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4172 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006

Polarization Stabilization in OpticalCommunications Systems

Mario Martinelli, Member, IEEE, Fellow, OSA, Paolo Martelli, Member, OSA, andSilvia Maria Pietralunga, Member, IEEE, Member, OSA

Abstract—The control of the state of polarization (SOP) of lightremains one of the open issues in optical communications. In par-ticular, the achievement of a stabilization of the SOP can find manyapplications in advanced optical communication systems: from themitigation of polarization-mode dispersion to the development ofnovel multilevel modulation formats. In this paper, theoretical andexperimental aspects of polarization stabilization are dealt with,and a novel algorithm to overcome the issues related to the prac-tical availability of finite-range birefringent components and tosolve the requirement for endless stabilization is also presented. Acomplete analysis of the control algorithm, based on the Jones ma-trix formalism, is also presented. The practical implementation ofthe polarization stabilizer is discussed, and experimental demon-strations based on liquid crystal and magnetooptical retardersare shown.

Index Terms—Optical communications, optical polarization,polarization control and stabilization.

I. INTRODUCTION

MASTERING of the state of polarization (SOP) of opticalsignals is an intrinsic issue to optical transmission and

processing, and it has been a recurrent subject of research inthe past years, leading to different solutions. The interest onthe subject was boosted in the 1980s by the attention paidto coherent detection as a means to obtain a high receiversensitivity [1]–[5], the visibility of the interference being higherthe more the signal and local oscillator SOPs matched. Aspolarization-independent erbium-doped fiber amplifiers wereestablished as the means for improving the quality of trans-mission on long-haul fiber links, polarization control no longerseems to be an issue. Nevertheless, even if the SOP ceased tobe considered a problem in itself for a long time, the statisticalnature of polarization evolution soon turned out to become aconcern: As the demand of high-capacitance long-haul fiberlinks started to increase, polarization-mode dispersion (PMD)began to compete with chromatic dispersion in limiting theproduct bit rate per distance [6]–[10]. At the end of the1990s, in the perspective of massive digital transmission at10 Gb/s and up to 40 Gb/s, architectures for PMD mitigationand compensation were conceived, including typically somemeans to actively track and affect the SOP of transmittedbeams [11]–[13].

Manuscript received May 11, 2006; revised August 31, 2006.M. Martinelli is with the Dipartimento di Elettronica ed Informazione,

Politecnico di Milano, 20133 Milan, Italy, and also with CoreCom, 20133Milan, Italy (e-mail: [email protected]).

P. Martelli and S. M. Pietralunga are with CoreCom, 20133 Milan, Italy(e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/JLT.2006.884228

The rush for huge transmission capacity was suddenlyslowed down by the crisis of the telecommunications market atthe beginning of this century, which made the more advancedand still expensive 40-Gb/s standard somewhat unnecessary; onthe other hand, the newly developed low-PMD fibers [14]–[16]seemed to fully cope with actual demand in terms of quality oftransmission. The combination of these two reasons resulted toa temporary lowering in the interest for polarization control.

However, recently, a new attention is being paid on theachievement of SOP control of transmitted optical signals. Theorigin of this new interest is somewhat complementary to ever-increasing optical signal bandwidth. Polarization is, in fact, afurther degree of freedom of the electromagnetic field, whichcan be used to label optical channels and to add dimensionalityto transmission. For instance, by simply introducing a levelof polarization-division multiplexing (PolDM) in an intensity-modulated link, the number of available optical channels couldbe multiplied by two [17]–[19]. Furthermore, the request formodulation formats to provide high spectral efficiency withinthe assigned channel bandwidth has risen; this has renewedthe interest for coherent detection [20], [21], which is the onlyscheme that allows a complete retrieval of optical field parame-ters. Last but not least, the perspective of massive introductionof new high-capacity single-link connections (at 40 Gb/s ormore) will finally restate the importance of polarization controlas a means to design effective PMD mitigators.

The control of the SOP is essentially an automatic controlissue, implying the realization of a feedback system, i.e., thepolarization controller, to transform the excursions of the inputSOP into a prerequired output SOP, which, in principle, canitself be a variable one. The term polarization stabilizer isalso generally used, with reference to an automated device thattransforms any input SOP into a defined stable output SOP. Itis, however, worth to remark that the terms polarization control,polarization stabilization, and polarization tracking are oftenmeant to be interchangeable.

The approach to polarization control can be pictured on thePoincaré sphere—the powerful representation of any SOP andits evolution [22], [23]. Any point on the surface of the spherecan be bijectively mapped to one SOP, which, therefore, canbe represented by two angular coordinates, i.e., 1) azimuth and2) elevation. Hence, the complete control of the output SOPrequires the setting of a pair of variables, which represents thefirst issue in the control strategy. In other words, algorithm mustbe developed in a two-dimensional (2-D) space and the solutionto the control problem is a minimum on a surface representa-tion. To operate in a 2-D space represents a significant increase

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MARTINELLI et al.: POLARIZATION STABILIZATION IN OPTICAL COMMUNICATIONS SYSTEMS 4173

of complexity, if not from the viewpoint of the control theory,for the practical implementation of the control algorithm. Actu-ally, once the algorithm is found, optical engineers must providethe optical components to implement it, therefore modifying theangular parameters of the SOP in accordance to the defined law.If the input SOP evolves periodically, one can choose amongmany available components. However, typically, light behavesdifferently, and the SOP to be processed evolves unpredictably.It exists one particularly unpleasant occurrence, i.e., when oneof the two angular coordinates, which represents the SOP on thePoincaré sphere, drifts monotonically. This actually happensquite often, e.g., with thermally induced evolution of the SOPin optical fibers. In this case, one would require the opticalcomponents acting on the SOP (and/or the related electronicactuators) to be unlimited and able to track the SOP evolutionwithout resetting and discontinuity, i.e., an endless and reset-free approach. A solution to this problem is represented by theuse of components with rotating birefringent axes [1], [2], [5],[24]–[28]. As an alternative, if common retarders having a finiterange of operation are used, a reset procedure is unavoidablyrequired. In this case, the reset is acceptable only if it doesnot introduce any perturbation on the SOP as perceived by theend-user [29]–[37]. Anyway, the polarization controller mustperform an endless tracking of the SOP.

In the present work, we have approached the SOP controlusing finite-range components. The problem to be solved canbe summarized in the following two issues:

1) algorithm/implementation scheme must be able to copewith the double-variable optimization;

2) algorithm/implementation scheme must perform end-lessly.

In this paper, we introduce a novel principle to achieve end-less polarization stabilization, which exploits the symmetriesof the Poincaré representation for the SOP to simplify the resetprocedure, also leading to a reduction in the number of controlvariables required, with respect to known solutions. Two practi-cal implementations of the principle are also presented, the firstone leading to modules based on linearly birefringent electroop-tical elements, whereas the second one relies on magnetoopticalFaraday rotators.

The working principle of the polarization stabilizer is firstdescribed at the aid of the Poincaré sphere representation inSection II. It is shown that the polarization stabilization requiresthe duplication of two elementary modules, namely 1) a firststage that leads to a partial stabilization of the polarization ona great circle of the Poincaré sphere and 2) a second stagethat brings the polarization free running on the great circleto a specific point, representing the output-stabilized SOP. InSection III, the proof of the validity of this approach to polar-ization stabilization is given in Jones matrix formalism togetherwith a schematic representation of the structure of the modules,as actually realized. As an experimental demonstration of howthe control algorithm operates with separate variables, in orderto stabilize the SOP, in Section IV, it is given a description ofthe practical implementations of the modules, which realize thefirst stage of the control, e.g., the locking of the SOP on theprincipal circle of the Poincaré sphere, and the experimentalresults are discussed.

Fig. 1. Poincaré sphere representation of the action of the VR of the firststage, acting on a generic input SOP, to lock it on the principal circle, by ratherincreasing or decreasing the birefringence retardation within its finite range.

Fig. 2. Visualization of the working principle of the polarization stabilizer byoperation of a first stage and a second stage, each driven by one independentcontrol variable. The first stage performs a first reduction of the polarizationdegrees of freedom by mapping the SOP evolution onto a great circle ofthe Poincaré sphere. The second stage operates on the processed SOP andeventually stabilizes it on a point of the great circle.

II. WORKING PRINCIPLE—POINCARÉ

SPHERE REPRESENTATION

In order to fix ideas, and without any loss of generality, letus consider the situation depicted in Fig. 1, where an inputSOP is represented as a point P , randomly wandering on thesurface of the Poincaré sphere and observed from the NorthPole (assumed to correspond to the left circular SOP |L〉). Thehorizontal, vertical, and diagonal (i.e., oriented at ±π/4 rad)states are also pointed out and are, respectively, indicated as|H〉, |V 〉, |Q〉, and | − Q〉.

Let us suppose that the input light describes the line illus-trated in Fig. 1, where P follows a random walk in movingfrom left to right, and let us suppose that the aim of the controlis to fix the input SOP at the state | − Q〉. The objective can bereached in two steps, as represented in Fig. 2.


Fig. 3. Poincaré sphere representation of the action of the SR of the firststage, acting to swap the input SOP to the opposite hemisphere of the Poincarésphere so that the VR can rewrap its retardation in order to compensate formonotonically increasing accumulation of birefringent retardation.

The first step is to map the trajectory on the surface of thesphere into a linear trajectory and, more precisely, to force theSOP on the great circle crossing the desired output state, e.g.,the circle Γ including the states | −Q〉, |L〉, and |Q〉 in Fig. 1.The device that provides this operation is the first stage ofthe stabilizer. If the light at the output is analyzed by a linearpolarizer set in |V 〉, a 3-dB loss in the intensity is experienced;in general, this single stage can provide a stabilization of thelight on any linear SOP, provided that 3 dB in attenuation isaccepted. The second step toward stabilization involves bring-ing the SOP from any point on the circle Γ to a specific pointof the line; in this case, the point | −Q〉 that represents therequired ouput-stabilized SOP. The device that implements thisoperation is the second stage of the stabilizer. This splittinghypothesis (theoretically permitted by the linearity nature ofthe polarization control operations as implemented by any bire-fringent element) leads to a splitting of the control algorithmitself, which eventually allows to overcome the complexity ofthe control strategy, as pointed out in Section I.

Let us now concentrate on the first objective, e.g., to bringthe light on Γ. If the point P drifts backward with respect to Γ(like P ′ in Fig. 1), a variable retarder (VR) of variable bire-fringence and whose axis is set to 0 rad with respect to the finalSOP to be achieved will provide the missing Θ′ retardation, andvice versa, i.e., if the point P drifts forward with respect to Γ(like P ′′ in Fig. 1), the same VR will act in order to reduceby Θ′′ the accumulated retardation. In summary, the action ofjust one VR is sufficient to track the SOP on the required greatcircle. Its control signal can be extracted by a linear polarizer,which is set in |V 〉, as detailed hereafter. Therefore, after theaction of VR, any variable input SOP will be mapped on Γ, as inFig. 3, and the precision is set by the tolerance of the feedbackcircuit. If the input SOP evolves monotonically toward or awayfrom Γ, the necessity for a reset mechanism arises, since VRcannot indefinitely increase (or decrease) its retardation in orderto compensate for the position of P .

The reset mechanism must act, so as to enable VR to rewrapitself, without altering the detected intensity as measured afterthe polarizer set in |V 〉. In order to solve this second issue, let

Fig. 4. Poincaré sphere representation of the second stage.

us insert another retarder on the light path, i.e., a switchingretarder (SR), whose axes are set at π/4 rad with respect tothose of VR and whose retardation can be switched between 0(OFF state) and π rad (ON state). SR topologically acts on theoptical field before the action of VR so that we will name theSR as VR1 and the VR as VR2, as they are located in sequencealong the optical path. SR switches whenever VR should over-come one of its retardation limits, either 0 or π rad, so as to keepVR within its operation range. In locking conditions, when theretardation introduced by VR equals either limits, the input SOPP itself is necessarily on the great circle Γ. Let S represent thetransformed SOP after VR2. As VR1 is switched ON, S willideally move on Γ around the H–V axis toward the orthogonalpoint S′ on the opposite hemisphere. As will be detailed inSection IV, if VR1 switching speed is orders of magnitudefaster (e.g., 100 times) than the operation speed of VR2, thepoint S actually moves into S ′ on Γ and, consequently, withoutaffecting the intensity detected after the polarizer set in |V 〉.At this point, let us imagine that the input SOP continues itsmonotonic drift in the same direction (a trend from left to rightin the left hemisphere in our example); the action of VR1 haschanged the reference system, causing the SOP representativepoint to move in the new hemisphere (the right one) with aninverted trend. In this way, VR2 can rewrap its action, andalso, the second issue can be solved. As far as VR2 reducesits action toward zero, VR1 is switched off, and the algorithmcan proceed indefinitely. The result is shown in Fig. 3, wherecascaded SR and VR, with birefringence axes mutually orientedat π/4 rad, act in locking S on Γ.

In summary, we have graphically demonstrated how to set apair of common finite-range retarders, so as to lock the pointrepresentative of a SOP on a great circle of the Poincaré sphere,preventing the reset procedure to act on the output intensity. Theaction of this first stage on a generic input SOP is such that thelinear output polarizer will provide a stable linear SOP with anintrinsic 3-dB loss.

The graphical representation of the evolution of the polariza-tion, as ruled by the second stage of the polarization stabilizer,is shown in Fig. 4. The second stage is analogous to the firstone based on a pair of retarders (VR3 playing the role of SRand labeled SR in Fig. 4, followed by VR4 labeled VR inFig. 4); its task is to bring a SOP, which moves on a well-determined great circle, to a fixed point of the circle itself. This


Fig. 5. Schematic representation of the single-stage stabilizer.

duty can be accomplished by means of a principle similar tothat exploited in the first stage. Namely, VR takes any pointS moving on Γ and stabilizes it on |Q〉. Again, SR switchesanytime VR would overcome one of its retardation range limits,either 0 or π. In correspondence to these values, S coincideswith either |Q〉 or | − Q〉, which are the eigenstates of SR.Therefore, the switching of SR does not affect the output state,and at the same time, it provides endless control by keeping theretardation of VR within its finite range. It is worth remarkingthat the second stage must work fast enough to track andcompensate all evolutions of S on Γ, both those that originatedby modifications in the input SOP and those that resulted fromthe switching of VR1.

III. WORKING PRINCIPLE—JONES MATRIX CALCULUS

As hinted in Section I and graphically demonstrated inSection II, from the fundamental principle of operation, twopolarization stabilizing devices can be derived. The simpler onewill be named single-stage configuration in the following andfeatures the locking of the SOP on a great circle of the Poincarésphere to be directly transformed into a stable SOP at constantpower and a 3-dB intrinsic loss by the addition of an outputlinear polarizer. The second device, in which no polarizingcomponent is actually used to directly modify the SOP of thetransmitted beam, is named double-stage configuration, whichis actually a sort of cascade of the first stage (but for the outputpolarizer) and a replica of itself. In both cases, the workingprinciple can be analyzed by means of the Jones matrix calculus[38], which also provides mathematical demonstration of thelocking conditions.

A. Single-Stage Configuration

Let us focus first on the single-stage configuration, asschematically depicted in Fig. 5.

All descriptions are referred to the horizontal axis as refer-ence axis just for mathematical convenience. It has to be statedthat any other orientation could be chosen, without any loss invalidity.

A generic time-dependent input field uin, which is repre-sented by the Jones vector and is defined as

uin =[

Ex · eiφx

Ey · eiφy

](1)

propagates through the cascade of VR1 and VR2 made by linearbirefringent elements. In (1), Ex,y are the x, y components ofthe field amplitude and φx,y the respective phases. VR1 haseigenaxes oriented at ±π/4 rad with respect to the laboratoryreference (horizontal axis) and is represented by the followingJones matrix:

M1 =

cos

π

4− sin

π

4sin

π

4cos

π

4

·[ ei

ϑ12 0

0 e−iϑ12

]· cos

π

4sin

π

4− sin

π

4cos

π

4

=

cos

ϑ1

2i sin

ϑ1

2

i sinϑ1

2cos

ϑ1

2

. (2)

VR2 has eigenaxes that are, respectively, horizontal and verticaland is represented by the following Jones matrix:

M2 =

exp(

iϑ2

2

)0

0 exp(−i

ϑ2

2

) . (3)

The action of the two VRs is represented by MI(ϑ1, ϑ2), whichreads as follows:

MI(ϑ1, ϑ2)

= M2(ϑ2) · M1(ϑ1)

=

exp(

iϑ2

2

)· cos

ϑ1

2i exp

(iϑ2

2

)· sin ϑ1

2

i exp(−i

ϑ2

2

)· sin ϑ1

2exp

(−i

ϑ2

2

)· cos

ϑ1

2

. (4)

Both retardations ϑ1,2 can, in principle, vary between 0 andπ. Actually, VR1 plays the role of SR, so that ϑ1 is switchedbetween the extremes in response to the control algorithm,whereas ϑ2 gradually varies to track the evolution of the SOPto be compensated.

The aim of the control loop is to obtain after VR2, startingfrom any input uin, an optical beam represented by a Jonesvector as

uI = MI · uin =

[E

(I)x · eiφ

(I)x

E(I)y · eiφ

(I)y

]=

[E

(I)x · eiφ

(I)x

E(I)y · ei

(φ

(I)x ±π/2

) ] (5)

where the constraint on the phases φ(I)x,y to obtain an elliptical

SOP with axes that are, respectively, horizontal and verticalhas been anticipated. The input to the feedback loop is a smallportion of uI, termed u′′

I , extracted by means of an unbalancedbeam splitter (BS), which preserves the SOP. Therefore, u′′

I =αIuI, where the coefficient αI represents the amplitude splittingratio of BS. The aim of the feedback loop is to minimize,ideally, to cancel, the error εI given by the absolute valueof the difference between the optical intensities detected afterthe polarization beam splitter (PBS) whose axes are oriented,respectively, at ±π/4 rad.

For the control to be endless, the key feature is to switchthe value of ϑ1 between either 0 and π rad, or vice versa, i.e.,


when the value of ϑ2 reaches a limit of its finite range (either0 or π rad); whenever the switching is not required, ϑ2 varieswithin its finite range, and the value of ϑ1 is kept constant, equalto either 0 and π rad. Hence, the error εI behaves as follows(see Appendix A):

εI(ϑ1 = 0, ϑ2) = 2α2I ExEy |cos(φy − φx − ϑ2)| (6a)

εI(ϑ1 = π, ϑ2) = 2α2I ExEy |cos(φy − φx + ϑ2| . (6b)

The condition εI = 0 leads to the following locking condition:

ϑ2|ϑ1=0 = ∓ π

2+ φy − φx (7a)

ϑ2|ϑ1=π = ± π

2+ φx − φy. (7b)

From (7a) and (7b), we obtain

ϑ2|ϑ1=0 = −ϑ2|ϑ1=π. (8)

The values ϑ2 = 0 and ϑ2 = π rad are solutions of (8). There-fore, it has been demonstrated that the switching of ϑ1 doesnot change the locking condition for ϑ2, whenever it reaches arange limit. Moreover, we can see from (8) that the switchingof ϑ1 inverts the sign of the locked value of ϑ2. Let us imaginethat the input SOP uin varies monotonically, and in order tocompensate for that, the locked value of ϑ2 needs to increaseuntil it eventually reaches the upper range limit of π rad. Atthis point, as ϑ1 switches, the magnitude of the locked valuefor ϑ2 decreases. Therefore, by switching ϑ1, the locked valueϑ2 remains within its limited range. An analogous argumentapplies to the case of ϑ2 decreasing until it reaches zero,eventually causing ϑ1 to switch.

To complete the demonstration that the control is endless,we have to verify that the condition εI = 0 also holds duringthe switching of the value of ϑ1, as ϑ2 remains fixed to eithera range limit. An expression for εI can be found, valid for bothcases and independent on ϑ1 (see Appendix A), as follows:

εI(ϑ1, ϑ2 = 0, π) = 2α2I ExEy |cos(φx − φy)| . (9)

The condition εI = 0 implies

φ(I)x − φ(I)

y = ±π

2(10)

as also detailed in Appendix A and already anticipated in (5).Since (9) is independent of ϑ1, then once the locking conditionis met, εI remains equal to zero during the switching of ϑ1.Hence, the endlessness of the control is verified.

The condition εI = 0 means that an output polarizer orientedat π/4 transmits half of the optical power of uI independentlyof the input SOP. In this way, a stable output SOP is endlesslygenerated, using just one control variable, at the expense of a3-dB loss.

B. Double-Stage Configuration

The double-stage configuration features the cascade of twosimilar modules. The first stage is equal to the single-stageconfiguration described in Section III-A apart from the absence

Fig. 6. Schematic representation of the second stage, composing the double-stage stabilizer.

of the output polarizer, so that the input of the second stage isrepresented by the Jones vector u′

I = (1 − α2I )

1/2 · uI, whichsatisfies the constraint (10) in case the first stage is lockedand, therefore, corresponds to an elliptical SOP with axes thatare, respectively, horizontal and vertical. The schematic of thesecond stage is given in Fig. 6. The second stage is controlledindependently of the first stage, and the aim of its control is toobtain a vertical output SOP in correspondence to a horizontalorientation of the polarizer in the feedback path. Alternatively,by setting the polarizer vertically, a horizontal output-stabilizedSOP would be obtained. Any fixed SOP could eventually beobtained from the output-stabilized SOP by using a pair ofsuitable birefringent retarders.

The state u′I is processed by the sequence of linearly birefrin-

gent VR3 and VR4. VR3 has eigenaxes that are, respectively,horizontal and vertical and is represented by the followingJones matrix:

M3 =

exp(

iϑ3

2

)0

0 exp(−i

ϑ3

2

) . (11)

VR4 has eigenaxes oriented at ±π/4 rad and is represented by

M4 =

cos

π

4− sin

π

4sin

π

4cos

π

4

·[ ei

ϑ42 0

0 e−iϑ42

]· cos

π

4sin

π

4− sin

π

4cos

π

4

=

cos

ϑ4

2i sin

ϑ4

2

i sinϑ4

2cos

ϑ4

2

. (12)

The Jones matrix representing the second stage can be ob-tained as

MII(ϑ3, ϑ4)

= M4(ϑ4) · M3(ϑ3)

=

exp(

iϑ3

2

)· cos

ϑ4

2i exp

(−i

ϑ3

2

)· sin ϑ4

2

i exp(

iϑ3

2

)· sin ϑ4

2exp

(−i

ϑ3

2

)· cos

ϑ4

2

. (13)

The optical beam after VR4 is represented by the Jones vectorgiven by uII = MII · u′

I. A small portion of this beam isextracted for creating the feedback loop by the BS2 preservingthe SOP, so as to obtain the Jones vector u′′

II = αII · uII. The


output beam is then described by u′II = (1 − α2

II)1/2 · uII. The

error signal at the input of the DSP controller of the secondstage εII is the intensity of u′′

II, as analyzed by a horizontalpolarizer. The objective of the control is to minimize εII, so asto bring u′′

II and, hence, the output u′II in a vertical SOP.

The necessary condition for an endless control at the secondstage is the same as for the first stage. VR3 is the switchingelement, whereas VR4 compensates for SOP excursions. There-fore, the value of ϑ3 is switched between either 0 and π rad,or vice versa, i.e., whenever the value of ϑ4 reaches a limit ofthe finite range between 0 and π rad. In the usual operatingcondition, ϑ4 takes values within its finite range, and the valueof ϑ3 is kept constant, equal to either extreme.

It can be demonstrated (see Appendix B) that for ϑ3 = 0, thecondition εII = 0 is equivalent to

tanφ(I)x · tan φ(I)

y + 1 = 0 (14a)

tanϑ4

2=

E(I)x cos φ

(I)x

E(I)y sin φ

(I)y

. (14b)

Equation (14a) is equivalent to (10), which is satisfied when thefirst stage is locked. Moreover, from (10), we have

sinφ(I)y = sin

(φ(I)

x ∓ π

2

)= ∓ cos φ(I)

x . (15)

The substitution of (15) in (14b) gives the following lockingcondition for ϑ4 in case of ϑ3 = 0:

ϑ4|ϑ3=0 = ∓2 atan

(E

(I)x

E(I)y

). (16)

It can also be demonstrated that for ϑ3 = π, the conditionεII = 0 is equivalent to


y + 1 = 0 (17a)

tanϑ4

2=

E(I)x sin φ

(I)x

E(I)y cos φ

(I)y

. (17b)

If the first stage is locked, then (10) and, equivalently, (17a)are satisfied. Moreover, it results to

cos φ(I)y = cos

(φ(I)

x ∓ π

2

)= ± sin φ(I)

x . (18)

The substitution of (18) in (17b) gives the following lockingcondition for ϑ4 in case of ϑ3 = π:

ϑ4|ϑ3=π = ±2 atan

(E

(I)x

E(I)y

). (19)

From comparison between (18) and (19), we can derive

ϑ4|ϑ3=0 = −ϑ4|ϑ3=π. (20)

Equation (20) is satisfied for ϑ4 equal to 0 and π rad. Therefore,it has been demonstrated that the switching of ϑ3 does notchange the locking condition for ϑ4, whenever ϑ4 reaches arange limit. We can also see from (20) that the switching of ϑ3

inverts the sign of the locked value of ϑ4. If the input SOP uin

varies endlessly in order to make the locked value of ϑ4 increas-ing until it reaches the upper range limit of π rad, the switchingof ϑ3 makes the locked value of ϑ4 decreasing. Therefore, theswitching of ϑ3 allows the locked value of ϑ4 to remain insideits limited range. An analogous argument applies to the caseof a locked value of ϑ4 decreasing until 0. To complete thedemonstration that the control is endless, we have to verifythat the error remains equal to zero during the switching ofthe value of ϑ3, while the value of ϑ4 remains constant, equalto either a range limit. In Appendix B, we demonstrate that

εII(ϑ3, ϑ4 = 0) = α2II

(1 − α2

I

)E(I)2

x (21)

which means that in case of ϑ4 = 0, the requirement for εII = 0is E

(I)x = 0. Since (21) is independent of ϑ3, the error remains

equal to zero during the switching of ϑ3 for any value assumedby ϑ3. In analogy, it can be demonstrated that

εII(ϑ3, ϑ4 = π) = α2II

(1 − α2

I

)E(I)2

y (22)

so that in case of ϑ4 = π, the requirement for εII = 0 is E(I)y =

0, and the error remains equal to zero during the switching ofϑ3, since it is independent of ϑ3. Therefore, the demonstrationthat the control of the second stage is also endless has beencompleted.

IV. EXPERIMENTAL RESULTS

In the present section, we show the experimental results thatverify the effectiveness of the novel endless scheme of polariza-tion stabilization proposed in this paper. The principle of opera-tion has been implemented in two ways, both using finite-rangecomponents: In the first implementation, liquid crystal birefrin-gent cells have been used, whereas the second realization makesuse of the Faraday magnetooptical effect. In both cases, free-space realizations using bulk optical components have beenobtained and used to process an optical beam at λ = 1.55 µm.

For the liquid crystal implementation, nematic liquid crystalcells (Meadowlark Optics Inc.) have been used, in which thevariable phase retardation between the fixed orthogonal eige-naxes is electrically tuned. These cells present a good precisionand repeatability of operation but have a response time of theorder of several milliseconds. For practical convenience, thereference axis is oriented at π/4 rad with respect to the verticalaxis of the laboratory, as in Section II. In this case, for the firststage, the eigenaxes of VR1 are, respectively, horizontal andvertical, whereas the eigenaxes of VR2 are diagonal, and thePBS splits the polarization components, respectively, horizon-tally and vertically. In this way, εI = 0 when uI is ellipticalwith diagonal eigenaxes, and the stabilized SOP after the outputpolarizer is set to |V 〉.

A digital control algorithm is implemented on a digital signalprocessor (DSP). At each step of the control algorithm, whichis of duration Tstep, while the phase retardation ϑ1 of VR1is kept constant, the retardation ϑ2 of VR2 is changed by asmall amount, with absolute value ϑstep. At the end of eachstep, the error εI is evaluated and, if it has increased, then thesign of the retardation variation to be applied in the followingstep is changed; otherwise, this sign is kept unaltered. To obtain


Fig. 7. Working test of the liquid crystal single-stage polarization stabilizer.Both the electrical signal given by the photodetection of the output opticalpower and the electrical signal controlling the phase retardation of VR1 areshown. The output SOP is linear vertical. The input SOP rotates uniformlyat about 0.1 rad/s on the great circle of the Poincaré sphere correspondingto the elliptical SOPs with axes that are, respectively, horizontal and vertical.The output power is stabilized after the feedback has started. It also shows theswitching of the phase retardation ϑ1 needed for the endless control.

endless stabilization, ϑ1 is switched between either 0 and π rad,or vice versa, i.e., whenever the value of ϑ2 should overcome arange limit. More precisely, if ϑ2 has reached a range limit andif in the following step it should overcome this range limit, thenthe ϑ2 variation is not executed, while at the same time, ϑ1 isswitched between either 0 and π rad, or vice versa, and the signof ϑstep variation is changed.

The liquid crystal realization of the single-stage polarizationstabilizer has been tested by cyclically varying the input SOPin three different ways, which correspond to the three types oftrajectory on the Poincaré sphere. The input free-space opticalbeam is linearly polarized by a fixed bulk polarizer. The first tra-jectory is a rotation on the Poincaré sphere equator, obtained byusing a uniformly rotating bulk half-wave plate, that generatesa linear SOP with uniformly varying azimuth angle. The othertwo trajectories are generated by means of a fixed bulk quarter-wave plate placed after the rotating half-wave plate and witheigenaxes suitably oriented. In the second case, the eigenaxes ofthe quarter-wave plate are, respectively, horizontal and vertical,so as to obtain a SOP uniformly rotating on the great circleof the Poincaré sphere, including |H〉, |V 〉, and |L〉, whichcorrespond to the elliptical SOPs with horizontal and verticaleigenaxes. In the third case, the eigenaxes of the quarter-waveplate are diagonal, so as to obtain a SOP uniformly rotating onthe great circle of the Poincaré sphere, including |Q〉, | −Q〉,and |L〉, which correspond to the elliptical SOPs with diagonalaxes. The effectiveness of the endless polarization stabilizationhas been experimentally verified, and a good performance isobtained in correspondence to all three cases. Fig. 7 reports theexperimental results in the second case.

The velocity of rotation of the input SOP on the great circleof the Poincaré sphere is about 0.1 rad/s. The duration Tstep of

Fig. 8. Experimental characterization of a variable Faraday rotator. Thepolarization rotation as a function of the driving current is plotted.

the control algorithm step is set to 20 ms, and the absolute valueof the variation ϑstep of the VR2 phase retardation at each stepis set to 4◦, that is, π/45 rad. It can be seen that the photode-tected output optical power is stabilized by the switching on ofthe feedback. In Fig. 7, the electrical control signal sent by thecontroller to the actuator of the switch VR1 is also shown.

As a second way to implement the stabilizer, magnetoopticalcomponents have been used. The interest for magnetoopticaldevices based on the Faraday effect has recently risen in viewof the realization of high-reliability and high-speed polarizationcontrol [39], [40]. In the description of the single-stage con-figuration, we have referred so far to two linearly birefringentVRs with respective eigenaxes mutually oriented at π/4 rad.It is worth remarking that for the invariance of the principleof operation with respect to rotations of the Poincaré sphere,the single-stage configuration can be realized by a circularlybirefringent VR, that is, a polarization rotator, followed by alinearly birefringent variable with diagonal axes, which, in turn,can be realized by a polarization rotator placed between twofixed quarter-wave plates mutually rotated of π/2 rad, both withhorizontal and vertical eigenaxes.

In order to test the magnetooptical realization of the stabi-lizer, we have used two variable polarization rotators at λ =1.55 µm based on the Faraday effect, produced by TDK Cor-poration. In such devices, the increase of the driving electricalcurrent in the range from about 9 to 26 mA causes a decreaseof the polarization rotation from π to π/2 rad. It is knownthat a polarization rotation of an angle ϕ/2 is equivalent to acircular birefringence with a phase retardation ϕ between thecircular eigenstates [41]. Therefore, the current variation from9 to 26 mA corresponds to a phase retardation from 2π to π rad.The limits of this range are identical to the limits of the rangebetween 0 and π rad so that the proof of the effectiveness of theproposed principle of operation retains its validity. An experi-mental characterization example of the polarization rotation asa function of the driving current is shown in Fig. 8.

Fig. 9 shows the optical output power transmitted by theoutput polarizer of the single-stage stabilizer in the state |V 〉.In this case, the input SOP rotates at about 120 rad/s on thegreat circle of the Poincaré sphere, representing the ellipticalSOPs with axes that are, respectively, horizontal and vertical. In


Fig. 9. Working test of the magnetooptical polarization stabilizer. Both theelectrical signal given by the photodetection of the output optical power and theelectrical signal controlling the phase retardation of VR1 are shown. The outputSOP is linear vertical. The input SOP rotates at about 120 rad/s on the greatcircle of the Poincaré sphere corresponding to the elliptical SOPs with axesthat are, respectively, horizontal and vertical. After the feedback has started, theoutput power is stabilized. It also shows the switching of the phase retardationϑ1 needed for the endless control.

open-loop operation, the output power varies sinusoidally; asthe feedback that starts the output power is stabilized 3 dBbelow the maximum. Fig. 9 also shows the control signal sentto the VR1 current driver, evidently periodically switchingbetween two limit values. Such a switching allows the VR2 po-larization rotation to remain in a finite range during the endlessoperation of the polarization stabilizer. In this experiment, thetime required by each step of the control algorithm is set to40 µs. Such a value has been chosen since it is slightly greaterthan the time required by the VR1 rotation switching. It isinteresting to note that this switching time is limited by theelectrical response of the current driver realized for our expe-riment and not by the intrinsic response of the Faraday effect.The absolute value of the variation of the VR2 polarizationrotation at each step is set to 1◦. In terms of VR2 phase retarda-tion, we obtain an absolute value of variation ϑstep equal to 2◦,corresponding to π/90 rad. The decrease of ϑstep allows the sta-bilizer to have a greater precision, but with less control speed.Furthermore, the tests of the polarizer stabilizer in correspon-dence to input SOPs rotating on the equator and on the great cir-cle, representing the elliptical SOPs with vertical and horizontalaxes, have led to effective endless polarization stabilization.

The single-stage stabilizer theoretically involves an intrinsic3-dB loss due to the action of the output polarizers. Further-more, we actually measured an additional 3.2-dB attenuation,which is caused by the use of a 50/50 BS to generate thefeedback signal. This loss could be reduced, for example, to0.2 dB by using a 95/5 BS. Further loss is due to the insertionloss of the output polarizer (0.1 dB) and that of the VRs(0.1 dB in the liquid crystal realization and 0.8 dB when usingFaraday rotators). The double-stage stabilizer does not involveany intrinsic 3-dB loss. In case of using two 95/5 BS, an overall

insertion loss of either about 0.5 or 2 dB can be expected whenusing, respectively, either liquid crystal cells or magnetoopticalrotators.

The experimental validation of the principle of SOPstabilization here reported has been obtained using continuouswave light. In case of light modulated at a high bit rate sothat the bit slot time Tbit is much less than the stabilizer steptime Tstep, the effectiveness of the polarization stabilizer is notaffected by the modulation. For example, a bit rate higher than 1Gb/s corresponds to Tbit less than 1 ns, which is several ordersof magnitude shorter than the characteristic time of the randompolarization fluctuations in a fiber optical link. Therefore, it ispossible to choose a Tstep much longer than Tbit and to employthe polarization stabilizer in optical communications systems.

V. CONCLUSION

In this paper, we have presented a novel endless scheme forstabilizing the SOP of an optical beam, which takes advantageof the constitutional symmetries of optical polarization, aseffectively visualized by means of the Poincaré sphere repre-sentation. The method employs finite-range birefringent com-ponents and relies on two independent one-variable-drivencontrol loops, to decouple the degrees of freedom of theevolving SOP, so as to eventually stabilize it. The principleof operation has been illustrated with the aid of the Poincarésphere, and the locking conditions for the stabilization of theSOP have also been retrieved using the Jones matrix calculus.

The stabilizer proposed here is characterized by the cascadeof two similar stages. Each stage comprises two variable bire-fringent retarders, namely: 1) a switching element and 2) aquasi-continuously variable one, both driven by the control loopof the stage. The first stage can also act by itself, leading toa very straight method to stabilize polarization by using justone control parameter and fixing, shall we say, one of the twoangular variables that describe the evolution of the SOP on thePoincaré sphere. In this case, however, in view of SOP stabiliza-tion, an output polarizer is needed, involving a 3-dB power loss.Experimental results reported here, which are related to thesingle-stage performance, confirm the validity of the presentapproach.

The double-stage configuration independently controls thetwo angular variables, representing the SOP on the Poincarésphere. Both stages act according to a principle of operationsimilar to that of the single-stage stabilizer and result into afixed output SOP ideally without any intrinsic loss. This is dueto the absence of polarizing elements on the output light path.Due this feature, while the single-stage configuration is suitablefor stabilizing the SOP before polarizing-dependent devices, oranytime a linear SOP is needed, the double-stage configurationappeals also to PMD compensation as well as to the exploitationof polarization for the labeling of optical channels, as in PolDMsystems.

APPENDIX A

Here, we present in detail the calculation of the error εI ofthe single-stage configuration in four particular cases, whichare meaningful in the analysis of the operating principle of the


single-stage polarization stabilizer. Subsequently, we demon-strate (10), which is satisfied by the Jones vector uI in case ofεI = 0.

Given P (π/4) and P (−π/4) as the Jones matrices of linearpolarizers oriented, respectively, at ±π/4 rad, the error εI as afunction of ϑ1 and ϑ2 can be formally written as

εI(ϑ1, ϑ2) = α2I

∣∣∣‖P (π/4) · MI(ϑ1, ϑ2) · uin‖2

−‖P (−π/4) · MI(ϑ1, ϑ2) · uin‖2∣∣∣ . (A1)

The symbol ‖ · ‖ represents the Euclidean norm of a vector,that is, the square root of the sum of the square modulus ofeach component. We consider the Jones matrix MI(ϑ1, ϑ2)expressed by (3) and the input Jones vector uin expressedby (4).

Let us calculate the error εI in case of ϑ1 = 0. It results to

P (π/4)·MI(ϑ1 = 0, ϑ2)·uin

=12

[1 11 1

]·[

exp(iϑ2/2) 00 exp(−iϑ2/2)

]·[

Ex ·eiφx

Ey ·eiφy

]

=12{Ex ·eiφx ·eiϑ2/2 + Ey ·eiφy ·e−iϑ2/2}·

[11

](A2)

‖P (π/4)·MI(ϑ1 = 0, ϑ2)·uin‖2

=12|Ex ·eiφx ·eiϑ2/2 + Ey ·eiφy ·e−iϑ2/2|2

=12(E2

x + E2y

)+ Ex ·Ey ·cos(φy − φx − ϑ2) (A3)

P (−π/4)·MI(ϑ1 = 0, ϑ2)·uin

=12

[1 −1−1 1

]·[


]·[

Ex ·eiφx

Ey ·eiφy

]

=12{Ex ·eiφx ·eiϑ2/2 − Ey ·eiφy ·e−iϑ2/2}·

[1−1

](A4)

‖P (−π/4)·MI(ϑ1 = 0, ϑ2)·uin‖2

=12|Ex ·eiφx ·eiϑ2/2 − Ey ·eiφy ·e−iϑ2/2|2

=12(E2

x + E2y

)− Ex ·Ey ·cos(φy − φx − ϑ2). (A5)

Substitution of (A3) and (A5) in (A1) gives

εI(ϑ1 = 0, ϑ2) = 2α2I · Ex · Ey · |cos(φy − φx − ϑ2)| . (A6)

Let us calculate the error εI in case of ϑ1 = π. It results to

P (π/4)·MI(ϑ1 = π, ϑ2)·uin

=12

[1 11 1

]·[

0 i exp(iϑ2/2)i exp(−iϑ2/2) 0

]·[

Ex ·eiφx

Ey ·eiφy

]

=i

2{Ex ·eiφx ·e−iϑ2/2+Ey ·eiφy ·eiϑ2/2}·

[11

](A7)

‖P (π/4)·MI(ϑ1 = π, ϑ2)·uin‖2

=12|Ex ·eiφx ·e−iϑ2/2+Ey ·eiφy ·eiϑ2/2|2

=12(E2

x+E2y

)+Ex ·Ey ·cos(φy − φx+ϑ2) (A8)

P (−π/4)·MI(ϑ1 = π, ϑ2)·uin

=12

[1 −1−1 1

]·[

0 i exp(iϑ2/2)i exp(−iϑ2/2) 0

]·[

Ex ·eiφx

Ey ·eiφy

]

=i

2{Ex ·eiφx ·e−iϑ2/2 − Ey ·eiφy ·eiϑ2/2}·

[−11

](A9)

‖P (−π/4)·MI(ϑ1 = π, ϑ2)·uin‖2

=12|Ex ·eiφx ·eiϑ2/2+Ey ·eiφy ·e−iϑ2/2|2

=12(E2

x+E2y

)− Ex ·Ey ·cos(φy − φx+ϑ2). (A10)


εI(ϑ1 = π, ϑ2) = 2α2I · Ex · Ey · |cos(φy − φx + ϑ2)| .

(A11)

Let us calculate the error εI in case of ϑ2 = 0. It results to

P (π/4)·MI(ϑ1, ϑ2 = 0)·uin

=12

[1 11 1

]·[

cos(ϑ1/2) i sin(ϑ1/2)i sin(ϑ1/2) cos(ϑ1/2)

]·[

Ex ·eiφx

Ey ·eiφy

]

=12{cos(ϑ1/2) + i sin(ϑ1/2)}

·{Ex ·eiφx + Ey ·eiφy}·[

11

](A12)

‖P (π/4)·MI(ϑ1, ϑ2 = 0)·uin‖2

=12{cos2(ϑ1/2) + sin2(ϑ1/2)

}·|Ex ·eiφx + Ey ·eiφy |2

=12(E2

x + E2y

)+ Ex ·Ey ·cos(φx − φy) (A13)

P (−π/4)·MI(ϑ1, ϑ2 = 0)·uin

=12

[1 −1−1 1

]·[


]·[

Ex ·eiφx

Ey ·eiφy

]

=12{cos(ϑ1/2) − i sin(ϑ1/2)}


1−1

](A14)

‖P (−π/4)·MI(ϑ1, ϑ2 = 0)·uin‖2

=12{cos2(ϑ1/2) + sin2(ϑ1/2)

}·|Ex ·eiφx − Ey ·eiφy |2

=12(E2

x + E2y

)− Ex ·Ey ·cos(φx − φy). (A15)


εI(ϑ1, ϑ2 = 0) = 2α2I · Ex · Ey · |cos(φx − φy)| . (A16)

Let us calculate the error εI in case of ϑ2 = π. It results to

P (π/4)·MI(ϑ1, ϑ2 = π)·uin

=12

[1 11 1

]·[

i cos(ϑ1/2) − sin(ϑ1/2)sin(ϑ1/2) −i cos(ϑ1/2)

]·[

Ex ·eiφx

Ey ·eiφy

]

=12{sin(ϑ1/2) + i cos(ϑ1/2)}

·{Ex ·eiφx − Ey ·eiφy}·[

11

](A17)


‖P (π/4)·MI(ϑ1, ϑ2 = π)·uin‖2

=12{sin2(ϑ1/2) + cos2(ϑ1/2)


=12(E2

x + E2y

)− Ex ·Ey ·cos(φx − φy) (A18)

P (−π/4)·MI(ϑ1, ϑ2 = π)·uin

=12

[1 −1−1 1

]·[

i cos(ϑ1/2) − sin(ϑ1/2)sin(ϑ1/2) −i cos(ϑ1/2)

]·[

Ex ·eiφx

Ey ·eiφy

]

=12{− sin(ϑ1/2) + i cos(ϑ1/2)}


1−1

](A19)

‖P (−π/4)·MI(ϑ1, ϑ2 = π)·uin‖2

=12{sin2(ϑ1/2) + cos2(ϑ1/2)


=12(E2

x + E2y

)+ Ex ·Ey ·cos(φx − φy). (A20)

The substitution of (A18) and (A20) in (A1) gives

εI(ϑ1, ϑ2 = π) = 2α2I · Ex · Ey · |cos(φx − φy)| . (A21)

The condition of error εI equal to zero is equivalent to havinga Jones vector uI representing an elliptical SOP with axes thatare, respectively, horizontal and vertical. Such a SOP has theproperty that the transmission through a polarizer oriented atπ/4 rad implies an intrinsic power loss of 3 dB. This conditioncan be expressed as

‖P (π/4) · uI‖2

‖uI‖2=

12. (A22)

It results to

‖P (π/4) · uI‖2 =14

∥∥∥∥[

1 11 1

]·[

E(I)x · eiφ

(I)x

E(I)y · eiφ

(I)y

]∥∥∥∥2

=14

∥∥∥∥{E(I)x · eiφ

(I)x + E(I)

y · eiφ(I)y

}·[

11

]∥∥∥∥2

=12

(E(I)2

x + E(I)2

y

)+ E(I)

x · E(I)y · cos

(φ(I)

x − φ(I)y

)(A23)

‖P (π/4) · uI‖2

‖uI‖2=

12

+E

(I)x · E(I)

y

E(I)2x + E

(I)2y

cos(φ(I)

x − φ(I)y

).

(A24)

From (A24), we deduce that (A22) is satisfied when

φ(I)x − φ(I)

y = ±π

2. (A25)

APPENDIX B

Here, we present in detail the calculation of the error εII ofthe second stage in four particular cases, which are meaningfulin the analysis of the operation principle of the double-stagepolarization stabilizer. The error εII is the square norm of the

Jones vector P (0) · u′′II, where P (0) is the Jones matrix of

a horizontal polarizer and can be expressed as the followingfunction of ϑ3 and ϑ4:

εII(ϑ3, ϑ4) = ‖P (0) · u′′II‖2

= α2I

(1 − α2

II

)‖P (0) · MII(ϑ3, ϑ4) · uI‖2 . (B1)

In this equation, we consider the Jones matrix MII(ϑ3, ϑ4)expressed by (13) and the Jones vector uI expressed by (5).

Let us calculate the error εII in case of ϑ3 = 0. It results to

εII(ϑ3 = 0, ϑ4)= α2

I

(1 − α2

II

)·∥∥∥∥[

1 00 0

]·[


]·[

E(I)x ·eiφ

(I)x

E(I)y ·eiφ

(I)y

]∥∥∥∥2

= α2I

(1 − α2

II

)×∣∣∣E(I)

x ·cos(ϑ4/2)·eiφ(I)x + iE(I)

y ·sin(ϑ4/2)·eiφ(I)y

∣∣∣2 .

(B2)

The condition of error εII(ϑ3 = 0, ϑ4) equal to zero is equiv-alent to the following equations:

E(I)x · cos(ϑ4/2) · cos φ(I)

x

− E(I)y · sin(ϑ4/2) · sin φ(I)

y = 0 (B3a)

E(I)x · cos(ϑ4/2) · sin φ(I)

x

+ E(I)y · sin(ϑ4/2) · cos φ(I)

y = 0. (B3b)

From (B3a), we obtain

tanϑ4

2=

E(I)x cos φ

(I)x

E(I)x sinφ

(I)x

. (B4)

The substitution of (B3a) in (B3b) gives


y + 1 = 0. (B5)

Let us calculate the error εII in case of ϑ3 = π. It results to

εII(ϑ3 = π, ϑ4)

=α2I

(1 − α2

II

)·∥∥∥∥[

1 00 0

]·[

i cos(ϑ4/2) sin(ϑ4/2)− sin(ϑ4/2) −i cos(ϑ4/2)

]·[

E(I)x ·eiφ

(I)x

E(I)y ·eiφ

(I)y

]∥∥∥∥2

=α2I

(1 − α2

II

)×∣∣∣iE(I)

x ·cos(ϑ4/2)·eiφ(I)x + E(I)

y ·sin(ϑ4/2)·eiφ(I)y

∣∣∣2 .

(B6)

The condition of error εII(ϑ3 = π, ϑ4) equal to zero is equiv-alent to the following equations:

− E(I)x · cos(ϑ4/2) · sinφ(I)

x

+ E(I)y · sin(ϑ4/2) · cos φ(I)

y = 0 (B7a)

E(I)x · cos(ϑ4/2) · cos φ(I)

x

+ E(I)y · sin(ϑ4/2) · sinφ(I)

y = 0. (B7b)


From (B7a), we obtain

tanϑ4

2=

E(I)x sinφ

(I)x

E(I)y cos φ

(I)y

. (B8)

Substitution of (B7a) in (B7b) gives


y + 1 = 0. (B9)

Let us calculate the error εII in case of ϑ4 = 0. It results to

εII(ϑ3, ϑ4 = 0)= α2

I

(1 − α2

II

)·∥∥∥∥[

1 00 0

]·[


]·[

E(I)x ·eiφ

(I)x

E(I)y ·eiφ

(I)y

]∥∥∥∥2

= α2I

(1 − α2

II

) ∣∣∣E(I)x ·eiϑ3/2 ·eiφ

(I)x

∣∣∣2= α2

I

(1 − α2

II

)E(I)2

x . (B10)

Let us calculate the error εII in case of ϑ4 = π. It results to

εII(ϑ3, ϑ4 = π)= α2

I

(1 − α2

II

)·∥∥∥∥[

1 00 0

]·[

0 i exp(−iϑ3/2)i exp(iϑ3/2) 0

]·[

E(I)x ·eiφ

(I)x

E(I)y ·eiφ

(I)y

]∥∥∥∥2

= α2I

(1 − α2

II

) ∣∣∣iE(I)y ·e−iϑ3/2 ·eiφ

(I)y

∣∣∣2= α2

I

(1 − α2

II

)E(I)2

y . (B11)

ACKNOWLEDGMENT

The authors would like to thank A. Barberis for the technicalsupport and A. Carrara and A. Schgoer (Siel Tre srl) for therealization of the DSP controller and for the implementation ofthe control algorithm.

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Mario Martinelli (M’83) was born in Mantova,Italy, in 1952. He received the Laurea degree innuclear–electronics engineering from the Politecnicodi Milano, Milan, Italy, in 1976.

He is a Full Professor in optical communica-tions with the Dipartimento di Elettronica ed Infor-mazione, Politecnico di Milano, and the Directorof CoreCom, Milan, Italy. He started his researchactivities in 1977 at the Centro Informazioni Studi edEsperienze (CISE) Laboratories, Segrate (near Mi-lan), Italy, where he joined the Quantum Electronics

Division. In 1980, he started new research activities regarding optical fibers,which led him to the position of Director of the Coherent Optics Department.In 1981, he was a Visiting Researcher with the London University College, Lon-don, U.K. In 1992, he was appointed as Professor of optical communications atthe Politecnico di Milano, where he activated the first Italian-related course; in1993, he founded the Photonic Laboratory at the Electronics and InformationDepartment; and in 1995, he was involved in the foundation of CoreCom, aResearch Consortium between Politecnico di Milano and Pirelli Cables andSystems, which was established to develop research in optical processing andphotonic switching. He is the author of more than 80 scientific papers inthe most important journals and of more than 100 communications presentedat international conferences. He is also the holder of 35 patents. He is theReviewer for all the main publications in the optics and photonics research area.

Prof. Martinelli was elected as a Fellow of the Optical Society of America in2004 for his contributions in the domain of optical communications and opticalfiber sensors and for the discovery of the Faraday mirror effect. He is a mem-ber of the IEEE/Lasers and Electro-Optics Society Executive Board (ItalianChapter), the Società Italiana di Ottica e Fotonica Executive Committee, andthe Executive Board of the Federazione Italiana di Elettrotecnica, Elettronica,Automazione, Informatica e Telecomunicazioni (AEIT) Group on Photonicsand Optoelectronics.

Paolo Martelli was born in Milan, Italy, in 1970. Hereceived the Laurea degree (cum laude) in electronicsengineering and the Ph.D. degree in informationengineering from the Politecnico di Milano, Milan,Italy, in 1998 and 2005, respectively.

He is currently a Researcher with CoreCom, Mi-lan, Italy. His major interests research are modelingand simulation of optical communications systems,advanced modulation formats, and control of thestate of polarization. He has coauthored more than20 scientific publications and three assigned patents.

Silvia Maria Pietralunga (A’96–M’01) received theLaurea degree (cum laude) in opto-electronics andthe Ph.D. degree in electronic and communicationengineering from the Politecnico di Milano, Milan,Italy, in 1993 and 1998, respectively.

After a period spent with the Photonic Laboratory,Politecnico di Milano, she joined CoreCom, Milan,Italy, in 1995, where she is currently a PrincipalInvestigator in the area of materials and devices. Shehas coauthored 42 publications in international re-views and seven assigned patents, covering the fields

of optical switching and wavelength conversion, electrooptics, magnetooptics,nonlinear optics, and photoconduction in semiconductors. She is also devoted toresearch in analysis and exploitation of the evolution of the state of polarizationof light in optical fibers and waveguides.

Dr. Pietralunga is a member of the IEEE/Lasers and Electro-Optics SocietyExecutive Board (Italian Chapter). She is also a member of the OpticalSociety of America and the Federazione Italiana di Elettrotecnica, Elettronica,Automazione, Informatica e Telecomunicazioni (AEIT).

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