1. Introduction

For years, manufacturers of optical fibers have tried to improve the manufacturing processes of optical fibers, to avoid imperfections in them [1–3]. However, the fibers exhibit internal disturbances such as core ellipticity and internal stress, producing unwanted birefringence. Fiber data sheets do not include the value of this residual birefringence. Therefore, it must be evaluated, since it introduces differences in the propagation constants of the orthogonal polarization modes, whose dispersion limits the performance of high bit rate communication systems [4], and modifies the state of polarization (SOP) of the signal, deteriorating the measurement accuracy of optical fiber sensors [5,6]. Moreover, in recent applications involving biological or medicine sensing based on the use of polarization effects, the internal architecture of single-mode fibers is modified [7,8]. Here again, their final birefringence requires to be measured using procedures adapted to each specific case.

Due to the vector nature of light, to study the physical effects that modify the performance of fiber devices, it is necessary to define a reference framework to be able to control and evaluate the SOP of the signals that intervene in an optical system. Moreover, to develop an accurate description of the problem, we should use a set of relationships coherent with each other. Concerning optical fibers, this set of equations relate the magnitudes and phases of the electric field components, the Jones vector, coherency matrix, Stokes parameters, Poincaré sphere, polarization ratio, and the azimuth and ellipticity of the polarization ellipse. This set of mathematical relations consistent with each other was developed in 1979 (Nebraska Convention) [9], and using them; we can describe the alignment between the fiber birefringence and the state of polarization of light.

Techniques developed to evaluate the birefringence of these low-birefringence fibers must consider that the characterization of the ellipticity change of the output radiation alone does not supply enough information on the birefringence parameters of the sample [10]. Therefore, these methods should take into account the evolution of polarization under real conditions. In the beginning, to describe the fiber birefringence, it was considered that linear birefringence was the dominant contribution, and they included circular birefringence only for twisted fibers. Another rarely mentioned contribution is of fast-axis rotation [1], a sort of low residual torsion [11,12]. Following these ideas, measurement techniques typically assumed the samples presented a residual linear birefringence. However, some authors proposed the fiber birefringence could be described as a hybrid (non-homogeneous) combination [13] of a linear retarder and a circular retarder [3] or as a homogeneous combination of linear and circular birefringence (elliptical retarder, [14,15]).

To select the type of birefringence to be used, using classical polarization optics, we find that for a short fiber length, attenuation can be considered negligible, and under this hypothesis, the birefringence matrix should be unitary; i.e., the fiber behaves as a retarder [15]. According to polarization optics, homogeneous retarders are the most straightforward description for polarization retardation. Using Mueller matrix description and the Cartesian coordinate system associated to Stokes vectors S1, S2, S3, these matrices correspond to rotation operators whose rotation axis would be located, using the Cartesian coordinate system associated to Stokes vectors S1, S2, S3, with its axis lying on 1) the S1-S2 plane, 2) the S3 axis, or 3) the rotation axis could form an angle different from 0° or 90° with the S1-S2 plane. Using the Poincaré sphere representation, it is easy to understand that the first rotation axis describes a linear birefringence, the second one, a circular birefringence, and the third one, an elliptical birefringence, being elliptical birefringence the general case of homogeneous retarders.

In 2005 Treviño et al. proposed an evaluation procedure allowing to verify if the fiber retardation corresponded to a homogeneous retarder [16]. It is possible to determine if the type of retardation presented by the fiber under study corresponds to a homogeneous retarder, using this method, avoiding to assume apriori knowledge of the fiber birefringence, as is usually done.

In this work, we verified that the fiber exhibited an elliptical birefringence. Under this scope, we present a simplification of the methods used to determine the polarization eigenmodes [14 and 17] of an elliptical retarder. In both references, to measure the elliptical angle of fiber birefringence, it is necessary at first to determine the azimuthal angle of the fast polarization axis. This critical step allows eliminating the effect of birefringence axes orientation on the measurement [6]. Furthermore, the evaluation of the ellipticity angle required an additional experiment based on the knowledge of the polarization eigenmode azimuth. The method we propose allows the simultaneous estimation of the angles related to the polarization eigenmode (ellipticity and azimuth) and the total retardation angle modulus-π using a single experiment, reducing experimental complexity and improving measurement precision.

Typically, birefringence evaluation is limited to the measurement of polarization beat length [18–20], an estimation of the distance over which the orthogonal polarization modes experience one complete cycle of change. Generally, to evaluate this fiber length, researchers use complex optical systems, involving various optical devices. So without a well-defined frame of reference, they cannot tell if the measurement is being affected by the additional optical devices. Furthermore, using these techniques, the orientation of the fast axis of the birefringence of the fiber is not known.

2. Method

The experiment consists in measuring the output SOP in an SMF-28e fiber sample by varying the azimuth angle of the linear input SOP from 0 to 180° (0.5° increments). As mentioned above, it is necessary to generate an SOP to control the input SOP to the fiber sample; to achieve this, we have the following components. In Section 1a), a tunable Hewlett Packard 8168C laser diode was used as a light source and following this, a polarization controller was placed. The ears of the controller were manipulated to generate a circular SOP. Behind, a polarizing prism (calcite crystal) was aligned with the system to modify the circular SOP into a linear SOP with constant power. In the experimental setup, both the prism polarizer and the coupler between the prism and the optical fiber (C2) are automatically controlled by mechanical assemblies that allow us to remove and reposition them in order to create the reference frame used by the polarization analyzer, keeping the alignment. Furthermore, the polarizing prism is fixed to an automatically operated rotation mount that controls the prism axis orientation, in order to define the azimuth angle of the input linear SOP. In Fig. 1, the dotted line indicates that light is coupled by air to the helically wound fiber optic sample shown in Section 2. Finally, the output SOP is measured with an Agilent 8509C polarization analyzer.

 

Fig. 1. Experimental configuration used to characterize a helically wound SMF-28e sample.

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Two samples of SMF-28e fiber of ∼1 m long with different jackets (0.9 Kevlar and 3 mm yellow cable), the SMF-28e fiber attenuation is <0.18 dB/km at a wavelength of 1550 nm. The fiber samples were wound helically, sample 1 in a coil of 5.2 cm in diameter and sample 2 in a coil of 7.3 cm, so that the birefringence throughout each fiber was uniform. To guarantee that the contribution of the polarization introduced by the torsion was negligible, we used a helix pitch equal to the diameter of the fiber cable [21]. Sample 1 and 2 were measured using two wavelengths; 1550 nm and 1560 nm. In all measurements the signal power was −32.2 dBm (power at the fiber input), it is worth mentioning that for each measurement, a reference frame was generated to ensure that the input SOP was linear.

2.1 Mathematical analysis

Assuming the fiber behaves as an elliptic homogeneous retarder and using Mueller calculus, the output state of polarization (SOP) is

In general, the azimuth angle of the fast birefringence axis of the retarder forms an angle $\theta$ with the laboratory reference system, a consideration not included in Refs [14] and [17] that simplifies our measuring procedure. Therefore, Eq. (1) must be

3. Measuring procedure

This article analyzes the evolution of the output SOP [Eq. (6)] when plotting Stokes vectors in the Poincaré sphere. When the azimuth angle of the linear input SOP varies from 0 to 180 °, the output SOP draws a great circle in the Poincaré sphere (Figs. 2–3), which intersects at two points the equator of the sphere of Poincaré. These crossing points correspond to linear output SOPs. The main objective of this work is to determine with a single experiment θ, μ, and δ. As in Refs [14,17], the first parameter to be calculated is θ (azimuthal angle of the fast birefringence axis), which is calculated through the following equation

We must consider the quadrant in which ${{\mathrm{\alpha}}_{\textrm{out}}}$ is located to avoid an incorrect result, e.g. ${{{{\textrm {S}}_2}} \mathord{\left/ {\vphantom {{{{\textrm {S}}_2}} {{{\textrm {S}}_1} ={-} {{{{\textrm {S}}_2}} \mathord{\left/ {\vphantom {{{{\textrm {S}}_2}} { - {{\textrm {S}}_1}}}} \right.} { - {{\textrm {S}}_1}}}}}} \right.} {{{\textrm {S}}_1} ={-} {{{{\textrm {S}}_2}} \mathord{\left/ {\vphantom {{{{\textrm {S}}_2}} { - {{\textrm {S}}_1}}}} \right.} { - {{\textrm {S}}_1}}}}}$, but they correspond to different values of the azimuth angle. Also, the angle sign (positive or negative) should be taken into account. If the angle evolves counterclockwise, it is considered as positive.

To obtain μ and δ with data collected from the same experiment used to determine θ, we use some specific points of the measured great circle.

To determine μ, we use the point at which ${\mathrm{\alpha}} = {\mathrm{\theta}}$. For this point, Eq. (6) is reduced to the following expression,

Finally, to determine $\delta$, we use a linear ${\textrm{S}_{\textrm{out}}}^{\prime}$, i.e. ${{\textrm {S}}_{\textrm{3out}}} = 0$, and from Eq. (6),

4. Results

The fiber samples (∼1 m long) used in this work were standard fibers (SMF-28e) with different jackets (0.9 μm Kevlar and 3 mm yellow cable). The output SOP for both fiber samples was measured for two different wavelengths (1550 nm, and 1560 nm) varying the azimuth of the input linear SOP from 0 to 180° (0.5° increment). Representing the output SOP on the Poincaré sphere, we obtained a circular path whose position depends on the value of the input azimuth angle (see Figs. 2 and 3). In all cases, these paths are major circles, since to obtain them, we applied a rotation operator [Eq. (1)] to the circle corresponding to the Poincaré sphere equator. In Figs. 2–3, the comparison between the experimental results and those calculated using the values determined for θ, μ, and δ modulus-π in Eq. (6) are shown. In the figures shown below, subscript E indicates experimental results and subscript S, theoretical results.

 

Fig. 2. Comparison between the simulation and the experimental result of the evolution of the output SOP in a SMF-28e fiber sample (900 µm jacket) using a linear input SOP. The sub-index E indicates experimental results, and sub-index S designates theoretical results.

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Fig. 3. Comparison between the simulation and the experimental result of the evolution of the output SOP in a SMF-28e fiber sample (3 mm jacket) using a linear input SOP. The sub-index E indicates experimental results, and sub-index S designates theoretical results.

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The birefringence parameters measured for sample 1 (900 µm jacket) were: for 1550 nm, θ =-3.99°, μ=1.51° and δ =-143.19°, and for 1560 nm, θ =-4.45°, μ = 2.08° and δ =-142.58°. While for sample 2 (3 mm jacket), the birefringence parameters were: for 1550 nm, θ =-8.49°, μ=10.28° and δ = 95°; for 1560 nm, θ =-8.63°, μ = 10.73° and δ = 93.49°.

Based on the results shown in Figs. 2 and 3, we can see that the coincidence between the experimental results and the simulation is excellent for the two signal wavelengths used. For samples 1 and 2, the azimuth angle of the fast birefringence axis, ellipticity, and retardation angle differ by less than 1° for both wavelengths. As expected, in both cases, these values are similar due to the low polarization mode dispersion of this type of fiber. However, we observe that using a larger diameter coil, the fluctuation of the signal decreases.

5. Conclusions

The non-destructive measurement procedure here proposed is simple and allows an accurate determination of the parameters required to characterize the birefringence of the fiber. Using the same set of data produced by an azimuthal scanning of the linear SOP of the input signal, it is possible to calculate the azimuth angle of the fast axis, ellipticity angle, and total retardation angle modulus-π. This outcome results from our theoretical description based on considering the misalignment of the retarder fast birefringence axis with the laboratory reference system, using relations that satisfy the Nebraska convention. The experimental verification performed for two commercial optical fiber samples, SMF-28e, proves the accuracy of this technique.