Analysis of clusters and uniformity of distributionof states of polarization on the Poincaré sphere

Krzysztof Perlicki

An analysis of the clusters and the uniformity of distribution of states of polarization on the surface of aPoincaré sphere generated by rotating wave plates is given. The analysis of clusters of the states ofpolarization is based on a spherical radial distribution function. For uniform analysis of the distribution,two methods are proposed. The first method is based on calculation of the correlation coefficient; thesecond method is based on calculation of the angles between pairs of the states of polarization on thePoincaré sphere. For polarization scramblers consisting of eight or more rotating wave plates, nonclus-tered and near-uniform distribution of states of polarization is obtained. © 2005 Optical Society ofAmerica

OCIS codes: 220.4840, 260.1440, 060.2300.

1. Introduction

Because of the random nature of polarization modecoupling in an optical fiber, several polarizationeffects may occur that lead to impairments in long-haul and high-bit-rate optical fiber transmission sys-tems. Such impairments include polarization modedispersion, polarization-dependent loss, polarization-dependent modulation, polarization-dependent gain,and polarization-dependent center wavelength.1Scrambling the states of polarization (SOPs) hasbeen shown to be a technique that can reduce polar-ization impairment or measurement uncertainly.2 Apolarization scrambler actively changes the SOPs byusing a polarization-modulation method. Severalscramblers based on different technologies are avail-able today; they include LiNbO3 based scramblers,resonant fiber coil based scramblers, and fibersqueezer based scramblers. In general, the polariza-tion scrambler configuration consists of rotating waveplates or phase-shifting elements. It is often neces-sary that the scrambler’s output SOPs be distributeduniformly on the entire Poincaré sphere. In this pa-per the effect of the rotating wave-plate polarizationscrambler3,4 configuration on the distribution of out-

put SOP’s is described. Analysis of clusters of thedistributions of SOPs on the surface of a Poincarésphere is based on an analysis of the so-called spher-ical radial distribution function. To the best of au-thor’s knowledge, this is the first application of theradial distribution function to analysis of the distri-butions of SOPs. For uniform analysis of the distri-bution two methods are compared. The first methodis based on calculation of the correlation coefficientbetween the reference pattern (uniform distribution)and a pattern that comprises a set of the SOPs gen-erated by the scrambler. The second method is basedon calculation of the angles between pairs of SOPs ona Poincaré sphere.

2. Theoretical Background

A. Analysis of Clusters

Analysis of clusters of SOPs on a Poincaré sphere isbased on the spherical radial distribution function.The spherical radial distribution function is a modi-fied form of the well-known plane radial distributionfunction. The plane radial distribution function isdescribed in Refs. 5–9, for example. The sphericalradial distribution function is defined as follows:

g(d) �KnAT

AnKT, (1)

where Kn is the number of pairs of points separatedby a given range of distances (in Fig. 1, betweendistances dn�1 and dn�1), An is the area of the sphere

The author (perlicki@tele.pw.edu.pl) is with the Institute of Tele-communications, Warsaw University of Technology, Nowowiejska15�19, 00-665 Warsaw, Poland.

Received 23 September 2004; revised manuscript received 25November 2004; accepted 29 November 2004.

0003-6935/05/214533-05$15.00/0© 2005 Optical Society of America

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between two circles (in Fig. 1, between circles cn andcn�1), KT is the total number of pairs of the points onthe sphere, and AT is the area of the sphere.

The great circle distance �dn� between two points�n1 and n2), whose coordinates are ��1, �1� and��2, �2�, is given by10

dn � 2R arcsin�sin2��1 � �2

2 �� sin(�1)sin(�2)sin2��2 � �1

2 ��1�2

, (2)

where R is the sphere’s radius (here R � 1).Equation (2) is the so called Haversine formula

(developed by Sinnott), which many authors recom-mend as better behaved for use over short dis-tances, for which very small angular differences areinvolved.11–14

Figures 2(a) and 3(a) show the spherical radialdistribution as a function of distance d for a theoret-ical random distribution and a theoretical clustereddistribution. For the random distribution [Fig. 2(b)]the value of g�d� is close to 1. For the clustered dis-tribution [Fig. 3(b)] the first peak on the g�d� curveprovides information about the parameters of theclusters (mean dimension and degree of the cluster-ing). The location of the lower peaks on the curveindicates the mean distance between clusters.

B. Analysis of Uniformity

For analysis of uniformity, two methods are com-pared. The first method is based on calculation of the

correlation coefficient between the SOP xR � UR andthe nearest SOP xk � Uk, where UR is the referencepattern (uniformly distributed SOPs) and Uk is a setof SOPs generated by the scrambler. In the case ofuniform distribution the algorithm described in Ref.15 was adopted.

The correlation coefficient is given by

� �Yxx*

�YxxYx*x*

, � � (�1, 1), (3)

where

Yxx* � det�i�1

n

xR,i, xk,i�� det

S1R,iS1k,i S2R,iS1k,i S3R,iS1k,i

S1R,iS2k,i S2R,iS2k,i S3R,iS2k,i

S1R,iS3k,i S2R,iS3k,i S3R,iS3k,i�,

(4)

Yxx � det�i�1

n

xR,i, xR,i�� det

S1R,iS1R,i S2R,iS1R,i S3R,iS1R,i

S1R,iS2R,i S2R,iS2R,i S3R,iS2R,i

S1R,iS3R,i S2R,iS3R,i S3R,iS3R,i�,

(5)

Yx*x* � det�i�1

n

xk,i, xk,i�� det

S1k,iS1k,i S2k,iS1k,i S3k,iS1k,i

S1k,iS2k,i S2k,iS2k,i S3k,iS2k,i

S1k,iS3k,i S2k,iS3k,i S3k,iS3k,i�,

(6)

where S1R,i, S2R,i, and S3R,i are the first, second, andthird Stokes parameters of the ith reference SOP andS1k,i, S2k,i, and S3k,i are the first, second, and thirdStokes parameters of the ith SOP generated by thescrambler, respectively.

The values of � � ��1, 1� correspond, respectively,to complete positive and complete negative correla-tion. If there is a perfect linear relationship with apositive slope between the reference pattern and thegenerated pattern, then we have a correlation coeffi-cient of 1. If there is a perfect linear relationship witha negative slope between the reference pattern andthe generated pattern then we have a correlationcoefficient of �1. A correlation coefficient of 0 meansthat there is no linear relationship between the ref-erence pattern and the generated pattern.

The second method is based on a statistical ap-proach. The statistics for testing uniformity againstalternative models are investigated. The test is basedon calculation of the angles between pairs of unitvectors (SOPs on the Poincare sphere)16:

Fig. 1. Example of the calculation of a spherical radial distribu-tion function; see Eq. (1). The great circle distance dn betweenreference points n1 and n2 is given by Eq. (2). The number of pointsfound in the area between circles cn and cn�1 is counted.

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�ij � arccos(S1iS1j � S2iS2j � S3iS3j), 1 � i j � n,(7)

where n is the number of unit vectors (number ofSOPs) and S1i, S2i, and S3i are the first, second, andthird Stokes parameters of the ith SOP and S1j, S2j,and S3j are the first, second, and third Stokes param-eters of the jth SOP, respectively.

The statistic (Fn) for testing uniformity against allalternative models is given by17

Fn �3n2 � � 4

n�i�1

n�1

j�i�1

n

�ij � sin(�ij). (8)

The hypothesis of uniformity is rejected in favor of allalternative models if the value of Fn is too large. If thevalue of Fn decreases, then the probability that thedistribution will be uniform will increase. In addition,if Fn 1.948, then there is effectively no evidencethat the distribution is not uniform.6

3. Simulation Results

In our theoretical experiment the distribution ofSOPs on the Poincaré sphere generated by a polar-ization scrambler is investigated. The scrambler con-figuration consists of several polarization elements:rotating half-wave plates ���4) and quarter-waveplates ���2�. The rotation angles are randomly dis-tributed between 0 and 2. The input state ofpolarization is horizontally linear. The number ofgenerated SOPs is 100, and the distance increment dis equal to �200.

In Fig. 4 the spherical radial distribution functionsfor polarization scramblers that consist of three���4, ��2, ��4� and four ���4, ��2, ��4, ��2� elementsare compared.

Figure 5 shows the spherical radial distributionfunctions for five ���4, ��2, ��4, ��2, ��4� and six���4, ��2, ��4, ��2, ��4, ��2� elements.

The distribution of SOPs on the Poincaré sphere isa clustered distribution for three rotating wave plates[Fig. 4(a)]. For four and more rotating wave plates

Fig. 2. Theoretical random distribution. (a) Distribution of SOPson the surface of the Poincaré sphere. (b) Spherical radial distri-bution function versus distance. For a random distribution thevalue of g�d� is close to 1.

Fig. 3. Theoretical clustered distribution. (a) Distribution ofSOPs on the surface of the Poincaré sphere. (b) Spherical radialdistribution function versus distance. For the clustered distribu-tion the first peak on the g�d� curve provides information about theparameters of the clusters. The location of the lower peaks on thecurve indicates the mean distance between clusters.

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[Figs. 4(b) and 5] a random distribution of SOPs onthe Poincaré sphere is obtained [the value of g�d�oscillates about 1]. One can attempt to estimate thedegree of clustering by determining the area betweenthe g�d� curve for a clustered distribution and g�d�� 1.18 Here the mean diameter of the clusters isdefined as the minimum value of the distance d forwhich g�d� � 1. The degree of clustering for Fig. 4(a)is 0.45 rad. In turn, the mean diameter of the clustersis 1.5 rad.

Figure 6(a) shows the value of the correlation co-efficient versus the number of elements. The value ofthe correlation coefficient increases if the number ofelements increases from three to eight. For eight andmore elements the value of the correlation coefficientis constant and equal to 0.75; a high correlation be-tween the reference pattern and the pattern gener-ated by polarization scrambler is obtained. Theseeight and more elements give near-uniform distribu-

tion. In Fig. 6(b) the value of Fn as a function of thenumber of elements is shown. The value of Fn de-creases if the number of elements increases fromthree to eight. For eight and more elements the valueof Fn is constant and equal to 1.4 �Fn 1.948�. Thereis a high probability that the distribution will beuniform for eight and more elements. The conclusionsfrom these two methods show good agreement.

4. Conclusions

In this paper an analysis of some properties of thedistribution of states of polarization on the surface ofa Poincaré sphere that was generated by rotatingwave plates has been presented. The spherical radialdistribution function has been shown to be a valuablemethod for determining the type of SOP distribution.The spherical radial distribution function can be usedto analyze the clustered distribution points (SOPs) ona Poincaré sphere and to estimate the cluster param-eters: the degree of the clustering and the clusterdiameter. A nonclustered distribution for four and

Fig. 4. Spherical radial distribution function versus distance for(a) three rotating wave plates and (b) Four rotating wave plates.For three rotating wave plates the distribution of SOPs on thePoincaré sphere is clustered distribution. For four rotating waveplates the value of g�d� oscillates about 1 (random distribution).

Fig. 5. Spherical radial distribution function versus distance for(a) five rotating wave plates and (b) Six rotating wave plates. Forfive and six rotating wave plates the value of g�d� oscillates about(random distribution).

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more rotating wave plates was shown. For uniformanalysis, two methods were presented. Similar con-clusions were reached with both. A near-uniform dis-tribution was shown for eight and more rotating waveplates. The conclusion from analysis of clusters anduniformity is the following: For polarization scram-blers consisting of eight or more rotating wave plates,a nonclustered and near-uniform distribution ofstates of polarization is obtained.

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Fig. 6. Effect of the number of rotating wave plates on (a) thevalue of the correlation coefficient and (b) the value of Fn. The valueof the correlation coefficient increases and the value of Fn decreasesif the number of elements increases from three to eight. From eightelements the correlation coefficient and Fn are constant.

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