JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Pauli-Algebraic Operators in Polarization Optics*CYNTHIA WHITNEY

Charles Stark Draper Laboratory, a Division of The Massachusetts Institute of Technology,Cambridge, Massachusetts 02139

(Received 2 November 1970)

Techniques for analysis and synthesis of linear optical systems involving polarization effects are relatedand extended through fuller exploitation of the algebra of complex 2X2 matrices.INDEX HEADING: Polarization.

The purpose of this paper is to provide a frameworkwithin which to relate the formalisms for polarizationoptics previously developed by many other authors, andto extend the scope of results obtainable. The purposeis served through a full and systematic exploitation ofcomplex 2 X 2 matrices.

The topics to which some contribution is here addedinclude the concatenation of optical operators, the polardecomposition of 2X2 matrices, the properties of non-perfect polarizers, the measurability of polarizationstates and optical operators, the decomposability ofarbitrary optical systems into sequences of standardones, and parallel combinations of operators.

The framework provided conveniently relates thewell-known Jones and Mueller formalisms and thespherical-trigonometric approach to polarization opticssuggested by the well-known Poincar6-sphere repre-sentation of polarized light. These important techniqueshave virtually supplanted vector-field algebraicmethods.1-lO

THE PAULI ALGEBRA

The set of all complex 2X2 matrices together withthe operation of matrix multiplication constitutes whatmathematicians call an algebra. Any member of thealgebra can be expanded in terms of four basis matrices,and the particular choice of basis used here comprisesthe identity matrix

and the vector matrix,

t = (cT1,a2,cr},with

ro 11 0 -in I 01aI=l aJ2= o0ff=L 1

The basis matrices a are the well-known Pauli spinmatrices. For this reason, the set of complex 2 X2matrices is here called the Pauli algebra. The vector-likesymbol a invites scalar-like dot products with conven-tional three vectors (scalar entries), exponentiation ofsuch dot products, etc. Such techniques are here calledPauli algebraic.

DESCRIPTION OF POLARIZATION STATES

Typically, the full description of radiation propa-gating in a given direction at a given location requires afrequency-dependent four-parameter construct. Twosuch constructs are commonly used. One is a 4X 1mtarix S of the real parameters introduced by Stokes"and the other is the hermitian 2 X2 coherency matrix Jintroduced by Parrent and Roman.'2 For the purposesof this paper, neither construct is completely satis-factory. The 8 is, of course, not a 2 X)2 matrix. AlthoughJ satisfies that criterion, it is related to the Stokesparameters' somewhat inconveniently, and is con-structed from x and y linear polarizations that are notthe best possible choice if we wish to use basis statesthat admit some special physical interpretation. Itseems convenient to introduce a third construct similarto J, but based on polarization states with a specialdynamic significance; namely, the definite, positive andnegative spin density demonstrated by Rossi"4 for statesof positive and negative helicity, or left and rightcircular polarization. From them, a 2X2 hermitianmatrix S is formed in the same manner as J is formed.The procedure involves correlation of entries in a 2XIconstruct. Most generally, the correlation can be per-formed with arbitrary time delay and followed byFourier decomposition to produce the hermitian S.

Let S be expressed in the Pauli-algebraic form'

S=s0ro+s -Fa.

Let the four numbers (sos} be called the matrix com-ponents of S. Similar notation, with matrices denotedby capital letters and components by the correspondinglower-case letters, is used throughout the paper.Further, let

s=ss,

where s is a scalar and § is a unit vector. The vector § iscalled the axis of the matrix S. Similar notation definingthree-vector axes for other matrices is used in theremainder of the paper. Hermiticity of S allows so, s,and § all to be real. The so represents total flux of signal.For polarized light, s=so. For this reason, s and se-sare generally interpreted as the fluxes of polarized, andunpolarized, natural constituents of the signal, respec-tively."l,'5-1 Thus only S matrices with so5 s occur.

1207

VOLUME 61, NUMBER 9 SEPTEMBER 1971

CYNTHIA WHITNEY

The work of Perrin"8 suggests that s be used to deter-mine a point on a unit sphere. Apart from reversal ofthe vertical poles, the sphere is the same as that usedby Poincary6.& In particular, the states of + and -

helicity and x and y linear polarization for propagationalong 2 appear as shown in Fig. 1. These special casesare summarized in the general relation between thespherical coordinates a and f of § and a classicalpolarization ellipse,

/2=tilt of ellipse from x axis

I tan[2(0j-)]J =minor axis/major axis.

The reverse problem of obtaining 8 directly from thepolarization ellipse has been taken up by Collett."0 Fromthe above remarks concerning so, s, and 5, it follows thatapart from a minus sign on S3, the matrix components{so,s) of S are equivalent to the well-known Stokesparameters. The minus sign arises because the S isbased on +, - spin density instead of right-, left-circular polarization.

Introducing the determinant ISI helps to clarifyanalogies between the state S in polarization optics andconstructs that occur* in relativity theory. Thedeterminant

lSI =s 02 -S 2

is the analog of the squared length of a Lorentz lineelement."'" 2 One of several ways to evaluate IS I is from

IcIo=SS =S-S,where the - operation is defined by

S =SOTO-SIU.

Thus S and S are analogs of contra- and covariant

FIG. 1. States of + and - helicity, x and y linear polarization,and arbitrary polarization on a modifiedjPoincare sphere.

four vectors. For polarized light, l S l is zero, so S and S8are singular and separate into direct products of 2>X 1constructs. These four 2 X 1 constructs are analogs of thefour kinds of spinors in relativity theory. 22

OPERATIONS ON POLARIZATION STATES

Let us consider the transformation of a polarizationstate induced by an optical system, with radiant energypropagating in given directions at given points beingthe system input and output. Since the input and outputare functions of time only, the system can be character-ized by a unit-impulse-response function that alsodepends on time only. Consideration is limited tooptical systems that can be treated as linear, thus ex-cluding those numerous optical phenomena that changefrequencies or depend on irradiances. The principles ofa linear theory have been discussed most extensively byParke," who shows that the relation between output,input, and response can be expressed in the time (0)domain as convolution, or in the frequency (X) domainas multiplication. The linear transformation appropriateto the co domain was stated by Soleillet24 and expressedas a 4X4 matrix operation on the 4X1 S by Perrin."The linear transformations have been studied exten-sively by Hans Mueller (unpublished) and are oftencalled Mueller transformations. Attention is herelimited to a subclass of the Mueller transformations,namely, those that can be expressed in terms of the2X2 hermitian matrix S with pre- and postmatrixmultiplication by elements of the Pauli algebra,

'= VS Vt.

If I v l Fy0, the above transformation is proportionalto a Lorentz transformation" from the restrictedLorentz group. The conditions so- 0 and so> s areautomatically preserved. If I V 0, then I S' j 0 andso's'Ž 0 for all S. Thus, in either case, the transforma-tion preserves physical relations between so and s.

COMPILING MULTIPLE OPERATIONS

The general rule for compiling operations on S is thesame as that discussed by Parrent and Roman' 2 foroperations on J, or that occurring in the Jones for-malism> 3'2 for operations on 2X I representations ofpolarized states. In these matrix formalisms, operationsare concatenated by matrix multiplication. Similarly,here, a sequence of N systems represented individuallyby V1, V2, . . . VNr is represented by the reverse-ordermatrix product.

V = VN VN-1.** V1-

As a special case, suppose V1 =V 2 =... JN, ThenV = VYo. Jones"7 shows a standard procedure for raisingVI to power N applicable if 7, is diagonable; that is,if there exists a matrix A such that A VA-' is diagonal.The Pauli-algebraic formalism suggests other proce-

1 208 Vol. 61

PAULI OPERATORS IN POLARIZATION OPTICS

dures that don't require that V be diagonable. If VI isnonsingular, it can be written

Vi=k exp(v-a),

where k is a suitable scalar and v is a complex threevector. Then

V =4N exp(Nv-U).

On the other hand, if VI is singular,

V1 = vo~ro+ v~ *(7)and

V= (2v-)N V,.

PROPERTIES SPECIFIED WITHPOLAR DECOMPOSITION

An optical operator is passive if it neither increasestotal flux nor creates phase correlations. Not every Vmatrix is passive, and the problem of characterizingthose that are has received previous attention.21 ' 33 Astandard eigenvalue-eigenvector approach33 is appli-cable to V matrices that are diagonable, particularlythose diagonable by transformations that are unitaryand therefore, as shown below, passive themselves. Butnot all V matrices fall into that class. A partial applica-tion of Pauli-algebraic techniques2 ' treats all non-singular V matrices, but again, some V matrices don'tfall into that class. Only a restriction on V much weakerthan those indicated above is justified: Since V and Vtboth occur as factors in the transformation of S, V canbe chosen with I VI real and nonnegative. The Pauli-algebraic techniques will here be used to determine theconditions that make any such V passive.

The fundamental step in analysis of V matrices ispolar decomposition of V into the form

V= UH,

where H is hermitian and U is unitary. The H and U arecalled polar factors of V because they are roughlymatrix analogs of the r and ei6 polar factors of a complexnumber. The main difference between matrix andscalar factors is that matrix factors do not generallycommute. When a matrix decomposition of the formV =H'U' is required, the factors have to be U' = U andH' = UHU-1, and generally H' differs from H.

The notion of polar decomposition is well known inlinear algebra, but does not seem to have been fullyapplied before in polarization optics. This may be due inpart to the fact that the mathematical description of polardecomposition is hedged with cautions about uniquenessin the case of noninvertible transformations. 3 4 '3 5 Theproblem is that generally only the positive part of anoninvertible transformation can be uniquely definedfor all inputs on which the transformation operates.However, the problem is not insurmountable, becauseit relates only to abstract transformations, and not toconcrete matrix representations. Uniqueness problems

are avoided here by (1) decomposing a matrix repre-sentation V and not an abstract transformation, and(2) requiring that the same H and U be regarded aspolar factors regardless of what the state input to V is.

The following well-known matrix relations are usedto secure the decomposition

(AB)t =BtAt

Ut=U-1Ht =H

(a-*r)(be r) =a*bao+i(aXb) -or.

(1)

(2)

(3)

(4)

First obtained is the hermitian factor H=hoaoo+h-cr.We form the hermitian matrix

W=VtV=woo0o+w C,

and substitute V= UH, obtaining

W H 2 = (hz2 +oz2)cro+21oh oa.

Comparison of the two expressions yields h= and(hothk) 2=woizw, from which

ho = 2t(Wo+W)1+ (Wo-W)I]

andhI=ff(wa+w)- (wo-w)fl.

From V and H, the unitary factor U=zeaao+u u canbe found. Because U is unitary, Uo is real and u isimaginary. Comparison of the expressions V= vouo+ va

and

UH = (uoho+u *h)ro+ (oh+hou+iuXh) or

indicates that

st=Re(v 0)/ho, u=Im(v)/Iho,

where Re and Im denote real and imaginary parts,respectively.

It is readily verifiable that a unitary matrix U doesnot affect the zero component of any state on which itoperates. Thus, the requirement that total flux neverbe increased by V implies only a restriction on itshermitian polar factor. Consider, therefore, S'=zHSH.The output flux is

so'= (ho12+ 2)so+2/oh s,

which for s = soh takes its maximum valueso' = (ho+h) 2sa.Clearly, a passive H matrix has ko+hz• 1. This conditionis equivalent to that obtained by Barakat. 2 ' A morecomplicated but equivalent criterion was obtained byJones28 with Hurwitz.26

It is also easily verifiable that a unitary matrix Udoes not alter the degree of polarization P=s/so of anystate on which it operates. The requirement that phasecorrelations not be created by V also restricts only thehermitian polar factor. With S'=HSH, the outputpolarized flux produced by unpolarized input flux

1209September 1971

CYNTHIA WHITNEY

S=soo- is s'=2ktolhso. Noncreation of phase correlationsrequires that no more than half the unpolarized inputbe converted to polarized flux. Thus a passive H matrixmust have 2 ho/-<1. Since the geometric mean of /toand hi is less than their arithmetic mean, the conditionalready obtained fulfills this requirement.

POLAR FACTORS AS OPTICAL INSTRUMENTS

The transformations induced by U and H matricesthat are passive can readily be associated with variousoptical instruments. Some problems of physical inter-pretation in the standard eigenvalue-eigenvectoralgebraic formalism have been treated by Marathay.36

The Pauli-algebraic parameters of the matrices areinterpreted in the present paper. We shall find it con-venient to consider some special cases of S that arepolarized, have unit total flux, and have axes alignedwith the axis fi of U or the axis fi of H; that is, statessuch as

S=U=aoo=fiur, or H±+=co-h-fa.

First consider a unitary matrix operator expressed as

U=exp[(-it/2)fi .&].

It induces rotation of § by a positive angle q about fi onthe Poincar6 sphere. Thus the transformation introducesa phase advance of angle 0 on U+. The correspondingoptical instrument is commonly called a rotor or phaseshifter. If ft coincides with axis 3, the instrument iscommonly called a rotator, because it rotates the planeof linear polarization by a positive angle q6. If ft lies inthe 1,2 plane, the instrument is commonly called a waveplate, or retarder, because it delays one state of linearpolarization with respect to the state opposite on thePoincar6 sphere.

Next, consider a hermitian matrix operator

H = zooo+h*.

The H+ and H- are simply scaled by factors (ho0+Iz)2

and (ho-h)2 , respectively. Since it favors state H±over H-, the instrument represented by H is here calleda general polarizer. If fi lies in the 1,2 plane, the instru-ment is a linear polarizer. If H is singular, it is a perfectpolarizer1 2 that annihilates H. Otherwise the instru-ment is a partial polarizer.

The term partial polarizer may be somewhat mis-leading, since the instrument can actually decrease thepolarization P=s/so of some inputs. Depolarizationcaused by selective absorption has been noted byMarathay3 6 for linear states and linear polarizers. Hegives a formula for P' as a function of input state andparameters of the polarizer with graphs for some specialcases. By use of the Pauli algebraic parameters, hisresults can be generalized to simple mathematicalstatements valid for arbitrary states and arbitrarypolarizers. It is convenient to work in terms of the

normalized determinant

D=4 1S! /[Tr(S)]2= 1 -p2

where Tr indicates matrix trace. The polarization P isdecreased by a polarizer if D is increased. A generalpolarizer scales any nonzero D by the factor

D'/D= { H I 2[Tr(S)]2 }/{ [Tr(HtS)9I ).

Note that the factor is independent of any componentsi. of s perpendicular to h. Furthermore, sincelHlo|,=HH-=HVI, it follows that an input state

proportional to H- renders the ratio equal to unity.Generally, nonsingular states with the si componentof s parallel to h satisfying

srl/so< -h/k,

produce scale factors greater than unity. These statesare depolarized by the instrument. In particular, con-sider an input state S given by (H-)2 . The output

S' =H(Hf)2H

is completely depolarized, and has flux l H 12.Depolarization by. selective absorption is to be

distinguished from depolarization by decorrelation,which occurs when the input beam is separated intoseveral beams that are differently operated upon andthen recombined incoherently. Such combinations areconventionally handled in the Mueller formalism.23

MEASURABILITY

At this point in the formal development, 2X2 matrixrepresentations of some critically important physicalconcepts have been introduced. We have the polariza-tion state, represented by the hermitian matrix S, theoptical-instrument operator represented by an arbitrarymatrix V, and, as special cases, the general polarizerrepresented by a hermitian matrix H, and the generalrotor represented by a unitary matrix U. It is importantto confirm the operational nature of these representa-tions by demonstrating the measurability of each interms of others, presumed given.

Consider, first, the polarization state S. Measurementschemes described in the literature sometimes involvesix measurements using perfect polarizers.17 But onlyfour measurements are really needed.'2 Furthermore,total flux so can be measured directly, and only theremaining three measurements require polarizers. Pauli-algebraic techniques permit us to see that these polar-izers need not be perfect. The output of a flux of ageneral polarizer

So' = (ho2 +h2 )so+2hoh s

uniquely fixes the component of s parallel to h as

mI = [so' - (ho2+h2)so]/2koh.

Three such measurements can determine s.

1210 Vol. 61

PAULI OPERATORS IN POLARIZATION OPTICS

The measurability of S matrices ensures that if a2X2 matrix representation for a system exists, it can beinferred from physical measurements. For a polarizer,the determination is trivial. Let unpolarized normalizedlight S=uo- be incident on the system; the output isS' =H 2, giving H=(S')1. For a general system, (S')provides only the hermitian polar factor H'= UHU-'.The unitary factor can be determined up to a finalrotation about §'=fi' from two inputs polarized alongaxes that are perpendicular in 1,2,3 space. We measurethe output fluxes and infer the two values of sit in theintermediate states USU-' that are inputs to H'. Thefinal rotation about fi' can easily be detected with onemeasurement of phase between fractions of the outputthat pass polarizers represented by the matrices H'+and H'-. However, if H' is singular, none of the outputpasses H'-, and it follows that the matrix representationU is not determined uniquely by the opticaltransformation.

THEOREMS

Hurwitz and Jones2 6 proved a set of theorems aboutthe decomposability of optical systems into sequencesof linear polarizers, rotators, and retarders. Onealternate proof of a theorem was given by Richartzand Hsii3 and some special decompositions werediscussed by Marathay.3 6 Use of Pauli-algebraictechniques permits the body of theorems to be extendedin scope.

(1) Any system represented by a matrix V is equiva-lent to one phase shifter represented by matrix Ufollowing a polarizer represented by matrix H. Thetheorem is equivalent to the polar decompositiontheorem V = UH.

r2

FIG. 3. Alternate synthesis of phase shifter.

(2) Any phase shifter U can be made from a rotatorfollowed by a retarder. This statement is equivalent totheorem I proved by Hurwitz and Jones,2 6 and byPoincar6.'9 The theorem follows from the Rodrigues-Hamilton theorem. 7 The axes fi, 3, and u,. of the shifter,rotator, and retarder are shown in Fig. 2 along with theassociated half-angles y =/2, a = rot/2, 3 =kret/2.

(3) Any phase shifter U can be made (nonuniquely)from two retarders. This theorem also follows from theRodrigues-Hamilton theorem. In Fig. 3, one possiblefirl and the corresponding uf,2, a and f3 are shown.

(4) Any polarizer H can be realized (nonuniquely)from a linear polarizer and two retarders. Let theretarders be inverse to one another, and let one precedeand the other follow the linear polarizer. Thus matricesU, and H, are sought such that i, and hi lie in the 1-2plane, and

H= UrHIU7-1 .

Clearly, hio=ho and hl=h are required. In Fig. 4, onepossible choice of fil and the associated fi, and q5 areshown.

(5) A system represented by any matrix V can besynthesized from one linear polarizer, two retarders,and a rotator. (This is a slightly generalized statementof theorem III proved by Hurwitz and Jones.2 6 ) Fromtheorem 1, V = UH. From theorem 4,

H = UrctiHiUretfl1 ,

and from theorem 2

UUretl = Uret2Urot-

Thus

FIG. 2. Synthesis of phase shifter.

September 1971 1211

V= Uret2U�-.tHiU,,.tl-'.

1 CYNTHIA WH1I.TNEY V 6

FIG. 4. Synthesis of general polarizer.

(6) A system represented by any matrix V can besynthesized from one linear polarizer and three re-tarders. The proof follows that of theorem 5, with thesubstitution

UUrcti` Uret2Urcta,

made from theorem 3.(7) A system V composed only of linear polarizers

and rotators can be replaced by a system containingone linear polarizer and one rotator. (This is a slightlygeneralized statement of theorem II proved by Hurwitzand Jones.26) Each linear polarizer or rotator is repre-sented by a matrix of the form

Vn7nov+oo Vn * a,

where Im(vo)=O and Im(vn)=(v,)3. The product oftwo such matrices is another such matrix. Thus,ultimately V is of the same form. Expressed in termsof its polar factors,

V= UNH= (oho+u h)ao+ (uoh+hou+iuXh) a.

Since u is imaginary, the polar factors must be suchthat u-h=0 and fl=3.

Other theorems that delimit the variety of passiveoptical systems that can be built from given fixedoptical elements can readily be proved. For instance

(8) Two retarders of fixed retardations 4A 40 cannotsynthesize a phase shifter represented by matrix Uwith ai lying outside the equatorial band

(ir-4u)/2< 0< (r+0j)/2.

(9) Two retarders of fixed retardations if and alinear polarizer cannot synthesize a polarizer repre-sented by matrix H with fi lying outside the equatorial

band

ir/2-4< 0<ir/ 2 +±.

Finally, theorems can be constructed concerningparallel as well as series combinations of instruments.For instance

(10) The parallel combination of any number of Vmatrices satisfying 1i(VtV)i~=ao preserves total fluxand induces depolarization by decorrelation. If, inparticular, the hermitian polar factors of all of thematrices have collinear axes, then basis inputs withaxes collinear to those axes are not themselves de-polarized; only coherence between them is reduced byreduction of the component of § perpendicular to theaxes, Si. The output flux is

-Tr(2%ViSV.t) = 'Tr[2i(VtV)iS]= 2 Tr(S),

equal to the input flux. Generally, let the product(VtV)i be recognized as H1 2, where Hi is the hermitianpolar factor of Vi. When there are only two V's-in theparallel combination, the H matrices have collinearaxes. The factor by which the parallel combination ofHl and H2 scales any s. is the sum of determinants,I HI +IH2 Ž. Let the matrix HI scale the basis inputs

HA by (h10 4) 2, respectively. Then H2 scales the sameinputs by 1-_(/ztih_)2, respectively. The determinantof H 2 is the geometric mean of these two scale factors,which is always less than their arithmetic mean,1-ho 2-h 2 , which in turn is less than 1-/i 0

2+h2=1 - I HI 1. Thus the scale factor on s± is less than unity.Thus the parallel combination reduces any coherencebetween the H± basis states, ensuring depolarizationby decorrelation in the combination of V's. When thereare more than two V matrices, generally let one Hmatrix play the role of H1 above, and let the sum ofsquares of all the others play the role of (H2)2. Again,depolarization is produced. If all of the fi axes arecollinear, then basis states with axes collinear to the iaxes are not depolarized.

DISCUSSION

This paper attempts to contribute to a unified theoryof 2X2 matrix operators in polarization optics by ex-tending the scope of previous 2X2 formulations. Wefind that matrices can be raised to powers convenientlyeven if they are not diagonable, and that polar factorscan be obtained even for matrices that are singular.Next, we find that depolarization by selective absorp-tion can be discussed conveniently even for polarizersthat select states that are not linear and that measure-ments can be performed accurately even with non-perfect polarizers. Finally, we find that some well-knowntheorems concerning the decomposition of arbitraryoptical systems into sequences of standard ones can begeneralized in various ways, and that a parallel com-bination, typical of the Mueller formalism, can beconsidered as well.

1212 Vol. 61

PAULI OPERATORS IN POLARIZATION OPTICS

The mathematical tool used in this paper is the Paulialgebra of complex 2X2 matrices, the main usefulfeatures of which include use of a generally complexscalar and three-vector set of matrix components,incorporation of the well-known formalism of vectorcalculus, use of polar decomposition, and incorporationof the spherical-trigonometric Rodrigues-Hamiltontheorem.

ACKNOWLEDGMENTS

The author is grateful to Prof. L. Tisza, of the MITPhysics Department, who gave her the opportunity toparticipate in the development of the Pauli algebra,and who suggested that it would complement othertechniques for the formal description of polarizationoptics.

Discussions with R. Var, J. D. Coccoli, H. L.Malchow, and J. Adlerstein of MIT are also gratefullyacknowledged.

The paper is dedicated to the memory of HansMueller of the MIT Physics Department.

REFERENCES

* This work was sponsored by the National Aeronautics andSpace Administration, in part through Contract NAS 9-4065 fromthe Manned Spacecraft Center and in part through ContractNAS 1-9884 from the Langley Research Center.

1 F. E. Wright, J. Opt. Soc. Am. 20, 529 (1930).2 U. Fano, J. Opt. Soc. Am. 39, 859 (1949).'M. Richartz and Hsien-yu Hsi, J. Opt. Soc. Am. 39, 136

(1949).4 G. A. Deschamps, Proc. IRE 39, 540 (1951).

I G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am.42, 49 (1952).

6 H. G. Jerrard, J. Opt. Soc. Am. 44, 634 (1954).S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 247 (1956).

8 S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 398 (1956).H. C. Ko, Proc. IRE 50, 1950 (1962).

10 W. A. Shurcliff and S. S. Ballard, Polarized Light (VanNostrand, Princeton, N. J., 1964).

11G. G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852);reprinted in Mathematical and Physical Papers (Cambridge U. P.,New York, 1901), Vol. III, p. 233.

12 G. B. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).13 R. W. Schmieder, J. Opt. Soc. Am. 59, 297 (1969).14 B. Rossi, Optics (Addison-Wesley, Reading, Mass., 1957).15 A. Blanc-Lapierre and P. Dumontet, Rev. Opt. 34, 1 (1955).16 H. A. Tolhoek, Rev. Mod. Phys. 28, 277 (1956).17 E. Wolf, Nuovo Cimento 13, 1165 (1959).18 F. Perrin, J. Chem. Phys. 10, 415 (1942).19 H. Poincar6, Thgorie Mathematique de la Luiniere, Vol. 2

(Carr6, Paris, 1892).20 E. Collett, Am. J. Phys. 36, 713 (1968).21 R. Barakat, J. Opt. Soc. Am. 53, 317 (1963).22 P. Rastall, Rev. Mod. Phys. 36, 820 (1964).23 N. G. Parke, J. Math. Phys. 28, 131 (1949).24 P. Soleillet, Ann. Phys. (10), 12, 33 (1929).25 R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).26 H. Hurwitz, Jr. and R. C. Jones, J. Opt. Soc. Am. 31, 493

(1941).27 R. C. Jones, J. Opt. Soc. Am. 31, 500 (1941).28 R. C. Jones, J. Opt. Soc. Am. 32, 486 (1942).29 R. C. Jones, J. Opt. Soc. Am. 37, 107 (1947).30 R. C. Jones, J. Opt. Soc. Am. 37, 110 (1947).11 R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948).32 R. C. Jones, J. Opt. Soc. Am. 46, 126 (1956).33 R. Menasse, Ph.D. thesis, MIT, 1955.34 P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed.

(Van Nostrand, New York, 1958).35 I. M. Gelfand, Lectures on Linear Algebra (Wiley-Interscience,

New York, 1961)..26 A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).37 E. T. Whittaker, Analytic Dynamics (Cambridge U. P.,

New York, 1964).

September 1971 1213