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
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 , enhancing the image contrast for superficial and deeper structures of tissues [2–4], quantifying the protein properties , and differentiating the normal and precancerous cells or revealing the border of cancer .
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 , symmetric decomposition , reverse polar decomposition , differential decomposition [18,19]. In these methods, the polar decomposition method proposed by Gil has been proved to be very useful . 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 .
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 . 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 . 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 . 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 . 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 . 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  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 shows that the depolarization matrix can 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.
2.1 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 a retarderand a depolarizer . In our analysis, we first consider the non-depolarizing case.
When medium is non-depolarizing homogeneous anisotropic, its Mueller matrix can be expressed as
To extract the linear and circular birefringence of the medium, can be further decomposed into a product of two matrices of linear birefringence and circular birefringence , Eq. (1) can then be written asEq. (2), we have
Here it should be pointed out that for homogeneous medium m is independent of z. Thus, can be readily integrated to obtain . In this case, m always commutes with M (that means commutes with , commutes with , and so on). On the other hand, if ,and can commute with each other, then each one of ,and can commute with any one of ,and . A combination of the above two points suggests that any two of ,and will commute with each other. However, and do not commute with each other in the polar decomposition and (). So the matrices ,and do not commute. The same analysis is valid for and .25], here, is the first value in , then, we have
Noting that the corresponding differential matrix has been related to the macroscopic Mueller matrix through its eigenvalues and eigenvectors :Equation (9) shows that can be completely determined from the eigenanalysis of differential matrix . 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 which describes the linear birefringence denoted by the parameters and:
By using Eq. (9), the effective Mueller matrix corresponding to can be expressed as
At the same time, the general form of the Mueller matrix for a linear retarder  isEquations (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:
Next, consider the differential matrix representing circular retardance effect alone:
With the same token, the effective Mueller matrix can be related to the corresponding differential matrix :
We haveEq. (18) becomes , only linear phase retardance is considered and circular retardance can be neglected;conversely, only the circular retardance is significant, we have . It is evident that, in general case, Eq. (18) should be employed.
For the differential Mueller matrix of the diattenuation,Eq. (20) can be explicitly rewritten as
whereand , is the element of the macroscopic Mueller matrix. , and . Note that the intensity transmittance is completely determined by the first row of the arbitrary Mueller matrix.
A diattenuation vector is usually used as a measure of the diattenuation and its direction, which is defined as 15]. On comparison of Eq. (23) with Eqs. (22) and (63), we have
It should be pointed out that the horizontal and the 45°diattenuation have been combined to describe the linear diattenuation.
2.2 Depolarizing homogeneous anisotropic media
For the depolarizing homogeneous anisotropic media, its Mueller matrix can be decomposed into a product of three matrices :30–34], can be expressed asEq. (28).
By decomposing the differential Mueller matrix of the macroscopic matrixinto 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 which should satisfy Eq. (28). To this end, can be decomposed intoEq. (30) [see the discussions of Eq. (64) in Appendix].
By differentiating both sides of Eq. (31), we have
Using , Eq. (32) can be rewritten as
Consider a linear homogeneous medium,
Equation (33) is then reduced toEq. (35) will be discussed in the next part.
To give specific expressions of and, a general differential matrix for depolarizing medium can be expressed as [18,24]23].
To separate the different properties and satisfy the form of in Eq. (28), , and are given by
The corresponding macroscopic Mueller matricesand can then be found:
The above analysis is valid only if the eigenvectors of the elementary differential matrices are different from each other. For differential matrixEq. (9) due to the multiple roots, consequently, this one can be treated as a exceptional case, by the following equation
Through integration, the last three equations yield constant values independent of z for and, which indicate the conservation of the horizonal, the 45°and the circular polarization preference. The first equation yields the linear decay of that is a function of the distance z19,25], the decay of can be neglected. So the corresponding Mueller matrix of Eq. (40) can be an identical matrix to .
By using the expressions of , and (see Eqs. (38) and (39)), can be expressed as the combination of the three matrices ()
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)Eq. (64), is too small to be considered, and can be replaced by 1, then the net depolarization coefficient defined in  can be expressed as
3. Physical realizability and discussions
To examine the physical realizability of the elementary matrices of , the general criterion derived by Ossikovski and Devlaminck  is used36] can be transformed into the examination of positive semidefiniteness of the coherency matrix (denoted as ). For this purpose, we consider the physical realizability of . The coherency matrix of the identity matrix is
Equation (47) can be reduced to
The corresponding coherency matrix of can be expressed as
As the consequence, is the same as which is obviously positive semidefinite with four eigenvalues of 1, 0, 0 and 0. Then, the coherency matrices for and are readily obtained
The eigenvalues of and are , , respectively. The first three eigenvalues of both coherency matrices are non-negative and the last ones of them tend to zero faster than and can be seen as zero. Similarly, when the differential matrix m is depolarizing, and should be verified to be positive semidefinite. This can be found by calculating the eigenvalues of and as, according to the following matrices
Since andare too small, multiplications of two of them can be treated as zero. In virtue of the complexity of both forms and , and can be decomposed as combinations of several basic coherency matrices which must be positive semidefinite each to satisfy the criterion. Coherency matrix of can be deposed as
It follows that the first two terms are proved to be positive semidefinite according to the eigenvalues ofand , the physical realizability of solely depends on the third part whose eigenvalues are , since andare too small, multiplications of two of them can be treated as zero. Like deposition Eq. (53),
To check the positive semidefiniteness of associated matrix, only the second term is discussed. The eigenvalues are easily obtained as , where, the first one is the largest one (one can always achieve the condition by adding to the coherency matrix above , where c is a constant) since and are negative ones, in general, all the ones can be closed to zero or positive.
Here, an experimental data  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 . 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 -diameter polystyrene microspheres, where exhibited the properties of diattenuation, depolarization and birefringence. The scattering coefficient and the glucose concentration coefficient measured were and 5M. For the usual method,
The effective differential matrix is then found by , whereand are the eigenvalues of , is a matrix whose columns are the eigenvectors of .
The depolarizing part and the determined part are
The corresponding polar decomposition of the Mueller matrix Eq. (55) consists of, and are given by
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.
The estimated results are in close agreement with the expected values in Eqs. (57) and (58) except for and. 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:
Note that the , and can be referred to analyze the general coefficients straightforwardly like , , , 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 , and 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.
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 including three matrices leads to the obtainment of the more specific depolarizing characteristics compared with the only matrix and implies the possibility to analyze the influence of the depolarization completely using polar decomposition method.
A description of the that consists ,, and as three parameters of birefringence is presented, :
Using Eqs. (9) and (19), the total diattenuation matrix can be given by, :
The description of the whole Mueller matrix of the depolarizer is given by,, :
Another possible order to obtain is , which is easily proved to be closely to the deposition , with the conditions that and or are approximately equal to one and zero, respectively. Whatever the order is, the submatrix can be treated as a symmetric matrix due to the minimal difference of the parameters at the symmetrical position.
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).
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