A comprehensive polarimetric characterization of asample is usually represented by the sample’s 4 3 4eal Mueller matrix. This matrix describes the mod-fication of polarization of light after interaction withsample in the general linear case. Indeed, the use
f the Mueller-matrix formalism becomes necessaryhenever the sample is partially or totally depolar-
zing, as in cases of rough surfaces and scatteringedia. Mueller-matrix ellipsometry can be consid-
red an extension of the classic and generalized el-ipsometry that is restricted to nondepolarizingamples.It is important, when one is using a Mueller-matrix
llipsometer ~MME!, to calibrate accurately the twoptical arms that consist of a complete polarization-
When this research was performed, the authors were with theLaboratoire de Physique des Interfaces et des Couches Minces,Unite Mixte de Recherche du Centre National de la RechercheScientifique, Ecole Polytechnique, 91128 Palaiseau, France. S.Poirier is now with Lore S. A., 11 boulevard Pershing, 75017, Paris,France.
Received 20 October 1998; revised manuscript received 22 Feb-ruary 1999.
3490 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999
state generator ~PSG! and a complete polarimeter.As defined in Ref. 1, “complete” means that the PSGand the polarimeter can generate and measure thefour states of polarization of light.
Usually the calibration of a polarimetric tool is atwo-step process. First, one precisely orients theoptical elements, a procedure that requires high-precision devices together with careful alignment.In a second stage, the behavior of the optical elementsmust conveniently be described and characterized byphysical properties. The behavior can be distortedby first-order failures, as in the case of a photoelasticmodulator.2 Thus one has to combine several mea-surements in different configurations to determinethe main values and the shortcomings. Those mea-surements are adapted to each optical setup. Theseaspects of the calibration process have already beenreviewed in the literature.3–5 Furthermore, one cannotice that usual calibration procedures are difficultto implement in situ because of the influence of theoptical elements that are necessarily included in thelight path ~filters, windows, lenses, mirrors, etc.!.
ll these elements can induce modifications of thetate of polarization of light.The eigenvalue calibration method ~ECM! consists
rst in extending the use of the matrix formalism tohe global experimental setup to be calibrated.
tmEMfef
sbAtnMppscurnrtstdtE
i
Three characteristic 4 3 4 real matrices are naturallyintroduced: W for the entrance arm ~PSG!, A for theexit arm ~polarimeter!, and M, the Mueller matrix ofthe sample. Then a basic measurement is the ma-trix product AMW, i.e., 16 coefficients. The ECMakes advantage of this matrix representation inaking extensive use of linear algebra. With theCM the entire calibration of the 32 coefficients of aME ~A and W! can be performed with only three or
our measurements of reference samples. The ref-rence samples used here are smooth, isotropic sur-aces and linear polarizers.
The ECM has three advantages: First, no as-umption has to be made about the system that iseing calibrated, except that it must be complete.ll the optical elements are automatically included in
he matrix representation, even if their behavior isot completely described; this enables the existingME to be simplified because the orientations and
ositions of the various elements do not have to berecisely adjusted. For example, the presence of atressed window will automatically be taken into ac-ount when the ECM is used. The procedure is alsoniversal because it depends only on the choice of theeference samples; first-order approximations are notecessary. Second, the few characteristics of theeference samples are completely determined duringhe calibration, without the need for subsidiary mea-urements: the Fresnel reflection coefficients andhe polarizer orientation and transmission factors areeduced from the eigenvalues of the measured ma-rices. If a spectroscopic MME is calibrated by theCM, W and A will depend on the wavelength l, and
the characteristics of the reference samples will bedetermined for each wavelength in the same manner.Third, the accuracy of the calibration procedure canbe precisely evaluated when the ECM is used, asdescribed below.
After a brief description of the Mueller-matrix for-malism we review the ways in which the various MMEconfigurations can easily be described with a matrixformalism. Then we describe the ECM in detail, be-ginning with the simplest configuration, in which theMME can produce measurements in a reflection-and-transmission configuration. The other practical situ-ation, in which the MME can produce measurementsonly in a reflection configuration for a limited numberof angles of incidence, is deduced directly from the firstcase. In the last part of the paper we display exper-imental results obtained during the calibration of amultiwavelength MME consisting of a coupled phasemodulator6 associated with a prism division-of-amplitude polarimeter7 at 458 and 633 nm.
2. Matrix Formalism for Polarimeters,Polarization-State Generators, and Mueller-MatrixEllipsometers
A. Mueller Matrices: Reference Samples
Throughout this paper the more general state of po-larization of a light beam is represented by its Stokesvector8–10 S 5 T~I, Q, U, V!, where T is the transpo-
sition operator. The linear interaction with a sam-ple is described by the four-by-four real Muellermatrix of the sample. As a consequence of the lin-earity, the overall matrix M of an optical system isobtained from the matrix product of its componentsMi:
M 5 MnMn21 . . . M2M1. (1)
Only two types of Mueller matrix that have per-fectly known theoretical forms are used in this paper;they are called the reference samples. The first onesare the linear polarizer matrices P~t, u!, where u des-gnates the orientation of the polarizer and t is the
transmission coefficient:
P~t, 0! 5t
2 31 1 0 01 1 0 00 0 0 00 0 0 0
4 ,
P~t, py4! 5t
2 31 0 1 00 0 0 01 0 1 00 0 0 0
4 . (2)
The Mueller matrices with other orientations areobtained by use of the rotation
P~t, u! 5 U~u!P~t, 0!U~2u!, (3)
where
U~u! 5 31 0 0 00 cos 2u 2sin 2u 00 sin 2u cos 2u 00 0 0 1
4 . (4)
In this paper, polarizer matrices need to be perfect.This requirement leads to the use of crystal polariz-ers ~Rochon, Glan, Wollaston, etc.! in the near-ultraviolet–near infrared range, which have anexcellent extinction ratio that is less than 1025. Di-chroic sheet polarizers for near-ultraviolet–near-infrared or grid polarizers for infrared cannot berepresented in this way with satisfactory accuracy.
The second class of theoretically well-known sys-tems corresponds to smooth isotropic samples. Thereflection on their surface can be decomposed into apartially polarizing effect ~C! and a phase-shiftingeffect ~D!, where C and D correspond to the usualellipsometric angles related to the Fresnel coeffi-cients @tan~C!exp~iD! 5 rpyrs in reflection#. The re-flection Mueller matrix R~t, C, D! of such a sample is
1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3491
2 2
f
ipsm
ri
Table 1. Composition of the Harmonic Vector for Four Modulation
H
3
@where t 5 0.5 ~urpu 1 ursu ! is the reflection coefficientor unpolarized light#
R~t, C, D! 5 t 3
31 2cos 2C 0 0
2cos 2C 1 0 00 0 sin 2C cos D sin 2C sin D0 0 2sin 2C sin D sin 2C cos D
4 .
(5)
B. Polarization-State Generator, Polarimeter, andMueller-Matrix Ellipsometer
To give a general presentation of the ECM, wepresent a short, nonexhaustive review of the varioustechniques used in the PSG–polarimeter setups.The objective is to get a unified formalism in whichfor all the setups the PSG and the polarimeter arerepresented by two characteristic 4 3 4 real matricesW and A. The calibration procedure will finally con-sist of determining their 2 3 16 coefficients.
1. Polarization-State GeneratorA PSG is a light source that is able to generate dif-ferent states of polarization. Only complete PSG’sthat can generate at least four independent states ofpolarization are considered in this paper. Their ma-trices W are necessarily invertible.
The simplest nonmodulated manual PSG consistsof various polarizing elements located successively ina light beam to generate a set of Stokes vectors.These Stokes vectors Si can be written as the productof the PSG matrix W and at least four input vectorsei:
Si 5 Wei. (6)
For example,11 using a polarizer successively at 0,45°, and 90° and a polarizer at 0° followed by aquarter-wave plate at 45° leads to ~where ti is thecoefficient of transmission!
W 512 3
t1 t1 t1 t2
t1 0 2t1 00 t1 0 00 0 0 t2
4 ,
e1 5 110002 , . . . , e4 5 1
00012 . (7)
In the case of a time-modulated PSG there is onlyone input vector, consisting of a set of harmonic func-tions:
S~t! 5 We~t!. (8)
Harmonic vector e~t! depends on the modulationtechnique.5,6,12–14 Some examples are given in Table1. A review paper1 describes most of these exam-ples.
492 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999
2. PolarimeterA polarimeter is dedicated to the measurement of anunknown Stokes vector. It can be described by acharacteristic matrix A that links the measuredntensity~ies! to the input Stokes vector. Only com-lete polarimeters, which are able to measure all thetates of polarization, are considered here. Theiratrices A are necessarily invertible.All the previous PSG setups can be used as pola-
imeters. In those cases, the direction of light isnverted: matrix A instead of W, output vectors f
instead of input vectors e, and a photodetector in-stead of a light source are used. The detected inten-sity is then related to Stokes vector S bymeasurement
Ii 5 TfiAS ~i 5 1, . . . 4!,
or
I~t! 5 Tf~t!AS. (9)
Another class of polarimeter that is different fromthose used in the former PSG techniques has beendescribed in the literature. It essentially includes afour-detector photopolarimeter15 and a division-of-amplitude polarimeter.7,16–18 These polarimetersuse four detectors instead of one, so four intensitiesI1–4 can be measured simultaneously. The Stokesvector is then directly related to these four intensitiesby
1I1
I2
I3
I4
2 5 AS. (10)
3. Mueller-Matrix EllipsometerA MME is composed of a complete PSG and a com-plete polarimeter. By combining measurements forthe sets of input and output vectors, the basic capac-ity of any MME is to measure the matrix product
Techniques
Low-frequency modulationFixed polarizer and rotating compensatora
e~t! 5 T~1, sin 2vt, cos 4vt, sin 4vt!Element axis, vt
Fixed polarizer and oscillating compensatorb
e~t! 5 T~1, sin vt, cos 2vt, sin 3vt!Element axis, c0 1 c1 sin~vt!
igh-frequency modulationDual phase modulatorc
e~t! 5 T@1, cos 2v2t, cos~v1 2 v2!t, sin~2v1 2 v2!t#Two phase modulations d1,2~t! 5 A1,2 sin v1,2t
Coupled phase modulatord
e~t! 5 T~sin vt, cos 2vt, sin 3vt, cos 4vt!One phase modulation d~t! 5 A sin~vt!
aRef. 16.bRef. 13.cRef. 5.dRef. 6 and 14.
o
w
ov
gntmobr
Matd
Mfh
ctcietid
s
pa
mt
wW
AMW, where M is the Mueller matrix of the sampleunder investigation. The experimental procedureand treatment of data depend on the PSG and thepolarimeter that are used. The simplest MME usesa nonmodulated PSG and polarimeter.19 AMW isbtained from the 16 intensities Iij associated with
input and output vectors i and j:
AMW 5 @Iij#i, j51. . .4, (11)
here
Iij 5 TfiAMWej.
With a modulated PSG and polarimeter AMW isbtained from the Fourier analysis of a single time-arying signal12:
I~t! 5 Tf~t!AMWe~t!. (12)
When a four-channel polarimeter is used, AMW isobtained from the measurement of four intensi-ties20,21:
1I1
I2
I3
I4
2~t! 5 AMWe~t!
or
1I1
I2
I3
I4
2i
5 AMWei. (13)
The configurations described above are of coursenonunique, and other arrangements can be proposedor found in the literature. The configurations thatuse a manual setup are often used to calibrate apolarimeter or a PSG, and two matrices are generallyinvolved, even if only one is needed for calibration.
Before going into the details of the calibration pro-cedure we point out that, although the matrices Aand W are most of the time 4 3 4 real matrices, theeneral form is n 3 4 for A and 4 3 m for W, whereand m are greater than 4. n . 4 means that more
han four input Stokes vectors are generated, and. 4 means that the output Stokes vectors are
verdetermined. For example, the 16-beam grating-ased division-of-amplitude polarimeter describedecently22 has a 4 3 16 characteristic matrix. In this
paper the indicated dimensions refer to 4 3 4 A andW matrices; nevertheless the ECM is fully compatiblewith other sizes, except that the dimensions of theintermediate matrices should be adapted.
3. Calibration Theory
We assume at first that the MME system providesthe measurement of the matrix product AMW for any
ueller matrix, including the identity matrix. It isssumed that the MME can work in both the reflec-ion and the direct-transmission configurations, asisplayed in Fig. 1. It is also assumed that the
ueller matrices M of the reference samples are per-ectly known. In what follows, we describe in detailow to deal with these limitations.The following conventions are used in this paper:
apital-letter matrices ~such as M and X! refer toheoretical matrices, whereas boldface italic lower-ase letters, such as ~amw!, correspond to the exper-mental measurements. Without experimentalrrors, the 4 3 4 measured matrix ~amw! is equal tohe matrix product AMW. If M is known, this equal-ty provides the basic information used by the ECM toetermine the matrices A and W.With respect to the matrix algebra, mathematical
upport can be found in many textbooks,23,24 and oneshould notice that the matrices A, W, and R~t, C, D!defined in Eq. ~5! are invertible, A and W because thePSG and the polarimeter are complete and R~t, C, D!because C is assumed to be different from zero and
y2. Consequently, their measured products ~aw!nd ~arw! are also invertible matrices.
A. Mathematical Construction
1. System of EquationsWe define a linear mapping HM from the set M4~R! ofthe real 4 3 4 matrices into itself. It depends on theparameter M, the Mueller matrix of a reference op-tical element, on ~amw!, the corresponding measure-
ent, and on ~aw!, the measurement in directransmission without samples:
HM:M4~R!3M4~R!,
X3MX 2 X~aw!21~amw!. (14)
HM has the property of containing W within its nullspace @HM~W! 5 0#, whatever the value of M, because
ithout experimental errors ~aw!21~amw! is equal to21MW. Moreover, inasmuch as W belongs to the
null space of all applications HM, it also belongs totheir intersection. Finally, a well-chosen set of ref-erence samples $M1, . . . , Mn%, as described in Sub-section 3.B below, can reduce the number of solutions
Fig. 1. Measurement configurations involved in the ECM. ~a!Direct transmission measurement without a sample: The corre-sponding measurement is aw. ~b! The reflection configuration:the corresponding measurement is amw.
1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3493
t
k
w
Mv
16
s
~tHd
ataP
mdmceqrEsr
3
to one: W is determined as being the unique solu-ion of
HM~X! 5 0, M [ $M1, . . . , Mn%. (15)
A is then obtained from the measured matrix product~aw!:
A 5 ~aw!W21. (16)
2. Solution of the SystemAssuming that an appropriate set of reference sam-ples $M% 5 $M1, . . . , Mn% has been chosen, it is usefulto adopt a matrix formulation to solve the linear sys-tem of Eq. ~15!. As a matter of fact, a 4 3 4 matrixX can be written as a vector X16 5 ~xi!i51
16 in anappropriate basis set $gi%i51. . .16 of M4~R! such thatthe 4 3 4 matrix X is equal to ¥i51. . .16 xi z gi. On thisbasis, the linear mapping HM is represented by a 16 316 real matrix HM, the 4 3 4 matrix HM~X! is equiv-alent to the product HMX16, and W, the solution of Eq.~15!, is represented by the vector W16 solution of
HMX16 5 0, M [ $M1, . . . , Mn%. (17)
The solution of the overdetermined linear system~17! by a least-squares method is given by the well-
nown relation24
KX16 5 0, (18)
here
K 5
TSHM1···HMn
HM1···HMn
D 5 THM1HM1 1 · · · 1 THMnHMn.
(19)
The matrix K is a positive symmetric real matrix,so it can be diagonalized. It has 15 non-null and 1null eigenvalue because Eq. ~18! has only one solu-tion, W16, which is the unique eigenvector associatedwith the null eigenvalue.
We have found the solution to the calibration prob-lem from a mathematical point of view but withouttaking into account the physical limitations: Prac-tically, the equality HM~W! 5 0 can never be exactlyverified. Because of the limited experimental preci-sion, W~aw!21~amw! is always slightly different from
W. This fact imparts peculiar properties to eigen-alues ~l1 . . . l16! of K. They are all different from
zero except l16, the smallest, which should theoreti-cally be null and which is close to zero in reality.One gets
K 5 TO3l1 0 · · · 0
· · ····
0 li
0···
· · ·0 · · · 0 l16
4O,
l1 . · · · . l15 .. l16 < 0, (20)
where O is the orthonormal matrix whose rows arethe eigenvectors of K. The null space of K is empty
494 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999
in practical circumstances, so the solution W is theeigenvector of K or the row of O associated with themallest eigenvalue:
W16 5 TO10···012 . (21)
Finally, W16 is written back as a 4 3 4 matrix Wand A is obtained with Eq. ~16! from the measuredmatrix product ~aw!. The values of W and A havebeen fully determined, apart from a multiplicationfactor a that is not important because W and A can bechanged into aW and a21A without consequences.
B. Choice of the Reference Samples Set $M%
1. General PropertiesIn practice, the samples used are restricted to thosewhose theoretical Mueller matrices have a perfectlywell-known structure. These matrices are functionsof intrinsic parameters such as the transmission orpartial polarizing coefficients defined in Eqs. ~2! and5!. Their choice is ruled by the necessity to reducehe intersection of the null spaces of the mappingsM, defined in Eq. ~14!, to get only one solution, whichetermines W.The first important and interesting property, dem-
onstrated in Appendix A, is that the choice of refer-ence samples is universal: It does not depend on Wnd A. This consequently emphasizes the utility ofhe present calibration method, which can addressny kind of technology within the limitation that theSG and the polarimeter be complete.In Appendix B we demonstrate that the number ofatrices that commute with the reference matrices
epends only on the multiplicity of eigenvalues of theatrices, as displayed in Eq. ~B2! below. A direct
onsequence of Eq. ~B2! is that using a single refer-nce sample yields at least four solutions. Conse-uently, to yield only one solution ~W!, the set ofeference samples must include at least two matrices.quation ~B2! and expressions ~B4! below demon-trate that using a reflection sample R~t, C, D! aloneestricts the number of solutions of Eq. ~15! to four
when C Þ 0 @py4# and D Þ 0 @p#. In the same man-ner, a consequence of Eq. ~B2! and expressions ~B3! isthat using a polarizer alone yields ten solutions. Fi-nally, one can verify that the combination of a polar-izer ~with u Þ 0 @py2#! and a reflection sample ~withC Þ 0 @py4# and D Þ 0 @p#! enables us to get only onesolution. The restriction on the polarizer angle is toavoid commutation with R~t, C, D!, which would notreduce the number of solutions:
W is the unique solution of HM~x! 5 0,
M [ $M% when $M% 5 $R~tR, C, D!, P~tP, u!%. (22)
lR
m
g
$
ttpocvpt
td~~
wu
Tmt
et
as
Tr
2. Choice of the Characteristics of the ReferenceSamplesWe have seen from expression ~22! that the use of aset of reference samples including at least a reflectionon a smooth sample and a polarizer, properly ori-ented, completely determines W and A. Neverthe-ess, one has to choose among the characteristics of~tR, C, D! and P~tP, u! those that will enable W to be
determined with the best precision. A theoreticalstudy of propagation of the experimental errors isbeyond the scope of this paper. Nevertheless, it isdemonstrated in Appendix C that it is necessary toensure a good balance among the 15 nonzero eigen-values of the matrix K. The eigenvalues of K beingsorted, as in Eq. ~20!,
the accuracy is maximum when l15yl1 is maximum.
(23)
The conditioning of K, as defined in expression ~23!,is of course affected by A and W, and the calibrationis less accurate if A and W are ill-conditioned matri-ces ~close to noninvertible matrices, i.e., the norms oftheir eigenvalues are not of the same order of mag-nitude!. This means that MME’s that provide equalsensitivity to all the Mueller matrices can be cali-brated with higher accuracy.
A numerical procedure that uses Powell’s methodbased on a steepest-descent algorithm24 and opti-
izes criterion ~23! provides the optimum values forthe characteristics of the reference samples. Whenonly two reference samples, R~tR, C, D! and P~tP, u!,are used, with well-balanced matrices W and A oneets
M% 5 $R~tR, C, D!, P~tP, u!%
is optimum when tR 5 0.71, C 5 py6 or 3py6,
u 5 6 py4, D 5 6 py2. (24)
The physical systems that provide such values arediscussed in Subsection 3.C./
3. Other Sets of Reference SamplesValues other than those displayed in Eq. ~24! could beobtained if more than the two necessary matriceswere used for calibration. In that case the precisionof the calibration would increase because of the in-crease in the overdetermination. Other sets of ref-erence samples also provide the needed properties.A particular set is worth describing because it en-ables us to calibrate a transmission-limited configu-ration ~see Fig. 1!. This set includes a phase shifter~a quarter-wave plate is optimum but not necessary!and a polarizer measured at 0° and 45°.
C. Experimental Determination of the Characteristics ofthe Reference Samples
In the theoretical discussion above we assumed aknowledge of the matrices of the reference samples.One could consequently argue that we have until nowmerely shifted the problem from the determination of
W and A to the determination of the characteristics ofhe reference samples. However, the advantage ishat the indetermination relies no longer on a com-lex system but on well-known matrices that dependn a few parameters. Moreover, these parametersan be experimentally determined because the eigen-alues of a matrix product are independent of theroduct order. Consequently one can identify theheoretical eigenvalues of M to the eigenvalues of the
product of measured matrices ~aw!21~amw!. Iden-ifying the eigenvalues of R~tR, C, D! and P~tP, u!,isplayed in expressions ~B3! and ~B4! below to belr1, lr2, lc1, lc2!, the real and complex eigenvalues ofaw!21~arw!, leads to
tp 5 trace @~aw!21~apw!#, tr 5 0.5~lr1 1 lr2!,
D 5 0.5 arg~lc1ylc2!, C 5 arctanÎlr1ylr2, (25)
where the trace is equal to the sum of the eigen-values.
Note that in identifying the two sets of eigenvaluesone cannot distinguish lr1 from lr2 and lc1 from lc2,
hich results in an indetermination of D and C val-es because permuting lc1 and lc2 changes the sign of
D and permuting lr1 and lr2 changes C into py2 2 C.hese indeterminations are of course easily removederely by comparison of the obtained values with the
heoretical ones.The orientation u of the polarizer cannot be recov-
red by identification of the eigenvalues becausehose of P~tp, u! displayed in expressions ~B3! do not
depend on u. A more physical approach consists innoticing that the orientation of the polarizer itself ismeaningless if it is not related to the reflection sam-ple, which is the basic reference and does not appearin ~aw!21 z ~apw!. Nevertheless, as soon as theother characteristics have been determined from theeigenvalues, u can be obtained from matrix K, definedin Eq. ~20!, which is a function of u as follows:
K~u! 5
TS HR
HP~u!DS HR
HP~u!D 5 TO~u!Fl1~u! 0
·· ·0 l16~u!
GO~u!.
(26)
If u has the correct value, all the eigenvalues of Kare nonzero except one, as displayed in Eq. ~20!. Onthe other hand, when we depart from the actualvalue, the null space of K is empty and the systemhas only a trivial solution, which is the null matrixbecause all the eigenvalues are nonzero. As a con-sequence, we obtain the value of u by minimizing themplitude of the smallest eigenvalue of K with re-pect to the 15 other eigenvalues:
u verifies thatl16~u!
l15~u!3 0. (27)
he polarizer is then automatically referenced to theeflection sample.
1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3495
w
m
w
F
o
r~o
aedtmqeoiet
3
D. Adaptation to the Reflection Configuration
We performed the operations above by assuming thatthe experimental setup was compatible with reflec-tion and direct-transmission measurements ~Fig. 1!,which is of course not always the case. In situMME’s, for example, are often limited to the reflec-tion configuration with a narrow angular range. Onthe other hand, polarimetric tools and especially po-larimeters are often highly sensitive to the angle ofincidence. Therefore, combining measurements inseveral configurations could generate experimentalerrors because of changes that would occur in matrixA. Consequently it is worth describing how the cal-ibration theory developed above can be adapted to areflection-limited configuration.
When the MME operates in a diffraction-limitedconfiguration, four measurements are required forcalibrating A and W instead of three: two reflec-tions, ~ar1w! and ~ar2w! with two different samplesR1 and R2; and two others, including a polarizer after~apar1w! and before ~ar1pbw!, the reflection on R1.W is then determined in the same manner in theintersection of the null spaces of HPb
and HReq:
HPb:X3PbX 2 X~ar1w!21~ar1pbw!,
HReq:X3ReqX 2 X~ar1w!21~ar2w!, (28)
here Pb~tb, ub! is a polarizer matrix as in the pre-vious case and Req is obtained from the reflection
Matrix Req has no physical meaning but has thestructure of a reflection matrix described by Eq. ~5!.
inally, the characteristics ~tb, ub, teq, Ceq, Deq! of thetwo matrices Pb and Req can be experimentally de-termined, as in Eqs. ~25!, from the eigenvalues of~ar1w!21~ar2w! and the trace of ~ar1w!21~ar1pbw!.
The characteristics of ~R1, R2! and the orientationf Pb must be chosen such that Req and Pb fulfill
conditions ~24! and that R1 is invertible ~usin 2C1u.. 0!.
The value of matrix A cannot be determined di-ectly from W as in the first case because the productaw! is no longer available. Moreover, it cannot bebtained from either ~ar1w! or ~ar2w! because the
experimental characteristics of R1 and R2 are un-known. As a consequence, A is calibrated like W byuse of the fourth measurement ~apar1w!: It is in the
496 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999
null space of
H9Pa:X3 XPa 2 ~apar1w!~ar1w!21X,
H9Req:X3 XReq 2 ~ar2w!~ar1w!21X. (30)
This system of equations is solved as in the previ-ous case, and the vector representation A16 of A isfound to be the eigenvector of K* associated with itssmallest quasi-null eigenvalue l169, as follows:
A16 5 TO*10···012 , (31)
where
K* 5 TH*PaH*Pa
1 TH*ReqH*Req
5 TO*Sl19 0·· ·
0 l169DO*
l19 . · · · . l159 .. l169 < 0. (32)
E. Error Control
1. DefinitionThe relative errors εW and εA in W and A, which aredue to experimental errors, are defined by
εW 5uW 2 WECMu
uWu, εA 5
uA 2 AECMuuAu
, (33)
where
uMu 5 S(i, j
Mij2D1y2
(34)
is the Frobenius norm, A and W are the actual matrixvalues, and AECM and WECM are those values deter-mined by the ECM. Usually these calibration errorsare unknown because of the lack of knowledge of thevalues of A and W. Nevertheless, one of the maindvantages of the ECM resides in its capability tostimate those errors by use of error estimators. In-eed, the reference samples were chosen such thathe null spaces of matrices K and K* contain only theatrices W and A ~see Subsection 3B.1!. Conse-
uently K and K* should have one, and only one, nulligenvalue. Yet, because of experimental errors,ne can find only small eigenvalues. Consequentlyt is possible to control the influence of errors with therror estimators εECM and εECM9 that reflect the con-rast between the null and the nonnull eigenvalues:
εECM 5 Îl16yl15, ε9ECM 5 Îl916yl915. (35)
The mathematical connection between the actualrelative errors εW and εA and their estimators εECMand ε9ECM is complex and beyond the scope of thispaper. On the other hand, a numerical simulation
r
Aaacve
e
t
leg
i
Table 2. Measured and Theoretical Values of the Characteristics of the
will efficiently demonstrate the relevance of theseestimators.
2. Numerical SimulationThe numerical simulation uses the measured valuesA and W of our polarimeter and PSG and the theo-etical Mueller matrices Pb, Pa, R1, and R2 of the
samples, obtained from definitions ~2! and ~5! andTable 2, to simulate the measured matrices ~amw! 5
MW. Then we take the experimental errors intoccount by adding random noise with a controlledmplitude uD~amw!u 5 0.005u~amw!u to these matri-es. Finally, the ECM is implemented, which pro-ides the calibration values AECM and WECM and therror estimators εECM and ε9ECM. In that case, as the
actual values A and W are known, calibration errorsεA and εW can be estimated and compared with therror estimators.Figure 2 displays the calibration errors εA and εW
for 50 calibrations with the same noise amplitude~0.5%!. The variations are due to the random struc-ure of the noise matrices. The average errors ^εA& 5
0.25 and ^εW& 5 0.29 are two times less than theinjected noise, which demonstrates that the ECM isan accurate and robust calibration method. Figure3 displays the ratio between the actual and the esti-mated errors. The values of the average ratios ^εAy
Fig. 2. Fifty numerical simulations of the calibration accuracy.The measurement errors are simulated by addition of randommatrices to the theoretical measurements. Relative errors in themeasurements are taken to be equal to 0.5% and result, with theECM, in average relative error ^εA& 5 0.25% in A and εW 5 0.29%n W.
ε9ECM& 5 0.69 and ^εWyεECM& 5 1.29 together with theimited data dispersion demonstrate that the errorstimators are meaningful. In using the ECM, oneets the values of the calibration matrices W and A,
which is the primary reason for using the ECM, andalso a confidence interval.
F. Calibration Algorithm
The overall calibration procedure is schematicallysummarized in Fig. 4.
Fig. 3. Fifty numerical simulations of the ratio between the ac-tual calibration errors ~εA, εW! in ~A, W! and the error estimatorsεECM and ε9ECM.
Fig. 4. ECM algorithm. Numbers in parentheses refer to thecorresponding equations in text.
1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3497
tttTwbli
3
4. Experimental Results
The ECM procedure was implemented to calibrate amultiwavelength MME.6 We first used a configura-ion compatible with transmission and reflection andhen a configuration restricted to reflection, which ishe present geometric configuration of the MME.he presentation of experimental results obtainedith our MME at two wavelengths ~458 and 633 nm!y use of the reflection-limited configuration will il-ustrate the effective efficiency and simplicity of us-ng the ECM.
A. Choice of the Calibration Samples
The two polarizers used in this experimental exerciseare Glan prisms. They have extinction coefficientssmaller than 1025 and can be considered perfect withrespect to commonly available precision.
The two reference samples R1 and R2 are chosensuch that Req, defined by Eq. ~29!, has overall char-acteristics close to the optimum values given in Eq.~24!. These characteristics depend on the wave-length and the angle of incidence. As a consequence,it is difficult to find universal samples that can beused in any configuration. Nevertheless, manysamples, eventually coated, can be used to fit anyspecial configuration. Moreover, their flexibilityand simplicity of use are considerably improved bythe self-consistency of the ECM: The characteristicsof the samples merely have to be roughly close to theoptimum to minimize error propagation; their exactvalues are determined during the calibration. Sat-isfying properties are obtained by use of a commonc-Si wafer associated with either chromium- orpalladium-coated glass. The dispersion of the theo-retical polarimetric properties Ceq and Deq, defined inEq. ~29!, are plotted in Figs. 5 and 6 for the cSi–Crand the cSi–Pd samples, respectively. Displaying asmall spectral dispersion25 for wavelengths under theoptical gap of silicon, they provide satisfying proper-ties in a broad spectrum ~at least 400–800 nm! and
Fig. 5. Equivalent polarimetric properties of cSi–Cr samples.The values are plotted as a function of the wavelength for severalangles of incidence.
498 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999
for a convenient range of angles of incidence ~64°–70°!.
Finally, the best experimental results presented inthis paper were obtained with c-Si- and Pd-coatedglass. As a matter of fact, the results of the calibra-tion with Cr mirrors were altered by stresses in thedeposited layer that induced perturbations in the op-tical response of this material. One of the solutions,to prevent surface anisotropy from inducing cross-polarization effects, lies in using uncoated bulk ma-terials, which would limit surface stresses. Thecalibration results presented in this paper were re-corded at an angle of incidence of 64°. R1 and R2correspond, respectively, to reflection on the Pd andon the c-Si sample.
B. Samples Characteristics
The characteristics of the reference samples are de-termined from Eqs. ~25!: The equivalent samplematrix Req 5 R1
21R2 from the eigenvalues of themeasured matrices ~ar1w!21~ar2w! and the trans-mission coefficients of the polarizers from the tracesof ~ar1w!21~ar1pbw! and ~ar1w!21~apar1w!. Thesevalues are compared in Table 2 with theoretical onesobtained from Ref. 25 and Eq. ~29!. The theoreticalvalues do not take into account the thickness of thePd layer and the native oxide layers and possibleroughness on the top of the reference samples. Thismay explain the small discrepancy, without practicalconsequences, between experimental and theoreticalvalues. The transmission coefficients of the polariz-ers increase with the wavelength because of the in-crease of the refractive index.
The geometrical orientations of the entrance andexit polarizers are determined from expression ~27!.Figure 7 shows that the geometric orientations of thepolarizers are easily determined without any ambi-guity. One gets ua 5 46.75° for the polarizer intro-duced between the sample and the polarimeter~apar1w! and ub 5 45.80° for the polarizer introduced
Fig. 6. Equivalent polarimetric properties of cSi–Pd samples.The values are plotted as a function of the wavelength for severalangles of incidence.
pbtcoslb
tbrmiMasr
1t
mmt
l
between the sample and the PSG ~ar1pbw!. As ex-ected, the orientations of the polarizers are found toe independent of the wavelength. This experimen-al determination is important on the one hand be-ause it increases the calibration precision and on thether hand because it obviates the need to use expen-ive high-precision rotating holders to orient the po-arizers accurately. The polarizers merely need toe roughly oriented at ;45°.
C. Error Control
The overdetermination of calibration matrices W andA allows us to get extra information. We have al-ready explained how the characteristics of the sam-ples are determined from eigenvalues of themeasured matrices. In the same manner, the studyof the eigenvalues of K and K* provides an importantestimation of the final accuracy of the calibration ~seeSubsection 3.E!.
The eigenvalues of K and K* are displayed in Fig.8. The control of their amplitudes is important fortwo reasons. The first is that the accuracy of thecalibration is affected by the balance among the non-null values ~see Subsection 3.B.2!. A lack of balancewould increase the error transmission, and a well-
Fig. 7. Determination of the orientations of the two polarizers.The correct orientations ub and ua minimize the ratios l16yl15 and916yl915.
Fig. 8. K and K* eigenvalues at 633 nm. They reflect the behav-ior of the calibration procedure ruled by the choice of the referencesamples and the accuracy of the basic measurements.
balanced calibration is obtained by use of samplesclose to the optimum values displayed in Eq. ~24!.The second reason is related to the necessity to pro-vide the values of the error estimators εECM and ε9ECMdefined in Eqs. ~35!:
εECM 5 0.34%,
ε9ECM 5 0.98%. (36)
These estimators are affected less by the choice ofhe calibration samples than by the accuracy of theasic measurements. Consequently this contrasteflects the ability to provide accurate measure-ents. A contrast of several orders of magnitude as
n the present case proves that the behavior of theME is satisfying. By using the estimator values
nd the results of the numerical simulation in Sub-ection 3.E one can estimate that the calibration er-or does not exceed 0.5%.
D. Accuracy of the Measurements
We evaluated the accuracy of this calibration proce-dure by measuring the Mueller matrix of a SiO2 layerdeposited by plasma-enhanced chemical-vapor depo-sition26 upon a monocrystalline ~111! silicon wafercleaned by standard procedure.27 The layer thick-ness estimated from in situ ultraviolet–visible spec-troscopic ~1.5–5-eV! phase-modulated ellipsometry is.04 6 0.005 mm, and the refractive index is close tohose of thermal silica, 1.465 6 0.005 ~at 458 nm! and
1.457 6 0.005 ~at 633 nm!. Figure 9 displays thevariations of the 16 Mueller-matrix coefficients as afunction of the angle of incidence for two wavelengths~458 and 633 nm!. The experimental values arecompared with the theoretical ones; the thickness ofthe SiO2 layer is fitted. The estimated thickness is1.0408 6 0.001 mm ~at 458 nm! and 1.0409 6 0.001
m ~at 633 nm!, which corresponds to the phase-odulated ellipsometry value. Finally, except for
he grazing incidences ~fi . 85°! for which the laserbeam section is larger than the sample, the maxi-mum relative error for the 16 Mueller-matrix coeffi-cients is also found to be less than 0.5%.
5. Summary and Conclusions
A new calibration procedure, the eigenvalue calibra-tion method, has been presented. It is devoted to thecomplex and critical calibration of polarization-stategenerators, polarimeters, and Mueller matrix ellip-someters. An optimized universal procedure basedon the extensive use of matrix algebra has been de-scribed. It relies on the measurements of three orfour reference samples whose few characteristics aredetermined during the calibration. It strongly facil-itates the implementation of a MME without theneed for precise orientation of the various elements.The numerous inherent defects of the optical ele-ments that constitute the polarization-state genera-tor and the polarimeter, including eventual filters,beam expanders, windows, or mirrors, are automat-ically taken into account by the ECM. Furthermore,
1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3499
p
s
S
6r
3
a meaningful error-control indicator evaluates theaccuracy of the calibration.
Until now the optical characterization of scatteringsamples has been limited by difficulties in carryingout MME measurements. It can be expected thatthe ECM will contribute to overcoming this experi-mental limitation and enable MME users to focus onthe physical interpretation of the measurements.
Appendix A: Choice of the Reference Sample
The proof that the choice of the reference sample doesnot depend on W and A, which are invertible matri-ces, is quite direct: Considering the change X* 5XW21 in the variables of the mapping HM, defined inrelation ~14!, neglecting the experimental errors~aw!21~amw! 3 W21MW, and noting that the map-
ings X3 HM~X! and X3 HM~X!W21 have the samenull space, one gets
HM~X! 5 0N @M, X*# 5 0, (A1)
where @M, # designates the Lie product or commuta-tor of M, which is also a mapping, like HM, from theet M4~R! into itself:
@M, #:M4~R!3M4~R!, X*3MX* 2 X*M. (A2)
Relation ~A1! shows that the solutions of Eq. ~15!are influenced only by matrices M, independently ofA and W.
Fig. 9. Measured Mueller matrices of a SiO2-coated c-Si substrat33 nm!. The coefficient M11 is the intensity reflectance, and the oesponse. The error between experimental and theoretical value
500 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999
Transforming the calibration problem in this man-ner has two advantages: The first is to get rid of theapparent dependence on A and W; the second is to geta well-established, extensively studied mathematicalformulation. As a matter of fact, W will be unam-biguously determined by Eq. ~15! if the only matrix X*that verifies
@M, X*# 5 0, @M [ $M% (A3)
is the identity matrix I4; then X* 5 I4 implies that X 5W.
Appendix B. Size of HM Null Space: Referenceample Eigenvalues
Relation ~A1! shows that the null space of HM and thecommutator of M have the same dimension. In thepresent case, in which the reference matrices definedin Eqs. ~3! and ~5! are 4 3 4 diagonalizable matrices,the dimension of the vector space that commutes withthem is a function of the multiplicity of their eigen-values. As a matter of fact, if P~x! is the character-istic polynomial of M,
P~x! 5 det~xI4 2 M! 5 )i51. . .m
~x 2 li!ai, (B1)
m is the number of eigenvalues li, and ai is theirmultiplicity, one can easily demonstrate ~by working
a function of angle of incidence ~fi! for two wavelengths ~458 andcoefficients, normalized by M11, display the complete polarimetricid curves! is less than 0.5%.
e asther
s ~sol
dd
r
n
o
in a diagonalization basis of M! that the dimension ofthe commutator null space is
(i51· · ·m
ai2. (B2)
Consequently the null space of HP contains 10 solu-tions because P~tp, u!, defined in Eq. ~3!, has twodistinct eigenvalues:
tp ~multiplicity, 1!, 0 ~multiplicity, 3! (B3)
and HR contains four solutions because R~tR, C, D!,efined in Eq. ~5!, has two real and two complexistinct eigenvalues:
2tR sin2~C!, 2tR cos2~C!;
tR sin~2C!exp~iD!, tR sin~2C!exp~2iD!. (B4)
Appendix C. Matrix K Conditioning
The basic necessary condition for matrix K, defined inEq. ~20!, is to have only one null, nondegeneratedeigenvalue, which implies that
l16yl15 ,, 1. (C1)
Without noise, this condition is always fulfilled be-cause l16 5 0. Without trying to give a general dem-onstration, we illustrate by a particular case that acondition to satisfy inequality ~C1! when experimen-tal errors occur is
l15yl1 .. error level. (C2)
We consider that the experimental error induces aelative perturbation on K less than ε:
uDKuyuKu # ε, (C3)
and note, using Eq. ~20! and the definition of theorm u u in ~34!, that
l1 # uKu # 4l1. (C4)
Then, considering a particular perturbation DK thatis proportional to identity matrix I16 to commute withK while verifying inequality ~C3!,
K 1 DK 5 K 1ε4
l1I16
5 tO3l1 1
ε4
l1
· · ·
l15 1ε4
l1
· · ·ε4
l1
4O,
(C5)
ne obtains for condition ~C1!
l15yl1 .. εy4. (C6)
We have verified that a necessary condition is thatthe eigenvalues of K have to be well balanced.
The authors thank R. Ossikovski of Jobin-YvonInstruments S.A., and B. Kaplan of the Laboratoirede Physique des Interfaces et des Couche Minces fortheir careful reading of the manuscript and for theirnumerous pertinent remarks.
References and Notes1. P. S. Hauge, “Recent developments in instrumentation in el-
lipsometry,” Surf. Sci. 96, 108–140 ~1980!, and referencestherein.
2. B. Drevillon, Progress in Crystal Growth and Characterizationof Materials ~Pergamon, Oxford, 1993!, and references therein.
3. P. S. Hauge, “Mueller matrix ellipsometry with imperfect com-pensators,” J. Opt. Soc. Am. 68, 1519–1528 ~1978!.
4. R. M. A. Azzam and A. G. Lopez, “Accurate calibration of thefour-detector photopolarimeter with imperfect polarizing opti-cal elements,” J. Opt. Soc. Am. A 6, 1513–1521 ~1989!.
5. R. C. Thompson, J. R. Bottiger, and E. S. Fry, “Measurementof polarized interactions via the Mueller matrix,” Appl. Opt.19, 1323–1332 ~1980!.
6. E. Compain and B. Drevillon, “High-frequency modulation ofthe four states of polarization of light with a single phasemodulator,” Rev. Sci. Instrum. 69, 1574–1580 ~1998!.
7. E. Compain and B. Drevillon, “Broadband division of ampli-tude polarimeter based on uncoated prisms,” Appl. Opt. 37,5938–5944 ~1998!.
8. S. Huard, Polarisation de la Lumiere ~Masson, Paris, 1994!.9. E. Collett, Polarized Light—Fundamentals and Application
~Marcel Dekker, New York, 1993!.10. J. C. Stover, Optical Scattering—Measurement and Analysis
~SPIE, Press, Bellingham, Wash., 1995!.11. W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller ma-
trices, and polarized scattered light,” Am. J. Phys. 53, 468–478 ~1984!.
12. R. M. A. Azzam, “Photopolarimetric measurement of the Muel-ler matrix by Fourier analysis of a single detected signal,” Opt.Lett. 2, 148–150 ~1978!.
13. R. M. A. Azzam, “Simulation of mechanical rotation by opticalrotation: application to the design of a new Fourier polarim-eter,” J. Opt. Soc. Am. 68, 518–521 ~1978!.
14. E. Compain and B. Drevillon, “Complete high-frequency mea-surement of Mueller matrices based on a new coupled-phasemodulator,” Rev. Sci. Instrum. 68, 2671–2680 ~1997!.
15. R. M. A. Azzam, “Arrangement of four photodetectors for mea-suring the state of polarization of light,” Opt. Lett. 10, 309–311~1985!.
16. R. M. A. Azzam, “Division-of-amplitude photopolarimeter~DOAP! for the simultaneous measurement of all four Stokesparameters of light,” Opt. Acta 29, 685–689 ~1982!.
17. K. Brudzewski, “Static Stokes ellipsometer: general analysisand optimization,” J. Mod. Opt. 38, 889–896 ~1991!.
18. S. Krishnan, “Calibration, properties, and applications of thedivision-of-amplitude photopolarimeter at 632.8 and 1523nm,” J. Opt. Soc. Am. A 9, 1615–1622 ~1992!.
19. W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller ma-trices, and polarized scattered light,” Am. J. Phys. 53, 468–478 ~1984!.
20. R. M. A. Azzam, “Mueller-matrix measurement using the four-detector photopolarimeter,” Opt. Lett. 11, 270–272 ~1986!.
21. S. Krishnan, “Calibration, properties, and applications of thedivision-of-amplitude photopolarimeter at 632.8 and 1523nm,” J. Opt. Soc. Am. A 9, 1615–1622 ~1992!.
22. Y. Cui and R. M. A. Azzam, “Sixteen-beam grating-based
1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3501
division-of-amplitude photopolarimeter,” Opt. Lett. 21, 89–91
2
ment of Physics, University of North Carolina, Raleigh, N.C.
2
3
~1996!.3. M. C. Pease, Methods of Matrix Algebra ~Academic, London,
1965!.24. A useful book on the computation of the ECM algorithms is
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vet-terling, Numerical Recipes in Pascal ~Cambridge U. Press,Cambridge, 1989!; books for other computer languages areavailable as parts of a series by the same publisher.
25. Dielectric functions are the library of D. E. Aspnes, ~Depart-
502 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999
27645!, except for that of Cr, which was measured in theauthors’ laboratory with a spectroscopic phase-modulated el-lipsometer.
26. P. Bulkin, N. Bertrand, and B. Drevillon, “Deposition of SiO2
in an integrated distributed electron cyclotron resonance mi-crowave reactor,” Thin Solid Films 296, 66–68 ~1997!.
7. W. Kern and D. A. Puotinen, “Cleaning solutions based onhydrogen peroxyde for use in silicon semiconductor technolo-gy,” RCA Rev. 31, 187–206 ~1970!.
