Recovery of the matrix operators in the similarity and congruency transformations: applications in polarimetry

Vol. 10, No. 4/April 1993/J. Opt. Soc. Am. A 719

Recovery of the matrix operators in the similarity andcongruency transformations: applications in polarimetry

Laurence J. November

National Solar Observatory, Sacramento Peak, Sunspot, New Mexico 88349

Received February 11, 1992; revised manuscript received September 14, 1992; accepted October 23, 1992

Formulas are presented for the recovery of the matrix operators in arbitrary-order similarity and congru-ency transformations. Two independent input and output matrix pairs exactly determine the similarity-transformation matrix operator, while three independent Hermitian-matrix pairs are required for thecongruency-transformation operator. The congruency transformation is the natural form for the quantumobservables of a multiple-element wave function, e.g., for polarized-light transfer: the recovery of the Jonesmatrix for a nondepolarizing device is demonstrated, given any three linearly independent partially polarizedinput Stokes states. The recovery formula gives a good solution even with large added noise in the test ma-trices. Combined with numerical least-squares methods, the formula can give an optimized solution for mea-sures of observation error. A more general operator, which includes the effect of isotropic depolarization, isdefined, and its recovery is demonstrated also. The recovery formulas have a three-dimensional geometric in-terpretation in the second-order case, e.g., in the Poincar6 sphere. It is pointed out that the geometric propertyis a purely mathematical property of quantum observables that arises without referring to spatial characteris-tics for the underlying wave function.

1. INTRODUCTION

It is usual in polarimetry to characterize a polarizationtransformation by a linear operator, a Mueller matrix.Thus an incoming polarization state, which is parameter-ized by a four real-element Stokes vector s = (so, s1, s2, S3)T,

is modified by matrix multiplication by a Mueller real4 X 4 matrix M to give an output Stokes vector s' = Ms;sT denotes the vector transpose, so is the total intensity, siis the intensity difference between a linear polarizer at 00and one at 9O, S2 is the intensity difference between lin-ear polarizers at 450 and 135°, and S3 is the right circularlypolarized intensity minus the left.' The matrix M can bederived experimentally given m different Stokes vectors sjwith corresponding measured Stokes vectors sj' countedbyj = 1, m by a standard matrix method. 2 Simply, the msimultaneous vector equations can be written as one ma-trix equation containing matrices with the Stokes vectorsin their columns:

[Si' S2' ... Sm'] = M[s1 S2 ... Sm],

and then solved for M directly:

M = [Sl' S2 ' ... sm']IS1 S2 ... Sm]I* (1.1)

Four independent Stokes vectors sj are always required,and for m = 4, S' denotes the square matrix inverse of S;for a general number of measurements m, S' should betaken to be the pseudoinverse for the optimum solution.

Unfortunately the linear operator representation forlight-state transformations is an overspecification. Ob-servationally overspecification is inconvenient, necessitat-ing more independent input states than would be neededotherwise. An overspecification can be difficult to inter-pret in a physical parameterization. Also, it is possible todefine physically unrealizable Mueller matrices that donot preserve the light-state condition: so'2 2 sl'2 +

S2 + S3 2, for all input Stokes states S.3 '4 Light transfer

through materials must be described as a transformationon the two-element wave function for the light state, whichis written here as a complex vector, e(t), a function oftime t, which is temporally coherent in its two indepen-dent elements. is the Hilbert analytic signal of the realelectric-field temporal function e(t). In its low-amplitudelimit every transformation of the wave function is repre-sented by a linear transformation, and any linear trans-formation of the wave function, or Jones transformation, 5

can be realized physically.The Stokes parameters are the quantum observables de-

fined by the averages of all the possible product pairs be-tween the wave-function components, which define thecoherency matrix for the Stokes vector6 ":

J(s) = ((t) () 6(t)t)t = 1 8s 82- 3]2 S2 + S3 So -S1

(1.2)

where 0 is the vector outer product and ( denotes thetime average. Then the complex 2 x 2 Jones matrix N,which transforms the two-element complex electromag-netic wave function ((t) with the linear transformation'(t) = N(t), transforms the corresponding coherency

matrix, J(s) = (e(t) 0 6(t))t, with the congruencytransformation

J(s') = NJ(s)Nt, (1.3)

where J(s') = ('(t) 0 V'(t)%; Nt denotes the complex con-jugate of N. The congruency transformation is less gen-eral than a linear transformation and gives a limited set ofpossible output states s' for an input state s. It is known tobe a good representation of nonpolarizing optical systems.

Given general sample Stokes input states sj with mea-surements s/', the recovery of the Jones matrix N fromthe congruency transformation is a nonlinear problem.

0740-3232/93/040719-21$05.00 0 1993 Optical Society of America


720 J. Opt. Soc. Am. A/Vol. 10, No. 4/April 1993

Jones'2 and Azzam and Bashara"314 have shown that threeindependent purely polarized input Stokes states sj aresufficient to determine a Jones matrix N uniquely.Purely polarized solutions are available, because a purelypolarized Stokes state can be represented by a temporallyconstant two-element wave function. In this case theproblem reduces to finding N in the linear transformatione' = N6, with an unknown phase factor for every giveninput . In Sections 2 and 3 I present formulas for therecovery of the similarity- and congruency-transformationmatrix operators of any order, given input, and given out-put matrix samples. These formulas represent a solutionto the coherent-light-transfer problem [Eq. (1.3)], giventhree partially polarized input Stokes states, as I discussin Section 4. In Section 5 a natural extension of thetransfer formula is developed to represent the more gen-eral case of isotropic depolarization. Isotropic depolariza-tion makes the recovery formalism applicable to nonidealimaging systems. I show that a Mueller matrix with itsdepolarization restricted to be isotropic can be recoveredfrom three independent Stokes states, too. The formal-ism is illustrated by a practical example in Section 6.

The mathematics of the nonlinear congruency transfor-mation, Eq. (1.3), is understood by the natural geometricproperty for the related similarity transformation."' 9

The similarity transformation is the form of Eq. (1.3) withN a unitary matrix: Nt = N-'. The similarity trans-formation does not map a matrix J(s) into any matrix J(s')as is possible with a linear transformation, i.e., a Muellertransformation. Rather, the eigenvalues of the matrixJ(s') must be the same as those of J(s), or equivalentlyso' = s, and sI'2 + 2'2 + S3,2 = S12 + 22 + S32. The ei-genvalue constraints are equivalent to metric constraintsin the Stokes parameter space (sI,S2,s 3 ). Thus the uni-tary operator N can be visualized as a rotation in thethree-dimensional parameter space, which preserves thevector amplitude. The inverse problem for the recoveryof N can be understood geometrically: the rotation thatmaps the vector (l,S2,s 3 ) into (',S2',S3') must have itsrotation axis on the perpendicular plane that bisects theline between (, 2, 3) and (sI', 2, s3'; the intersection oftwo such independent planes defines that axis uniquely.The axis defines the eigenvectors for N, and the rotationalamplitude defines its eigenvalues. Thus two independentpairs of Stokes states sj and sj', j = 1, 2, must determinethe common similarity-transformation operator. This re-sult is valid for nonunitary similarity transformationsand generalizes to arbitrary order and to non-Hermitianinput and output matrices, as I describe in Section 2.

A relationship between the nonunitary congruencytransformation, Eq. (1.3), and the similarity transforma-tion necessarily exists, because by the singular-valuedecomposition of N we have

N =U[ 1 ° Ovt (1.4)

The theorems on matrix recovery are developed for ar-bitrary order, though only the formalism for second orderis useful in the polarimetric inverse problem. It is an in-teresting property that the familiar three-dimensionalgeometric group arises as a purely mathematical propertyfor the quantum observables of a two-element wave func-tion, i.e., Eq. (1.3). This feature of the second-order solu-tion can be put into the larger context of geometries ofgeneral order that exist with the observables associatedwith higher-order wave functions. I return to a prelimi-nary discussion of the natural geometric property of thetransformations for the quantum observables of a multiple-element wave function in the conclusion, Section 7.

2. RECOVERY OF THE SIMILARITY-TRANSFORMATION MATRIX OPERATOR

In this section I prove that two nonzero pairs of complexmatrices (A, A) and (B, B'), related by the nonsingularsimilarity transformation of order n 2 2,

A' = NAN-',

B' = NBN-' (2.1)

determine the common transformation matrix N uniquely,except for a scalar multiplier, as long as A and B satisfy acondition of independence. For independence it is alwaysnecessary that A and B have independent eigenvectors.For n < 4 the matrices are independent if and only ifdet(AB - BA) 0.

The eigenvalue decomposition for the matrix A is writ-ten in matrix form as follows:

.AlA

A = XA

0

0

kA2A XA-1

kA ,

where AjA are the eigenvalues of A and XA contains thesimilarly ordered eigenvectors of A in its columns. Sub-stituting this expression for A and the eigenvalue decom-position for A into Eqs. (2.1) gives

A1A'

XA'

0

A2A'

0

XAr-1

AnA

AlA

= NXA

0

A2A

0

(NXA)_1,

AnA

where AjA' are the eigenvalues for A and XA' is the matrixof ordered eigenvectors for A. It is always possible toorder the eigenvalues so that AjA' = AjA. As long as all theeigenvalues in A are distinct, the corresponding eigenvec-tors can be related uniquely also, and we have

where U and V are unitary matrices and (rl, r2) are posi-tive real numbers, the matrix singular values.2 0 The con-gruency transformation can be understood as a rotation,followed by an anisotropic rescaling of two Stokes intensi-ties, followed by a second rotation.

a1

XA' = NXA

0

0

a 2

an

(2.2)



where aj are arbitrary complex scalar multipliers for theeigenvectors. When we apply the determinant rule forthe multiplication of matrices, we know that no aj canbe zero, since the matrices of eigenvectors XA' and XAare nonsingular by definition and N is nonsingular byassumption.

Equation (2.2) has a three-dimensional geometric inter-pretation when N is second order and unitary and A isHermitian. In that case XA and XA' are unitary, andthe unknown coefficients in the vector (a,,aO2) must bephase factors:

Wil/willW21/W2 1'

Wnl Wnl'

W12 /W12 '

W22 /W2 2'

Wn2/W.2'

* * WlnlWln

... W2n/w2n'

... Wnn/Wnn'

'a,/,B, aE/,62

a 2/,81 a 2 /,32

an/161 an/132

N XA' Eexp(-iO)0 exp(io)l]A

The matrix is a vector outer product that is separable bysingular-value decomposition (svd):

for the real variable 0. The formula must confine the pos-sible principal axes for N to the perpendicular plane thatbisects the line from the A parameters to the A' parame-ters in their real parameter space. The orientation of theeigenvector axis in the bisecting plane must be a functionof 0.

A second independent relation is needed to determine N.We have, like Eq. (2.2), for the general B with distincteigenvalues:

a/,B1

an/18,

ai /fl32

a2/132

an/162

... ai/113n

... a2/fn

... an/13n

svd Cl a2/Ua

an/Ua

( ~)lUb 82Ub 2.nUb

(2.6)

1'P

XB' = NXB 132

0

where XB and XB' are the matricesB and B', respectively, for similarl:,Qj are arbitrary complex scalar muvectors. Eliminating the commorEqs. (2.2) and (2.3) gives the relatic

a,

0

a2

= (XAf

or, denoting the products W' = XA'-

a1

0

0 '1

a2 W = W

anj 0

When the matrix elements W' =are used, and the matrix products cequated in every element, we obtain

0 The decomposition is unique, except for an arbitrary com-plex scalar ua. Clearly the vector outer product on the

(2.3) right-hand side of Eq. (2.6) gives the matrix on the left

uniquely. Also, the complex svd of a matrix that has onlyone nonzero real positive singular value (uub) is decom-posed into the coefficient vectors on the right, uniquely;

of the eigenvectors for see Appendix A. The svd defines both coefficient vectorsy ordered eigenvalues- in Eq. (2.5) from an overdetermined system when n > 2.Itipliers for the eigen- If there is noise in the matrix of ratios Wkl/Wkl', then then nonsingular N from coefficient vectors associated with the largest singularon value are also optimum solutions for equal weighting of all

the matrix elements, as I prove in Appendix B. Thereforethe complex svd applied to the matrix, Eq. (2.5), deter-mines the coefficient vector (a,, a 2 ,... an), except for acomplex scalar multiplier, and this coefficient vector de-termines N, except for a scalar multiplier, by substitutioninto Eq. (2.2).

The generalization of Eq. (2.4), N 0W' = WNb, for arbi-13' 0 trary unknown matrices N, and Nb is underdetermined.

XB) /32 Clearly, for any choice of Nb, an N, can always be foundXB) . ' that is a solution, i.e., N, = WNbW'-'. Even for the more

specific form, Nb a diagonal matrix and N, a general ma-0 On trix, every choice of Nb is still a solution. Even if there is

only one 2 X 2 diagonal submatrix of N, that is not a di-

'X,' and W = Xf'1X,, agonal matrix, the system is underdetermined, as we cansee by expanding the product as matrices of submatrices.For the unknown matrices Na and Nb to be diagonal in

0 Eq. (2.4), it is necessary that both matrices A and B have

132 distinct eigenvalues. Therefore if two matrices A and B.. (2.4) cannot be formed (by linear combinations) having all dis-

tinct eigenvalues, recovery of the matrix operator N isprecluded.

For some XA' and XB' the product W' = XA''lXB' can= (Wkl') and W = (Wkl) contain zero values. When there are any zeros in W', the)n each side of Eq. (2.4) svd method presented in Eq. (2.6) is not applicable. How-(as long as all Wkl' 0 0) ever, a well-defined ratio aC/13, in Eq. (2.5) serves to deter-

... a 1 /,B

... a2/16.

..a, /1 -(2.5)



mine one 13,k in terms of the first aj, which can be taken asarbitrary; then a is defined with another well-definedratio az/Pk, etc. For n < 4, if there are at least twononzero elements in each row and in each column of W',then a set of well-defined relations in Eq. (2.5) existsthat is sufficient to determine both coefficient vectors(a .... an) and ( ,... 1), except for a scalar multiplier.For order n 4 there are some arrangements with twononzero elements in each row and column, for which thecoefficient vectors are still not determined.

The only indeterminate case for n < 4 is when one col-umn or row of W' = XA'_'XB' contains only one nonzeroelement. If W' has a column with only one nonzero ele-ment, then from XA'W' = XB' we know that one column ofXA' occurs in XB' also. I show in the following that XA'and XB' with a degenerate column must satisfy the com-mutation relation

det(A'B' - B'A') = 0 (2.7)

for matrices A and B' with the matrices of eigenvectorsXA' and XB', respectively. If one row of W' contains onlyone nonzero element, then, from XB'T-1WPT = XA T-1, weknow that one column in XA'T-1 is the same as one columnin XB' T. In this case Eq. (2.7) is also satisfied, when thetranspose of Eq. (2.7) is taken. Substitution of the simi-larity transformations [Eq. (2.1)] shows that the commuta-tion condition, Eq. (2.7), is equivalent to det(AB - BA) =0 for a nonsingular N. Therefore W' contains a rowor a column with only one nonzero element only whendet(AB - BA) = 0. As long as det(AB - BA) 0 forn < 4 a unique solution for N must exist, except fora scalar multiplier. The proof of the theorem is nowcomplete.

Commutator relations, such as Eq. (2.7), are used to testfor repeated columns in matrices in the theory of matri-ces.2 '123 If an eigenvector x is common to both A and B,it is an eigenvector of the commutator (AB - BA), since(AB - BA)x = AXA - BxAA = xA, and the eigenvalueA = AAA - AAA = 0. Thus in this case the determinantof the commutator, det(AB - BA), must also be zero.Also, if a matrix A or a matrix B has repeated eigenvalues,then two of its eigenvectors are not unique, and theyproject according to the input vector to give det(AB -BA) = 0.

To summarize, two pairs of matrices (A, A) and (B, B'),which are similarly transformed as in Eq. (2.1), give aunique determination of a nonsingular N except for a com-plex scalar multipler by the following formula:

1

N =XA'

0

0

- '

XA and XA' are the eigenvector matrices of A and A', re-spectively, for similarly ordered eigenvalues. The complexparameters (ala 2 ... an) can be defined in most cases,except for a complex scalar multiplier, by the svd:

W11/W11' W12/W12' *-

W21/W21' W2 2/W2 2 ' *-

Wnl/wnl' Wn2/Wn2' ...

ai/u.svd) a!/Ua- (U,.Ub) . (

an/Uaj

WlnlWln'

W2 n /W2n

Wnn/Wnn'

(1 1

kPlUb 2Ub(2.9)

where Wk, = XA_1XB, (Wkl') = XA''lXB', ua and Ub are arbi-trary complex scalar multipliers, and UaUb is the real sin-gular value. With noise in the measurement, the largestsingular value must be selected.

Numerical tests verify the recovery formula in applica-tions with noisy sample matrices. Figure 1 shows threedifferent measures of error in the recovered matrix N, oforder n = 2, which was defined from matrices Ak andAk', k = 1, 2. The measure of error for the upper panel isthe rms in the real components in the matrix differenceN - N. For the difference Nr - N, we normalized N, toconsider its degeneracy with multiplication by an arbi-trary scalar. The other two measures of error for thesecond and third panels are defined by the norm X12:

Xi2 = ( >- IA'- NrAN, -1112,22n) , Jk

for n, the order of the system. Al,2 denotes the sum ofsquares of the matrix real components for the secondpanel; AId = det(A)J in denotes a determinant measurefor the third panel. The divisor specifies the number ofreal components that enter into the sum, so that A'Y2 repre-sents an average error for the set of input and ouputmatrices.

The average rms in the real components in N, - N (up-per panel) is -0.15 less than the component rms IXrI(middle panel). This difference is consistent with the factthat N, contains half as many total real components as thetwo matrices Ak' that determine it; i.e., 0.15 log(V'2).The meaning of the zero in the log of the determinantmeasure, when JXdl equals the rms of the added noise, isnot obvious. However, the smaller variance in XdI indi-cates that it is the most favorable error measure of thethree for the recovery formula. This should not be sur-prising, as the determinant is a type of average, whereasthe other two measures reflect the largest value in thematrix components. The worst case (upper dotted curves)in every example with added noise of less than 10-4 wasdetermined by the sample matrices N that were the closestto being singular in the set. The recovery formula cannotbe considered useful when the total noise is of order 1, thesame amplitude of the matrix elements. In the worstcases this noise level is found with >1% added noise andcan be traced to input matrices A of similar eigenvalues.

Figure 2 shows a similar test applied to matrix samplesof general order n. The relative matrix error N, - N isshown as a function of the log of the added noise. As theadded noise approaches amplitude 1, the amplitude of the


1...JGnUb


4lmatrix error

3 _....... .. ... ... .. ... .. . .

__________ ______…__----- --- ----- _

1 __ ...

4 |I g Is I.

IX, I rms3. ........ .. ... ...

..... ... .. ... ... ..

2

1________-___ -_----

4 i i

1Xd I determinant

3

2.. ,,,,,, ,,~~~~~~~~~~~~~. .. ,,'".. ."""..

1~~~~~~~~~. .. . ., _ _ - _ _- _---- -- - - - - - - - - - - - - - _

-------------------------------0g----.......................................................A=~~~~~ ~

-5 4log(noise)

-_5 -Z

Fig. 1. Noise sensitivity in the recovery of the second-order matrix operator. The similarity-transformation matrix operator Nr wasrecovered from two matrix pairs, A, and A,', k = 1,2. A was filled with random components in the range ±1 and A,' = NAkN-' plusnoise. N was a complex matrix filled with random numbers in the range ±1 and selected to avoid a small determinant. In the upperpanel the relative rms error in the components of N, - N is shown as a function of the rms of the added noise. In the lower two panels,component rms and determinant measures of the matrix difference A,' - NrAkN -j are shown as functions of the rms of the added noise.Each filled square shows the average of the logio of the relative error; dashed curves show ±1 standard deviation in the log, and dottedcurves indicate the worst and the best cases from 100 test combinations of matrices N, Al, and A2. The figure demonstrates the stabilityof the similarity-transformation recovery formula.

matrix components, the relative error in the worst case-goes down, since the recovery formula preserves the ma-trix amplitude. These figures demonstrate the numericalstability of the recovery formula for any matrix order.

Gantmacher 21 uses a discussion of the general problemof the recovery of the similarity-transformation matrixoperator to introduce the numerical problem of the eigen-value matrix determination. He describes computationalalgorithms based upon matrix-element interchange meth-ods for the eigenvalue decomposition for N but does notenter into the matrix-operator recovery problem.

3. RECOVERY OF CONGRUENCY-TRANSFORMATION MATRIX OPERATOR

In this section I prove that three nonzero matrix pairs,(A,A'), (B,B'), and (C,C'), where A, B, and C are

Hermitian and related by the nonsingular congruencytransformation of order n 2 2,

A' = NANt,

B' = NBNt,

C' = NCNt, (3.1)

are sufficient and necessary, except in one case, to deter-mine a general transformation matrix N except for a phasefactor, as long as a condition of independence among A,B, and C is satisfied. For n < 4, the condition of inde-pendence is det(ABCBAC - CBACAB) 0; B denotesthe adjoint of a matrix B. For matrix independence it isnecessary that A, B, and C be linearly independent intheir elements, but linear independence is sufficient alsoonly for n = 2. The one case in which three matrix pairs

0x. )

E

_En

Eo

-10

IC

0

-I0

25aC

0


.

-1I-6


are not necessary is when the eigenvalues of A can be re-lated to those of A by a single scalar multipler; then Nmust be a matrix of unitary type N` X Nt, and two pairs(A, A) and (B, B') are sufficient to determine N uniquely,except for a phase factor, by the similarity-transformationrecovery formula of Section 2.

The eigenvalue decomposition for the Hermitian matrixA can be written in the following matrix form:

AMA

A = XA

0

0

A2A XA t

A.A

since XA is unitary (XAt = XA-1) and det(XA) = 1 by as-sumption. Substitution of this relation and the eigenvalue

4

3

2

0

4

1' %I-

0E

E,

cP0

3.

2

0

4

3

2

0

-6 -5

matrix relation for A into the congruency-transformationrelation [Eq. (3.1)] for A gives:

A1A'

XA'

0

A2A'

0

XA't

A.A'

AMA

= NXA

0

0

'A2A (NXA)',

AnA

(3.2)

where XA' is the matrix of ordered eigenvectors for A andAjA' are its eigenvalues.

If one nonzero scalar multiplier y exists such thatAjA' = AjA for all j, then the eigenvectors of the two sides

-4log(noise)

-3 -2 -1

Fig. 2. Noise sensitivity in the recovery of matrix operators of different order. The similarity-transformation matrix operator Nr wasrecovered from matrices Ak and A,', k = 1,2. A was filled with random components in the range ±1 and A,' = NAkN-' plus noise. Nwas a test matrix filled with random components in the range ±1 and selected to avoid a small determinant. In each panel the rms errorin the components of N - N is shown as a function of the log1o of the rms of the added noise for the specified order. The average of thelog of the relative rms error is shown as a filled square; +1 standard deviation in the log is plotted with dashed curves, and the worst andthe best cases are shown with dotted curves from 100 test combinations of matrices N, Al, and A 2.

- order 3

------------------ --- --- -"-- -Z-------.- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . ..............- - -

order 5

------- U.-.---.------

I . .. . . .. . I. -* .............. -........... ............ .... .... .. * .....................

- order 7

* .. . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

… ___________- --- …-._-__._

I I I I , I


I I I I I

-7 ........

I I I

. ............. ..... . ... .. .. .... .. . . .

I I I I


of Eq. (3.2) must be the same by the uniqueness of theeigenvalue decomposition for a matrix A with distincteigenvalues. Relating determinants gives yn = Idet(N) 2.

Relating eigenvectors leads to N' = y4N'. Applicationof the similarity-transformation matrix operator recoveryformula of Section 2 shows that two matrix pairs deter-mine N except for a phase factor, when A and B satisfythe condition of independence for the similarity-transfor-mation recovery formula.

Conversely, if the eigenvalues of A and A cannot all beequated with a single scalar factor, then the eigenvalue de-compositions on the right- and left-hand sides of Eq. (3.2)are not the same. In this case a unique condition on thematrices of eigenvectors cannot be obtained. However, acombination of two of Eqs. (3.1) can always be formed,which gives the same eigenvalue decomposition on bothsides of the equation; we combine Eqs. (3.1) and choose Aand B in the following way:

A'B'= det(N)2NABN',

= XABwlC XABattW = XAB'CXAB t ,

a,

0

O'c *-I0 a

a2 W =W

Ofn 0

a 2*-1

0

a( *35

(3.5)

If the matrix products in Eq. (3.5) and the matrix ele-ments for W' = (kl'),W = (), where W' # 0, are re-combined we find that

wl1/wil'W2 1IW21

Wnl Wnl"

W12 /W1 2'

W22 /W22 '

Wn2/Wn2

(3.3)

where B = det(B)B-' is the adjoint matrix. The adjointis used rather than the inverse, because it remainswell defined even when det(B) = 0. The adjoint Bcan be obtained by substitution of the product1kB.... Aj-1,BAj+lB... AnB for each of the eigenvalues AjBof B. In the derivation of Eq. (3.3), I applied the determi-nant relation from Eqs. (3.1), det(B') = Idet(N) 2det(B).

The matrix combination, Eq. (3.3), is a similarity trans-formation for N nonsingular, and so the eigenvalues ofAfB' must always be the same as those of Idet(N)2AB.When the eigenvalues are distinct and are ordered in thesame way, the corresponding eigenvector relation mustbe satisfied:

a,

XAB' = NXAB

0

0

a 2

an

... Win/Wln'

... W2n/W2n'

... Wnn/Wnn'

affl Clg* *--alai ala2a2al* a 2a2* ...

na* t nO2* ...

The matrix is a Hermitian vector outer product:

... a1an*

-- anon*afla2*

evd) a2/U a,*

Ofn/U

a 2*

U *

... *) (3.6)

(3.4)

where XAB' is the matrix of eigenvectors of AB' and XAB

is the matrix of eigenvectors of AB. The factors a arethe arbitrary nonzero complex scalar multipliers for theeigenvectors.

The third of Eqs. (3.1) determines (a,, a 2 ... an). Sub-stitution of N from Eq. (3.4) gives

a 2

0

(X! *-I

*-

- XABlCXABt1

0

a 2*-1

or, if the matrix products are denoted by the new symbols

The complex eigenvalue decomposition (evd) gives(ala 2,...an)/u and (a1,a2 ,... an)*/u* and the real eigen-value u* uniquely. It therefore gives (al, a 2,... an)IuI/u,leaving ul/u as an undetermined phase factor. Then sub-stitution back into Eq. (3.4) gives N uniquely except for anarbitrary phase factor.

The most general form of Eq. (3.5), NaW' = WNat-', foran arbitrary unknown matrix Na is underdetermined, be-cause this relation defines a general congruency transfor-mation. Even if there is only one 2 X 2 submatrix in Nathat is not a diagonal matrix, Na is underdetermined, ascan be seen by expanding the product as matrices of sub-matrices. Therefore, if no matrix AB with distinct ei-genvectors can be formed (by reordering the matrix pairsor by forming linear combinations), the recovery of thecongruency-transformation matrix operator is precluded.

The two diagonal matrices are determined exceptfor a scalar multiplier as in Eq. (2.4). Thus the sys-tem, Eq. (3.5), overdetermines the coefficient vector (a,,a 2 ... an), since it appears in both diagonal matrices. Forpractical considerations, to minimize noise effects oneshould select the largest eigenvalue.


alan*a2an*

aa,*

alai*a2al*

anal*


With sparse zeros in the matrix, W', the coefficient vectorcan still often be recovered with the procedure describedin Section 2. If the matrix W' = XAB'_1CIXAB`t- has acolumn or a row containing only one nonzero element, thecoefficient vector cannot be recovered. For a columnwith only one nonzero element (XAB')W' = (C'XAB't_1),and one column of XAB' degenerate with one column ofC'XAB' t ', we have

Al 0

det XAB' A2

0 An

A,'

X AB C1XAB~

0

Al'

- CXAB't1

0

0

A 2 ' XAB'tC

Ant

0

A2 '

Ant

matrices A, B, and C, linearly independent in their ele-ments, are always necessary for finding a solution for N.

For the second-order case there are no matrices ABwith repeated eigenvalues that are not proportional tothe identity, and so for all these matrices B a A. If theHermitian matrices (aA + bB) and C have one eigenvectorin common, then they have two, and so in the second-ordercase there are no matrices A, B, and C that satisfyEq. (3.8) that do not also satisfy Eq. (3.9). Therefore forthe second-order case any three mutually linearly indepen-dent matrices A, B, and C are also sufficient for obtaininga solution for N. The proof of the theorem is now complete.

To summarize the recovery formula, three nonzeromatrix pairs (A, A), (B, B'), and (C,C'), all Hermitian andrelated by the congruency transformation [Eqs. (3.1)], de-termine the nonsingular N, except for a phase factor, bythe following formula:

11

N = XAB'

0

a2 'I0

XAB-1,

an1

(3.10)

Al

X XAB'tCXABI

0

A2

where XAB' and XAB are the matrices of eigenvectors of0 (3.7) AB' and AB, respectively, for similarly ordered eigenval-

ues. The coefficient vector (a,, a2 ,... an) is determinedexcept for a phase factor by the eigenvalue decomposition:

for any values of the variables Aj and A/'. The adjoint formfor C', C' is used, so that Eq. (3.7) is applicable in the casewhen det(C') = 0. The same relation, Eq. (3.7), representsthe case of a degenerate row in XAB' and C'XAB't_ 1. If Ajare the eigenvalues of AB, and Aj' are the eigenvalues ofthe transpose conjugate BA, then

det(A'B'C'B'A'C' - C'B'A'C'A'B') = 0;

then, application of the congruency-transformation rela-tions, Eqs. (3.1), gives

det(ABCBAC - CBACAB) = 0. (3.8)

This condition of matrix dependence also suffices to indi-cate repeated eenvalues in AB, because if the combina-tion matrix AB has repeated eigenvalues, then BA =(AB)t must have repeated eigenvalues also. The con-dition, Eq. (3.8), always indicates matrix dependence,and for order n < 4 it also is the only condition for matrixdependence.

Of course, the determinant of the commutator, Eq. (3.8),is zero, if the commutator itself is zero:

ABCBAC - CBACAB = 0. (3.9)

It is easily demonstrated that this relation is satisfied ifB A or if C A, or if C B. Algebraic verificationwhen C = aA + bB, for real coefficients a and b and forthe Hermitian C, requires more persistence and relies onmultiplication through on one side repeatedly to removeadjoints. Other functional forms, deduced from the powerrelation B A", C Am, or C B', for m $ 1, do not sat-isfy the commutation relation, Eq. (3.9). Therefore three

Wnl/Wnl'

W211W21'

Wnl /Wnl

W12 /W12 '

W22 /W2 2 '

Wn2/Wn21

Wln/Wln'

W2n/W2n'

Wnn/Wnn'(a,* a *ai/uevd a2/U-=-> (U U*)

an/u

where (Wkl) = XABiCXABti and (Wkl') = XAB''C'XAB' t ;u is an arbitrary complex scalar multiplier, and uu* is thereal eigenvalue.

Of course, the recovery of the congruency-transforma-tion matrix operator can also be applied to a simultaneousmatrix system without specific Hermitian symmetry. Anarbitrary matrix A can always be decomposed intothe sum of two nonzero Hermitian matrices, A+ and A_such that A = A+ + iA_, where A+ = (1/2)(A + At) andA_ = (1/2i)(A - At). The congruency transformationpreserves the Hermitian symmetry, so the relationA' = NANt gives both relations: A+' = NA+Nt andA-' = NA-Nt. The congruency transformation of anon-Hermitian matrix thus provides two Hermitian pairsfor the recovery formula.

The determinant of N can be obtained from any ofEqs. (3.1) separately, i.e., Idet(N)12 = det(A)/det(A), aslong as det(A) # 0. A more useful formula is obtained bysumming Equations (3.1):

Idet(N)12 = det(aA + bB + cC) (3.12)

(3.11)


a,*U*


where one can often find arbitrary real coefficients a, b,and c to avoid a zero denominator.

Figures 3 and 4 demonstrate the stability of thecongruency-transformation recovery formula, Eqs. (3.10)and (3.11), with added noise. The measure of error for theupper panel in each figure is the rms in the real compo-nents of the matrix difference N,. - N. For the errorN,. - N, we normalized N,. to consider its degeneracy withmultiplication by an arbitrary phase factor. The lowertwo panels in each figure plot two other measures of errordefined by the norm XI2:

A"2 = i JAk' -NrAkN,.t I 2

where Al,2 denotes the sum of squares of the matrix realcomponents for the second panel and Aid = Idet(A)l I/n is adeterminant measure for the third panel.

4

L-..

a)

E cU)

3

2

0.

4

3

, IC

0-

2.

11

0

4

3

IC_ X

2

C

-6

The average error in the matrix real components ofN,. - N is approximately 0.25 less than IXI (middle panel),which is consistent with the fact that N,. is derived fromthree matrices, each with added noise in all of itscomponents, i.e., 0.25 log(</). For the congruency-transformation recovery formula, the determinant mea-sure of noise (third panel) is only a little more favorable.Figure 4 demonstrates that the average error in Nr - Ndoes not increase with increasing matrix order n.

4. JONES-MATRIX RECOVERY INCOHERENCY-MATRIX TRANSFORMATIONS

The congruency transformation [Eq. (1.3)] is known to bea good description of the transformation of the Stokes pa-rameters through all nondepolarizing optical systems.Even though it is not so general as the Mueller linear

-4 -3log(noise)

*-----------------------------------------) -- - - - - -... .................................... ............ ....... - - -

.. .. . . .. . . .- .j . sl .lo . . . .

Fig. 3. Noise sensitivity in the recovery of the second-order matrix operator. The congruency-transformation matrix operator N, ofsecond order was recovered from three matrix pairs Ak, Ak', k = 1, 3. Ak was a complex Hermitian matrix of random components in therange ±1 and the corresponding Ak' = NAkNt plus noise. N was a complex matrix filled with random components in the range +1 andwas selected to avoid a small determinant. In the upper panel the relative rms in the components of the error Nr - N is shown as afunction of the added noise. In the lower two panels two other measures of the matrix difference, Ak' - N,AkN t, are plotted as func-tions of the added noise. Each filled square shows the average of the logio of the rms; dashed curves show ±1 standard deviation in the logof the rms, and dotted curves indicate the worst and the best cases from 100 test combinations of N and Ak, for k = 1, 3. The figuredemonstrates the stability of the congruency-transformation recovery formula with added noise.

matrix error

a a 0 a 0 E a a a a a a a N N 0 a 0 0 E a a 0 0 M a a N a a M a 0 a a 0 0 0 a .. - -

a 0

.. . ..... ... .... ... .......I I

I I If


II I

.......

... I I I I

... ., , , I ... .'.. - ., '.. .. ...

I- a a M N a M a a a a N a 0 N a 0 E E s N a N N a a N a a 0 0 a 0 a N M a

--------------------------------------------------

'17......... --I I I.................................................................................................7

I I Il l I

I I

l

_1

-Z. I- 5

lx, I rms

I141 determinant


xtI-a.0

o oE c

E

0a

I I . I

_

3

2- �

l -_UU U U U UU UU UU UU U U U UU UU UU UU U U U *

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

0

- order 5

-… _ __________ -

* ..... ............. " " . . g .g.... __ _ _ _ _ _ _ _ _ _ _ _-_ _ _ _-~~~~~~~ -- _

I I II

- order

:-------------------_ --- - ----- - - _

-- - - -- - - - - - - - - - - - - - - - - -- - - - - -. ----- - - - -

.. ...... . - .

-b. -4 _xlog(noise)

-2 -1

4

1

3

2

1

0~

4

3,

2

1

0.

Fig. 4. Noise sensitivity in the recovery of congruency-transformation matrix operators of different order. The congruency-transformation matrix operator Nr was recovered with three matrix pairs, Ak and Ak'. Ak was a complex Hermitian matrix with randomcomponents in the range ±1 and Ak' = NAN t plus noise. N was a complex matrix filled with random components in the range ±1 andselected to avoid a small determinant. In each panel the logl, of the relative rms in the components of the error N, - N is plotted as afunction of the log of the rms of the added noise for the specified order. The average of the log of the relative rms error is plotted as afilled square, ±1 standard deviation in the log is indicated by the dashed curves, and the worst and the best cases are indicated by thedotted curves from 100 test combinations of N and matrices Ak.

order 3

.... . I

transformation algebra in the four Stokes parameters, thelimited degrees of freedom can be an advantageous prop-erty, because these optical systems can be characterizedwith reference to fewer than four independent polarizationsources. Thus it is possible in some cases to avoid thewell-known uncertainties in the ellipsometry that refersto both polarizers and wave plates, such as wavelengthand temperature dependence of wave-plate retardationand dichroism, alignment of principal axes in rotationazimuth, alignment of principal axes with wave-platecrystal faces, and composite-plate effects from surfacecoatings.4 24 29 Precise sources can be obtained in natu-ral polarizing processes, which may provide three inde-pendent polarization states by rotation.30

The recovery formula from Section 3 [Eqs. (3.10) and(3.11)] can be applied directly to obtain the Jones matrixfor an unknown device. Figures 5 and 6 show the rmserror in the components of the matrix N,. - N, determined

in numerical experiments by applying the congruency-transformation recovery formula to simulated experimentswith added noise. In every case a Jones matrix N, is de-fined from input Stokes states Sk and corresponding out-put states Sk', where J(sk') = NJ(sk)Nt plus noise. Whenthe difference N,. - N is formed, we normalize N,. to con-sider its degeneracy with multiplication by an arbitraryphase factor. Figure 5 demonstrates a number of practi-cally useful combinations of purely and partially polarizedinput states.

Figure 6 shows numerical experiments for different de-vice types. Optical devices are classified by their eigen-value decomposition 3 1 :

N = [+ e1 ]a0 exp(-ip/2)a( exp(ip/2) 4.1)

(4.1)

. . - .- - - I .

I


I

I I

.......... .1-

I

I

- 5


where p is the retardation, a+ and a- are the real absorp-tion coefficients, and e+ and e_ are the constant complexeigenvectors for the two principal axes. The eigenvec-

2

1

0

-1

-22

I I

3 linear polarizers

tors e+ have corresponding Stokes coherency matrices,J(s±) = e+ C) 6.t, which define the Stokes three-elementvectors (Slt,S2±,S3±)T as the principal axes of the device,

I I I I I I

-@4@ . - --. ~ .4Mwss ~ ~~ -.8SS~

I z2 linear w zero

I J I I I I I

- He b / ado I-.. .*D .s -. ..\- '\ ' 'mM U 19,1 '2 linear w circua

:U ___.. .

,~~~.. U *s-I

20 40 60 80 100 120 140 160 180span of polarizers (degrees)

0.1 .0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9polarization fraction

Fig. 5. Noise sensitivity in the recovery of the Jones matrix in transformations of the Stokes coherency matrix. A Jones matrix N, wasrecovered from three input polarization states Sk and corresponding output states Sk', where J(sk') = NJ(sk)Nt plus noise of rms 10-2. Nwas a test Jones matrix defined with random components in the range ±1. The rms in the components of N,. N is shown as a functionof the input polarization set. The average of the logio of the rms is shown as a filled square; ±1 standard deviation in the log of the rms isshown by dashed curves, and the worst and the best cases are shown by dotted curves from 10 Jones matrices N tested. In the top threepanels the first two input states (A and B of Section 3) were pure linear polarizations separated in orientation by the span in degrees withSo = 1. For the bottom panel the first two input states were partial linear polarization states with the specified polarization fraction andseparated by 520. The third input state (C of Section 3) was the linear polarization state between the two for the top panel, an unpolar-ized state for the second panel, a circularly polarized state for the third panel, and a completely unpolarized state for the bottom panel.

I-

0EEn

0

0

-1

-21

2

1.

0

.. ., , . , ., '..'z I'"

, _ _ , s > . . . _ _ - '

- . An_/ . zI ; d

ir ~ U M IL~U %-1

-21

2

1

0

-1

-2

2 partial w zero

- . . ..

- _ - U. .U. .. _

I I I 1 '1 . . . .......


I I I I I

730 J. Opt. Soc. Am. A/Vol. 10, No. 4/April 1993 Laurence J. November

-0.5' I I retarders

-1

-0.5 -8> oto oaies .. H

-1.

........... ~ ~ ~ ~ ~ ~ ~ ~ .. II

-2

-3.5

0dich, prti polartzlr

~~~\ MM / 1 : . :.

0'""~~ | Ut,\ '' U'd L --- -34A- --tit.dichroic crystals ..

........... m m -1 .- ~ u -...

aritrary d mchricecs

"J 40 61

span of polarizers vdegrees)

2g6.NiewtdifrndeietpsThJoemarxNwareoeefrmtreip5plrztosttssancorsodn

-3 -~ ~ ~ ~ -----

-3.5 - ~ ~U-- - * mU~

20 40 60 80 1Q0 120 140 160 180span of polarizers (degrees)

Fig. 6. Noise with different device types. The Jones matrix N, was recovered from three input polarization states Sk and correspondingoutput states sk', where J(sk') NJ(sk)N t plus noise of rmis 10-4. N was a test Jones matrix defined by random physical parameters.The rms in the components of N,. - N is shown as a function of the span in degrees of three input linear polarization states. The averageof the log of the rms is denoted by a filled square; ±1 standard deviation in the log of the rms is shown by dashed curves, and the best andthe worst cases are shown by dotted curves from 10 Jones matrices N tested. In each panel a different type of Jones matrix N was used;they are, from the top, retarders, partial linear polarizers, dichroic crystals, tilted dichroic crystals as explained in the text, and arbitrarymatrices.


where SO±2 = S1±2 + 2±2 + S3+2. All special devices haveorthogonal principal axes, which correspond to oppositelydirected Stokes vectors, s + and s_, in the Poincare sphere.Retarders are unitary matrices N. a+ = 1, with oppositeprincipal axes in the linear polarization plane of thePoincare sphere. Partial polarizers are Hermitian ma-trices N: a+ = 1 and a_ s 1, p = 0, with opposite prin-cipal axes in the linear polarization plane. Dichroiccrystals and tilted dichroic crystals have a+ = 1 anda- 1, with p # 0. With tilted dichroic crystals the op-posite principal axes are not confined to the linear polar-ization plane as they are for plain dichroic crystals.Figure 6 shows that the noise in the recovered matrix isconsistently smaller for matrix types that are always non-singular, i.e., retarders.

The congruency-transformation recovery formula,which does not make equal use of the three matrix pairs,does not give the optimum matrix operator with addednoise. Thus numerical least-squares methods that inprinciple provide the solution that reproduces the test datamost accurately may be better. In practical problems therecovery formula can be used to give a good initial guess,which is always necessary for least-squares iteration pro-cedures to succeed. When more than three observationsare available, the average of similar Stokes states in threedifferent groups gives independent Stokes input and out-put states with reduced noise for the recovery formula.The transformation [Eq. (1.3)] is linear in the Stokes vec-tors s and s', and so averages of input and output vectorsgive valid Stokes-vector pairs.

The least-squares x2 is defined as

X= > IJ(sk') - NJ(sk)N t l,2,4m k.1

where sk and Sk' are the m input and output Stokes vec-tors, IJ(s),2 denotes the sum of the squares of the realcomponents in the matrix, and N,. is the Jones matrix tobe determined. The congruency-transformation recoveryformula is first applied to Stokes vectors averaged in threegroups to obtain an initial guess for N.. Then the itera-tion procedure defined in Appendix C gives the Jones ma-trix that minimizes x2 locally.32

In practical problems the least-squares iteration alwaysgives the optimum N., which minimizes x2 for the mea-surements, if the iteration converges. Convergence islargely dependent on the quality of the initial guess andthe noise in the measurements. In practice, I find thatthe procedure in Appendix C is usually convergent, givenan initial guess from the recovery formula with less than0.5% measurement noise. With more than 0.5% measure-ment noise, more than three Stokes sampels are needed.With 100 measurement samples, the procedure usuallyconverges as long as the noise per measurement is lessthan 5%.

where 0 denotes the outer product,

1 0 0 1

C= 1 1 0 0 -1

V2_ 0 1 1 0

0 i -i O

[nl,N* n12 N* 1N 0 N = n 2 N* n22N*JI'

and C- = Ct; the 2 X 2 matrix of 2 X 2 matrices isequivalent with the 4 X 4 matrix of the elements. TheMueller matrix M has a one-to-one correspondence withthe matrix of dual products of the Jones matrix N; N canbe recovered from M by svd as follows:

nllfin*

nl2 nll*

n2lnll*n22n,,*

njjn12* nlin2l* njjn22*f l n f l , 2 f l h l f l ~ f l n fl.

svd) n12un/Ua il- (UaUb) n2 1/u, ( u)

n2l /Ua Ub

n22/Ua

nl12* n2 l* n22*

Ub ub Ub /(5.2)

The variables un and ub are arbitrary complex scalars, anduub is the real singular value. Thus any Mueller matrix

M for a nondepolarizing device also determines the corre-sponding Jones matrix N uniquely except for a phase fac-tor. The larger number of degrees of freedom for M cancontain effects such as depolarization. If one takes thevectors associated with only the largest singular value inthe decomposition equation (5.2), one obtains the Jonesmatrix nearest to the given Mueller matrix from the theo-rem from Appendix B.

The effect of isotropic depolarization is represented bythe following diagonal Mueller matrix:

M1

0

1 - d0

1 - d1 - d

where d is the real depolarization factor, 0 ' d - 1.Depolarization does not have a corresponding propercongruency-transformation operator, but it can be definedby a different nonlinear operator D[J(sk)], which affectsonly the eigenvalues of the coherency matrix J(sk):

J(Sk') = D[J(Sk)]

1 Ik + (1 - d)Pk2 0

0 Xk'1

Ik - (1 - d)Pk

(5.3a)

where5. EXTENSIONS FOR ISOTROPICDEPOLARIZERS

The Mueller matrix for a nondepolarizing device can beobtained from the Jones matrix for a device directly33 :

M = C(N ) N*)C-1

J(sk) = 2 Xk + PkIk - Pk

(5.3b)

Now we use the notation Ik to represent the Stokes total in-tensity s0* to avoid overly complicated indices; Pk2 = s12 +

S2k2

+ S3k2 defines the polarized intensity for the Stokes


(5.1)


state Sk; Xk is the matrix of ordered eigenvectors for J(sk).The serial effect of depolarization Da, followed by theJones-matrix congruency transformation N, followed bythe second depolarization Db, is defined by the serialoperator:

J(sk') = Db[ND.[J(sk)]N t]. (5.4)

The depolarization operator Db has the inverse:

Once the depolarization factors are known, the Jonesmatrix N that transforms depolarization-correctedcoherency matrices for the relation Db-l[J(sk')] =

NDa[J(Sk)]Nt can be obtained by applying the congruency-transformation recovery formula, Eqs. (3.10) and (3.11).Thus three independent polarization experiments withthe Stokes vectors Sk and Sk', k = 1, 3, are usually suffi-cient for determination of the Mueller matrix M for theisotropic depolarizer, which is defined by the product

M 1 - db1

0.1

x 1- da

0

Db'[J(Sk')]

1 [Ik' + Pk/(l - db) 0 12 L o Ik - Pk/(l - db) J

where db is the depolarization factor for Db. If the deter-minants from both sides of Db-'[J(sk')] = ND[J(Sk)]N t

are related, one obtains

,k2

Ik 2 p d 2 = Idet N 2(1k2-(1 - da)2Pk2), (5.5)

where d,, is the depolarization factor for the operator Dn.Three relations, Eq. (5.5) for k = 1, 2,3, for nonsingular

J(sk) and J(sk') determine both depolarization factors daand db, as well as the determinant det NI:

1- 2 (I32I212 - 22I3 2)P1 ' 2 + (12I312 - 3

2 11'2 )P2'2 + (221112 - 2I ,2)p312(1 a)2 (p2

2 p32

- p32 p2' 2)1l' 2 + (p 3

2 p1'2 - p 12p 3 '2 )12 '2 + (p 2 p2

2-P22PI )I3

(1 - d )2 - [I12 - (1 - d)2P 12]P2

2 [22 _ (1 -d)2p22]pl,2[I12 - (1 - da)2Pi 2 ]I2' 2

- [22 - (1 -d)2p22]lt2

In practice a number of singular conditions exist for theformulas; these conditions correspond to zeros in the de-nominators. For example, all three input states must notbe pure, but the formulas usually give a good solutionwhen at least one input state is partially polarized. It isnot possible to average different Stokes input and outputstates as with the congruency transformation, becausethe transformation for isotropic depolarization, Eq. (5.4),is not linear in the Stokes vectors. However, it is possibleto average orthogonal polarization states, proportional in(Si, S2, S3), since these have the same eigenvector matricesX. Thus having measurements for the two orthogonalpurely polarized states is equivalent to having measure-ments for the partially polarized intermediate states, too.With more than three input states, Eq. (5.5), with k count-ing the input state, gives an overdetermined linear systemin (1 - d) 2, 1/(1 - db)2 , and in Idet N 2, which can be op-timally solved by using the pseudoinverse.

The case in which the Jones matrix N is of unitary typemust be treated separately. The depolarization operatorD[J(s)] modifies the eigenvalues of J(s) only, and the uni-tary N does not change the eigenvalues of J(s), and soD[NJ(s)N t] = ND[J(s)]N t in this case. The depolariza-tion operator, D[], commutes with the unitary N. Thedepolarizations in the transformation, Eq. (5.4), combineto give one depolarization d, where (1 - d) = (1 - da)(1 - db), leaving da and db not determined independently.Therefore, for the case of unitary N, we can set da = 0and determine the combined depolarization degree withonly the second relation, Eq. (5.6).

0

-db [C(N ®9 N*)C-1]1 - db

0

1 - da

1 - da

(5.7)

With more than three polarization samples the recoveryformula can be applied to three linear combinations ofdepolarization-corrected states as above. The formulas,Eqs. (5.6), are verified in numerical tests such as those il-lustrated in Fig. 7. Even though the solutions appear to berather good, the ambiguity. in the solution for a unitary N

(5.6)

is difficult to distinguish in an automatic algorithm. Thiscase becomes a problem when there is added noise in themeasurement of a device that may be almost unitary.

6. PRACTICAL EXAMPLE

It is now well established that the two Mueller matricesmeasured by Howell34 for his collimator and for hisintensity-measuring radiometer are not realizable, as theyviolate the light-state condition so 2 -_ s2 + 2'2 + 312 forpossible input states.3 4 The congruency transformationof the coherency matrix must always give a realizableStokes transformation. The isotropic depolarization op-erator gives a realizable Stokes transformation, if thedepolarization is greater than zero. Thus it is interestingto consider Howell's measurements as examples of the for-malism presented here.

Howell presents the following purely polarized input tohis collimator:

0.5

0.5S = )

0

0.5

0

0.5

0

0.5

-0.5Sb= 0

0

0.5

0Sd =

0.5

(6.1)



4

3

2

1

I..u01

X WL.n[0

0aEo

0

4

3

2

1

0~

-5.5 -5 -4.5 -4 -a5sIogenoise)

-3 -2.5 -2 -1.5 -1

Fig. 7. Noise with recovery of the isotropic depolarizer Mueller matrix from three Stokes states. Numerical experiments were per-formed with three input independent Stokes states (two independent linear and one unpolarized) on Mueller matrices with depolarization[see Eq. (5.7)]. The rms error in the elements of the inferred Mueller matrix is plotted as a function of the rms of the added measurementnoise. The upper panel shows the results for depolarizing (unitary) retarders defined with one random depolarization in the range 0 to0.1. For the lower panel random matrices with two depolarizations each in range 0 to 0.1 were tested. A filled square denotes the aver-age of the log of the rms; dashed curves are the ±1 standard deviation in the log of the rms, and dotted curves the worst and the best casesfrom 20 Mueller matrices tested.

From these input states he obtained the following partiallypolarized output states:

0.3993

Sa 0.39010.0604

-0.0152

0.4391

0.02050.4286

0.0309

0.4496

-0.4404Sb -0.0310 ,

0.0769

0.4553

, = -0.0712Sd - 0.0147

0.4282

output Stokes states or linear combinations gives Jonesmatrices for Howell's collimator. All combinations givesimilar results within 5% rms accuracy per measuredvalue. The determination that excludes the pair (Sb, Sb')

gives the most self-consistent solution with an accuracy of3.8%. With a Jones matrix derived in this way as an ini-tial guess, the least-squares solution from the procedurein Appendix C has an rms accuracy of 1.9% per measuredvalue. That Jones matrix is written as a seven-parameterphysical description with the decomposition formula,Eq. (4.1):

a = 0.956, a_ = 0.872, p = 2.27°,

He then sent these states into his radiometer and obtainedas output

0.3488

,, _ 0.2057-0.0665-0.0464

0.3947-0.1192

0.2889

0.0560

0.4111

-0.26300.1049

0.1704

0.4392

SI _ -0.1218Sd" = 0.0333

0.3924

Applying the congruency-transformation recovery for-mula given in Eqs. (3.10) and (3.11) to any three input and

*1-0.396

S+ = 0.208

0.894

1

0.597

= -0.643-0.478

(6.4)

The normalized eigenvectors s+ are expressed as pureStokes states. A small retardation was characteristic ofall these solutions. The near identity of the system is ap-parent with only a 9% difference in the anisotropic absorp-tions a+.

Applying the recovery formula, Eqs. (3.10) and (3.11),to various combinations of three measurements or lin-ear combinations gives consistent Jones matrices forHowell's radiometer with a 5-10% rms accuracy. Theleast-squares procedure gives a solution with 4.3% rms

- retarders

.. ,,,. UiUr' ............. U..........U.....U...i....i.I.I- … ~~.-. … .

I I - arbitrary matrices

-ss,/>_,-ssa a s, a_ a _ > a ~ ",s . %% -#% ..... . - .;...-.. / mp

. - - %. - - . _ _/

iUUUU i.U U U..* **I_ _U _ _U _ .U



accuracy per measured value with the following physicalparameters:

a = 0.927, a_ = 0.789, p = -3.21°,

1

0.048s+ = 0.540

0.840

1

0.799s_ = 0.579

-0.161

(6.5)

The radiometer, too, has a matrix that is close to theidentity matrix with small retardation and anisotropicabsorption.

Depolarization effects can be considered with Eqs. (5.6).The computation tends to be somewhat unstable withnoisy data applied to the near-unitary device matrices.However, for the radiometer, if one fixes da = 0 and com-

0.6 l0

040.2 \\

.0

-o

putes db alone, one obtains a value that is possible. Apply-ing the recovery formula to the depolarization correctedStokes states with the least-squares procedure gives thefollowing solution for the radiometer, which has a 4.4% rmsaccuracy:

db = 0.139, a, = 0.968,

a = 0.793, p = -3.06°,

1.0

-0.015

S+= 0.529

0.848i

1.0

0.846

- 0.483-0.225

(6.6)

The solution is quite similar to the one without the de-polarization correction, Eqs. (6.5), although it contains asubstantial depolarization degree.

daFig. 8. rms accuracy for Howell's collimator as a function of depolarization. For every value of the depolarizations da and db the rmsaccuracy per measurement is shown for the optimum Jones matrix. The Jones matrix is derived from the congruency-transformationrecovery formula improved with the least-squares iteration procedure applied to Howell's Stokes measurements; sk and sk', Eqs. (6.1) and(6.2), corrected for depolarization. The contour interval is 0.01, and the minimum value is 0.011, which occurs at da = 0.5, db = -0.9.However, only solutions in the upper-right-hand quadrant are physically realizable.



.0

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8d

Fig. 9. Rms accuracy for Howell's radiometer as a function of depolarization. The rms accuracy per measurement is shown for the opti-mum Jones matrix as a function of the depolarizations da and db. In this case the congruency-transformation recovery formula improvedby the least-squares iteration procedure was applied to Howell's Stokes measurements: Sk' and Sk", Eqs. (6.2) and (6.3). The contourinterval is 0.01 and the minimum value is 0.0176 at da = 0.53, db = -0.9.

All these solutions, Eqs. (6.4)-(6.6), are physically real-izable, but the relatively noisy and small sample makesthe solution somewhat ill determined and not criticallydifferent from physically unrealizable solutions. Thisproperty is illustrated in Figs. 8 and 9, which show therms accuracy per measured value in the Jones matrixderived from the measurements corrected for the specifieddepolarizations da and db. For the collimator, Fig. 8, theoptimum solution has a 1.1% rms per measured value butat the physically unrealizable values da = 0.50 and db =

-0.90. The best physically realizable solution, da 2 0 anddb 2 0, occurs at da = 0.03 and db = 0, with rms 1.9%.The physical parameters for that solution are essentiallyunchanged from the solution given by Eqs. (6.4) for da =db = 0. For the radiometer from Fig. 9 the minimum rmsis 1.8% at the physically unrealizable da = 0.53 and db =

-0.90. The best physically realizable solution occurs atda = 0.20 and db = 0, where the rms is 3.6%. That solu-

tion has the parameterization

da = 0.325, db = O,

a+ = 0.963, a_ = 0.911, p = -6.24°,

1.0

0.399S+=0.389

0.830

1.0

0.429S_= 0.897

0.105

(6.7)

The solution exhibits nearly isotropic absorption andlarger retardation, but the axes are far from oppositepolarization states, as we would expect for a simple physi-cal device.

Negative depolarization is one source of difficulty inMueller matrices defined from a standard procedure, i.e.,Eq. (1.1). Mueller matrices defined by Eq. (5.7) for the col-limator solution given by Eqs. (6.4), or with the small de-



polarizations d, = 0.03, db = 0, are both within 4.9% rmsper matrix element of Howell's Eq. (4). (A typographicalerror in his paper has been corrected: the matrix ele-ment M31, which reads as 0.913, should have been 0.0913.)With a small unphysical depolarization, d negative, theagreement improves a little. With the large unphysicaldepolarization da = 0.5 and db = -0.9, the solutions differby 17.5%. The radiometer solutions given by Eqs. (6.5)and (6.6), or solutions with depolarizations da = 0.20 anddb = 0, give Mueller matrices that differ from Howell'sby =14.6%.

7. CONCLUSION

Congruency transformations are the natural form fortransformations of the observables in quantum-mechanicalsystems. The set of quantum observables is contained inthe time-averaged mutual coherence, or Hermitian co-herency matrix Jn = ((t) 0 6(t)t)t, for an n-element wavefunction e(t) that is a function of time t. In general thewave function (t) consists of n independent complexvector elements, representing coherent electromagneticfluctuations, each defined as the Hilbert analytic signal ofa real electric-field temporal function. The coherencymatrix contains the independent time averages of wave-function product pairs as n2 real observables. It is decom-posed into the eigenvalue matrix form

'So + Pi

Jn =1Xn

0

So + P2

0

x- 1 ,

So + Pa

(7.1)

where so is the parameter that multiplies the identity inJn. The n eigenvalues are the real positive measurableintensities o + Pk, with Pk signed metric-type functionsof the parameters. The eigenvectors in the matrix X areorthogonal.

A linear transformation of the order-n wave functionby the operator N, e(t) = N-, can be expanded by singular-value decomposition:

N = U

0

r2 . V-1,

rnj

The similarity transformation by the unitary V- on Jnmodifies the eigenvector matrix X, which preserves theorthogonality of the eigenvectors without affecting theeigenvalues of Jn. The transformation represents ann-vector rotation in the (n2 - l)-parameter space definedby all the parameters Of Jn except so. The rotation by V-leaves the parameter so unchanged. The multiplicationby the diagonal matrices represents anisotropic scaling inn particular orthogonal vectors in the parameter space.The U similarity transformation is a second rotation.

Therefore linear transformations of a wave function oforder n naturally lead to geometric transformations in an(n2 - l)-parameter space of the corresponding quantumobservables without referring to spatial characteristicsfor the underlying wave function. The higher-order solu-tions have properties that are qualitatively similar to thesecond-order solution, which leads to rotations in thethree-dimensional Poincar6 sphere space.

The eigenvector constraint leads to recovery formulasthat provide a practical tool in polarimetry. It is possibleto determine the Jones matrix for an unknown device,given three (and in one case two) independent generalpartially polarized sources as input, with a precise deter-mination of the Stokes parameters for the correspondingoutput light states. The Jones-matrix solution may bepreferable to the standard Mueller representation becauseit is constrained by fewer measurements and gives anexact correspondence to a complete physical parameteriza-tion in the coherent-light-transfer process. I have shownthat the method is stable with added noise of any ampli-tude and can be used to provide an initial guess for a least-squares iteration procedure to be used to optimize thesolution for a particular measure of error.

Depolarization effects are not a part of the coherenttransformation algebra defined by linear operations onthe light-state wave function. However, isotropic depolar-ization can be added to the transformation formalism asan additive operator that depends on the eigenvalues ofthe light-state coherency matrix. Thus isotropic depolar-ization preserves the coherency matrix form, Eq. (7.1). Ageneralization to depolarization that is not isotropic wouldintroduce a nonlinear dependence of the depolarizationoperator on the eigenvectors X.

(7.2)

where U and V are unitary matrices and rl, r 2 , ... rn arepositive real scalars. A linear transformation of the wavefunction has a corresponding congruency transformationof the quantum observables Jn = NJnNt and is expandedas follows:

0

r2 v-lx

rnj

1il= U

n0

So + Pi

x

0

So + P2

0 r

X-1 v r 2

Pa 0

0

rnj

APPENDIX A: COMPLEX SINGULAR-VALUEDECOMPOSITION

Any matrix A can be written as the product of aHermitian matrix times a unitary matrix. Since the ei-genvalues of the Hermitian matrix are real, and its matrixof eigenvectors unitary, a square matrix A of order n canalways be decomposed into the following form:

A r Vt X At0

A=U Vt (Al)

0

U-. where rj are real and U and V are complex unitary matri-ces.910 Equation (Al) is the svd of A.

There are numerical recipes that determine the real-(7.3) matrix svd. 5 To generalize to the complex case, one can



write the complex product a,3 = y in a 2form:

X 2 real-matrix

a,. as [ b, bi L Cr Ci]

L-ai a L-bi b L-Ci CS

matrix of product pairs:

am, l t62 ... a 1,6

a 2f,1 a2j3 2 ... a2f8nW = + ,an anf2 * an/3E

af ,{31 a.,8 2 ...- ani3n

where

a = a, + iai,

/3 = b, + ibi,

y = c, + ii

for the real components a,, a, b, bi, c., and ci. Thus thecomplex n X n svd, Eq. (Al), has the equivalent 2n X 2nreal svd form, written as follows:

where E is a matrix of random elements all small com-pared to any akf8l-

The svd of W is written in general as

'risvd r

W ='U 200

0

Vt

rn]

(B2)

for unitary U and V and r > r2 > ... rn. A measure forW to be a matrix of product pairs is x2 , defined as follows:

E all, alli1-aii a,

[Ullr~lli] ... Fl0= -mU11i Ur L r,

0

U1,,. Vi,~~ .x L 1i 11

(B3)x = E IWkm - ruklvml*I Xk,m

0

(A2)

where W = (wkm) and the vectors (ull, u 21 , ... Unl)T and(v ,v21, ... ,Vni)T are the first columns of U and V, respec-tively. With no noise, E = 0, the zero matrix, and ak =

ukl/ra,3r = Vlm*/rb for arbitrary complex scalars rin and rbsuch that rairb = rl; in this case X2 = 0.

With noise in the matrix W E in Eq. (Bi) is not zero, andthe measure of error X2 will not be zero. The derivativeof X2 with respect to each conjugate, Ukl* can be written invector form as follows:

Numerical methods with repeated singular values tendto converge poorly. We observe that the matrix product

a, + e a br + bi 1 Crl Cil[ -ai a, - -bi b, - eJ [-ci 2 Cr2J

gives an approximation for the complex y accurate to e2 :

'= 1/2 (r + Cr2) + (i/2)(cil - Ci2) = y + 0(E2). (A3)

When all the complex elements of A are written as real2 X 2 matrices with the small perturbations ±e in its twodiagonal elements, we obtain a form for the svd with dis-tinct singular values. The modified form gives an ap-proximation for the complex matrices U, V, and thesingular values rj, with an error of order e2. The methodgeneralizes without difficulty to nonsquare complex ma-trices, following standard svd definitions.2' 0

APPENDIX B: OPTIMUM DECOMPOSITIONOF MATRICES OF PRODUCT PAIRS

The svd of a matrix of product pairs W gives a determina-tion of the product coefficients, as I described in Section 2.If there is small noise in the matrix of product pairs, thenit can be shown that the coefficient vectors associatedwith the largest singular value are also optimum solutionsfor equal weighting of the matrix elements. We define the

ax2/auu,* u 1 1

ax2

/aU 2 1* U21= -ri(W -r 0 ("'

aX2/aUnl* Unl

Vl1

x V21

Vn1

V12 ... Vnl*)

(B4)

The coefficient vector (1, V22, ... Vnl)T, being the firstcolumn of the unitary V, satisfies the relation:

Vl ' 01Vt V = . V1

Vn1J OJ

Vi)

V21* . Vn1i*) V2

where the vector product in parenthesis on the right-handside is a scalar. Thus, for W defined in Eq. (B2),

Vi 1

Vn1J

V21*Un1= r . X9 (Vll*

kUnl

Vil

U n1

With no Uk1 zero,

U21 (V* - * V21W - 9V1*V 21'* .. Vnl ) = 0. 5Un Vn1

(Bi)



Therefore from Eq. (B4) we see that ax 2 /auk1* = 0, orthe complex-conjugate relation, aX2 /aukl = 0. This prop-erty for the derivatives of x2 with respect to each coeffi-cient, Vl, can be demonstrated similarly, thereby provingthat the svd solution is a minimum in the error X2 definedin Eq. (B3).

APPENDIX C: CONGRUENCY-TRANSFORM-MATRIX OPTIMIZATION

With a close approximation for the congruency-transformation matrix operator, a linearized estimatorcan optimize that solution for a particular measure oferror. Such optimization procedures are useful in realapplications in which there is noise or the system is over-sampled. We assume that we are given a square matrix Nof order n that is close to the optimum congruency-transformation matrix operator, given the set of m inputand output matrices Ak and Ak', k = 1, m. We define X2

as a measure of accuracy for N:

1 mX 2nM Z Ak' - NAkNtIJ, (Cl)

2n k-(

where the modulus squared of a matrix, 1 ,2 denotes thesum over all the squared real components. Small pertur-bations in the elements of the matrix N change x2. Wetake D as an n X n matrix of small perturbations; then,to first order in the perturbation, we have

X2 N+D = E A'- NAkNt- DAkNt- NAD t I,2 .2n m k-1

(C2)

It is convenient to define the operator vec(D) to repre-sent the n2 -element vertical column vector containing theelements of a matrix D:

did2l

vec

Ldnl

The following identities for general n X n matrices can beproved for vec and 0:

vec(DX) = (XT l)vec(D),

vec(XD) = (1 0 X)vec(D),

vec(D T) = ((i-)/nJ+(i-)%n+)vec(D)

(W Y)(X Z) = (WX) 0 (YZ), (C3)

where 1 is the order-n identity matrix, denotes then2 X n2 matrix containing the Kronecker delta in itselements, L J is the integer floor, and a%b denotes a modu-lar b.

We can express x2, Eq. (C2), in the form of a vector dot

product with the vec() operator:

1 In

X |N+D = [ek + vec(DAkNt) + vec(NAkD t )] t2 nm k-+

x [ek, + vec(DAkNt) + vec(NA,,D)], (C4)

where

ek = vec(A' - NANt).

Applying vec() identities gives equivalently:

XI21 N+D =nm- E [ekt + vec(D)tBkt + vec(D) T Ckt ]n + k+1

x [ek, + Bk vec(D) + Ck vec(D)*], (C5)

where

Bk = [(N*AkT) 0 1],

Ck = [1 (0 (NAk)][8i,(i-l)/nJ+(i-l)%n+l].

X is a local minimum when ax2 /adij = 0 and when theindependent conjugate condition aX2/adij* = 0 are bothsatisfied for every matrix element dij of D. We can writethe derivative ax 2/adij* from Eq. (C5) in the following par-ticular way:

ax2

adij N+D

=2 {ad( Bk'[ek + Bk vec(D) + Ck vec(D)*]

+ a vec(D) Ck[ek*+ Bk* vec(D)* + Ck* vec(D)]}

The product is a complex scalar because the horizontalvector on the left multiplies a vertical column vector onthe right in each term. I have written Eq. (C6) so thateach vector on the left is zero in all its elements exceptthe one corresponding to the dij matrix element, where itis one. Thus the vertical column vector of partial deriva-tives can be written simply as

veefdX)v 2n +N+D

2nE f Bk'[ek + Bk vec(D) + Ck, vec(D)*]

+ CkT[ek* + Bk* vec(D)* + Ck* vec(D)]}.

vec(D) and vec(D)* can be factored out of the sum:

r + P vec(D) + Q vec(D)* = 0,

where

(C6)

mr = E (Bktek + Ckrek*),k=1

P = E (Bk Bk + CkTCk*),k-l

Q = E (Bk Ck + CknBk ).k=1

Then the simultaneous equations a 2/adij = 0 and ax2/adij* = 0 can be solved explicitly:

vec(D) = (Q* P*Q 4'P)i'(P*Q-r - r). (C7)

The improved solution is N' = N + D. Repeated applica-tion of Eq. (C7) converges quadratically to a matrix N thatminimizes X2 in Eq. (Cl).


d12 ... din'd22 ... d2n

dn2 ... dnn


ACKNOWLEDGMENTS

The svd routine for real matrices, given by Press et al.,35 isthe basis for the complex svd estimation defined inAppendix A. Many of the algebraic derivations wereperformed with the help of, and verified by using, MATHE-MATICA by Wolfram.36

The National Optical Astronomy Observatories are op-erated by the Association of Universities for Research inAstronomy, Inc., under cooperative agreement with theNational Science Foundation.

REFERENCES

1. G. G. Stokes, "On the composition and resolution of streamsof polarized light from different sources," Trans. CambridgePhilos. Soc. 9, 399-416 (1852).

2. R. M. A. Azzam, I. J. Elminyawi, and A. M. El-Saba, "Gen-eral analysis and optimization of the four-detector photo-polarimeter," J. Opt. Soc. Am. A 5, 681-689 (1988).

3. S. R. Cloude, "Conditions for the physical realisability ofmatrix operators in polarimetry," in Polarization Considera-tions for Optical Systems II, R. L. Caswell, ed., Proc. Soc.Photo-Opt. Instrum. Eng. 1166, 177-185 (1989).

4. M. Sanjay Kumar and R. Simon, "Characterization of Muellermatrices in polarization optics," Opt. Commun. 88, 464-470(1992).

5. R. C. Jones, 'A new calculus for the treatment of optical sys-tems I. description and discussion of the calculus," J. Opt.Soc. Am. 31, 488-503 (1941).

6. N. Wiener, "Generalized harmonic analysis," Acta Math. 55,117-258 (1930).

7. E. Wolf, "Two-beam interference with partially coherentlight," J. Opt. Soc. Am. 47, 895-902 (1957).

8. G. B. Parrent, Jr., "On the propagation of mutual coherence,"J. Opt. Soc. Am. 49, 787-793 (1959).

9. G. B. Parrent, Jr., and P. Roman, "On the matrix formulationof the theory of partial polarization in terms of observables,"Nuovo Cimento 15, 370-388 (1960).

10. R. Barakat, "Theory of the coherency matrix for light of arbi-trary spectral bandwidth," J. Opt. Soc. Am. 53, 317-323(1963).

11. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon,Oxford, 1970).

12. R. C. Jones, 'A new calculus for the treatment of opticalsystems VI. Experimental determination of the matrix,"J. Opt. Soc. Am. 37, 110-112 (1947).

13. R. M. A. Azzam and N. M. Bashara, "Polarization transferfunction of an optical system as a bilinear transformation,"J. Opt. Soc. Am. 62, 222-229 (1972).

14. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polar-ized Light (North-Holland, Amsterdam, 1977).

15. H. Poincar6, La Lumijre, Cours de Physique Math6matique

(1892), Chap. XII, "Polarisation rotatoire-theorie de M.Mallard," pp. 275-301.

16. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading,Mass., 1950).

17. H. Takenaka, 'A unified formalism for polarization optics byusing group theory," Nouv. Rev. Opt. 4, 37-41 (1973).

18. S. R. Cloude, "Group theory and polarization algebra," Optik75, 26-36 (1986).

19. S. R. Chin, "Differential coupled mode analysis and thePoincare sphere," Appl. Opt. 28, 1661-1665 (1989).

20. R. Barakat, "Conditions for the physical realizability of polar-ization matrices characterizing passive systems," J. Mod.Opt. 34, 1535-1544 (1987).

21. F. R. Gantmacher, The Theory of Matrices (Chelsea,New York, 1959).

22. A. S. Householder, The Theory of Matrices in NumericalAnalysis (Dover, New York, 1964).

23. G. Strang, Linear Algebra and Its Applications (Academic,New York, 1980).

24. C. A. Hollingsworth, W E. Kaiser, and W T. Granquist,"Theory of measurement of flow birefringence by use of thesenarmont compensator. II. Effect of quarter-wave plateerror and polychromatism," J. Opt. Soc. Am. 54, 633-637(1964).

25. E. Schmidt, "Precision of ellipsometer Measurement," J. Opt.Soc. Am. 60, 490-494 (1970).

26. R. M. A. Azzam and N. M. Bashara, "Unified analysis of el-lipsometry errors due to imperfect components, cell-windowbirefringence, and incorrect azimuth angles," J. Opt. Soc.Am. 61, 600-607 (1971).

27. D. E. Aspnes, "Measurement and correction of first-ordererrors in ellipsometry," J. Opt. Soc. Am. 61, 1077-1085 (1971).

28. M. A. Thiel, "Error calculation of polarization measure-ments," J. Opt. Soc. Am. 66, 65-67 (1976).

29. P. S. Hauge, "Mueller matrix ellipsometry with imperfectcompensators," J. Opt. Soc. Am. 68, 1519-1528 (1978).

30. L. J. November, "Exploiting spatial transformations of thelight state for precise polarimetry," in Polarization Analysisand Measurement, R. A. Chipman and D. H. Goldstein, eds.,Proc. Soc. Photo-Opt. Instrum. Eng. 1746, 76-87 (1992).

31. R. A. Chipman, "Polarization analysis of optical systems, II,"in Polarization Considerations for Optical Systems II, R. L.Caswell, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 79-94 (1989).

32. L. J. November, "Determination of the Jones matrix for theSacramento peak vacuum tower telescope," Opt. Eng. 28,107-113 (1989).

33. R. Simon, "Mueller matrices and depolarization criteria,"J. Mod. Opt. 34, 569-575 (1987).

34. B. J. Howell, "Measurement of the polarization effects on aninstrument using partially polarized light," Appl. Opt. 18,809-812 (1979).

35. W H. Press, B. P. Flannery, S. A. Teukolsky, and W T. Vetter-ling, Numerical Recipes: The Art of Scientific Computing(Cambridge U. Press, Cambridge, 1986).

36. S. Wolfram, Mathematica, A System for Doing Mathematicsby Computer, 2nd ed. (Addison-Wesley, New York, 1988).


Documents

Recovery of the matrix operators in the similarity and congruency transformations: applications in polarimetry