In this work we analyze the birefringence matrix developed for a twisted fiber in order to identify the basic optical effects that define its birefringence. The study was performed using differential Jones calculus. The resultant differential matrix showed three independent types of birefringence: circular, linear at 0 degrees and linear at 45 degrees (Jones birefringence). We applied this birefringence matrix to the description of the output state of polarization measured for three commercial fibers that due to its higher rigidity present stronger birefringence changes when twisted. The torsion applied to the erbium-doped fiber samples varied from 0 to 1440 degrees.
©2013 Optical Society of America
In a series of studies performed on twisted single-mode optical fibers we have found that the effect of torsion is far more complex than the initial proposal of twist-induced circular birefringence developed by Ulrich and Simon in 1979 . In an earlier investigation, making use of classical polarization optics we demonstrated that for a very low torsion (~4 rad/m) the fiber could be represented as the concatenation of multiple segments with the same residual anisotropy (elliptical) whose birefringence axes rotate uniformly along the fiber length. Under this scope, the birefringence matrix is equal to the product of two operators, the elliptical rotator describing the residual anisotropy of the fiber, multiplied by a circular rotator whose rotation angle is equal to the twist angle . Since these matrices are rotation operators describing gyration around two different axes, the fiber no longer behaves as a homogeneous retarder; i.e. it cannot become circularly birefringent.
For larger rotation angles, and working with standard telecommunication fibers, we found that the effect included not only a rotation of the birefringence axes but also a photoelastic contribution. Due to photoelastic effect, the retardation angle varied linearly with the applied torsion while fiber ellipticity remained constant . In these experiments, we analyzed the evolution of the output polarization state with the applied torsion using the Stokes parameters (S1, S2, S3 vs. applied twist). We obtained nearly sinusoidal curves whose frequencies and amplitudes remained fixed for positive and negative twist angles. The frequency and amplitude of parameters S1 and S2, had the same values; while the frequency of parameter S3 was higher, and its oscillation amplitude lower. Due to the fact that the output polarization state was maintained on the surface of the Poincaré sphere, the effect of torsion was described in terms of unitary rotation matrices with different orientations on their gyrating axes and/or responses to torsion: 1) A rotation matrix associated with rotation of birefringence axes, which depends on the applied torsion, 2) A rotation matrix with a constant delay, and, 3) The matrix of an elliptical retarder in which delay between polarization eigenmodes depends on torsion. Such semi-empiric model predicted with high accuracy the results obtained using single-mode fibers for optical communications (Corning SMF-28 and SMF-28e) , even in the case of fibers having residual torsion (Nufern 1060-XP) . Nevertheless, this model does not predict observed changes of the polarization state in single-mode erbium-doped fibers (EDFs).
In order to extend the application of the model offered in , it was translated into Jones differential matrices , which allowed identifying the origin of observed changes and facilitated its numerical adjustment. The present work submits an amplified form of the model, which makes use of the linear birefringence component known as Jones birefringence. We should point out that we used short fiber samples (< 2 m) whose absorption was negligible, therefore, along this work signals are considered as fully polarized.
This work also discusses the adjustment obtained within values predicted by the model and experimental data obtained using single-mode fibers doped with low and medium erbium concentrations. As a form of comparison, in this study we include some aspects of the twist-induced polarization behavior of a standard fiber (SMF-28e) used in communications.
2. Matrix differential analysis of a twisted-fiber
In 1948 Jones published an investigation in which he described the evolution of the polarization state of a plane wavefront that propagates through an anisotropic medium. The differential matrices used in his infinitesimal description were derived from the birefringence matrix of the optical medium M (expressed as a 2 × 2 Jones matrix) using the relation:
2.1 Jones matrix (MT) of a twisted-fiber
It has been shown recently that the birefringence of a twisted-fiber can be described using the matrix :Eq. (2) the fast birefringence axis of the fiber is rotated by an angle θ with respect to x axis of the laboratory reference frame, β is a fixed rotation angle between the system coordinates and the fiber birefringence axes, and bτ describes the geometric rotation of the principal axes due to torsion, b is a constant with a value close to one unit. R is used to represent a matrix of rotation that in the space of the Poincaré sphere takes place around the axis crossing the poles of the sphere (it should be noted that in this space, rotation angles are doubled and measured from S1 axis). In Eq. (2), Mτ(δτ,σ) is the matrix of an elliptical retarder with eigenmode ellipticity ε (here described in the space of the Poincaré sphere by the angle σ = π/2 - 2ε) and zero azimuthal angle. The retardation δτ induced between eigenmodes by the applied torsion is:Eq. (3) has the form: δτ = c0 z + c Tz.
Assuming the birefringence axes of the anisotropic medium and the laboratory system are aligned (θ = 0) and using Jones’s matrices (they are defined for a right orthogonal coordinate system to be able to follow ), Eq. (2) is reduced to:
2.2 Differential matrix NT of a twisted fiber
Considering that for any normal section of the fiber, z is its coordinate along the fiber axis, with the monochromatic wave propagating to the positive z, and that only the twist angle τ and the retardation angles δ0 and δτ are functions of z, we obtain substituting (4) in (1), NT, the differential matrix of a twisted fiberEq. (5) can be broken down into three elemental matrices:Eqs. (5) and (6) are typical of a twisted fiber . Applying Jones’s notation  we may define,Eq. (6), we find that matrix NT may be written as:
In order to understand the birefringence changes introduced when an elliptical retarder is twisted, the differential matrix of a fiber with elliptical birefringence is calculated below, and compared with the mathematical description of Eq. (10).
2.3 Differential matrix NΤ of a fiber with elliptical birefringence
Jones’s matrix Me describing a fiber with elliptical retardation (using again a right coordinate system) is:
Applying Eq. (1), we find that the differential matrix of a fiber with elliptical birefringence is:
The first term corresponds to circular birefringence and the second term to linear birefringence. The coefficients indicate the relative values of such birefringences: and . Comparing Eqs. (10) and (12) we notice that the description of linear birefringence requires two components (shown in Fig. 1) only when dealing with a twisted fiber.
From relations (7) and (8) we observe that for a fixed torsion, the geometric sum of the linear birefringence components generates a linear birefringence (βl) whose fast axis in general does not match axes x, y or ± 45°. Therefore, the applied torsion produces a change of the value of the azimutal angle of the linear birefringence fast axis. The resulting linear birefringence coefficient is (Ac cos2ε)/2.
It should be pointed out that despite such simplification, since the coefficient of circular birefringence of a set value of torsion is -A(2b + sin2ε)/2, the linear and circular birefringence components of the twisted medium do not correspond to the components of a matrix N of elliptical birefringence [Eq. (12)]. Furthermore, expressing the result in terms of Jones’s matrices M, shows the product of a rotation matrix with a circular birefringence coefficient (β + bτ) (where β is a constant), multiplied by an elliptical birefringence matrix.
2.4 Mueller matrix of a twisted fiber with two linear birefringence components: a deduction from its Jones matrix
In this section we describe the result developed in the previous section, in terms of a Mueller matrix. To be able to use the Mueller matrix of an elliptical retarder developed by C. Tsao , we used a left coordinate system . Therefore, we base our description on the following Jones’s matrix of elliptical birefringence:
In this unitary matrix [Eq. (13)], Γ = δ is equal to δτ /2 modulo 2π, and linear birefringence has two components: one running along the horizontal axis and another one at 45 degrees (Fig. 1). The transformation rule to solve Mueller matrix (MMe), is :
For simplicity, we work with a (3 × 3) version of the matrix, and based on Fig. 1, we rewrite Eq. (17) substituting the inverse ratio of total retardation coefficient Γ to one of its components (along the vertical axis or on the horizontal plane) in terms of angle σ. We also relate the components of linear birefringence through angle σl:
Therefore, for a linearly-polarized entry signal with azimuth angle φ in the physical space, Stokes vector Se-out at the elliptical retarder exit-point is:
The use of the additional birefringence associated with the rotation matrix R(β + bτ) that multiplies matrix MR in Eq. (4) (as mentioned in the last paragraph in section II.3), reveals that Stokes vector Sout at the exit-point of the twisted fiber is:
3. Experimental results
In this work, following the experimental procedure reported in , we study the effect of torsion on the birefringence of three commercial erbium-doped fibers which are listed in Table 1: Sample F1 = Photonetics EDOS 230, sample F2 = Fibercore 1500E and sample F3 = INO 402 K5. We selected these fibers because their higher rigidity introduces stronger birefringence changes when twisted.
We found that an important difference between the polarization performance of doped and undoped twisted fibers is their dependence on the magnitude of the applied twist. For undoped fibers the polarization evolution described a curve that oscillated around a fixed parallel of the Poincaré sphere, keeping almost the same amplitude and frequency . This polarization evolution curve corresponds to a medium with a fixed ellipticity and fixed azimuth angle. By contrast, in Fig. 2 we present the curves depicted by the output polarization state for two twisted erbium-doped fibers (F1 and F3).
For these fibers the oscillation does not take place around the same parallel and the oscillation frequency and amplitude depend on the applied twist. To describe this behavior it is necessary that the linear and circular components of the elliptical birefringence vary in an independent manner. This condition is satisfied in our model with the introduction of Jones birefringence.
The model we present in this work has been used to predict the evolution of the output polarization state for the three commercial EDFs we work with. Two examples of the numerical fitting results obtained for fiber F1 using Eq. (24) are shown in Fig. 3 for two signal wavelengths (1530 and 1560 nm). Figures 3(a) and 3(c) show the numerical results obtained for Stokes parameters due to the applied twist scanning (τa = 0 to 1440°). Figures 3(b) and 3(d) present the output SOP evolution on the Poincaré sphere for a 600° to 1440° scanning of the applied twist τa (τa = τ(L), where L is the fiber length). The azimuth angles of the input linear polarizations are φ = 60° for 1530 nm and φ = 0° for 1560 nm. Stokes parameters labeled “out” have been obtained using the theoretical model and those labeled “n” are experimental values.
For fibers F2 and F3 the experimental results show a very irregular variation for low torsion values (τa < 500°) as we can observe in Figs. 4(a), 4(b), 4(d), 4(e), 5(a), 5(b), 5(d), and 5(e). This behavior has been previously reported for a standard single mode fiber with residual torsion . Since the EDFs we study in this work also present a different response to right and left torsions, we know these fibers have a residual torsion. Here, we must mention that the results we discuss correspond to twists with the same handedness of the residual torsion.
In this work we model the evolution of the output polarization state [Eq. (24)] considering that twist-induced birefringence is the dominant effect (the parameters we used produce a good fitting for the higher values of torsion). Therefore, the initial irregular variations of Stokes parameters associated with uneven modifications of the waveguide are not predicted by our model. The parameters used to fit the results shown in Figs. 3–5 are shown in Table 2.
4.1 Numerical fitting
The numerical fitting procedure requires the determination of parameters: c, b, β and also, initial values (when there is no torsion applied to the fiber) for: the retardation between polarization eigenmodes (δ0), the ellipticity angle of the circular birefringence component (σ0) and the azimuth angle of the linear birefringence component (σl0).
Since the evolution of angles σc and σl must be independent, to model them we defined coefficients cc and cl such that
The values determined for c and b for EDFs studied in this work and standard fibers reported in  and  are presented in Table 3. We can see that, as expected, geometrical parameter b shows values very close to unity for doped and undoped fibers. We should remember that product bτ describes the rotation of the fast birefringence axis produced by the applied torsion .
Parameter c remains close to 0.9 for standard fibers and assumes different and lower values for erbium-doped fibers. We hypothesize it must be connected with the specific waveguide structure, although further information is required to relate c to the waveguide profile and glass composition.
In Table 4 we present the values determined for wavelength dependent parameters (θ, and β) and its range of variation (Δθ and Δβ) for the three erbium-doped fibers studied in this work.
Angle θ is the fiber’s fast axis azimuth angle. Its spectral variation has no parallel for bulk anisotropic media, but since we modeled the evolution of the output polarization state considering that twist-induced birefringence is the dominant effect, we can propose that torsion is the origin of the spectral dependence of angle θ. Nevertheless, we should mention that the spectral variation of the azimuth angle of the fast birefringence axis (θ) has also been reported in  for fibers F1 and F3 when they were kept straight. To understand the spectral behavior of θ, we must take into account the previously mentioned unequal changes observed for low twists (τa < 500°), and the different response observed for the right and left torsions measured in this work. Under this scope, the spectral dependence of angle θ obtained for non-twisted erbium-doped fibers can be explained in terms of the presence of a residual torsion (introduced during the manufacturing process). In the presence of a residual torsion a cold twist introduces a change in the index of refraction of the glass host that gives rise to a spectral dependence of some parameters such as θ. Furthermore, for standard optical fibers parameter θ kept constant for all the input azimuth angles used for each signal wavelength (φ = 0-150 °, with a 30 ° step) [3,4]; while for erbium-doped fibers it was φ-dependent. When the azimuth angle of the input linear polarization varied, we obtained a fixed value for θ only for some wavelengths: for fiber F1 from 1540 to 1560 nm; for Fiber F2, for 1540, 1550 and 1560 nm and for fiber F3 from 1525 to 1565 nm.
In addition, parameter β showed no wavelength or geometric dependence in standard optical fibers. Its value (180°) was constant for Corning SMF-28e  and nufern 1060XP . In the present study β is not only wavelength dependent but also varies with the orientation of the azimuth angle of the input linear polarization (φ). The variation obtained for β follows that of angle θ (orientation of the fast birefringence axis). This higher sensitivity to optical alignment can be produced by fabrication tolerances of the waveguide structure, although further information is required to confirm this hypothesis.
In Table 5 we compare the values we determined for parameters related with the anisotropy of the fibers (cc, σ0, cl, σl0 and δ0) and its range of variation (Δcl, Δσl0 and Δδ0) for the three EDFs we study in this work and the standard fibers investigated in  and . The elliptical birefringence of standard optical fibers was described using angle σ, related with the total elliptical birefringence. The value of angle σ was wavelength dependent.
For erbium-doped fibers we describe its elliptical birefringence in terms of the circular and linear birefringence components using σ = σ0 + cc⋅τa and σl = σl0 + cl⋅τa, respectively. σ0 and cc keep constant for each fiber and the wavelength dependency is observed just for σl0. cl exhibits spectral variation only for fiber F2. Here, it is interesting to remark that our value of the Jones birefringence component is much higher than the values mentioned by other authors [9–12].
The initial retardation angle between polarization eigenmodes δ0 used for numerical fitting showed (in  and in this work) spectral dependence for all the fibers with the exception of nufern 1060XP .
4.2 Birefringence changes induced by torsion
The evolution of the azimuth angle of the output state of polarization is faster for twisted erbium-doped fibers than for undoped fibers. This result is in agreement with the notion that EDFs birefringence is higher than SMF birefringence. An example of this behavior is shown in Fig. 6, where the evolution obtained for a 1530 nm signal using an undoped fiber (SMF28e, 1.75 m) is compared with that measured for erbium-doped fiber F2 (Fibercore 1500E, 1.51 m).
From the measurements analysis we conclude that the evolution of the output polarization state due to the applied torsion is different for optical fibers doped with similar erbium concentrations (F1 and F3, Figs. 3 and 5). This result reveals there are other contributions, such as waveguide structure and residual torsion that dictate the birefringence response of erbium doped fibers.
Torsion of fibers introduces a stronger modification of polarization properties in erbium-doped fibers than in undoped single-mode fibers. The description of the polarization performance of EDFs requires a model that includes a linear birefringence component at 45° (Jones birefringence) in addition to a residual elliptical birefringence with one linear component (at 0°) and one circular component. All these birefringence components are twist-dependent.
We have found that Jones birefringence induced by twisting the birefringent media studied in this work (optical fibers) is several orders of magnitude greater than previously reported values of Jones birefringence induced by electric and/or magnetic fields in bulk birefringent media.
As a final remark, we should mention that in erbium-doped optical fibers birefringence properties are mainly linked to the glass host composition and the fiber geometrical structure than to the erbium concentration.
We would like to thank Miguel Farfán Sánchez for his help with data collection and to Conacyt for the economical support. This work was sponsored by project SEP-CONACYT, CB-2010-01-155121.
References and links
2. D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun. 282(5), 830–834 (2009). [CrossRef]
3. D. Tentori, A. García-Weidner, and C. Ayala-Díaz, “Birefringence matrix for a twisted single-mode fiber: Photoelastic and geometrical contributions,” Opt. Fiber Technol. 18(1), 14–20 (2012). [CrossRef]
4. D. Tentori and A. Garcia-Weidner, “Right- and left-handed twist in optical fibers,” submitted for publication.
5. R. C. Jones, “A new calculus for the treatment of optical systems. VII properties of the N-matrices,” J. Opt. Soc. Am. 38(8), 671–685 (1948). [CrossRef]
6. C. Tsao, Optical Fibre Waveguide Analysis (Oxford University, 1992).
7. P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. 96(1-3), 81–107 (1980). [CrossRef]
8. D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun. 271(1), 73–80 (2007). [CrossRef]
9. A. Rizzo and S. Coriani, “Birefringences: A challenge for both theory and experiment,” Adv. Quantum Chem. 50, 143–184 (2005). [CrossRef]
10. V. A. De Lorenci and G. P. Goulart, “Magnetoelectric birefringence revisited,” Phys. Rev. D Part. Fields Gravit. Cosmol. 78(4), 045015 (2008). [CrossRef]
12. M. Izdebski, W. Kucharczyk, and R. E. Raab, “Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence,” J. Opt. Soc. Am. A 18(6), 1393–1398 (2001). [CrossRef] [PubMed]