## Abstract

For what is the first time to our knowledge, we study the performance of Gaussian pulse transmission over an ultrahigh polarization-mode-dispersion (PMD) fiber. In the experiment the Gaussian pulse breaks into a series of deformed pulses; this phenomenon is attributed to the walk-off of the deformed pulses caused by ultrahigh PMD. The simulation and transmission experiment were performed with fiber with a PMD coefficient of 237.95 ps/km^{1/2}. The result of the simulation agrees well with that of the experiment.

©2003 Optical Society of America

## 1. Introduction

The increasing demands of bandwidth drive most telecommunication operators toward the deployment of large-capacity transmission systems. When chromatic dispersion and loss in optical fiber no longer limit the performance of high-data-rate transmission systems, polarization mode dispersion (PMD) becomes the key factor that blocks updating high-performance optical transmission systems and networks. In high-bit-rate optical communication systems beyond 10 Gbps, signal distortion caused by PMD introduces a major limitation on transmission distance [1–3]. PMD is a distortion arising from unwanted birefringence in optical fibers. Real fibers are never perfect and have slight asymmetries or other perturbations that give rise to degeneracy, which leads to two polarization states with slightly different phases and velocities, a phenomenon known as PMD [1,4,5]. In addition, PMD is a statistically random quantity, for the birefringence of a fiber changes randomly along a fiber link and is time dependent. In practice, PMD represents the major impairment for high-bit-rate systems; it results in pulse broadening and distortion, which in turn leads to system performance degradation [5–7].

Although present-day fibers have a 0.5-ps/km^{1/2} PMD coefficient, many of the fibers installed before the 1990s have relatively high-PMD coefficients (for example, more than 10 ps/km^{1/2}), so their PMD is larger than 200 ps when light pulses are transmitted over 400 km [5,6]. Gaussian pulses may propagate through high-PMD components, such as those with PMD compensation based on nonlinear fiber Bragg gratings; for example, nonlinear fiber Bragg gratings written by polarization-maintaining fiber have a PMD higher than 300 ps. Therefore, for updating optical fiber networks to high-bit-rate systems, it is necessary to study the effects of transmission over high-PMD fibers. We simulate Gaussian pulse transmission over ultrahigh-PMD fiber (UHPF) by solving the coupled nonlinear Schrödinger equations [8,9]. The simulated result shows that the Gaussian pulses become deformed when they propagate through UHPF. For what is the first time to our knowledge, we carry out a PMD experiment by using a fiber with the PMD coefficient of 237.95 ps/km^{1/2}, and we find that the result of the experiment agrees well with that of the simulation.

## 2. Theoretical background

Poole and Wagner [4] introduced a phenomenological model of PMD in long optical fiber in which two orthogonal input and output principal states of polarization were identified in a way analogous to the eigenmodes of an uncoupled system, which is a useful tool for studying PMD. However, fiber PMD is a stochastic variable, which is affected by temperature, environment, and time. In the simplest birefringence model [9] it is assumed that the birefringence strength *δβ* is fixed and the orientation angle *φ* is driven by the white-noise process. In other words, the fiber is modeled by dividing it into *m* segments, each of which is characterized by a local differential group delay *τ*_{i}
or the local birefringence strength *δβ*, with axes of given orientation angle *φ* (here assumed to be wavelength independent). The Jones matrices *J*, which consist of the Jones matrices *B*_{i}
and *R*_{i}
at the angular frequency *ω*, corresponding to the *i*th delay element and rotation, respectively, are given by [8,9]

where
${w}_{i}=\frac{\sqrt{\frac{3\pi}{8}}\sqrt{\delta {z}_{i}}\delta \beta \omega}{4}$
,*δz*_{i}
is the length of the *i*th segment, and *B*_{i}
is the birefringence matrix of length *δz*_{i}
. We can study the PMD of optical fiber by using Eq. (1). The coupled nonlinear Schrödinger equation can describe the effect of PMD on a light pulse when it is transmitted through optical fiber with PMD, which is given as [9]

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+{n}_{2}{k}_{0}\left[\frac{5}{6}{\mid A\mid}^{2}A+\frac{1}{6}\left({A}^{+}{p}_{3}A\right){p}_{3}A+\frac{1}{3}\left({A}^{+}{p}_{2}A\right){p}_{2}{A}^{*}\right]=0,$$

where A=(*A*_{x}* A*_{y}
)
^{t}
is a column vector with elements *A*_{x}
and *A*_{y}
, which are the complex envelopes of the two polarization components. The *z* coordinate measures distance along the optical fiber axis, while the *t* coordinate represents retarded time—the time relative to the moving center of the signal; * designates complex conjugation. *β*
_{2} is the fiber group-velocity dispersion (GVD), with the assumption ${\beta}_{2}^{x}$≈${\beta}_{2}^{y}$=*β*
_{2}. The nonlinear Kerr coefficient is *n*
_{2}. The wave number is represented by *k*
_{0}=2π/*λ*, where *λ* is the wavelength of light in vacuum. The specific group delay per unit length is represented by

where *ω*
_{0} is the center frequency of the transmitted light. The matrix is as follows:

The *p*_{i}
(*i*=0–3) are represented by Pauli’s matrices:

Therefore we can simulate the effects of PMD on the transmitted pulse by using Eqs. (1)–(5).

## 3. Numerical simulation and experiment

Computer simulations have been conducted to determine the effects of ultrahigh PMD. The simulated parameters are determined according to the fibers that we used in the experiment. The length of the optical fiber is 1.18 km. The fiber dispersion is -102 ps/(nm km), which is measured by CD400 EG&G. The effective area is 15.23 µm^{2}, which is measured by NR9200 EXFO. The nonlinear factor *n*
_{2} is 3.2×10^{-20} m^{2}/W. The coefficient of the high-PMD fiber is 237.95 ps/km^{1/2}, which is measured with an HP 8509B polarization light-wave analyzer. The full width at half-maximum (FWHM) of the light pulse is 24.3 ps. Thus the FWHM of the light pulse is narrower than those used in 20 Gbps transmission. The input power is 10 dBm. The signal wavelength is 1550 nm.

The coupled nonlinear Schrödinger equation is solved with a split-step operator technique. The PMD is simulated by a discrete random waveplate model (DRWM). The angle *φ* rotates randomly; *δβ* may also vary randomly, but here we assume that *δβ* is a constant. The input power is 50:50 for fast axis versus slow axis. Thirty simulations, in which the input pulse FWHM is 24.3 ps, were performed; the simulations show that the FWHM becomes 27.6 ps after 50 m in UHPF (UHPF); the FWHM is 31.7 ps after 100 m in UHPF; the transmitted pulse begins to break up slightly after 200 m UHPF and break up severely after 400 m.

In order to compare theory with the experiment. The transmission over 400m and 1.18 km ultra-high PMD fiber has been numerically studied. The two simulated results are shown in Fig. 1. Fig. 1(a) shows the Gaussian pulse transmission over 400m ultra-high PMD fiber; light pulses begin to break up. Fig. 1(b) shows that the Gaussian pulse transmission over 1.18km high-PMD fiber, the pulses are deformed badly. We cannot find what the original pulses are from Fig. 1(b).

The experiment has been done by using ultra-high PMD fiber also. Fig 2 shows the measured fiber PMD. The coefficient of high-PMD fiber is 237.95ps/km1/2, which is measured by a HP 8509B polarization mode dispersion analyzer. The transmitted light source is a Santec TSL-210 tunable wavelength laser. The output power is 10 dBm. The output pulses are compressed to 24.3 ps. Then the light pulses are transmitted over ultra-high PMD fiber. The transmitted pulses are detected by a Tektronix CSA 803A Communication Signal Analyzer. The experimental result is shown in Fig. 3.

Figure 3(b) shows the Gaussian pulse transmission over 400 m ultra-high PMD fiber; light pulses begin to break up. Fig. 3(c) shows that for Gaussian pulse transmission over 1.18km high-PMD fiber, the pulses are deformed badly. Just as in the simulated results, we cannot find the original pulses from the Fig. 3(c).

The results of the simulation agree with those of the experiment from Fig. 1 and Fig. 3. Why are the Gaussian pulses deformed so badly? The input LP01 mode comprises two orthogonal components ${\mathit{\text{HE}}}_{11}^{x}$ and ${\mathit{\text{HE}}}_{11}^{y}$. They mix together and we don’t distinguish them if the transmitted link doesn’t have PMD. On the contrary, if the transmitted link has PMD, the two orthogonal components ${\mathit{\text{HE}}}_{11}^{x}$ and ${\mathit{\text{HE}}}_{11}^{y}$ will have different velocities and produce walk-off, see Fig 4. The Fig 4 only shows the visual walk-off pulse, which doesn’t consider the stochastically coupled-phase. The walk-off is very fast when the transmitted link’s PMD is very high; and this will make that the walk-off pulse becomes two independent pulses. Then, the two independent pulses break into four independent pulses, which in turn break into 8 independent pulses, and the same process continues. At last we can see the deformed walk-off pulses in the end of transmitted link.

## 3. Conclusion

To our knowledge, the experiment reported here is the first ever tried to study Gaussian pulses transmission over ultra-high PMD fiber. The simulation and experiment have been carried out using a PMD efficient with 237.95ps/nm.km1/2. The numerical result of simulation agrees with that of the experiment. The broken pulse due to walk-off is observed.

## Acknowledgments

This work supported by “863” Project of China (2001AA122042), the National Science Foundation of China (69903001), and “Xiao Rencai” Fund of Beijing Jiaotong University (2003RC011).

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