Polarimetric characterization of light and media

47 Pages·2007·0.75 MB·English

by  Gil J. J.

Eur. Phys.J. Appl.Phys. 40, 1–47 (2007) THE EUROPEAN
DOI:10.1051/epjap:2007153
PHYSICAL JOURNAL
APPLIED PHYSICS
Polarimetric characterization of light and media
Physical quantities involved in polarimetric phenomena
J.J. Gila
Universidad deZaragoza, ICE c/ Pedro Cerbuna12, 50009 Zaragoza, Spain
Received: 11 June 2007 / Received in finalform: 16 July 2007 / Accepted:26 July 2007
Published online: 3 November2007 – (cid:1)c EDP Sciences
Abstract. Anobjective analysis is carried out of thematricial models representingthepolarimetric prop-
ertiesoflightandmaterialmedialeadingtotheidentificationanddefinitionoftheircorrespondingphysical
quantities,usingtheconceptofthecoherencymatrix.Forlight,casesofhomogeneousandinhomogeneous
wavefront areanalyzed,andamodelfor3Dpolarimetricpurityisconstructed.Forlinearpassivematerial
media, a general model is developed on the basis that any physically realizable linear transformation of
Stokes vectors is equivalent to an ensemble average of passive, deterministic nondepolarizing transforma-
tions.Throughthisframework,therelevantphysicalquantities,includingindicesofpolarimetricpurity,are
identified and decoupled.Some decompositions of thewhole system intoa set of well-defined components
are considered, as well as techniques for isolating the unknown components by means of new procedures
forsubtractingcoherencymatrices.Theseresultsandmethodsconstituteapowerfultoolforanalyzingand
exploiting experimentaland industrial polarimetry. Some particular application examples are indicated.
PACS. 42.25.Ja Polarization – 42.68.Mj Scattering, polarization
1 Introduction – The propagation direction is fixed, whereas the po-
larization ellipse fluctuates; in consequence, the light
Aproperdescriptionofthepolarizationpropertiesofelec- beamispartiallypolarized.Itiswell-knownthatthese
tromagnetic waves relies on the concept of the coherency states are polarimetrically equivalent to a statistical
matrix. This mathematical formulation is applicable re- mixture of two pure states with orthogonal polariza-
gardless the particular band of the electromagnetic spec- tions. We will show that mixed states are also equiv-
trum considered. In this paper we will frequently refer to alent to statistical mixtures of non-orthogonal pure
“light”but,exceptforparticularcaseswhichareidentified states.
from the context, this term can be substituted by “elec- – Both the propagation direction and the polarization
tromagnetic radiation”.In fact there are many important ellipse fluctuate. As we will see, an objective measure
industrial and research subjects involving polarization of of the degree of directionality can be defined, which is
electromagnetic radiation beyond the optical range such always higher than (or equal to) the objective mea-
as microwaves,X-rays and gamma-rays. sure of the stability of the polarization ellipse. These
Given a fixed point in space, the electric field of an states require a particular treatment, different to that
electromagnetic wave describes an ellipse (the polariza- the commonly used for light with a fixed direction of
tion ellipse) in a plane perpendicular to the propagation propagation.
direction.Forthe idealcaseofmonochromaticlight,both
polarization ellipse and propagation direction are fixed. In general, concerning the optical band of the electro-
Ingeneral,polychromaticlightbehavesasmonochromatic magnetic spectrum, the time of measurement T (i.e. the
fortime intervalslongerthanthe naturalperiodT0 ofthe response time of the detector) is much higher than the
wave and shorter than the coherence time τ, so that the coherence time. Typical values of the indicated time in-
following different situations can be distinguished: tervals are the following: T0 ≡ 1/ν0 ∼= 10−15s (ν0 being
– Boththepropagationdirectionandthepolarizationel- t1h0e−9cesn(cid:1)traτl(cid:1)fre1q0u−e4nscyanodf tThe∼=sp1e0c−t4rasl[1p,r2o]fi.lTehoufst,hfeorwmavoes)t,
lipse arestable;inconsequence,the lightbeamis fully
purposestheassumptionofquasimonochromaticityisjus-
polarized.For mostpurposes,andfroma polarimetric
tified.
point of view, monochromatic totally polarized light
In a second-order optics approach, the Stokes param-
can be considered equivalent to monochromatic light.
eters [3] provide a complete mathematical characteriza-
a e-mail: [email protected] tion of the different states of polarization. The physical
Article published by EDP Sciences and available at http://www.edpsciences.org/epjap or http://dx.doi.org/10.1051/epjap:2007153
2 The European Physical Journal Applied Physics
meaningofthesefourrealparametershasbeenstudiedin thechangesproducedontotallypolarizedlight.Theinter-
depth by several authors [4–7]. actiontransformsthe“inputpolarizationellipse”,charac-
When a material medium is irradiated by an electro- terized by a complex number that contains information
magnetic wave,molecular electric chargesare set in oscil- about the corresponding azimuth and ellipticity, into a
latory motion by the electric field of the wave, producing reflected “output polarization ellipse”. The ratio between
secondaryradiation,sothattheoveralleffectsofthecom- thesenumbersresultsintwoellipsometricrealparameters.
bined basic interactions (scattering in its essential sense) Ellipsometrictechniquesprovideveryfastin-situandnon-
[8] results in refracted, reflected, diffracted or scattered destructive control of several industrial processes. Never-
light. The effects of linear passive media can be repre- theless,itshouldbenotedthatthe term“ellipsometry”is
sented by linear transformations of the electric field vari- also frequently used as a synonym for “polarimetry”and,
ables.By “passive”we refer to the propertyof notampli- hence, refers to general techniques for measuring polari-
fying the light intensity. metric properties.
Depending on: (1) the nature and particular condi- In general, interactions not only produce changes in
tions of the linear interaction; (2) the spectral profile of the ellipsometric parameters, but also cause selective
the incident light beam; (3) the chromatic properties of changes in the transmitted intensity as well as depolariz-
the material sample, and (4) the specific polarizing prop- ingeffects,sothat,inthese cases,acomplete polarimetry
ertiesofthesample,thewavesemergingfromdifferentho- is required. For the measurement of all the sixteen ele-
mogeneous parts of the material target can have different ments of a Mueller matrix, the material sample is placed
degrees of mutual coherence. Coherent interactions can betweenageneratorandananalyzerofpolarizationstates.
be represented through the Jones formalism [9], whereas The generator must be arranged into at least four inde-
forthegeneralcaseofincoherentinteractionsofpolychro- pendentconfigurationsforeachone ofwhichthe analyzer
matic light, the Stokes-Mueller model is required. must in turn be disposed into at least four independent
configurations.Usually, the generatorincludes a totalpo-
Concerning the transformations of the state of polar-
larizer followed by a retarder placed in front of the light
izationduetotheinteractionofthewavewithlinearmate-
source, whereas the analyzer includes a retarder followed
rial media, Jones introduced a simple method applicable
by a total polarizer before the detector. By means of ro-
for deterministic interactions [9]. In this model, the po-
tating wave-plates,Pockelscells or otherkinds ofcontrol-
larization states are represented by two-component com-
lable retarders, these polarimeters can be automatically
plex “Jones vectors” and the medium is characterized by
a 2×2 complex “Jones matrix”. operated [20–28].
The analyzer described, used alone, can be used as a
The classic contributions of authors such
Stokespolarimeter,enablingthe Stokesparametersofthe
as Soleillet [10], Perrin [11], Mueller [12] and
incoming light to be measured [29]. Other interesting de-
Parke III [13,14], have led to a more general model,
vicesarebasedontheuseofpolarizationgratings[30,31].
inwhichthe actionofmediaisrepresentedthroughlinear
transformations of the Stokes parameters. Although the Different kinds of polarimeters are used in severaldif-
4×4 real matrices characterizing these transformations ferent environments such as remote sensing [32]; scat-
are usually called “Mueller matrices” it should be noted tering applications [8,33,34]; ITER & Tokamak [35];
thatitwasSoleilletwhofirstintroducedthisconcept[10]. Astronomy [36], etc.
The lecture notes where Mueller used these matrices For some applications,spectralpolarimetry andinter-
remained unpublished except for a brief note [12]. The ferometry are combined into a whole device [35–40].
connection between the Jones and the Soleillet-Mueller Polarimetric techniques are powerful tools for the
models was studied by Parke III [13,14]. An interesting study andanalysisofmaterialsamples because,giventhe
compilation of several seminal works on polarization specific interactionconditions (reflection,refraction,scat-
optics can be found in reference [15]. tering...) and given the spectral and spatial properties
Some works have dealt with the case of determinis- of the light probe, up to sixteen independent parameters
tic nondepolarizing media (polarimetrically pure media), can be measured. These measurable quantities are the el-
leading to clear physical interpretations of the seven in- ements of the Mueller matrix, which characterize the po-
dependent parametersthat appear in this case [16,17].In larimetricpropertiesof the sample with respectto the in-
the generalcase,acomplete polarimetriccharacterization teraction conditions mentioned. Nevertheless, important
ofthemediumrequiresconsideringuptosixteenindepen- limitations of these techniques arise from the absence of
dent parameters. acomplete understandingofthe informationcontainedin
Given the essentialnature of polarizationphenomena, theMuellermatrix.Thephysicalinformationisstructured
polarimetry is a general-purpose technique which can be in a complicated manner and, thus, the obtainment of a
appliedina greatvarietyofindustrial,medicalandscien- setofsixteenquantitieswithclearphysicalmeaningisnot
tific environments. Thus, different interaction conditions straightforward.In fact, as we will see, some new param-
can be arranged in order to obtain information about eters must be introduced in order to reach the indicated
material samples from polarimetric measurements. These objective.
techniqueshavebeenwidelyusedformanyyears.Onepar- Consequently, the mathematical characterization
ticular technique, called “ellipsometry” [18,19], is based of Mueller matrices is a key question in the exploita-
on determining some properties of material samples from tion of powerful polarimetric techniques. In general, the
J.J. Gil: Polarimetric characterization of light and media 3
values of the elements of Mueller matrices are restricted As usual in polarization optics, under this assump-
byseveralquadraticandbilinearconstraininginequalities, tion of quasimonochromaticity,it is very advantageousto
which have been the main subject of several papers. use the analytic signal representation, where the compo-
In this review, we consider the mathematical descrip- nentsaredescribedthroughtheirrespectivecomplexvari-
tion of the polarimetric properties of light and media ables [6].
throughaunifiedmodelbasedontheconceptofcoherency Let us consider the analytical signal representations
matrices. The measurable quantities arise as the coeffi- ηx(t), ηy(t) of the two mutually orthogonal components
cients of the expansion of the coherency matrix in a set of the electric field in a plane perpendicular to the direc-
ofHermitiantrace-orthogonalmatricesconstitutedbythe tion of propagation
generatorsoftheSU(n) groupplus then×nidentity ma-
trix. The physical parameters with direct physical mean- ηx(t)=Ex(t)+iE˜x(t) =Ax(t)ei(u(t)+βx(t)),
itnitgieas.re defined from the corresponding measurable quan- ηy(t)=Ey(t)+iE˜y(t) = Ay(t)ei(u(t)+βy(t)), (3)
The different sections of this paper include relevant
results quoted in the literature, as well as some new re- where,E˜x(t),E˜y(t),aretheHilberttransformsofthereal
components of the electric field, and u(t)=k¯z−ω¯t.
sults, which are treated on the basis of the unified frame-
These twocomponentscanbe arrangedasthe compo-
work indicated [41].Thus, for example, in the light of the
nents of the following 2×1 complex vector
respective “indices of purity”, the polarimetric purity of
3D states of polarizedlightis analyzed, as well as the po- (cid:3) (cid:4) (cid:3) (cid:4)
larimetric purity of media. Moreover, the retarding, po- η(t)= η1 =ei(u(t)+βx(t)) Ax(t) ,
larizinganddepolarizing propertiesofmaterialmedia are η2 Ay(t)ei(βy(t)−βx(t))
characterizedthroughasetofsixteenindependentparam- (4)
eters with direct physical meaning. We will find that ten and,byavoidingtheglobalphasefactor(withoutphysical
of these parameters are physically-invariantquantities. meaning), the “instantaneous Jones vector” is defined as
The following sections consider successivelythe polar- (cid:3) (cid:4)
ization states of plane waves, the 3D states of polariza- ε(t)= Ax(t) (5)
tionandthe polarimetricpropertiesofmaterialmedia.In Ay(t)eiδy(t)
general,when possible, the habitual notations and names
have been preserved. In some cases, particular notations where δy(t)=βy(t)−βx(t) is the relative phase.
have been introduced for the sake of clarity. This vectorincludesallthe informationrelativetothe
temporal evolution of the electric field. It is called “in-
stantaneous” because the amplitudes and relative phase
2 Polarized light are time-dependent variables. In the case of a polychro-
matic wave,the instantaneousJones vectorhas slow time
Given a point in space,the state of polarizationof a light dependence with respect to the coherence time, so that,
beam that propagates in a fixed direction z is given by for time intervals shorter than the coherence time, the
the temporal evolution of the electric field of the electro- polarization ellipse can be considered constant. For time
magnetic wave,which lies in a plane perpendicular to the intervalshigherthanthecoherencetimeofthelightwave,
propagationdirection.LetE(z,t)be theelectricfieldata the instantaneous Jones vector can vary resulting in par-
point z, at time t, of a monochromatic plane wave prop- tially polarized light.
agating in an isotropic medium, and let (e1,e2,e3) be a Thus, when the following quantities have not time de-
referencebasisoforthonormalvectorsalongtherespective pendence
axes XYZ. The components of the electric field are
Ay(t)
Ex(z,t)=Axcos(kz−ωt+βx) Ax(t) =constant, δy(t)=constant, (6)
Ey(z,t)=Aycos(kz−ωt+βy) (1)
so that the electric field describes a stable well-defined
where k = 2π/λ0 (λ0 is the wavelength for the vacuum); ellipse (Fig. 1), the Jones vector is defined as [9]
ω is the angular frequency, ω = 2πν (ν is the natural (cid:3) (cid:4)
farreequtheencrye)s;peβcxt,ivβeyaamrepltihtuedreess.pective phases, and Ax,Ay ε= AAxyeiδy . (7)
In the case ofquasimonochromaticlight,i.e. when the
bandwidth ∆ν is very narrow with respect to the central It should be noted that the conditions given by equa-
frequency ν¯ of the spectral profile, the components of the tion(6)arecompatiblewithintensityfluctuations.Infact,
electric field can be expressed as totally polarizedlight maintains the azimuthand elliptic-
(cid:1) (cid:2) ity of the polarizationellipse fixed whereas the size ofthe
Ex(z,t)=Ax(t)cos(cid:1)k¯z−ω¯t+βx(t)(cid:2) ellipse fluctuates resulting in a “mean intensity” over the
Ey(z,t)=Ay(t)cos k¯z−ω¯t+βy(t) (2) measurement time. Moreover, slow time variations of the
Jones vector with respect to the measurement time can
where k¯,ω¯ are the respective mean values of k,ω. be represented by this model [42].
4 The European Physical Journal Applied Physics
Due to its statisticalnature,Φ is characterizedby the
fact that its two eigenvalues are non-negative. These two
constraints are a complete set of necessary and sufficient
conditions for a Hermitian matrix Φ to be a coherency
matrix, i.e. to represent a particular state of polarization
of a light beam.
The statistical properties of Φ appear explicitly when
its elements are written in terms of the corresponding
standard deviations σ0,σ1 and the complex degree of co-
herence µ (cid:3) (cid:4)
Φ= σµ02∗σ0σ1 µσσ012σ1 . (10)
As some authors have pointed out [2,46] the normalized
matrix
Φ
Φˆ ≡ (11)
trΦ
is the corresponding density matrix, which contains in-
Fig. 1. (Color online) The polarization ellipse.
formation about the populations and coherences of the
polarization states.
We see that the Jones vector provides a basic model
for representing the state of polarizationwhen light is to-
2.2 The Stokes parameters
tally polarized. The state of polarization of the coherent
superposition of totally polarized light beams is given by
The coherency matrix Φ can be expressed as a linear ex-
the sum of the corresponding Jones vectors.
pansion,with realcoefficients, in the following basis,con-
Inthecaseofpartiallypolarizedlight,theazimuthand
stituted by the three Paulimatrices plus the identity ma-
(or) ellipticity of the polarization ellipse varies during a
trix
typical measurementtime. In consequence, the treatment
(cid:3) (cid:4) (cid:3) (cid:4)
ofpartiallypolarizedlightrequiresadifferentmathemati-
10 1 0
calmodelinorderto takeintoaccountallthe parameters σ0 = 01 , σ1 = 0−1 ,
that characterize completely the state of polarization. (cid:3) (cid:4) (cid:3) (cid:4)
01 0−i
σ2 = , σ3 = . (12)
10 i 0
2.1 The coherency matrix
It should be noted that, although the habitual notations
The “coherency matrix” (or polarization matrix) Φ [4, forthe standarddeviations(σ0,σ1)andforthe Paulima-
6,13,43–45] of a light beam contains all the measurable trices (σi) have been preserved, this should not lead to
information about its state of polarization (including in- confusionbecauseσiarematrices(bold),whereasthevari-
tensity). This Hermitian 2×2 matrix is defined as ances are scalar quantities.
(cid:3) (cid:4) This set of linearly independent matrices σi consti-
Φ=(cid:3)ε(t)⊗ε+(t)(cid:5) = (cid:3)(cid:3)εε12((tt))εε∗1∗1((tt))(cid:5)(cid:5) (cid:3)(cid:3)εε12((tt))εε∗2∗2((tt))(cid:5)(cid:5) (8) ttmrua(tσetrsiiσcaenjs).a=Tdhe2qeδuyijaa,treaenbdHasesiarsmtifsiotfryiatnσh2ieσ=sip=Dacσe(1+io,;f1)t2r,a×[Dce2-(o1Hr,te1hr)mobgioetiinanangl
whereεistheinstantaneousJonesvectorwhosetwocom- the 2×2 unity matrix]. Thus, these matrices are unitary
ponents are the analytic signals of the electric field of and, except for σ0, are traceless. This basis allows Φ to
the wave; ε+ is the transposed conjugated vector of ε; ε∗ be expressed as the following linear expansion [4,47]
i
represents the complex conjugate of εi; ⊗ stands for the (cid:6)3
Kronecker product, and the brackets indicate time aver- Φ= 1 siσi (13)
aging over the measurement time 2
i=0
1 (cid:5)T where the real coefficients si are given by
(cid:3)X(t)(cid:5)= lim X(t)dt. (9)
T→∞T si =tr(Φσi), i=0,1,2,3. (14)
0
Therefore, Φ is a covariance matrix whose elements are The “Stokes parameters” s0,s1,s2,s3 are measurable
quantities which satisfy the restrictions given by the in-
the second-order moments of the zero-mean analytic sig-
equalities
nstaaltsioεni(atr)y; ain=d 1er,g2o.dUicn,dtehretbhreaacksseutsmcpatnioanlttehrantattihveeylyabree s0 (cid:2)0, s20 (cid:2)s21+s22+s23, (15)
considered as ensemble averaging of ε⊗ε+, where ε are which are equivalent to the non-negativity of the co-
simple realizations. herency matrix Φ.
J.J. Gil: Polarimetric characterization of light and media 5
These parameters, arranged as a 4×1 vector, consti-
tute the “Stokes vector” s. Although in this section we
will use the Stokesvector,we alsointroduce the following
alternativenotationinordertocomparesomeexpressions
with others that will appear in later sections
(cid:3) (cid:4) (cid:3) (cid:4)
S= s00 s01 ≡ s0 s1 . (16)
s10 s11 s2 s3
2.3 The purity criterion
Let us consider now the Euclidean norms of Φ and S
⎡ ⎤
1/2
(cid:6)1 (cid:11) (cid:1) (cid:2)(cid:12) (cid:11) (cid:1) (cid:2)(cid:12)
(cid:6)Φ(cid:6)2 ≡⎣ |φij|2⎦ = tr Φ+Φ 1/2 = tr Φ2 1/2,
i,j=0
(cid:13) (cid:14)1/2 Fig. 2. (Color online) Poincar´e sphere.
(cid:6)3 (cid:11) (cid:1) (cid:2)(cid:12)
(cid:6)S(cid:6) ≡ s2 = tr STS 1/2. (17)
2 i
i=0 Inthesecond-orderapproach,thePoincar´esphere(Fig.2)
Moreover,takingintoaccountthatΦisapositivesemidef- provides a useful geometric representation of all possible
states of polarization. Taking into account the character-
inite Hermitian matrix, we can define the following norm
istic properties of the Stokes parameters, they define a
(cid:6)Φ(cid:6) ≡trΦ=(cid:15)(cid:15)(cid:15)√Φ(cid:15)(cid:15)(cid:15)2. (18) 4D cone, whose generatrix is the s0-axis. The cut of this
0 2 cone with the plane s0 = 1 lead to the 3D representa-
tion given by the Poincar´esphere, which is a solid sphere
Itiseasyto showthatthe abovenormssatisfythe follow-
of radius 1 where all states have an intensity equal to
ing relations
1. Points on the surface represent totally polarized states
(cid:6)Φ(cid:6)2 = 1(cid:6)S(cid:6)2, (19) (pure states) and points inside represent partially polar-
2 2 2 ized states (mixed states). The origin represents second-
(cid:6)Φ(cid:6)0 =s0, (20) order unpolarized light.
1(cid:6)Φ(cid:6)2 (cid:1)(cid:6)Φ(cid:6)2 (cid:1)(cid:6)Φ(cid:6)2. (21)
2 0 2 0 2.5 Decomposition of mixed states
For pure states, (cid:6)Φ(cid:6)2 = (cid:6)Φ(cid:6)2. This equality consti-
2 0 Fromequation(22),the following“trivialdecomposition”
tutesanobjectivepuritycriterion,whereastheotherlimit
of s as a convex sum of a pure state and an unpolarized
(cid:6)Φ(cid:6)2 = 1(cid:6)Φ(cid:6)2 occurs in the case of unpolarized light
2 2 0 state can be immediately obtained [6,44]
(equiprobable mixture of states).
(cid:16) (cid:17) (cid:3) (cid:16) (cid:17) (cid:16) (cid:17)(cid:4)
1 1 1
s=I =I P +(1−P) . (24)
Pu u 0
2.4 Physical quantities characterizing the polarized
light
Anotheralternativedecompositionwithdifferentphysical
meaning is [44]
The Stokes vector can be written as
(cid:16) (cid:17) (cid:16) (cid:17) (cid:3) (cid:16) (cid:17) (cid:16) (cid:17)(cid:4)
s= IIPu , (22) s=I P1u =I 1+2P u1 + 1−2P −1u . (25)
where the main physical magnitudes: (1) the intensity I; This“spectraldecomposition”,representedgraphicallyin
(2)thedegreeofpolarizationP;and(3)theunitaryvector Figure 3, expresses a mixed state as a convex linear com-
u that determines univocally the azimuth ϕ (0(cid:1)ϕ<π) bination of two orthogonal pure states (their respective
and the ellipticity angle χ (−π/4(cid:1)χ(cid:1)π/4) of the “av- Jonesvectorsareorthogonal),whichhavethe samedirec-
erage polarization ellipse”, appear explicitly tioninthePoincar´esphereandarerepresentedbyantipo-
(cid:1) (cid:2) dal points (these states have the same azimuth and their
s2+s2+s2 1/2 ellipticities differ in π/2).The originofthe name spectral
I =s0, P ≡ 1 2 3 , decompositionis derivedfromthe fact that the above de-
⎛ ⎞ s⎛0 ⎞
compositionofaStokesvectorisequivalenttothespectral
u≡ I1P ⎝ss12⎠= I1P ⎝ccooss22χχcsions22ϕϕ⎠. (23) decoNmevpeorstihtieolensso,fgitivsecnoraremspixoenddisntgatceo,htehreernecyexmisattrinixfi.nite
s3 sin2χ possibilities for decomposing it as a convex sum of two
6 The European Physical Journal Applied Physics
(cid:3) (cid:4)
I 1+Pcos2ϕcos2χ P(sin2ϕcos2χ−isin2χ)
Φ= , (28)
2 P(sin2ϕcos2χ+isin2χ) 1−Pcos2ϕcos2χ
Fig.3.(Coloronline)Spectraldecompositionofamixedstate. Fig. 4. (Color online) Arbitrary decomposition of a mixed
state intoa convex combination of two purestates.
pure states
(cid:16) (cid:17) Figure 4 illustrates the arbitrary decomposition by
1 meansofthecorrespondingrepresentationinthePoincar´e
s=I =I(cx+(1−c)y);
Pu sphere.
(cid:16) (cid:17) (cid:16) (cid:17)
1 1
x≡ , y≡ , (26)
v w
2.6 Physical invariants and degree of polarization
where v,w are linearly independent unitary vectors, and
Although the models of the Stokes vectors and the
0<c<1.
Poincar´esphereareparticularlyappropriateforrepresent-
Given a mixed state s, any pure state x can be con-
ing the above decompositions,it is useful to return to the
sidered as a component. Once x is chosen, the coefficient
coherencymatrixmodelbecause,aswillbeshowninlater
c andthe remaining pure componenty are univocallyde-
sections, this model can be generalized in order to repre-
termined
sent the 3D states of polarization, as well as the polari-
c= 2(11−−PPu2Tv), 0<c<1; wi = Pu1i−−ccvi. (27) metTrihcee2ffe×ct2scoofhleinreenacrymmedatiari.xΦ canbewritteninterms
of the physical parameters I, P, χ, ϕ
Provided s is a mixed state, it is straightforwardto show
see equation (28) above
that the condition 0<c<1 is always satisfied.
To illustrate this “arbitrarydecomposition” by means and, from this expression, it is straightforward to repro-
of an explicit example, let us consider the mixed state duce the trivial decomposition.
s = (1,1/2,0,0)T and let us chose the pure state x = The diagonalizationof Φ leads to the spectral decom-
T position,expressedintermsofcoherencymatrices[44,49].
(1,0,1,0) as a component. We find that c = 3/8 and
The quantities I andP are invariantin the sense that
y=(1,4/5,−3/5,0)T.
they remain unchanged under unitary transformations of
Althoughinalatersectionwewillseeademonstration
the coherency matrix [50] and, hence, they are invariant
ofthearbitrarydecompositionof3×3coherencymatrices,
with respect to changes of the reference system XY. In
whichcanalsobeappliedton×ncoherencymatrices,we
fact, I and P are directly related with the eigenvalues
have considered it useful to include this demonstration of Φ [44]
in terms of Stokes vectors, because it provides a more
intuitive view of this kind of decomposition. 1 1
The existence ofdifferentpossibledecompositionsofa λ0 = 2I(1+P), λ1 = 2I(1−P);
mtoixitesdimstpaltiecainttioonpsuirnetshteatpeosshsaibslepa“rttaircguelatrdiemcopmorptoasnicteiodnus”e I =λ0+λ1=trΦ, P = λ0tr−Φλ1. (29)
used in radar polarimetry, where the spectral decomposi-
tion is usually considered to be the only way to achieve In agreement with the statistical nature of Φ, which is a
this [48]. covariance matrix (and, hence, is a positive semidefinite
J.J. Gil: Polarimetric characterization of light and media 7
Hermitian matrix), λ0,λ1 are non-negative. Pure states Therefore, S(2) is characterized univocally by P and
are characterized by rank-1 polarization matrices (only decreases monotonically as P increases. The maximum
one nonzero eigenvalue, P =1), whereas rank-2 polariza- S(2) = ln2 corresponds to P = 0, whereas the minimum
tion matrices correspond to mixed states (P <1). S(2) = 0 is reached for P = 1, (i.e. when light is totally
Appropriate expressions of P that will be useful for polarized, regardless of its spectral profile) [2].
future considerations are We observe that S is dimensionless. Its relation with
(cid:22) (cid:1) (cid:2) (cid:23)1/2 (cid:22) (cid:23)1/2 the concept of “specific radiation entropy” introduced by
2tr Φ2 2(cid:6)Φ(cid:6)2
P(2) = (trΦ)2 −1 = (cid:6)Φ(cid:6)22 −1 , (30) PBlaarnakk,aftofromrunlatpeednceixlsploicfitrlaydbiaytivoonn[L5a3]u,ehaansdbeexetnensdtueddibedy
0
⎛ ⎞ by Barakat and Brosseau [54], who have expressed the
(cid:25)1 1/2
⎜⎜i,j=0s2ij ⎟⎟ (cid:22)tr(cid:1)STS(cid:2) (cid:23)1/2 ntworomvaolnizNedeusmpeacnifincernatdroiaptiieosn. entropy as the difference of
P(2) =⎜⎝ s2 −1⎟⎠ = s2 −1 , (31) Moreover, from the von Neumann entropy, Brosseau
00 00
andBicout[56]havedefineda“polarizationtemperature”
whichcanbeexpressedasamonotonicdecreasingfunction
where the subscript (2) has been added in order to com- of the degree of polarization.
pare these expressions with other that will appear con- Althoughadetailedstudyofthedifferentmeasuresre-
cerning higher order coherency matrices. latedwithentropyandtherelationsamongthemfallsout-
As another interesting way of studying P, Wolf, in sidethe scopeofthis review,otherrelevantapproachesto
a classic paper [6], showed that there always exist two polarization entropy are briefly discussed in Sections 3.6.
orthogonal reference directions such that the degree of and 3.7.
coherence µ reaches its maximum value, which coincides As we will see in later sections, the concept of “polar-
with P. ization entropy” [57], when related with the depolarizing
There are various random distributions which corre- effects of a material medium, is commonly used to ana-
spond to unpolarized light. As Ellis and Dogariu have lyze measurements and images [58]by remote sensing, by
shown [51], the measurement of the correlations of the lidar detection and by synthetic aperture radar polarime-
Stokes parameters allows us to distinguish between these try (SAR Polarimetry) in order to detect spatial hetero-
differenttypesofunpolarizedlight.Moreover,forthecase geneityinagreatvarietyofterrestrialandoceanicsurface
of non-Gaussian light, a quantitative determination of targets [59]. It also will be shown that a meaningful pa-
the degree of polarization of type-II unpolarized light by rameter defined as the degree of polarimetric purity can
means of the distance between the polarization distribu- be used as a quantity alternative to entropy.
tion and the uniform distribution associated with type-I
unpolarized light, has been introduced by Luis [52].
2.8 Scope of the coherency matrix model and some
extensions
2.7 Polarization entropy
It is opportune to consider now the concept of the The Stokes parameters are defined through second-order
von Neumann entropy S applied to electromagnetic moments of the field variables. Therefore, the statistical
waves[2,46,49,53–55], which, in terms of the density ma- nature of light underlies in the mathematical description
trix Φˆ is defined as of the states of polarization of light, so that the study
(cid:27) (cid:28)
S =−tr ΦˆlnΦˆ . (32) of the fluctuations of the field variables in different rel-
evant cases is an important subject. For the cases of
Thisquantityisameasureofthedifferenceintheamount non-Gaussian probability distributions of these variables,
of information between a pure state and a mixed state higherordermoments mustbe consideredand,hence,the
(both with the same intensity) and can also be ex- Stokes parameters do not provide a complete description
pressed as of the state of polarization. Several authors have dealt
(cid:6)1 (cid:27) (cid:28) with this matter, so that a significant amount of knowl-
S =− λˆilnλˆi (33) edge has been achieved [5,7,60–65]. A good compilation,
i=0 includingseveralcontributionsoftheauthor,canbefound
where λˆi = λi/trΦ are the eigenvalues of Φˆ. Thus, in in a book by Brosseau [2].
the case of 2×2 coherency matrices, the von Neumann The measurement of the polarization properties of
light beams with non-Gaussian spectral profiles should
entropy can be written as a function of the degree of po-
requires particular measurement arrangements involving
larization P [54]
interferometers and fast detectors.
(cid:29) (cid:16) (cid:17)
1 1 Coherence properties have a close relation with polar-
S =S(2)(P)≡− (1+P)ln (1+P) ization description, and several authors have dealt with
2 2
(cid:16) (cid:17)(cid:30) this important subject [45,66–73].
1 1
+ (1−P)ln (1−P) . (34) A particular definition of Stokes parameters at the
2 2 wavelet scale has been introduced by Castan˜eda [74].
8 The European Physical Journal Applied Physics
These parameters take into account the correlation prop- 3 3D Polarized light
ertiesofthe randomelectromagneticfieldonthe aperture
plane. Thetwo-dimensionalformalismconsideredintheprevious
section is valid when the propagation direction of light
AgeneralizationoftheStokesparametersofarandom is constant, which is the commonest physical situation of
electromagnetic beam has been introduced by Ellis and interestinpolarimetry.Inthemostgeneralcase,thethree
Dogariu [71] in the space-time domain and has also been components of the electric field vector of the light wave
studiedbyKorotkovaandWolf[75]inthespace-frequency should be considered in order to describe the evolution of
domainonthe basis ofthe Wolf’s unified theoryofcoher- the end of the electric field vector (which determines the
ence and polarization [76]. Whereas the usual Stokes pa- polarization state).
rametersdependononespatialvariable,these “two-point Thissectiondealswiththeconstructionofamodelfor
Stokes parameters”, which can be defined in the space- ageneraldescriptionofthepolarizationpropertiesoflight,
frequency domain from the cross-spectral density matrix applicable to waves whose direction of propagationis not
that characterizes the correlations at two points, depend stable in time. Thus,from the extensionofthe concept of
on two spatial variables and contain additional informa- the Jones vector, the 3D polarization matrix, as well as
tion about the coherence properties. the 3D Stokes parameters, is defined. The main physical
quantities characterizingthe 3Dstates of polarizationare
Another relevant approach to polarization description
obtainedandanalyzed,withspecialattentionbeing given
isthemodelof“spatial-angularStokesparameters”devel-
to the description of polarimetric purity by means of two
oped by Luis [77,78]. Using Wigner functions [79], these
invariant parameters.
parameters are introduced for generalized rays including
spatial and angular dependence and allow their evolution
during propagation to be considered.
3.1 3D Jones vector
Both the above-mentioned generalizations have their
Let us consider a quasimonochromatic wave of arbi-
correspondingcounterpartin Mueller representations,i.e.
trary form, propagating in an isotropic medium, and let
Mueller matrices transforming two-point Stokes parame-
ters [80,81] and spatial-angular Mueller matrices trans- (e1,e2,e3) be a reference basis of orthonormal vectors
along the respective axes XYZ. Given a point r in space,
forming spatial-angular Stokes parameters [78].
theanalyticsignalsofthethreecomponentsoftheelectric
Theaboveparagraphsdealwithaclassicaldescription field of the electromagnetic wave can be arranged as the
of polarization. The Stokes parameters have been used 3×1 complex vector
traditionallytodescribecertainquantumsystems[46,82]. ⎛ ⎞ ⎛ ⎞
Nevertheless, it should be noted that a quantum descrip- η1 ηx(t)
tionofpolarization,performedbymeansofreplacingclas- η(t)=⎝η2⎠=⎝ηy(t)⎠
sical amplitudes by bosonic operators, results in some η3 ηz(t)
⎛ ⎞
properties that cannot be fully described by the classi- Ax(t)
calStokesparameters[83].Thus,forinstance,the electric =ei(u(t)+βx(t))⎝Ay(t)ei(βy(t)−βx(t))⎠ (35)
field vector never describes a definite ellipse (due to un- Az(t)ei(βz(t)−βx(t))
certainty relationships); there are states with P = 0 that
cannotberegardedasunpolarized,andP doesnotreflect whereu(t)=k¯·(r/|r|)−ω¯t,k¯beingthemeanwave-vector
the lack of perfect polarization for any quantum state. at the point r.
Thecorresponding“3DinstantaneousJonesvector”is
The quantum descriptionofpolarizationstates is nec-
defined as
essary for the study and characterization of coherent,
⎛ ⎞ ⎛ ⎞
squeezed, number, and phase states, and for the design
ε1(t) Ax(t)
of experiments in order to show fundamental properties ε(t)=⎝ε2(t)⎠=⎝Ay(t)eiδy(t)⎠. (36)
andapplicationsofquantumtheorysuchasentanglement, ε3(t) Az(t)eiδz(t)
complementarity, information processing and cryptogra-
phy, teleportation and Bell inequalities.
In this expression, the possible time dependence of am-
plitude and phases of the three components of the wave
The characterization of states of polarization through
are indicated. Since only the phase differences have phys-
probabilitydistributionsonthePoincar´esphereallowsthe
definition of the quantum degree of polarization as the ical meaning, δx(t) has been taken as reference, so that a
global phase factor has been avoided.
distance betweenthe correspondingpolarizationdistribu-
Fortotally-polarizedsates,thefollowingconditionsare
tion and the uniform distribution representing unpolar-
satisfied
ized light. This definition can be particularized as a trace
distance between the coherency matrix and the identity Ay(t) Az(t)
matrix (unpolarized light), and is an important and open =constant, =constant,
Ax(t) Ax(t)
subjectthat providesresultsapplicable to the description
of quantum field states [83–86]. δy(t)=constant, δz(t) =constant, (37)
J.J. Gil: Polarimetric characterization of light and media 9
⎛ ⎞
(cid:2)A2x(t)(cid:3) (cid:2)Ax(t)Ay(t)e−iδy(t)(cid:3) (cid:2)Ax(t)Az(t)e−iδz(t)(cid:3)
R=⎝(cid:2)Ax(t)Ay(t)eiδy(t)(cid:3) (cid:2)A2y(t)(cid:3) (cid:2)Ay(t)Az(t)ei(δy(t)−δz(t))(cid:3)⎠. (40)
(cid:2)Ax(t)Az(t)eiδz(t)(cid:3) (cid:2)Ay(t)Az(t)e−i(δy(t)−δz(t))(cid:3) (cid:2)A2z(t)(cid:3)
and the generalized Jones vector is matrix”, i.e. to represent a particular state of polariza-
⎛ ⎞ tion of a light beam. An equivalent set of conditions is
Ax derived from the fact that the Hermitian matrix R has
ε=⎝Ayeiδy⎠. (38) three nested non-negative principal minors [88].
Azeiδz The statistical properties of R arise clearly when its
Itiseasytoverifythat,inthiscase,thedirectionofprop- elementsrij (i,j =0,1,2)arewrittenintermsofthecor-
agation remains constant and the end of the electric field responding standard deviations σi and degrees of coher-
vector describes an ellipse perpendicular to the direction ence µij,
of propagation. The 2D model is then easily reproduced
by taking the direction of propagation as the Z reference rij ≡µijσiσj, (41)
axis, so that the third component of the 3D Jones vector
is cancelled. where σi2 =(cid:3)εiε∗i(cid:5)=rii, µij =rij/σiσj, i.e.
⎛ ⎞
3.2 3D coherency matrix σ02 µ01σ0σ1 µ02σ0σ2
R=⎝µ∗01σ0σ1 σ12 µ12σ1σ2⎠. (42)
Severalauthors have dealt with the 3D description of the µ∗02σ0σ2 µ∗12σ1σ2 σ22
polarization state. From a chronological point of view, it
is worth emphasizing the work of Roman [87], who de-
fined a set of generalized Stokes parameters from a par- Fromthe non-negativityofthree nestedprincipalminors,
ticular basis of 3 × 3 matrices. These matrices are not wecanwrite a setofnecessaryandsufficientexplicitcon-
trace-orthogonaland, consequently, do not constitute the ditions for R to be a covariance matrix and, hence, to be
bestoptionforbuilding abasis.As Fanopointedout[46], a coherency matrix representing3D states of polarization
a convenientbasisfor n×n coherencymatrices is a setof
Hermitian trace-orthogonal operators. Thus, for our pur- r00 (cid:2)0; (43a)
poses,anappropriatebasisisthatconstitutedbythe3×3
identity matrix together with the eight Gell-Mann matri-
ces. We will see that this choice, based on the generators
of the SU(3) group, is consistent with a generalization of 1(cid:2)ρ01; (43b)
the polarization algebra to n×n coherency matrices.
The so-called “coherency matrix” [6,43–45]or “polar-
ization matrix”, which contains all measurable informa-
tion about the state of polarization (including intensity) detR=1+2ρ01ρ12ρ02cos(β01+β12−β02)
ofanelectromagneticwave,isdefinedasthefollowing3×3 −ρ2 −ρ2 −ρ2 (cid:2)0 (43c)
01 12 02
Hermitian matrix
R=⎛(cid:3)ε(t)⊗ε+(t)(cid:5) ⎞ where ρij ≡|µij|,µij ≡ρeβij.
(cid:3)ε1(t)ε∗1(t)(cid:5) (cid:3)ε1(t)ε∗2(t)(cid:5) (cid:3)ε1(t)ε∗3(t)(cid:5) An interesting geometric interpretation of the 3D co-
=⎜⎝(cid:3)ε2(t)ε∗1(t)(cid:5) (cid:3)ε2(t)ε∗2(t)(cid:5) (cid:3)ε2(t)ε∗3(t)(cid:5)⎟⎠, (38) herencymatrixhasbeenpresentedbyDennis[89]through
thedecompositionof Rintoarealsymmetricpositivedef-
(cid:3)ε3(t)ε∗1(t)(cid:5) (cid:3)ε3(t)ε∗2(t)(cid:5) (cid:3)ε3(t)ε∗3(t)(cid:5) inite matrix (interpreted as the moment of inertia of the
ensemble, and represented by means of the correspond-
whoseelementsarethesecond-ordermomentsofthezero-
ing ellipsoid) and a real axial vector (corresponding to
meananalyticsignalsεi(t); i=1,2,3,so thatR is a co- the mean angular momentum of the ensemble). Another
variancematrixand,therefore,R iscompletelycharacter-
kind of representation of 3D states of polarization (but
izedbythefactthatitisapositivesemidefiniteHermitian
only valid for pure states), based on the Majoranasphere
matrix [88].
representation has been presented by Hannay [90] who
The expression of R in terms of the amplitudes and
has studied its relation with the Berry phase. Moreover,
relative phases of the field components is
Saastamoinen and Tervo [91] and Ellis and Dogariu [92]
see equation (40) above have introduced geometric representations of the invari-
As we have observed, R is characterized by the fact that ants of R in the eigenvalue space.
its three eigenvalues are non-negative. These three con- As a basis for the expansionof R, let us now consider
straintsareacompletesetofnecessaryandsufficientcon- the following set of Hermitian, trace-orthogonal matrices
ditions for a Hermitian matrix R to be a “3D coherency constituted by the Gell-Mann matrices plus the identity
10 The European Physical Journal Applied Physics
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
(cid:31) (cid:31)
1 00 01 0 0 01
ω00 ≡⎝0 10⎠ ω01 ≡ 3⎝10 0⎠ ω02 ≡ 3⎝0 00⎠
2 2
0 01 00 0 1 00
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
(cid:31) (cid:31) (cid:31)
0 −i0 1 0 0 0 00
ω10 ≡ 3⎝i 0 0⎠ ω11≡ 3⎝0−10⎠ ω12 ≡ 3⎝0 01⎠
2 2 2
0 0 0 0 0 0 0 10
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
(cid:31) (cid:31)
0 0−i 0 0 0 1 0 0
ω20 ≡ 3⎝0 0 0 ⎠ ω21 ≡ 3⎝0 0−i⎠ ω22 ≡ √1 ⎝0 1 0 ⎠. (44)
2 i 0 0 2 0 i 0 2 0 0−2
(cid:31) (cid:31)
3 3
q00 ≡r00+r11+r22 q01 ≡ (r01+r10) q02 ≡ (r02+r20)
2 2
(cid:31) (cid:31) (cid:31)
3 3 3
q10 ≡i (r01−r10) q11 ≡ (r00−r11) q12 ≡ (r12+r21)
2 2 2
(cid:31) (cid:31)
3 3 1
q20 ≡i (r02−r20) q21 ≡i (r12−r21) q22 ≡ √ (r00+r11−2r22) (46)
2 2 2
⎛(cid:1) √ √ (cid:2) (cid:1)√ √ (cid:2) (cid:1)√ √ (cid:2)⎞
⎜ 2q0(cid:1)0√+ 6q11√+ 2q(cid:2)22 (cid:1) 6q√01−i 6√q10 (cid:2) (cid:1)√6q02−i√6q20(cid:2)⎟
R= 61⎜⎝ (cid:1)√6q01+i√6q10(cid:2) 2q(cid:1)00√− 6q11√+ 2(cid:2)q22 (cid:1) 6q12−√i 6q21(cid:2) ⎟⎠. (47)
6q02+i 6q20 6q12+i 6q21 2q00−2 2q22
matrix is also included in the basis. This choice differs from
those used in the cited references, as well as in refer-
see equation (44) above
ence [96].
The notation used for these matrices is justified for the
Obviously, these options have no physical consequences
sake of simplicity as well as to emphasize the symmetry
and do not affect the results expressed in terms of invari-
in some future mathematical expressions.
antquantities(i.e.derivedfromtheeigenvaluesof R),but
As occurs with the basis used for the expansionof the
2×2coherencymatrices,the abovematricesωij areHer- leadto differentexpressionsfor the 3DStokesparameters
mitian ωij = ω+ij; trace-orthogonal tr(ωijωkl) = 3δikδjl, andAfosrwtehewirillresleaet,iothnes wnoitrhmoatlihzeartipoanrapmerefoterrms.ed is consis-
and satisfy ω2 = D(1,1,1). Thus, they are unitary and,
ij tentwithageneralizedmodelforthepolarimetricproper-
except for the identity matrix ω00, they are traceless. ties of light and media based on n-dimensional coherency
The basisconstituted byωij allowsRto be expressed matrices.
as the following linear combination [2]
(cid:6)2
R= 1 qijωij, qij =tr(Rωij), (45) 3.3 3D Stokes parameters
3
i,j=0
According to the definitions introduced in the previous
where the nine real coefficients qij can be properly called section, the 3D Stokes parameters take the form
the “3D Stokes parameters”. see equation (46) above
Inordertocomparethisexpansionwiththatpresented
by other authors, the following observations should be andthecoherencymatrixcanbeexpressedintermsofqij
as follows
taken into account:
see equation (47) above
– for the sakeof symmetry and simplicity in the mathe-
matical relations, the order chosen for the Gell-Mann ItisobviousthatthePoincar´esphererepresentationisnot
matrices differs from that used by Gell-Mann and by applicable to 3D states of polarization. The constraining
several authors [93–95]. Except for the change ω11 ↔ inequalities between qij (derived from the non-negativity
ω22, the order coincides with the one used by of the eigenvalues of R) are of greater complexity than
Brosseau [2]; that of the 2D Stokes parameters.(cid:1)In(cid:2)fact, these re(cid:1)stric(cid:2)-
– the Gell-Mannn matrices have been normalized in or- tionsinvolves,notonlytrRandtr R2 ,butalsotr R3 .
der to ensure that their weight (Euclidean norm) co- Nowweexaminethephysicalmeaningofthe3DStokes
incide with that of the 3×3 identity matrix, which parameters.To do this, we first consider the reduction to