 2 ln 2S −1 , which takes into account only the white and flicker noises.The Lorentzian linewidths and flicker noise levels of all lasers under test can be easily estimated using parameter selection in approximation curves.The only exception is OEW, which has such a high linear drift rate a in relation to its ∆ν L and S −1 , that only upper estimation of ∆ν L and S −1 is possible.The precise values of ∆ν L and S −1 should be evaluated using a higher probe pulse rate, which in the case of using a 1 km fiber spool could be up to f p = 100 kHz.The measurement results of all tested lasers are shown in Table 2.In addition, Table 2 includes results obtained from optical heterodyning [17] and DSHI [12].The obtained results coincide with each other.The slight difference in linear drift rates for the OEW and INV measured by different approaches can be explained by the different self-warming conditions of the lasers during their operation.

Discussion
As mentioned above, the temperate fluctuations of fiber spool affect the accuracy of long-term frequency measurements.During our measurements, the temperature in the thermal-insulated box was measured using a thermocouple.Thermocouple temperature drift did not exceed 0.05 degrees per 1 h, which corresponds to the measurement error of the average linear frequency drift rate equal to or less than 25 kHz/s.It should be noted that the temperature fluctuations of the fiber should be less than that of the thermocouple due to the fact that the fiber spool consists of many layers, which makes the fiber spool more heat-resistant.
The phase change δΦ measured using DP-ϕOTDR is proportional to the delay time τ d .In DP-ϕOTDR, τ d is regulated by AOM and can be easily tuned in the range from the pulse duration time τ p up to the time that the frequency drifts cannot be neglected.In the case of the usage of AOM as an intensity modulator, τ p cannot be less than the sum of the rise and fall times or approximately ~50-100 ns.If shorter values of τ p and τ d are required, an electro-optic I/Q modulator, instead of a conventional AOM, can be used in DP-ϕOTDR.Thus, by varying τ d and f p , it is possible to adjust the range of measured laser carrier frequency changes δv.This favorably distinguishes the proposed method from the approaches of delayed homo-and self-heterodyning, where a change in the fiber length of the interferometer arm is required to change the time delay τ d .
While measuring with DP-ϕOTDR, the phase changes between two successive measurements must be much less than 2π for correct phase unwrapping, otherwise the phase change calculation algorithm described in [32] may lead to errors.The maximum values of a, ∆ν L , S −1 , S −2 , at which δΦ = 2π in DP-ϕOTDR with τ p = 200, τ d = 300 ns, and ∆t = 4 ms are shown in Table 3.One can see that even a conventional DFB diode laser with laser linewidth ~10 MHz can be measured using the DP-ϕOTDR technique.
As it was shown in the OEW case, measuring the Lorentzian linewidth < 10 Hz can be complicated when there are high long-term frequency drifts (linear drift or random walk).The minimum measurable value of the Lorentzian linewidth ν L min can be calculated by equating the values the Allan variances of DP-ϕOTDR linear drift and white frequency noise at τ = ∆t = 4/ f p : For example, for a = 1 MHz/s and f p = 8 kHz, the minimum measurable value of the Lorentzian width is ν L min ≥ 1 Hz.For a more accurate estimate, it is necessary to take into account the flicker noise as well.
Comparing tested lasers, it is clear that INV and OEW have the lowest values of Lorentzian linewidths (~30 and ~10 Hz, respectively), which makes their use in fiber optic interferometric or distributed acoustic sensors the most promising.INV has the flicker noise level in order of magnitude more than that of OEW.The flicker noise in fiber lasers does not have the fundamental origin, and we believe that it can be suppressed by troubleshooting technical issues or with the use of additional frequency discriminators and electrical feedback [34].
At the same time, despite RIO having a 1 kHz Lorentzian linewidth, an order of magnitude worse than OEW and INV, it has the lowest linear frequency drifts.As frequency drifts cause difficulties with long-term analysis, RIO is the most suitable among tested lasers for fiber optic sensors where low frequency signals are analyzed (e.g., distributed temperature [35] or seismic [36] measurements).
Despite the fact that ITLA PurePhotonics PPCL200 has a lower Lorentzian linewidth than most of the other ITLAs in the telecom industry, its application in fiber optic interferometric or distributed sensors is complicated because of high-frequency random walks.All of the above is also true for the DFB diode laser ILNP-249.However, according to our knowledge, ILNP-249 is a distributed feedback diode laser with the narrowest Lorentzian linewidth.

Conclusions
A new method for measuring frequency stability and phase noise of highly coherent lasers based on the phase-sensitive reflectometry is proposed and experimentally investigated.It is shown that when the tested laser output is used in DP-ϕOTDR, it is possible to measure both the long-term stability of the laser frequency (for example, linear drifts and random walks) and short-term stability (white and flicker noise) through analysis of the phase information of DP-ϕOTDR traces for the fiber spool placed in the thermal-insulated box.The significant advantage of the proposed method over heterodyning is that there is no need to use a reference laser with a higher frequency stability.This makes it possible to measure the characteristics of laser frequency stability in a wide wavelength range.The advantages of the suggested measurement method over homodyne and self-heterodyne methods lie in the simplicity of adjusting the delay time τ d , which with variation in probe pulse rate ensures high sensitivity and measurement accuracy for both long-term and short-term stability of the laser carrier frequency.
Comparison of the measurement results for ITLA obtained by the proposed method and by the method of optical heterodyning (with RIO as the reference laser) shows their coincidence within measurement accuracy.
It has been shown both using numerical simulations and experiments that the proposed method is convenient for separating the contributions of various types of frequency noise.As an example, it is shown that the ITLA and DFB lasers are characterized by a random frequency walk, and the values of the parameter S −2 were estimated, which turned out to be equal to 10 12 and 10 11 Hz 3 , respectively.The RIO, OEW, and INV lasers are characterized by linear drifts with rates ~70, ~300, and ~500 kHz/s, respectively.Using DP-ϕOTDR, the Lorentz linewidths of orders 10 1 -10 4 Hz and flicker noise levels of orders 10 5 -10 10 Hz 2 were successfully estimated.

Figure 1 .
Figure 1.Schematic of an experimental setup for testing the concept of measuring the freque variation in single-frequency lasers.

Figure 1 .
Figure 1.Schematic of an experimental setup for testing the concept of measuring the frequency variation in single-frequency lasers.

Figure 2 .
Figure 2. (a-c): Waterfalls of the serial phase differences in rad, obtained by DP-φOTDR with RIO as a laser source for different values of pulse duration p τ and time delay d τ , where the horizon- tal axes correspond to coordinates along the fiber and the vertical axes correspond to time.(d-f): Corresponded averages along the fiber phase differences (left Y-axis), white phase noise (purple lines, left Y-axis) calculated by applying high-pass filter with 10 Hz cut-off frequency to phase difference data, and the laser frequency drift in MHz (black lines, right Y-axis) calculated using (7) and additionally applying low-pass filter with 10 Hz cut-off frequency.Notice that in Figure 2d-f, left

Figure 2 .
Figure 2. (a-c): Waterfalls of the serial phase differences in rad, obtained by DP-ϕOTDR with RIO as a laser source for different values of pulse duration τ p and time delay τ d , where the horizontal axes correspond to coordinates along the fiber and the vertical axes correspond to time.(d-f): Corresponded averages along the fiber phase differences (left Y-axis), white phase noise (purple lines, left Y-axis) calculated by applying high-pass filter with 10 Hz cut-off frequency to phase difference data, and the laser frequency drift in MHz (black lines, right Y-axis) calculated using (7) and additionally applying low-pass filter with 10 Hz cut-off frequency.Notice that in (d-f), left Y-axes are fixed; meanwhile, right Y-axes are different because of the division by τ d in conversion Formula (7).

Figure 3 .
Figure 3.The dependencies of RMS of white phase noise for different combinations of pulse duration and time delay

are evaluated and shown in Figure 4 .Δ = 2
. The set of data for the RIO for time delays d τ = 300 and 600 ns at pulse repetition rate fp = 1000 Hz, downsampled to 250 Hz, was experimentally measured, and Earlier in our previous works[12,17], using both heterodyning and DSHI, the Lorentzian linewidth and flicker noise level of tested RIO were measured and evaluated as L ν kHz and S−1 = 2 × 10 6 Hz 2 , respec- correspond to the experimental data, and the solid and dashed lines represent numerical simulation results.The dotted lines for different combinations fp and d

Figure 3 .
Figure 3.The dependencies of RMS of white phase noise for different combinations of pulse duration and time delay s Φ = s Φ (τ p , τ d ): (a) for different τ p and fixed τ d = 1200 ns; and (b) for different τ d and fixed τ p = 200 ns.Red rectangles correspond to the experimental data, black dots correspond to the simulation results, and blue lines fit s Φ = 2π∆ν L τ d 2 / τ d + τ p /2 .

Figure 4 .
Figure 4.The DP-φOTDR Allan deviations for the RIO laser at different delay times d τ and prob- ing frequencies fp.The dots correspond to the experimental data, and the solid and dashed curves correspond to the data obtained from the numerical simulations.The black dotted lines correspond to Formula (9) for different combinations d τ and fp at L ν Δ = 2 kHz, and black dashed line describes

Figure 4 .
Figure 4.The DP-ϕOTDR Allan deviations for the RIO laser at different delay times τ d and probing frequencies f p .The dots correspond to the experimental data, and the solid and dashed curves correspond to the data obtained from the numerical simulations.The black dotted lines correspond to Formula (9) for different combinations τ d and f p at ∆ν L = 2 kHz, and black dashed line describes linear frequency drift with a rate of a = 70 kHz/s.

Figure 5 .
Figure 5. (a) Waterfall of the phase difference obtained by DP-φOTDR with ITLA as a laser source; the horizontal axis corresponds to coordinates along the fiber, and the vertical axis corresponds to time; (b) the laser carrier frequency drift measured with DP-φOTDR (purple line) and optical heterodyning with RIO as the reference laser (black line); (c) Allan deviations of laser carrier frequency measured with DP-φOTDR (purple line) and optical heterodyning method (black line); and the dashed line marks the Allan deviation of the random frequency walk with the level S−2 = 1.2 × 10 12 Hz 3 .

Figure 5 .
Figure 5. (a) Waterfall of the phase difference obtained by DP-ϕOTDR with ITLA as a laser source; the horizontal axis corresponds to coordinates along the fiber, and the vertical axis corresponds to time; (b) the laser carrier frequency drift measured with DP-ϕOTDR (purple line) and optical heterodyning with RIO as the reference laser (black line); (c) Allan deviations of laser carrier frequency measured with DP-ϕOTDR (purple line) and optical heterodyning method (black line); and the dashed line marks the Allan deviation of the random frequency walk with the level S −2 = 1.2 × 10 12 Hz 3 .

Figure 6 .
Figure 6.The DP-ϕOTDR Allan deviations for different lasers for probe pulse rates f p = 1 and 8 kHz (right and left sides, respectively): f p = 1 kHz was used to estimate linear drift rates and random walk levels, and f p = 8 kHz was used to estimate Lorentzian linewidths and flicker noise levels.Approximations: linear drift (dashed lines), random walk (dash-dotted lines), and Lorentzian linewidth with flicker noise (dotted curves).

Table 1 .
Information about the lasers.

Table 1 .
Information about the lasers.

Table 2 .
Measurement results of frequency instability of various lasers measured using DP-φOTDR in comparison with heterodyning and DSHI.

Table 2 .
Measurement results of instability of various lasers measured using DP-ϕOTDR in comparison with heterodyning and DSHI.

Table 3 .
The maximum values of linear drift rate, Lorentzian linewidth, flicker noise, and random walk levels when phase change does not exceed 2π between two measurements (∆t = 4 ms) for DP-ϕOTDR with τ p = 200 and τ d = 300 ns.