# Polarization Mode Dispersion

This is a continuation from the previous tutorial - ** origins of chromatic dispersion**.

### Overview

In the early 1990s, the deleterious effects of PMD were first reported in transmission of analog signals over Hybrid Fiber Coax (HFC) networks. The transmission distances in these networks were not long (typically <50 km), but the analog signals (unlike digital ones) were very sensitive to small levels of impairment that can be generated by unwanted dispersion.

Here, it was found that chromatic dispersion, which is deterministic and can be compensated, was not the only impairment, but that PMD, coupled with source laser frequency chirp, was also playing a critical role.

Since that time, fiber PMD has been improved far beyond the limits where this analog limitation is observed (for systems with small source frequency chirp). However, digital signal transmission rates increased, and PMD was found to be a limiting factor in long-haul transmission.

Modern state-of-the art optical fiber for transmission is capable of carrying signal line rates of 10–40 Gbps over distances of thousands of kilometers without serious degradation due to PMD.

Dispersion compensating fibers and some other specialty fibers pose a PMD concern because of the difficulty in manufacturing a small fiber core with an optically uniform circular cross-section.

In this case, careful manufacturing controls and measurement are required to produce a reliable product. A fair amount of work has been devoted to PMD compensation.

This generally is possible when the PMD is due to a fixed component (such as a LiNb modulator). For the fiber generated PMD, the compensation problem is much more difficult, because dynamic adjustments must be made on a per-channel basis (for each transmission wavelength).

This tutorial will not attempt to cover the full spectrum of PMD topics. The intent is rather to give a basic understanding of the phenomenon and a few references for the interested reader. The requirements imposed by high-speed digital transmission are addressed in later tutorials.

### Background

The first concept to be understood is that of lightwave polarization, because PMD originates when different light polarizations travel at different average speeds in the optical fiber.

The polarization of light (or any electromagnetic wave) is a description of how the electric field vector of the wave varies in time at a fixed position in the fiber.

In general, the tip of the electric field vector traces out an elliptical shape in a plane transverse to the fiber axis as time evolves. This elliptical polarization can, in special cases, degenerate to a line or circle (linear or circular polarization). Details can be found in most elementary texts on electromagnetic fields or optics.

The speed that light will travel in an optical fiber is a function of the effective index of refraction. This effective index can be a function of polarization (leading to PMD) and wavelength (leading to chromatic dispersion).

This phenomenon is familiar from the optical concept of birefringence (typically due to index of refraction variation with lightwave electric field orientation in a crystalline material).

Confusion often arises over the difference between differential group delay (DGD) and PMD. It has been shown mathematically that for any fixed wavelength and length of optical fiber, there are two principle states of polarization (PSP).

These states possess, among other properties, the maximal and minimal transit time through the fiber of any input state of polarization. The difference in transit time between these states, over a given length of fiber, is the DGD.

The DGD is generally a function of wavelength. In a uniform birefringent material, there are fast and slow axes corresponding to the orientation of the two PSPs. In this case, the PSPs are independent of wavelength and remain invariant along the direction of light propagation (because the fiber cross-section does not vary).

Light that is input in any polarization other than the PSPs suffers a periodic evolution in polarization state along the propagation direction. This spatial period depends on the wavelength and birefringence strength and is called the beat length.

The DGD builds up through the material in a linear fashion, with a net delay that is directly proportional to the propagation distance in the material. In the case of an optical fiber, the birefringence is generally not fixed but fluctuates along the fiber length because of variations in core transverse geometry or core mechanical stress for an overview of the many physical reasons for this birefringence).

In this case, the PSPs vary along the fiber length and are wavelength dependent. For any fixed set of fiber terminals (input and output positions along the fiber), there is a unique set of PSPs. Because of this, the DGD (at a fixed wavelength) builds up along the fiber in a complicated way, instead of simple linear growth.

An example is given by the simulation results shown in Fig. 2.11.

Here, four erratic curves show the buildup of DGD in a group of fibers with slightly different distribution of birefringence along their length. Note that the DGD is almost as likely to decrease as to increase as the lightwave propagates down the fiber.

The fiber length over which the polarization ‘‘randomizes’’ is termed the field correlation length. This correlation length can be understood in terms of a fictitious experiment: launch a lightwave along a local PSP into each of an ensemble of equivalent fibers and determine the average position in these fibers where the optical power in the polarization mode orthogonal to that launched is within \(1/e^2\) of the launch power.

Note that the lightwave and the birefringence may randomize independently and have different correlation lengths. The PMD is defined as the average (linear or rms) of the DGD over all realizations of the system (which may vary in time, wavelength, or birefringence pattern in the fiber).

The net PMD (or DGD) obtained after transit through a length of fiber is called the PMD value (or DGD value) with units typically in picoseconds. The rate of increase in PMD (or DGD) along a length of fiber is known as the PMD (or DGD) coefficient with typical units of picosecond per kilometer (for very short fibers) or picosecond/km^{1/2} for long fibers.

Because any polarized lightsource consists of a range of wavelengths, after propagation through a length of fiber a range of polarizations will be present, one for each wavelength.

In this way, a source will depolarize because of its own finite line width (the PMD required for depolarization is roughly equal to the source coherence time).

### Modeling and Simulation

The evolution of the lightwave polarization is most conveniently studied in terms of Stokes vectors in Poincaré space. The three-dimensional normalized Stokes vector, \(\bar{s}=(s_1,s_2,s_3)\), consists of combinations of the time-harmonic transverse electric field components for the lightwave:

\[\tag{2.26}s_1=\frac{|E_x|^2-|E_y|^2}{|E_x|^2+|E_y|^2},\qquad{s_2}=\frac{2\text{Re}\{E_xE_y^*\}}{|E_x|^2+|E_y|^2},\qquad{s_3}=\frac{2\text{Im}\{E_xE_y^*\}}{|E_x|^2+|E_y|^2}\]

and polarization-dependent loss (PDL), which describes the case in which the PSPs have unequal attenuation, is neglected.

The polarization at any point in the fiber can be determined from the launch polarization by the Muller Matrix (which, in the absence of PDL, is just a rotation matrix) \(\overline{\overline{R}}(z,\omega)\), where \(\omega\) is the angular frequency of the lightwave and \(z\) is the distance traveled along the fiber from the launch point

\[\tag{2.27}\bar{s}(z,\omega)=\overline{\overline{R}}(z,\omega)\bar{s}(0,\omega)\]

The evolution of \(\bar{s}\) in frequency and axial position can be found by differentiating Eq. (2.27) with respect to \(\omega\) and \(z\), respectively. Eliminating the launch polarization from the resulting equations gives

\[\tag{2.28a}\frac{\partial}{\partial\omega}\bar{z}(z,\omega)=\left[\frac{\partial}{\partial\omega}\overline{\overline{R}}(z,\omega)\right]\overline{\overline{R}}^{-1}(z,\omega)\bar{s}(z,\omega)\equiv\overline{\Omega}(z,\omega)\times\bar{s}(z,\omega)\]

\[\tag{2.28b}\frac{\partial}{\partial{z}}\bar{z}(z,\omega)=\left[\frac{\partial}{\partial{z}}\overline{\overline{R}}(z,\omega)\right]\overline{\overline{R}}^{-1}(z,\omega)\bar{s}(z,\omega)\equiv\overline{W}(z,\omega)\times\bar{s}(z,\omega)\]

where the operator identities for the polarization dispersion vector (PDV), \(\overline{\Omega}\), and the material birefringence, \(\overline{W}\), are

\[\tag{2.29a}\overline{\Omega}(z,\omega)\times\equiv\left[\frac{\partial}{\partial\omega}\overline{\overline{R}}(z,\omega)\right]\overline{\overline{R}}^{-1}(z,\omega)\]

\[\tag{2.29b}\overline{W}(z,\omega)\times\equiv\left[\frac{\partial}{\partial{z}}\overline{\overline{R}}(z,\omega)\right]\overline{\overline{R}}^{-1}(z,\omega)\]

The cross-product model works because with no PDL, any change in \(\bar{s}\) can only result in a change in orientation (magnitude is fixed at unity). Thus, \(\bar{s}\) and its derivatives are perpendicular.

A dynamical equation for the PDV can be found from Eqs. (2.28a and 2.28b):

\[\tag{2.30}\frac{\partial}{\partial{z}}\overline{\Omega}(z,\omega)=\frac{\partial}{\partial\omega}\overline{W}(z,\omega)+\overline{W}(z,\omega)\times\overline{\Omega}(z,\omega)\]

Under the assumption of a suitably narrow source spectral width, the PDV can be written as a power series in frequency:

\[\tag{2.31}\overline{\Omega}(z,\omega)\approx\overline{\Omega}(z,\omega_0)+(\omega-\omega_0)\frac{\partial}{\partial\omega}\overline{\Omega}(z,\omega_0)+\ldots\]

where the terms on the right-hand side of Eq. (2.31) are the first and second-order PDVs.

Using Eq. (2.31) in Eq. (2.30), one can obtain equations for each PDV order. In particular, \(|\overline{\Omega}(z,\omega_0)|\) is the first order DGD and \(\frac{\overline{\Omega}(z,\omega_0)}{|\overline{\Omega}(z,\omega_0)|}\) is the slow PSP at position \(z\).

Early models considered the fiber to be a stack of wave plates with Gaussian distributed birefringence components and fixed axial length. This model requires a physical average over optical wavelength or random birefringence to obtain the PMD.

Another approach is to directly form stochastic differential equations that can be solved for expectation values of the polarization dispersion vector. The stochastic equation approach is the most commonly used today.

From a variety of models, the PMD as a function of fiber length is found:

\[\tag{2.32}PMD=\sqrt{2}h\frac{\partial\Delta}{\partial\omega}\sqrt{e^{-z/h}+z/h-1}\]

where the PMD is defined as the rms value of the magnitude of the stochastic PDV, \(h\) is the fiber correlation length, and \(\Delta\) is the expectation value of the birefringence magnitude.

This equation shows the expected linear growth of PMD for short fiber lengths leading to growth of PMD depending on the square root of the fiber length at long lengths (when the DGD due to short segments of fiber add statistically to previous segments).

### Control of PMD in Fiber Manufacturing

Asymmetric stresses or noncircular core geometries are difficult to completely eliminate in optical fiber manufacturing. Even worse, such non-uniformities in fiber cross-section may tend to be rather uniform along the fiber length. This causes the resulting birefringence to be deterministic on possibly kilometer-length scales. The resulting PMD can be unacceptably large.

In addition, the fiber becomes very sensitive to environmental conditions (temperature, cabling stresses, or movement). Thus, it is very hard to predict the PMD performance for the end-user.

The origin of noncircular core geometry is usually related to non-uniform materials or processes in the preform manufacture. Fibers made with the MCVD, OVD, or PCVD methods require the collapse of a hollow glass cylinder at some point in manufacture of the core.

Careful control of the collapse process must be maintained to avoid introduction of excess ovality. The MCVD and PCVD processes, most capable for fabrication of complex index profiles, use starting tubes that must be specified to be highly circular and of uniform wall thickness to avoid resulting geometrical imperfections in the resulting fiber.

A source of both geometrical and stress non-uniformity is the presence of trapped vapor bubbles in the preform, which translate to airlines in the optical fiber. Various methods have been devised to screen performs for bubbles and fiber for airlines.

In spite of careful process control, there is inevitably some non-uniformity to the fiber cross-section, in either the geometry or the stress profile. The only known practical solution to this problem is to spin the fiber during the draw process.

The most effective way to spin the fiber is to use a device to rotate the fiber just below the draw furnace. This causes the fiber to rotate, as a rigid body, with the molten glass at the preform tip accommodating the deformation (and relieving most stresses).

As the glass transitions to a solid, the spin is ‘‘frozen-in’’ with little residual stress. The transmission effect is to average out the azimuthal non-uniformities experienced by the lightwave.

For spinning to be effective, it must suitably randomize the deterministic birefringence and do so on a length scale less than a beat length (otherwise the lightwave polarization will tend to follow the spin as in the case of a weakly twisted polarization maintaining fiber).

Optimal spin parameters have been studied theoretically (beyond the notion of making several fiber spins per beat length), although in practice little improvement is found because of random birefringence effects and production uncertainties. Spinning can be accompanied by mechanical twist.

This term is used to describe the case where the fiber is twisted after the glass has solidified. Here, the twist is accompanied by an elastic stress, which results in a circular birefringence, because of the stress-optic effect and unwanted DGD.

Generally, a small amount of mechanical twist lowers DGD in an unspun fiber, while the DGD in a well-spun fiber is always increased by mechanical twist. The sensitivity of fibers to twist is relatively independent of fiber type.

Mechanical twist levels, as small as one twist per meter, can noticeably degrade the PMD performance of modern fibers. Twist control can be maintained by ensuring proper functioning of the spin device and alignment of the draw/spin/takeup process. Some process monitoring is required to maintain low twist production.

### Measurement of PMD

PMD measurement is complicated by several factors including the statistical nature of the phenomenon and its sensitivity to the surrounding environment (temperature or mechanical stress changes).

It is useful then to describe measurement of spun and unspun fiber, the fiber environment during the measurement, and the way the DGD averaging is accomplished.

The standardized reference test method is based on Jones Matrix Eigenanalysis (JME). Test sets based on this method measure DGD over a limited wavelength range.

Other test methods include those based on interferometry and wavelength scanning (see list of useful standards at the end of this section for information on these methods).

From the foregoing discussion, it is clear that the level of mechanical twist in the fiber must be accurately assessed before making claims of the intrinsic fiber PMD.

In addition, spool-based measurements (even large diameter or collapsible spools) are unreliable on unspun fiber because inadvertent mode coupling occurs and has an enormous effect on the PMD (PMD generally appears much lower than it really is).

Figure 2.12 shows some measurements that indicate that this PMD difference can be as large as an order of magnitude.

The spooled geometry provides several fiber stresses, which lead to mode coupling. These are bending, lateral stress (caused by winding tension), and fiber crossovers.

These effects occur on a scale length of tens of centimeters, much smaller than the correlation length of a typical unspun fiber. Because of this, mode mixing occurs and the fiber PMD is lowered.

When the fiber is unspooled (e.g., during the cabling operation), it returns to its former, weakly coupled, state and the PMD once again is large. Modern well-spun fiber can have very low PMD, often well below 0.02 ps/km^{1/2}.

Accurate measure of PMD for this fiber is difficult because spool effects usually raise the PMD (because of fiber bending and tension), and instrument bandwidths are generally insufficient to reduce the measurement uncertainty to an acceptable value.

Figure 2.13 shows the PMD difference obtained on a group of non-zero dispersion-shifted fibers (NZDFs) when measured on a typical shipping spool (elevated PMD) and on a large diameter collapsible spool with manual disturbance.

Fortunately, spun fiber is less susceptible to small random external effects, so some approximate methods can be employed. One is a loose winding of fiber on a large (generally >30-cm) diameter spool. Depending on fiber design, the resulting fiber crossovers may or may not mask the true fiber PMD.

Other methods include spreading the fiber on a large flat surface (usually >10m diameter) or measuring the fiber in a quiescent cable design such as a low fiber count central core.

With 20-km lengths of fiber (or cable), one can lower fundamental statistical measurement uncertainties to below 30%. To reduce this uncertainty further, manual disturbances have been used to randomize the birefringence and obtain more independent DGD samples.

Because no two fibers (or cables) from a given manufacturer are exactly alike, either among a production run or when comparing the same fiber in the factory and the field, there is doubt as to the actual quality of the product the end-user receives.

One standardized means of specification is to use the link design value (LDV or PMD_{Q}). This statistical specification creates virtual links by randomly selecting and concatenating \(M\) fibers from the production fiber PMD distribution, then determining the probability of a maximum PMD being exceeded.

A typical specification may state that the LDV for a given fiber product is 0.04 ps/km^{1/2} for \(M=20\) and \(Q=10^{-3}\). This says that when 20 fibers are randomly concatenated to form links, only 0.1% (\(Q^*\)100) of these links will have PMD above 0.04 ps/km^{1/2}.

Clearly, a specification that requires more fibers to be used (larger \(M\)) or higher fraction of fibers exceeding the specification (high \(Q\)) is weaker than a low \(M\), low \(Q\) specification.

The computation of LDV is often done using the Monte Carlo technique, but for reasonably smooth distributions (with single maxima), an analytical method due to Jacobs provides accurate results.

The concatenation rule for PMD coefficients (note that this does not work for DGD) is

\[\tag{2.33}\text{PMD}_\text{total}=\sqrt{\frac{\sum_{k=1}^M\text{PMD}_k^2L_k}{\sum_{k=1}^ML_k}}\]

where \(\text{PMD}_k\) and \(L_k\) are the PMD coefficient and length of the \(k\)th fiber in the concatenation and \(M\) is the total number of fibers.

### Fiber-to-Cable-to-Field PMD Mapping

Ensuring good PMD performance in cable requires adequate process control in fiber and cable production, as well as sufficient measurement to sample production populations.

A particularly useful measurement is one that follows fibers through the cabling process and installation, thereby producing a mapping function that, on an individual fiber basis, compares changes in PMD due to cabling and installation.

Figure 2.14 shows a typical mapping between uncabled and cabled fiber for NZDF fiber in a central core cable. Manual disturbance is very helpful in reducing uncertainties in these measurements.

Figure 2.15 shows the results of a blind test run on a cable with two identical tubes of 12 unique fibers.

In Fig. 2.15a, the PMD for the corresponding fibers in each tube do not appear related. After 10 manual disturbances, Fig. 2.15b shows that the measurement uncertainty has been reduced to the point where the identical fiber pairs are obvious.

In field tests and in fiber production, manual disturbance of the sample is usually not possible. In these cases, a large number of samples are used (with the longest lengths possible) to estimate the PMD mapping.

Although this type of measurement runs the danger of confusing DGD and PMD, the results have been quite successful. An example is the data obtained for a 40 Gb/sec network built by MCI.

Useful International Electrotechnical Commission (IEC) documents on PMD system and measurement include the following:

- IEC 60793-1-48: Optical fibres—Part 1–48: Measurement methods and test procedures—Polarization mode dispersion
- IEC 61280-4-4: Fibre optic communication subsystem basic test procedures—Part 4–4: Cable plants and links—Polarization mode dispersion measurement for installed links
- IEC 61290-11-1: Optical amplifier test methods—Part 11–1: Polarization mode dispersion—Jones matrix eigenanalysis method (JME)
- IEC 61290-11-2: Optical amplifiers—Test methods—Part 11–2: Polarization mode dispersion parameter—Poincaré sphere analysis method
- IEC/TR 61282-3: Fibre optic communication system design guides—Part 3: Calculation of polarization mode dispersion
- IEC/TR 61282-5: Optical amplifiers—Part 5: Polarization mode dispersion parameter—General information
- IEC/TR 61282-9: Fibre optic communication system design guides—Part 9: Guidance on polarization mode dispersion measurements and theory.

The next tutorial discusses about ** microbending loss**.