Soe-Mie F. Nee Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. A 1651

Error analysis for Mueller matrix measurement

Soe-Mie F. Nee

Naval Air Warfare Center, Research Department, China Lake, California 93555-6100

Received January 16, 2003; revised manuscript received April 11, 2003; accepted April 15, 2003

The linear errors of Mueller matrix measurements are formulated for misalignment, depolarization, and in-correct retardation of the polarimetric components. The measured errors of a Mueller matrix depend not onlyon the imperfections of the measuring system but also on the Mueller matrix itself. The error matrices fordifferent polarimetric systems are derived and also evaluated for the straight-through case. The error matrixfor a polarizer–sample–analyzer system is much simpler than those for more complicated systems. The gen-eral error matrix is applied to null ellipsometry, and the obtained errors in ellipsometric parameters c and Dare identical to the errors specifically derived for null ellipsometry with depolarization.© 2003 Optical Societyof America

OCIS codes: 120.2130, 120.5410, 120.4800, 260.5430.

1. INTRODUCTIONMeasurements of Mueller matrices have received increas-ing attention in recent years as polarizations are exam-ined for more complicated samples such as rough sur-faces, anisotropic materials, and others. There havebeen many different methods for measuring Mueller ma-trices, such as dual rotating compensator polarimetry,1–5

null ellipsometry,6,7 and the Stokes method, which usesintensities for polarimetric components set at specificpositions.8,9 Measurements of Mueller matrices and el-lipsometric parameters suffer from the errors of imperfec-tion and misalignment, and the error analyses for thesemeasurements are complicated.2,5,6,10,11

Errors of Mueller matrix measurement and ellipsomet-ric measurement arise from the assumptions that all com-ponents are perfect and all alignments are perfect. Ageneral error analysis for Mueller matrix measurement isdeveloped here, and the first-order errors for misalign-ment, depolarization, and retardation of the polarimetriccomponents with unpolarized incident light are derived.Higher-order errors and errors caused by more compli-cated systems can be derived following the principles andprocedures established. This error analysis can be ap-plied to many polarimeters or ellipsometers, and it is use-ful in designing a polarimeter that can minimize the un-avoidable errors. Section 2 presents the basic principlesfor the error analysis for Mueller matrix measurement.The error matrix is derived for a polarizer–sample–analyzer (PSA) system in Section 3, and for a polarizer–compensator source system or a compensator-analyzer de-tecting system in Section 4. Section 5 gives the errormatrices for different systems for the straight-throughcase. Section 6 demonstrates the application of the errormatrix to the case of null ellipsometry. Section 7 con-cludes the investigation.

2. BASIC PRINCIPLESErrors exist in a Mueller matrix measurement because ofimperfect components and misalignments. Regardless of

the methods of measurement, some errors are inherentbecause we assume that all components and conditionsare ideal while in fact they are not. The intensity at thedetector can be expressed as

I 5 D – M – X, (1)

where M is the 4 3 4 Mueller matrix of a sample, X is the1 3 4-column Stokes vector for polarized light incidenton the sample, and D is the 4 3 1-row Stokes vectorsensed by the detection system. We use – to representmultiplication between matrices of different orders re-sulting in a lower order product. Different arrangementsof the source arm give different X, and different arrange-ments of the detector arm give different D. Let Xo andDo be the ideal Stokes vectors for the source and detectionsystems, respectively; d X and d D are the deviations ofthese Stokes vectors from their ideal vectors Xo and Do .The subscript o denotes the ideal case for a matrix or vec-tor. The measured intensity given by Eq. (1) can be ex-pressed as

I 5 ~Do 1 d D! – M – ~Xo 1 d X! 5 Do – M* – Xo , (2)

where M* is the measured Mueller matrix as if all com-ponents were perfect. In general, the error source vectord X can be expressed as an error matrix d mx operated onthe ideal source vector Xo , i.e., d X 5 d mx – Xo . Thesame applies to the detection system, i.e., d D5 Do – d md . The apparent Mueller matrix M* affectedby imperfections of all components is then

M* 5 ~1 1 d md!M~1 1 d mx! 5 M 1 d Mx 1 d Md .(3)

The combined error of a Mueller matrix due to all imper-fections is

d M 5 d Mx 1 d Md 5 Md mx 1 d mdM. (4)

The errors of a measured Mueller matrix come from thecouplings between the Mueller matrix M and the errormatrices for both the source system and the detection sys-tem. For the straight-through case, M is a unit matrixand the error matrix is then

1652 J. Opt. Soc. Am. A/Vol. 20, No. 8 /August 2003 Soe-Mie F. Nee

d Mo 5 d mx 1 d md . (4a)

The Mueller matrix for the straight-through case is agood check for the error matrices d mx and d md .

An ellipsometer can be configured in different arrange-ments, such as a PSA system or a polarizer-compensator-sample-analyzer (PCSA) system. If we denote the com-pensator in front of the analyzer as Q, we also have PSQAand PCSQA systems. These systems can be decomposedinto the source system and the detector system. TheMueller matrix for the detection system is the transposeof the Mueller matrix for the source system with similarcomponents. We need only evaluate d mx ; then d md canbe obtained from the transpose with the change of P to Aand C to Q. In the following sections, d mx will be de-rived for different systems to the first-order errors; thend M can be computed according to Eq. (4).

3. ERROR MATRICES FOR POLARIZER–SAMPLE–ANALYZER SYSTEMSA. Misalignment ErrorsTo derive the errors in Mueller matrices, we need to usethe principal Mueller matrix with depolarization for thepolarimetric components. To simplify the algebraicevaluations, we choose a special principal Mueller matrixto represent the nearly perfect components of a polarim-eter. This matrix is6,7,12–15

Here D and Dv are, respectively, the depolarization andcross-polarized depolarization, c and D are the ellipsomet-ric parameters, and the transmittance T 5 T8(1 1 D).A good polarizer has c 5 90° and a good wave plate hasc 5 45°. For a perfect polarizer, c 5 90° and D 5 Dv5 0. The principal Mueller matrix for a perfect polar-izer is16–19

Po 5 TF 1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0G . (6)

We drop the transmittance T for the convenience of ma-trix evaluation. The zeros of the polarizer and analyzerin an ellipsometer are referred to the principal axes of asample. Let R(P) be the rotation matrix with a rotationangle P. The Mueller matrix for a polarizer with itstransmission axis oriented at an angle P is Po(P)5 R(2P)PoR(P). An unpolarized light source is as-sumed. The ideal Stokes vector for light incident on thesample is

Xo 5 Po~P ! – So , (7a)

So 5 ~1, 0, 0, 0 !T. (7b)

M 5 T8F 1 1 D 2cos 2c 0 0

2cos 2c 1 1 D 2 2Dv 0 0

0 0 sin 2c cos D sin 2c s

0 0 2sin 2c sin D sin 2c c

The superscript T denotes the transpose of a matrix. Thetranspose of a row matrix is a column matrix. Misalign-ment of a polarizer is equivalent to the rotation of the po-larizer by a small amount dP. The Stokes vector forlight incident on a sample is

X 5 P~P 1 dP ! – So

5 R~2dP !Po~P !R~dP ! – So 5 R~2dP ! – Xo , (8)

where R(dP) 5 1 1 d R(dP) with

d R~dP ! 5 F 0 0 0 0

0 0 2dP 0

0 22dP 0 0

0 0 0 0G . (9)

The misalignment error matrix for the source system isd mx 5 d R(2dP), and for the detector system is d md5 d R(dA).

B. Depolarization ErrorsBy taking c 5 90° in Eq. (5), the principal Mueller ma-trix for a good polarizer with depolarization is6,12,20

P~0 ! 5 F 1 1 Dp 1 0 0

1 1 1 Dp 2 2Dvp 0 0

0 0 0 0

0 0 0 0G . (10)

Here Dp and Dvp are, respectively, the depolarization andcross-polarized depolarization of a polarizer. For a polar-izer with its transmission axis oriented at an angle P, theStokes vector incident on the sample for an unpolarizedlight source is

X 5 P~P 1 dP ! – So

5 R~2dP ! – ~1 1 Dp , cos 2P, sin 2P, 0!T

5 R~2dP !F 1 1 Dp 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1G – F 1

cos 2Psin 2P

0G . (11)

The error matrix caused by polarizer depolarization forunpolarized incident light is

dDp 5 FDp 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0G . (12)

The error matrix caused by polarizer depolarization andmisalignment is then

G . (5)

in D

os D

Soe-Mie F. Nee Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. A 1653

d mx 5 d R~2dP ! 1 dDp 5 FDp 0 0 0

0 0 22dP 0

0 2dP 0 0

0 0 0 0G .

(13a)

Similarly, for the analyzing system,

d md 5 d R~dA ! 1 dDa 5 FDa 0 0 0

0 0 2dA 0

0 22dA 0 0

0 0 0 0G .

(13b)

The errors of a Mueller matrix caused by depolarizationand misalignment of the polarizer and analyzer are givenby Eq. (4) with

For the straight-through case, the error matrix caused bymisalignments and depolarization for a PSA system withunpolarized incident light is

d Mo 5 FDa 1 Dp 0 0 0

0 0 2dA 2 2dP 0

0 2dP 2 2dA 0 0

0 0 0 0G .

(15)

From the straight-through test for a PSA system, it ishard to separate the errors between the polarizer and theanalyzer.

4. ERROR MATRIX FOR THEPOLARIZER–COMPENSATOR SYSTEMLet C denote the angle for the fast axis of a phase re-tarder or compensator. The principal Mueller matrixC(C, t) for a phase retarder with a retardation t along itsfast axis (C 5 0) is6,7,13,14

d Mx 5 Md mx 5 FDpM00 2dPM02 22dPM01 0

DpM10 2dPM12 22dPM11 0

DpM20 2dPM22 22dPM21 0

DpM30 2dPM32 22dPM31 0G , (14

d Md 5 d mdM 5 F DaM00 DaM01 DaM02 D2dAM20 2dAM21 2dAM22 2d

22dAM10 22dAM11 22dAM12 22

0 0 0

C~0, t!

5 TcF 1 1 Dc 0 0 0

0 1 1 Dc 2 2Dvc 0 0

0 0 cos t sin t

0 0 2sin t cos t

G . (16)

Dc and Dvc are, respectively, the depolarization and thecross-polarized depolarization for the compensator. Theprincipal Mueller matrix Co(0, t) for a perfect retarderhas Dc 5 Dvc 5 0 in Eq. (16). Again, drop Tc for the con-venience of derivation. Error by retardation dt can betreated in a similar way to the case of rotation because

C~0, t 1 dt! 5 C~0, dt!Co~0, t! 5 @1 1 d c~dt!#Co~0, t!,

d c~dt! 5 FDc 0 0 0

0 Dc 2 2Dvc 0 0

0 0 0 dt

0 0 2dt 0G . (17)

For an unpolarized source with So given by Eq. (7b), theStokes vector passing through a polarizer compensatorsystem is

X 5 C~C 1 dC, t 1 dt!P~P 1 dP ! – So

5 ~1 1 d mx!Co~C, t!Po~P ! – So , (18)

d mx 5 d R~2dC ! 1 R~2C !d c~dt!R~C !

1 R~2C !Cod R~dC 2 dP !Co21R~C !

1 dDp . (19a)

Co in Eq. (19a) stands for the abbreviation of Co(0, t).On the right-hand side of Eq. (19a), the first term is dueto the misalignment of the wave plate, the second term isdue to the off-quarter-wave retardation and depolariza-tion of the wave plate, the third term is due to the mis-alignment of both the polarizer and the wave plate, andthe fourth term is due to the depolarization of the polar-izer. These terms are given by

03

23

13G . (14b)

a)

aM

AM

dAM

0

d R~2dC ! 5

0 0 0 0

0 0 22dC 0

0 2dC 0 0, (19b)

1654 J. Opt. Soc. Am. A/Vol. 20, No. 8 /August 2003 Soe-Mie F. Nee

F0 0 0 0

GR~2C !d c~dt!R~C ! 1 dDp 5 F Dc 1 Dp 0 0 0

0 ~Dc 2 2Dvc!cos2 2C 1/2~Dc 2 2Dvc!sin 4C 2dt sin 2C

0 1/2~Dc 2 2Dvc!sin 4C ~Dc 2 2Dvc!sin2 2C dt cos 2C

0 dt sin 2C 2dt cos 2C 0

G , (19c)

R~2C !Cod R~dC 2 dP !Co21R~C ! 5 2~dC 2 dP !F 0 0 0 0

0 0 cos t 2cos 2C sin t

0 2cos t 0 2sin 2C sin t

0 cos 2C sin t sin 2C sin t 0

G . (19d)

From the above equations, we see that d mx caused by de-polarization of a wave plate occurs at the elements at theupper left corner and in the inner square, and the off-quarter-wave retardation error occurs at the last columnand the last row, as given by Eq. (19c). The misalign-ment errors occur at the last column, the last row, and theelements of 12 and 21, as given by Eqs. (19b) and (19d).d md can be obtained from Eqs. (19) by replacing P with2A, C with 2Q, subscript p with a, and subscript c withq.

For a quarter-wave plate in the source arm, t 5 90°1 dtc, and the error matrix of the source system at C5 0u90° 1 dC and C 5 645° 1 dC are, respectively,

d mx~C 5 0u90° !

5 F Dc 1 Dp 0 0 0

0 Dc 2 2Dvc 22dC 72~dC 2 dP !

0 2dC 0 6dtc

0 62~dC 2 dP ! 7dtc 0

G ,

(20a)

d mx~C 5 645° !

5 F Dc 1 Dp 0 0 0

0 0 22dC 7dtc

0 2dC Dc 2 2Dvc 72~dC 2 dP !

0 6dtc 62~dC 2 dP ! 0

G .

(20b)The symbol u stands for ‘‘or.’’ For a compensator–analyzer detection system, we have results similar to Eqs.(20a) and (20b). If we let the orientation angle and the

retardation of the compensator in the detector arm be Qand 90° 1 dtq , respectively, the error matrices for the de-tector systems are

d md~Q 5 0u90° !

5 F Dq 1 Da 0 0 0

0 Dq 2 2Dvq 2dQ 72~dA 2 dQ !

0 22dQ 0 6dtq

0 62~dA 2 dQ ! 6dtq 0

G ,

(21a)

d md~Q 5 645° !

5 F Dq 1 Da 0 0 0

0 0 2dQ 6dtq

0 22dQ Dq 2 2Dvq 72~dA 2 dQ !

0 7dtq 62~dA 2 dQ ! 0

G ,

(21b)

Adding a compensator to either the source or the detectorsystem makes the associated errors much more compli-cated.

5. ERROR MATRICES FOR STRAIGHT-THROUGH CASESIn Subsection 3.B the Mueller error matrix for thestraight-through case in a PSA system is given by Eq.(15). For a PCSA system with a quarter-wave plate atangle C, the Mueller error matrix for the straight-throughcase is

Soe-Mie F. Nee Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. A 1655

d Mo 5 FDc 1 Dp 1 Da 0

0 ~Dc 2 2Dvc!cos2 2C

0 22~dA 2 dC ! 1 ~Dc 2 2Dvc!sin 2C cos 2C

0 dt sin 2C 1 2~dC 2 dP !cos 2C

0 0

2~dA 2 dC ! 1 ~Dc 2 2Dvc!sin 2C cos 2C 2dt sin 2C 2 2~dC 2 dP !cos 2C

~Dc 2 2Dvc!sin2 2C dt cos 2C 2 2~dC 2 dP !sin 2C

2dt cos 2C 1 2~dC 2 dP !sin 2C 0

G . (22a)

For a PCSA system with a quarter-wave plate placed at C 5 0u90° and 645°, the Mueller error matrices are, respectively,

d Mo~0u90° ! 5 FDp 1 Da 1 Dc 0 0 0

0 Dc 2 2Dvc 2~dA 2 dC ! 72~dC 2 dP !

0 22~dA 2 dC ! 0 6dtc

0 62~dC 2 dP ! 7dtc 0G , (22b)

d Mo~645° ! 5 FDp 1 Da 1 Dc 0 0 0

0 0 2~dA 2 dC ! 7dtc

0 22~dA 2 dC ! Dc 2 2Dvc 72~dC 2 dP !

0 6dtc 62~dC 2 dP ! 0G . (22c)

For a general PCSQA system, the error matrix is muchmore complicated. The elements of the Mueller errormatrix for no sample are

dM00 5 Dp 1 Da 1 Dc 1 Dq ,

dM01 5 dM02 5 dM03 5 dM10 5 dM20 5 dM30

5 dM33 5 0,

dM11 5 ~Dc 2 2Dvc!cos2 2C 1 ~Dq 2 2Dvq!cos2 2Q,

dM12 5 2dQ 2 2dC 1 ~Dc 2 2Dvc!sin 2C cos 2C

1 2~dC 2 dP !cos tc 1 ~Dq

2 2Dvq!sin 2Q cos 2Q 1 2~dA 2 dQ !cos tq ,

dM13 5 2dM31 5 2dtc sin 2C 2 2~dC

2 dP !cos 2C sin tc 1 dtq sin 2Q

2 2~dA 2 dQ !cos 2Q sin tq ,

dM21 5 2dC 2 2dQ 1 ~Dc 2 2Dvc!sin 2C cos 2C

2 2~dC 2 dP !cos tc 1 ~Dq

2 2Dvq!sin 2Q cos 2Q 2 2~dA 2 dQ !cos tq ,

dM22 5 ~Dc 2 2Dvc!sin2 2C 1 ~Dq 2 2Dvq!sin2 2Q,(23)

dM23 5 2dM32 5 dtc cos 2C 2 2~dC

2 dP !sin 2C sin tc 2 dtq cos 2Q 2 2~dA

2 dQ !sin 2Q sin tq .The measured Mueller error matrix for the straight-through case is a check for the errors caused by differentsources. However, the measured errors with a sampleare not the same as the errors for the no-sample case.

6. APPLICATION TO NULL ELLIPSOMETRYTo demonstrate how to use the formulas in the previoussections, we apply this analysis to null ellipsometry. Wewill evaluate the errors for a PCSA null ellipsometry andcompare with the known results. A null ellipsometermeasures the c and D of a sample by finding the nullangles for both polarizer and analyzer. The four-zone av-erage can cancel the misalignment errorseffectively.6,10,11,17 The ideal four-zone positions of P, C,and A and the four-zone average are listed in Table 1.Since null ellipsometry does not measure the full Muellermatrix, we may use the principal Mueller matrix for M inEq. (4). The principal Mueller matrix for a sample can beexpressed as6,7,12–15

Table 1. Ideal Null Positions of Polarizer Angle P and Analyzer Angle A for the Four Zonesin Null Ellipsometry

Zone C (deg) P (deg) A

1 245 245 1 D/2 c

2 45 245 2 D/2 c

3 245 45 1 D/2 2c

4 45 45 2 D/2 2c

Average D 5 (P1 2 P2 1 P3 2 P4)/2, c 5 (A1 1 A2 2 A3 2 A4)/4

1656 J. Opt. Soc. Am. A/Vol. 20, No. 8 /August 2003 Soe-Mie F. Nee

M 5 TF 1 Px 0 0

Px 1 2 2Dv 0 0

0 0 Py Pz

0 0 2Pz Py

G , (24a)

Px 5 2P cos 2c,

Py 5 P sin 2c cos D,

Pz 5 P sin 2c sin D,

D 5 1 2 P 5 Du 1 Dv . (24b)

Here, P is the degree of polarization, Px and Pz are thelinear and circular polarization, respectively, and Py isthe preserved part of polarization.

If we use Eq. (20b) for d mx and Eq. (13b) for d md andsubstitute them with Eq. (24a) into Eq. (4), the error ma-trix is given by

‘‘NA’’ stands for the elements not used by the principalMueller matrix. Since there is no Q in the detector sys-tem, we do not use the last row. dPy and dPz are ob-tained from dM22 and dM23 , respectively. Because thenull angle of the analyzer is determined by the secondand third rows of the matrix, we do not use the first rowfor Px . dPx is obtained from dM10 . The errors for Px ,Py , and Pz are then

dPx 5 Px~Dc 1 Dp!,

dPy 5 ~Dc 2 2Dvc!Py 6 2~dC 2 dP !Pz ,

dPz 5 72~dC 2 dP !Py . (26a)

For the four-zone average, the 6 signs for C 5 645° inEqs. (26a) are averaged to zero. The misalignment er-rors have been cancelled by the four-zone average, thusleaving only the depolarization errors. The surviving Eq.(26a) after the four-zone average becomes

dPx 5 Px~Dc 1 Dp!,

dPy 5 ~Duc 2 Dvc!Py ,

dPz 5 0. (26b)

Here Duc and Dvc are the co- and cross-polarized depolar-ization for the compensator, and their sum is the depolar-ization Dc according to the last equation of the set of Eqs.(24b).

The inverse equations for c and D of Eq. (24b) are

tan D 5 Pz /Py ,

tan 2c 5 ~Py2 1 Pz

2!1/2/Px . (27)

d M 5 FDc 1 Dp 1 Da DaPx NA

Px~Dc 1 Dp! 0 NA

NA NA ~Dc 2 2Dvc!Py 6 2~dC 2

NA NA 62~dC 2 dP !Py 2 ~Dc 2

The errors for D and c can be evaluated as

dD 5 sin D cos D~dPz /Pz 2 dPy /Py!

5 2~Duc 2 Dvc!sin D cos D. (28a)

dc 5 1/2 sin 2c cos 2cS PzdPz 1 PydPy

Py2 1 Pz

22

dPx

PxD

5 1/4 sin 4c@~Duc 2 Dvc!cos2 D 2 ~Dc 1 Dp!#.

(28b)

The above errors are the same as Eqs. (24) and (25) of Ref.6 by neglecting all second-order errors in them. Equa-tions (4), (13), and (19) can be used to evaluate the linearerrors for other kinds of ellipsometers and Muellerom-eters.

7. CONCLUSIONSThe linear errors in Mueller matrix measurements havebeen formulated for the error sources of misalignment,depolarization, and incorrect retardation of the polarimet-ric components. The errors of a measured Mueller ma-trix come from the couplings between the Mueller matrixand the error matrices for both the source system and thedetection system. In other words, the errors of the Muel-ler matrix measured for a sample are not the same as theerrors for a no-sample case. The error matrices for sys-tems of PSA, PCSA, PSQA, and PCSQA have been de-rived. The error matrix for a PSA system is much sim-pler than that of a PCSQA system. The error matricesfor different systems are evaluated for the straight-through case. The general error matrix is applied to nullellipsometry, and the obtained errors in c and D are iden-tical to the results of the error analysis specifically de-rived for null ellipsometry with depolarization. The ap-plication to null ellipsometry demonstrates the use andaccuracy of the general error analysis. This general er-ror analysis can be used to evaluate error matrices forother ellipsometers or Muellerometers and to design a po-larimeter that can minimize the errors of misalignment,depolarization, and incorrect retardation.

ACKNOWLEDGMENTThis research was supported under the Naval In-HouseLaboratory Independent Research Program of the NavalAir Warfare Center at China Lake, California.

Soe-Mie F. Nee may be reached as follows: telephone,760-939-1425; fax, 760-939-6593; e-mail, soemie.nee@navy.mil.

NA

NA

!Pz 72~dC 2 dP !Py

c!Pz 62~dC 2 dP !Pz

G . (25)

dP

2Dv

Soe-Mie F. Nee Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. A 1657

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