## Abstract

Using the Poincaré sphere and wavelength scanning it is possible to determine if the fiber birefringence corresponds to that of a linear, circular or elliptical retarder, as well as to obtain an approximate measurement of the polarization beatlength. This method is useful for low birefringence single-mode fibers. It is applied to erbium-doped fibers.

© 2005 Optical Society of America

## 1. Introduction

Real fibers present various internal perturbations, such as core ellipticity and internal stress that cause characteristic differences of the propagation constants of the orthogonal polarization modes. For short distances where statistical polarization mode coupling can be neglected, the polarization properties of single mode fibers are often described by its phase birefringence, characterized by the birefringence beat length *L*_{b}
. In addition to this parameter, to be able to describe the evolution of polarization along the single-mode fiber it is almost generally assumed that the fiber residual birefringence is linear [1–3]. Linear birefringence is considered to be so dominant that circular birefringence or fiber twist are completely neglected and, only when the fiber is twisted it is assumed that in addition to the residual linear birefringence there is a circular birefringence contribution. When the twisting rate is high it is considered that circular birefringence can become dominant [4,6]. Another approach used by some authors is to treat the fiber as an elliptic retarder [7–10]; i.e. to assume that the fiber presents simultaneously linear and circular birefringence. Circular birefringence can be produced by optical activity and/or axial rotation (twist).

In this work we use the Mueller matrix formalism and wavelength scanning to identify the type of retarder suitable to describe the fiber birefringence, as well as to determine the approximate value of the polarization beat length. Wavelength scanning has been previously used to evaluate the beat length of high birefringence fibers with a low birefringence dispersion [11,12]. We work with low birefringence fibers with high birefringence dispersion (erbium-doped fibers). We present experimental results of the assessment of the residual anisotropy of two commercial erbium-doped fiber samples.

## 2. Cutback and wavelength scanning

In the experiment, the input polarization is fixed (it is usually linear). When the length of the fiber is varied (cutback technique) or the input signal wavelength is modified (wavelength scanning method) the output polarization state moves along some trajectory on the Poincaré sphere [11–13]. The phase change introduced by the birefringence is

where Δ*n* is the birefringence (linear, circular, or elliptical), λ is the signal wavelength and *s* is the fiber length. For Φ=2π the input state of polarization is restored, and the length *s=L*_{b}
associated to this phase change is the polarization beat length.

Equation (1) shows us that cutback and wavelength scanning methods produce similar results if the wavelength dependence of birefringence can be neglected. To satisfy this requirement in this work we use closely spaced signal wavelengths to perform the measurements [14]. In this case wavelength scanning is a better option because in practice it is easier to keep the same orientation of the fiber. In addition to it, it is a non-destructive technique.

## 3. Birefringence assessment using the Poincáre sphere

Since the characterization of the ellipticity of the output radiation alone does not supply enough information on the birefringence parameters of the sample [15], numerous analytical and graphic methods have been devised, based on the evolution of polarization under different conditions. Among them, the graphic methods permit a clear insight, even in complex situations. In this section, we simulate the trajectory described on the Poincaré sphere by the output polarization state, when the signal input polarization is linear (Fig. 1). We assume that the cutback technique or the wavelength scanning method have been used to modify the phase change [Eq. (1)].

To work with the Poincaré sphere we represent the electric field using Stokes vectors and the fiber birefringence is described using Mueller matrices. A linearly polarized signal from a monochromatic light source is launched into the single-mode fiber sample. At the fiber output, the Stokes vector of the signal is

where **M** is the Mueller matrix (fast axis azimuth equal to zero) that describes the fiber birefringence and the input linearly polarized signal with azimuth angle φ is

where *t* indicates transpose.

The explicit form of matrix **M** is given in Table I for each type of retarder. Substituting Eq. (3) and the proper Mueller matrix in relation (2), we find that for each type of retarder the output state of polarization is given by the relations shown in Table II. Equations (4) to (6) in Table II were used to simulate the path described on the Poincaré sphere when the fiber length or the signal wavelength are modified; i.e. when the phase retardation changes. The results we obtained are shown in Figs. 2 to 5.

*Linear retarder*. As the phase shift γ between the two polarization modes of a fiber sample with linear birefringence is increased, only components *S*
_{2} and *S*
_{3} in Eq. (4) are modified. They describe a circle of radius |sin 2φ|, centered on (*S*
_{1},*S*
_{2},*S*
_{3})=(cos2φ,0,0); i.e. it lies on a plane perpendicular to the axis labeled as *S*
_{1}. When φ=±45° the radius takes its maximum value; while for φ=0° or ±90°, the radius of the circle is equal to zero for any value of the retardation angle γ (i.e. for any fiber length or for any signal wavelength). Fig. 2 presents the results obtained for different orientations of the input polarization state. We used continuous lines for φ=5°, 10°, 15°, 20°, 25°, 30° and 45°, and dotted lines for φ=65°, 70°, 75°, 80° and 85°. In particular, for φ=0°, **S**
_{out}=(1,1,0,0)^{t} and, when φ=±90°, **S**
_{out}=(1,-1,0,0)^{t}. These two opposite points on the Poincaré sphere are the intersections of the symmetry axis of any trajectory with the Poincaré sphere and correspond to the principal polarization states for a linear retarder with azimuth angle equal to zero [17].

*Circular retarder*. For a circular retarder, as the retardation δ is increased, the output Stokes vector describes a major circle that matches the equator of the Poincaré sphere (Fig. 3). We can notice from Eq. (4) that *S*
_{3}=0 and ${S}_{1}^{2}$+${S}_{2}^{2}$=1 for any value of φ. Hence, this path does not depend on the orientation of the linear input polarization state of the signal with respect to the sample. The symmetry axis of this trajectory crosses through the north and south poles of the sphere (principal polarization states for a circular retarder) [17].

*Elliptical retarder*. For an elliptical retarder the path described on the Poincaré sphere by the polarization state of the signal as it propagates along the fiber, is also a circle (fig. 4). These circular paths share a common axis of symmetry. To analyze these trajectories [Eq. (5)] it is convenient to use the rotated axes (*S*
_{1}’, *S*
_{2}, *S*
_{3}’), where the rotation angle with respect to the *S*
_{2} axis is (π/2+σ). In terms of the rotated coordinates, the position on the Poincaré sphere of the output state of polarization is

As the signal propagates along the fiber, the change in the retardation angle δ modifies components *S*
_{2} and *S*
_{3}’ in Eq. (7). Since ${S}_{2}^{2}$+*S*
_{3}’^{2} is constant, the trajectory is a circle. The radius of the circular path is

The inclination of the axis of symmetry of these circular paths depends only on the value of σ (*tan*σ is the linear to circular retardation ratio [10].) Since for a given fiber sample the value of σ remains constant, we can see from Eq. (8) that for a specific fiber, the radius of each circular trajectory will be determined by the azimuth angle of the input linear polarization. This result is illustrated in Fig. 4. In this figure we can notice that when the input signal is linearly polarized, the minimum value for these radii is obtained for φ=0° and using Eq. (8) we get that *r*_{min}
=|cosσ|. For the simulation in fig. 4, σ=10° and, we used continuous lines for radii ranging from 0° to 40° (10° steps) and dotted lines when φ varied from 70° to 85° (5° steps). Figure 5 illustrates the change in the value of the minimum radius (φ=0°) when the elevation angle ζ, between the symmetry axis and the equator plane, varies from to 5° to 30°. We can also notice from Eq. (8) that for φ=45° the trajectory would be a major circle.

## 4. Experiment

Using the set-up shown in Fig. 1, carefully aligned following the procedure reported in reference 18, we scanned the sampling wavelength from λ=1511nm to 1571nm using 6nm steps. The erbium-doped fiber sample was kept straight without introducing any torsion. The elongation was controlled using a ~2Nt weight [18]. We used two commercial single-mode erbium-doped fibers: Photonetics EDOS 103 (length 1.63 m) and INO-NOI 402K5 (length 1.61 m).

The path described by the output Stokes vectors obtained for each wavelength scanning experiment is shown as a curve (continuous or dotted) in Figs. 6 to 9. In these figures we used the same color for equivalent orientations of the input linear polarization azimuth angle (such as 0°, 180°, 360°). In Fig. 6 the results for φ=10° to 40° (10° steps) are represented by continuous lines and we used dotted lines for input azimuth angles between 190° and 220° (10° steps). When the experiment was finished the sample was removed from the optical set up and allowed to hang free, holding it from one connector. Every time the same fiber end was placed at the input position.

The lightwave polarization analyzer requires an operation wavelength to define the reference frame (laboratory system used to define the azimuth angle of the polarization state); since we are working with several optical wavelengths within a wide spectral band, to minimize systematic errors we defined the reference frame using a wavelength in the center of our working range (λ=1541nm). The degree of polarization of the output signals varied between 97% and 100%, hence the depolarizing effect of absorption through the subsequent fluorescent emission (typical of erbium-doped fibers) is negligible. Despite the fact that the wavelength scanning results shown in Figs. 6 and 8 for some azimuth angles of the input linear polarization show clearly that both erbium-doped fibers behave as elliptical retarders, these results do not match the theory. A closer analysis of the trajectories followed by the output state of polarization shows that the paths described on the Poincaré sphere are not plane curves, as can be seen in Figs. 7 and 9, comparing segments with the same color.

The simulation results shown in section 3 indicate that for linear retardation the elevation angle is ζ=0° and the minimum value of the radius is zero (for φ=0°). For circular retardation, ζ=90° and for any value of the azimuth angle of the input linear polarization φ, the trajectory is a major circle. For elliptical retardation 0°≤ζ≤90° and the minimum value of the “circular path”, given by Eq. (8) is different from zero. In this work, to determine the elevation angle ζ we take into account that the circular path described on the Poincaré sphere is normal to a line (axis of symmetry) that forms an angle with the plane of the equator equal to ζ; where *tan*(*ζ*/2) is the ellipticity [15]. The cross product of the Stokes vectors of two consecutive sampling wavelengths, obtained using the same input linear polarization, **L**
_{1,2}(φ)=**S**(λ_{1},φ)×**S**(λ_{2},φ), is normal to the symmetry axis of the circular path. Hence, the angle between **L**
_{1,2}(φ) and S_{3} axis is equal to the elevation angle ζ, between the symmetry axis and the plane of the equator (Fig. 5).

For the INO NOI sample $\overline{\zeta}$ =-3.1°±2.4°, and for the Photonetics sample $\overline{\zeta}$ =86°±1.1°. We can notice in Figs. 6 to 9 that the apparent inclination of the axis of symmetry in each one of these figures is different to these values. This effect is produced by birefringence dispersion. When a wide spectral range is used, the contribution of the wavelength dependence of birefringence is no longer negligible.

The results obtained for the average value of the elevation angle, calculated using consecutive values of the sampling wavelength are shown in Figs. 10 and 11 for INO NOI and Photonetics samples, respectively. The quasiperiodic behavior of the elevation angle is similar to the result reported by Heffner [19] using a different measuring system, a different type of fiber (Corning PRSM designed for use at 1300 nm, length 1167.8 mm) a different wavelength scanning band (1466–1570 nm), and different data processing. This behavior is probably consequence of birefringence dispersion, although further research is needed to verify this assumption. To verify the validity of results obtained with the method here proposed we used Eq. (8) and the average value determined for the inclination angle of the INO NOI sample to predict the radii of the circular paths for some of the angles used in this work. The results shown in Fig. 12 indicate that in practice it is valid to assume that birefringence dispersion is negligible when the evaluation is performed using closely spaced light wavelengths [17].

In regard with beatlength evaluation, a 2π circular angle produces a full circular path on the Poincaré sphere. From Fig. 6 we can see that scanning the sampling signal from 1511 nm to 1571 nm produces a circular angle smaller than π for a 1.61 m fiber length. Since the Poincaré sphere is a 2-sphere, the phase change is smaller than π/2; i.e. *L*_{b}
>4_{s}(λ_{2}/λ_{1}-1)≈24 cm for the INO NOI sample. For the same wavelength interval the Photonetics sample covers a circular angle smaller than 40°, hence *L*_{b}
>1.1m.

The average value of the elevation angle was also used to determine the ratio of the linear to the circular birefringence components, *tan* σ [10]. The precision of this evaluation is poor. For ζ=π/2 - σ=3.1°±2.4° (INO NOI, sample) it varies from -82 to -10. These values indicate us that the linear retardation is at least 10 times higher than the circular retardation, and their signs are opposite. For the Photonetics sample circular birefringence is dominant. The circular to linear birefringence ratio varies from 11 to 19. These values indicate us that the circular retardation is at least 11 times larger than the linear retardation, and they have the same sign.

## 5. Conclusion

Previous to the birefringence characterization of single-mode erbium-doped fibers, it is necessary to identify the type of retardation that better describes the polarization optics of a fiber sample and to determine an approximate value for the birefringence beat-length. In this work we present the theoretical basis used to perform this assessment. In particular we propose a methodology based on the Poincaré sphere, and we apply it to two single-mode commercial erbium-doped fiber samples.

## Acknowledgments

This work was supported by Conacyt-DAIC, project G37000-E and by the scholarship granted to Fernando Treviño Martínez by PROMEP-UANL-256.

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