The rotating quarter-wave method is the simplest ofa class of techniques for measuring the four-elementStokes vector, in which the unknown polarizationstate of the radiation under measurement is timemodulated.1–3 The method is best described in Ref. 1(p. 103). Let us consider a radiation beam, partially orcompletely polarized, with an unknown polarizationstate, passing through a rotatable quarter-wave plateand then through a fixed linear polarizer followed bya detector (Fig. 1). We introduce a Cartesian refer-
ence system whose z axis is parallel to the direction ofthe radiation beam wave vector k, and the x axisdirection is parallel to the axis of the fixed linearpolarizer and let S � �S0, S1, S2, S3� be the Stokesvector representing the polarization state of the ra-diation in the given reference system. When thequarter-wave plate axis is rotated at an angle � withrespect to the reference x axis, the Stokes vector ofthe radiation incident on the detector is
SD��� � LP · QW��� · S, (1)
where
LP �12 �
1 1 0 01 1 0 00 0 0 00 0 0 0
�, (2a)
L. Giudicotti ([email protected]) is with the Diparti-mento d’Ingegneria Elettrica, Università di Padova, Via Gradenigo6�a, 35131 Padova, Italy and is with M. Brombin at the ConsorzioRFX, Associazione EURATOM-ENEA Sulla Fusione, Corso StatiUniti 4, 35127 Padova, Italy.
Received 28 August 2006; accepted 10 December 2006; posted 10January 2007 (Doc. ID 72817); published 23 April 2007.
are the Mueller matrices of the linear polarizer set to0° and of the quarter-wave plate set at the angle �.1,4
The intensity of the radiation at the detector is
ID��� � �SD�0 �12�S0 � S1 cos2 2� � S2 sin 2� cos 2�
� S3 sin 2��
�12 S0 �
14 S1 � S3 sin 2� �
14 S1 cos 4�
�14 S2 sin 4�. (3)
This expression can be considered the truncated Fou-rier series of a function of the angle �, with a periodof 2:
ID��� �a0
2 � �n�1
2
�a2n cos 2n� � b2n sin 2n��, (4)
where the coefficients are
a0 �1�
�0
�0�2
ID���d�, (5a)
a2n �1�
�0
�0�2
ID���cos 2n�d� n � 1, 2, (5b)
b2n �1�
�0
�0�2
ID���sin 2n�d� n � 1, 2, (5c)
and �0 is an arbitrary initial angle. By a comparisonof Eqs. (3) and (4) we find
a0 � S0 �12 S1, (6a)
a2 � 0, (6b)
b2 � �12 S3, (6c)
a4 �14 S1, (6d)
b4 �14 S2, (6e)
so that the Stokes vector elements are given by
S0 �2�
�0
�0�2
ID����12 � cos 4�d�, (7a)
S1 �4�
�0
�0�2
ID���cos 4�d�, (7b)
S2 �4�
�0
�0�2
ID���sin 4�d�, (7c)
S3 � �2�
�0
�0�2
ID���sin 2�d�. (7d)
In the practical application of the method, the Stokesvector is usually measured by rotating the quarter-wave plate at a constant angular frequency so that� � t and measuring the detector signal componentsat the frequencies 2 and 4 with a signal analyzer.The technique is suitable for measuring a time-dependent Stokes vector, whose changes occur on atime scale of � �� 2�. For this reason it is widelyused in many applications of real-time ellipsometrysuch as the study of film growth dynamics.3
A similar technique, also based on the polarizationmodulation provided by a rotating quarter-wave plate,has been proposed for the diagnostics of the internalmagnetic fields in a fusion plasma by multichord far-infrared (FIR) polarimetry.5 In this case selected ele-ments of the plasma Mueller matrix are determined byFourier analysis of the detector signal, providing in-formation on the time evolution of the spatial profilesof the plasma electron density and internal magneticfield.4
However, the method is mostly used to measure sta-tionary polarization states. In this case the quarter-wave plate is rotated (for example, by a stepper motor)at N � 1 discrete angular positions �i �i � 0, N� uni-formly spaced on a complete turn, and for each posi-tion the detector signal is recorded. Let Ii � ID��i� bethe recorded samples; then by using the trapezoidal
Fig. 1. Rotating quarter-wave method for the measurement of theStokes vector.
These expressions take into account that the lastmeasurement IN � ID��N� � ID��0 � 2� is a repetitionof the initial measurement I0 � ID��0�.
We have used this method, along with a more con-ventional ellipsometric technique, for an experimen-tal study on the way surface depositions attributableto exposure to a tokamak plasma affect the polariza-tion properties of metallic mirrors and corner cuberetroreflectors in the FIR �� � 118.8 �m�.6–8 Thisstudy is part of the research effort for the design of aFIR polarimeter for the diagnostic of the electrondensity and magnetic field spatial profiles in the fu-ture ITER (International Thermonuclear Experimen-tal Reactor) plasma fusion experiment.9
In the use of this method we encountered two prob-lems. The first is the correct determination of mea-surement errors on the Stokes vector elements andon the other parameters derived from them. Theseshould be determined from the experimental vari-ances �i
2 of the measured Ii that can be measured byrepeatedly sampling the detector signal at each an-gular position. The second problem is related to thephysical consistency of the measured Stokes vector S.In fact the Stokes vector elements of any physicalradiation beam must always satisfy the condition
P ��S1
2 � S22 � S3
2�1�2
S0� 1, (9)
where P is the degree of polarization. However, owingto experimental errors this condition is often not sat-isfied. This occurred quite frequently in our experi-ment, where owing to the fluctuations of the FIRlaser output power and to the relatively high noiseof the room temperature pyroelectric detectors, themeasurement error on the recorded signal was notnegligible, especially when the polarization degree ofthe radiation to be measured was close to 1. Unfor-tunately, when the measured data do not satisfy theabove condition and no estimate of measurement er-rors is available, whether to keep or reject the resultsseems to be somewhat arbitrary. Moreover, in theseconditions the above theory does not give any indica-tion about how to derive a result consistent with Eq.(9). In this case the more reasonable choice seems tobe to discard the data and repeat the experiment.This may be a limitation of this analysis method.
The correct determination of measurement errors,however, would be straightforward. In fact, Eqs.(8) represent a linear transformation mapping theN � 1 measured intensities Ii to the four elements Sk
of the Stokes vector S. Therefore the measurementerrors of the Sk are described by their covariancematrix that can be calculated from the measuredvariances �i
2 by the conventional propagation errortheory.10 However, the problem of the determinationof measurement errors and of the physical consis-tency of the measured Stokes vector can be solvedsimultaneously by an analysis technique based on aleast-squares best fit.
2. Data Analysis by a Least-Squares Best Fit
Both the above problems can be simultaneouslysolved by a data analysis technique based on a least-squares best fit. By this method, which uses the sameset of recorded data, the calculation of measurementerrors on the measured Sk and on the parametersderived from them, can be carried out in a simplebut rigorous way. In addition, a way for obtaining aStokes vector that always satisfies Eq. (9) is immedi-ately recognized. Let us first consider the implemen-tation of the method without considering Eq. (9). Westart by constructing the 2 function as
2 � �i�0
N 1
�i2�Ii � ID��i, S� 2, (10)
where the quantities �i2 are the variances of the mea-
sured Ii determined by repeatedly sampling the de-tector signal, and ID��i, S� is the expected signal atthe position �i as given by Eq. (3). The four elementsof the Stokes vector are found as the set of values thatminimizes the 2. These are calculated by solving thefour equations:
� 2
�Sk� 0, k � 0, 3. (11)
Setting wi � 1��i2 and Ri � Ii � ID��i, S�, these equa-
tions can be written
�i�0
N
wiRi � 0, (12a)
�i�0
N
wiRi cos2 2�i � 0, (12b)
�i�0
N
wiRi sin 2�i cos 2�i � 0, (12c)
�i�0
N
wiRi sin 2�i � 0. (12d)
Since the fitting function ID��i, S� is a linear functionof the parameters Sk to be determined, the system ofEqs. (12) always has a solution, and this solution isunique.10,11 The system can be written in matrix formand its solution, i.e., the four parameters Sk, can becalculated by a matrix inversion.10 However, in lightof the extension of the method presented in Section 4,we have found it more convenient to use a numericalprocedure for nonlinear minimization.12
It is also useful to represent the measured Stokesvector in the form:
where P, �, and � are the degree of polarization, theazimuth, and the ellipticity angles, respectively. Wewill call them the polarization parameters. They arecalculated from the four elements of S as
P ��S1
2 � S22 � S3
2�1�2
S0, (14a)
� �
12 arctan�S2�S1� for S1 � 0 and S2 � 0
12 arctan�S2�S1� � for S1 � 0 and S2 � 0
12 arctan�S2�S1� �
2for S1 � 0
2for S1 � 0 and S2 � 0
(14b)
�12 arcsin S3
�S12 � S2
2 � S32�1�2�. (14c)
For S1 � S2 � 0 � is undetermined. Eqs. (14) give0 � � � and ��4 � � �4 according to the usualconventions.1
The solution provided by this method is more gen-eral than the previous one. First the best-fit methodtakes into account the fact that the data may havedifferent measurement errors. In addition in the so-lution provided by Eqs. (8) the data Ii need to be takenat regular intervals, spanning a complete period ofthe fitting function, whereas in the best-fit approachthey can be taken at regular or irregular intervalscovering any angular region. In fact it is possible todemonstrate that Eqs. (8) are a special case of Eqs.(12), and when the Ii are taken at regular intervals on
a complete rotation and the �i2 are all equal, the two
sets of equations become the same expressions.
3. Calculation of the Measurement Errors
The best-fit method also provides a solid theoreticalframework for the calculation of the uncertaintiesof the Stokes vector elements and of the polariza-tion parameters derived from them. From the stan-dard theory of least-squares best fit, the standarddeviations of the measured Sk are the square roots ofthe diagonal elements of the covariance matrix C ofthe fitted parameters.10,11 In our case the measured Ii
are independent of each other, and the matrix C isgiven by
C � ��1, (15)
where � is the curvature matrix that represents thecurvature of the 4D 2 surface in correspondence toits minimum. � is a symmetric matrix whose ele-ments are given by
�k,l �12� � 2
�Sk�Sl
S. (16)
From Eq. (10) we find
Once the covariance matrix of the S elements isknown, it is possible to calculate the measurementerrors associated with the polarization parameters.To this purpose let us indicate the polarization pa-rameters as the three elements of the variablep � �p0, p1, p2� � �P, �, � that is related to the mea-sured S by the (nonlinear) transformation repre-sented by Eqs. (14). Then according to the standarderror propagation, the measurement errors on p arecalculated from those of S by a linear expansion ofthe transformation Eqs. (14) around the fitted S andthe covariance matrix of the p parameters is10
and the derivatives are again evaluated in correspon-dence to the measured S. From Eqs. (14) we find
Therefore from Eqs. (15)–(20) the measurement er-rors both on the elements of the Stokes vector and onthe polarimetric parameters P, �, and � can be cal-culated in a consistent way.
4. Data Analysis by a Constrained Best Fit
The least-squares best-fit analysis can be further ex-tended to solve the problem of the physical consis-tency of the measured Stokes vectors. In this case Eq.(9) can be used as a constraint to force the best-fitprocedure to find the physically consistent Stokesvector most compatible with the measured data. Weshow now that, although Eq. (9) is expressed by aninequality constraint, the solution found in this wayalways has P � 1. In fact, the constraint of Eq. (9)restricts the search for the minimum of the 2 withina region of the 4D S space whose boundary is definedby the condition P � 1. We will call it the allowedregion. Suppose now that we have a set of data forwhich the unconstrained fit gives a solution withP � 1, outside the allowed region. We know from thelinear nature of the problem that this solution isunique or, in other words, that this minimum, al-though located outside the allowed region, is the onlylocal minimum of the 2. In this case it can be shownthat the lowest value of the 2 within the allowedregion will always be located on its boundary.13
Therefore when the unconstrained best fit gives anonphysical Stokes vector with P � 1, the same datacan be analyzed again including the inequality con-straint Eq. (9). The solution, however, will alwayshave P � 1, i.e., will be a Stokes vector correspondingto a completely polarized radiation. From these con-siderations it follows that from the practical point ofview the inequality constraint Eq. (9) can be replacedby the equality constraint P � 1, i.e., by
S0 � �S12 � S2
2 � S32�1�2, (21)
resulting in a considerable simplification of the best-fitprocedure. For this particular problem the more con-
venient procedure is to use Eq. (21) to express S0 as afunction of S1, S2, and S3 in Eq. (3):
ID���, Sk� �12��S1
2 � S22 � S3
2�1�2 � S1 cos2 2�
� S2 sin 2� cos 2� � S3 sin 2� , k � 1, 3.(22)
This is a straightforward application of the tech-nique known as the method of elements,10 which issimpler than the better-known method of Lagrangemultipliers. Now the 2 is a function of S1, S2, and S3only, and the problem is reduced to an unconstrainedminimization in three dimensions. Putting Ri � Ii
� ID���i, Sk�, k � 1, 3, the solution is found by solvingthe system
�i�0
N
wiRicos2 2�i �S1
�S12 � S2
2 � S32�1�2�� 0,
(23a)
�i�0
N
wiRisin 2�i cos 2�i �S2
�S12 � S2
2 � S32�1�2�� 0,
(23b)
�i�0
N
wiRi�sin 2�i �S3
�S12 � S2
2 � S32�1�2�� 0.
(23c)
Contrary to the unconstrained best fit, the depen-dence of the fitting function Eq. (22) from the fitparameters is no longer linear, and therefore a nu-merical method for nonlinear optimization is re-quired to find the solution. On the other hand, thenonlinear method can also be used for the linear case,and therefore, except for the different fitting functionand for the number of parameters, the same proce-dure can be used in both cases. Note also that unlikein the linear case the uniqueness of the solution is nolonger guaranteed so that there can be two or more
Stokes vectors, all with P � 1, all equally compatiblewith the experimental data. Once the solution isfound, the measurement errors on the three Stokesvector elements and on the polarimetric parametersare calculated as in Eqs. (15) and (16). In this case theelements of the curvature matrix are
�11� �12 �
i�0
N
wi12�cos2 2�i �
S1
�S12 � S2
2 � S322
� Ri
S22 � S3
2
�S12 � S2
2 � S32�3�2�, (24a)
�22 � �12 �
i�0
N
wi12�sin 2�i cos 2�i �
S2
�S12 � S2
2 � S322
� Ri
S12 � S3
2
�S12 � S2
2 � S32�3�2�, (24b)
�33 � �12 �
i�0
N
wi12��sin 2�i �
S3
�S12 � S2
2 � S322
� Ri
S12 � S2
2
�S12 � S2
2 � S32�3�2�, (24c)
�12 � � �21 � �12 �
i�0
N
wi12�cos2 2�i �
S1
�S12 � S2
2 � S32
� �sin 2�i cos 2�i �S2
�S12 � S2
2 � S32
� Ri
S1S2
�S12 � S2
2 � S32�3�2�, (24d)
�23 � � �32 � �12 �
i�0
N
wi12�sin 2�i cos 2�i �
S2
�S12 � S2
2 � S32
� ��sin 2�i �S3
�S12 � S2
2 � S32
� Ri
S2S3
�S12 � S2
2 � S32�3�2�, (24e)
�13 � � �31 � �12 �
i�0
N
wi12�cos2 2�i �
S1
�S12 � S2
2 � S32
� ��sin 2�i �S3
�S12 � S2
2 � S32
� Ri
S1S3
�S12 � S2
2 � S32�3�2�, (24f)
and the matrix D� of the �S1, S2, S3� → ��, � trans-formation is
D � � � �12
S2
S12 � S2
2
12
S1
S12 � S2
2 0
�12
S1S3
�S12 � S2
2�
12
S2S3
�S12 � S2
2
12
�S12 � S2
2
S12 � S2
2 � S32�.
(25)
5. Detection of Systematic Errors
One of the circumstances that may result in the mea-surement of a nonphysical Stokes vector is the pres-ence of systematic errors in the angular positions �i.These represent the angle between the axes of therotatable quarter-wave plate and the fixed linear po-larizer. In conventional visible or near-infrared ellip-sometry, the effects of errors in both the retardationand orientation of the wave-plate compensators havebeen extensively studied.14,15 The retardation errorsof quarter-wave plates are expected to scale inverselywith the operational wavelength, and in the FIR theycan be considered negligible.16 On the other hand,even when a good-mechanical-quality, step-motor-driven, rotation stage is used to rotate the quarter-wave plate, systematic errors on the axis orientationcan build up as a result of poor encoder performancein continuous back-and-forth movements. Thereforethe initial angular settings need to be periodicallychecked by a separate calibration procedure, usingadditional elements in the optical train such as anadditional linear polarizer. However, the rotatingquarter-wave method can be extended to take intoaccount and correct occasional misalignments of thequarter-wave-plate axis.
Let us assume that the quarter-wave-plate axishas an initial misalignment � with respect to theaxis of the linear polarizer so that the measuredangles �i are all affected by a systematic error. Thenthe true angle is �� � � � �, and the detector signal inEq. (3) is
ID��� �12 S0 �
14 S1 � S3 sin�2�� � ��
�14 S1 cos�4�� � �� �
14 S2 sin�4�� � �� .
(26)
This expression can be written
ID��� �12 S0 �
14 S1 �
12 S3 sin 2� cos 2�
�12 S3 cos 2� sin 2�
�14�S1 cos 4� � S2 sin 4��cos 4�
�14�S2 cos 4� � S1 sin 4��sin 4�. (27)
The Fourier coefficients of Eq. (4) are related to theStokes vector elements and the angle � by
Since � is a small correction angle, we can well as-sume ��4 � � � �4. Then from Eqs. (28b) and(28c) we find
� �12 arctan�a2
b2. (29)
The Stokes vector elements Sk can also be calculatedfrom Eqs. (28). However, a better strategy is to re-move the systematic error, calculating the correctangles �i� � �i � �, and process the data again withone of the previous methods. Note that a direct effectof a misalignment error is that the conventional Fou-rier analysis yields a2 � 0. Therefore it is possible todetect (and remove) this systematic error by simplyinspecting the value of the a2 obtained, provided themeasured radiation has S3 � 0, i.e., is elliptically orcircularly polarized. This is a useful technique forperiodically checking the initial calibrations, withoutresorting to specific measurements on known polar-ization states. Also the calculation of higher-orderFourier coefficients, which are expected to be all null,may be useful as a general consistency check. Anexample of misalignment detection and removal willbe given in the next section.
6. Examples of Stokes Vector Measurements
These analysis methods have been used in a seriesof experiments for the characterization in the FIR�� � 118 �m� of the polarimetric properties of metal-lic mirrors whose surfaces have been exposed to a
tokamak plasma.6,7 The objective of these tests was toidentify possible sources of errors in the measure-ment of the poloidal magnetic field spatial profile byFIR polarimetry in the future ITER fusion experi-ment.9
The experimental setup is shown in Figs. 2 and 3.The radiation source is a CO2-pumped, CH3OH FIRlaser, whose output is a linearly polarized beam at� � 118.8 �m. Specific polarization states are gener-ated by a polarization state generator (PSG) consti-tuted by a half-wave plate, a linear polarizer, and aquarter-wave plate, mounted on precision rotationstages. The polarization state of the radiation re-flected by the mirror is measured by the quarter-wave plate mounted on a step-motor-driven rotationstage, a fixed linear polarizer, a focusing parabolicmirror, and a room-temperature pyroelectric FIR de-tector. The detector signal is measured as a functionof the angular orientation of the quarter-wave plateand is normalized to an identical reference detectorsampling the laser output by means of a 45°, 25 �mthick, kapton film beam splitter, to take into accountfluctuations of the laser power output.17 A TPX17 lens
Fig. 2. Schematic of the Stokes polarimeter for the polarimetriccharacterization of plasma-exposed mirrors. LP, linear polarizer;M, mirror; BS, beam splitter; HW, half-wave plate; QW, quarter-wave plate; RQW, rotating quarter-wave plate.
Fig. 3. (Color online) Top view of the Stokes polarimeter for thepolarimetric characterization of plasma-exposed mirrors.
Fig. 4. (Color online) Example of data analysis for a Stokes vectormeasurement. The two waveforms show the reconstruction of thedetector signal obtained with the Fourier technique (FT) and witha four-parameter, unconstrained best fit (BF4P).
focuses the beam to a spot of �4 mm diameter on themirror surface, allowing the measurement of selectedareas of the exposed surface. The FIR beam is inten-sity modulated at 3 kHz by a mechanical chopper onthe pump CO2 beam. The detector outputs are pro-cessed by simple electronic cards that implement thebasic functions of a lock-in amplifier, and their outputsignals are recorded by a digital oscilloscope. All themeasurements are controlled by a LabVIEW soft-ware interface. Measurements are fully automatic,and data are taken every 5° for a 180° or 360° rotationof the quarter-wave plate. For each angular position,5000 samples of both signals are recorded and stored.These data are later processed to calculate the fourStokes parameters and the corresponding values ofP, �, and �.
Figure 4 shows an example of a Stokes vector mea-surement. The values of the polarimetric parametersare shown in Table 1. Data analysis by the Fouriermethod in this case gives a slightly overpolarizedStokes vector �P � 1.03�. However, in the absence ofthe determination of measurement errors it is notpossible to accept or reject the result. The analysis ofthe same data with the four-parameter best fit, yieldsa physically consistent Stokes vector with P � 0.98 asthe consequence of taking into account individual er-rors for each measured data. The calculation of themeasurement errors on the polarization parametersP, �, and � shows that the results of both methods areconsistent and that the radiation polarization state iscompatible with that of a completely polarized radi-ation beam, linearly polarized at �15°, as roughlyexpected.
A second example is shown in Fig. 5 and Table 2. Inthis case the Fourier analysis yields a remarkablyoverpolarized Stokes vector �P � 1.12� and a similarresult is obtained also with the four-parameter bestfit. For this set of data the measurement error on P isstill compatible with a fully polarized Stokes vector sothat it is reasonable to force the analysis by applyingthe constrained, three-parameter best fit. This givesa completely polarized Stokes vector with essentiallythe same values of � and � as the previous methods.Depending on the particular application one may notbe satisfied by the above procedure and may want torepeat the measurements. However, it is clear thatfor a specific data set the technique gives the resultsmost compatible with the experiment.
In Fig. 6 and Table 3 we show an example of de-tection and cancellation of a systematic error. Figure
Fig. 5. (Color online) Another example of data analysis for aStokes vector measurement. Waveforms show the reconstructionof the detector signal obtained with the Fourier technique (FT);with a four-parameter, unconstrained best fit (BF4P); and finallyby a three- parameter, constrained best fit (BF3P).
Table 1. Polarimetric Parameters Obtained from the Data of Fig. 4 byUsing Different Analysis Methods
Fourier Best Fit, Four Parameters
P 1.03 0.98 0.09� 166.0° 165.5° 2.5°� �1.4° �1.3° 2.0°
Table 2. Polarimetric Parameters Obtained from the Data of Fig. 5 byDifferent Analysis Methods
Fig. 6. (Color online) Example of data analysis with subtractionof a systematic error. Upper panel: measured data and fitted wave-forms. The significant value of the cos 2� component suggests thatthere is a systematic error � in the measured data angular posi-tions �i. The analysis of Section 5 yields � � 6.26°. Lower panel: thesame data with new, corrected angular positions �i� � �i � � andthe fitted waveforms.
6(a) shows the data taken on an elliptically polarizedbeam and the reconstructed waveform obtained fromthe Fourier analysis. The cos 2� component is alsoplotted, suggesting that the data may be affected bya significant systematic error � in the angular posi-tion �i. This has been determined to be � � 6.26° bythe technique in Section 5. Figure 6(b) shows theresults of the analysis of the same set of data in whichthe angular positions �i� � �i � � have been corrected.Note that the Fourier and four-parameter best-fitmethods give essentially the same results. These ex-amples demonstrate that the analysis techniques ofthe previous sections are effective for the analysis ofdata in Stokes polarimetry.
7. Determination of Mueller Matrices
In this section we discuss the previous techniques inthe light of the problem of the experimental determi-nation of physically consistent Mueller matrices. Inthe following we will call a Stokes vector for whichP � 1 a physical Stokes vector (PSV). Only PSVsrepresent physical beams of radiation. In the sameway we will call a physically consistent Mueller ma-trix (PMM) a matrix whose output in correspondencewith any input PSV is again a PSV. Exactly as forexperimental Stokes vectors, measurement errorsoften result in experimental Mueller matrices thatare not PMMs. The problem of the determination ofPMMs has been considered by many authors18–23 andremains an important issue in polarimetry. Based ona theoretical result by Cloude24,25 a technique hasbeen developed by which an experimental Muellermatrix can be postprocessed to recover a PMM, sup-posedly removing the effect of measurement errors.1However, the technique of Cloude is based on thealgebraic properties of Mueller matrices, and it isnot guaranteed to yield the PMM most compatiblewith the experimental data. In the following we willpresent some considerations on the possibility ofconstraining an experimental Mueller matrix to be aPMM by constraining the Stokes vectors measuredduring their determination to be PSVs. We first recallthat a sufficient condition for a Mueller matrix to bea PMM is that it be a PMM for any completely po-larized PSV.18 This is based on the fact that anypartially polarized PSV can be decomposed into thesum of two, orthogonal, completely polarized PSVsand that the sum of two PSVs is also a PSV. Thereforeour analysis for the determination of experimentalPMMs can be restricted to the case in which the inputis a completely polarized, generic PSV.
Let us consider a simple case. Many optical sys-tems can be represented by a Mueller matrix of theABCD type1:
M � �A B 0 0B A 0 00 0 C D0 0 �D C
�. (30)
Now the condition for which this type of matrix is aPMM is that the four elements obey
A2 � B2 � C2 � D2. (31)
To measure an ABCD matrix, it is convenient to usean input beam linearly polarized to 45°, representedby the Stokes vector
Sin ��1010�. (32)
Then the M-transformed output Stokes vector will be
Sout � MSin ��ABCD�. (33)
Therefore it is sufficient to constrain Sout to be a PSVto ensure that the experimental ABCD matrix will bea PMM. Right (or left) circularly polarized radiationcan also be used for the same purpose.1
However, for more general Mueller matrices theproblem is not so simple. Let us discuss the case ofa general 4 � 4 Mueller matrix. The conventionalmethod of determining the 16 elements of a generalMueller matrix consists of measuring the polariza-tion state of the output radiation in correspondencewith the four standard input Stokes vectors26:
Sin1 ��1100�, Sin2 ��
1�100�, Sin3 ��
1010�, Sin4 ��
1001�.
(34)
It is possible to show that requiring that the fourcorresponding output Stokes vectors be PSVs is onlya necessary but not a sufficient condition for a genericMueller matrix to be a PMM. Consider, in fact, thefollowing Mueller matrix27:
formed beam never exceeds the input intensity. Theoutput Stokes vectors corresponding to the four inputStokes vectors in Eq. (34) have P � 0.979, 1, 1, and 1,respectively, and they all are PSVs. However, M isnot a PMM. To demonstrate this we have calculatedthe M-transformed Stokes vector in correspondenceto a set of completely polarized input Stokes vectors,spanning all the polarization states on the Poincarésphere. Figure (7) shows the contour plot of the func-tion representing the polarization degree P of theoutput as a function of the � and � angles of the inputbeam. It clearly shows that there are large regions ofinput polarization states for which P � 1. Thereforethe condition that the four basic Stokes vectors in Eq.(34) be transformed into PSVs, as it can be experi-mentally imposed, is only a necessary but not a suf-ficient condition for a Mueller matrix to be a PMM.Additional conditions are clearly requested.
Some questions naturally arise from these consid-erations: Do classes of Mueller matrices exist withless than 16 nonnull elements or with symmetryproperties that can be constrained to be PMMs in theway shown above? And for a generic 16-elementMueller matrix, what additional conditions should beimposed to ensure it is a PMM? Is it possible to in-troduce them as constraints on the results of exper-imental measurements? These are questions thatdeserve further investigations.
8. Conclusions
In this work we have shown that in the measurementof the polarization state by the rotating quarter-wavemethod, some limitations of the conventional Fourieranalysis technique can be overcome by adopting dataanalysis methods based on a least-squares best fit. Inthis way it is possible to consistently determine theexperimental uncertainties on the elements of themeasured Stokes vector and on the polarization pa-rameters derived from it, also taking into account the
fact that the measured data may have different mea-surement errors. In addition we have shown that thebest-fit technique can be constrained in such a way toalways yield a physically consistent Stokes vector.The analysis techniques can be extended to includealso the determination and subtraction of possiblesystematic errors on the angular measurement posi-tions. These techniques have been used in the anal-ysis of data in experiments for the characterizationof the polarimetric properties of metallic mirrors ex-posed to a plasma, and they have been shown to bereliable and effective.
Finally, some considerations on the possibility ofusing the above techniques for the determination ofphysically consistent Mueller matrices have been pre-sented, raising new questions that deserve furtherinvestigation.
References and Notes1. D. Goldstein, Polarized Light (Dekker, 2003).2. P. S. Hauge, “Recent developments in instrumentation in
ellipsometry,” Surf. Sci. 96, 108–140 (1980).3. R. W. Collins, “Automatic rotating element ellipsometers:
calibration, operation, and real-time applications,” Rev. Sci.Instrum. 61, 2029–2062 (1990).
4. S. E. Segre, “A review of plasma polarimetry—theory and meth-ods,” Plasma Phys. Controlled Fusion 41, R57–R100 (1999).
5. S. E. Segre, “Determination of both the electron density andthe poloidal magnetic field in a tokamak plasma from polari-metric measurements of phase only,” Plasma Phys. Control.Fusion 38, 883–888 (1996).
6. L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual,“Experimental investigation of polarization effects in the FIRrange of deposited layers on mirrors,” Final Report of contractTW3-TPDS-DIADEV D2, EFDA (European Fusion Develop-ment Agreement), 2005 (unpublished).
7. L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, andE. Zilli, “Far-infrared polarimetric characterization of metallicmirrors exposed to a tokamak plasma,” Rev. Sci. Instrum. 77,123504 (2006).
8. V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. ShevchenkoA. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G.Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I.Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov,“Simulation of environment effects on retroreflectors in ITER,”Rev. Sci. Instrum. 76, 083502 (2004).
9. A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti,E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S.Prunty, M. Spillane, and C. Walker, “Poloidal polarimeter forcurrent density measurements in ITER,” Rev. Sci. Instrum. 75,4694–4701 (2004).
10. S. Brandt, Statistical and Computational Methods in DataAnalysis (North-Holland, 1976).
11. P. R. Bevington and D. K. Robinson, Data Reduction and ErrorAnalysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).
12. All data analyses have been performed by the mathematicalpackage MathCad (www.mathcad.com). MathCad files for allprocedures in this paper are available from the authors.
13. C. S. Beightler, D. T. Phillips, and D. J. Wilde, Foundations ofOptimization, 2nd ed. (Prentice-Hall, 1979).
14. P. S. Hauge, “Mueller matrix ellipsometry with imperfect com-pensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
15. D. H. Goldstein and R. A Chipman, “Error analysis of a Muel-ler matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
16. H. R. Jerrard, “The calibration of quarter-wave plates,” J. Opt.Soc. Am. 42, 159–165 (1952).
Fig. 7. Contour plot of the degree of polarization P of theM-transformed Stokes vector as a function of the � and � angles ofthe input Stokes vector. Input polarization states span the entirePoincaré sphere.
17. Kapton is the commercial name of a polyimide film producedby DuPont. TPX (polymethylpentene) is a FIR transmittingplastic material.
18. A. B. Kostinski, C. R. Givens, and J. M. Kwiatkowski, “Con-straints on Mueller matrices of polarization optics,” Appl. Opt.32, 1646–1651 (1993).
19. C. R. Givens and A. B. Kostinski, “A simple necessary andsufficient condition on physically realizable Mueller matrices,”J. Mod. Opt. 40, 471–481 (1993).
20. E. S. Fry and G. W. Kattawar, “Relationship between elementsof the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
21. R. Bakarat, “Conditions for the physical realizability of polar-ization matrices characterizing passive systems,” J. Mod. Opt.34, 1535–1544 (1987).
22. C. Brosseuau, C. R. Givens, and A. B. Kostinski, “Generalized
trace condition on the Mueller–Jones polarization matrix,” J.Opt. Soc. Am. A 10, 2248–2251 (1993).
23. J. J. Gil, “Characteristic properties of Mueller matrices,” J.Opt. Soc. Am. A 17, 328–334 (2000).
24. S. R. Cloude, “Group theory and polarisation algebra,” Optik(Stuttgart) 75, 26–36 (1986).
25. S. R. Cloude, “Conditions for the physical realisability of ma-trix operators in polarimetry,” Proc. SPIE 1166, 177–185(1989).
26. B. J. Howell, “Measurement of the polarization effects of aninstrument using partially polarized light,” Appl. Opt. 18,809–812 (1979).
27. This matrix is not experimental; it has been obtained by per-turbating with random noise the Mueller matrix of a combi-nation of half-wave and quarter-wave plates.
