## Abstract

The bandwidth limitations of Poole’s higher-order polarization-mode dispersion (PMD) interpretation are examined. Correlations and errors related to the truncation of the PMD Taylor series are determined by analysis and simulation. As the PMD order increases, the effective bandwidth of the Poole representation is found to grow slowly beyond the bandwidth of the principal state applicable to first-order PMD.

©2003 Optical Society of America

## 1. Introduction

Polarization mode dispersion (PMD) causes system impairment in optical fiber transmission depending on the signal bandwidth and on the PMD vector *τ⃗*(*ω*) of the fiber (see, e.g., [1,2]). When there are significant variations of *τ⃗*(*ω*) as a function of optical frequency *ω*, then higher-order PMD effects need to be considered. A question that is often discussed in the field asks how much increase in effective bandwidth is provided by including higher orders in the PMD representation, and there is considerable variety in the opinions expressed about the proper answer to this question. The answer is, of course, of crucial import for the understanding and design of higher-order PMD emulators as well as PMD compensators, for assessing systems penalties due to higher-order PMD and other PMD issues.

Figure 1 shows the frequency variation of the measured output PMD vector *τ⃗*(*ω*) of a fiber [3]. Here, the PMD vector,

defines the polarization of the slow principal state of polarization (PSP) via the unit Stokes vector *p̂* and the differential group delay (DGD), Δ*τ*. In the widely used Poole interpretation [4,1,2], higher-order PMD is represented by higher-order PMD vectors. These vectors, such as the second-order vector, *τ⃗*_{ω}
, are obtained by successive differentiation of *τ⃗*(*ω*) as indicated by subscripts *ω* or superscript (*n*). The higher-order PMD vectors are the coefficients in the Taylor expansion of *τ⃗*(*ω*) near the carrier *ω*
_{0},

where *ω=ω*
_{0}+Δ*ω*, and Δ*ω* is the frequency deviation from the carrier.

It is the aim of this paper to examine the bandwidth or, more precisely, the spectral range near *ω*
_{0}, over which this higher-order PMD representation can be effectively applied. The well understood [2] effective range for first-order PMD is the bandwidth of the principal state, Δ*ω*_{PSP}
, related to the mean DGD
$\stackrel{}{\Delta \tau}$
of the fiber by

Three aspects of Δ*ω*_{PSP}
are well known (see, e.g., [2,5]): (1) The first order PMD vector *τ⃗*(*ω*) is essentially constant inside the bandwidth of the principal state, i.e. over the range *ω*
_{0}±Δ*ω*
_{PSP}/2; (2) An uncompensated system meets commonly adopted criteria for systems outage if the signal bandwidth does not exceed Δ*ω*_{PSP}
; (3) Higher-order PMD effects are usually negligible when the signal bandwidth is smaller than Δ*ω*_{PSP}
.

The question before us is related to one that is often asked: “Can a higher-order description of PMD increase the effective bandwidth of the Poole representation, or is it just a better description within Δ*ω*_{PSP}
?” While lacking precision, this question opens an important issue. A close examination of this issue addresses the very meaning of the higher-order PMD interpretation and of the effectiveness of higher-order PMD emulation or compensation based on that representation. We shall discuss answers to these questions from three different points of view: 1) examination of the root mean square (rms) magnitudes of the terms of the Taylor expansion, 2) correlation between the vector predictions of the truncated Taylor expansion and the actual PMD vector, and 3) the examination of the mean square magnitude of the (vector) errors made by those predictions.

At this point we should mention that there exist alternate representations of higher-order PMD, including the representations proposed by Bruyère *et al.* [6–9] and those by Eyal *et al.* [10–12]. To our knowledge, the limitations of these alternatives have not yet been fully analyzed.

Scaling frequency with Δ*ω*_{PSP}
, we use a normalized frequency deviation from the carrier, *μ*, defined by

One has, approximately,
$\mu \approx \frac{1}{2}\frac{\Delta \omega}{\Delta {\omega}_{\mathit{PSP}}}$
. Note that our definition of *μ* multiplies the *μ* - parameter used in Ref. [4] by the frequency deviation Δ*ω*. This parameter greatly simplifies the mathematical expressions needed for our discussion. For example, the normalized auto correlation [13,14] of the first-order PMD vector takes the simple form

As indicated, the normalization is the mean square DGD,
$\Delta {\tau}_{\mathit{rms}}^{2}=\u3008{\tau}^{2}\u3009=\left(\frac{3\pi}{8}\right)\stackrel{}{\Delta {\tau}^{2}}$
. Together with experimental results, this auto correlation function has been the basis for the definition of Δ*ω*_{PSP}
(see, e.g., [2]), and it will play an important role in our discussion of the bandwidth of higher-order PMD. The subscript provides distinction from the higher-order correlation functions to be discussed below.

## 2. Magnitude of Taylor series terms

The results of Shtaif *et al.* [14] allow a determination of the root-mean-square magnitudes of the higher-order terms in the Taylor series in Eq. (2) [5]. They find the mean square magnitudes 〈*τ⃗*(*n*)·*τ⃗*(*n*)〉 of the higher-order PMD vectors, *τ⃗*
^{(n)}, on the basis of the first-order correlation of Eq. (5). Using these results, one determines that the rms magnitudes of the terms, normalized by the rms DGD, Δ*τ*_{rms}
, scale like the powers of the normalized frequency, *μ*,

where the subscript (*n*+1) refers to the PMD order. Examples for the lower order magnitudes are *M*
_{1}=1, *M*
_{2}=*μ*, *M*
_{3}=*μ*
^{2},
${M}_{4}=\sqrt{\frac{5}{6}}{\mu}^{3}$
.

Figure 2 shows the rms magnitude *M*_{n}
of the six lowest-order terms as a function of the frequency deviation Δ*υ* for a mean DGD of
$\stackrel{}{\Delta \mathit{\tau}}=1\phantom{\rule{.2em}{0ex}}\mathit{ps}$
where Δ*υ*_{PSP}
=Δ*ω*_{PSP}
/2*π*=125 *GHz*. The full bandwidth of first-order PMD is characterized by the range *ω*
_{0}±Δ*ω*_{PSP}
/2 as discussed before. In the figure, this limit occurs at Δ*υ*=62.5 *GHz* or *μ*≈1/4(0.246). Note that, at that first-order bandwidth limit, the wings of the signal experience a 2^{nd}-order PMD term having an rms value that is 25% that of the first-order component. When the signal bandwidth is doubled to 2Δ*υ*_{PSP}
, or a limit of *μ*≈1/2 i.e. Δ*υ*=Δ*υ*_{PSP}
=125 *GHz*, there is strong presence of third-order effects and a significant appearance of higher orders. Another doubling of the bandwidth to 4Δ*υ*_{PSP}
with a limit of Δ*υ*=2Δ*υ*_{PSP}
, or *μ*≈1, shows the strong appearance of at least six PMD orders. The growth of the rms magnitudes suggests that inclusion of higher PMD orders enlarges the bandwidth over which the Taylor series representation is meaningful. Following the example set by first-order PMD, we define the bandwidth limit of n^{th}-order PMD to be the frequency deviation at which the rms value of the (n+1)^{th}-order term reaches 25% that of the first-order term. This bandwidth grows only slowly with the number (*n*+1) of included orders, increasing approximately as
$\sqrt{\left(n+1\right)}$
. Note that the above bandwidth estimate based on Shtaif’s results applies to the magnitudes of the PMD vectors terms only. The direction of these vectors and their conditional probabilities should be an important factor in determining effective bandwidth and will be included in the improved analyses to be given in the next two sections.

## 3. Correlation of Taylor series prediction

The analyses used for this and the next viewpoint have a close mathematical relationship with the first-order correlation function of Eq. (5). Like the latter, they include higher-order PMD effects, as well as the relative direction of the higher-order PMD vectors, and their conditional probabilities. In this section, we consider the prediction made for the PMD vector *τ⃗*(*ω*
_{0}+Δ*ω*) at a frequency *ω*
_{0}+Δ*ω* by using a truncated Taylor series of the form

containing the higher-order PMD vectors evaluated at *ω*
_{0} and truncated to exclude PMD

orders higher than *n*. Next, we consider the correlation between this prediction and the actual vector *τ⃗*(*ω*
_{0}+Δ*ω*). In analogy with Eq. (5), this correlation function is defined as the normalized average of the dot product between the actual and the predicted vectors,

where the subscript *n* indicates the highest PMD order included in *τ⃗*_{pred}
. The average is
performed over different fiber realizations of equal mean DGD, and is normalized with
respect to 〈*τ*
^{2}〉. From this definition and using Eq. (5) one can derive recursion relations and power series expressions for *C*_{n}
. These are listed in the appendix.

Figure 3 shows a plot of the lowest six correlation functions as a function of frequency for a mean DGD of
$\stackrel{}{\Delta \mathit{\tau}}=1\phantom{\rule{.2em}{0ex}}\mathit{ps}$
. As the *C*_{n}
functions are normalized with respect to the mean square DGD their value at ω_{0} is unity. For large frequency deviations the *C*_{n}
decay to zero as expected. Note that *C*
_{1} stays within 3 percent of unity inside the first-order bandwidth limit of 62.5 GHz, i.e. *μ*≈1/4. Using a ±3 percent change in correlation as a criterion for the higher orders, we note that the prediction based on the next higher orders extends the bandwidth limit to 64 GHz (*μ*≈0.251), 133 GHz *(μ*≈0.522), 239 GHz (*μ*≈0.941) and 225 GHz (*μ*≈0.886) respectively. In the figure we also note that the higher-order correlation functions overshoot the unity value at frequency deviations larger than the allowed bandwidth before finally decaying to zero.

To gain a simple physical picture for the cause of the overshoots consider the diagram of Fig. 4 for the second-order prediction *τ⃗*_{pred}
=*τ⃗*(*ω*
_{0})+*τ⃗*_{ω}
Δ*ω* of the PMD vector *τ⃗*(*ω*
_{0}+Δ*ω*) at *ω*
_{0}+Δ*ω*. We know that the mean squares of *τ⃗*(*ω*
_{0}) and *τ⃗*(*ω*
_{0}+Δ*ω*) are the same and equal to 〈*τ*
^{2}〉. We also know that the perpendicular second-order component *τ⃗*
_{ω⊥} dominates statistically over the component *τ⃗*
_{ω‖} parallel to the first order vector *τ⃗*(*ω*
_{0}) [2]. From the right-angle triangle of the figure it follows that a first estimate of the mean square magnitude of the prediction is 〈${\tau}_{\mathit{\text{pred}}}^{2}$〉=〈*τ*
^{2}〉+〈${\tau}_{\omega \cong}^{2}$〉Δ*ω*
^{2}. For small frequency deviations, therefore, we would expect an overshoot of the predicted magnitude proportional to Δ*ω*
^{2}, and the same for the second-order correlation function. An improvement of this hand-waving argument is given in Appendix B, showing good agreement with the exact results for small frequency deviations.

To summarize the viewpoint of the higher-order correlation functions, we conclude that the useful bandwidth provided by predictions based on the truncated Taylor series increases with the PMD order included. However the increase is slow, as in the earlier analysis. As we shall see below, the increase is also non-monotonic, owing to oscillations of the higher-order correlation functions around unity.

## 4. Error of Taylor series prediction

Our third point of view uses as a bandwidth criterion the average error in magnitude made by the prediction of the truncated Taylor series of Eq. (7),

More precisely, it is the mean square magnitude of the vector difference between the prediction and the actual PMD vector, a measure that is, perhaps, the most meaningful of our three criteria. Here, *E*_{n}
is normalized with respect to 〈τ^{2}〉. Detailed inspection of the two

definitions reveals that the *E*_{n}
and the *C*_{n}
functions are closely related. For the first PMD order, for example, we have

Other relationships between *E*_{n}
and *C*_{n}
are listed in the appendix. As before, we can derive recursion relations and power series expressions for the higher-order *E*_{n}
functions from the above definition. We list those also in Appendix A. Figure 5 shows a plot of the lowest four *E*_{n}
functions as a function of frequency based on this theory and on simulations.

The simulation results are based on frequency averages of virtual fibers containing 200 birefringent sections. The frequency averages encompassed a bandwidth of 200 times Δ*υ*_{PSP}
. The frequency averages were subsequently averaged over 1000 instantiations. Thus, there were approximately 33,000 independent samples for the low-order results [2]. To avoid periodicity in the simulated PMD spectrum the sections were assigned random birefringence with uniform distribution of a width equaling 30% of the mean. The sections were rotated randomly relative to each other over the full Poincaré sphere with a distribution suggested by Menyuk [15]. For frequency offsets less than 5 times Δ*ν*_{PSP}
, we find agreement to better than 0.5% between the results of these simulations and the theoretical predictions. The distributions underlying *E*_{n}
and *C*_{n}
become wider as the frequency offset increases and as *n* increases, requiring increasing sample sizes to maintain accuracy. In the figure we note that the error function *E*
_{1} for first-order PMD reaches a value of 0.059 at the first-order bandwidth limit of Δ*υ*_{PSP}
/2=62.5 GHz (*μ*≈0.25). Using the same error value to define the half bandwidth limit for the higher orders, we find that the inclusion of second-order PMD in the Taylor series more than doubles the bandwidth limit to 129 GHz (*μ*≈0.51), the inclusion of third-order PMD extends the limit further to 168 GHz (*μ*≈0.66), and the inclusion of the

fourth order yields 196 GHz (*μ*≈0.77). In terms of the full bandwidth of the PSP, this means that the first-order bandwidth Δ*ω*_{PSP}
is approximately doubled to 2.1Δ*ω*_{PSP}
when second-order PMD is included. Addition of third-order PMD increases the bandwidth to 2.7Δ*ω*_{PSP}
, and a subsequent addition of fourth-order PMD provides a bandwidth of 3.1Δ*ω*_{PSP}
.

Figure 6 shows the trend of the bandwidth increase up to the 30^{th} PMD order. As the order increases, more PMD orders have to be added to add another Δ*ω*_{PSP}
-unit to the bandwidth. Approximately, one gets a full bandwidth of 2Δ*ω*_{PSP}
by including two PMD orders, 3Δ*ω*_{PSP}
for four orders, 4Δ*ω*_{PSP}
for seven orders, 5 Δ*ω*_{PSP}
for 12 orders, and 6Δ*ω*_{PSP}
by including 19 PMD orders. Improving the bandwidth becomes increasingly difficult for the higher orders. For PMD orders less than 20, a prime-number rule applies: the number of orders that must be added for each subsequent addition of a Δ*ω*_{PSP}
bandwidth unit is the subsequent prime number (i.e., 1, 2, 3, 5, 7). See Table 1.

## 5. Conclusions

We have examined the effective bandwidth increase of the higher-order Taylor Series PMD description from three different points of view. Figure 7 shows the full bandwidth obtained by each of the bandwidth measures for truncated Taylor series of orders 1 through 20. We find that an indication of the bandwidth to be expected from the inclusion of higher PMD orders can be obtained from the correlation (*C*_{n}
in Fig. 7) between the actual PMD vector and the prediction by a truncated Taylor series. However, since the correlation functions oscillate with frequency, bandwidths obtained from them do not increase smoothly with expansion order. This deficiency does not appear in bandwidths obtained using either the rms magnitudes of the Taylor series terms
(*M*_{n}
in Fig. 7) or the mean square magnitude of the vector error (*E*_{n}
in Fig. 7) made by the truncated Taylor series prediction. The latter two measures of expansion error yield remarkably
similar bandwidths given their disparate origins. This similarity can be understood by realizing
that *E*_{n}
is the mean square length of the vector that is the sum of the Taylor series terms of index greater than *n* while *M*
_{n+1} is the rms length of the first term in that sum. To the extent that the first term dominates the summation, *E*_{n}
will nearly equal ${M}_{n+1}^{2}$. The relationship between *E*_{n}
and ${M}_{n+1}^{2}$ is provided in Appendix C.

We have obtained analytical results and made simulations of these mean errors. We find that the effective bandwidth of the PMD description is doubled by the inclusion of second-order PMD relative to the first-order bandwidth of the principal state. Inclusion of four PMD orders triples the bandwidth. Adding even higher orders is found to be decreasingly efficient in improving bandwidth.

## Appendix A: C- and E-functions of arbitrary PMD order

The general definitions for the correlation functions *C*_{n}
of the truncated Taylor series prediction for *τ⃗*(*ω*
_{0}+Δ*ω*) and for the corresponding error function *E*_{n}
for arbitrary PMD order (*n*+1) are given in Eqs. (8,9). The C- and E-functions can be derived from these definitions and the appropriate derivatives of Eq. (5). Here we list the key results for the general recursion formulas and the power series expressions valid for arbitrary PMD order.

The recursion formula for the correlation function of any order is

where *C*
_{1} is the first-order correlation function of Eq. (5). The power series for the correlation function of order (*n*+1) is

where *ν*
_{min} is an integer obtained by rounding downwards,
${v}_{min}=\mathrm{int}\left(\frac{n+4}{2}\right).$
.
The leading terms of the correlation functions of the three lowest orders are
${C}_{1}=1-\frac{1}{2}{\mu}^{2}\dots $
,
${C}_{2}=1+\frac{1}{2}{\mu}^{2}\dots $
,
${C}_{3}=1+\frac{1}{2}{\mu}^{4}\dots $
.
A recursion formula for the error functions E is

The corresponding power series expression for the E-function of order (*n*+1) is

The leading term of this expansion is

Examples for the three lowest orders are
${E}_{1}={\mu}^{2}-\frac{1}{3}{\mu}^{4}\dots $
,
${E}_{2}={\mu}^{4}-\frac{5}{12}{\mu}^{6}\dots $
,
${E}_{3}=\frac{5}{6}{\mu}^{6}-\frac{7}{20}{\mu}^{8}\dots $
Among other useful relationships between the *E*_{n}
and the *C*_{n}
functions we list *E*
_{2}=2+*μ*
^{2}-2*C*
_{2}, *E*
_{3}=2+*μ*
^{4}-2*C*
_{3},
and
${E}_{4}=2-\frac{1}{3}{\mu}^{4}+\frac{5}{6}{\mu}^{6}-{2C}_{4}$
.

## Appendix B: Estimate of overshoot in correlation

Here, we follow up on the discussion of section 3 and provide a more refined estimate for the overshoot of the second-order correlation function *C*
_{2} at small frequency deviations Δ*ω*. We refer to Fig. 4. and attempt to estimate the magnitude of *τ⃗*_{pred}
from the parallel and perpendicular components of *τ⃗*_{ω}
, i.e. *τ⃗*
_{ω‖} and *τ⃗*
_{ω⊥}. We have from the right-angle triangle that
〈${\tau}_{\mathit{\text{pred}}}^{2}$〉=(Δ*τ*+*τ*
_{ω‖}Δ*ω*)^{2}+${\tau}_{\omega \perp}^{2}$Δ*ω*
^{2}.
Averaging over virtual fiber instantiations and realizing that 〈*τ⃗·τ⃗*_{ω}
〉=〈*τ⃗·τ⃗*
_{ω‖}+*τ⃗·τ⃗*
_{ω⊥}〉=〈*τ⃗·τ⃗*
_{ω‖}〉 vanishes [14] we find that
〈${\tau}_{\mathit{\text{pred}}}^{2}$〉=〈*τ*
^{2}〉+(〈${\tau}_{\omega \perp}^{2}$〉+〈${\mathrm{\tau}}_{\omega \perp}^{2}$〉)Δ*ω*
^{2}.
From studies of PMD statistics [4,2] we know this becomes
$\u3008{\tau}_{\omega \parallel}^{2}\u3009+\u3008{\tau}_{\omega \perp}^{2}\u3009=\frac{1}{3}{\left(\u3008{\tau}^{2}\u3009\right)}^{2}$
.
Using the definition for *µ* given in Eq. (4) we get for the normalized mean square prediction
〈${\tau}_{\mathit{\text{pred}}}^{2}$〉/〈*τ*
^{2}〉=1+*μ*
^{2}.

For small *µ*, the rms value of the normalized prediction is approximately
$1+\frac{1}{2}{\mu}^{2}$
. Now we refer to Fig. 4 and assume that the direction of the predicted *τ⃗*_{pred}
agrees closely with that of the actual vector *τ⃗*(*ω*
_{0}+Δ*ω*) at *ω*
_{0}+Δ*ω*. The normalized average of that vector is unity. The correlation function is the average of the dot product between *τ⃗*_{pred}
and *τ⃗*(*ω*
_{0}+Δ*ω*), whence we arrive at the estimate
${C}_{2}=1+\frac{1}{2}{\mu}^{2}$
.
This happens to be exactly the leading term for small frequency deviations we derived by more rigorous method in Appendix A.

## Appendix C: Relation between E_{n} and M_{n}

Comparing the power series for *E*_{n}
given in Eq. (A.4) and the definition of *M*_{n}
in Eq. (6), one finds that *E*_{n}
can be expressed as a sum of squares of the *M*_{n}
-functions of higher order

with the coefficients

The coefficients of the first two leading terms are *m*
_{1}=1 and *m*
_{2}=*-n*/(*n*+2).

## Acknowledgements

The authors are grateful to Curtis Menyuk and Lynn Nelson for valuable discussions. Hong Chen thanks Professor Greg Pottie for support and stimulating discussions.

## References and links

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