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Vectorial Pauli algebraic approach in polarization optics.

I. Device and state operators

Tiberiu Tudor

Faculty of Physics, University of Bucharest, P.O. Box MG-11, 0771253 Bucharest-Magurele, Romania

Received 3 October 2008; accepted 13 January 2009

Abstract

This paper inscribes on the line of the efforts (sketched in the Introduction) in elaborating theoretical approachesalternative to the traditional Jones and Mueller matrix calculi in polarization optics. The more abstract, compact andelevated forms of linear algebra are not fully exploited yet in the polarization optics. A vectorial and pure operatorialPauli algebraic approach to the interaction between the polarized light and the polarization optical systems is given.This is the most compact, adequate and elegant calculus corresponding to the well-known geometric handling of thepolarization states and their interaction with the polarization devices on the Poincare sphere. In this first paper, wededuce the Pauli algebraic vectorial forms of the operators corresponding to the orthogonal and nonorthogonalpolarization devices and to all the states of light polarization. In the next paper we shall give the vectorial Paulialgebraic analysis of the interaction between the whole hierarchy of these devices and the various forms of polarizedlight.r 2009 Elsevier GmbH. All rights reserved.

Keywords: Light polarization; Poincare sphere; Quantum operators; Pauli algebra

1. Introduction

In any theoretical approach to polarization optics weare faced with three main problems: the description of theoptical polarization states (SOPs), the description of theoptical polarization systems (anisotropic devices or media)and the analysis of the action of these systems on thepolarization states. The vast bibliography on this subject,covering almost two centuries of researches is presented,somewhat complementary, in some of the fundamentaltextbooks or review articles in the field, e.g. [1–6].

From a mathematical viewpoint, the interaction linearpolarization systems – SOPs is a question of linear algebra,more precisely one of noncommutative (nonabelian) linear

e front matter r 2009 Elsevier GmbH. All rights reserved.

eo.2009.01.004

ess: ttudor@nipne.ro.

algebra, and there are some languages (matrix, pureoperatorial, group-theoretical) in which this algebra canbe wrapped, e.g. [7,8].

A first characteristic of our approach is that it is apure operatorial (‘‘non-matrix’’, ‘‘coordinate-free’’) one.

In polarization optics the great majority of the papersand of the textbooks are written in the Jones andMueller formalisms, that is in matrix representationsof the linear algebra. The group-theoretical approach,which has been spectacularly developed in the lastdecade [9–11], is also handled in the 2� 2 and 4� 4matrix representations of the involved groups, corre-sponding in fact to the Jones and Mueller languages,respectively.

The matrix method has the computational advantageof following automatically a fixed, early learned and

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well-known routine, but a disadvantage of principle:it works with blind collections of numbers, associatedmore or less arbitrarily to the described reality (SOPsand polarization systems). Referring concretely to theJones and Mueller formalism, Shurcliff [1], in hisfundamental textbook, says: ‘‘The investigation followsa fixed routine in which little thought is required beyondlooking up the vectors and matrices in a table andperforming the standard multiplication operations’’.

A first major blow was given to the dominant positionof the matrix form of linear algebra in physics by Dirac’sPrinciples of Quantum Mechanics [12]: ‘‘There is thesymbolic method, which deals directly in an abstractway with the quantities of fundamental importance andthere is the method of coordinates or representations,which deals with sets of numbers corresponding to thesequantitiesy . The second of these has the advantagethat the kind of mathematics required is more familiarto the average studenty . The symbolic methodhowever seems to go more deeply into the nature ofthings. It enables one express the physical law in a neatand concise way.’’ ([12], Preface to the first edition).

Nowadays Dirac’s ‘‘bra-ket’’ pure vectorial (‘‘coordi-nate-free’’) and pure operatorial (‘‘matrix-free’’) languageis largely adopted in quantum mechanics, but initiallythere was a general reluctance to accept it.

A radical viewpoint concerning the coordinate andmatrix languages in physics and even in mathematics isexpressed by David Hestenes in his Oersted Medal

Lecture 2002, Reforming the Mathematical Language of

Physics [13]: ‘‘The insistence of conceiving a vector as alist of number of coordinatesy is a kind of conceptualvirus, because it impedes development of a more generaland powerful concept of vector. I call it the coordinate

virus!y The entire physics curriculum, including mostof the textbooks, is infected with this virus’’. Hesteneshas revitalized Grassmann’s and Clifford’s ‘‘geometricalgebra’’ as a unifying coordinate-free and matrix-freelanguage of physics. For a long time the physicalcommunity ignored or even dismissed Hestenes’ work:‘‘I learned to be careful about when, where and how Ipresented my work to other physicists, because thereaction was invariable dismissive as soon as theydetected deviation from standard practice or beliefs’’[13]. But in the last decades a whole thinking movementhas been growing up, which has led to a reformulationof the classical and quantum mechanics, electrody-namics and theory of relativity in a coordinate andmatrix-free language based on Clifford algebra [14–16].The extent of the field may be gleaned from severalwebsites [17–19].

In polarization optics, the pure operatorial (‘‘coordi-nate-free’’ or ‘‘matrix-free’’) approaches are somewhatisolated. Beginning with his doctoral thesis [20],Fedorov [21,22] has developed an invariantiv languagein polarization optics in the real physical space and in

the frame of electromagnetic theory of light. The term‘‘coordinate-free’’ appears explicitly in the Preface of[22]. Marathay [23,24] used mainly a pure operatoriallanguage in two of his papers, in analyzing some of theproperties of the polarization device operators on acomutatorial basis.

Simmons and Guttmann’s [5] singular textbookStates, Waves and Photons; A Modern Introduction to

Light introduces firmly Dirac’s language in polarizationoptics, but with a large tolerance to the coordinateand matrix representations too. Brosseau [25] appliedthe Dirac formalism to the study of the interaction ofpolarized radiation with polarization devices. A sys-tematic treatment of the polarization optical devices inthis language is given in [26–29].

Although tributary to the matrix language (even inthe titles), some very important papers of Lu andChipman [30], Barakat [31,32] and Brosseau [33], handlethe spectral, polar and the singular-value decomposi-tions and their consequences in polarization optics inoperatorial terms. Following the evolution of the papersof such important authors, one can see how, actually,the polarization theory is pushed towards an invariantivlanguage. This fact has a very objective motivation: theobservables (intensity, gain, degree of polarization) areinvariants [2–6,34].

The second characteristic of our approach is that it isa vectorial Pauli algebraic one.

Well-known, besides the description of the polarizedlight in the real physical space, there is anotherapproach, in the abstract Hilbert space of the polariza-tion states [35,36]. The isomorphism of this space withthe real unit ball S3

1 [2,3] underlies the famous, intuitiveand effective Poincare representation of the polarizationstates [1–6].

The Pauli algebra is the most widespread of the variousmathematical tools (spherical trigonometry, quaternionicalgebra, turns, Clifford algebra) of handling rotations inR3 and particularly on the Poincare sphere. Thereforeit was largely adopted in the polarization theory (e.g.[3,37,38]).

It is Whitney [39] who introduced a Pauli algebraicpure operatorial (non-matrix) approach in analyzingsome device (‘‘instrument’’) operators, and in general-izing the Jones and Hurwitz’ theorems concerning thecomposability of polarization optical systems. Shenoted: ‘‘the well-known Jones and Mueller formalismsand the spherical-trigonometric approachy have vir-tually supplanted vector-field algebraic methods.’’

More recently, non-matrix descriptions of the statesof polarized light (SOPs) and of the operators of the‘‘canonical’’ [4] polarization devices have been givenin the quaternionic language [40,41] and in a Cliffordalgebraic approach [42].

Quaternionic algebra is isomorphic, with minordifferences, to the Pauli algebra, and both can be

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embedded in the Clifford algebra Cl3. From these threealgebraic ‘‘dialects’’, the Pauli algebra is by far morefamiliar to the physicists.

Pauli algebra can be handled in a very compact andeffective, vectorial, form. The theory of the transfer ofthe SOPs through the polarization devices has not takenso far advantage of the mathematical virtues of thisform. Moreover, in the whole field of the polarizationtheory, elements of the vectorial Pauli algebra occuronly sporadically in the bodies of the standard scalarand generally matrix approaches.

In this and a companion paper we will give asystematic and coherent pure operatorial treatmentof the SOPs, polarization systems and their interaction,in the frame of the vectorial Pauli algebra and of thePoincare geometric representation.

We have to emphasize that the Pauli algebraicapproach is intimately connected with the widespreadgeometric handling of the polarization problems on thePoincare sphere. The Poincare sphere S2

1 (more gen-erally the Poincare ball S3

1) provides the most effectiveand at the same time the most profound geometricrepresentation of the SOPs and their interactions withpolarization systems: it joints the advantages of theintuitiveness of a R3 representation with that of workingin a space topologically isomorphic with the polariza-tion state space.

The Pauli algebra, namely in its compact form wepropose here – on the one hand pure operatorial andon the other hand vectorial – is the most economic andelegant form of handling movements on the Poincaresphere or in the Poincare ball. It gives a rigorousmathematical basis to the intuitive Poincare geometricalhandling of the polarization states and of their interac-tions with the polarization devices.

In this first paper we shall deduce the vectorial Paulialgebraic forms of the operators of the linear polariza-tion systems and of the SOPs. In the next paper we shallanalyze their interactions.

2. Pauli algebraic expansions of the 2� 2

operators

2.1. Pauli algebraic expansion and the Pauli axis of

a 2� 2 operator

Let us consider a linear operator A defined on aunitary space of dimension two over the field of complexnumbers C1. In the following we shall refer to theseoperators as ‘‘two-dimensional operators’’ or ‘‘2� 2operators’’, as they are shortly called sometimes in theliterature [43]. Obviously, this abbreviation is tributaryto the matrix representation of the operators (whichpays no role in our approach). The manifold of these

operators forms a group, usually called the GL(2, C)group, under the operation of multiplication.

The operators of the various polarization (‘‘non-image forming’’ [2]) systems (devices, more generallylinear anisotropic media) are two-dimensional opera-tors. The space over which they are defined isthe bidimensional complex space of the spinors(Jones-vectors in the matrix language) which representthe SOPs.

The 2� 2 operators themselves form a quadridimen-sional vectorial complex space, and can be expanded invarious bases of four 2� 2 operators. One of these basesis that of the Pauli operators si defined by themultiplication rule [44]:

sisj ¼ s0dij þ ieijksk; i; j ¼ 1; 2; 3, (1)

where dij and eijk are the Kronecker and the Levi-Civitasymbols respectively, and s0 is the unit 2� 2 operator.This equation defines completely the Pauli algebra; the(various [3,5]) matrix representations of the si operatorsis irrelevant for the core of this algebra.

In the Pauli basis any 2� 2 operator may be expandedin the form:

A ¼ a0s0 þ a1s1 þ a2s2 þ a3s3, (2)

where ai are some, generally complex, scalars. Inthe case of the Hermitian operators corresponding tothe SOPs these scalars are real, and a1, a2, a3 have anintuitive meaning in the frame of the Poincare repre-sentation of the states: they are the projections of thePoincare unit vector of the SOP on the three orthogonalaxes in the reference system of which the Poincaresphere is represented and are known under the nameof Stokes parameters of the state.

Here we refer to the whole class of linear operatorsdefined on a bidimensional space over the field of complexnumber, so that a1, a2, a3 are generally complex. We shallextend for them the denomination of Stokes coefficients(parameters) of the operator A.

Labeling by r a vectorial operator of components s1,s2, s3 – the Pauli vectorial operator – Eq. (2) may bewritten in a condensed form:

A ¼ a0s0 þ a � r , (3)

where a(a1, a2, a3,) is in general a complex vector. Weshall call this vector the Pauli vector of the operator A

and the corresponding unit vector a=jjajj – the Pauli axis

of the operator [45]. For a Hermitian operator (e.g.corresponding to a SOP), the Pauli axis coincides withthe real Poincare unit vector (of the SOP), but in thegeneral case it is a complex vector, defined in the moreabstract, C3 space. This correspondence to a R3 vectorfor the particular case of the Hermitian operators is asolid support for our intuition in going further in C3.

The Pauli operator r being completely defined byEq. (1), Eq. (3) tells us that every 2� 2 operator is

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completely defined by its Pauli axis and (as we shall see,only in irrelevant details) by the scalar a0. Eq. (3) is afirst bridge between the Hilbert space of the 2� 2operators and the (complex, indeed, but) simplevectorial space of their Pauli axes. These two spacesare isomorphic and

�

the characteristic features of various kinds of 2� 2operators can be transposed in some characteristicfeatures of their Pauli axis, as we shall see in thepresent paper, � the action of the operators of various systems (in

polarization optics – various devices or anisotropicmedia) on the operator of the state (here SOP) can bemapped in various movements of the Pauli axis of theoperator of the state in the Pauli ball S1

3C. For anintuitive grasp of this correspondence it is gratifyingto say that, if the operator of the system (e.g.polarization device) is unitary or Hermitian (ingeneral: normal), the corresponding movement maybe reduced to one in the Poincare S1

3R ball. Here theindices C and R stand for complex and real,respectively. This mapping will be analyzed for theoptical polarization case (polarization states andsystems) in a further, companion, paper.

I have formulated the above aspects of the correspon-dence operators – Pauli axes in quite general terms (states,systems) because this theory may be applied, point bypoint, for any other two-state systems [46], e.g. in rayoptics [47], two-beam interferometry [48], multilayer optics[49], laser optics [50], two-mode squeezed states of light[51]. All the two-state systems have the same underlyingalgebra, as was demonstrated in the frame of the group-theoretical approach in the above quoted papers.

2.2. The commutator of the 2� 2 operators: normal

and nonnormal operators

In the following we shall make largely use of the Pauliexpansion of the product of two 2� 2 operators. Let usconsider a second operator:

B ¼ b0s0 þ b � r. (4)

Then, with Eq. (3), we get:

AB ¼ a0b0 þ a0b � rþ b0a � rþ ða � rÞðb � rÞ, (5)

and by means of Dirac’s relation:

ða � rÞðb � rÞ ¼ a � bþ iða� bÞ � r, (6)

one obtains:

AB ¼ ða0b0 þ a � bÞs0 þ ða0bþ b0aÞ � rþ iða� bÞ � r .

(7)

This equation is a second important bridge betweenthe Hilbert space of the 2� 2 linear operators and the C3

space of their Pauli vectors, in the same sense as Eq. (3):various relationships or characteristic features of theseoperators can be straightforwardly transposed in thecorresponding relationships or features of their Paulivectors, reaching this way a direct geometrical repre-sentation.

Let us mention first a quite general result that followsfrom this relation. The commutator of two 2� 2operators, A and B, is determined by (the outer productof) their Pauli vectors:

½A;B� ¼ 2iða� bÞ � r . (8)

In other words two operators commute when theirPauli vectors are collinear:

a� b ¼ 03a ¼ lb3a � b, (9)

where l is a complex scalar and the sign � standshere for ‘‘congruent’’ or ‘‘equivalent’’ (in the sense ofcolinearity).

We shall now apply these basic results in establishingthe Pauli algebraic condition of the normality of a 2� 2operator, more precisely for transposing the conditionof normality of an operator in a characteristic feature ofits Pauli vector.

One of the necessary and sufficient conditions of thenormality of an operator is the commutativity with itsadjoint. In our case and notations:

½A; Ay� ¼ 0, (10)

where

Ay ¼ an

0s0 þ an � r. (11)

With Eqs. (3) and (4) in Eqs. (8) and (9) the operatorialcondition of normality of an operator, Eq. (10), reduces toa condition referring to its Pauli axis:

a� an ¼ 0, (12)

i.e. the two complex-conjugate vectors a and a* must becollinear:

an � a3an ¼ la (13)

with l a complex number.This condition means that, apart from a complex

scalar factor, the Pauli vector of a normal operatorreduces to a real vector:

a ¼ eiar. (14)

Hence, with Eq. (3), the Pauli expansion of a normal

operator is

N ¼ eia0 ja0js0 þ eiar � r, (15)

where r is a R3 vector, a0 is a real scalar modulo 2p and ais a real scalar modulo p.

Labeling by m the modulus of r, and by n thecorresponding unit vector, i.e. the Pauli axis of the

ARTICLE IN PRESST. Tudor / Optik 121 (2010) 1226–12351230

operator, Eq. (15) may be written as

N ¼ eia0 ja0js0 þ eiamn � r . (16)

The Pauli axis of any normal operator is of the form:

a

jjajj¼ eian, (17)

where n is a real unit vector.Hence the Pauli axis of a normal operator is reducible

to a real unit vector. In other words, by a suitableprocessing, the Pauli axis of a normal operator, Eq. (17),may be brought into the real subspace R3 of the C3.

On the contrary, the Pauli axes of the nonnormaloperators are irreducible complex vectors.

It is worth stressing once again that passing throughthe algebraic bridge (Eq. (8)) we have transposed theoperatorial condition of normality of an operator (Eq.(10)) in a characteristic feature of its Pauli axis, Eq. (17):we have passed from the Hilbert space of the 2� 2operators in the C3 space of their Pauli axes, which isisomorphic to the first one.

3. System and state operators in polarization

optics

In the following we shall particularize the expansion(16) for some operators widespread in polarizationoptics: unitary operators (corresponding to the variouskinds of retarders), Hermitian operators, in particularsqueeze operators and projectors (corresponding on theone hand to various kinds of polarizers, on the otherhand to various states of polarized light).

3.1. System operators

3.1.1. Unitary operators. Birefringent systems, phase-

shifters

If N ¼ U is a unitary operator:

UUy ¼ I, (18)

(where I ¼ s0 is the unit operator) with Eq. (16) oneobtains:

ðeia0 ja0js0 þ eiamn � rÞ ðe�ia0 ja0js0 þ e�iamn � rÞ

¼ ðja0j2 þ m2Þs0 þ 2ja0jmn � r cosða� a0Þ ¼ s0, (19)

wherefrom:

ja0j2 þ m2 ¼ 1, (20)

2ja0jm cosða� a0Þ ¼ 0. (21)

From Eq. (21), we get:

a� a0 ¼p2

modulo p; (22)

hence:

eiða�a0Þ ¼ �i. (23)

With this restriction in Eq. (16), we get:

U ¼ eia0ðja0js0 � imn � rÞ. (24)

In polarization optics the unitary operators arehandled in an exponentiated form. We can reach itstraightforwardly by noticing that Eq. (20) may befulfilled if we put:

ja0j ¼ cosd2; m ¼ sin

d2. (25)

With (25) in (24) the Pauli algebraic expansion of themost general unitary operator may be written in theform:

U ¼ eia0 s0 cosd2� in � r sin

d2

� �¼ eia0e�iðd=2Þn�r: (26)

The Pauli axis of a unitary operator

a

jjajj¼ �in (27)

is a pure imaginary vector, i.e. it is situated in the I3

subspace of the C3: in 2 I3.On the other hand, I3 is isomorphic with R3, ðI3 ¼ iR3

Þ

so that, making abstraction of the factor 7i, the Pauliaxis of a unitary operator may be looked as a real unitvector – the Poincare axis – as it is usually treated.

In polarization optics the unitary operators describebirefringent systems (media, particularly devices: thephase-shifters, retarders of various kinds – linear,circular, and elliptical). Referring to Eq. (26), for sucha system d is the phase shift introduced by the systembetween its eigenstates, a0 – an isotropic phase shift andn – the Poincare unit vector of the system, correspond-ing to its major eigenstate (the advanced eigenstate) [3].

3.1.2. General Hermitian operators: orthogonal dichroic

systems and partial polarizers

If N ¼ H is a Hermitian operator:

H ¼ Hy, (28)

with Eq. (16) one obtains:

eia0 ja0js0 þ eiam n � r ¼ e�ia0 ja0js0 þ e�iamn � r, (29)

wherefrom:

a0 ¼ 0 modulo p; a ¼ 0 modulo p, (30)

so that the general Pauli algebraic form of a Hermitianoperator is:

H ¼ ja0js0 � m n � r, (31)

i.e. – a well-known fact – all the Stokes coefficients of aHermitian operator are real.

This operator may be put in an exponential formclosely analogous to that of the unitary operator,

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Eq. (26). That form was obtained straightforwardlybecause the unitary operators are automatically unim-odular; Eq. (20) is, in fact, a condition of unimodularity.This observation suggests the way we can adopt here.

The general Hermitian operator (Eq. (31)), maybe written as the product of the square root of itsdeterminant with the corresponding unimodular Her-mitian operator. Hence we can reduce the problem offinding the Pauli exponential expression of a generalHermitian operator to that of a Hermitian unimodularoperator. The only difference between them consist in ascalar factor.

For a unimodular Hermitian operator, with Eq. (31),we get:

DetH ¼ ja0j2 � m2 ¼ 1, (32)

equation which can be fulfilled by putting:

ja0j ¼ coshZ2; m ¼ sinh

Z2. (33)

Coming back to Eq. (31) with Eq. (33), the Pauliexpansion of a unimodular Hermitian operator may beput in the form:

H ¼ s0 coshZ2� n � r sinh

Z2¼ e�ðZ=2Þn�r. (34)

The unimodular Hermitian operators are largely used –under the name of squeeze or boost operators – in thegroup-theoretical approach to the problems of two-statesystems (including polarization), especially in the quasir-elativistic approach to these problems [9–11,47–51]. Theypertain to the group SL(2, C), which is locally isomorphicto the O(3, 1) Lorentz group.

The corresponding expansion of a general Hermitianoperator may be written in a form similar to Eq. (26) as

H ¼ er s0 coshZ2� n � r sinh

Z2

� �¼ ere�ðZ=2Þn�r . (35)

The Pauli axis of a Hermitian operator is a realvector, it is situated in the R3 subspace of the C3.Obviously, it reduces to the Poincare axis of theoperator, a notion which is well-known in the particularcase of Hermitian operators constituted by the projec-tors (corresponding to ideal polarizers in polarizationoptics).

It is gratifying to note an interesting complementaritybetween the Hermitian and unitary operators: theirPauli axes are situated in the complementary subspaces,R3 and I3 of the complex space C3.

In polarization optics Hermitian operators corre-spond to the orthogonal dichroic systems [10] (media,particularly devices: diattenuators, diamplifiers orsqueeze dichroic devices – amplifiers on one channel,attenuator on the other [52]). Referring to Eq. (35), forsuch a device/medium, n is the Poincare axis of its majoreigenstate (the eigenstate of the higher transmittance)and Z and r are some coefficients of anisotropic and

isotropic transmittance respectively. More precisely:

er ¼ eðZ1þZ2Þ=2, (36)

and

eZ ¼ eZ1�Z2 , (37)

are the isotropic and the relative amplitude transmit-tances, respectively, of the dichroic system, with eZ1 andeZ2 its principal (eigen-)transmittances. The coefficientsZ1 and Z2 may be, each of them, positive as well asnegative. For fixing the ideas, in the case of diattenua-tors both are negative and consequently the overalltransmittance is subunitary.

Unlike the unitary operators which can represent onlysystems, the Hermitian operators can represent alsostates (in polarization optics – SOPs). We shall discussthis aspect in Section 4.1.

3.1.3. Projectors: orthogonal ideal polarizers

An important special Hermitian operator, corre-sponding in polarization optics to the ideal polarizers,is the orthogonal projector. It is a singular operator.

For establishing its Pauli algebraic expansion wecan start with the general form of a Hermitian operator,Eq. (31).

The idempotency, characteristic for an orthogonalprojector

P2¼ P, (38)

implies with Eq. (31):

ðja0j2 þ m2Þs0 � 2ja0jmn � r ¼ ja0js0 � mn � r, (39)

wherefrom:

ja0j ¼12; m ¼ 1

2, (40)

where we have taken only the positive solution for m,corresponding to its signification of modulus of r.Further again with Eq. (31):

P ¼ 12ðs0 þ n � rÞ . (41)

Well-known a projector corresponds in polarizationoptics to an ideal polarizer [3,5]. Eq. (41) describesoperatorially the ideal polarizers of Poincare axis n.

3.1.4. Nonnormal operators: nonorthogonal composite

polarization devices

Generally a polarization device or a polarizationarrangement is composed by a series of elementary(‘‘canonical’’) devices: homogeneous polarizers, retar-ders, rotators [1–6].

The 2� 2 operators of these canonical devices arenormal operators: their eigenvectors are orthogonal.Therefore and in this sense, the canonical devices may becalled also orthogonal devices [27,45]. The canonicalretarders are represented by unitary operators (26).The canonical polarizers are represented by Hermitian

ARTICLE IN PRESST. Tudor / Optik 121 (2010) 1226–12351232

operators (35). In the representations (26) and (35) ofthe unitary and Hermitian operators, both the unitvectors n and m are real. They are the Poincare axes ofthe two operators. Both the operators (26) and (35) areevidently normal operators.

A series of orthogonal devices (each characterized bya normal operator) gives rise generally to a nonortho-gonal device (nonorthogonal eigenvectors, nonnormaloperator) [30,28]. The simplest example is the nonho-mogeneous circular polarizer formed by sandwichingtogether a linear polarizer and a quarter-wave plate.

We have established the general form of a nonnormal2� 2 operator in Section 2.2. Here we want to put thisoperator in an exponentiated form similar to Eqs. (26)and (35). A way to reach this goal is by using the polardecomposition of the operator [7].

Any operator (normal or nonnormal) may beexpressed by the polar decomposition as the productof a Hermitian positive semi-definite operator and aunitary operator. Particularly by means of various pairsof operators (26), (35) we can build up the whole groupGL(2, C):

A ¼ HmðZÞUnðdÞ ¼ erþia0eðZ=2Þm�re�iðd=2Þn�r . (42)

This is the most general Pauli algebraic exponentiatedform of a (generally nonnormal) 2� 2 operator.

Referring to our physical problem, Eq. (42) meansthat any composed polarization device can be reducedto (can be conceived as) an orthogonal succession of apartial polarizer and a retarder (both generally elliptic).In this way its synthetic birefringent and dichroicproperties can be separated [30]. Subtle discussionsconcerning the physical aspects of this separability aregiven in [6,53–56]. In connection with these aspects Ihave used above the term ‘‘synthetic’’.

Let us now expand the general Pauli expression (42)of a GL(2, C) operator, by means of Eqs. (26) and (35):

A ¼ erþia0 s0 coshZ2þm � r sinh

Z2

� �s0 cos

d2� in � r sin

d2

� �.

¼ erþia0 s0 coshZ2cos

d2þ m sinh

Z2cos

d2� in cosh

Z2sin

d2

� �� r

�

�im � ns0 sinhZ2sin

d2þ ðm� nÞ � r sinh

Z2sin

d2

�

¼ erþia0 coshZ2cos

d2� in �m sinh

Z2sin

d2

� �s0

�

þ m sinhZ2cos

d2� in cosh

Z2sin

d2þ ðm� nÞ sinh

Z2sin

d2

� �� r

.

(43)

This is a general Pauli algebraic linear expansion of aGL(2, C) operator.

Herefrom we can obtain another Pauli algebraiccondition of normality of A.

Till now we have used the left polar decomposition,Eq. (42). A right decomposition may be equally used.In general, these two decompositions are different (the

pairs of unit vectors m, n, differ in the two polardecompositions) and the two relevant factors in Eq. (42)do not commute.

One of the properties of the normal operators isthat their right and left polar decompositions coincide.In other words, the polar decomposition of a normaloperator is unique and invertible (e.g. [7]).

Working out similarly to (43) the right polardecomposition corresponding to (42), putting the pre-vious condition of normality and avoiding the trivialcases Z ¼ 0, d ¼ 0, one gets that the operator is normalif and only if:

m� n ¼ 03m ¼ ln3m � n, (44)

(with l a generally complex factor of modulus 1.) This isanother Pauli algebraic necessary and sufficient condi-tion of the normality of a 2� 2 operator: the Poincareaxes of the ‘‘operatorial modulus’’ and of the ‘‘phasefactor’’ of the operator A given by Eq. (42) are parallel(coincide).

The same conclusion was recently obtained by aquaternionic reasoning [11].

Referring to the problem of polarization devices, thePoincare axes of synthetic dichroism and of syntheticbirefringence for an orthogonal polarization devicecoincide (Eq. (44)) whereas for a nonorthogonal devicethey are different (Eq. (42)).

4. State operators

4.1. Projectors: polarization density of pure states

(completely polarized light)

As we have seen in Section 3.1, the general Paulialgebraic form of a projector is (41). As polarizationsystem operator, a projector describes an orthogonalideal polarizer. On the other hand, such an expressioncan describe also a state, namely a pure state, that inpolarization optics means totally polarized light [36,57].It is easy to realize this fact taking into considerationthat such a state can be obtained by means of an idealpolarizer (41) of Poincare axis n.

In the following we shall label the (polarizationdensity) operators of the SOPs by J, as a reminiscenceof the usual notation in optics for the coherence matrix(which is in fact a matrix representation of thepolarization density operator). Thus

J ¼ 12ðs0 þ s � rÞ , (45)

is the Pauli vectorial algebraic expansion of the densityoperator of a pure SOP of Poincare unit vector (axis) s.

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4.2. General Hermitian operators: polarization

density of mixed states (partially polarized light)

In what concerns the mixed states (partially polarizedlight), we can build up the general Pauli algebraic formof their density operators starting from Eq. (45) andby using the ‘‘polarization – form dichotomy’’ [1]:any partially polarized SOP may be represented as anincoherent superposition of two orthogonal totallypolarized SOPs. Incoherent superposition means addi-tivity at the level of intensities, more general at the levelof density operators (coherency matrices in the matrixrepresentation of the operators). Hence for a mixed stateof polarization we may write:

J ¼ I1J1 þI2J2 ¼ I112ðs0 þ s � rÞ þI2

12ðs0 � s � rÞ,

(46)

where I1 and I2 are the intensities of the twocomponents and we have taken into consideration thatorthogonal states have antipodal Poincare axes. We getfurther

J ¼1

2½ðI1 þI2Þs0 þ ðI1 �I2Þs � r�

¼1

2ðI1 þI2Þ s0 þ

I1 �I2

I1 þI2s � r

� �¼

1

2Iðs0 þ ps � rÞ.

(47)

Here I1 þI2 ¼ I is the total intensity of the light and

p ¼I1 �I2

I1 þI2(48)

is (by definition in the frame of the ‘‘polarization – formdichotomy’’ [1]) the degree of polarization of the mixedSOP. Hence the vectorial Pauli algebraic expansion ofthe polarization density operator of a partially polarizedSOP with intensity I, degree of polarization p andPoincare axis s is

J ¼ 12Iðs0 þ ps � rÞ . (49)

l 9If the state is normalized to unit intensity (49)becomes:

J ¼ 12ðs0 þ ps � rÞ. (50)

We have to remark, that in characterizing a mixedstate (partially polarized light), the polarization degree p

and the Poincare axis of the state s appear alwaystogether, in the product ps, which is the Poincarevector of the state. While the top of the Poincare axis sof the state lies on the Poincare sphere (of radius 1) S2

1,

the top of its Poincare vector ps lies in the Poincare ballS31, on the sphere of radius p,S2

p.

5. Conclusions

We have given above a vectorial pure operatorial(matrix-free) description of the operators correspondingto all the linear polarization systems (devices/media) andto all the polarization states of light. These operatorsbeing defined on a bidimensional unitary space over thefield of complex numbers, their specific and adequatealgebra is the Pauli algebra. We have established herethe vectorial Pauli algebraic expansions for the wholehierarchy of 2� 2 linear operators.

It is worth emphasizing that although the subjectwhich we refer to is a very specific one – the interactionof polarized light with the polarization (‘‘non-image-forming’’) systems – the field of physical applications ofthese results is much larger: it covers all the ‘‘two-state’’(i.e. ‘‘two-level’’, ‘‘two-beams’’, ‘‘four-pole’’) systems.They may be applied as well in two-beam interferome-try, in multilayer optics, ray optics, analysis of states oflight with orbital angular momentum, squeezed statesof light, and, of course, spin 1/2 and two-level atomproblems. Therefore I have used firstly a versatilealgebraic terminology and only secondly I have appliedit to our specific problem. This is reflected in thesubtitles and structures of Sections 3.1 and 4.1.

A central main result of this paper is that any 2� 2linear operator is characterized by a generally complexunit vector, which we have called its Pauli axis. Inpolarization optics we are familiarized with the Poincareaxis of a SOP (whose density operator is Hermitian) andwith the Poincare axis of a retarder (whose deviceoperator is unitary). These operators are normal and, byconsequence, their Pauli axes are reducible to real unitvectors, so that they may reach a R3 Poincarerepresentation. As we have seen the Pauli axes of thenonnormal operators (e.g. corresponding to variouscomposite polarization devices) cannot be reduced toreal R3 vectors, they are irreducible complex C3 vectors.The Pauli axis of a nonnormal operator may beconceived, if we want, as a generalization of thePoincare axes of the Hermitian and unitary operatorsencountered in polarization optics.

In this paper we have established some bridgesbetween, on the one hand, the Hilbert space of theoperators of the states or of the systems and, on theother hand, the C3 space of their Pauli axes.

Further, if we will act with a system operator(e.g. polarization device operator) on a state operator(e.g. polarization density of a SOP), this action shallbe mapped onto a displacement of the Pauli axis of thestate operator in the C3 space of the Pauli axes of thestates. This will be the subject of the forthcoming paper.

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