Polarization degree of optical waves with non-Gaussian probability density functions: Kullback

relative entropy-based approach

Philippe RéfrégierPhysics and Image Processing Group, Fresnel Institute, Centre Nationale de la Recherche, Unité Mixte de Recherche,

6133, Scientifique, École Généraliste d’Ingénieurs de Marseille, Domaine Universitaire de Saint-Jérôme, 13397Marseille Cedex 20, France

Received December 17, 2004

The definition of degree of polarization for non-Gaussian partially polarized light is analyzed. A generalframework based on the Kullback relative entropy is developed, and properties that enlighten the physicalmeaning of the degree of polarization are established. In particular, it is shown how the degree of polariza-tion is related to the measure of proximity between probability density functions and to the measure of dis-order provided by the Shannon entropy. © 2005 Optical Society of America

OCIS codes: 260.5430, 030.0030, 030.4280.

The polarization of light is a fundamental concept inoptics and is of interest in several active technologi-cal fields such as imagery, telecommunication, medi-cine, and instrumentation. In many applications thevariation of the degree of the polarization state isanalyzed and its precise definition and properties incases of non-two-dimensional Gaussian optical wavesis thus of great interest from both theoretical andpractical points of view.1–3 In this Letter the new defi-nition of the degree of polarization of partially polar-ized light introduced in Ref. 3 is analyzed and itsproperties in relation to the standard definition arediscussed.4 It will be shown that the Kullback rela-tive entropy5,6 (KRE) allows one to obtain a unifiedframework for both approaches and thus providesphysically meaningful interpretations.

The KRE plays an important role in several areas.For example, in statistical physics it is proportionalto the free energy (see Ref. 6 for a simple introduc-tion). In quantum optics it allows one to characterizethe degree of entanglement of light.7 In this Letter itis shown that the KRE, and thus the different defini-tions of the degree of polarization, directly providemeasures of disorder through the Shannon entropy.5,6

This analysis leads to a simple interpretation of theevolution of the degree of polarization in nonlinearmedia.

Classically, in dimension two, the electric field E= sE1 ,E2dT of speckled coherent electromagnetic ra-diation is represented by a two-dimensional (2D)complex random vector4 with coherency matrix G, de-fined by

G = F kuE1u2l kE1E2*l

kE2E1*l kuE2u2l G , s1d

where the symbol k¯l denotes ensemble averagingthat can correspond to different types of physical av-eraging depending on the considered application. Thedegree of polarization is defined by

P = S1 − 4detfGg

trfGg2 D1/2

, s2d

where detfGg (trfGg) is the determinant (the trace) ofG. In Ref. 3, to obtain a simple relation between thedegree of polarization and the entropy, the followingnew definition of the degree of polarization is pro-posed:

fP8g2 = 1 − expfSsP̃dg/s4ped, s3d

where SsP̃ad is the Shannon entropy of the probabil-

ity density function (PDF) P̃a of the electric field inpolar coordinates on the Poincaré sphere. In Carte-sian coordinates, the Shannon entropy is defined by5

SsPd=−ePsEdlnfPsEdgdE, where e¯dE standsfor complex 2D integration. In polar coordinatessf ,u ,rd on the Poincaré sphere, one has SsP̃d=−eP̃sf ,u ,rdlnfP̃sf ,u ,rdgdfdudr, which is differentfrom SsPd since the entropy of a continuous randomvariable is not invariant when one changes the pa-rameterization. It is proposed in the following thatboth definitions of Eqs. (3) and (2) are bijective func-tions of two different KREs that are simply related toeach other, and physical consequences are analyzed.

Let PasEd and PbsEd denote the PDF of two opticalwaves. The KRE between these two PDFs isKsPa iPbd=ePasEdlogfPasEd /PbsEdgdE. It can beshown5 that KsPa iPbdù0 and that KsPa iPbd=0 if andonly if Pa=Pb. Thanks to the Sanov theorem,5 theKRE has a simple interpretation. Indeed, let us con-sider an experiment in which we observe N indepen-dent measurements of the electric field of a partiallypolarized light distributed with PDF Pb and covari-ance matrix Gb. Let us also assume that the measure-ments are performed with a precision q with q4

!detsGbd so that one can consider PDFs instead ofthe probability laws obtained after quantization. Onecan determine the number of times NsEd that thevalue E is observed with precision q on each compo-

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nent. Of course, the density QsEd=NsEd / sN q4dconverges to the probability density PbsEd whenN→ +` and q→0. However, when N becomes largebut is still finite, the Sanov theorem lets us knowthat the probability of observing QsEd=PasEd isPNsPa uPbd.expf−N KsPa iPbdg.8

Let us now consider a Gaussian optical wavedefined by its PDF PGa

sEd=1/ sp2 detfGagd3expf−E†Ga

−1Eg (where the symbol † denotes theHermitian conjugate) and a totally depolarizedGaussian light with the same intensity I0 and thuswith a covariance matrix G0 proportional to the iden-tity matrix. One can show2 that Eq. (2) leads to

Pa2 = 1 − expf− KsPGa

i PG0dg. s4d

Thus for Gaussian waves one has PNsPGauPG0

d.f1−Pa

2gN.The standard definition of the degree of polariza-

tion given by Eq. (2) takes into account only second-order statistics (i.e., covariance matrices). If Pa is thePDF of a non-Gaussian wave of covariance matrix Ga,the relation between the standard definition of thedegree of polarization and the KRE between Pa andPG0

is not described by Eq. (4). The nonlinear degreeof polarization defined by

fPaNLg2 = 1 − expf− KsPa i PG0

dg s5d

is an immediate generalization of Eq. (4) to a non-Gaussian wave of PDF Pa. We show below that Pa

NL

corresponds to the definition of Eq. (3). However, thisdemonstration will be easier after some fundamentalproperties of KsPa iPG0

d have been demonstrated. Ingeneral KsPa iPcdÞKsPa iPbd+KsPb iPcd, but one hasthe following property.

Property A. Let Pa be a PDF with covariance matrixGa, PGa

be a Gaussian PDF with the same covariancematrix Ga, and PG be a Gaussian PDF with covari-ance matrix G but the same intensity as Pa. Then,

KsPa i PGd = KsPa i PGad + KsPGa

i PGd. s6d

Indeed, it is easy to show9 that KsPGaiPGd

=logsdetfGg /detfGagd+trfGaG−1g−2. Further, onecan write KsPa iPGd=−SsPad+WsPad with SsPad=−e lnfPasEdgPasEddE and WsPad=−e lnfPGsEdg3PasEddE. A direct calculus leads to WsPad=logsp2 detfGgd+trfGaG−1g and thus KsPa iPGd=−SsPad+logsp2 detfGgd+trfGaG−1g. One thus obtainsin particular KsPa iPGa

d=−SsPad+logsp2 detfGagd+2.The combination of these equations proves Eq. (6).

If one considers the PDF PG0sEd of a totally depo-

larized light with the same intensity as PGa, Property

A leads to

KsPa i PG0d = KsPa i PGa

d + KsPGai PG0

d. s7d

Moreover, since KsPGaiPGdù0, one sees that

KsPa iPGdùKsPa iPGad. Thus KsPa iPGd is minimal if

KsPGaiPGd=0 and therefore if G=Ga. Property A thus

shows that considering KsPGaiPG0

d in the definition ofPa

2 instead of KsPa iPG0d consists of looking at the

KRE between PG0and the nearest Gaussian PDF, in

the KRE meaning, to Pa. This result is illustrated inFig. 1(a) and leads to the following property.

Property B. One necessarily has

fPaNLg2 ù fPa

2g s8d

and fPaNLg2= fPa

2g if and only if the wave has a Gauss-ian PDF.

Indeed, using Eqs. (4), (5), and (7), one getsfPa

NLg2=1−bNL2 f1−Pa

2g with bNL2 =expf−KsPa iPGa

dg.Since KsPa iPGa

dù0, bNL2 ø1. One thus gets f1−Pa

2gù1− fPa

NLg2 or Pa2ø fPa

NLg2. Further, Pa2= fPa

NLg2 im-plies that bNL

2 =1 or, in other words, KsPa iPGad=0,

which means that Pa=PGa(i.e., Pa is Gaussian). Us-

ing these results, it can appear natural to define thefollowing degree of non-Gaussianity: fGag2=1−bNL

2 .In particular, Property B shows that for a non-

Gaussian PDF, one has fPaNLg2.0 when Pa

2=0. Thedefinition of nonlinear degree of polarization given byEq. (5) thus leads to a nonnull degree of polarizationfor non-Gaussian isotropic waves that can be prob-lematic for some applications.

Let us now analyze the relation between the KREand the Shannon entropy. In general KsPa iPbdÞSsPbd−SsPad but one can show the following prop-erty [see Fig. 1(b)].

Property C. With the same notations as in PropertyA, one has

KsPa i PGad = SsPGa

d − SsPad,

KsPa i PG0d = SsPG0

d − SsPad,

KsPGa i PG0d = SsPG0

d − SsPGad. s9d

Indeed, for any Gaussian PDF one has SsPGd=logsp2 detfGgd+2. One can write KsPa iPGa

d=−SsPad+UsPad with UsPad=−e lnfPGa

sEdgPasEddE. A directcalculus shows that UsPad=SsPGa

d, which proves the

Fig. 1. Schematic illustrations of the consequences of (a)Property A and (b) Property C.

May 15, 2005 / Vol. 30, No. 10 / OPTICS LETTERS 1091

first equality. In the same way, one can writeKsPa iPG0

d=−SsPad+VsPad with VsPad=logsp2 detfG0gd+efE+G0

−1EgPasEddE. However, G0 is equal tosI0 /2dMId, where MId is the identity matrix in dimen-sion two and I0 is the intensity of the wave. SinceefE+EgPasEddE=I0, one deduces that VsPad=SsPG0

d,which proves the second equality. The third equalityis a direct consequence of the second.

Since KRE is necessarily positive or equal to 0, onethus has

SsPad ø SsPGad ø SsPG0

d. s10d

One can illustrate these properties by considering anexample of a non-Gaussian partially polarized wavewith uniformly distributed intensities for both coor-dinates of the field. Its PDF can be easily written as afunction of the modulus square r1= uE1u2, r2= uE2u2,and of the phase w1, w2 of electric field E sincePasr1 ,r2 ,w1 ,w2d= s1/16p2I1I2dR2I1

fr1gR2I2fr2g, where

Rusxd=1 if 0øxøu and Rusxd=0 otherwise. It is easyto see that KsPa iPGa

d=2 lnse /2d and thus fGag2=1−4/e2, which leads to

Pa2 = 1 −

4I1I2

sI1 + I2d2 = S I1 − I2

I1 + I2D2

,

fPaNLg2 = 1 −

4

e2

4I1I2

sI1 + I2d2 .

It can be verified for this example that fPaNLg2

. fPa2g.

Let us now show that PaNL defined in Eq. (5) indeed

corresponds to the definition of Eq. (3). Let P̃G0be the

PDF of the isotropic Gaussian wave of unit intensityand SsP̃G0

d be its entropy in polar coordinates on the

Poincaré sphere. One has3 SsP̃G0d=lns4ped and thus

fPa8g2=1−expfSsP̃ad−SsP̃G0dg. Since the KRE and the

difference in entropies are invariant with change invariables, one can use Property C and thus SsP̃ad−SsP̃G0

d=KsP̃a i P̃G0d=KsPa iPG0

d, where Pa and PG0de-

note the PDF in Cartesian coordinates of the electricfield. The equation of fPa8g2 is thus identical to Eq. (5).

It is interesting to note that if one considers, as inRef. 3, propagation of light in a nonlinear medium,which conserves the intensity and the entropy, Paand Pa

NL lead to simple physical interpretations. In-deed, Property C shows that one must have constantKsPa iPG0

d during the evolution. One can thus deducewith Property A that KsPa iPGa

d+KsPGaiPG0

d has to beconstant but transfer of disorder from Gaussianity todepolarization can appear [i.e., SsPG0

d−SsPGad can de-

crease if SsPGad−SsPad can increase]. In other words,

one can observe depolarization (i.e., Pa can decrease)if non-Gaussianity appears (i.e., Ga increases) since1− fPa

NLg2= f1−Ga2gf1−Pa

2g.In summary, two different definitions of the degree

of polarization have been discussed, and it has beenshown that they correspond to differences of entro-pies that are due to different sources of disorder. Patakes into account only anisotropy properties of thesecond-order statistics (i.e., the covariance matrix).Pa

NL allows one to overcome this problem but does notmeasure only the order that is due to anisotropy,since it also includes a measure of non-Gaussianitydescribed by Ga.

The author thanks A. Picozzi, F. Goudail, and thePhyti team. P. Réfrégier’s e-mail address isphilippe.refregier@fresnel.fr

References

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2. Ph. Réfrégier, F. Goudail, P. Chavel, and A. Friberg, J.Opt. Soc. Am. A 21, 2124 (2004).

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