## Abstract

We derive an analytical expression for differential group delay of spun and twisted fibers, which should provide valuable guidance for optimization of such parameters to produce low polarization mode dispersion fiber.

©2003 Optical Society of America

## 1. Introduction

By the removal of one obstacle after another, use of optical fiber transmission has made tremendous progress over the past few years. However, polarization mode dispersion (PMD) has recently emerged as one of the major limitations in long-haul wavelength division multiplexing (WDM) systems that operate at bit rates of 40 Gbits/s and beyond. PMD mitigation techniques have attracted much attention in recent years [1]. Typically, the proposed PMD mitigation schemes can be divided into two main approaches: (1) electrical or optical PMD compensation [2,3], and (2) reduction of PMD in new fibers [4,5]. The former tends to be a solution for already installed fibers, but a method to develop the ultralow PMD fibers is essential for PMD mitigation in future system installations.

As is known, the PMD of fibers is a linear function of length for a short distance, whereas for a long distance it scales as the square root of length that is due to random coupling between two polarizations. Recent studies have shown that PMD could be reduced by spin during the fiber drawing process [4] or subsequently by twist [5]. Otherwise, the performance of variable spin rates shows better PMD behavior compared with that of constant spin rates, and periodic spin functions are preferred. An accurate analytical form has been developed for the reduction of PMD subject to a constant spin or twist rate. For fibers with a variable spin or twist profile, the analytical form has not yet been reported so most of these results must be obtained numerically and empirically. It is desirable to develop an analytical theory to obtain physical insight as to how to reduce PMD of fibers by means of spin and twist.

Chen *et al.* [6] derived an approximate analytical expression of PMD for spun fibers by using the coupled-mode theory, and they ignored twist. However, twist differs from spin inasmuch as it introduces an additional circular birefringence on fibers owing to the photoelastic effect. Beginning with the dynamic polarization dispersion vector (PDV) equation [7], an analytical expression for PMD of fiber with arbitrary spin and twist rates is derived based on perturbation theory. To test the validity of our analytical model, we compared the analytical results with the numerical simulations using various spin and twist profiles. The results show that a maximum PMD reduction can be achieved with optimal periodic spin profiles. In this case, the PMD along the fiber is periodic. For fiber twisting, a particular twist rate exists that can maximally reduce PMD for a certain amount of fiber birefringence and length. However, if the twist rate were large, the PMD would increase linearly.

## 2. Theoretical Background

Based on a principal state of polarization (SOP) model, PMD characteristics can be completely represented by the Stokes vector PDV $\overrightarrow{\Omega}$ . The direction of the PDV is the direction of the fast principal SOP, and its length is a differential group delay (DGD). Evolution of the PDV along the fiber can be related to the fiber microscopic birefringence through the dynamic vector equation [7]:

where *z* represents the position along the fiber, *ω* represents the angular frequency, and $\overrightarrow{\beta}$
is the three-dimensional local birefringence vector (LBV) of the fiber. The primes designate derivatives with respect to *z*. Note that Eq. (1) is written in a fixed reference frame.

In general, LBV $\overrightarrow{\beta}$
(*z*,*ω*) of an unspun fiber with just linear birefringence can be assumed to be *z* independent with magnitude *β*_{l}
(*ω*) and can be expressed as [*β*_{l}
(*ω*), 0, 0]. From Eq. (1) we can easily obtain the analytical result of the PDV of a linear birefringence fiber: $\overrightarrow{\Omega}$
(*z*, *ω*)=[*β*_{ω}*z*,0, 0], where
${\beta}_{\omega}=\frac{d{\beta}_{l}\left(\omega \right)}{\mathit{d\omega}}$
is the derivative of *β*_{l}
(*ω*) with respect to *ω*. When frequency dependence properties of the mode field are omitted, *β*_{ω}
can be approximated by
${\beta}_{\omega}\approx \frac{{\beta}_{l}\left(\omega \right)}{\omega}$
. Accordingly, the magnitude of $\overrightarrow{\Omega}$
(*z*,*ω*), i.e., the DGD Δ*τ* along fibers, can be expressed as Δ*τ*=*β*_{ω}*z*. It is straightforward to observe that DGD of unspun fibers grows linearly with the fiber length.

When the spin profile *a*(*z*) and externally applied twist rate *γ* are added to a fiber with initial linear birefringence *β*_{l}
, the LBV can be expressed as [8]

where *g* is the rotation coefficient. It is worthwhile to note that twist differs from spin because spin is applied to the fiber by oscillation during the drawing operation, whereas twist is applied when the fiber is already solidified and cool, which results in an additional circular birefringence *gγ*. The LBV can be simplified by rotation of the reference frame (on the Poincaré sphere) at the rate of 2*a*′(*z*)+2*γ*. This introduces an additional apparent circular birefringence *β*_{rc}
=2*a*′(*z*)+2*γ*. The LBV is transformed as follows:

Therefore, in this new rotating reference frame, Eq. (1) is expressed by

where *gω* denotes differentiation of *g* with respect to *ω* and Ω
_{k}
(*k*=1, 2, 3) are the components of $\overrightarrow{\Omega}$ in the new reference frame. Eq. (4) is a differential equation group with variable coefficients, and its exact analytical solution is at present intractable. To simplify our analysis, under the condition *β*_{l}
≪1, i.e., assuming the intrinsic linear birefringence is a small parameter, the perturbation theory can be applied to solve Eq. (4) approximately. Here we omit the mathematical details and present just the analytical results as

where
$V\left(z\right)=\mathrm{exp}\left\{A\left(z\right)\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\right\}$
and
$A={\int}_{0}^{z}[\left(g-2\right)\gamma -2a\text{'}\left(\xi \right)]d\xi $
. *V*^{T}
represents the transpose of the *V* matrix. From the results it is straightforward to observe that the spin does not contribute to Ω_{3}, which is determined only by twist and thus grows linearly. Consequently, we can easily obtain the DGD of the spun and twisted fiber as

where
$C\left(t\right)=\sqrt{{\beta}_{\omega}^{2}+{g}_{\omega}^{2}{\gamma}^{2}{\beta}_{l}^{2}{t}^{2}}$
and cos *φ*=*β*_{ω}
/*C*(*t*). Eq. (6) describes the DGD evolution along the fiber and allows us to have insight into the physical spin and twist of fiber. Note that the DGD of the spun and twisted fiber is determined mainly by these three parameters, namely, the initial linear birefringence, the twist rate, and the spin profile, which includes spin magnitude and the period of the alternating spinning process.

## 3. Results and Discussion

The polarization effects vary along the length of a long fiber owing to manufacture variations and externally applied mechanical stress. These perturbations are usually random and serve to couple power between the two polarization modes. As mode coupling can reduce PMD, various techniques are often introduced to increase intrinsic mode coupling, such as artificial spin during the fiber drawing operation or twist when the fiber is already solidified. By use of the model described above, we have studied PMD mitigation for fibers with various spin profiles and twists.

To test our results we compared our analytical results based on Eq. (6) with the numerical solutions of Eq. (4) for a variety of spin and twist rates. As an example, Fig. 1(a) shows the comparison of the fiber with a sinusoidal spin profile. This spin rate function is depicted in the inset of Fig. 1(a) and can be expressed as

Fig. 1(b) shows the comparison results of fiber with a triangular asymmetrical spin rate profile. The inset of Fig. 1(b) represents the curve of the triangular period spin function, and its form can be written as

where *a*
_{0}, *p* represent the spin magnitude and the period, respectively, and *r* indicates the asymmetrical properties of the triangular spin profile. It is assumed that the fiber has an initial beat length of 15 m, rotation coefficient *g* is set to be 0.14, and *g*_{ω}
=0.09*g*/*ω*. The excellent agreement between them confirms the validity of our approximate analytical model. As twist and spin are two main techniques to produce low PMD fibers, we prefer to study the effects of twist and spin on PMD reduction.

#### 3.1 PMD Reduction by Twist

If we consider a twisted fiber with null spin, as in Eq. (6), *A* is simplified as *A*=(*g*-2)*γz*. Fig. 2 shows the variation of DGD versus the external twist rate for two fiber samples with different initial linear birefringence. We also compare our results based on the perturbation theory with the analytical solution derived by Schuh *et al.* [5]. Three comments are important. First, note that a twist rate exists to make maximum DGD reduction. Its value was determined by the initial linear birefringence of fiber and the dispersion of rotation coefficient *g*_{ω}
. Second, DGD decreases with the increase of twist rate at low twist rates because mode coupling was strengthened between two polarizations. At high twist rates, DGD increases linearly with the twist rate because of twist-induced circular birefringence. This instability puts the twist method at a disadvantage because DGD could increase immediately when twist is a little larger than the best twist for the minimum PMD condition. Finally, we found that, when the twist rate is low, our results are inexact and differ significantly from that of Schuh *et al.* compared with that in a high twist case. In contrast, the difference also increases with the initial linear birefringence because our analytical model is based on the perturbation theory and tends to be valid only when *β*_{l}
is small. With the improvement of fiber manufacturing techniques in recent years, most intrinisic birefringence of fibers is small enough to meet this requirement.

#### 3.2 PMD Reduction by Spin

Spin on fiber is preferable for PMD reduction with respect to twist for it does not introduce the undesired circular birefringence. Therefore, it is worthwhile to study the properties of spun fibers and to understand the details of PMD reduction by spin.. If we neglect the twist on the fiber, Eq. (6) can be simplified to

where we substitute spin profile $A=-{\int}_{0}^{z}2a\text{'}\left(\xi \right)d\xi $ for that in Eq. (6). The result is the same as that reported in Ref. 6. We defined the PMD reduction factor (RF) as the ratio between DGD of spun fiber and that of the same fiber without spin. It is expressed as

It is straightforward to observe that the RF is independent of the fiber intrinsic birefringence. This would simplify the fiber design and negate the need for different birefringence to optimize the spin profile. In addition, with the periodic spin profile, the DGD of the fiber with length *n*·*p* can be written as

It is clear that the DGD increases linearly with the fiber length in a global analysis, although there are some concomitant oscillations in one local period. Furthermore, note that the maximum PMD reduction can be achieved when ${\int}_{0}^{p}$ exp(*i*·*A*)*dt*=0 i.e., the phase-matching condition is satisfied. Taking the triangular spin function described in Eq. (8) as an example, we can obtain fiber DGD from Eq. (9):

where the linear term is

$$+\mathit{exp}\left(\frac{-i{a}_{0}p}{2}\right)\mathit{erf}\left(\sqrt{\frac{-i{a}_{0}\left(p-2r\right)}{2}}\right)\left]\right\}$$

and the oscillating term is

where *erf* is the error function defined as
$\mathit{erf}\left(z\right)=\frac{2}{\sqrt{\pi}}{\int}_{0}^{z}{e}^{-{t}^{2}}\mathit{dt}$
. It can be seen that the DGD is determined mainly by the linear term *L*(*n*). Figure 3 illustrates the evolution of DGD along the fiber with an initial beat length of 15 m and a triangular spin profile with different sets of parameters. We found that a periodic DGD can be achieved with optimum spinning parameters or else the DGD grows linearly with the fiber length.

For a spin rate profile with a 1-m period, according to Eq. (12) the locus of (*a*,*r*) for optimal PMD reduction is shown in Fig. 4. We can observe that there are several couples of (*a*
_{0},*r*)that satisfy the phase-matching conditions for maximum PMD reduction. In addition, the optimum condition is determined mainly by the spin magnitude and is insensitive to the value of parameter *r*. Galtarossa *et al.* [9] also discussed the phase-matching conditions for an asymmetrical spin rate function. However, we believe that the conditions presented by Galtarossa *et al.* are incorrect because spin profile *A* does not have the odd harmonic properties as reported in their paper. We also believe that the constraint conditions for spin periods reported in Ref. 9 are not error free. By neglecting the oscillating term, we can express the RF as *RF*=|*L*(1)|. Figure 5 illustrates variation of the PMD RF with respect to spin magnitude and parameter *r*. We note that the RF generally decreases gradually when the spin magnitude increases, although there are some oscillations embedded in the dominant decreasing trend.

From a practical point of view, installed fibers with an optimal spin design can be twisted simultaneously by external shear forces. From Eq. (6) we note that phase-matching conditions are destroyed when twist is added. With the same fiber initial birefringence and spin parameters, Fig. 1(b) illustrates the effect of twist on DGD evolution along the fiber length in comparison with that in Fig. 3(a). We have shown that DGD grows with the fiber length instead of oscillating with it. The increased rate was determined mainly by an external twist rate.

## 4. Conclusion

Based on the perturbation theory, an analytical expression has been derived for DGD of spun and twisted fiber. Using the analytical results we reported the details for maximum fiber PMD reduction by twist and spin. The design of ultralow PMD fiber would provide valuable guidance for optimization of spin and twisted parameters.

## Acknowledgments

This research was supported by the National Hi-tech Research 863 Project of China (No. 2001AA120204).

## References and links

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