We present the magneto-optic method to measure the local birefringence of single-mode fibers. We use this method to study birefringence of various telecommunication fibers submitted to external twist. Analysis of measurements gives access to the linear and circular part of the birefringence. This allows to evaluate the stress-optic coefficient g with a good accuracy for each fiber.
© 2003 Optical Society of America
Transmission optical systems capacities over 1 Tb/s have been obtained at the end of 90’s . In this aim, fiber characteristics such as chromatic dispersion or effective area have been optimized . Polarization mode dispersion (PMD) has become, in this context, one essential parameter that degrades transmission systems performances. In industrial environment, attention is paid on process parameter to obtain low PMD fibers : ovality, stress and drawing conditions are the three major parameters to control. Fiber PMD also strongly depends on the way the fiber is conditioned. In fact, in both process and conditioning steps, unwanted external twist is usually applied on the fiber. It induces polarization mode coupling and circular birefringence  and causes the PMD to be different from the initial one with beneficial impact of mode coupling and detrimental one due to the circular birefringence dependence with wavelength . It is then of great importance to characterize and quantify the impact of twist on circular birefringence of a given fiber.
In this paper, we present a method to measure the twist-induced circular birefringence of single-mode optical fibers using the magneto-optic effect. We apply this method to extract the stress-optic coefficient of highly-doped fibers. Theoretical background is given in Section 2. The measurement method is presented in Section 3. Experimental results are presented and discussed in Section 4.
2. Theoretical background
A real single-mode fiber always exhibits linear birefringence |nx -ny |. This quantity corresponds to the difference between effective indices of the two orthogonally-polarized eigen-modes of the fiber. The birefringence is induced by geometric anisotropy of the core, residual stress or local bending [5, 6]. In the following, we will consider the birefringence to be the difference between propagation constant of the two orthogonal modes Δβl =2π|nx -ny |/λ (rad/m) where λ is the wavelength of light in vacuum.
Initial studies on externally-twisted single-mode fibers  have established the relationship between the twist rate and the corresponding induced birefringence. This birefringence is of the circular type (identical to optical activity) and its amount Δβc (rad/m) is given by Δβc =gγ where γ is the twist rate (rad/m) and g the stress-optic coefficient. It is generally admitted that typical values of g are between 0.14 and 0.16, depending on the core dopants [3, 7]. The twist also induces a rotation of the local birefringence axes leading to a coupling between polarization modes [3, 8].
These two effects of twist can be taken into account through a vectorial model that allows to study polarization mode dispersion in spun and/or twisted fibers . In order to find the magnitude of the vector birefringence in a birefringent fiber submitted to twist, it is generally convenient to consider a framework rotating at the twist rate γ. In this context, an input polarization state evolves periodically in an elliptical birefringent medium whose birefringence Δβ (rad/m) is :
The birefringence Δβ includes a linear part (Δβl ) and a circular part. In the circular part, gγ is the physically induced value of the circular birefringence whereas 2γ accounts for the rotation of the local axes and is comparable to a “geometrical value”of the circular birefringence . The key point is that Δβ, given by Eq. (1), is the local birefringence that can be extracted from our measurements.
The beat length Lb of the fiber is related to the birefringence Δβ through the relation :
Beat-length extraction techniques, like polarization-sensitive optical frequency-domain reflectometry  or polarization-sensitive optical time-domain reflectometry , have been recently used to measure circular birefringence in long single-mode fibers. In these experiments, the aim was to evaluate the impact of the stress-induced birefringence on PMD rather than to extract the stress-optic coefficient g. In this paper, we propose to use, in short fibers, a magneto-optic beat-length extraction technique to measure g with a good accuracy on highly doped fibers. These fibers present a Ge-doped central core with refractive index percent Δn relative to silica ranging from 0.9 to 1.6 % (see table 1). Different Ge-doped cores are considered here to evaluate the doping impact on the stress-optic coefficient.
3. The experimental setup
The beat length of the fibers is measured using a magneto-optic method, based on the Faraday effect. This method has already been successfully used to measure birefringence of polarization-maintaining fiber [10, 11], rare-earth-doped fiber  or photonic crystal fiber . The setup of the experiment is shown in figure 1. The principle of the method has already been detailed by the authors in references [12, 13]. In the following we will only summarize the basic principle.
The source is a linearly-polarized laser diode operating at 1555 nm. The half-wave plate allows to control the polarization state injected into the fiber. A coil (about 1000 turns over 1 cm long) produces an AC axial magnetic field of about 14×103 A/m and can be translated along the longitudinal direction of the fibre over a total length of 98 cm. The fiber is kept straight in a thin capillary tube in order to avoid bends and to get rid of mechanical perturbations when the coil is moving. The output signal is detected through a polarizer appropriately oriented. The signal is observed on an oscilloscope through a lock-in amplifier referenced with the AC bias current that produces the magnetic field. The signal evolves periodically as the coil moves along the fiber. The period is related to the beat length Lb of the fiber. Since the position of the coil changes linearly with time, Lb is directly extracted from the recorded signal. This local and non-destructive method permits, with our experimental apparatus, to measure beat lengths from ~1 cm to ~1 m with a good accuracy.
To study the influence of the stress-induced birefringence on the beat length of the fiber, a uniform twist is applied between points A and B over a total length of lt =1.37 m. For the tested fiber, beat length is measured at different twist rates.
4. Experimental results
Examples of signal detected are shown in Fig. 2, for fiber F1 (Δn=1.6 %), in the three following cases : 0, 4, and 8 turns of twist, applied between points A and B. We first note that each signal consists in a regular sinusoidal curve. The beat length is the period of these curves and is quoted in the figure. The second point is that the value of the beat length varies when the fiber is twisted, as expected from relation (1). In order to determine the evolution of the birefringence of the fiber as a function of the twist rate, we performed experiments from -21 turns to +24 turns for this fiber. Figure 3 represents the birefringence Δβ (calculated from Lb , using Eq. (2)) versus γ=2πN/lt where N is the number of turns.
The first comment is that, as expected by the theory, the birefringence of the fiber increases with the twist rate. The second point is that the minimum of birefringence does not occur at zero twist. This is due to an initial twist γ 0 induced when the fiber is positioned in the experimental apparatus.We have verified this point by removing and re-positioning the fiber and have obtained similar curves but with different values for the minimum-birefringence twist rate This proves that the value of γ 0 depends only on the installation of the fiber. The curve of Fig. 3 can give three informations : the intrinsic linear birefringence Δβl , the stress-optic coefficient g and the initial twist γ 0. To extract these parameters this curve must be fitted by Eq. (1) where γ0 is taken into account, i.e. :
Using a least squares method to fit experimental data of Fig. 3 with Eq. (3) we find the following parameters for fiber F1 : Δβl =51.86±0.14 rad/m, g=0.147±0.008 and γ 0=4.37±0.14 rad/m. Precision for these parameters is obtained using a boot-strap method  and is given for a 95% confidence interval. Bootstrap method allows to estimate a statistical distribution of the fitting parameters and thus their precision measurement. The fitting function of Δβ, also plotted in Fig. 3, shows the very good adequacy between relation (3) and our experimental data. The value of γ 0 has no real signification since it depends on the experimental configuration. The value of g is in good agreement with the theoretical prediction  and with the values of g usually taken [7, 9]. We note that g is extracted with a good precision by our method.
Following the same procedure than for fiber F1, we have performed the measurements for fibers F2, F3 and F4. Fibers F1 and F2 differ from fibers F3 and F4 by the refractive index percent Δn of their Ge-doped core (see table 1). The evolution of the local birefringence Δβ is represented in Fig. 4 for each fiber as a function of γ′=γ-γ 0. The parameters extracted from the fit of Δβ are summarized in table 1.
Results given in table 1 show that the refractive index percent Δn of the Ge-doped core influences the value of the intrinsic linear birefringence Δβl : high values of Δn increase the value of Δβl which is a result generally admitted . Taking into account the accuracy of our method and the range of values of Δn, it seems that Δn has no real impact on the value of the stress-optic coefficient g.
The function f=|(g-2)γ′| is also plotted in Fig. 4 for g=0.14. This function represents the asymptotes of Δβ when γ′ tends towards infinity, accordingly with Eq. (3). We note that fibers with low birefringence Δβl (F3 and F4) are closer to the asymptote than other fibers (F1
and F2). Indeed, for low-birefrigence fibers, the term |(g-2)γ′| becomes predominant in Eq. (3) with only a few amount of twist rate.
It is also interesting to note that the higher the birefringence Δβl is, the higher the twist rate γ can be, for extracting the beat length in good conditions. Indeed, it has been shown in reference  that the amplitude of the measured sinusoidal signals of Fig. 2 is proportional to the ratio of the square of the linear part of the birefringence over the cube of the elliptical (total) birefringence, i.e. Δ /Δβ 3. First, this ratio tends towards zero when γ increases. This leads to a degradation of the signal detected in the experiment and, therefore, limits the maximum number of turns of applied twist. Secondly, the ratio tends all the more rapidly towards zero since Δβl is low. This explains why, with a low-birefringence fiber, the values of applied twist rate are limited.
Note that a method of measuring the stress-optic coefficient g has been proposed by A. M. Smith  who found g to be smaller than that theoretically predicted . However, this method requires that a linear birefringence is minimal. On the contrary, the method proposed in this paper requires some linear birefringence for the magneto-optic effect to be visible. The method works even for highly birefringent fibers.
In summary, we have proposed an accurate method to measure the stress-optic coefficient of single-mode fibers. First, we have presented the magneto-optic method to measure the local birefringence of single-mode fibers submitted to external twist. We have verified that the theoretical expression of the birefringence versus the twist rate fits the experimental data. Then, we were able to extract from measurements pertinent parameters such as the linear intrinsic birefringence and the stress-optic coefficient of fibers, with a good accuracy.
Using this method, we have shown that, for the fibers tested, the dopant concentration has a negligible influence on the stress-optic coefficient while it increases the value of the intrinsic linear birefringence of fibers.
Provided the tested fiber is single-mode, this method can be used at any wavelength. It can therefore be used to study the wavelength dependence of g in order to evaluate the impact of the twist-induced birefringence on PMD.
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