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We propose a novel method to improve the spatial resolution of Brillouin optical time-domain reflectometry (BOTDR), referred to as synthetic BOTDR (S-BOTDR), and experimentally verify the resolution improvements. Due to the uncertainty relation between position and frequency, the spatial resolution of a conventional BOTDR system has been limited to about one meter. In S-BOTDR, a synthetic spectrum is obtained by combining four Brillouin spectrums measured with different composite pump lights and different composite low-pass filters. We mathematically show that the resolution limit, in principle, for conventional BOTDR can be surpassed by S-BOTDR and experimentally prove that S-BOTDR attained a 10-cm spatial resolution. To the best of our knowledge, this has never been achieved or reported.

The spectrum of Brillouin scattering in optical fiber shifts in proportion to changes in the strain and temperature of the fiber. Brillouin optical time-domain analysis (BOTDA) [

To improve the spatial resolution of both techniques, it is necessary to narrow the pulse; however, the linewidth of the observed spectrum then becomes wider and makes measurement of the spectral shift difficult. For this reason, the spatial resolution has been limited to about one meter in both techniques [

For BOTDA, Bao

For the construction of pump light from two elements, the authors of [

On the other hand, the pump light of BOTDR plays a single role, since the BOTDR utilizes spontaneous scattering, so that it is difficult to attain high resolution by the same idea as for BOTDA. For resolution improvement of BOTDR, Koyamada

In this paper, we propose a novel method, referred to as synthetic BOTDR (S-BOTDR), to improve the spatial resolution of BOTDR. In this method, a synthetic Brillouin spectrum is constructed by combining several spectrums obtained by BOTDR measurements with different pump lights and low-pass filters. The pump lights and low-pass filters are composed of short and long elements with phase differences. We mathematically show that the resolution limit in principle for conventional BOTDR could be overcome by S-BOTDR and experimentally prove that 10-cm spatial resolution, which is much smaller than the conventional BOTDR limit, is attained by S-BOTDR.

The remainder of this paper is organized as follows: Section 2 presents the mathematical formulation of BOTDR and its performance limit in principle. In Section 3, the proposed method, called S-BOTDR, is described. Sections 4 and 5 are devoted to the evaluation of the proposed method through simulations and experiments, respectively. Finally, we conclude our paper in Section 6.

A BOTDR system is shown in

Brillouin scattering in optical fiber is described by the following equations [_{p}, _{s} and _{g} is the light speed in the fiber, _{B} is the linewidth of the acoustic wave spectrum and _{B}(

Random force _{s}| ≪ |_{p}|, the RHS of _{p}(_{s}(_{p} and _{eff} is the effective core area of a fiber and Γ = Γ_{B}/2 is set.

The solution to last section's BOTDR equations can be represented analytically. For simplicity, assuming that the fiber loss is small, we set

First, the solution to _{f} is the length of the fiber.

The backscattered light returned to the input end of an optical fiber in BOTDR is represented as:
_{g}_{Y}_{2} = _{Y} κ_{1}.

The Brillouin spectrum obtained by one measurement of BOTDR is represented as:

Since

We can calculate the expectation of

We note that

Though the ideal Brillouin spectrum is the Lorentzian spectrum, the observed Brillouin spectrum is spread by the PSF. Therefore, it is desirable for the PSF to have a shape close to a two-dimensional

To overcome the resolution limitation of conventional BOTDR described in the previous section, a synthetic spectrum approach, which we refer to as synthetic BOTDR (S-BOTDR), is proposed.

We consider a short pulse, _{1}(_{2}(_{[}_{a,b}_{]} is the definition function of interval [_{1} and _{2} are the pulse widths of two pulses and _{0}.

Similar to the PPP-BOTDA method [_{1} is set as less than the spatial resolution to be attained and _{2} is taken as sufficiently longer than the acoustic wave lifetime, 2/Γ_{B}, say about 9 ns, to improve frequency resolution. A low-pass filter is also composed of two elements, each of which is a matched filter of short or long pulses:

By using the above elements, a pump light and a low-pass filter are composed as:
_{kl}

Therefore, among the terms of the RHS of

To extract only element
_{j}, ϕ_{j}^{(}^{j}^{)}(^{(}^{j}^{)}(^{(}^{j}^{)}(^{(}^{j}^{)}(_{j}, j

This problem is equivalent to finding _{j}, ϕ_{j}, c_{j}, j

The PSF by the synthetic approach is represented as:

For two pulses of _{1} is sufficiently small and _{2} is sufficiently large, the PSF is approximated as:

The PSFs for a conventional BOTDR and S-BOTDR are plotted in ^{(}^{j}^{)}, ^{(2)} − Ψ^{(1)} and Ψ^{(4)} − Ψ^{(3)}, and is completely suppressed with the use of all PFS elements.

As shown in the analysis of the previous section, the synthetic Brillouin spectrum has the ideal property of being close to a Lorentzian spectrum. An S-BOTDR system for obtaining the synthetic Brillouin spectrum is constructed as shown in

The ideal property described in the previous section is for the expectation of the synthetic Brillouin spectrum. Achieving this property requires averaging or summing of the spectrums obtained by a number of measurements.

The pump lights of S-BOTDR are composed of short and long pulses and include parameters _{0} and _{0} is the difference of pulse start times and

_{0} If the short pulse is included in the long pulse, _{0} ≤ _{2}_{1}, the integral of _{0} affects the concentration of the PSF near the origin. The degree of concentration is estimated by the integral:
_{0}) approaches one for small

The values of _{0}) are plotted for different _{0}'s in _{0} = (_{2} − _{1})/2, the short pulse is located at the center of the long pulse; for _{0} = 0, the short pulse is at the edge of the long pulse, and the medium case is _{0} = (_{2} − _{1})/4. From _{0} = (_{2} − _{1})/2.

_{peak} and
_{opt} are shown in

The SFR of S-BOTDR is obtained by substituting _{1} and _{2} are plotted in _{1} for fixed _{2}. On the other hand, BFS estimation error variance is inversely proportional to SFR [_{1}.

Numerical simulations were performed to verify the S-BOTDR and compare it with the conventional BOTDR. The condition of the optical fiber is shown in

Conventional BOTDR (

Pulse width is

S-BOTDR (

Pulse widths are _{1} = 1 ns for a short pulse and _{2} = 50 ns for a long pulse. The amplitude ratio is r = 0.066. The short pulse is located at the center of the long pulse.

All results were obtained by summing 2^{12} trials. For conventional BOTDR with a 10-ns pulse in

The simulation was performed using the exact values of the input parameters. In an actual situation, these parameters are, however, subject to errors. As in S-BOTDR four measurements are combined, the input light amplitude fluctuation and phase-setting errors become issues of concern. Among them, the light amplitude fluctuation becomes negligible by averaging many replicate measurements. The errors in the phase differences between short and long elements are, on the other hand, anticipated. We performed a series of additional simulations in which the phase differences were subject to random errors. The results clearly demonstrated that they did not affect the spatial resolution, while the BFS estimation error increased only slightly, even if phase errors as high as 40 degrees were applied.

We performed BOTDR and S-BOTDR experiments using a test fiber. We constructed a test fiber by connecting two kinds of fibers: A and B (^{16} in all cases. The experimental result for BOTDR is shown in

We performed S-BOTDR experiments using the following combination of short and long elements for the pump light and the low-pass filter:

_{1} = 7.8 ns, _{2} = 60 ns,

_{1} = 2.7 ns, _{2} = 50 ns,

_{1} = 1.6 ns, _{2} = 50 ns,

_{1}becomes small. This is because the estimation error variance is nearly proportional to 1/

_{1}, as is described in Section 3.4. Therefore, some compromise is required to select small

_{1}. The synthetic spectrums at the 5-m strain interval are shown in

We showed that conventional BOTDR has, in principle, a resolution limit and proposed a synthetic spectrum approach, referred to as S-BOTDR, to overcome this limit. S-BOTDR uses composite pump lights and low-pass filters that are composed of short and long elements with different phases. Four BOTDR measurements are performed using four specific pairs of phases. By combining four Brillouin spectrums with specific weights, an ideal spectrum is obtained. The spectrum is close to the Lorentzian spectrum, and the resolution limit, in principle, disappears. We experimentally evaluated S-BOTDR and proved that we attained a 10-cm resolution that exceeds the resolution limit of conventional BOTDR.

A portion of the results presented in this paper was obtained during a joint project with Mitsubishi Corporation, with financial support of Japan Oil, Gas and Metals National Corporation (JOGMEC). This support is gratefully acknowledged.

In this appendix, we derive _{B}(_{f}]. The integral interval can then be replaced by [−∞, ∞], and we have:
_{B}(_{g}

The frequency integral of the PSF _{T} and radius Δ_{T} of Ψ_{T}(^{2} norm, and _{h}_{h}_{F}(_{f̂}_{ĥ}, σ_{f̂}_{ĥ}_{f}_{f}_{f}_{h}

The synthetic spectrum of S-BOTDR _{s}(^{(}^{j}^{)}(_{j}_{s}(^{(}^{j}^{)}(^{(}^{j}^{)}(_{B}(_{g}^{(}^{j}^{)} (

On the other hand, _{peak}, the mean of the spectrum at the peak, is approximated by

The maximization of the SFR of

The authors declare no conflicts of interest.

Block diagram of Brillouin optical time-domain reflectometry (BOTDR).

Pump lights and matched filters: (

Illustration of _{kl}, k, l_{kl}_{11}; (_{12}; (_{21}; and (_{22}.

Point spread functions: (_{1} = 1 ns; and _{2} = 50 ns).

Composition of the synthetic point spread function of S-BOTDR.

Block diagram of S-BOTDR.

Composition of the synthetic spectrum of S-BOTDR.

_{0}): the integral of the point spread function (PSF).

Signal-to-fluctuation ratio for various _{1} and _{2}.

Simulation condition: (

Simulation results of conventional BOTDR with D = 10 ns: (

Simulation results of conventional BOTDR with

Simulation results of S-BOTDR with _{1} = 1 ns and _{2} = 50 ns: (

The fiber under testing used in experiment: (

Experimental BOTDR results: (

Experimental S-BOTDR results: (

Numerical examples of _{opt}.

_{1} [ns] |
_{2} (ns) | |||
---|---|---|---|---|

| ||||

1 | 0.0896 | 0.0750 | 0.0656 | 0.0589 |

2 | 0.1499 | 0.1254 | 0.1096 | 0.0985 |

3 | 0.2020 | 0.1690 | 0.1477 | 0.1327 |

4 | 0.2493 | 0.2085 | 0.1823 | 0.1638 |

5 | 0.2930 | 0.2451 | 0.2143 | 0.1925 |

6 | 0.3342 | 0.2795 | 0.2444 | 0.2195 |

7 | 0.3731 | 0.3121 | 0.2728 | 0.2451 |

8 | 0.4102 | 0.3431 | 0.3000 | 0.2695 |

9 | 0.4458 | 0.3728 | 0.3260 | 0.2929 |

10 | 0.4800 | 0.4014 | 0.3510 | 0.3153 |