# A Novel Method of Spectra Processing for Brillouin Optical Time Domain Reflectometry

^{1}

^{2}

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## Abstract

**:**

## 1. Theoretical Background

## 2. A Novel Method Theory

_{0}+ I ∙ Δf, P

_{i}], where i is a sample number in spectrum, varying from 0 to 2N, f

_{0}—minimal frequency of the spectrum, Δf—frequency scanning step, determined by the hardware discretization, and P

_{i}—registered backscattering power density at frequency f

_{i}= f

_{0}+ I ∙ Δf (Figure 1a). P

_{i}consists of two parts—the useful signal itself and the noise, P

_{i}= P

^{s}

_{i}+ P

^{n}

_{i}. In the absence of the noise component, i.e., with P

_{ni}= 0, finding the maximum power density would give the center frequency of the BGS f

_{b}(BFS) up to a sampling error, which is Δf/4 in average. However, it was shown in [15] that the presence of even small noise (SNR < 20 dB) leads to the fact that the error is mainly determined by the noise in the spectrum and weakly decreases with decreasing Δf.

_{i}= P

_{2N}

_{− i}and “backward and shifted” signal P

^{″}(k) as P

^{″}

_{i}= P′

_{i}

_{− k}, considering that P

^{″}= 0 if [I − k] is out of [0, 2N] range (Figure 1b). Here k is the signal shift, which can take all kinds of integer values from −2N to 2N.

^{″}signals could be described as follows:

^{s}and P

^{″}

^{s}to each other are, the larger it is. By plotting the dependence of X on the shift value k (Figure 1c) and determining at what shift k

_{0}value of X reaches its maximum, one can obtain the frequency corresponding to the BGS maximum P

^{s}: f

_{b}= f

_{0}+ (N − k

_{0}/2) · Δf, accurate to sampling error.

^{n}, different from zero, the second, third, and fourth terms (let us call them “parasitic terms”) in expression (1), will not be exactly zero, which, in turn, may give an additional error in determining BFS f

_{b}. However, there are two considerations in favor of the new method. First, an increase in the noise component should not lead to such catastrophic consequences as described in [15] where finding for the maximum of the signal P (FindMax) presented—a single burst of the noise component at a certain frequency does not lead to a significant change in the parasitic terms and, moreover, to their strong dependence on k. Secondly, when reducing the frequency scanning step while maintaining the range (i.e., simultaneously decreasing Δf and increasing N in such a way that (2N + 1) ∙Δf = const), the parasitic components should increase only proportionally to N

^{0.5}, like any random walks, while the “useful” first term increases faster, proportionally to N. Thus, it is reasonable to assume that, at least with a low signal-to-noise ratio and a small (more detailed) frequency scanning step, the proposed method could give good results, as required initially. Of course, both of the above considerations require experimental verification.

^{″}is given by [16]:

_{P}is a signal dispersion. Since all values in this expression excepting <P · P

^{″}> are independent from the shift k, the cross-correlation r will have the maximum value at the same k

_{0}as the convolution X—finding the convolution maximum is equivalent to finding the cross-correlation maximum. Therefore, the authors propose the name “backward correlation method” for the new algorithm.

## 3. Numerical Simulation

_{b}of the BGS was randomly selected. For the given scanning step and SNR, a spectrum was generated in accordance with [17]:

_{n}—amplitude of the noise component. ${P}_{i}^{n}$ was chosen randomly equiprobably from range R:

_{i}—FindMax), by traditional method applied after low-pass filtering of original spectrum (averaged), as well as in a described way (by the “backward correlation method”). The difference between the actual value of BFS and those found was a single-measurement error for each of the methods. Then a new center frequency for each spectrum was selected and the process repeated. Over 10,000 spectra were generated. The results for the frequency scanning step of 1 MHz are shown in Figure 2. We have also added data for Lorentzian fitting technique, according to [18].

## 4. Experiment

^{−4}for FindMax and 9.1 × 10

^{−4}for the “backward correlation” method) indicate that with this SNR, the new method determines the BFS more accurately.

_{x}and n

_{y}—the refractive indices of the slow and fast axis of the fiber at a certain point, respectively; λ—radiation wavelength; f

_{x}and f

_{y}—BFSs of the slow and fast axis at a certain fiber spatial point; and V

_{x}and V

_{y}—average speeds of sound in the slow and fast axis of the fiber. It should be noted that the use of exact values of V

_{x}and V

_{y}is an important condition for high-quality birefringence observation. These values were calculated using the core refractive index of the single-mode fiber, which is the output of one of the polarizers.

^{−4}, which coincides with the fiber passport, while the algorithm with finding the maximum shows errors of up to 30–40% of the total values amount. There are also four areas where two methods demonstrate similar patterns of birefringence changes; but there are even more fragments on the graph where the data diverge significantly—perhaps this is due both to local variations of the SNR on the spectra corresponding to different fiber coordinates, and to the appearance of significant artifacts on them, significantly changing the symmetry of the spectral components.

## 5. Discussion and Future Work

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A schematic diagram combined using real data, describing the new method operation principle: (

**a**) forward spectrum with Brillouin frequency shift (BFS) detected using FindMax procedure; (

**b**) backward spectrum with its shifted copy; and (

**c**) backward correlation function with its BFS.

**Figure 2.**Comparison of different methods for finding the maximum of the Brillouin gain spectra (BGS) for frequency scanning step 1 MHz: Absolute error in finding the frequency (MHz) versus the signal-to-noise ratio (SNR) of the original spectrum, dB. Data on Lorentzian fitting is taken from [18].

**Figure 4.**Comparison of the results of the two methods on a real Brillouin optical time domain analyzers (BOTDA) trace.

**Figure 5.**An experimental set-up for obtaining the distribution of birefringence in a PM optical fiber.

**Figure 8.**Distribution of modal birefringence along the length of an anisotropic optical fiber. Marked by circles are the four regions where the birefringence changes for both methods have details which look alike.

Parameter | Units | Value |
---|---|---|

Core diameter | μm | 7.0 |

Numerical aperture (average value for the both axes) | - | 0.22 |

Attenuation at 1550 nm | dB/km | <1.5 |

Birefringence measured by the spectral method [18] | 8.0 × 10^{−4} | |

h-parameter | m^{−1} | <10^{−5} |

**Table 2.**A comparison of all discussed methods obtained from data simulated with the 1 MHz scanning step (SS).

Method/Parameter | Findmax | Averaged Data | “Backward Correlation” | Lorentzian Fitting |
---|---|---|---|---|

BFS Error (MHz) at: | ||||

SS: 1 MHz; SNR: 15 dB | 1.7 | 1.8 | 0.5 | 0.2 ^{1} |

SS: 1 MHz; SNR: 10 dB | 2.5 | 3.0 | 0.7 | 0.6 ^{1} |

SS: 1 MHz; SNR: 5 dB | 3.4 | 4.2 | 1.6 | 1.9 ^{1} |

SS: 1 MHz; SNR: 3 dB | 3.9 | 5.2 | 2.0 | 2.4 ^{1} |

Other Parameters: | ||||

Spectra non-symmetry influence | low | low | medium | high |

Estimated computational costs | low | low | medium | medium/high ^{2} |

Spectra non-symmetry influence | high | high | medium | high |

Overall low SNR influence | high | high | low | low |

^{1}Approximate value from Figure 9 at [18].

^{2}Depends on the exact algorithm.

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**MDPI and ACS Style**

Barkov, F.L.; Konstantinov, Y.A.; Krivosheev, A.I.
A Novel Method of Spectra Processing for Brillouin Optical Time Domain Reflectometry. *Fibers* **2020**, *8*, 60.
https://doi.org/10.3390/fib8090060

**AMA Style**

Barkov FL, Konstantinov YA, Krivosheev AI.
A Novel Method of Spectra Processing for Brillouin Optical Time Domain Reflectometry. *Fibers*. 2020; 8(9):60.
https://doi.org/10.3390/fib8090060

**Chicago/Turabian Style**

Barkov, Fedor L., Yuri A. Konstantinov, and Anton I. Krivosheev.
2020. "A Novel Method of Spectra Processing for Brillouin Optical Time Domain Reflectometry" *Fibers* 8, no. 9: 60.
https://doi.org/10.3390/fib8090060