Optical Time-Domain Reflectometry Methods For Measurement Of Distribution Of Excess Optical Fiber Length In Fiber-Optic Cable Loose-Tubes
In order to ensure the constancy of the fiber curvature in loose-tubes, the systems of control automation and control of excess length in the manufacture of loose-tubes are used [7–9]. However, the studies  have shown that during the subsequent technological operations of cable manufacture, the distribution of the fiber curvature in the loose-tube cable tubes can vary. These phenomena determine the urgency of the problem of measuring the distributions of excess fiber length in the loose-tubes of the finished product, i. e., delivery length of the cable.
Measurements of distributions of excess fiber length estimates along the cable can be performed indirectly based on measurements of mechanical stresses in the fiber, polarization characteristics, or attenuation of the fiber. Optic time-domain reflectometric methods are of the particular interest.
The authors of the paper  demonstrate the possibility of detecting mechanical stresses on bends using BOTDR – brillouin pulse optical time-domain reflectometer. At the same time, [12, 13] focus on the existing problems of measuring mechanical stresses at the fiber bends using BOTDR. The main reason limiting the mass application of brillouin optical time-domain reflectometers is their high cost. According to the authors, limited number of optical time-domain reflectometers is in operation up to date. The commercially available polarization pulse optical time-domain reflectometers (POTDR) for measuring PMD have low distance resolution and a fairly high cost. Specialized kits for the analysis of distributions of excess fiber length are not known from the results of measurements of polarization characteristics.
From the point of view of mass application, the most interesting are the reflectometric methods of excess fiber length estimation based on the measurement of the losses in the fiber, which can be implemented with the help of widely applied conventional pulsed backscattering optical time-domain reflectometers (OTDR).
It should be noted that in the design of fiber and cable structures, bend losses are tend to be minimized. Therefore, the approach under consideration has significant limitations. In particular, it is not applicable when using fibers with reduced bend loss for OC (ITU and T G 657 recommendation). But since the standard stepped fibers (ITU and T G 652 recommendation) are still widespread, it can be of interest for a number of practical applications.
The estimates of the fiber bending radius in the loose-tube, the curvature and the excess length are unambiguously related.
The fiber curvature is uniquely related to the bending radius R [5, 6]:
For the helical fiber trajectory (Fig. 1) in the loose-tube, the excess fiber length is determined [6, 14] as
Here, the fiber length is
where rm, p is the radius and pitch of the helicoid, correspondingly. For rm<<R, it can be approximated to assume that
Another method is known for analyzing the excess fiber length in optical cable  and it is based on fiber attenuation measurements. In accordance with this method, measurements of the backscattering characteristic of the optical fiber in the cable are performed at several wavelengths λi in advance. Based on the results of the analysis of these characteristics, the distribution of estimates of the increase in losses due to the fiber bending is determined along the length of the cable, Δα(z, λi). To exclude changes in the fiber attenuation non-connected to bends, the dependence of the Rayleigh losses on the wavelength is applied, namely, their proportionality to 1/λ4. The method for processing the backscattering characteristics for determining the bend loss in  is not disclosed, but it is obvious that it is based on the dependence of these losses on the wavelength and bending radius, and in particular on the increase in fiber bend loss with increasing wavelength and decreasing bending radius [16, 17]. The bending radius, depending on the bending loss, is calculated according to the formula :
where η(λi) is the fiber parameter, and RC is the value of the bending radius, beyond which the bend loss can be neglected. The parameters η(λi) and RC are determined either experimentally or calculated .
It is suggested in [18–20] to localize in the cable sections with areas with increased fiber curvature to compare the results of measurements of the fiber backscattering characteristics, performed at positive and low negative temperatures. Thus, in , a method for measuring the distributions of excess fiber length in a loose-tube for the delivery length of the optical cable is suggested, consisting in the fact that the reel with the cable delivery length is exposed to low negative temperature, the fiber backscattering characteristics are measured at positive temperature and at low negative temperature, followed by determination of the corresponding distributions of fiber attenuation ratio along the cable. Then, the distribution of attenuation due to fiber bends is determined as the difference in attenuation ratios measured at negative and positive temperatures. The estimates of the bending radii are determined further at the appropriate negative temperature, solving the equation:
where Cg(λ), g(λ) are the parameters determined either experimentally or calculated. Expression (12) 7 is in fact the form of writing the well-known Marcuse formula for bend losses in a fiber with a stepped profile of refractive index . Knowing the distribution of the fiber bending radii in the loose-tube along the length of the cable, the distributions of the curvature and excess fiber length are determined according to (1) and (4).
To calculate the estimates of the fiber bending radius in the loose-tube, using the well-known bend loss values, one can also use well-known Hanson formula  to determine Δα(λ, Т). It is well known [1–2, 6] that changes in excess length are directly proportional to temperature changes:
where T, Tn is the current and some reference temperature values, respectively; ΔαT is the absolute difference of the coefficients of linear expansion of the material of the loose-tube and quartz glass. Therefore, we obtain:
where A, B are the parameters determined experimentally or theoretically.
Thus, we have three methods for determining excess fiber length in the optical cable loose-tube according to the results of measurements of fiber bend losses: method 1, based on formulas (5) and (6), method 2, based on formulas (5) and (7), method 3 using formula (9). It is of interest to estimate the potential for determining the distribution of excess fiber length from the results of fiber attenuation measurements, as well as the correctness and inaccuracy of the listed methods.
As a criterion for evaluating the method, it is suggested to use the linear nature of the dependence of the excess fiber length in the loose-tube on temperature. And, accordingly, the errors of the methods are estimated as the approximation error by the linear dependence of the curve obtained experimentally. To implement this approach, a reel with optical cable was placed in a climatic chamber where it was exposed to temperature cycles. The optical cable with a length of 577 m with standard stepped optical fibers such as SMF‑28 was exposed to testing. The loose-tube cable consisted of four loose-tubes with four fibers each. The temperature profile is given in Fig. 2. At each step, after the reel was exposed at a set temperature for at least 4 hours, the backscattering characteristics of the cable fiber were measured. As an example, separate characteristics are given in Fig. 3.
The attenuation ratios were determined as a result of processing the backscattering characteristics by a conventional technique based on least-squares approximation. The changes in the fiber attenuation ratios at the bends were calculated as the difference between the found values of the attenuation ratios at the corresponding negative temperature (–30 °C, –40 °C, –50 °C, –60 °C) and positive temperature of 20 °C. Then, using one of the methods described above the estimates of the excess fiber length were obtained, and the experimental curves for the dependence of the excess fiber length in the loose-tube on temperature were plotted. The experimental curves were approximated by a linear relationship and the error of this approximation was estimated, which was assumed to be the error of the method.
The final results showed that the experimental curve plotted using method 1 cannot be approximated by linear dependence. The examples of experimental dependences obtained using method 2 and method 3 are given in Fig. 4, 5. The error of method 2 did not exceed 1–5% in all cases. The error of method 3 did not exceed 10–15% in all cases.
Thus, the obtained results demonstrate the possibility of estimating the distributions of excess fiber length in loose-tubes along the cable length by conventional optical time-domain reflectometers based on the results of attenuation measurements. Preference should be given to method 2, the error of which does not exceed units of percent. For a rough estimate, method 3 can also be used, which does not require the solution of a nonlinear equation, and provides an error of up to 10–15%.
However, it is obvious that to obtain correct results, it is necessary to reduce the errors of attenuation measurements at bends. In order to achieve this, in particular, the measurements should be performed at lower temperatures at a longer wavelength. Hence, it follows that this method cannot be applied to optical cables with the fibers corresponding to G.657 recommendations. For cables with such fibers, the development of specialized OTDRs is required, for example, based on the relationship between the curvature and the polarization characteristics of the fibers.
 Editor’s note: In the classification of optical fiber, the standard rule is the recommendation of International Telecommunication Union (ITU), "T" in "ITU-T" means Telecommunications (International Telecommunication Union – Telecommunication standardization sector).
G.657 "Characteristics of single-mode optical fiber and cable insensitive to macro-bend losses, for use in access networks"
G. 652 "Characteristics of single-mode optical fiber and cable". Fibers of this type are optimized for transmission at a wavelength of 1310 nm.