Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects

Mueller-matrix ellipsometry using thedivision-of-amplitude photopolarimeter:a study of depolarization effects

Shankar Krishnan and Paul C. Nordine

A fully automated Mueller-matrix ellipsometer with a division-of-amplitude photopolarimeter as thepolarization-state detector is described. This device achieves Mueller-matrix ellipsometry by measuringthe Stokes parameters of reflected light as a function of the fast axis C of a quarter-wave retarder, which,in combination with a fixed linear polarizer, determines the polarization state of incident light. Thereflected Stokes parameters were Fourier analyzed to give the 16 elements of the Mueller matrix. Weinvestigated depolarization of polarized light on reflection from rough, heterogeneous, and anisotropicsurfaces by obtaining measurements on rolled aluminum and plant leaves. The results demonstrate (1) avariation of degree of polarization of reflected light with the input polarization state, (2) the precision withwhich the measured matrices describe the depolarization results, (3) effects of surface anisotropy (rollingdirection) on depolarization and cross polarization by reflection from aluminum surfaces, and (4) largevalues and differences in the depolarization effects from conifer and deciduous leaves. Depolarization oflight reflected by the aluminum surfaces was most sensitive to the angle between the plane of incidenceand the rolling direction when the incident Stokes parameters S1, S2, and S 3 were equal.

1. Introduction

Rough surfaces are known to be depolarizing.' Onemay account for this fact by using the Mueller-Stokesformalism for the interpretation of ellipsometric datainstead of the Jones matrix representation, which isapplicable only to nondepolarizing systems. Mea-surements of the Mueller matrix are needed for abetter understanding of the scattering behavior ofrough surfaces and have been enhanced and simpli-fied with the development of four-detector polarim-eters2 4 capable of instantaneous measurement of thefour Stokes parameters of light. We describe meth-ods and results for Mueller-matrix ellipsometry ondepolarizing and anisotropic surfaces performed witha division-of-amplitude photopolarimeter (DOAP).The DOAP is a device originally conceived by Az-zam4 5 that has been implemented in our laboratory

When this study was done the authors were with Intersonics,Inc., 3453 Commercial Avenue, Northbrook, Illinois 60062. Theauthors are now with Containerless Research, Inc., 910 UniversityPlace, Evanston, Illinois 60201.

Received 5 April 1993; revised manuscript received 28 October1993.

0003-6935/94/194184-09$06.00/0.it 1994 Optical Society of America.

as described in a recent publications in which calibra-tion and conventional ellipsometric measurementswere presented. In this paper we extend this re-search to fully automated Mueller-matrix ellipsom-etry (MME) in which the DOAP is used as thepolarization-state detector.

There is a small collection of experimental Mueller-matrix data available on rough and heterogeneoussurfaces. The most comprehensive study was thedoctoral dissertation of Ramsey7 who used a dualrotating compensator ellipsometer and measured theMueller matrices of rough and smooth steel surfaces.Williams8 developed a mathematical model, based onone of Ramsey's matrices obtained on an extremelyrough steel specimen, that considered depolarization,cross-polarization, and ellipticity effects. Brosseau9

and Cloude 0 have also discussed the analysis ofMueller-matrix data. Bickel et al. 1 described theMueller-Stokes representation of surface scatteringand a general approach to use of this formalism.

In two recent publications Azzam et al.' 2" 3 de-scribed the use of a four-detector photopolarimeter(FDP) for the measurement of Mueller matrices onmetallic and grating surfaces. The FDP and DOAP,although optically different, are mathematicallyequivalent, and the methodology for MME is identical

4184 APPLIED OPTICS / Vol. 33, No. 19 / 1 July 1994

for both devices. They provide nearly instantaneousmeasurement of the four Stokes parameters of arbi-trarily polarized light, which speeds the measure-ment of Mueller matrices. The DOAP is an imagingdevice and permits accurate measurements on diverg-ing light beams produced by rough and inhomoge-neous surfaces. The operating principles for thesepolarimeters and their use for measuring Muellermatrices are reviewed below.

A. Polarimetry with the DOAP and FDP

The DOAP and FDP instruments are characterizedby a 4 x 4 instrument matrix .9 that describes thetransformation of the incoming Stokes vector S intoan intensity vector I measured at the four detectors.Mathematically this transformation is written as

I= .S, (1)

where I and S are column vectors (4 x 1). Methodsfor accurately determining 9 were described in ear-lier publications.6.'4 When F is known the Stokesvector of light collected by the instrument can bededuced through the equation

S= 5V-1I. (2)

B. Mueller-Matrix Ellipsometry

In the application of ellipsometry to specular andisotropic surfaces, the corresponding Jones matrix isdiagonal.'5 A measurement of the ellipsometric pa-rameters, 4, and A, is adequate for the determinationof the optical properties. However, the Jones matrixis not useful when the surfaces of interest are depolar-izing. In these conditions the Mueller matrix Aprovides a more complete description of polarizedlight scattering from the surface.

The Mueller matrix of a surface describes the linearrelationship between the Stokes vectors, Si and Sr, ofincident and reflected light:

Sr= ' Si. (3)

In this equation Si and Sr are column vectors (4 x 1).The Mueller matrix X' is a 4 x 4 real matrix thatdepends on the angle of incidence, wavelength, theoptical properties of the reflecting surface, the indexof refraction of the ambient, and the orientation ofthe reflecting surface with respect to the plane ofincidence, if the surface is anisotropic.

For an isotropic specular reflector, A' can be ex-pressed in terms of the ellipsometric parameters, 4and A, as follows' 5 :

When surfaces are not smooth and isotropic, nonzerovalues are obtained for elements in the 2 x 2 off-diagonal blocks. Also the terms in the 2 x 2 diagonalblocks that would be numerically equal in Eq. (4)differ in the anisotropic case.

Previous MME's7 ,'6 employed multiple rotatingretarders or polarization modulators' 7 in the incidentand reflected light paths. The current instrumentincorporates the mechanical and analytical simplicityof the DOAP and requires only that the incidentpolarization state be varied. In principle a minimumof four linearly independent incident states Si areneeded, but in practice the matrix elements aretypically extracted through a least-squares analysis ofmeasurements with a larger set of incident polariza-tion states. This method is mechanically and analyti-cally simpler than other Mueller-matrix measure-ment techniques in which two (imperfect) source andanalyzer quarter-wave retarders are synchronouslyrotated.7

The procedure for measuring the Mueller matrixused in this research is similar to that described byHauge'8 and used by Azzam et al.'3 The incidentpolarizer was fixed with its azimuth P equal to zero,and the fast axis azimuth C of a quarter-wave re-tarder was permitted to rotate through at least 1800,which produced linear, circular, and elliptical polariza-tion states that are incident on the specimen. Thereflected Stokes vector S was recorded, and thecomponents of S were fitted to a five-coefficientFourier series in 2C and 4C. The derived Fouriercoefficients provided all the elements of the desiredMueller matrix.

Hauge'8 gives an explicit expression for the re-flected Stokes parameters of Sr as a function of thefast axis azimuth C:

Sr,i = Ajo + Ai 2 cos 2C + Bi2 sin 2C + Ai4 cos 4C

+ Bi4 sin 4C. (5)

In this equation the subscript r on the left-hand sideof the equation corresponds to reflection, and thesubscript i is the row index of the matrix. TheFourier coefficients are functions of the elements ofthe Mueller matrix A'. Applying the equation fourtimes for each of the four Stokes parameters, wedetermine the elements of Xij (ij = 0, 1, 2, 3) from

-cos(2q)

1

0

0

0 sin(2qj)cos(A)

0 -sin(2*)sin(A)

0

0

sin(2+)sin(A)

sin(2+)cos(A\)

1 July 1994 / Vol. 33, No. 19 / APPLIED OPTICS 4185

1

-cos(2*)

0

0

(4)

-

the expressions given below'8:

Mio =Ajo - Ai4,

Mil = 2Ai4

Mi2-= 2Bi4)

Mi3 = Bi2- (6)

The Fourier coefficient Ai2 does not appear in theseexpressions since its value is 017, but it can be used asa consistency check during the data reduction process.Imperfections that occur in the retarder used togenerate the incident polarization states are ac-counted for in a manner described by Hauge16 andAzzam et al. 13

2. Experiment

The system used for conducting Mueller-matrix mea-surements was similar to that described in an earlierpublication.6 The DOAP consisted of a ZnS beamsplitter coated with a quarter-wave thickness of MgF2and inclined at 700 with respect to the incomingbeam. The components of light reflected and trans-mitted at this beam splitter were subject to differentphase and amplitude shifts. The reflected and trans-mitted beams were incident on separate beam-splitting Glan-Thompson polarizing prisms whosetransmission axes were at 450 with respect to theplane of incidence. Four silicon photodetectors wereused to measure the light fluxes. A position-sensi-tive detector placed ahead of the ZnS beam splitterwas used for accurate and reproducible alignment ofthe instrument. The optics were designed so thatlight reflected from the specimen surface was simulta-neously imaged onto the position-sensitive detectorand the four detectors.

The DOAP was placed on a Gaertner L119X ellip-someter table, and measurements were possible atany desired angle of incidence. A Compaq Deskpro286 laboratory computer was employed for dataacquisition, for data reduction, and to operate thepolarization-state generator through a stepping mo-

tor controller. Phase-locked detection of the outputvoltages was used to improve the signal-to-noiseratio, and a 16-bit analog-digital converter providedhigh precision in the digitized signal. A schematicdiagram of the system is shown in Fig. 1.

The light source consisted of a 633-nm He-Ne laserthat was acousto-optically modulated at 25 kHz whoseoutput was first transmitted through an optical at-tenuator. The attenuator comprised a manually ro-tated linear polarizer followed by a second linearpolarizer with a quarter-wave plate cemented ontothe output end so that the outgoing beam wascircularly polarized. The attenuator was followed bya linear polarizer and a quarter-wave retarder thatdetermined the polarization state of light incident onthe specimen. These components were located onindividual motorized rotation stages. The labora-tory computer was programmed to operate the rota-tion stages so that any combination of polarizer andquarter-wave retarder fast axis azimuths was easilyobtained. In conducting MME we fixed the polarizerazimuth at P = 0 and the quarter-wave retarder fastaxis azimuth C was rotated in 100 steps. A singlematrix could be determined in 3-5 s; signal averagingwas employed so that the measurement period wastypically 20 s.

The retarder fast axis was prealigned in a mannerdescribed by Azzam et al. 13 The imperfections in theretarder were measured in a manner similar to thatdescribed by Azzam and Lopez.14 The results in anearlier publications served as a basis for determiningthe retarder imperfection parameters f and g, whichwere determined to be 0.496 and - 0.003, respectively.The DOAP was calibrated in a manner previouslydescribed. 6

Aluminum and plant-leaf surfaces were investi-gated. The aluminum was a rolled sheet of analuminum-5.6 wt. % magnesium alloy that was sus-pected to have a thin (20-50-A) magnesium oxidelayer on the surface. We studied the anisotropy ofthis material by taking measurements as a functionof the angle between the plane of incidence and the

rter Wave Retarders

Spec Cn Rotatabler/ Polarizer 4Th<k-s...Specimen

At I A | <a \St~~~~age

Fig. 1. Schematic diagram of the system used for Mueller-matrix ellipsometry.


rolling direction. The rms roughness was separatelydetermined by profilometry to be 1020 ± 80 nm witha slope of 0.099. Mueller-matrix measurements wereobtained at an angle of incidence equal to 400 for adeciduous maple leaf and for a group of yew needleswith the needles aligned parallel to the plane ofincidence. The three materials scattered some inci-dent laser light out of the plane of incidence.

Measurements on the aluminum sheet were ob-tained at an incidence angle of 500. The specimenswere mounted on a rotation stage so that it waspossible to vary the angle between the rolling direc-tion and the plane of incidence that contained theincident and reflected beams and the surface normal.That is, the specimens could be rotated about thenormal to the reflecting surface.

3. Results

The use of MME to study surface anisotropy on rolledaluminum surfaces is presented in which variationsin the degree of polarization occur with respect to therolling direction and the incident polarization state.The application of MME to the study of scattering bydeciduous and conifer leaf surfaces is also presented.All measurements were conducted at the 633-nmwavelength of a He-Ne laser.

We experimentally determined the degree of polar-ization P by the measured Stokes parameters byusing the following equation:

_ (S12 + S22 + S32)1/2

so \ I

For any given incident polarization state, P wasalso calculated from the derived Mueller matrix.The matrix was multiplied by the incident Stokesvector to yield the reflected Stokes vector. We thenobtained the value of P from Eq. (7) using thecalculated reflected state. The comparison of experi-mental and calculated values of P confirmed theaccuracy of the measurements and that the measure-ments were self-consistent.

The Stokes vector for one of the incident states thatwas used to measure the Mueller matrix was {1 1 0 01,i.e.,p-polarized light. For this case it was possible tocalculate the p-s cross polarization Xp,, which wouldbe zero for an isotropic reflector. The value of Xp isgiven by the equation

ideal reflector [see Eq. (4)] and calculate tp and A fromthe following equations:

1 1_- M01- M10~1 = -Cos' 2

( M23 - M32A& = tan~M + M)

(9)

(10)

A. Rolled Aluminum Sheet

The results obtained for rolled aluminum sheets arepresented in the following three subsections. Muel-ler matrices obtained at selected angles between therolling direction and the plane of incidence are pre-sented first. The rolling direction was uncertain by

50. In Subsection 3.A.2 we show how the degree ofpolarization for reflected light varied with the inci-dent polarization state. The measured results arealso compared with the values derived from theMueller matrices. In Subsection 3.A.3 we presentthe effects on the measured ellipsometric parameters,qj and A, of the anisotropy present in aluminum.The orientation of the aluminum was taken to be 00(parallel) when the rolling direction was in the planeof incidence and lay along the propagation directionof the incident light. The antiparallel case thenoccurred for an angle of 1800.

1. Mueller MatricesFigure 2 presents the normalized Stokes parametersof reflected light as a function of the polarization stateof incident light determined by the retarder fast axisC. Each panel gives results obtained at three angles,00, 450, and 90°, of the rolling direction with respect tothe plane of incidence. The anisotropic properties ofthe aluminum surface are demonstrated by a depen-dence of the Stokes parameters on this angle.Differences occur in the depolarization characteris-tics for right- and left-handed incident states. Wecan see these differences by comparing, for example,the results for C = 450 and C = 135. Anisotropycould also be observed in the unequal depolarizationresponse for light incident parallel and antiparallel tothe rolling direction.

The rms deviation of the larger diagonal elementsof the derived matrices was +0.005. The matrixobtained on a specimen at 00 (parallel) to the rollingdirection was determined to be

(8)(So - S/2(SO + Si)/2

Approximate or apparent values of and A werealso calculated from the derived Mueller matrices,i.e., when the values in the 2 x 2 off-diagonal blockswere close to zero.7 The procedure was to use theaverage of the three sets of matrix element pairs,which would be identical functions of * and A for an

1.0000

-0.1416

.40 = 0.0044

-0.0155

-0.1505 0.0009 0.02031

0.9890 -0.0049 0.0009

0.0035 -0.7840 0.5762

0.0297 -0.5443 -0.7865

(11)

The matrix obtained on a specimen oriented withthe rolling direction at 1800 (antiparallel) to the plane


of incidence was

1.0000 -0.1550 -0.0016 0.02001

-0.1374 0.9838 -0.0156 0.0016

-180= 0.0088 -0.0065 -0.7879 0.5488

-0.0105 0.0207 -0.5065 -0.8277

(12)

We see that the pairs of matrix elements (MO, andM10, M22 and M 3 3 , M 2 3 and -M 3 2 ) that would benumerically equal for an isotropic, nondepolarizingreflector are more nearly equal for the 0° case than for1800.

Nearly equal matrices were derived for light inci-dent from opposite directions perpendicular to therolling direction and exhibited slightly larger valuesof the elements in the 2 x 2 off-diagonal blocks thanfor the parallel direction. The matrix obtained for aspecimen oriented with the rolling direction at 900 tothe plane of incidence was

1.0000 -0.3718 0.0259 0.02801

-0.3772 1.0001 0.0066 -0.0100

=-0.0009 0.0131 -0.6275 0.6437

-0.0237 0.0213 -0.6425 -0.6603

(13)

The largest values for matrix elements that wouldbe zero in the isotropic, nondepolarizing case wereobserved for light incident at intermediate angles,which can be seen in the matrix obtained on aspecimen oriented with the rolling direction at 450 tothe plane of incidence given below:

1.0000

-0.2612

45 = 0.0930

0.0271

-0.2572 -0.1018 0.0310

0.9898 0.0194 0.0379

-0.0195 -0.7375 0.5989

0.0544 -0.5904 -0.7452

(14)

Nondepolarizing optical systems have Mueller ma-trices that obey the relationship Tr(4fTO)/[4(MO)2 ]= 1, where Tr is the trace. Values of this propertywere in the 0.95-0.99 range for aluminum, 0.29 foryew- and 0.65 for maple-leaf surfaces. This testshowed that the materials examined were depolariz-ing.

2. Source Polarization EffectsIn Figs. 3-6 we plot the degree of polarization of thereflected light versus the retarder fast axis C forseveral angles between the rolling direction and planeof incidence. The curves through the data in thefigures are the P(C) functions calculated from thederived Mueller matrices.

Figure 3 shows that the P(C) functions for reflec-

0.8

0.6

cn 0.4

0.2

0

-0.2

-0.4

0.8

0.6

0.4

0.2ciN 0

-0.2

-0.4

-0.6

-0.8

-1

1

0.8

0.6

0.4

0.2

e 0

-0.2

-0.4

-0.6

-0.8

-1

0 20 40 60 80 100 120 140 160 180

Retarder Fast Axis, C, Degrees(a)

0 20 40 60 80 100 120 140 160 180

Retarder Fast Axis, C, Degrees(b)

.0

.45.90

r0-

0 20 40 60 80 100 120 140 160 180

Retarder Fast Axis, C, Degrees(c)

Fig. 2. Plots of the normalized Stokes parameters: (a) Sj, (b) S2,(C) S3 versus the retarder fast axis C in the polarization-stategenerator. The three curves in each panel were obtained forindicated angles of 00, 450, and 90° between the rolling directionand the plane of incidence.

tion parallel and antiparallel to the rolling directionare quite different. We expected this result becausethe Mueller matrices were different for the twodirections. The P(C) functions calculated from theMueller matrices are in good agreement with themeasured data, which implies that the data areself-consistent and that accurate determinations ofthe Mueller matrices were obtained.


.

0.9900

0 30 60 so 1800 ~ ~ ~ ~ ~ m

-MM 180

0 30 60 90 120 150 180Retarder Fast Axis, C, degrees

Fig. 3. Plots of the degree of polarization P versus the retarderfast axis C in the polarization-state generator for angles of 00 and1800 between the rolling direction and the plane of incidence. MMis the Mueller matrix.

Figure 4 shows that the P(C) functions for reflec-tion of light incident from either side perpendicular tothe rolling direction (90° and 2700) are similar, whichis in agreement with the behavior expected because ofthe rolling symmetry. However, these data are notas self-consistent with the derived Mueller matricesas for the measurements at 0° and 1800. Figures 5and 6 show that the intermediate cases of reflection at+45° to the rolling direction are less similar thanwould be expected for symmetry reasons. Thesedifferences may result from a greater sensitivity toerror in locating the orientation of the rolling direc-tion.

The results illustrate that the degree of polariza-tion for light reflected from rough surfaces maydepend on the incident polarization state, which wasobserved previously by Azzam.'9

Figures 7 and 8 present values of the degree ofpolarization as a function of the angle between therolling direction and the plane of incidence for inci-dent states with C = 110° and C = 00, which areincident states for which the effects of orientation

1.02I.

0

*E 10CL

ID

) 0.98

0.960 30 60 90 120 150

Retarder Fast Axis, C, degrees

180

Fig. 4. Plots of the degree of polarization P versus the retarderfast axis C in the polarization-state generator for angles of 900 and2700 between the rolling direction and the plane of incidence.

0.

0

0.9

0.960 30 60 90 120 150 180



were maximum or minimum, respectively. Thejagged lines in the figures connect the points calcu-lated from the Mueller matrices. A sinusoidal varia-tion is evident in both figures with a minimum in thedegree of polarization approximately normal to therolling direction.

The incident state of P = 0 and C = 1100 hasnormalized incident Stokes parameters that are ap-proximately equal: S = 0.586, S2 = 0.492, S3 =

-0.643. ForP = C = 0° they are Si = 1.00 and S2 =0.492, S3 = 0. In the former case approximatelyequal weighting occurs for all elements of the Muellermatrix in determining the reflected Stokes param-eters. Because the degree of polarization p for thereflected light is calculated from the sum of thesquares of the outgoing Stokes parameters [see Eq.(7)], it is more sensitive to the values in the off-diagonal blocks of the Mueller matrix in the formercase. For the latter case the terms in the off-diagonal blocks contribute less to the degree of polar-ization. If the incident Stokes parameter magni-tudes were to be chosen as S = S 2 = S 3 = 0.577,perhaps the largest variations would be observed,

* 1351.01 A 225

0.9-MM 135--M 225 * i

0

a-

5'0.97

0.95 . . . . . . . .0 30 60 90 120 150 180




1.01 1.02

1.02011

0C)11a. 1

.9

.0C0a 0.98

D

0.960 90 180 270

Angle From Rolling Direction

0.2

jXa

76 0.15

Co0CL

W 0.1

a)ID

0

0360

Fig. 7. Plot of the degree of polarization P versus the anglebetween the rolling direction and the plane of incidence with theretarder fast axis C set to 110°.

because this choice gives maximum weight to theeffect of off-diagonal elements of the Mueller matrixon the degree of polarization.

Figure 9 presents thep-s cross polarization (Xp,) ascalculated from the measurements for incident lightwith C = 0. The cross polarization was as large as0.12 and a maximum at angles -450 to the rollingdirection.

Note that the differences between measured valuesof P and those calculated from the Mueller matricesare all less than 0.01, which is close to the precisionwith which the measurements are made. This agree-ment indicates the overall self-consistency of themeasurements. Since the values of depolarizationcalculated from the matrices and those measured forparallel and antiparallel incidence agree to +0.003 orbetter, it is suspected that the occurrence of valuesfor P as large as 1.01 result from slight errors in theinstrument calibration or measurements.

3. EllipsometryThe aluminum sheet exhibited apparent ellipsomet-ric parameters, * and A, derived from each Mueller

1.02a1106it0.

15.o

aIL 0.98

a)c

0

0.9

0 90 180 270Angle From Rolling Direction

FX~

360

Fig. 8. Plot of the degree of polarization P versus the anglebetween the rolling direction and the plame of incidence with theretarder fast axis C set to 0°.

0 90 180 270 360

Angle From Rolling Direction

Fig. 9. Plot of the degree of p-s cross polarization Xps versus theangle between the rolling direction and the plane of incidence.

matrix, which varied with the angle between therolling direction and plane of incidence. The resultsare illustrated in Fig. 10 in which A, A, and the rollingangle are plotted in a three-dimensional fashion withprojections in three planes. The data in this plotmay be used to locate the rolling direction moreaccurately.

B. Leaf Surfaces

Depolarization of reflected light was greater for yewthan for maple leaves. This result is consistent withthe findings of Kalshoven and Dabney20'21 that depo-larization signatures may be used as a means forremote sensing of plant species in a canopy. Thematrix obtained for the maple leaf, at an angle ofincidence of 40° and a wavelength of 633 nm, is given

Fig.10. Three diensional plot of tho apparent values for $ andA and the rolling angle projected in three planes.


by

maple [ 1.0000 -0.5555 -0.0239 -0.02771

-0.5805 0.7655 -0.0196 0.0012

0.0636 -0.0726 -0.4061 0.1270

-0.0679 0.0141 -0.1715 -0.3476

(15)

The matrix for the yew, measured at the same angleand wavelength, is given by

[ 1.0000 -0.1521 -0.0019 -0.0199

-0.2172 0.2230 -0.0043 0.0205

yew = 0.0262 0.0053 -0.0868 0.0435

-0.0795 0.0104 -0.0543 -0.0298

(16)

Figure 11 presents the degree of polarization oflight reflected from the leaves as a function of theretarder fast axis C. The solid curves give the valuespredicted by the derived Mueller matrices. The de-gree of polarization is observed to be a maximum forboth types of leaf at C 450 and 135°, where circularpolarization occurs for the incident states. Thereare differences between the left- and right-handedstates, and the minima do not occur at C = 900 but areshifted in opposite directions for the experimentalpoints and matrix results. These effects indicatethat optical rotation may occur in the interaction oflight with the leaves.

The maximum degree of polarization for lightreflected from leaves occurs when S, = 0 in theincident light beam and the minimum occurs for S, =1 (C = 0°, 90°), which can be understood as a result ofthe larger reflectivity at oblique incidence for s- thanfor p-polarized light. For s-polarized light the firstsurface reflects a greater fraction of the incident

0.8

C0 0.6

0

004

{a 0.2 0 nYew

0 I.I, . . 1, 10 30 60 90 120 150 180


Fig. 11. Plots of the degree of polarization P versus the retarderfast axis C in the polarization-state generator for maple and yewleaves.

beam so that the collected light includes a smallercontribution of scattering from multiple internal leafsurfaces, which causes depolarization. Thus, if theincoming Stokes vector was set to be 1 -1 0 0}(s polarized), it is expected that the greatest degree ofpolarization in reflected light would be obtained.

C. Errors

The precision with which the depolarization valuescalculated from the Mueller matrices described theexperimental measurements was somewhat greaterthan can be expected from the precision in the actualmeasurements. This difference of precision is evi-dent in the cross-polarization results. The errors inthe values of Xp, should have been of the order of+0.05, assuming that the Stokes parameters weremeasured to ± 0.003.

It therefore appears that the high degree of self-consistency observed in the experimental results ispartly an artifact of the method used to measure thematrix itself. In fact, the polarization-state genera-tor was operated in this study so that all inputpolarization states were on a limited segment of thePoincar6 sphere surface. It is expected that when amore varied set of input states is used, the degree ofself-consistency would be reduced. This reductionrequires rotation of the polarizer as well as thequarter-wave plate in the polarization-state generator.It would be essential to provide a fifth detector tomonitor the incident intensity, which was essentiallyconstant for the conditions used in this study.

4. Concluding Remarks

In this study the DOAP has been used in conjunctionwith an automated polarization-state generator toachieve automated Mueller-matrix ellipsometry.The DOAP simplifies the methodology to obtainMueller matrices by measuring all four Stokes param-eters for reflected light as a function of the incidentpolarization state. The present facility providesMueller-matrix measurements in 3-5 s. Even fastermeasurements would be obtained if a polarization-state generator with electronic phase modulationwere used.

Rolled aluminum surfaces of 1000-nm rms rough-ness exhibited anisotropy, which was evident in theorientation dependence of the Mueller matrices, theapparent values of the ellipsometric parameters, andthe values of the degree of polarization and thecross-polarization of reflected light. It was demon-strated that the degree of polarization of reflectedlight depended also on the incoming polarizationstate for rolled aluminum as well as deciduous andconiferous leaves. The degree of polarization of lightreflected by plant-leaf surfaces was largely explainedas the result of greater external and smaller internalscattering by the leaves for s- than for p-polarizedlight.

This research was partially supported by a NationalScience Foundation Phase II Small Business Innova-tive Research award. We thank D. P. DeWitt and


Mark Pellerin of Purdue University for providing thespecimens, roughness data, and assistance in themeasurements on aluminum. We also thank JamesKalshoven of NASA's Goddard Space Flight Centerfor suggestions regarding laser depolarization byvegetation.

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Documents

Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects