## Abstract

We describe a new time-domain method for determining the vector components of polarization-mode dispersion from measurements of the mean signal delays for four polarization launches. Using sinusoidal amplitude modulation and sensitive phase detection, we demonstrate that the PMD vector components measured with the new method agree with results obtained from the more traditional Müller Matrix Method.

© Optical Society of America

## 1. Introduction

Polarization-mode dispersion (PMD) phenomena in lightwave systems are the collective
result of birefringence in the transmission path. They lead to pulse distortion and
system penalties, particularly for high bit-rate transmission. The PMD of a fiber is
commonly characterized [1,2] by two specific orthogonal states of polarization called the
principal states of polarization (PSP’s) and by the differential group
delay (DGD) between them. These entities can be encapsulated in the three-component
Stokes vector,
$\overrightarrow{\tau}$
=Δ*τ*
$\overrightarrow{p}$
, where $\overrightarrow{p}$
is a unit Stokes vector pointing in the
direction of the slower PSP, and the magnitude,
Δ*τ*, is the DGD. The vector
$\overrightarrow{\tau}$
(*ω*) usually
varies with optical angular frequency, ω. Typical DGD values encountered
in transmission systems range between 1 ps and 100 ps. Note that the
$\overrightarrow{\tau}$
(*ω*) vector
that we use to characterize PMD here is defined in a right-circular Stokes space
where right circular polarization corresponds to Stokes parameter
*S*
_{3}=+1. The vector
$\overrightarrow{\tau}$
(*ω*) is
closely related to Poole’s $\overrightarrow{\Omega}$
(ω) vector [1], which is defined for left-circular Stokes space (see
Appendix).

Measurement methods that determine PMD vectors with sufficient frequency resolution and signal-to-noise ratio (SNR) include Jones Matrix Eigenanalysis (JME) [3] and the Müller Matrix Method (MMM) [4,5]. Both of these frequency-domain techniques require differentiation with respect to frequency of measured matrix data. Note that PMD vector information is required for the determination of second- and higher-order PMD. When only DGD information is needed, other methods can be employed such as the technique using sinusoidal amplitude modulation described by Williams [6]. Here we discuss a time-domain method for measuring PMD vectors, the Polarization-Dependent Signal Delay Method (PSD Method) [7]. After describing the method, we present experimental results and compare them to data obtained by more traditional methods, and finally discuss some advantages offered by the method.

## 2. Theory

Figure 1 shows an example of the delay imposed on a pulse as it traverses a system. The green curve in Fig. 1 shows results similar to those that would be obtained when applying a delay measurement to a fiber span having +124 ps/nm of dispersion and no PMD. By taking the derivative of the measured delay with respect to wavelength, the chromatic dispersion can be determined. Any PMD present in the fiber can increase or decrease the delay by as much as half the DGD. The red curves in Fig. 1 show examples of the maximal and minimal delays for a fiber span having a dispersion of +124 ps/nm and a mean PMD of 35 ps. The upper curve is the delay that would be observed when launching a pulse with polarization aligned with $\overrightarrow{\tau}$ (ω) at each wavelength, while the lower red curve would be observed when launching a pulse aligned with -$\overrightarrow{\tau}$ (ω) at each wavelength. The delay measured for a fixed launch polarization will occupy the region bounded by the red curves. The markers in Fig. 1 show the delay measured for the specific launch polarization, Ŝ1, an axis of the Poincaré sphere in Stokes space. It can be seen that at some wavelengths, the delay for this launch is nearly equal to that of the fast PSP, while at other wavelengths, it is nearly equal to that of the slow PSP. Usually a launched polarization is a combination of the two PSP’s, so the delay lies somewhere between the two extrema. In fact, it can be shown that the delay is a power-weighted average of the two extreme delays, where the weights are the fractions of power launched into the two PSP’s. This is one of the arguments leading to Eq. (1) below.

#### 2.1 Moments

The PSD method uses the polarization dependence of the pulse propagation time to determine the PMD vector. To extract PMD vector information from measurements such as those shown by the blue markers in Fig. 1, the PSD method uses a result of Mollenauer, et al. [8], relating $\overrightarrow{\tau}$ , the launch polarization, and the mean signal delay,

Here $\overrightarrow{s}$
and $\overrightarrow{\tau}$
are, respectively, the Stokes vectors of the light and the PMD vector at the
fiber input. The mean signal delay, τ
_{g}
, is defined by the first moment of the pulse envelope in the time
domain [8–10], while τ
_{o}
is the polarization-independent delay component. More precisely, τ
_{g}
is expressed as the difference between the normalized first moments at
the fiber output and input,

where z is the distance of propagation in the fiber. Here
*W*=∫*dt*
$\overrightarrow{E}$
^{†}
$\overrightarrow{E}$
=∫*d*ω*Ẽ*
^{†}
*Ẽ*
is the energy of the signal pulse represented by the complex field vector
$\overrightarrow{E}$
(*z*,*t*) with
Fourier transform
*Ẽ*(*z*,ω), and
*W*
_{1}(*z*)=∫*dtt* $\overrightarrow{E}$
^{†}
$\overrightarrow{\tau}$
=*j*∫*d*ω*Ẽ*
^{†}
*Ẽ*
_{ω}
is the first moment. Eq. (1) assumes that τ
_{o}
and $\overrightarrow{\tau}$
(ω) do not vary significantly
over the bandwidth of the signal. If this approximation is not valid, Eq. (1) can be generalized by expressing the right-hand
side in an integral over *ω* [9,11].

Mean signal delay measurements at four appropriately chosen input polarizations
allow the determination of τ
_{o}
and the vector $\overrightarrow{\tau}$
. For example,
consider the case of the four launch polarizations coinciding with the
Poincaré sphere axes, *Ŝ*
_{1},
-*Ŝ*
_{1},
*Ŝ*
_{2}, and
*Ŝ*
_{3}, with corresponding mean signal
delay measurements τ_{g1},
τ_{g(-1)},
τ_{g2}, and
τ_{g3}. Then τ
_{o}
and the vector components of $\overrightarrow{\tau}$
follow from Eq. (1):

where *i*=1,2,3. Eq. (3) can be generalized to any four non-degenerate input
polarization launches $\overrightarrow{s}$
_{i}
(*i*=1,2,3,4) that span Stokes space. The corresponding signal
delay measurements, *τ*_{gi}
, can be grouped
into a four-dimensional (4-D) vector $\overrightarrow{\tau}$
_{g}
=(τ_{g1},τ_{g2},τ_{g3},τ_{g4}),
and similarly, the desired delay components can be grouped into a 4-D vector
$\overrightarrow{\tau}$
_{4D}=(τ_{0},τ_{1},τ_{2},τ_{3}).
The four launches together with Eq. (1) generate the required four linear equations for the
four unknowns represented by $\overrightarrow{\tau}$
_{4D}. Thus, $\overrightarrow{\tau}$
_{g}
=*X*$\overrightarrow{\tau}$
_{4D}, where
*X* is a 4×4 matrix containing the components of
the launched polarizations,

Here *s*_{ij}
is the jth component of polarization launch
$\overrightarrow{s}$
_{i}
. The equivalent of Eq. (3) is then
$\overrightarrow{\tau}$
_{4D}=*X*
^{-1}$\overrightarrow{\tau}$
_{g}
, and therefore τ
_{o}
and the other vector components of
$\overrightarrow{\tau}$
_{4D} can be determined by inverting the matrix
*X* at each frequency of interest.

#### 2.2 Sinusiodal modulation

Instead of determining moments from pulse-shape measurements, the signal delays, τ
_{gi}
, can also be obtained by observing phase shifts of transmitted
sinusoidal signals and benefiting from the precision of sensitive
phase-detection techniques. (Note that Williams [6] has used sinusoidal modulation for the determination of
the scalar DGD.) For most purposes, it suffices to only measure changes in τ
_{gi}
with polarization and optical frequency and to not resolve the
ambiguity presented by the use of a signal with periodic modulation. For a
sinusoidal signal, with the assumption of frequency-independent PSP’s
and DGD, we can use the dot-product rule [11] to determine the power fractions,
1/2(1+$\overrightarrow{p}$
·$\overrightarrow{s}$
)
and
1/2(1-$\overrightarrow{p}$
·$\overrightarrow{s}$
),
coupled into the slow and fast PSP’s, respectively. The phase of the
detected sinusoidal output intensity, ϕ_{out}=ω
_{m}
τ_{ϕ}, defines a signal delay
τ_{ϕ} that obeys the relation

where ω
_{m}
is the angular modulation frequency. For small ω
_{m}
, the signal delays τ_{ϕ} (defined by
sinuosidal phase) and τ
_{g}
(defined by momenta) are approximately equal. When the four launch
polarizations coincide with the Poincaré sphere axes,
*Ŝ*
_{1},
-*Ŝ*
_{1}, *Ŝ*,
and *Ŝ*
_{3}, Eq. (3) is still valid for sinusoidal modulation: ${\tau}_{o}=\frac{1}{2}\left({\tau}_{\varphi 1}+{\tau}_{\varphi \left(-1\right)}\right)$. Recalling that
${p}_{1}^{2}$+${p}_{2}^{2}$+${p}_{3}^{2}$=1,
expressions for τ*Δ* (the magnitude of
$\overrightarrow{\tau}$
) and the components,
*p*_{i}
, of the unit vector pointing in the
direction of $\overrightarrow{\tau}$
can also be determined
and solved exactly.

Like Eq. (3), Eq. (5) can be generalized to any four non-degenerate input
polarization launches
$\overrightarrow{\tau}$
_{i}
(*i*=1,
*a*, *b*, *c*) that span Stokes
space. We orient the input Stokes space so that
$\overrightarrow{s}$
_{1}=*Ŝ*
_{1}.
Then, with
$\overrightarrow{s}$
_{a}
=(*a*
_{1},
*a*
_{2}, *a*
_{3}),
$\overrightarrow{s}$
_{b}
=(*b*
_{1},
*b*
_{2}, *b*
_{3}), and
$\overrightarrow{s}$
_{c}
=(*c*
_{1},
*c*
_{2}, *c*
_{3}), we measure
the corresponding delays τ_{ϕ1},
τ_{ϕa},
τ_{ϕb}, and
τ_{ϕc}. We first express
*Ŝ*
_{1},
*Ŝ*
_{2}, and
*Ŝ*
_{3}, in terms of
$\overrightarrow{s}$
_{a}
,
$\overrightarrow{s}$
_{b}
, and
$\overrightarrow{s}$
_{c}
,

where the coefficients *α*_{i}
,
*β*_{i}
, and
*γ*_{i}
are obtained from

Note that $\overrightarrow{s}$
_{a}
,
$\overrightarrow{s}$
_{b}
, and
$\overrightarrow{s}$
_{c}
should not lie in a common plane.
If $\overrightarrow{s}$
_{a}
,
$\overrightarrow{s}$
_{b}
, and
$\overrightarrow{s}$
_{c}
are coplanar, for purposes of
analysis, rotate to a different Stokes space with
*Ŝ*
_{1}’ aligned with one of
the other launch polarizations.

Substituting
*Ŝ*
_{1}=*α*
_{1}
$\overrightarrow{s}$
_{a}
+*β*
_{1}
$\overrightarrow{s}$
_{b}
+*γ*
_{1}
$\overrightarrow{s}$
_{c}
into Eq. (5) gives

leading to a transcendental equation for τ
_{o}
:

A first trial for τ_{0} can be obtained by linearizing Eq. (9),

After solving for τ_{0} and using
${p}_{1}^{2}$+${p}_{2}^{2}$+${p}_{3}^{2}$=1,
an expression for the magnitude, Δτ, can be found:

The components, *p*_{i}
, of the PMD vector,
$\overrightarrow{\tau}$
, can then be determined from

Although the above procedure will yield any computational accuracy desired, it is
often not necessary. For small modulation frequencies, ω
_{m}
, we can approximate tan
(*x*)⋍*x* in Eq. (5) and use the procedure outlined above (i.e. Eq. (3–4)). These linear expressions are valid
for sinusoidal modulation to within 6% as long as ω
_{m}
Δτ<π/4. For instance, for
the peak DGD we observed here, 88 ps, better than 6% accuracy will be obtained
for modulation frequencies, f
_{m}
=ω
_{m}
/2π, less than 1.5 GHz.

## 3. Experiment

We implemented the PSD Method using sinusoidal modulation and the apparatus shown in Fig. 2. A tunable laser provided the signal light, which was
amplitude modulated by a LiNbO_{3} modulator typically operating at 1 GHz
with an extinction ratio of 1/3. A wavemeter monitored the carrier wavelength. Three
of the four launch polarizations were sequentially provided by a portion of a
commercial polarization analyzer instrument containing three linear polarizers that
could be switched into the beam. The fourth polarization was set by polarization
controller 2 (PC2) and obtained by switching all three polarizers out of the signal
path. In principle, the fourth polarization could have been provided by a polarizer
as well. The phase of the modulation was measured with a network analyzer. The
apparatus contained a polarimeter to allow PMD measurements by the MMM to verify the
accuracy of the PSD method. The polarimeter also provided a convenient monitor when
adjusting PC2.

The launch polarizations were three linear polarization states at
0°(=*Ŝ*
_{1}), 60.4°,
and 120.6°, and right-circular polarization,
*Ŝ*
_{3}. Figure 3 shows the wavelength dependence of the signal delays
for these four inputs in a fiber span with 35-ps mean DGD. The span included an 8-km
length of high-PMD dispersion compensating fiber and a 54-km length of standard
fiber to compensate the chromatic dispersion to a residual dispersion of
+124 ps/nm at 1542 nm. The fiber’s PMD causes the polarization
dependence of the signal delays, and the variations in these relative signal delay
curves originate from the change of direction and magnitude of
$\overrightarrow{\tau}$
(ω) as well as from the underlying chromatic
dispersion. Because the signal delays through the 62-km fiber span are about 300
µs, relative signal delays have been plotted in Fig. 3, by subtracting a reference delay from all measured
data. The *Ŝ*
_{1}, 60.4°, and
120.6° delay curves shown in Fig. 3 have been corrected for the ≈2 ps
additional delay caused by the thickness of the polarizers. The data in Fig. 3, however, have not been corrected for drift caused by
changes in the optical length of the fiber during the course of the measurement.

## 4. Results and Discussion

Figure 4 shows both the input PMD vector
$\overrightarrow{\tau}$
(ω) and the relative
polarization-independent delay, τ_{0R}, for the
fiber span, obtained from the data of Fig. 3. We used the linear approximation to the tangent, i.e. Eq. (4) and ${\overrightarrow{\tau}}_{4D}={X}^{-1}{\overrightarrow{\tau}}_{\varphi}$, where
τ̃_{ϕ}=(τ_{ϕ1},τ_{ϕ2},τ_{ϕ3},τ_{ϕ4}).
Here, τ_{0R} is the polarization-independent
delay through the fiber span at each wavelength compared to the delay at the center
wavelength, 1542.0 nm. The modulation frequency was 1.0 GHz, providing ω
_{m}
Δτ=0.55 at the maximum DGD. The measurement points
were separated by 0.02 nm. We show a scan of only 1 nm so that the magnitude and
components of $\overrightarrow{\tau}$
(ω) are
discernible. The drift in fiber length was tracked and corrected by performing a
delay measurement at the center wavelength every five steps. Also shown in Fig. 4 is an MMM measurement of the input
$\overrightarrow{\tau}$
(ω) (see Appendix) for the
same fiber span. This measurement, using an MMM step-size of 0.03 nm and 0.01-nm
interleave steps, was taken immediately after the PSD measurement. The two
techniques agree to within a maximum discrepancy of 2 ps for the DGD and 4 ps for
the components, where some of the discrepancy is caused by drift of the
fiber’s PMD vector during the measurements. We also computed
$\overrightarrow{\tau}$
(ω) from the PSD data using
the exact expression in Eq. (5). For this measurement using 1-GHz modulation frequency,
the largest deviation of the linear expression from the exact expression was 0.80
ps.

To further investigate the range of applicability of the linear approximation to the
tangent in Eq. (5), we measured signal delays for the same fiber span
using a modulation frequency of 3 GHz. Figure 5 shows a comparison of
$\overrightarrow{\tau}$
(ω) and
τ_{0R} computed from the 3-GHz data
using the exact expression and the linear approximation. For this case, ω
_{m}
Δτ=1.66 at the maximum DGD. The largest difference
between the two DGD curves was 2.79 ps at 1542.14 nm, corresponding to a 5.3%
difference. Note that with a 3-GHz modulation frequency, the modulation sidebands
are spread out over 48 pm, so there is averaging of the signal delay over this
bandwidth.

The PSD method has a number of advantages over the well-known frequency-domain techniques for PMD vector measurement. Generally, the PSD technique benefits from the accuracy of sensitive phase detection compared to the accuracy of a polarimeter. The wavelength step is created electronically by the modulator and is accurately known. In addition, the accuracy of the PSD method is not dependent on the measured difference between two optical frequencies. Because the PMD vector is determined at the wavelength where the measurements are made, the PSD method could eventually allow continuous monitoring of the PMD at a specific wavelength. It also offers more simplicity as the receiver end requires only electronics and not a polarimeter. One disadvantage of the PSD method is that the drift of the fiber length must be corrected. A second disadvantage is that four polarization launches at each optical frequency are required, instead of two launches as with the JME or MMM. Although the PSD method does not yield the fiber rotation matrix, adding a polarimeter at the output would allow this determination, without sacrificing the other advantages of the PSD method.

## 5. Conclusion

We have described a time-domain method for determining first-order PMD vectors by measuring polarization-dependent signal delays. The method also provides a way of accurately measuring chromatic dispersion in the presence of PMD, to our knowledge the only method available for this today. Experimental results show that the PMD vectors obtained by the method agree with those obtained by frequency-domain techniques. No frequency differentiation is necessary for first-order PMD vectors, however, the data appear smooth, allowing differentiation of the vector components to determine second-order PMD vectors as in the Müller Matrix Method.

## 6. Appendix

In the main text of this paper we require the measurement of input PMD vectors
for comparison with the PSD method, and we use right-circular Stokes space
(where *S*
_{3}=+1 for right-circular
polarization), conforming with the traditional optics literature and the
available measurement instrumentation. The discussion of the Müller
Matrix Method (MMM) in Ref. [5], on the other hand, conforms with the PMD literature in
using left-circular Stokes space (where
*S*
_{3}=+1 for left-circular polarization) and
provides an algorithm for the PMD vector at the fiber output. This appendix
provides a bridge between the two.

The right-circular PMD vector, $\overrightarrow{\tau}$ , is aligned with the slow PSP, while Poole’s left-circular PMD vector, $\overrightarrow{\Omega}$ , is aligned with the fast PSP. As a consequence, the relation between the two as expressed by the components of $\overrightarrow{\tau}$ is [11]

The labels R and L in the equation numbers refer to right- and left-circular
Stokes space, respectively. The MMM measures the rotation matrices
*R*
_{0} and *R*
_{+}
at two adjacent optical frequencies ω_{0} and
ω_{+}=ω_{0}+Δω [5]. It then determines the finite difference rotation

at the fiber output. Here *R̃* denotes the transpose of
*R*. From this *R*
_{Δ} the
MMM extracts the output PMD vector
$\overrightarrow{\tau}$
(ω) or the corresponding
$\overrightarrow{\Omega}$
(ω), where
ω=ω_{0}+Δω/2.

Using the label s for the corresponding input quantities one can express the
input PMD vector, $\overrightarrow{\tau}$
_{s}
, as

where *R*(ω) is the fiber’s rotation matrix
at ω given by

with ${R}_{\Delta}=\sqrt{{R}_{\Delta}}\sqrt{{R}_{\Delta}}$. The finite difference rotation
*R*
_{Δs} at the fiber
input is the matrix transform of *R*
_{Δ},

which simplifies with Eq. (A4) to

(Note the symmetry between *R*
_{Δ} and
*R*
_{Δs}.) The PMD
vectors $\overrightarrow{\tau}$
and
$\overrightarrow{\tau}$
_{s}
are extracted from Eqs. (A2) and (A6) following the MMM procedure sketched in the following.
Note that the expressions for *R*
_{Δ} and
*R*
_{Δs} have the same
form for both right- and left-circular Stokes space, while their actual
components are different for the two spaces.

The extraction procedure uses the conventional definitions for the PMD vectors, i.e.

where $\overrightarrow{p}$
and $\overrightarrow{q}$
are the
Stokes vectors of the principal states. The procedure uses the rotational form
for *R*
_{Δ}, which is different for the two
spaces,

where *I* is the 3×3 unit matrix, $\overrightarrow{r}$
$\overrightarrow{r}$
is a dyadic, and
$\overrightarrow{r}$
× is the cross-product operator [5,11]. Here ϕ is the rotation angle and
$\overrightarrow{r}$
the rotation axis. From Eqs. (A8.R) and (A8.L) the cross-product operators for
$\overrightarrow{r}$
follow as

from which the components of $\overrightarrow{r}$ are determined. The final steps of the MMM identify the DGD as

and the principal state vectors $\overrightarrow{p}$ and $\overrightarrow{q}$ as

Note the minus sign in Eq. (A11.L). Due to a misprint this minus sign was
omitted in Ref. [5] (without affecting the data reported). When that paper
was written, the authors were unaware that two different definitions of Stokes
space were in use. Thus, the formulas of Ref. [5] are consistent with Poole’s definition, i.e.
they are left-circular formulas. Yet, the data were measured with a commercial
polarimeter giving right-circular Stokes vectors and, therefore, right-circular
finite difference rotation matrices *R*
_{Δ}.
When extracting $\overrightarrow{r}$
from
*R*
_{Δ}, the minus signs in the two
left-circular formulas, Eqs. (A9.L) and (A11.L), cancel. As a consequence, the experimental
results reported in Ref. [5] are data for the right-circular PMD vector
$\overrightarrow{\tau}$
(and not for $\overrightarrow{\Omega}$
as stated).

## References and links

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