Method of interpreting Mueller matrix of anisotropic medium

Y. Chang; W. Gao

doi:10.1364/OE.27.003305

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 $M_{Δ}$ shows that the depolarization matrix $M_{Δ}$ 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. 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 $M_{D},$ a retarder $M_{R}$ and a depolarizer $M_{Δ}$ . 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 = M_{R} M_{D} .

To extract the linear and circular birefringence of the medium, $M_{R}$ can be further decomposed into a product of two matrices of linear birefringence and circular birefringence $M_{R} = M_{L R} M_{C R}$ , Eq. (1) can then be written as

M = M_{L R} M_{C R} M_{D},

where

M_{L R}

and

M_{C R}

represent the Mueller matrices of the linear retarder and the circular retarder. By taking the derivatives of the both sides of Eq. (2), we have

\frac{d M}{d z} = \frac{d M_{L R}}{d z} M_{C R} M_{D} + M_{L R} \frac{d M_{C R}}{d z} M_{D} + M_{L R} M_{C R} \frac{d M_{D}}{d z} .

Based on the definition of the differential Mueller matrix $\frac{d M}{d z} = m M$ [23],Eq. (3) can be rewritten as

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}),

where

m_{L R}

,

m_{C R}

and

m_{D}

represent the differential matrices of linear birefringence, circular birefringence and diattenuation, corresponding to

M_{L R}

,

M_{C R}

and

M_{D}

. In the limit of

z \to 0

, the polarization state of the incident light encounters no influence. So we have

M |_{z_{0} = 0} = M_{L R} |_{z_{0} = 0} = M_{C R} |_{z_{0} = 0} = M_{D} |_{z_{0} = 0} = I .

On substitution of Eq. (5) in Eq. (4), the differential matrix at any location can be expressed as the sum of $m_{L R}$ , $m_{C R}$ and $m_{D}$ :

m |_{z_{0} = 0} = m_{L R} + m_{C R}^{} + m_{D}^{} .

Here it should be pointed out that for homogeneous medium m is independent of z. Thus, $\frac{d M}{d z} = m M$ can be readily integrated to obtain $M = \exp (m z)$ [23]. In this case, m always commutes with M (that means $m_{D}$ commutes with $M_{D}$ , $m_{C R}$ commutes with $M_{C R}$ , and so on). On the other hand, if $m_{L R}$ , $m_{C R}$ and $m_{D}$ can commute with each other, then each one of $m_{L R}$ , $m_{C R}$ and $m_{D}$ can commute with any one of $M_{L R}$ , $M_{C R}$ and $M_{D}$ . A combination of the above two points suggests that any two of $M_{L R}$ , $M_{C R}$ and $M_{D}$ will commute with each other. However, $M_{C R}$ and $M_{L R}$ do not commute with each other in the polar decomposition $M = M_{Δ} M_{R} M_{D}$ and $M = M_{R} M_{D}$ ( $M_{R} = M_{L R} M_{C R}$ ). So the matrices $m_{L R}$ , $m_{C R}$ and $m_{D}$ do not commute. The same analysis is valid for $m_{d e t}$ and $m_{d e p}$ .

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]:

m_{i n i t i a l} = [\begin{matrix} α & β & γ & δ \\ β & α & μ & ν \\ γ & - μ & α & η \\ δ & - ν & - η & α \end{matrix}],

where

α

denotes the parameter of the isotropic absorption,

β

and

γ

are the linear dichroism parameters along the x-y laboratory axes and the 45°axes,

δ

is the parameter of the circular dichroism,

η

and

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

m_{i n i t i a l, 11} I

from the initial differential matrix

m_{i n i t i a l}

when

α

is small and this operation has no influence on the other properties [25], here,

m_{i n i t i a l, 11}

is the first value in

m_{i n i t i a l}

, then, we have

m = [\begin{matrix} 0 & β & γ & δ \\ β & 0 & μ & ν \\ γ & - μ & 0 & η \\ δ & - ν & - η & 0 \end{matrix}] .

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

M = W d i a g (\exp (σ_{0} z), \exp (σ_{1} z), \exp (σ_{2} z), \exp (σ_{3} z)) W^{- 1},

where

σ_{0}, σ_{1}, σ_{2}

and

σ_{3}

are the eigenvalues of

m

, and the columns of the orthogonal matrix

W

are the respective eigenvectors of

m

. Equation (9) shows that

M

can 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

m_{L R}

which describes the linear birefringence denoted by the parameters

η

and

ν

:

m_{L R} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & ν \\ 0 & 0 & 0 & η \\ 0 & - ν & - η & 0 \end{matrix}] .

By using Eq. (9), the effective Mueller matrix $M_{L R}$ corresponding to $m_{L R}$ can be expressed as

M_{L R} (η, ν) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{η^{2} + ν^{2} \cos (i \sqrt{- η^{2} - ν^{2}})}{η^{2} + ν^{2}} & \frac{η ν (\cos (i \sqrt{- η^{2} - ν^{2}}) - 1)}{η^{2} + ν^{2}} & - \frac{ν i \sin (i \sqrt{- η^{2} - ν^{2}})}{\sqrt{- η^{2} - ν^{2}}} \\ 0 & \frac{η ν (\cos (i \sqrt{- η^{2} - ν^{2}}) - 1)}{η^{2} + ν^{2}} & \frac{ν^{2} + η^{2} \cos (i \sqrt{- η^{2} - ν^{2}})}{η^{2} + ν^{2}} & - \frac{η i \sin (i \sqrt{- η^{2} - ν^{2}})}{\sqrt{- η^{2} - ν^{2}}} \\ 0 & \frac{ν i \sin (i \sqrt{- η^{2} - ν^{2}})}{\sqrt{- η^{2} - ν^{2}}} & \frac{η i \sin (i \sqrt{- η^{2} - ν^{2}})}{\sqrt{- η^{2} - ν^{2}}} & \cos (i \sqrt{- η^{2} - v^{2}}) \end{matrix}] .

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

M_{L R} (φ, θ) = [\begin{matrix} 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 \end{matrix}],

where

φ

is the linear phase retardance in radian,

θ

is 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 = i \sqrt{- η^{2} - v^{2}}, \cos 2 θ = - \frac{η i}{\sqrt{- η^{2} - ν^{2}}}, \sin 2 θ = \frac{ν i}{\sqrt{- η^{2} - ν^{2}}} .

Next, consider the differential matrix $m_{C R}$ representing circular retardance effect alone:

m_{C R} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & μ & 0 \\ 0 & - μ & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] .

With the same token, the effective Mueller matrix $M_{C R}$ can be related to the corresponding differential matrix $m_{C R}$ :

M_{C R} (μ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos μ & \sin μ & 0 \\ 0 & - \sin μ & \cos μ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] .

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

M_{C R} (δ_{c}) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos δ_{c} & \sin δ_{c} & 0 \\ 0 & - \sin δ_{c} & \cos δ_{c} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] .

We have

δ_{c} = μ,

where

δ_{c}

represents 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 $M_{R}$ [15] (also see Appendix),

R = \cos^{- 1} (\frac{t r (M_{R})}{2} - 1) = \cos^{- 1} (\frac{(\cos μ + 1) (1 + \cos (i \sqrt{- η^{2} - ν^{2}}))}{2} - 1),

where

t r (M_{R})

represents the trace of

M_{R}

. When

μ \to 0

, Eq. (18) becomes

R = φ = i \sqrt{- η^{2} - v^{2}}

, only linear phase retardance is considered and circular retardance can be neglected;conversely, only the circular retardance is significant, we have

R = δ_{c} = μ

. It is evident that, in general case, Eq. (18) should be employed.

For the differential Mueller matrix of the diattenuation,

m_{D} = [\begin{matrix} 0 & β & γ & δ \\ β & 0 & 0 & 0 \\ γ & 0 & 0 & 0 \\ δ & 0 & 0 & 0 \end{matrix}] .

After straightforward calculation [see the formula Eq. (63) in Appendix], the effective Mueller matrix $M_{D}$ can be expressed in terms of the parameters of $m_{D}$ , while the general form of $M_{D}$ can be expressed as

M_{D} = T_{u} [\begin{matrix} 1 & \vec{D^{T}} \\ \vec{D} & m_{3} \end{matrix}],

m_{3} = a_{1} I + b_{1} (\hat{D} {\hat{D}}^{T}) = {(1 - {| D |}^{2})}^{1 / 2} I + [1 - {(1 - {| D |}^{2})}^{1 / 2}] \hat{D} {\hat{D}}^{T},

where

m_{3}

is a

3 \times 3

sub-matrix,

I

is the

3 \times 3

identity matrix,

\hat{D} = \vec{D} / | \vec{D} |

denotes the unit vector along

\vec{D}

,

T_{u}

is the transmittance for incident light and

M_{D}

is symmetric with respect to the main diagonal axis. By noting these facts, Eq. (20) can be explicitly rewritten as

M_{D} = [\begin{matrix} 1 & B_{12} & B_{13} & B_{14} \\ B_{12} & c + \frac{b B_{12}^{2}}{a} & \frac{b B_{12} B_{13}}{a} & \frac{b B_{12} B_{14}}{a} \\ B_{13} & \frac{b B_{12} B_{13}}{a} & c + \frac{b B_{13}^{2}}{a} & \frac{b B_{13} B_{14}}{a} \\ B_{14} & \frac{b B_{12} B_{14}}{a} & \frac{b B_{13} B_{14}}{a} & c + \frac{b B_{14}^{2}}{a} \end{matrix}],

where $B_{12} = \frac{m_{01}}{m_{00}}, B_{13} = \frac{m_{02}}{m_{00}}$ and $B_{14} = \frac{m_{03}}{m_{00}}$ , $m_{i j} (i = 0, j = 0, 1, 2, 3)$ is the element of the macroscopic Mueller matrix. $a = B_{12}^{2} + B_{13}^{2} + B_{14}^{2}$ , $b = 1 - \sqrt{1 - a}$ and $c = \sqrt{1 - a}$ . Note that the intensity transmittance is completely determined by the first row of the arbitrary Mueller matrix.

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

\vec{D} = [\begin{matrix} D_{H} \\ D_{45} \\ D_{C} \end{matrix}] = [\begin{matrix} B_{12} \\ B_{13} \\ B_{14} \end{matrix}],

where

D_{H}

,

D_{45}

,

D_{C}

and

D_{}

represent horizontal, 45°linear, circular, and total diattenuation respectively [15]. On comparison of Eq. (23) with Eqs. (22) and (63), we have

D_{L} = \sqrt{D_{H}^{2} + D_{45}^{2}} = \frac{\sqrt{(- β^{2} - γ^{2})} \tan (i \sqrt{β^{2} + δ^{2} + γ^{2}})}{\sqrt{β^{2} + δ^{2} + γ^{2}}} = \sqrt{B_{12}^{2} + B_{13}^{2}} .

D_{C} = B_{14} = - \frac{δ i \tan (i \sqrt{β^{2} + δ^{2} + γ^{2}})}{\sqrt{β^{2} + δ^{2} + γ^{2}}} .

D = \sqrt{D_{H}^{2} + D_{45}^{2} + D_{C}^{2}} = \sqrt{D_{L}^{2} + D_{C}^{2}} = | \vec{D} | = \pm i \tan (i \sqrt{β^{2} + δ^{2} + γ^{2}}) .

It should be pointed out that the horizontal and the 45°diattenuation have been combined to describe the linear diattenuation $D_{L}$ .

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 = M_{Δ} M_{R} M_{D},

where

M_{R}

,

M_{D}

and

M_{Δ}

represent a retarder, a diattenuator and a depolarizer, respectively. According to the experimental results [30–34],

M_{Δ}

can be expressed as

M_{Δ} = [\begin{matrix} 1 & 0 & 0 & 0 \\ p_{1} & e_{1} & 0 & 0 \\ p_{2} & 0 & e_{2} & 0 \\ p_{3} & 0 & 0 & e_{3} \end{matrix}],

where

p_{1}, p_{2}

and

p_{3}

are the components of the polarizance vector

\vec{p}

that characterizes the polarizing capability of the depolarizer,

e_{1}

and

e_{2}

are the parameters of the horizontal or vertical and the 45°linear depolarization, and

e_{3}

is 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 $m_{Δ}$ of the macroscopic matrix $M_{Δ}$ into 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 $M_{Δ}$ which should satisfy Eq. (28). To this end, $M$ can be decomposed into

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},

where

M_{Δ - α_{1} α_{2} α_{3}, β γ δ}, M_{Δ - - β - γ - δ}

and

M_{Δ - μ ν η}

represent 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. For

M_{Δ - μ ν η}

, we have

M_{Δ - μ ν η} = M_{Δ - μ} M_{Δ - ν η} = M_{Δ - ν η} M_{Δ - μ},

where

M_{Δ - μ}

and

M_{Δ - ν η}

represent 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

M_{Δ - μ} M_{Δ - ν η}

and

M_{Δ - ν η} M_{Δ - μ}

in Eq. (30) [see the discussions of Eq. (64) in Appendix].

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

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} .

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

\begin{array}{l} \frac{d M}{d z} = \frac{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} \frac{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} \\ \frac{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} \frac{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} \frac{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} \frac{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} \frac{d M_{D}}{d z} . \end{array}

Using $\frac{d M}{d z} = m M$ , Eq. (32) can be rewritten as

\begin{array}{l} 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}) . \end{array}

Consider a linear homogeneous medium,

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 .

Equation (33) is then reduced to

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},

where

m_{d e t}

and

m_{d e p}

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 $m_{Δ - α_{1} α_{2} α_{3,} β γ δ}, m_{Δ - - β - γ - δ}, m_{Δ - μ}$ and $m_{Δ - ν η}$ , a general differential matrix for depolarizing medium can be expressed as [18,24]

m_{d e p} = [\begin{matrix} 0 & - β^{'} & - γ^{'} & - δ^{'} \\ β^{'} & α_{1} & μ^{'} & ν^{'} \\ γ^{'} & μ^{'} & α_{2} & η^{'} \\ δ^{'} & ν^{'} & η^{'} & α_{3} \end{matrix}] = m_{Δ - α_{1} α_{2} α_{3,} β γ δ} + m_{Δ - - β - γ - δ} + m_{Δ - μ} + m_{Δ - ν η},

where

α_{1}, α_{2}

and

α_{3}

are the parameters of the anomalous isotropy depolarization,

β^{‘}, γ^{’}

and

δ^{‘}

,

μ^{‘}, η^{’}

and

ν^{‘}

are 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 $M_{Δ}$ in Eq. (28), $m_{Δ - α_{1} α_{2} α_{3}, β γ δ}$ , $m_{Δ - μ}$ and $m_{Δ - ν η}$ are given by

m_{Δ - α_{1} α_{2} α_{3}, β γ δ} = [\begin{matrix} 0 & 0 & 0 & 0 \\ β^{'} & α_{1} & 0 & 0 \\ γ^{'} & 0 & α_{2} & 0 \\ δ^{'} & 0 & 0 & α_{3} \end{matrix}], m_{Δ - μ} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & μ^{'} & 0 \\ 0 & μ^{'} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], m_{Δ - ν η} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & ν^{'} \\ 0 & 0 & 0 & η^{'} \\ 0 & ν^{'} & η^{'} & 0 \end{matrix}] .

The corresponding macroscopic Mueller matrices $M_{Δ - α_{1} α_{2} α_{3}, β γ δ}, M_{Δ - μ}$ and $M_{Δ - ν η}$ can then be found:

M_{Δ - α_{1} α_{2} α_{3}, β γ δ} = [\begin{matrix} 1 & 0 & 0 & 0 \\ \frac{β^{'} (\exp (α_{1}) - 1)}{α_{1}} & \exp (α_{1}) & 0 & 0 \\ \frac{γ^{'} (\exp (α_{2}) - 1)}{α_{2}} & 0 & \exp (α_{2}) & 0 \\ \frac{δ^{'} (\exp (α_{3}) - 1)}{α_{3}} & 0 & 0 & \exp (α_{3}) \end{matrix}], M_{Δ - μ} = [\begin{matrix} 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 \end{matrix}] .

M_{Δ - ν η} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{η^{' 2} + ν^{' 2} \cos (i \sqrt{η^{' 2} + ν^{' 2}})}{η^{' 2} + ν^{' 2}} & \frac{η^{'} ν^{'} (\cos (i \sqrt{η^{' 2} + ν^{' 2}}) - 1)}{η^{' 2} + ν^{' 2}} & - \frac{ν^{'} i \sin (i \sqrt{η^{' 2} + ν^{' 2}})}{\sqrt{η^{' 2} + ν^{' 2}}} \\ 0 & \frac{η^{'} ν^{'} (\cos (i \sqrt{η^{' 2} + ν^{' 2}}) - 1)}{η^{' 2} + ν^{' 2}} & \frac{ν^{' 2} + η^{' 2} \cos (i \sqrt{η^{' 2} + ν^{' 2}})}{η^{' 2} + ν^{' 2}} & - \frac{η^{'} i \sin (i \sqrt{η^{' 2} + ν^{' 2}})}{\sqrt{η^{' 2} + ν^{' 2}}} \\ 0 & - \frac{ν^{'} i \sin (i \sqrt{η^{' 2} + ν^{' 2}})}{\sqrt{η^{' 2} + ν^{' 2}}} & - \frac{η^{'} i \sin (i \sqrt{η^{' 2} + ν^{' 2}})}{\sqrt{η^{' 2} + ν^{' 2}}} & \cos (i \sqrt{η^{' 2} + ν^{' 2}}) \end{matrix}] .

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

m_{Δ - - β - γ - δ} = [\begin{matrix} 0 & - β^{'} & - γ^{'} & - δ^{'} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}],

where

m_{Δ - β, - γ, - δ}

fails to satisfy Eq. (9) due to the multiple roots, consequently, this one can be treated as a exceptional case, by the following equation

\frac{d S}{d z} = m S,

where S is the Stokes vector including

S_{0}, S_{1}, S_{2}

and

S_{3}

. A set of differential equations are then obtained

\begin{array}{l} 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. \end{array}

Through integration, the last three equations yield constant values independent of z for $S_{1}, S_{2}$ and $S_{3}$ , which indicate the conservation of the horizonal, the 45°and the circular polarization preference. The first equation yields the linear decay of $S_{0}$ that is a function of the distance z

S_{0} = S_{00} - (β^{'} S_{1} + γ^{'} S_{2} + δ^{'} S_{3}) z,

where

S_{00}

is 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

S_{1}, S_{2}

and

S_{3} \leq 1

,

β^{‘}, γ^{’}

and

δ^{‘}

are really small [19,25], the decay of

S_{0}

can be neglected. So the corresponding Mueller matrix of Eq. (40) can be an identical matrix to

I

.

By using the expressions of $M_{Δ - α_{1} α_{2} α_{3}, β γ δ}, M_{Δ - μ}$ , $M_{Δ - ν η}$ and $M_{Δ - - β - γ - δ}$ (see Eqs. (38) and (39)), $M_{Δ}$ can be expressed as the combination of the three matrices ( $z = η^{' 2} + ν^{' 2}$ )

\begin{array}{l} 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}) = \\ [\begin{matrix} 1 & 0 & 0 & 0 \\ \frac{β^{'} (\exp α_{1} - 1)}{α_{1}} & \exp α_{1} & 0 & 0 \\ \frac{γ^{'} (\exp α_{2} - 1)}{α_{2}} & 0 & \exp α_{2} & 0 \\ \frac{δ^{'} (\exp α_{3} - 1)}{α_{3}} & 0 & 0 & \exp α_{3} \end{matrix}] [\begin{matrix} 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 \end{matrix}] [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{η^{' 2} + ν^{' 2} \cos (i \sqrt{z})}{z} & \frac{η^{'} ν^{'} (\cos (i \sqrt{z}) - 1)}{η^{' 2} + ν^{' 2}} & - \frac{ν^{'} i \sin (i \sqrt{z})}{\sqrt{z}} \\ 0 & \frac{η^{'} ν^{'} (\cos (i \sqrt{z}) - 1)}{z} & \frac{ν^{' 2} + η^{' 2} \cos (i \sqrt{z})}{z} & - \frac{η^{'}^{'} i \sin (i \sqrt{z})}{\sqrt{z}} \\ 0 & - \frac{ν^{'} i \sin (i \sqrt{z})}{\sqrt{z}} & - \frac{η^{'}^{'} i \sin (i \sqrt{η^{' 2} + ν^{' 2}})}{\sqrt{z}} & \cos (i \sqrt{z}) \end{matrix}] . \end{array}

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)

M_{Δ_{1}} = [\begin{matrix} 1 & \vec{0} \\ \vec{p_{1}} & m_{Δ_{1}} \end{matrix}], m_{Δ_{1}}^{T} = m_{Δ_{1}},

here,

m_{Δ_{1}}

is a

3 \times 3

submatrix contains depolarization properties,

{\vec{p}}_{1}

depends on the polarizing capability of the medium. According to Eq. (64),

i \sin (i μ^{'}) \frac{η^{'} ν^{'} (\cos (i \sqrt{η^{' 2} + ν^{' 2}}) - 1)}{η^{' 2} + ν^{' 2}}

is too small to be considered, and

\cos (i \sqrt{η^{' 2} + ν^{' 2}})

can be replaced by 1, then the net depolarization coefficient defined in [15] can be expressed as

Δ_{1} = 1 - \frac{| t r (m_{Δ_{1}}) |}{3} = 1 - \frac{| t r (M_{Δ_{1}}) - 1 |}{3} = 1 - \frac{(\exp (α_{1}) + \exp (α_{2})) \cos (i μ^{'}) + \exp (α_{3}) \cos (i \sqrt{η^{' 2} + ν^{' 2}}))}{3}, 0 \leq Δ_{1} \leq 1.

Note that the depolarization capability and power of a depolarizer can be indicated by

Δ_{1}

,

e_{1}, e_{2}

and

e_{3}

can be obtained by diagonalizing

m_{Δ_{1}}

,

{\vec{p}}_{1}

can be straightforwardly extracted from the first column of the

M_{Δ_{1}}

.

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 + Δ z) = (I + m_{d e t} Δ z + m_{d e p} Δ z) M (z),

where

I

is the identity matrix, and

Δ z

is 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 + m_{d e t} Δ z + m_{d e p} Δ z)

(denoted as

C (I + m_{d e t} Δ z + m_{d e p} Δ z)

). For this purpose, we consider the physical realizability of

m_{L R}

. The coherency matrix of the identity matrix is

C (I) = [\begin{matrix} 1 & {\vec{0}}^{T} \\ \vec{0} & O_{3} \end{matrix}] .

Equation (47) can be reduced to

M (z + Δ z) = (I + m_{L R} Δ z) M (z) .

The corresponding coherency matrix of $m_{L R}$ can be expressed as

C (m_{L R}) = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] .

As the consequence, $C (I + m_{L R} Δ z)$ 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 $m_{C R}$ and $m_{D}$ are readily obtained

C (m_{C R}) = [\begin{matrix} 0 & 0 & 0 & - \frac{μ i}{2} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{μ i}{2} & 0 & 0 & 0 \end{matrix}], C (m_{D}) = \frac{1}{2} [\begin{matrix} 0 & β & γ & δ \\ β & 0 & 0 & 0 \\ γ & 0 & 0 & 0 \\ δ & 0 & 0 & 0 \end{matrix}] .

The eigenvalues of $C (I + m_{C R} Δ z)$ and $C (I + m_{D} Δ z)$ are $λ_{1} = 0, λ_{2} = 0, λ_{3} \to 1, λ_{4} \to \frac{μ^{2} Δ z^{2}}{4}$ , $λ_{1} = 0, λ_{2} = 0, λ_{3} \to 1, λ_{4} \to - \frac{(β^{2} + γ^{2} + δ^{2}) Δ z^{2}}{4}$ , respectively. The first three eigenvalues of both coherency matrices are non-negative and the last ones of them tend to zero faster than $Δ z$ and can be seen as zero. Similarly, when the differential matrix m is depolarizing, $C (I + m_{Δ - μ} Δ z), C (I + m_{Δ - ν η} Δ z), C (I + m_{Δ - α_{1} α_{2} α_{3}, β γ δ} Δ z)$ and $C (I + m_{Δ - - β - γ - δ} Δ z)$ should be verified to be positive semidefinite. This can be found by calculating the eigenvalues of $C (I + m_{Δ - μ} Δ z)$ and $C (I + m_{Δ - ν η} Δ z)$ as $λ_{1} = 0, λ_{2} = 1, λ_{3} \to \frac{μ^{‘} Δ z}{2}, λ_{4} \to - \frac{μ^{’} Δ z}{2}$ , $λ_{1} = 0, λ_{2} = 1, λ_{3} \to \frac{Δ z \sqrt{η^{‘ 2} + ν^{’ 2}}}{2}, λ_{4} \to - \frac{Δ z \sqrt{η^{‘ 2} + ν^{’ 2}}}{2}$ according to the following matrices

C (m_{Δ - μ}) = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{μ^{'}}{2} & 0 \\ 0 & \frac{μ^{'}}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], C (m_{Δ - ν η}) = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{ν^{'}}{2} \\ 0 & 0 & 0 & \frac{η^{'}}{2} \\ 0 & \frac{ν^{'}}{2} & \frac{η^{'}}{2} & 0 \end{matrix}] .

Since $Δ z, μ^{‘}, η^{'}$ and $ν^{'}$ are too small, multiplications of two of them can be treated as zero. In virtue of the complexity of both forms $C (I + m_{Δ - α_{1} α_{2} α_{3}, β γ δ} Δ z)$ and $C (I + m_{Δ - - β - γ - δ} Δ z)$ , $m_{Δ - α_{1} α_{2} α_{3}, β γ δ}$ and $m_{Δ - - β - γ - δ}$ can be decomposed as combinations of several basic coherency matrices which must be positive semidefinite each to satisfy the criterion. Coherency matrix of $m_{Δ - - β - γ - δ}$ can be deposed as

C (m_{Δ - - β - γ - δ}) = \frac{1}{4} [\begin{matrix} 0 & - β^{'} & - γ^{'} & - δ^{'} \\ - β^{'} & 0 & - i δ^{'} & i γ^{'} \\ - γ^{'} & i δ^{'} & 0 & - i β^{'} \\ - δ^{'} & - i γ^{'} & i β^{'} & 0 \end{matrix}] = \frac{1}{4} ([\begin{matrix} 0 & - β^{'} & - γ^{'} & - δ^{'} \\ - β^{'} & 0 & 0 & 0 \\ - γ^{'} & 0 & 0 & 0 \\ - δ^{'} & 0 & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & - i δ^{'} & 0 \\ 0 & i δ^{'} & 0 & 0 \\ 0 & 0 & 0^{'} & 0 \end{matrix}] + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i γ^{'} \\ 0 & 0 & 0 & - i β^{'} \\ 0 & - i γ^{'} & i β^{'} & 0 \end{matrix}]) .

It follows that the first two terms are proved to be positive semidefinite according to the eigenvalues of $C (m_{D})$ and $C (m_{Δ - μ})$ , the physical realizability of $C (I + m_{Δ - - β - γ - δ} Δ z)$ solely depends on the third part whose eigenvalues are $λ_{1} = 0, λ_{2} = 0, λ_{3} \to \frac{Δ z \sqrt{β^{‘ 2} + γ^{’ 2}}}{4}, λ_{4} \to - \frac{Δ z \sqrt{β^{‘ 2} + γ^{’ 2}}}{4}$ , since $Δ z, μ^{‘}, η^{'}$ and $ν^{'}$ are too small, multiplications of two of them can be treated as zero. Like deposition Eq. (53),

C (m_{Δ - α_{1} α_{2} α_{3}, β γ δ}) = \frac{1}{4} [\begin{matrix} 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} \end{matrix}] = \frac{1}{4} ([\begin{matrix} 0 & β^{'} & γ^{'} & δ^{'} \\ β^{'} & 0 & - i δ^{'} & i γ^{'} \\ γ^{'} & i δ^{'} & 0 & - i β^{'} \\ δ^{'} & - i γ^{'} & i β^{'} & 0 \end{matrix}] + [\begin{matrix} 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} \end{matrix}]) .

To check the positive semidefiniteness of associated matrix $C (I + m_{Δ - α_{1} α_{2} α_{3}, β γ δ} Δ z)$ , only the second term is discussed. The eigenvalues are easily obtained as $λ_{1} = \frac{(α_{1} + α_{2} + α_{3}) Δ z}{4}, λ_{2} = \frac{(α_{1} - α_{2} - α_{3}) Δ z}{4}, λ_{3} = \frac{(- α_{1} + α_{2} - α_{3}) Δ z}{4}, λ_{4} = \frac{(- α_{1} - α_{2} + α_{3}) Δ z}{4}$ , where, the first one is the largest one (one can always achieve the condition $λ_{1} > 0$ by adding $C (c I) = c C (I)$ to the coherency matrix above [35], where c is a constant) since $α_{1}, α_{2}$ and $α_{3}$ are 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 $2.0 μm$ -diameter polystyrene microspheres, where exhibited the properties of diattenuation, depolarization and birefringence. The scattering coefficient and the glucose concentration coefficient measured were $μ_{S} = 0.6$ ${mm}^{-1}$ and 5M. For the usual method,

M = [\begin{matrix} 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 \end{matrix}] .

The effective differential matrix is then found by $m = W_{1} d i a g (I n τ_{0} z, I n τ_{1} z, I n τ_{2} z, I n τ_{3} z) W_{1}^{- 1}$ [19], where $τ_{0}, τ_{1}, τ_{2}$ and $τ_{3}$ are the eigenvalues of $M$ , $W_{1}$ is a matrix whose columns are the eigenvectors of $M$ .

m = [\begin{matrix} -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 \end{matrix}] = m_{d e t} + m_{d e p} .

The depolarizing part and the determined part are

m_{d e t} = [\begin{matrix} 0 & β & γ & δ \\ β & 0 & μ & ν \\ γ & - μ & 0 & η \\ δ & - ν & - η & 0 \end{matrix}] = [\begin{matrix} 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 \end{matrix}] .

m_{d e p} = [\begin{matrix} 0 & - β^{'} & - γ^{'} & - δ^{'} \\ β^{'} & α_{1} & μ^{'} & ν^{'} \\ γ^{'} & μ^{'} & α_{2} & η^{'} \\ δ^{'} & ν^{'} & η^{'} & α_{3} \end{matrix}] = [\begin{matrix} 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 \end{matrix}] .

The corresponding polar decomposition of the Mueller matrix Eq. (55) consists of_{$M_{Δ}$}, _{$M_{R}$} and _{$M_{D}$} are given by

M_{Δ} = [\begin{matrix} 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 \end{matrix}], M_{R} = [\begin{matrix} 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 \end{matrix}], M_{D} = [\begin{matrix} 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 \end{matrix}] .

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,

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.

The estimated results are in close agreement with the expected values in Eqs. (57) and (58) except for _{$γ, β^{‘}$}and_$δ^{'}$. 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:

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 _{$m_{d e p}$}

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Fig. 1 Relationships between $M_{R}$ and $m_{d e t}$ .

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Fig. 2 Relationships between $M_{D}$ and $m_{d e t}$ .

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Fig. 3 Relationships between _{$M_{Δ}$}and _{$m_{d e p}$}.

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Note that the _{$M_{Δ}$}, $M_{R}$ and _{$M_{D}$} can be referred to analyze the general coefficients straightforwardly like _{$Δ_{1}$}, _{$| \vec{D} |$}, _$R$, 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 $M_{Δ}$ , $M_{R}$ and _{$M_{D}$} 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 _{$M_{Δ}$}including three matrices leads to the obtainment of the more specific depolarizing characteristics compared with the only matrix _{$M_{Δ}$}and implies the possibility to analyze the influence of the depolarization completely using polar decomposition method.

Appendix

A description of the $M_{R}$ that consists $η$ ,_$ν$, and $μ$ as three parameters of birefringence is presented, $x = - η^{2} - ν^{2}$ :

M_{R} (η, ν, μ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{η^{2} + ν^{2} \cos (i \sqrt{x})}{- x} \cos μ + \frac{η ν (\cos (i \sqrt{x}) - 1)}{x} \sin μ & \frac{η^{2} + ν^{2} \cos (i \sqrt{x})}{- x} \sin μ - \frac{η ν (\cos (i \sqrt{x}) - 1)}{x} \cos μ & - \frac{ν i \sin (i \sqrt{x})}{\sqrt{x}} \\ 0 & \frac{η ν (\cos (i \sqrt{x}) - 1)}{- x} \cos μ + \frac{ν^{2} + η^{2} \cos (i \sqrt{x})}{x} \sin μ & \frac{η ν (\cos (i \sqrt{x}) - 1)}{- x} \sin μ - \frac{ν^{2} + η^{2} \cos (i \sqrt{x})}{x} \cos μ & - \frac{η i \sin (i \sqrt{x})}{\sqrt{x}} \\ 0 & \frac{ν i \sin (i \sqrt{x})}{\sqrt{x}} \cos μ - \frac{η i \sin (i \sqrt{x})}{\sqrt{x}} \sin μ & \frac{ν i \sin (i \sqrt{x})}{\sqrt{x}} \sin μ + \frac{η i \sin (i \sqrt{x})}{\sqrt{x}} \cos μ & \cos (i \sqrt{x}) \end{matrix}] .

Using Eqs. (9) and (19), the total diattenuation matrix _{$M_{D}$}can be given by, _{$y = β^{2} + δ^{2} + γ^{2}$}:

M_{D} (β, γ, δ) = [\begin{matrix} 1 & - \frac{β i \tan (i \sqrt{y})}{\sqrt{y}} & - \frac{γ i \tan (i \sqrt{y})}{\sqrt{y}} & - \frac{δ i \tan (i \sqrt{y})}{\sqrt{y}} \\ - \frac{β i \tan (i \sqrt{y})}{\sqrt{y}} & \frac{δ^{2} + γ^{2} + β^{2} \cos (i \sqrt{y})}{y \cos (i \sqrt{y})} & \frac{β γ (\cos (i \sqrt{y}) - 1)}{y \cos (i \sqrt{y})} & \frac{β δ (\cos (i \sqrt{y}) - 1)}{y \cos (i \sqrt{y})} \\ - \frac{γ i \tan (i \sqrt{y})}{\sqrt{y}} & \frac{β γ (\cos (i \sqrt{y}) - 1)}{y \cos (i \sqrt{y})} & \frac{δ^{2} + β^{2} + γ^{2} \cos (i \sqrt{y})}{y \cos (i \sqrt{y})} & \frac{δ γ (\cos (i \sqrt{y}) - 1)}{y \cos (i \sqrt{y})} \\ - \frac{δ i \tan (i \sqrt{y})}{\sqrt{y}} & \frac{β δ (\cos (i \sqrt{y}) - 1)}{y \cos (i \sqrt{y})} & \frac{δ γ (\cos (i \sqrt{y}) - 1)}{y \cos (i \sqrt{y})} & \frac{β^{2} + γ^{2} + δ^{2} \cos (i \sqrt{y})}{y \cos (i \sqrt{y})} \end{matrix}] .

The description of the whole Mueller matrix of the depolarizer is given by, $z = η^{' 2} + ν^{' 2}$ , $z_{1} = \cos (i \sqrt{z}), z_{2} = \sin (i \sqrt{z}), z_{3} = \cos (i μ^{'}), z_{4} = \sin (i μ^{'})$ :

\begin{array}{l} 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}) = \\ [\begin{matrix} 1 & 0 & 0 & 0 \\ \frac{β^{'} (e^{α_{1}} - 1)}{α_{1}} & e^{α_{1}} (z_{3} \frac{η^{' 2} + ν^{' 2} z_{1}}{z} - i z_{4} \frac{η^{'} ν^{'} (z_{1} - 1)}{z}) & e^{α_{1}} (z_{3} \frac{η^{'} ν^{'} (z_{1}) - 1)}{z} - i z_{4} \frac{ν^{' 2} + η^{' 2} z_{1})}{z}) & e^{α_{1}} (- z_{3} \frac{ν^{'} i z_{2}}{\sqrt{z}} - z_{4} \frac{η^{'} z_{2})}{\sqrt{z}}) \\ \frac{γ^{'} (e^{α_{2}} - 1)}{α_{2}} & e^{α_{2}} (z_{3} \frac{η^{'} ν^{'} (z_{1} - 1)}{z} - i z_{4} \frac{η^{' 2} + ν^{' 2} z_{1}}{z}) & e^{α_{2}} (z_{3} \frac{ν^{' 2} + η^{' 2} z_{1}}{z} - i z_{4} \frac{η^{'} ν^{'} (z_{1} - 1)}{z}) & e^{α_{2}} (- z_{4} \frac{ν^{'} z_{2}}{\sqrt{z}} - z_{3} \frac{η^{'} i z_{2}}{\sqrt{z}}) \\ \frac{δ^{'} (e^{α_{3}} - 1)}{α_{3}} & - e^{α_{3}} \frac{ν^{'} i z_{2}}{\sqrt{z}} & - e^{α_{3}} \frac{η^{'} i z_{2}}{\sqrt{z}} & e^{α_{3}} z_{1} \end{matrix}] \end{array}

Another possible order to obtain _{$M_{Δ - μ ν η}$}is _{$M_{Δ - ν η} M_{Δ - μ}$}, which is easily proved to be closely to the deposition _{$M_{Δ - μ} M_{Δ - ν η}$}, with the conditions that _{$\cos (i \sqrt{η^{' 2} + ν^{' 2}})$} and _{$i \sin (i μ^{'}) \frac{η^{'} i \sin (i \sqrt{η^{' 2} + ν^{' 2}})}{\sqrt{η^{' 2} + ν^{' 2}}}$}or _{$i \sin (i μ^{'}) \frac{ν^{'} i \sin (i \sqrt{η^{' 2} + ν^{' 2}})}{\sqrt{η^{' 2} + ν^{' 2}}}$}are approximately equal to one and zero, respectively. Whatever the order is, the _{$3 \times 3$} submatrix _{$m_{Δ_{1}}$}can 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).

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Parameters	Relationships
$ν$	$M_{R} (2, 4)$
$η$	$M_{R} (3, 4)$
$\cos μ$	$(M_{R} (3, 3) + M_{R} (2, 2)) / 2$
$\sin μ$	$(M_{R} (2, 3) - M_{R} (3, 2)) / 2$

Parameters	Relationships
$α_{1}$	$I n (M_{D} (2, 2))$
$α_{2}$	$I n (M_{Δ} (3, 3))$
$α_{3}$	$I n (M_{Δ} (4, 4))$
$η^{’}$	$M_{Δ} (4, 3) / \exp (α_{3})$
$ν^{’}$	$M_{Δ} (4, 2) / \exp (α_{3})$
$\sin μ^{’}$	$(M_{Δ} (2, 3) + M_{Δ} (3, 2)) / (\exp (α_{1}) + \exp (α_{2}))$
$γ^{‘}$	$M_{Δ} (3, 1) α_{2} / (\exp (α_{2}) - 1)$
$δ^{‘}$	$M_{Δ} (4, 1) α_{3} / (\exp (α_{3}) - 1)$
$β^{’}$	$M_{Δ} (2, 1) α_{1} / (\exp (α_{1}) - 1)$

Parameters	Relationships
$ν$	$M_{R} (2, 4)$
$η$	$M_{R} (3, 4)$
$\cos μ$	$(M_{R} (3, 3) + M_{R} (2, 2)) / 2$
$\sin μ$	$(M_{R} (2, 3) - M_{R} (3, 2)) / 2$

Parameters	Relationships
$α_{1}$	$I n (M_{D} (2, 2))$
$α_{2}$	$I n (M_{Δ} (3, 3))$
$α_{3}$	$I n (M_{Δ} (4, 4))$
$η^{’}$	$M_{Δ} (4, 3) / \exp (α_{3})$
$ν^{’}$	$M_{Δ} (4, 2) / \exp (α_{3})$
$\sin μ^{’}$	$(M_{Δ} (2, 3) + M_{Δ} (3, 2)) / (\exp (α_{1}) + \exp (α_{2}))$
$γ^{‘}$	$M_{Δ} (3, 1) α_{2} / (\exp (α_{2}) - 1)$
$δ^{‘}$	$M_{Δ} (4, 1) α_{3} / (\exp (α_{3}) - 1)$
$β^{’}$	$M_{Δ} (2, 1) α_{1} / (\exp (α_{1}) - 1)$

Method of interpreting Mueller matrix of anisotropic medium

Abstract

1. Introduction

2. Theory

2.1 Non-depolarizing homogeneous anisotropic media

2.2 Depolarizing homogeneous anisotropic media

3. Physical realizability and discussions

4. Conclusion

Appendix

Funding

References

Cited By

Figures (5)

Tables (3)

Equations (64)

Optics Express

Parameters	Relationships
$β$	$M_{D} (1, 2)$
$γ$	$M_{D} (1, 3)$
$δ$	$M_{D} (1, 4)$

Parameters	Relationships
$β$	$M_{D} (1, 2)$
$γ$	$M_{D} (1, 3)$
$δ$	$M_{D} (1, 4)$