Endless Tracking Polarization Controller
by Kazuhiro Ikeda *, Takeshi Takagi *, Tatsuya Hatano *,Hajime Kazami *, Yu Mimura * and Hiroshi Matsuura *As a technology for increasing the capacity to meet growing communications
demand in recent years, dense wavelength-division multiplexing (D-WDM) is
being implemented in tandem with achieving higher transmission rates on each channel. At nextgeneration transmission rates of 40 Gbps, polarization mode dispersion (PMD) is thought to be
the main factor in the deterioration of transmission characteristics. There is thus a need for endless tracking polarization controllers as constituents of PMD compensators. Here we report thedevelopment of an endless tracking polarization controller using variable Faraday rotators(VFRs), which offers superior optical characteristics and the promise of higher reliability.

ABSTRACT1.

INTRODUCTION

In response to the need for larger capacities in long-haul
optical digital transmission, greater channel density has
been achieved by wavelength division multiplexing(WDM), and the bit rate for each channel is also being
increased.
At 40 Gbps, which is in the next generation of transmission rates, polarization mode dispersion (PMD), which
does not present a problem at bit rates of 10 Gbps andbelow, can impose significant limitations on the distanceover which transmission is possible. Further, at the distances served by 10-Gbps transmission, which is currently
becoming the mainstream technology, there are concernsthat using fibers installed in previous years may, depending on the amount of PMD, cause problems in terms ofsystem configuration.
In recent years there have been many programs tostudy PMD compensators 1)~3) . In PMD compensators,
however, changes in distributed birefringence--induced
either by increasing the core eccentricity of single-mode
fibers or by means of internal stress due to changes in lateral pressure or temperature--give rise to a phenomenon
known as differential group delay (DGD), in which two
orthogonal polarization modes, which should, in theory, bedegenerating in fact diverge.
In PMD compensators, a polarization controller isrequired to track changes in polarization due to distributed
birefringence in the transmission path 1)~3). These changes
in polarization are, due to variations in the transmission
path environment, random with respect to time, with theresult that in order to maintain the compensated state, it is*

Optical Components Development Dept., FITEL-Photonics
Lab.

necessary that the polarization controller track continuously all changes in polarization without becoming saturated(referred to in this paper as “endless tracking”). In particular, when a phase shifting device is used, attention mustbe paid to its control due to the limited range of phaseshifting, a problem that has already been investigated 4),5).
In wave plates that are mechanically rotated or waveguide-type polarization controllers using Lithium niobate(LN), which move in an equivalent manner 6), it is the optical axis (eigen polarization axis) that is rotated, rather
than the phase shift, so that there is no need to pay attention to saturation, but there are problems with the
response speed in the former case and optical characteristics in the latter.
Phase-shifting devices can all be configured in the fiber,and fiber squeezers having superior optical characteristics
have been widely studied in the past 4),5), but since they
apply direct lateral pressure to the fiber they raise theproblem of reliability as an in-line device.
Accordingly we report the development of an endlesstracking polarization controller using variable Faraday
rotators (VFRs), which offers superior optical characteristics and the promise of higher reliability. In the process ofdevice design, it was necessary to investigate two issues:(1) whether the range of phase shifting of the VFR, being
less than that of the fiber squeezer, was adequate or not;and (2) the control method and tracking stability. However,
it is difficult to investigate these two issues if some of thepolarization conversions of the polarization controller are
only discussed as examples on the Poincare sphere, as inpreviously published work. In the present work, we used
expression 7) of the rotation group to express the polarization conversion itself, and since we presented a mathematical treatment of the two issues above, we will also
report on our methodology.

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Endless Tracking Polarization Controller

2.

POLARIZATION CONTROLLER USING
VFRs

Table 1 ranks the main characteristics of the opticaldevices that constitute a polarization controller.
Lithium niobate (LN), as also used in modulators, iscapable of operating at response speeds in the submicrosecond order, but its insertion loss and PDL aregreater than the other devices. Liquid crystal has a slow
response speed (millisecond order) and generally has
problems with respect to reliability. The fiber squeezer
applies lateral pressure to the fiber directly by means of apiezoelectric device, and as an all-fiber device has
extremely low insertion loss, but there are problems inachieving reliability for the stress applying section.
The variable Faraday rotator, on the other hand has anadequate response speed of several hundred microseconds, and since it comprises a garnet crystal, it can provide optical characteristics (insertion loss, PDL, etc.) andlevels of reliability that are of the same order as conventional passive modules, and therefore constitutes a morebalanced polarization controller than the alternatives.
Figure 1 shows the operating characteristics of the VFR.
As shown in Figure 1a), the VFR of itself is, so to speak, avariable circular phase shifter that causes the polarized
light to rotate. On the Poincare sphere it possesses a
rotating function with the S3 axis as the axis of rotation,
but with this alone, arbitrary polarization conversion is
impossible. But by arranging 1/4 wave plates (QWPs)
before and after the VFR so that their optical axes are
orthogonal (see Figure 1b)), it becomes possible to configure a variable linear phase shifter that can vary the phase
Table 1

Optical devices that constitute a polarization controller.
LiquidcrystalFibersqueezer: Satisfactory: FairLN

VFR

Response speed
Reliability
Insertion loss
PDL
Notes

: Superiora)Polarization rotation
proportional to magnetic field

Magnetic field
Garnetcrystal

S3

b)

3.

ANALYSIS OF POLARIZATION CONVERSION BY ROTATION-GROUP PARAMETERS

3.1

Representing Polarization Conversion in
Parameter Space
It is well known that light in the fully polarized state can be
represented as a point on the Poincare sphere, and polarization conversion can be thought of as a path from point
to point. Thus it is easy to understand polarization conversion intuitively by following the paths actually traveled bylight sequentially as trajectories on the Poincare sphere,
but there are an infinite number of such trajectories andthey depend on the polarization state of the input light.
Reaching a systematic understanding of the conversion
structure configured by a polarization controller requires
an understanding of polarization conversion as a rotational conversion from point to point on a Poincare sphere.
That is to say, a polarization controller is a collection ofrotational conversions made by the variable parameters(phase, eigen axis angle), and these are understoodmathematically as a rotation group 7).
This can be confirmed by the fact that the Jones matrix
generally used in calculations of polarization corresponds
to a two-dimensional representation (unitary matrix) of therotation group. That is to say the Jones matrix may generally be given as[a
U = -b–S2
S1

1/4-wave plate (QWP)
Phase difference proportional
to magnetic field

Figure 1

difference between orthogonal linear polarization modes.
This can be easily understood if the Jones matrix for thisconfiguration is analyzed. A variable linear phase shifter
having the arrangement of optical axes shown in Figure1b) will have the function of rotating on axis S2 on the
Poincare sphere.
By combining these variable circular phase shifters andvariable linear phase shifters alternately in multiple
stages, it becomes possible to configure a polarization
controller having an arbitrary conversion function and thecapability of endless tracking.
This phase shifting capability is the result of varying themagnetic field applied to the garnet crystal using a magnetic circuit. The phase shifting range of VFRs currently in
use is π, or 2π in terms of the angle of rotation on the
Poincare sphere. In the discussion of the polarization controller using fiber squeezers previously referred to 4),5) 4π isheld to be necessary, and it is necessary to investigate
whether a device using VFRs can provide endless tracking in the shifting range, and whether its stability is adequate.

Operating characteristics of VFR.

b
–
a

]

–

–

aa+ bb = a 2+ b 2 = 1(1)

Since a and b here constitute complex numbers, we
may restate for a =κ-iν and b = -µ-iλ, yielding[ ] [ ] [ ] [ ]

1 0
0 1
0 -i
1 0
U = κ 0 1 -iλ 1 0 -iµ i 0 -iν 0 -1
= κ E-iλS1 -iµS2 -iνS3

κ 2 +λ 2 +µ 2 +ν 2 = 1

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Endless Tracking Polarization Controller

where: S1, S2 and S3 are Pauli spin matrices.
As will be understood from Equation (2), all polarization
conversions may be defined in terms of four parameters(κ, λ, µ and ν) forming a Jones matrix, and the functionsof the polarization controller are determined by the rangeof values of these parameters, or, in other words, by thesize of the subgroup.
The relationship between the four parameters above (κ,λ , µ and ν ) and rotation in Stokes space may be
expressed as

κ = cosω2,λ = α sinω2,µ = β sinω2ω,ν = γ sin

2

P ( α sin( ω /2), β sin( ω /2), γ sin( ω /2) )νµλ

S: κ≥0S': κ≤0

Figure 2

Parameter space.

a) Stokes space (Poincare sphere)

ν

ν

Sin

C

3.2 Conditions for Arbitrary Conversion
Let us first consider a single set of input and output polarized light. In Figure 3a) input polarized light Sin and output
polarized light Sout are defined in Stokes space. It will be
understood that the rotational conversion at which
Sin→Sout is realized is the totality of conversions having an
axis of rotation (α, β and γ ) facing toward the orientation
on great circle C intersecting the mid-plane of Sin and Sout
and the Poincare sphere, and having the angle of rotation

µ

ω
Sout

Sin

The parameter space shown in Figure 2 is introduced
for the sake of a visual presentation of the range of subgroups of the polarization controller. That is to say, using
λ, µ and ν as a basis for the totality of polarization conversions shown in Equation (2), it is represented by two 3dimensional spaces distinguished by the sign of κ. In sucha case, we can see from Equation (3) that the totality ofpolarization conversions can be expressed by two spherical surfaces extending over the two 3-dimensional spaces
and the points within them. (spheres S and S' are connected by surfaces). Since this expression is a representationof conversion itself, it makes possible both the direct visualization of the conversion structure made by the polarization controller, and a discussion that need not takeaccount of the input polarized light.
Using this parameter space we delineate in the following the conditions for satisfying arbitrary conversion andendless tracking, for the purpose of discussing whether
the former is possible, and the latter is stable.

λC(α , β , γ )(3)where: α, β and γ indicate the orientation of the axes ofrotation, and ω indicates the angle of rotation.

ν

Sin
O

C
Sout

S

Figure 3

µ

Sout

λ
b) Parameter space

λ

µ

S'

Conversion corresponding to a pair of input and output polarization states.

Saturated!
Range for variability ofphase difference

Phase difference(variable, controlled)

Controlled

Eigen optical axisa)Figure 4

Range of variability ofb)phase difference(unwinding, freed from control)

Control procedure for endless tracking (unwinding).

required for Sin→Sout. Figure 3b) shows this as parameter
space.
From Figure 3b) it will be understood that the collectionof conversions that realize S in →S out is a closed curve
extending over S and S' (an ellipse of which the orientation at the mid-point of Sin and Sout is the major axis). In
effect, to realize Sin→Sout, it is sufficient that the subgroupof the polarization controller intersect with the ellipse representing Sin→Sout at at least one point.
Since arbitrary conversion is the function of realizing all
Sin→Sout, the subgroup of the polarization controller intersecting with all ellipses at at least one point constitutes thecondition for whether arbitrary conversion is possible ornot.
3.3 Conditions for Endless Tracking
Before discussing whether endless tracking is possible ornot, we will set forth a control method that avoids saturation using a phase-shifting device with a limited range ofphase shifting. Figure 4 shows the control procedure.
First of all, the phase shifting devices are so arranged
as to be orthogonal alternately to the eigen optical axes,
and the phase difference of each device is controlled within the range of variability. If one of these reaches the limitof the range of variability (e.g. reaches the maximum
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Endless Tracking Polarization Controller

no longer move in a direction such as to further increase
the phase difference. Accordingly the device must be
freed from normal control and must “unwind” in the direction of the center of the range of variability of phase difference. See Figure 4 b). Since this unwinding operation will
obviously affect the polarization state of the output light,
the amount of unwinding is decreased and the amount ofvariation in polarization caused by it must be compensated for by the other phase shifting devices. In other words,
to enable endless tracking even when a phase shifting
device becomes saturated, it is necessary that arbitrary
conversion be realizable using several stages other thanthe unwinding stage.
If we consider the condition for endless tracking in termsof parameter space, it is that the subgroups formed by theremaining devices continue to intersect with all ellipses(satisfy the conditions of arbitrary conversion), when thephase differences of the various phase shifting devices
that make up the polarization controller are forciblychanged, i.e., unwound.

4.

Linear phase shifter
VFR
1/4 wave plate (QWP)

Unwindable

Arbitrary Polarization
Controller

Arbitrary Polarization
Controller
Unwindable?

Figure 5

Unwindable

5-stage polarization controller with VFRs.

θ center=0

CONFIGURATION OF AN ENDLESSTRACKING POLARIZATION CONTROLLER

4.1 5-stage Polarization Controller
In a configuration comprising two phase shifting devices, it
is obvious that when one of them saturates, only theremaining one can be controlled, and the conditions forendless tracking are not satisfied. Not only that, but, if thesubgroups are represented in parameter space, even theconditions for arbitrary conversion are not satisfied. Thusto realize arbitrary conversion it is necessary to have atleast three phase shifting devices, the eigen axes of which
are orthogonal to each other. Even when using a VFR
with range of phase shifting π, the same will be obvious
using a parameter space analysis. It is also possible usingreflection to configure an arbitrary polarization controller
using two VFRs 8).
Thus to provide the function of endless tracking it isnecessary to have at least four phase shifting devices-three for arbitrary conversion and one to compensate forthe unwinding. With four devices, however, the unwinding
operation is complicated, and it is recommended that fivephase shifting devices be used 4),5).
Let us then consider first a 5-stage polarization controller using VFRs. Figure 5 shows the configuration of a
5-stage polarization controller using VFRs. The operational characteristics of the constituent parts are as previously described in Section 2.
In the example shown here, we see that when one of
the four VFRs other than the one in the center unwinds,
three or more VFRs in the remaining section can effectarbitrary conversion, so that unwinding operation can beprovided. The result of calculations of the subgroups produced by the four remaining VFRs when the center one
unwinds are shown in Figure 6. Figure 6 shows the subgroups formed by the angle of rotation of the other VFRs

θ center=π/2

θ center=πθ others=0~2π

Figure 6

S

S'

Subgroups during unwinding at the center VFR.

0~2π when the angle of rotation θcenter in Stokes space
due to the center VFR is fixed.
From Figure 6 it will be seen that the center VFR is also
capable of unwinding, and it will be seen that, in effect, ina polarization controller of the configuration shown in
Figure 5 all VFRs are capable of unwinding, so that endless tracking is possible.
In the foregoing discussion, however, only one VFR iscapable of unwinding, and it is not possible for two or
more VFRs to unwind simultaneously. This polarization
controller must therefore be controlled in such a way that
only one VFR provides unwinding, and this places limitations on the stability of the polarization controller.
That is to say when two VFRs exceed the limits of therange of variation of phase shifting, only one is permitted
to provide unwinding operation, and it is therefore possiblethat the two VFRs will hand off unwinding operation to

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Endless Tracking Polarization Controller

Linear phase shifter

S1

ω1

ω2

ω3

Output polarization
states

28.5deg/s

ω4

ω5

ω6

Polarization
controller
λ/4

λ/2

λ/4

2ms
Is inner product 1?

1ms

Input polarization
states
9.4deg/s 18.2deg/s

Feedback control
31.3deg/s

VFR

QWP

Figure 7

6-stage polarization controller with VFR.

S

S1

10.4deg/s

S'

Figure 9a) No offset

ω 2, ω 4 = 0b) With offset

ω 2, ω 4 = π/4

Figure 8others = 0~2π

others = π/4~9π/4

Simultaneous saturation of two VFRs.

each other, remaining at the extremity of the range. In
such a case there will be conversions that the polarization
controller cannot cover, and tracking stability will be lost.
4.2 6-stage Polarization Controller
At first sight, it might be proposed, as a means of improving stability, that the range of phase shifting be increased
thereby reducing the chance of reaching its limits, but for
the VFRs currently in use this is not desirable, given thedesign of the magnetic circuit. Accordingly it was decided
to configure a polarization controller with one more VFR.
Figure 7 shows the 6-stage configuration.
Let the angles of rotation in Stokes space for each of
the VFRs be ωi, where i is from 1 to 6. At such a time
there are a number of possible combinations of two VFRs
causing a problem when saturating simultaneously, andFigure 8 shows the subgroups that can be formed at theremaining VFRs when ω 2 and ω 4 reach the minimum
phase difference and saturate. Two cases are shown: withno offset applied to the range of variability of the angle ofrotation, and with an offset of π/4.
According to Figure 8, when no offset is applied to thevariable range of the angle of rotation, the conditions forarbitrary conversion are no longer satisfied and it is
impossible to perform tracking with respect to the conver-

λ/4

λ/2

λ/4

Target polarization
states

20.0deg/s

Configuration schema of simulation.

sion of specific input and output of polarized light. Whenconsidered in the way described below, it is an obvious
result. In other words since saying that the angles of rotation of ω2 and ω4 are 0 means that no conversion is performed, with the result that ω1 through ω5 are equivalent toa continuum of five linear phase shifters, and since this is
the same as a single linear phase shifter, it in effect functions only like a 2-stage polarization controller. Arbitrary
conversion is therefore impossible. When, on the other
hand, an offset is applied, parameter space is filled, andthe conditions for endless tracking are satisfied. The
above has also been confirmed for other combinations of
two VFRs.
In the final analysis it will be understood that by using a
6-stage configuration and also applying an offset, it is possible to allow two VFRs to saturate simultaneously, thereby improving tracking stability.
4.3 Simulation
Simulations were conducted to verify the conclusions
arrived at up to this point, and Figure 9 shows the schema
used.
Arbitrary polarization of the input light was produced in aconfiguration consisting of a 1/4 wave plate, 1/2 wave
plate and 1/4 wave plate, rotating at speeds of 9.4, 28.5
and 18.2 deg/s respectively. This light was input to apolarization controller having the configuration shown in
Figure 7, and each VFR was subjected to sequence control. The polarized light output from the polarization controller was then compared with an arbitrary target polarized light produced in the same way as the input polarized
light (speeds of 10.4, 31.3 and 20.0 deg/s respectively),and feedback was applied to the polarization controller sothat its inner product was 1. The times allocated to calculation and comparison of the inner product and to feedback control were 2 ms and 1 ms respectively.
For each of the VFRs 1 step was set at 5.73 deg, while
an unwinding step was 1.15 deg. Only one VFR would
unwind, and when another VFR saturated while one wasunwinding, the VFR that saturated later was unwoundwhile the first VFR to unwind was returned to normal control.
Figure 10 plots the points at which the value of the innerproduct was 0.95 or less over a 4-hour period of tracking

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Endless Tracking Polarization Controllera) 6-stage polarization controller (0.25π offset)

Inner product

1

0.95

0.9

0.85

0.8
0

10

20

Times
a) Visual appearanceb) Results of measurementof output polarizationb) 6-stage polarization controller (no offset)

Inner product1

Number of VFRs

0.95

1.37 dB

PDL

0.9

0.04 dB

Power consumption
0.85

0.6 W per VFR (max)< 200 µs

Response time
Dimensions

0.8
0

20

40

60

90×30×9.5 mmc) Specifications

Times

Figure 11 Prototype polarization controller.

c) 5-stage polarization controller
1

Inner product6

Insertion loss

0.95

0.9

0.85

0.8
0

50

100

Times

Figure 10 Results of simulation (inner product < 0.95).

for three types of polarization controller--6-stage with offset, 6-stage with no offset, and 5-stage.
The number of points at which the inner product was
0.95 or less clearly decreases, in the order of 5-stage, 6stage with no offset, and 6-stage with offset. Also in termsof the minimum value of inner product, there was anincrease in the order of 5-stage, 6-stage with no offset and6-stage with offset. Thus it was confirmed that the 6-stage
polarization controller with offset offered the greatest stability.

5.

PROTOTYPE POLARIZATION CONTROLLER

A prototype polarization controller actually using VFRs
was produced as shown in Figure 11. The VFRs hadphase shifting range with angles of rotation (in Stokes
space) of approximately π /4~9 π /4, forming a 6-stage
polarization controller with offset.
The results of measurements of output polarized light
were obtained using a polarimeter, with a current of 0~100mA applied to each VFR (other VFRs at 0 mA). The factthat the circular trajectories from the various VFRs do notintersect orthogonally is due to an offset being applied.
Low values were realized for insertion loss (1.37 dB)
and PDL (0.04 dB), and it was possible to reduce the size

Figure 12 PMD compensator.

to 90×30×9.5 mm.
Currently we are in the process of producing a prototype
PMD compensator incorporating the polarization controller
described here, and are verifying its operational characteristics (see Figure 12).

6.

CONCLUSION

We have developed a polarization controller with an endless tracking function that makes use of variable Faraday
rotators (VFRs) having superior optical characteristics andhigh reliability. At the design stage we introduced rotationgroup parameters to facilitate discussion of the tracking
stability due to the range of phase shifting of the VFRs
and the number of VFR stages, and set forth a methodology for achieving a systematic understanding of the conversion structure made by the polarization controller. By thismethodology it was also shown that a 6-stage polarization

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Endless Tracking Polarization Controller

controller with an offset applied to the phase shifting range
would in theory operate in a stable manner, and this was
confirmed by simulation. A prototype of this polarization
controller using actual VFRs was also produced, resulting
in a device that is compact and offers good optical characteristics.
It is anticipated that it will be possible through the use ofthis polarization controller to obtain a PMD compensatorthat operates in a stable manner.
Further, it is expected that in the next generation of optical transmission systems there will be increasing demandfor control of the polarization of signal light, not only in
PMD compensators but in non-linear devices, coherent
systems and the like. We feel confident that the methodology described here for polarization conversion and theadoption of polarization controllers based on VFRs offers
a promising way to solve this kind of polarization problem.

REFERENCES
1) T. Takahashi et al.; Automatic compensation technique for timewise fluctuation polarization mode dispersion in in-line amplifiersystems, Electron. Lett., Vol.30, No.4, pp.348-349, 1994.
2) F. Roy et al.; A simple dynamic polarization mode dispersion compensator, OFC’99, TuS4-1, p275, 1999.
3) H. Ooi et al.; Automatic polarization-mode dispersion compensation in 40-Gbit/s transmission, OFC’99, WE5-1, p86, 1999.
4) N. G. Walker et al.; ENDLESS POLARISATION CONTROLUSING FOUR FIBRE SQUEEZERS, Electron. Lett., Vol.23, No.6,
p290, 1987.
5) H. Shimizu et al.; Highly Practical Fiber Squeezer Polarization
Controller, J. of Lightwave Technol., Vol.9, No.10, p1217, 1991.
6) F. Heismann; Analysis of a Reset-Free Polarization Controller for
Fast Automatic Polarization Stabilization in Fiber-optic
Transmission Systems, J. of Lightwave Technol., Vol.12, No.4,
p690, 1994.
7) T. Yamauchi; Rotation Group and Its Expression, Iwanami Shoten
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8) K. Ikeda et al.; An Arbitrary Polarization Controller Using
Variable Faraday Rotator, General Convention of IEICE, B-10-112,
2002. (in Japanese)
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Convention of IEICE, C-3-31, 2002. (in Japanese)10) K. Ikeda; Arbitrary Polarization Controller with Variable Faraday
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