February 15, 2010 / Vol. 35, No. 4 / OPTICS LETTERS 559

Analytic inversion of the Mueller–Jonespolarization matrices for homogeneous media

Oriol Arteaga* and Adolf CanillasFEMAN Group, Departament de Física Aplicada i Òptica, IN2UB, Universitat de Barcelona, Barcelona 08028, Spain

*Corresponding author: oarteaga@ub.edu

Received September 23, 2009; revised January 8, 2010; accepted January 11, 2010;posted January 20, 2010 (Doc. ID 117648); published February 11, 2010

In this Letter we present the equations to calculate the six independent polarization effects of an arbitrarynormalized Mueller–Jones matrix corresponding to homogenous media. A comparison between this methodand other inversion procedures is discussed, and the application of the analytic inversion to experimentalMueller matrices is illustrated. © 2010 Optical Society of America

OCIS codes: 260.5430, 260.1440, 120.2130.

Light propagation through a homogeneous mediumwith arbitrary absorptive and refractive anisotropy isone of the most common situations one can find whenstudying complex materials with polarimetric tech-niques. This problem was first addressed by Jones [1]by introducing path derivatives of the Jones matrix,and later Azzam [2] extended this solution to Muellercalculus. These works have permitted one to calcu-late the Mueller–Jones matrix (the nondepolarizingMueller matrix), which describes an optical systemwith combined birefringence and dichroism. Nowa-days these results have important applications in thespectroscopy of oriented molecules, the analysis ofbiological materials, the characterization of opticalcomponents, and the study of crystal optics. Ratherthan the simple measurement of the Mueller matrixin these applications, the objective is the determina-tion of the optical properties of the sample. Thus aninversion method to obtain these properties from themeasured Mueller matrix is required.

The purpose of this Letter is to present an analyticsolution to the inversion problem of a Jones or aMueller–Jones matrix corresponding to a homoge-neous media with combined arbitrary absorptive andrefractive anisotropies. For many simpler situations,for example media presenting only linear birefrin-gent anisotropy or media that exhibits only circulardichroism, the inversion problem is almost trivial.However, when combined amplitude and phaseanisotropies are present the situation is more com-plex, because the optical effects are coupled togetherand, to our knowledge, no rigorous equations havebeen published to make possible an analytic inver-sion in this more general case. Quite frequently thesesituations are studied with matrix product decompo-sition techniques that allow one to separate the am-plitude anisotropies to the phase anisotropies. Themost used is the polar decomposition [3], which canbe applied to any Mueller matrix, but it cannothandle situations with noncommuting optical effects,i.e., situations in which if the diattenuating effectsare separated from the retarding effects into two dif-ferent matrix terms their final effect depends on theorder in which these two matrices are multiplied.The recently introduced pseudopolar decomposition[4] uses a modified algorithm that handles these situ-

ations better.

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The general 4�4 Mueller matrix contains 16 inde-pendent parameters, whereas the 2�2 Jones com-plex matrix contains only eight. These parameterscan be chosen so that they correspond to the eightphysical measurements one can make in an opticalsystem with a given light path [1], namely, mean ab-sorption �k�, mean refraction ���, circular birefrin-gence (CB), circular dichroism (CD), and two param-eters each for linear birefringence (LB, LB�) andlinear dichroism (LD, LD�). The definitions for theseeight effects are presented in Table 1, and further in-formation in the notation we use can be found in [4].

If the optical system represented by the Muellermatrix does not depolarize, only seven out of the 16elements of the Mueller matrix are independent, be-cause nine nonlinear identities exist among the 16matrix elements [5]. The number of independent pa-rameters in a Mueller–Jones matrix is seven insteadof the eight of a Jones matrix, because Mueller ma-trices cannot represent absolute phase ���. In thecase of normalized Jones or Mueller–Jones matrices,the number of independent parameters is six (CD,CB, LD, LB, LD�, and LB�).

For convenience we will express the elements ofthe Jones matrix in polar form:

J = �j00 j01

j10 j11� = ei�00� r00 r01e

i��01−�00�

r10ei��10−�00� r11e

i��11−�00�� . �1�

Table 1. Symbols Used and Definitions

Effect Symbol Definitiona

Isotropic phase retardation � 2�nl /�0

Isotropic amplitude absorption k 2��l /�0

(x–y) linear dichroism LD 2���x−�y�l /�0

(x–y) linear birefringence LB 2��nx−ny�l /�0

45° linear dichroism LD� 2���45−�135�l /�0

45° linear birefringence LB� 2��n45−n135�l /�0

Circular dichroism CD 2���−−�+�l /�0

Circular birefringence CB 2��n−−n+�l /�0an, refractive index; �, extinction coefficient; l, path

length through the medium; �0 vacuum wavelength oflight. Subscripts specify the polarization of light as x, y, 45°

to the x axis, 135° to the x axis, circular right �, or left �.

2010 Optical Society of America

560 OPTICS LETTERS / Vol. 35, No. 4 / February 15, 2010

The Jones matrix elements, as given in Eq. (1), cor-responding to a given Mueller–Jones matrix can becalculated using the following equations [6]:

r00 = ��m00 + m01 + m10 + m11�/2�1/2, �2�

r01 = ��m00 − m01 + m10 − m11�/2�1/2, �3�

r10 = ��m00 + m01 − m10 − m11�/2�1/2, �4�

r11 = ��m00 − m01 − m10 + m11�/2�1/2, �5�

ei��01−�00� =m02 + m12 − i�m03 + m13�

��m00 + m10�2 − �m01 + m11�2�1/2 , �6�

ei��10−�00� =m20 + m21 + i�m30 + m31�

��m00 + m01�2 − �m10 + m11�2�1/2 , �7�

ei��11−�00� =m22 + m33 + i�m32 + m23�

��m00 + m11�2 − �m10 + m01�2�1/2 . �8�

There are no physical restrictions on the values ofthe Jones matrix elements. The effects of Table 1 areincorporated into the general Jones matrix as [1,4]

J = e−i/2�cosT

2−

iL

Tsin

T

2

�C − iL��

Tsin

T

2

−�C + iL��

Tsin

T

2cos

T

2+

iL

Tsin

T

2� , �9�

where T=L2+L�2+C2, 2��− ik�, LLB− iLD, L�LB�− iLD�, and CCB− iCD.

For any Mueller–Jones matrix Eqs. (2)–(8) can beused to calculate r00, r01, r10, r11, ei��01−�00�, ei��11−�00�,and ei��11−�00�. These factors can be identified with theparameterized general Jones matrix in Eq. (9):

Table 2. Comparison of Methods to Ana

Case Mueller Matrix Generatio

1 � 1 −0.0291 −0.0111 0.0295−0.0029 0.9993 −0.0190 0.0049−0.0093 0.0188 0.9957 0.08020.0305 −0.0073 −0.0804 0.9963

� CD=0.030CB=−0.01LD=0.028LB=−0.08LD�=0.01LB�=0.00

2 � 1 −0.5074 0.2220 −0.5061−0.6388 0.7389 0.0525 0.5444−0.3932 −0.0750 0.6392 −0.42120.0156 −0.3780 0.2740 0.4684

� CD=0.020CB=0.077LD=0.828LB=0.480LD�=−0.5LB�=0.60

aThe analytic inversion gives the original parameters usb

Pseudopolar decomposition have been calculated with two c

�cosT

2−

iL

Tsin

T

2

�C − iL��

Tsin

T

2

−�C + iL��

Tsin

T

2cos

T

2+

iL

Tsin

T

2�

= K� r00 r01ei��01−�00�

r10ei��10−�00� r11e

i��11−�00�� , �10�

where K is a complex constant that can be deter-mined combining the matrix elements of Eq. (10) andusing the identity cos2 T/2+sin2 T/2=1:

K = �r00r11ei��11−�00� − r01r10e

i��01−�00�ei��10−�00��−1/2. �11�

Once K is known, the determination of LB, LB�,CB, LD, LD�, and CD from Eq. (10) becomes straight-forward:

LB = R�i�r00 − r11ei��11−�00���, �12�

LB� = R�i�r01ei��01−�00� + r10e

i��10−�00���, �13�

CB = R��r01ei��01−�00� − r10e

i��10−�00���, �14�

LD = − I�i�r00 − r11ei��11−�00���, �15�

LD� = − I�i�r01ei��01−�00� + r10e

i��10−�00���, �16�

CD = − I��r01ei��01−�00� − r10e

i��10−�00���, �17�

where =TK / �2 sin�T/2��, T=2 cos−1�K�r00+r11ei��11−�00�� /2�, and the symbols R and I respec-tively denote the real and imaginary parts.

As an example of the method described here, wehave inverted two Mueller–Jones matrices that weregenerated using some representative values of CD,CB, LD, LB, LD�, and LB�. These values along withthe results obtained from the polar and pseudopolardecompositions are displayed in Table 2. The analyticinversion offers an exact solution to the problem, i.e.,we obtain again the same parameters used for the

Homogeneous Mueller–Jones Matrices

nalytic Inver.a Polar Decomp. Pseudopolar Decomp.b

CD=0.0295 CD=0.0300CB=−0.0189 CB=−0.0189LD=0.0291 LD=0.0289LB=−0.0805 LB=−0.0805LD�=0.0111 LD�=0.0102LB�=0.0061 LB�=0.0061

CD=−0.3127 CD=0.0337CB=0.0725 CB=0.0746LD=0.7871 LD=0.8428LB=0.4525 LB=0.4481LD�=−0.4335 LD�=−0.5172LB�=0.5599 LB�=0.5639

r the generation of the Mueller-Jones matrices.

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orrection terms and 100 iterations.

February 15, 2010 / Vol. 35, No. 4 / OPTICS LETTERS 561

generation, while the results on the others methods,specially for the polar decomposition, largely dependon the magnitude of the effects and/or on their non-commutativity. In Case 1 all the optical effects aresmall and the noncommutative effects (that alwaysare second- and superior-order effects) can be omit-ted, thus obtaining a satisfactory result by any of thethree methods. It is interesting to note that here thepseudopolar decomposition, although it is an approxi-mate method, is able to reproduce exact results withan accuracy up to the forth decimal. Case 2 repre-sents a situation in which much bigger effects arepresent and, in such a way, that they generate sig-nificant noncommutativity between the diattenuat-ing and retarding effects. Here the pseudopolar de-composition offers a much better approximation tothe exact result provided by the analytic inversionthan the polar decomposition, which fails to find ac-curate results, in particular for CD.

Nevertheless, a polarimetric experiment can neverprovide an exact Mueller–Jones matrix because ofsample depolarization effects and measurement un-certainties. The polarization factor � can be calcu-lated from the experimental Mueller matrix as [6]

� =��

ijmij

2� − m002

3m00

, �18�

where �=1 for totally polarized light and �=0 for acompletely unpolarized beam.

To apply the analytic inversion method describedin the Eqs. (12)–(17) to analyze experimental Muellermatrices we use the approach based on the sum de-composition introduced by Cloude [7] to obtain a non-depolarizing Mueller (i.e., a Mueller–Jones) matrixestimation for a depolarizing experimental matrix.According to this decomposition any physically real-izable Mueller matrix M can be written as

M = �0MJ0 + �1MJ1 + �2MJ2 + �3MJ3, �19�

where MJi are Mueller–Jones matrices that can becalculated as described in [8] and �i are the eigenval-ues of the Cloude’s coherency matrix associated to M.If M is a Mueller–Jones matrix one has �0��1=�2=�3=0. For a experimentally determined M it is com-mon to have �0��1, �2, �3, and then the summand�0MJ0 can be considered as the nondepolarizing esti-mate for the measured M.

Figure 1 shows the values of �, CD, and CB ob-tained from experimental Mueller matrices for a wa-ter solution of J-aggregates of the pseudocyanine dyein a 0.1 mm path length cuvette. This sample can bewell described as a homogeneous anisotropic mediumthat contains linear birefringence and dichroism, ow-ing to the oriented elongated aggregates, and as wellas circular dichroism and birefringence, owing to thechirality of the involved molecules.

Measurements of the experimental Mueller matri-ces were taken spectroscopically in the 450–610 nmrange with a transmission two-modulator general-ized ellipsometer, as described in [9]. For the studiedsample there exists a band of significant depolariza-

tion around 570 nm that is caused by light scatteringby aggregates of electronically interacting chro-mophores. At each wavelength the nondepolarizingestimate ��0MJ0� of the experimental Mueller matrixwas calculated and was inverted using Eqs. (14) and(17) to obtain CB and CD. As a comparison, the CDand CB, obtained using a product decomposition bythe method described in [4], are also shown. The de-polarization of this sample is not well handled by theproduct decomposition, and the analytic inversionmethod gives CB values that are more Kramers–Kronig consistent with a CD bisignated band.

The polarization effects of a homogeneous mediumcontained in a Mueller–Jones matrix have beenshown to be easily revealed by means of simple ana-lytic equations. In contrast to other matrix productdecomposition methods this inversion offers exact re-sults for any Mueller–Jones matrix. In the case of ex-perimentally determined Mueller matrices thismethod can also be used if a nondepolarizing esti-mate of the experimental Mueller matrix is previ-ously found.

The authors acknowledge financial support fromAYA2006-1648-C02-01. O. A. also acknowledges fi-nancial support from FPU AP2006-00193. We thankJ. M. Ribó for motivating the research.

References

1. C. R. Jones, J. Opt. Soc. Am. 38, 671 (1948).2. R. M. A. Azzam, J. Opt. Soc. Am. 38, 1756 (1978).3. S.-Y. Lu and R. A. Chipman, J. Opt. Soc. Am. A 13,

1106 (1996).4. O. Arteaga and A. Canillas, J. Opt. Soc. Am. A 26, 783

(2009).5. R. Barakat, Opt. Commun. 38, 159 (1981).6. R. A. Chipman, Handbook of Optics, Vol. 2, 2nd ed.

(McGraw-Hill Professional, 1994), Chap. 22.7. S. R. Cloude, Proc. SPIE 1166, 177 (1989).8. S. Savenkov, V. Grygoruk, R. Muttiah, K. Yushtin, Y.

Oberemok, and V. Yakubchak, J. Quant. Spectrosc.Radiat. Transf. 110, 30, see Appendix B (2009).

9. O. Arteaga, A. Canillas, R. Purrello, and J. M. Ribó,Opt. Lett. 34, 2177 (2009).

Fig. 1. (Color online) Results of the analytic inversion(solid curve) when applied to spectroscopic series of experi-mentally determined Mueller matrices. Results for thepseudopolar decomposition (dashed curve) are also pre-sented for comparison.