Abstract

The differential Mueller matrix is an important concept for analyzing the polarization properties of an optically homogeneous anisotropic sample, both nondepolarizing and depolarizing. In this work, we present a new method of interpreting Mueller matrix of anisotropic medium based on the relationships that exist between the components of a differential Mueller matrix and the polar components of the corresponding macroscopic Mueller matrix, and the necessary conditions are determined that guarantee the physical realizability of the generating matrices. Finally, a group of the experimental data of a sample from the literature with some known polarization properties was used to demonstrate the analysis. The work is helpful for obtaining new insights or new interpretations of the measured Mueller matrix of the medium.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, considerable attention has been paid to the measurements of the changes of polarization state of light for obtaining information about polarization properties of the medium such as the birefringence, depolarization and diattenuation. In fact, polarization properties of medium have been applied in many fields, such as monitoring the glucose level in human [1], enhancing the image contrast for superficial and deeper structures of tissues [2–4], quantifying the protein properties [5], and differentiating the normal and precancerous cells or revealing the border of cancer [6].

Several methods have been presented to describe the interaction of polarized light with medium, including Jones formalism, Poincaré sphere method and Muller matrix mechanism [7–12]. Jones formalism can be used to describe the changes of polarization states of light propagating in tissues but cannot describe partially polarized light. Poincaré sphere method, on the other hand, is a three-dimensional representation of polarization states that allows for a intuitive depiction of the variation of polarization phenomena. In the Mueller formalism, a Mueller matrix is employed to represent the influences of the tissue on the polarization state of the incident light and it can describe both polarizing and depolarizing properties of the medium [13,14]. In the present analysis, Muller matrix mechanism is used.

To extract the specific polarization properties from the measured Mueller matrix, several decomposition methods have been proposed, including Lu-Chipman’s polar decomposition [15], symmetric decomposition [16], reverse polar decomposition [17], differential decomposition [18,19]. In these methods, the polar decomposition method proposed by Gil has been proved to be very useful [20]. In 1996, the polar decomposition method has been extended to a more general medium by Lu-Chipman who decomposed the measured Mueller matrix into a product of three component matrices, a retarder, a diattenuator and a depolarizer [15].

Since the matrix multiplication is usually not commutative, the order of the component matrices in the Lu-Chipman method may have significant influence on the results when analyzing the polarization properties of the sample using Mueller matrix. In fact, it has been shown that different orders result in different values of the polarization properties [21,22]. Furthermore, the relationships between the values of each element in the decomposed macroscopic matrices and the polarization properties of the medium such as depolarization and birefringence are ambiguous. It is clear that such relations are desirable for the assessment and interpretation of the measured Mueller matrix of media.

In 1987, Azzam introduced the concept of the differential Mueller matrix to describe the local effects of the medium on the polarization state of the light propagating through it [23]. Based on the properties of the differential Mueller matrix, several differential methods of decomposing the Mueller matrix have been proposed, which can be regarded as complementary ones to the polar decomposition methods mentioned above and enable one to obtain more information about the depolarization.

By using group theory, the relationships between the differential matrix and the set of transformation generators were derived and 16 differential matrices were resolved corresponding to the properties of depolarizing anisotropic media [24]. By using two differential matrices, one for non-depolarizing properties of the mean values, and the other for depolarizing properties of the uncertainties respectively, Devlaminck found a way of decomposing the differential Mueller matrix [25]. By performing 6 differential matrices, including 7 parameters for characterizing non-depolarizing behavior and 9 parameters for describing depolarizing properties, Germer proposed a differential decomposition based on the normal Mueller matrix rather than the convex sum of Muller-Jones matrix [26]. By using the eigenvalue decomposition to avoid the invalidity of the differential matrix formalism with non-positive eigenvalues, Villiger and Bouma obtained a derivation of the differential Mueller matrix [27]. However, until now, relationships between the parameters derived from the differential matrix methods and the parameters calculated from the polar decomposition method can hardly be found, to the best of our knowledge. Although the study in [28] provided links between these two decomposition methods, only few parameters and coefficients were determined, specific correspondences among each element of the two formalisms are not given.

In this paper, the parameters, the coefficients and the specific correspondences between the polar decomposition and the differential decomposition are considered theoretically and practically. Both non-depolarizing and depolarizing homogeneous anisotropic media are considered. Parameters of the linear and the circular birefringence, the diattenuation and the depolarization are extracted from both decomposition methods. Coefficients are then derived to describe the characteristics of birefringence, diattenuation, and depolarization. Analysis of the characteristics of the components of the differential depolarizing matrix and the corresponding macroscopic Mueller matrix MΔshows that the depolarization matrix MΔcan be decomposed into another four elementary differential matrices. Then, conditions necessary for guaranteeing the physical realizability of those matrices are explored. Finally, a group of literature data is selected to validate the relationships and elucidate the implications of the results.

2. Theory

2.1 Non-depolarizing homogeneous anisotropic media

To determine the polarization properties of a medium represented by a measured Mueller matrix, both non-depolarizing and depolarizing, in polar decomposition, it is, in general, decomposed into three elementary matrices, a diattenuator MD, a retarderMRand a depolarizer MΔ. In our analysis, we first consider the non-depolarizing case.

When medium is non-depolarizing homogeneous anisotropic, its Mueller matrix can be expressed as

M=MRMD.

To extract the linear and circular birefringence of the medium, MRcan be further decomposed into a product of two matrices of linear birefringence and circular birefringence MR=MLRMCR, Eq. (1) can then be written as

M=MLRMCRMD,
where MLRand MCRrepresent the Mueller matrices of the linear retarder and the circular retarder. By taking the derivatives of the both sides of Eq. (2), we have

dMdz=dMLRdzMCRMD+MLRdMCRdzMD+MLRMCRdMDdz.

Based on the definition of the differential Mueller matrixdMdz=mM [23],Eq. (3) can be rewritten as

mM=(mLRMLR)MCRMD+MLR(mCRMCR)MD+MLRMCR(mDMD),
where mLR,mCRand mDrepresent the differential matrices of linear birefringence, circular birefringence and diattenuation, corresponding to MLR,MCRand MD. In the limit of z0, the polarization state of the incident light encounters no influence. So we have

M|z0=0=MLR|z0=0=MCR|z0=0=MD|z0=0=I.

On substitution of Eq. (5) in Eq. (4), the differential matrix at any location can be expressed as the sum of mLR,mCRand mD:

m|z0=0=mLR+mCR+mD.

Here it should be pointed out that for homogeneous medium m is independent of z. Thus, dMdz=mM can be readily integrated to obtain M=exp(mz) [23]. In this case, m always commutes with M (that means mD commutes with MD, mCR commutes with MCR, and so on). On the other hand, if mLR,mCRand mD can commute with each other, then each one of mLR,mCRand mD can commute with any one of MLR,MCRand MD. A combination of the above two points suggests that any two of MLR,MCRand MD will commute with each other. However, MCR and MLR do not commute with each other in the polar decomposition M=MΔMRMD and M=MRMD(MR=MLRMCR). So the matrices mLR,mCRand mD do not commute. The same analysis is valid for mdetand mdep.

To find the explicit expressions for the differential matrix components on the right side of Eq. (6), we start from the general form of the differential matrix of non-depolarizing media [24]:

minitial=[αβγδβαμνγμαηδνηα],
where αdenotes the parameter of the isotropic absorption, βand γare the linear dichroism parameters along the x-y laboratory axes and the 45°axes, δ is the parameter of the circular dichroism, ηand ν are the parameters of the linear birefringence along the x-y laboratory axes and the 45°axes, μ is the parameter of the circular birefringence. To simplify the analysis, the parameter of the isotropic absorption can be removed by subtracting minitial,11Ifrom the initial differential matrixminitial when αis small and this operation has no influence on the other properties [25], here, minitial,11is the first value in minitial, then, we have

m=[0βγδβ0μνγμ0ηδνη0].

Noting that the corresponding differential matrix has been related to the macroscopic Mueller matrix through its eigenvalues and eigenvectors [19]:

M=Wdiag(exp(σ0z),exp(σ1z),exp(σ2z),exp(σ3z))W-1,
whereσ0,σ1,σ2andσ3are the eigenvalues of m, and the columns of the orthogonal matrix Ware the respective eigenvectors of m. Equation (9) shows that Mcan be completely determined from the eigenanalysis of differential matrix m. In the following analysis, the relationships that exist between the components of the differential matrix (see Eq. (8)) and the components of the corresponding macroscopic Mueller matrix (see Eq. (2)) will be derived. First, consider the mLRwhich describes the linear birefringence denoted by the parameters ηandν:

mLR=[0000000ν000η0νη0].

By using Eq. (9), the effective Mueller matrix MLRcorresponding to mLR can be expressed as

MLR(η,ν)=[10000η2+ν2cos(iη2ν2)η2+ν2ην(cos(iη2ν2)1)η2+ν2νisin(iη2ν2)η2ν20ην(cos(iη2ν2)1)η2+ν2ν2+η2cos(iη2ν2)η2+ν2ηisin(iη2ν2)η2ν20νisin(iη2ν2)η2ν2ηisin(iη2ν2)η2ν2cos(iη2v2)].

At the same time, the general form of the Mueller matrix for a linear retarder [29] is

MLR(φ,θ)=[10000cos22θ+sin22θcosφcos2θsin2θ(1cosφ)sin2θsinφ0cos2θsin2θ(1cosφ)sin22θ+cos22θcosφcos2θsinφ0sin2θsinφcos2θsinφcosφ],
where φis the linear phase retardance in radian, θis the orientation angle. Equations (11) and (12) are the different expressions for the same property of the medium. So each value at the same position of the expressions of Mueller in Eqs. (11) and (12) must be equal. Based on this fact, the relationships between the two forms of the parameters of linear phase retardance and orientation angle can be derived:

φ=iη2v2,cos2θ=ηiη2ν2,sin2θ=νiη2ν2.

Next, consider the differential matrix mCR representing circular retardance effect alone:

mCR=[000000μ00μ000000].

With the same token, the effective Mueller matrix MCRcan be related to the corresponding differential matrix mCR:

MCR(μ)=[10000cosμsinμ00sinμcosμ00001].

On comparison of Eq. (15) with the general expression of the Mueller matrix for a circular retarder [29],

MCR(δc)=[10000cosδcsinδc00sinδccosδc00001].

We have

δc=μ,
where δcrepresents the circular phase retardance.

For macroscopic Mueller matrix, a retardance coefficient R is a measure of the effective rotation angle in radians and is related to the complete retardance matrix MR [15] (also see Appendix),

R=cos1(tr(MR)21)=cos-1((cosμ+1)(1+cos(iη2ν2))21),
where tr(MR) represents the trace of MR. Whenμ0, Eq. (18) becomes R=φ=iη2v2, only linear phase retardance is considered and circular retardance can be neglected;conversely, only the circular retardance is significant, we have R=δc=μ. It is evident that, in general case, Eq. (18) should be employed.

For the differential Mueller matrix of the diattenuation,

mD=[0βγδβ000γ000δ000].

After straightforward calculation [see the formula Eq. (63) in Appendix], the effective Mueller matrix MD can be expressed in terms of the parameters of mD, while the general form of MDcan be expressed as

MD=Tu[1DTDm3],
m3=a1I+b1(D^D^T)=(1|D|2)1/2I+[1(1|D|2)1/2]D^D^T,
where m3is a 3×3 sub-matrix, Iis the3×3 identity matrix, D^=D/|D| denotes the unit vector along D, Tuis the transmittance for incident light and MD is symmetric with respect to the main diagonal axis. By noting these facts, Eq. (20) can be explicitly rewritten as

MD=[1B12B13B14B12c+bB122abB12B13abB12B14aB13bB12B13ac+bB132abB13B14aB14bB12B14abB13B14ac+bB142a],

whereB12=m01m00,B13=m02m00and B14=m03m00, mij(i=0,j=0,1,2,3)is the element of the macroscopic Mueller matrix. a=B122+B132+B142, b=11aand c=1a. Note that the intensity transmittance is completely determined by the first row of the arbitrary Mueller matrix.

A diattenuation vector D is usually used as a measure of the diattenuation and its direction, which is defined as [15]

D=[DHD45DC]=[B12B13B14],
where DH,D45,DC and Drepresent horizontal, 45°linear, circular, and total diattenuation respectively [15]. On comparison of Eq. (23) with Eqs. (22) and (63), we have

DL=DH2+D452=(β2γ2)tan(iβ2+δ2+γ2)β2+δ2+γ2=B122+B132.
DC=B14=δitan(iβ2+δ2+γ2)β2+δ2+γ2.
D=DH2+D452+DC2=DL2+DC2=|D|=±itan(iβ2+δ2+γ2).

It should be pointed out that the horizontal and the 45°diattenuation have been combined to describe the linear diattenuationDL.

2.2 Depolarizing homogeneous anisotropic media

For the depolarizing homogeneous anisotropic media, its Mueller matrix can be decomposed into a product of three matrices [15]:

M=MΔMRMD,
where MR,MD and MΔrepresent a retarder, a diattenuator and a depolarizer, respectively. According to the experimental results [30–34], MΔ can be expressed as
MΔ=[1000p1e100p20e20p300e3],
where p1,p2and p3 are the components of the polarizance vector p that characterizes the polarizing capability of the depolarizer, e1and e2 are the parameters of the horizontal or vertical and the 45°linear depolarization, and e3is the parameter of the circular depolarization. Note that a simplification process has been used in the derivation of Eq. (28).

By decomposing the differential Mueller matrix mΔ of the macroscopic matrixMΔinto a sum of 3 components, different kinds of depolarization behaviors of the medium can be revealed. It is then desirable to find the forms of the corresponding components of MΔ which should satisfy Eq. (28). To this end, M can be decomposed into

M=MΔMLRMCRMD=MΔα1α2α3,βγδMΔ-β-γ-δMΔμνηMLRMCRMD,
where MΔα1α2α3,βγδ,MΔ-β-γ-δand MΔμνηrepresent the depolarization Mueller matrices of the anomalous isotropy and a part of the anomalous dichroism, the remaining part of the anomalous dichroism, and the anomalous birefringence. ForMΔμνη, we have
MΔμνη=MΔμMΔνη=MΔνηMΔμ,
whereMΔμand MΔνηrepresent the depolarization Mueller matrices of the anomalous circular birefringence and the anomalous linear birefringence. It can be shown that there is little influence on the result with different orders of the MΔμMΔνηand MΔνηMΔμin Eq. (30) [see the discussions of Eq. (64) in Appendix].

Using the decomposition expression in Eq. (30), Eq. (29) can be rewritten as

M=MΔα1α2α3,βγδMΔ-β-γ-δMΔμνηMLRMCRMD=MΔα1α2α3,βγδMΔ-β-γ-δ(MΔμMΔνη)MLRMCRMD.

By differentiating both sides of Eq. (31), we have

dMdz=dMΔα1α2α3,βγδdzMΔ-β-γ-δMΔ-μνηMLRMCRMD+MΔα1α2α3,βγδdMΔ-β-γ-δdzMΔ-μνηMLRMCRMDMΔα1α2α3,βγδMΔ-β-γ-δd(MΔμ)dzMLRMCRMD+MΔα1α2α3,βγδMΔ-β-γ-δd(MΔνη)dzMLRMCRMD+MΔα1α2α3,βγδMΔ-β-γ-δMΔ-μνηdMLRdzMCRMD+MΔα1α2α3,βγδMΔ-β-γ-δMΔ-μνηMLRdMCRdzMD+MΔα1α2α3,βγδMΔ-β-γ-δMΔ-μνηMLRMCRdMDdz.

Using dMdz=mM, Eq. (32) can be rewritten as

mM=(mΔα1α2α3,βγδMΔα1α2α3,βγδ)MΔ-β-γ-δMΔ-μνηMLRMCRMD+MΔα1α2α3,βγδ(mΔ-β-γ-δMΔ-β-γ-δ)MΔ-μνηMLRMCRMD+MΔα1α2α3,βγδMΔ-β-γ-δ(mΔ-μMΔ-μ)MΔ-νηMLRMCRMD+MΔα1α2α3,βγδMΔ-β-γ-δMΔ-μ(mΔ-νηMΔ-νη)MLRMCRMD+MΔα1α2α3,βγδMΔ-β-γ-δMΔ-μνη(mLRMLR)MCRMD+MΔα1α2α3,βγδMΔ-β-γ-δMΔ-μνηMLR(mCRMCR)MD+MΔα1α2α3,βγδMΔ-β-γ-δMΔ-μνηMLRMCR(mDMD).

Consider a linear homogeneous medium,

M|z0=0=MΔα1α2α3,βγδ|z0=0=MΔβγδ|z0=0=MΔ-μ|z0=0=MΔ-νη|z0=0=MLR|z0=0=MCR|z0=0=MD|z0=0=I.

Equation (33) is then reduced to

m|z0=0=(mΔα1α2α3,βγδ+mΔβγδ+mΔ-μ+mΔ-νη)+(mLR+mCR+mD)=mdep+mdet,
where mdetand mdep denote the deterministic differential matrix and the depolarizing one. To guarantee the physical realizability of the basic differential matrices, coherency matrices of each differential matrix in Eq. (35) will be discussed in the next part.

To give specific expressions of mΔα1α2α3,βγδ,mΔβγδ,mΔ-μandmΔ-νη, a general differential matrix for depolarizing medium can be expressed as [18,24]

mdep=[0β'γ'δ'β'α1μ'ν'γ'μ'α2η'δ'ν'η'α3]=mΔα1α2α3,βγδ+mΔβγδ+mΔ-μ+mΔ-νη,
whereα1,α2and α3are the parameters of the anomalous isotropy depolarization, β,γandδ, μ,ηand νare the parameters for the different properties of anomalous dichroism and anomalous birefringence. The total differential Mueller matrix is regarded as the sum of the elementary differential matrices [23].

To separate the different properties and satisfy the form of MΔin Eq. (28), mΔα1α2α3,βγδ, mΔ-μand mΔ-νηare given by

mΔα1α2α3,βγδ=[0000β'α100γ'0α20δ'00α3],mΔ-μ=[000000μ'00μ'000000],mΔ-νη=[0000000ν'000η'0ν'η'0].

The corresponding macroscopic Mueller matricesMΔα1α2α3,βγδ,MΔμandMΔνη can then be found:

MΔ-α1α2α3,βγδ=[1000β'(exp(α1)1)α1exp(α1)00γ'(exp(α2)1)α20exp(α2)0δ'(exp(α3)1)α300exp(α3)],MΔ-μ=[10000cos(iμ')isin(iμ')00isin(iμ')cos(iμ')00001].
MΔ-νη=[10000η'2+ν'2cos(iη'2+ν'2)η'2+ν'2η'ν'(cos(iη'2+ν'2)1)η'2+ν'2ν'isin(iη'2+ν'2)η'2+ν'20η'ν'(cos(iη'2+ν'2)1)η'2+ν'2ν'2+η'2cos(iη'2+ν'2)η'2+ν'2η'isin(iη'2+ν'2)η'2+ν'20ν'isin(iη'2+ν'2)η'2+ν'2η'isin(iη'2+ν'2)η'2+ν'2cos(iη'2+ν'2)].

The above analysis is valid only if the eigenvectors of the elementary differential matrices are different from each other. For differential matrix mΔ-β-γ-δ

mΔβγδ=[0β'γ'δ'000000000000],
where mΔβ,γ,δfails to satisfy Eq. (9) due to the multiple roots, consequently, this one can be treated as a exceptional case, by the following equation
dSdz=mS,
where S is the Stokes vector including S0,S1,S2andS3. A set of differential equations are then obtained

dS0/dz=β'S1γ'S2δ'S3dS1/dz=0dS2/dz=0dS3/dz=0.

Through integration, the last three equations yield constant values independent of z for S1,S2andS3, which indicate the conservation of the horizonal, the 45°and the circular polarization preference. The first equation yields the linear decay of S0that is a function of the distance z

S0=S00(β'S1+γ'S2+δ'S3)z,
whereS00is the first parameter of the Stokes vector for the incident light, which implies a depolarization mechanism that exists gradual decrease of the total intensity, therefore, incident light polarized in any polarization state can eventually become vanish after propagating a sufficient distance. However, for a thin and layered medium, the effect can be treated as a loss of the total intensity. Also, since the values of S1,S2and S31, β,γand δare really small [19,25], the decay of S0can be neglected. So the corresponding Mueller matrix of Eq. (40) can be an identical matrix to I.

By using the expressions of MΔα1α2α3,βγδ,MΔμ,MΔνη and MΔ-β-γ-δ(see Eqs. (38) and (39)), MΔ can be expressed as the combination of the three matrices (z=η'2+ν'2)

MΔ=MΔα1α2α3,βγδMΔ-β-γ-δMΔμνη=MΔα1α2α3,βγδMΔ-β-γ-δ(MΔμMΔνη)=[1000β'(expα11)α1expα100γ'(expα21)α20expα20δ'(expα31)α300expα3][10000cos(iμ')isin(iμ')00isin(iμ')cos(iμ')00001][10000η'2+ν'2cos(iz)zη'ν'(cos(iz)1)η'2+ν'2ν'isin(iz)z0η'ν'(cos(iz)1)zν'2+η'2cos(iz)zη''isin(iz)z0ν'isin(iz)zη''isin(iη'2+ν'2)zcos(iz)].

For further expansion, Eq. (64) in Appendix can be referred. To determine the related parameters and coefficients, the most general expression for a depolarizer is given that is similar to the definition of Eq. (28)

MΔ1=[10p1mΔ1],mΔ1T=mΔ1,
here, mΔ1is a 3×3 submatrix contains depolarization properties, p1 depends on the polarizing capability of the medium. According to Eq. (64), isin(iμ')η'ν'(cos(iη'2+ν'2)1)η'2+ν'2is too small to be considered, and cos(iη'2+ν'2) can be replaced by 1, then the net depolarization coefficient defined in [15] can be expressed as
Δ1=1-|tr(mΔ1)|3=1-|tr(MΔ1)1|3=1-(exp(α1)+exp(α2))cos(iμ')+exp(α3)cos(iη'2+ν'2))3,0Δ11.
Note that the depolarization capability and power of a depolarizer can be indicated by Δ1, e1,e2 and e3can be obtained by diagonalizing mΔ1, p1 can be straightforwardly extracted from the first column of the MΔ1.

3. Physical realizability and discussions

To examine the physical realizability of the elementary matrices of m, the general criterion derived by Ossikovski and Devlaminck [35] is used

M(z+Δz)=(I+mdetΔz+mdepΔz)M(z),
where I is the identity matrix, andΔzis the incremental thickness. With the assumption of the physical realizability of Mueller matrix M(z), the criterion of physical realizability stated by Cloude [36] can be transformed into the examination of positive semidefiniteness of the coherency matrix (I+mdetΔz+mdepΔz)(denoted as C(I+mdetΔz+mdepΔz)). For this purpose, we consider the physical realizability of mLR. The coherency matrix of the identity matrix is

C(I)=[10T0O3].

Equation (47) can be reduced to

M(z+Δz)=(I+mLRΔz)M(z).

The corresponding coherency matrix of mLR can be expressed as

C(mLR)=[0000000000000000].

As the consequence, C(I+mLRΔz) is the same as C(I) which is obviously positive semidefinite with four eigenvalues of 1, 0, 0 and 0. Then, the coherency matrices for mCR and mD are readily obtained

C(mCR)=[000μi200000000μi2000],C(mD)=12[0βγδβ000γ000δ000].

The eigenvalues of C(I+mCRΔz) and C(I+mDΔz) are λ1=0,λ2=0,λ31,λ4μ2Δz24, λ1=0,λ2=0,λ31,λ4(β2+γ2+δ2)Δz24, respectively. The first three eigenvalues of both coherency matrices are non-negative and the last ones of them tend to zero faster than Δz and can be seen as zero. Similarly, when the differential matrix m is depolarizing, C(I+mΔ-μΔz),C(I+mΔ-νηΔz),C(I+mΔα1α2α3,βγδΔz)and C(I+mΔβγδΔz) should be verified to be positive semidefinite. This can be found by calculating the eigenvalues of C(I+mΔ-μΔz)and C(I+mΔ-νηΔz)asλ1=0,λ2=1,λ3μΔz2,λ4μΔz2, λ1=0,λ2=1,λ3Δzη2+ν22,λ4Δzη2+ν22 according to the following matrices

C(mΔ-μ)=[000000μ'200μ'2000000],C(mΔ-νη)=[0000000ν'2000η'20ν'2η'20].

Since Δz,μ,η'andν'are too small, multiplications of two of them can be treated as zero. In virtue of the complexity of both forms C(I+mΔα1α2α3,βγδΔz)and C(I+mΔβγδΔz), mΔα1α2α3,βγδand mΔβγδcan be decomposed as combinations of several basic coherency matrices which must be positive semidefinite each to satisfy the criterion. Coherency matrix of mΔβγδ can be deposed as

C(mΔβγδ)=14[0β'γ'δ'β'0iδ'iγ'γ'iδ'0iβ'δ'iγ'iβ'0]=14([0β'γ'δ'β'000γ'000δ'000]+[000000iδ'00iδ'00000'0]+[0000000iγ'000iβ'0iγ'iβ'0]).

It follows that the first two terms are proved to be positive semidefinite according to the eigenvalues ofC(mD)and C(mΔ-μ), the physical realizability of C(I+mΔβγδΔz)solely depends on the third part whose eigenvalues are λ1=0,λ2=0,λ3Δzβ2+γ24,λ4Δzβ2+γ24, since Δz,μ,η'andν'are too small, multiplications of two of them can be treated as zero. Like deposition Eq. (53),

C(mΔα1α2α3,βγδ)=14[α1+α2+α3β'γ'δ'β'α1-α2-α3-iδ'iγ'γ'iδ'-α1+α2-α3-iβ'δ'iγ'iβ'-α1-α2+α3]=14([0β'γ'δ'β'0-iδ'iγ'γ'iδ'0-iβ'δ'iγ'iβ'0]+[α1+α2+α30000α1-α2-α30000-α1+α2-α30000-α1-α2+α3]).

To check the positive semidefiniteness of associated matrixC(I+mΔα1α2α3,βγδΔz), only the second term is discussed. The eigenvalues are easily obtained as λ1=(α1+α2+α3)Δz4,λ2=(α1α2α3)Δz4,λ3=(α1+α2α3)Δz4,λ4=(α1α2+α3)Δz4, where, the first one is the largest one (one can always achieve the condition λ1>0 by adding C(cI)=cC(I)to the coherency matrix above [35], where c is a constant) since α1,α2and α3are negative ones, in general, all the ones can be closed to zero or positive.

Here, an experimental data [1] for forward scattering measurement is used to illustrate the relationships proposed since the applicability of the differential decomposition to all the reflection configuration measurements cannot be guaranteed [35]. Noting that relations are general in the sense that they are applicable to any anisotropic sample, either nondepolarizing or depolarizing. So in principle, any experimental values can be used in this context. However, for clarity, we choose the measured data of a medium with known and completed polarization and depolarization properties to demonstrate our results. The chiral turbid medium was prepared using aqueous suspension of 2.0μm-diameter polystyrene microspheres, where exhibited the properties of diattenuation, depolarization and birefringence. The scattering coefficient and the glucose concentration coefficient measured were μS=0.6mm-1 and 5M. For the usual method,

M=[10.0260.0440.0390.0290.9620.1440.0470.0020.1260.9750.0260.0390.0190.1150.936].

The effective differential matrix is then found by m=W1diag(Inτ0z,Inτ1z,Inτ2z,Inτ3z)W1-1 [19], whereτ0,τ1,τ2and τ3are the eigenvalues of M, W1 is a matrix whose columns are the eigenvectors of M.

m=[-0.00110.02370.0487-0.04050.0286-0.0292-0.1457-0.04670.00080.1292-0.01770.0303-0.04070.01260.1223-0.0684]=mdet+mdep.

The depolarizing part and the determined part are

mdet=[0βγδβ0μνγμ0ηδνη0]=[00.02620.0247-0.04060.02620-0.1374-0.02960.02470.13740-0.046-0.04060.02960.0460].
mdep=[0β'γ'δ'β'α1μ'ν'γ'μ'α2η'δ'ν'η'α3]=[0-0.00250.0240.00010.0025-0.0281-0.0083-0.0170-0.024-0.0083-0.01660.0763-0.0001-0.01700.0763-0.0673].

The corresponding polar decomposition of the Mueller matrix Eq. (55) consists ofMΔ, MR and MD are given by

MΔ=[10000.0080.9760.010.0210.0230.010.9820.0730.0090.0220.0730.941],MR=[100000.990.1380.02700.1360.990.04800.0330.0440.998],MD=[10.0260.0440.0390.0260.9980.0010.0010.0440.0010.9990.0010.0390.0010.0010.999].

To examine the method proposed, the elements of the differential matrix obtained through the polar decomposition method should be equivalent to the elements in Eqs. (57) and (58). Using the derivations in Appendix.

η=-0.048;ν=-0.027;μ=-0.1374;β=0.026;γ=0.044;δ=-0.039,
α3=-0.0608;α2=-0.0182;α1=0.0243;η=0.0776;ν=0.0234;μ=-0.0102;β=0.008;γ=-0.023;δ=-0.009.

The estimated results are in close agreement with the expected values in Eqs. (57) and (58) except for γ,βandδ'. The differences may be due mainly to the measurement errors and the small values. It can be seen that the results in Eqs. (60) and (61) are obtained and extracted straightforwardly from comparing the three matrices in Eq. (59) and in Appendix, through neglecting some minimal parts or elements, the relationships are illustrated as Tables 1-3 and Figs. 1-3, the coefficients related with corresponding tables are in bold type:

Tables Icon

Table 1. Relationships between MR and mdet

Tables Icon

Table 2. Relationships between MD and mdet

Tables Icon

Table 3. Relationships between MΔand mdep

 

Fig. 1 Relationships between MR and mdet.

Download Full Size | PPT Slide | PDF

 

Fig. 2 Relationships between MD and mdet.

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Fig. 3 Relationships between MΔand mdep.

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Note that the MΔ, MRand MD can be referred to analyze the general coefficients straightforwardly like Δ1, |D|, R, but the other specific coefficients related to the polarization and depolarization properties cannot be extracted immediately. However, from the tables and figures, one can directly obtain the values of the parameters and the coefficients of the polarization properties, such as the linear dichroism along the x-y laboratory axes and the 45°axes, the circular dichroism, the linear birefringence along the x-y laboratory axes and the 45°axes, the circular birefringence, the anomalous isotropic absorption and the anomalous depolarization properties only by analyzing the polar decomposed components of macroscopic Muller matrix. Thus, the relationships presented in this paper can be complementary explanations for the polar decomposition of the Mueller matrix. To compare the proposed method and the polar decomposition method and demonstrate the total process of the outcome parameters, flow charts are given below:

Where the meanings of the parameters and the coefficients in Figs. 4 and 5 are the same as the ones mentioned in the former parts. The process of polar decomposition of a Mueller matrix is clearly shown in Fig. 4, the direction of the arrows represents the decomposed order. The decomposed process of our method is given in Fig. 5. On comparison of the two methods, we can conclude that the proposed method using only MΔ, MR and MD has fewer steps to get the same parameters or coefficients of the known polarization properties, and the red blocks in Fig. 4 (same as the black ones in Fig. 5) can be canceled in this case, therefore, our method also provides a solution to simplify the polar decomposition of Mueller matrix.

 

Fig. 4 The flow chart showing the outcome parameters and coefficients of the polar decomposition method. The coefficients in the red dotted blocks represent three different polarization properties.

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Fig. 5 The flow chart showing the outcome parameters and coefficients of our method based on the relationships of the components between the polar decomposition and the differential Mueller matrix method. The coefficients in black blocks can be derived or obtained from the yellow blocks using parameters of the differential Mueller matrix.

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4. Conclusion

In this work, a comparative evaluation of the parameters and the polarization coefficients derived via the differential matrix decomposition and the polar decomposition and relationships between them are investigated. Related expressions and definitions are given. Physical realizability and necessary conditions of the generating differential matrices are derived. The practical usage and the feasibility of the relationships are illustrated with an experimental Mueller matrix from a literature. The results are in well agreement with the theoretical analysis. This method is appropriate for the media in which several polarization effects exist simultaneously. The general method performed by seven differential matrices described in this paper enables one to obtain the optical properties of either depolarizing or non-depolarizing media, and provides us with a solution to simplify the polar decomposition of the Mueller matrix and a new insight for a vast range of applications. The decomposition of MΔincluding three matrices leads to the obtainment of the more specific depolarizing characteristics compared with the only matrix MΔand implies the possibility to analyze the influence of the depolarization completely using polar decomposition method.

Appendix

A description of the MRthat consists η,ν, and μas three parameters of birefringence is presented, x=η2ν2:

MR(η,ν,μ)=[10000η2+ν2cos(ix)xcosμ+ην(cos(ix)1)xsinμη2+ν2cos(ix)xsinμην(cos(ix)1)xcosμνisin(ix)x0ην(cos(ix)1)xcosμ+ν2+η2cos(ix)xsinμην(cos(ix)1)xsinμν2+η2cos(ix)xcosμηisin(ix)x0νisin(ix)xcosμηisin(ix)xsinμνisin(ix)xsinμ+ηisin(ix)xcosμcos(ix)].

Using Eqs. (9) and (19), the total diattenuation matrix MDcan be given by, y=β2+δ2+γ2:

MD(β,γ,δ)=[1βitan(iy)yγitan(iy)yδitan(iy)yβitan(iy)yδ2+γ2+β2cos(iy)ycos(iy)βγ(cos(iy)1)ycos(iy)βδ(cos(iy)1)ycos(iy)γitan(iy)yβγ(cos(iy)1)ycos(iy)δ2+β2+γ2cos(iy)ycos(iy)δγ(cos(iy)1)ycos(iy)δitan(iy)yβδ(cos(iy)1)ycos(iy)δγ(cos(iy)1)ycos(iy)β2+γ2+δ2cos(iy)ycos(iy)].

The description of the whole Mueller matrix of the depolarizer is given by,z=η'2+ν'2, z1=cos(iz),z2=sin(iz),z3=cos(iμ'),z4=sin(iμ'):

MΔ=MΔα1α2α3,βγδMΔ-β-γ-δ(MΔμMΔνη)=[1000β'(eα11)α1eα1(z3η'2+ν'2z1ziz4η'ν'(z11)z)eα1(z3η'ν'(z1)1)ziz4ν'2+η'2z1)z)eα1(z3ν'iz2zz4η'z2)z)γ'(eα21)α2eα2(z3η'ν'(z11)ziz4η'2+ν'2z1z)eα2(z3ν'2+η'2z1ziz4η'ν'(z11)z)eα2(z4ν'z2zz3η'iz2z)δ'(eα31)α3eα3ν'iz2zeα3η'iz2zeα3z1]

Another possible order to obtain MΔμνηis MΔνηMΔμ, which is easily proved to be closely to the deposition MΔμMΔνη, with the conditions that cos(iη'2+ν'2) and isin(iμ')η'isin(iη'2+ν'2)η'2+ν'2or isin(iμ')ν'isin(iη'2+ν'2)η'2+ν'2are approximately equal to one and zero, respectively. Whatever the order is, the 3×3 submatrix mΔ1can be treated as a symmetric matrix due to the minimal difference of the parameters at the symmetrical position.

Funding

National Natural Science Foundation of China (NSFC) (61275198, 60978069); Key Special Projects of “Major Scientific Instruments and Equipment Development” of the National Key Research and Development Plan, Ministry of Science and Technology, P. R. China (2017YFF0107100).

References

1. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14(1), 190–202 (2006). [CrossRef]   [PubMed]  

2. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002). [CrossRef]   [PubMed]  

3. S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000). [CrossRef]   [PubMed]  

4. W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999). [CrossRef]   [PubMed]  

5. B. Jirgensons, Optical Rotatory Dispersion of Proteins and Other Macromolecules, (Springer-Verlag,1960).

6. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001). [CrossRef]   [PubMed]  

7. G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. 40(22), 3810–3821 (2001). [CrossRef]   [PubMed]  

8. N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004). [CrossRef]   [PubMed]  

9. S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003). [CrossRef]   [PubMed]  

10. A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005). [CrossRef]   [PubMed]  

11. C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006). [CrossRef]   [PubMed]  

12. J. F. de Boer, C. K. Hitzenberger, and Y. Yasuno, “Polarization sensitive optical coherence tomography - a review [Invited],” Biomed. Opt. Express 8(3), 1838–1873 (2017). [CrossRef]   [PubMed]  

13. R. A. Chipman, Handbook of Optics, (2nded., M. Bass, Ed.), Chap. 22.

14. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, (Wiley, 1998).

15. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]  

16. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009). [CrossRef]   [PubMed]  

17. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007). [CrossRef]   [PubMed]  

18. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36(12), 2330–2332 (2011). [CrossRef]   [PubMed]  

19. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36(10), 1942–1944 (2011). [CrossRef]   [PubMed]  

20. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

21. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29(19), 2234–2236 (2004). [CrossRef]   [PubMed]  

22. T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012). [CrossRef]   [PubMed]  

23. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4matrix calculus,” J. Opt. Soc. Am. 68(12), 1756–1767 (1978). [CrossRef]  

24. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36(13), 2429–2431 (2011). [CrossRef]   [PubMed]  

25. V. Devlaminck, “Physical model of differential Mueller matrix for depolarizing uniform media,” J. Opt. Soc. Am. A 30(11), 2196–2204 (2013). [CrossRef]   [PubMed]  

26. T. A. Germer, “Realizable differential matrices for depolarizing media,” Opt. Lett. 37(5), 921–923 (2012). [CrossRef]   [PubMed]  

27. M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39(7), 1779–1782 (2014). [CrossRef]   [PubMed]  

28. S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012). [CrossRef]   [PubMed]  

29. J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017). [CrossRef]   [PubMed]  

30. N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009). [CrossRef]   [PubMed]  

31. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009). [CrossRef]  

32. X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010). [CrossRef]   [PubMed]  

33. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011). [CrossRef]   [PubMed]  

34. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008). [CrossRef]   [PubMed]  

35. R. Ossikovski and V. Devlaminck, “General criterion for the physical realizability of the differential Mueller matrix,” Opt. Lett. 39(5), 1216–1219 (2014). [CrossRef]   [PubMed]  

36. S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttg.) 75, 26 (1986).

References

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  1. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14(1), 190–202 (2006).
    [Crossref] [PubMed]
  2. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
    [Crossref] [PubMed]
  3. S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
    [Crossref] [PubMed]
  4. W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
    [Crossref] [PubMed]
  5. B. Jirgensons, Optical Rotatory Dispersion of Proteins and Other Macromolecules, (Springer-Verlag,1960).
  6. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
    [Crossref] [PubMed]
  7. G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. 40(22), 3810–3821 (2001).
    [Crossref] [PubMed]
  8. N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
    [Crossref] [PubMed]
  9. S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
    [Crossref] [PubMed]
  10. A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
    [Crossref] [PubMed]
  11. C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
    [Crossref] [PubMed]
  12. J. F. de Boer, C. K. Hitzenberger, and Y. Yasuno, “Polarization sensitive optical coherence tomography - a review [Invited],” Biomed. Opt. Express 8(3), 1838–1873 (2017).
    [Crossref] [PubMed]
  13. R. A. Chipman, Handbook of Optics, (2nded., M. Bass, Ed.), Chap. 22.
  14. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, (Wiley, 1998).
  15. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996).
    [Crossref]
  16. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009).
    [Crossref] [PubMed]
  17. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007).
    [Crossref] [PubMed]
  18. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36(12), 2330–2332 (2011).
    [Crossref] [PubMed]
  19. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36(10), 1942–1944 (2011).
    [Crossref] [PubMed]
  20. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).
  21. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29(19), 2234–2236 (2004).
    [Crossref] [PubMed]
  22. T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
    [Crossref] [PubMed]
  23. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4matrix calculus,” J. Opt. Soc. Am. 68(12), 1756–1767 (1978).
    [Crossref]
  24. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36(13), 2429–2431 (2011).
    [Crossref] [PubMed]
  25. V. Devlaminck, “Physical model of differential Mueller matrix for depolarizing uniform media,” J. Opt. Soc. Am. A 30(11), 2196–2204 (2013).
    [Crossref] [PubMed]
  26. T. A. Germer, “Realizable differential matrices for depolarizing media,” Opt. Lett. 37(5), 921–923 (2012).
    [Crossref] [PubMed]
  27. M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39(7), 1779–1782 (2014).
    [Crossref] [PubMed]
  28. S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
    [Crossref] [PubMed]
  29. J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
    [Crossref] [PubMed]
  30. N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
    [Crossref] [PubMed]
  31. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
    [Crossref]
  32. X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
    [Crossref] [PubMed]
  33. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
    [Crossref] [PubMed]
  34. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
    [Crossref] [PubMed]
  35. R. Ossikovski and V. Devlaminck, “General criterion for the physical realizability of the differential Mueller matrix,” Opt. Lett. 39(5), 1216–1219 (2014).
    [Crossref] [PubMed]
  36. S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttg.) 75, 26 (1986).

2017 (2)

J. F. de Boer, C. K. Hitzenberger, and Y. Yasuno, “Polarization sensitive optical coherence tomography - a review [Invited],” Biomed. Opt. Express 8(3), 1838–1873 (2017).
[Crossref] [PubMed]

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

2014 (2)

2013 (1)

2012 (3)

T. A. Germer, “Realizable differential matrices for depolarizing media,” Opt. Lett. 37(5), 921–923 (2012).
[Crossref] [PubMed]

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

2011 (4)

2010 (1)

2009 (3)

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009).
[Crossref] [PubMed]

2008 (1)

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (2)

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14(1), 190–202 (2006).
[Crossref] [PubMed]

2005 (1)

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

2004 (2)

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29(19), 2234–2236 (2004).
[Crossref] [PubMed]

2003 (1)

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

2002 (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

2001 (2)

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. 40(22), 3810–3821 (2001).
[Crossref] [PubMed]

2000 (1)

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

1999 (1)

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

1996 (1)

1987 (1)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

1986 (1)

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttg.) 75, 26 (1986).

1978 (1)

Alexandrakis, G.

Arce-Diego, J. L.

Azzam, R. M. A.

Backman, V.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Badizadegan, K.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

Bouma, B. E.

Bouma, G. J.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Buddhiwant, P.

Busch, D. R.

Butler, J.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Cerussi, A. E.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttg.) 75, 26 (1986).

Darling, C. L.

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

Dasari, R. R.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

de Boer, J. F.

De Martino, A.

Devlaminck, V.

Dimofte, A.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Elson, D. S.

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

Faris, G. W.

Feld, M. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Finlay, J. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Fried, D.

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

Georgakoudi, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Germer, T. A.

Ghosh, N.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
[Crossref] [PubMed]

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14(1), 190–202 (2006).
[Crossref] [PubMed]

Gil, J. J.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

Goudail, F.

Groner, W.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Guo, X.

Gupta, P. K.

Gurjar, R. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Guyot, S.

Hanna, C. F.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Harris, A. G.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Hitzenberger, C. K.

Hsiang, D.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Huynh, G. D.

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

Ince, C.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Itzkan, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Jacques, S. L.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Jakubowski, D.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Kantor, S.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Khalil, O. S.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Kumar, S.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

Lee, K.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Li, R.-K.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Li, S. H.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Lo, Y.-L.

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

Lu, S. Y.

Manhas, S.

Messmer, K.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Morio, J.

Nadeau, R. G.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Ortega-Quijano, N.

Ossikovski, R.

Patterson, M. S.

Perelman, L. T.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Pham, T. T.

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

Purwar, H.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

Qi, J.

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

Ramella-Roman, J. C.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

Roman, J. R.

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Shah, N.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Singh, J.

Swami, M. K.

Tromberg, B. J.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Villiger, M.

Vitkin, I. A.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
[Crossref] [PubMed]

Weisel, R. D.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Wilson, B. C.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Winkelman, J. W.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Wood, M. F. G.

X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
[Crossref] [PubMed]

Yasuno, Y.

Yeh, S. J.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Zhu, T. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Appl. Opt. (2)

Biomed. Opt. Express (1)

J. Appl. Phys. (1)

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

J. Biomed. Opt. (8)

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
[Crossref] [PubMed]

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

J. Biophotonics (2)

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Lasers Surg. Med. (1)

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Nat. Med. (2)

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (8)

Optik (Stuttg.) (2)

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttg.) 75, 26 (1986).

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

Phys. Med. Biol. (1)

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Other (3)

B. Jirgensons, Optical Rotatory Dispersion of Proteins and Other Macromolecules, (Springer-Verlag,1960).

R. A. Chipman, Handbook of Optics, (2nded., M. Bass, Ed.), Chap. 22.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, (Wiley, 1998).

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Figures (5)

Fig. 1
Fig. 1 Relationships between M R and m d e t .
Fig. 2
Fig. 2 Relationships between M D and m d e t .
Fig. 3
Fig. 3 Relationships between M Δ and m d e p .
Fig. 4
Fig. 4 The flow chart showing the outcome parameters and coefficients of the polar decomposition method. The coefficients in the red dotted blocks represent three different polarization properties.
Fig. 5
Fig. 5 The flow chart showing the outcome parameters and coefficients of our method based on the relationships of the components between the polar decomposition and the differential Mueller matrix method. The coefficients in black blocks can be derived or obtained from the yellow blocks using parameters of the differential Mueller matrix.

Tables (3)

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Table 1 Relationships between M R and m d e t

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Table 2 Relationships between M D and m d e t

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Table 3 Relationships between M Δ and m d e p

Equations (64)

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M = M R M D .
M = M L R M C R M D ,
d M d z = d M L R d z M C R M D + M L R d M C R d z M D + M L R M C R d M D d z .
m M = ( m L R M L R ) M C R M D + M L R ( m C R M C R ) M D + M L R M C R ( m D M D ) ,
M | z 0 = 0 = M L R | z 0 = 0 = M C R | z 0 = 0 = M D | z 0 = 0 = I .
m | z 0 = 0 = m L R + m C R + m D .
m i n i t i a l = [ α β γ δ β α μ ν γ μ α η δ ν η α ] ,
m = [ 0 β γ δ β 0 μ ν γ μ 0 η δ ν η 0 ] .
M = W d i a g ( exp ( σ 0 z ) , exp ( σ 1 z ) , exp ( σ 2 z ) , exp ( σ 3 z ) ) W - 1 ,
m L R = [ 0 0 0 0 0 0 0 ν 0 0 0 η 0 ν η 0 ] .
M L R ( η , ν ) = [ 1 0 0 0 0 η 2 + ν 2 cos ( i η 2 ν 2 ) η 2 + ν 2 η ν ( cos ( i η 2 ν 2 ) 1 ) η 2 + ν 2 ν i sin ( i η 2 ν 2 ) η 2 ν 2 0 η ν ( cos ( i η 2 ν 2 ) 1 ) η 2 + ν 2 ν 2 + η 2 cos ( i η 2 ν 2 ) η 2 + ν 2 η i sin ( i η 2 ν 2 ) η 2 ν 2 0 ν i sin ( i η 2 ν 2 ) η 2 ν 2 η i sin ( i η 2 ν 2 ) η 2 ν 2 cos ( i η 2 v 2 ) ] .
M L R ( φ , θ ) = [ 1 0 0 0 0 cos 2 2 θ + sin 2 2 θ cos φ cos 2 θ sin 2 θ ( 1 cos φ ) sin 2 θ sin φ 0 cos 2 θ sin 2 θ ( 1 cos φ ) sin 2 2 θ + cos 2 2 θ cos φ cos 2 θ sin φ 0 sin 2 θ sin φ cos 2 θ sin φ cos φ ] ,
φ = i η 2 v 2 , cos 2 θ = η i η 2 ν 2 , sin 2 θ = ν i η 2 ν 2 .
m C R = [ 0 0 0 0 0 0 μ 0 0 μ 0 0 0 0 0 0 ] .
M C R ( μ ) = [ 1 0 0 0 0 cos μ sin μ 0 0 sin μ cos μ 0 0 0 0 1 ] .
M C R ( δ c ) = [ 1 0 0 0 0 cos δ c sin δ c 0 0 sin δ c cos δ c 0 0 0 0 1 ] .
δ c = μ ,
R = cos 1 ( t r ( M R ) 2 1 ) = cos - 1 ( ( cos μ + 1 ) ( 1 + cos ( i η 2 ν 2 ) ) 2 1 ) ,
m D = [ 0 β γ δ β 0 0 0 γ 0 0 0 δ 0 0 0 ] .
M D = T u [ 1 D T D m 3 ] ,
m 3 = a 1 I + b 1 ( D ^ D ^ T ) = ( 1 | D | 2 ) 1 / 2 I + [ 1 ( 1 | D | 2 ) 1 / 2 ] D ^ D ^ T ,
M D = [ 1 B 12 B 13 B 14 B 12 c + b B 12 2 a b B 12 B 13 a b B 12 B 14 a B 13 b B 12 B 13 a c + b B 13 2 a b B 13 B 14 a B 14 b B 12 B 14 a b B 13 B 14 a c + b B 14 2 a ] ,
D = [ D H D 45 D C ] = [ B 12 B 13 B 14 ] ,
D L = D H 2 + D 45 2 = ( β 2 γ 2 ) tan ( i β 2 + δ 2 + γ 2 ) β 2 + δ 2 + γ 2 = B 12 2 + B 13 2 .
D C = B 14 = δ i tan ( i β 2 + δ 2 + γ 2 ) β 2 + δ 2 + γ 2 .
D = D H 2 + D 45 2 + D C 2 = D L 2 + D C 2 = | D | = ± i tan ( i β 2 + δ 2 + γ 2 ) .
M = M Δ M R M D ,
M Δ = [ 1 0 0 0 p 1 e 1 0 0 p 2 0 e 2 0 p 3 0 0 e 3 ] ,
M = M Δ M L R M C R M D = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ μ ν η M L R M C R M D ,
M Δ μ ν η = M Δ μ M Δ ν η = M Δ ν η M Δ μ ,
M = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ μ ν η M L R M C R M D = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( M Δ μ M Δ ν η ) M L R M C R M D .
d M d z = d M Δ α 1 α 2 α 3 , β γ δ d z M Δ - β - γ - δ M Δ - μ ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ d M Δ - β - γ - δ d z M Δ - μ ν η M L R M C R M D M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ d ( M Δ μ ) d z M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ d ( M Δ ν η ) d z M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η d M L R d z M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R d M C R d z M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R M C R d M D d z .
m M = ( m Δ α 1 α 2 α 3 , β γ δ M Δ α 1 α 2 α 3 , β γ δ ) M Δ - β - γ - δ M Δ - μ ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ ( m Δ - β - γ - δ M Δ - β - γ - δ ) M Δ - μ ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( m Δ - μ M Δ - μ ) M Δ - ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ( m Δ - ν η M Δ - ν η ) M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η ( m L R M L R ) M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R ( m C R M C R ) M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R M C R ( m D M D ) .
M | z 0 = 0 = M Δ α 1 α 2 α 3 , β γ δ | z 0 = 0 = M Δ β γ δ | z 0 = 0 = M Δ - μ | z 0 = 0 = M Δ - ν η | z 0 = 0 = M L R | z 0 = 0 = M C R | z 0 = 0 = M D | z 0 = 0 = I .
m | z 0 = 0 = ( m Δ α 1 α 2 α 3 , β γ δ + m Δ β γ δ + m Δ - μ + m Δ - ν η ) + ( m L R + m C R + m D ) = m d e p + m d e t ,
m d e p = [ 0 β ' γ ' δ ' β ' α 1 μ ' ν ' γ ' μ ' α 2 η ' δ ' ν ' η ' α 3 ] = m Δ α 1 α 2 α 3 , β γ δ + m Δ β γ δ + m Δ - μ + m Δ - ν η ,
m Δ α 1 α 2 α 3 , β γ δ = [ 0 0 0 0 β ' α 1 0 0 γ ' 0 α 2 0 δ ' 0 0 α 3 ] , m Δ - μ = [ 0 0 0 0 0 0 μ ' 0 0 μ ' 0 0 0 0 0 0 ] , m Δ - ν η = [ 0 0 0 0 0 0 0 ν ' 0 0 0 η ' 0 ν ' η ' 0 ] .
M Δ - α 1 α 2 α 3 , β γ δ = [ 1 0 0 0 β ' ( exp ( α 1 ) 1 ) α 1 exp ( α 1 ) 0 0 γ ' ( exp ( α 2 ) 1 ) α 2 0 exp ( α 2 ) 0 δ ' ( exp ( α 3 ) 1 ) α 3 0 0 exp ( α 3 ) ] , M Δ - μ = [ 1 0 0 0 0 cos ( i μ ' ) i sin ( i μ ' ) 0 0 i sin ( i μ ' ) cos ( i μ ' ) 0 0 0 0 1 ] .
M Δ - ν η = [ 1 0 0 0 0 η ' 2 + ν ' 2 cos ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 η ' ν ' ( cos ( i η ' 2 + ν ' 2 ) 1 ) η ' 2 + ν ' 2 ν ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 0 η ' ν ' ( cos ( i η ' 2 + ν ' 2 ) 1 ) η ' 2 + ν ' 2 ν ' 2 + η ' 2 cos ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 η ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 0 ν ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 η ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 cos ( i η ' 2 + ν ' 2 ) ] .
m Δ β γ δ = [ 0 β ' γ ' δ ' 0 0 0 0 0 0 0 0 0 0 0 0 ] ,
d S d z = m S ,
d S 0 / d z = β ' S 1 γ ' S 2 δ ' S 3 d S 1 / d z = 0 d S 2 / d z = 0 d S 3 / d z = 0.
S 0 = S 0 0 ( β ' S 1 + γ ' S 2 + δ ' S 3 ) z ,
M Δ = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ μ ν η = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( M Δ μ M Δ ν η ) = [ 1 0 0 0 β ' ( exp α 1 1 ) α 1 exp α 1 0 0 γ ' ( exp α 2 1 ) α 2 0 exp α 2 0 δ ' ( exp α 3 1 ) α 3 0 0 exp α 3 ] [ 1 0 0 0 0 cos ( i μ ' ) i sin ( i μ ' ) 0 0 i sin ( i μ ' ) cos ( i μ ' ) 0 0 0 0 1 ] [ 1 0 0 0 0 η ' 2 + ν ' 2 cos ( i z ) z η ' ν ' ( cos ( i z ) 1 ) η ' 2 + ν ' 2 ν ' i sin ( i z ) z 0 η ' ν ' ( cos ( i z ) 1 ) z ν ' 2 + η ' 2 cos ( i z ) z η ' ' i sin ( i z ) z 0 ν ' i sin ( i z ) z η ' ' i sin ( i η ' 2 + ν ' 2 ) z cos ( i z ) ] .
M Δ 1 = [ 1 0 p 1 m Δ 1 ] , m Δ 1 T = m Δ 1 ,
Δ 1 = 1 - | t r ( m Δ 1 ) | 3 = 1 - | t r ( M Δ 1 ) 1 | 3 = 1 - ( exp ( α 1 ) + exp ( α 2 ) ) cos ( i μ ' ) + exp ( α 3 ) cos ( i η ' 2 + ν ' 2 ) ) 3 , 0 Δ 1 1.
M ( z + Δ z ) = ( I + m d e t Δ z + m d e p Δ z ) M ( z ) ,
C ( I ) = [ 1 0 T 0 O 3 ] .
M ( z + Δ z ) = ( I + m L R Δ z ) M ( z ) .
C ( m L R ) = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] .
C ( m C R ) = [ 0 0 0 μ i 2 0 0 0 0 0 0 0 0 μ i 2 0 0 0 ] , C ( m D ) = 1 2 [ 0 β γ δ β 0 0 0 γ 0 0 0 δ 0 0 0 ] .
C ( m Δ - μ ) = [ 0 0 0 0 0 0 μ ' 2 0 0 μ ' 2 0 0 0 0 0 0 ] , C ( m Δ - ν η ) = [ 0 0 0 0 0 0 0 ν ' 2 0 0 0 η ' 2 0 ν ' 2 η ' 2 0 ] .
C ( m Δ β γ δ ) = 1 4 [ 0 β ' γ ' δ ' β ' 0 i δ ' i γ ' γ ' i δ ' 0 i β ' δ ' i γ ' i β ' 0 ] = 1 4 ( [ 0 β ' γ ' δ ' β ' 0 0 0 γ ' 0 0 0 δ ' 0 0 0 ] + [ 0 0 0 0 0 0 i δ ' 0 0 i δ ' 0 0 0 0 0 ' 0 ] + [ 0 0 0 0 0 0 0 i γ ' 0 0 0 i β ' 0 i γ ' i β ' 0 ] ) .
C ( m Δ α 1 α 2 α 3 , β γ δ ) = 1 4 [ α 1 + α 2 + α 3 β ' γ ' δ ' β ' α 1 - α 2 - α 3 - i δ ' i γ ' γ ' i δ ' - α 1 + α 2 - α 3 - i β ' δ ' i γ ' i β ' - α 1 - α 2 + α 3 ] = 1 4 ( [ 0 β ' γ ' δ ' β ' 0 - i δ ' i γ ' γ ' i δ ' 0 - i β ' δ ' i γ ' i β ' 0 ] + [ α 1 + α 2 + α 3 0 0 0 0 α 1 - α 2 - α 3 0 0 0 0 - α 1 + α 2 - α 3 0 0 0 0 - α 1 - α 2 + α 3 ] ) .
M = [ 1 0.026 0.044 0.039 0.029 0.962 0.144 0.047 0.002 0.126 0.975 0.026 0.039 0.019 0.115 0.936 ] .
m = [ -0 .0011 0 .0237 0 .0487 -0 .0405 0 .0286 -0 .0292 -0 .1457 -0 .0467 0 .0008 0 .1292 -0 .0177 0 .0303 -0 .0407 0 .0126 0 .1223 -0 .0684 ] = m d e t + m d e p .
m d e t = [ 0 β γ δ β 0 μ ν γ μ 0 η δ ν η 0 ] = [ 0 0 .0262 0 .0247 -0 .0406 0 .0262 0 -0 .1374 -0 .0296 0 .0247 0 .1374 0 -0 .046 -0 .0406 0 .0296 0 .046 0 ] .
m d e p = [ 0 β ' γ ' δ ' β ' α 1 μ ' ν ' γ ' μ ' α 2 η ' δ ' ν ' η ' α 3 ] = [ 0 -0 .0025 0 .024 0 .0001 0 .0025 -0 .0281 -0 .0083 -0 .0170 -0 .024 -0 .0083 -0 .0166 0 .0763 -0 .0001 -0 .0170 0 .0763 -0 .0673 ] .
M Δ = [ 1 0 0 0 0.008 0.976 0.01 0.021 0.023 0.01 0.982 0.073 0.009 0.022 0.073 0.941 ] , M R = [ 1 0 0 0 0 0.99 0.138 0.027 0 0.136 0.99 0.048 0 0.033 0.044 0.998 ] , M D = [ 1 0.026 0.044 0.039 0.026 0.998 0.001 0.001 0.044 0.001 0.999 0.001 0.039 0.001 0.001 0.999 ] .
η = - 0. 048 ; ν = -0 .027 ; μ = - 0.1374 ; β = 0.026 ; γ = 0.044 ; δ = - 0.039 ,
α 3 = - 0.0608 ; α 2 = -0 .0182; α 1 = 0.0243 ; η = 0.0776 ; ν = 0.0234 ; μ = - 0.0102 ; β = 0.008 ; γ = - 0.023 ; δ = - 0.009.
M R ( η , ν , μ ) = [ 1 0 0 0 0 η 2 + ν 2 cos ( i x ) x cos μ + η ν ( cos ( i x ) 1 ) x sin μ η 2 + ν 2 cos ( i x ) x sin μ η ν ( cos ( i x ) 1 ) x cos μ ν i sin ( i x ) x 0 η ν ( cos ( i x ) 1 ) x cos μ + ν 2 + η 2 cos ( i x ) x sin μ η ν ( cos ( i x ) 1 ) x sin μ ν 2 + η 2 cos ( i x ) x cos μ η i sin ( i x ) x 0 ν i sin ( i x ) x cos μ η i sin ( i x ) x sin μ ν i sin ( i x ) x sin μ + η i sin ( i x ) x cos μ cos ( i x ) ] .
M D ( β , γ , δ ) = [ 1 β i tan ( i y ) y γ i tan ( i y ) y δ i tan ( i y ) y β i tan ( i y ) y δ 2 + γ 2 + β 2 cos ( i y ) y cos ( i y ) β γ ( cos ( i y ) 1 ) y cos ( i y ) β δ ( cos ( i y ) 1 ) y cos ( i y ) γ i tan ( i y ) y β γ ( cos ( i y ) 1 ) y cos ( i y ) δ 2 + β 2 + γ 2 cos ( i y ) y cos ( i y ) δ γ ( cos ( i y ) 1 ) y cos ( i y ) δ i tan ( i y ) y β δ ( cos ( i y ) 1 ) y cos ( i y ) δ γ ( cos ( i y ) 1 ) y cos ( i y ) β 2 + γ 2 + δ 2 cos ( i y ) y cos ( i y ) ] .
M Δ = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( M Δ μ M Δ ν η ) = [ 1 0 0 0 β ' ( e α 1 1 ) α 1 e α 1 ( z 3 η ' 2 + ν ' 2 z 1 z i z 4 η ' ν ' ( z 1 1 ) z ) e α 1 ( z 3 η ' ν ' ( z 1 ) 1 ) z i z 4 ν ' 2 + η ' 2 z 1 ) z ) e α 1 ( z 3 ν ' i z 2 z z 4 η ' z 2 ) z ) γ ' ( e α 2 1 ) α 2 e α 2 ( z 3 η ' ν ' ( z 1 1 ) z i z 4 η ' 2 + ν ' 2 z 1 z ) e α 2 ( z 3 ν ' 2 + η ' 2 z 1 z i z 4 η ' ν ' ( z 1 1 ) z ) e α 2 ( z 4 ν ' z 2 z z 3 η ' i z 2 z ) δ ' ( e α 3 1 ) α 3 e α 3 ν ' i z 2 z e α 3 η ' i z 2 z e α 3 z 1 ]

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