Extraction of anisotropic parameters of turbid media using hybrid model comprising differential- and decomposition-based Mueller matrices

Chia-Chi Liao; Yu-Lung Lo

doi:10.1364/OE.21.016831

1. Introduction

The polarization properties of the light scattered from turbid media are of significant interest in the biological, biomedical and biochemistry fields due to their potential application for therapeutic or diagnostic detection purposes. Accordingly, many different methodologies have been proposed for characterizing the optical properties of turbid samples. For example, Prahl et al. proposed a method for measuring the absorption coefficient, scattering coefficient and anisotropy factor of bovine muscle, human tissue and polyurethane using a single integrating sphere [1, 2] or two integrating spheres [3]. Many other techniques have also been proposed for measuring the absorption coefficient and scattering coefficient of tissue, including time-domain diffuse reflectance [4–7], frequency-domain diffuse reflectance [8–10], spatially-resolved steady-state diffuse reñectance [11, 12], opto-acoustics [13], digital microradiography [14], and so on.

Measuring the Stokes vectors of the light transmitted through turbid optical samples and calculating the corresponding Mueller matrix is a well established technique for characterizing the anisotropic properties of optical samples. Cameron et al. [15–17] proposed a Mueller matrix-based imaging approach for estimating the scattering coefficient contribution in complex mixtures of polystyrene spheres as a function of the particle size. Wang et al. [18, 19] used a Monte Carlo model and a Mueller matrix approach to approximate the propagation of polarized light in turbid media in accordance with either a single-scattering assumption or a double-scattering assumption. Vitkin et al. [20–23] proposed a method based on Mueller matrix decomposition for extracting the linear birefringence (LB), circular birefringence (CB), linear dichroism (LD) and depolarization (Dep) of complex turbid media. However, the methods proposed in [15–23] do not enable all of the effective optical parameters of scattering media to be determined. Hence, Pham and Lo [24] proposed a comprehensive decoupled analytical technique for extracting full-range measurements of all the LB, CB, LD, circular dichroism (CD), linear depolarization (L-Dep), and circular depolarization (C-Dep) properties of turbid media. When using the Mueller matrix decomposition method, the LB, CB, LD, CD and Dep components of the optical sample must be processed in strict sequential order. It may not suitable for each actual sample containing multiple properties. In [25], this problem was resolved by means of a differential 4 x 4 Mueller matrix formalism. Ossikovski [26] and Ortega-Quijano and Arce-Diego [27] used the differential Mueller matrix formalism to explore the anisotropic properties of depolarizing optical media. Ortega-Quijano and Arce-Diego [28] applied the same method to characterize the anisotropic properties of optical samples under reflection and backscattering conditions, respectively. However, in the differential calculation process, an ambiguity exists in the value of the angle in the trigonometric functions for the phase retardation and optical rotation angle of the sample, respectively. As a result, not all of the optical parameters can be measured over the full range.

Accordingly, the present study proposes a hybrid model comprising both the Mueller matrix decomposition method and the differential Mueller matrix formulism for extracting optically anisotropic properties of turbid media in LB/CB/Dep, CB/CD/Dep and LB/LD/Dep cases over the full range. Also, the case for simultaneously extracting all LB, CB, LD, CD, and Dep properties of turbid media is studied. The validity of the proposed method is demonstrated by means of both numerical simulations and experimental investigations involving a chitosan-glucose compound sample containing polystyrene microspheres (for LB/CB/Dep testing) and two ferrofluidic samples (for CB/CD/Dep and LB/LD/Dep testing), respectively. In general, the results show that the proposed model successfully resolves the respective limitations of the Mueller decomposition method and the differential Mueller matrix formalism, and therefore has significant potential for real-world therapeutic and diagnostic applications.

2. Mueller-stokes parameter measurement method

Figure 1 presents a schematic illustration of the experimental setup used in the present study to determine the Mueller matrix of turbid optical media. Note that P and Q are a polarizer and a quarter-wave plate, respectively, and are used to produce various linear and circular polarization lights.

Fig. 1 Schematic illustration of experimental Mueller-Stokes measurement system.

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The output Stokes vector in Fig. 1 has the form

S_{o u t p u t} = {[\begin{matrix} S_{0} \\ S_{1} \\ S {}_{2} \\ S_{3} \end{matrix}]}_{o u t p u t} = M_{s a m p l e} S_{i n p u t} = [\begin{matrix} M_{11} & M_{12} & M_{13} & M_{14} \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ M_{41} & M_{42} & M_{43} & M_{44} \end{matrix}] {[\begin{matrix} {\hat{S}}_{0} \\ {\hat{S}}_{1} \\ \hat{S} {}_{2} \\ {\hat{S}}_{3} \end{matrix}]}_{i n p u t}

where M_sample is the Mueller matrix of the sample and S_input is the input Stokes vector. In the present study, the sample is illuminated with four different input polarization lights, namely three linear polarization lights (i.e.,

{\hat{S}}_{0^{0}} = {[\begin{matrix} 1, & 1, & 0, & 0 \end{matrix}]}^{T}

,

{\hat{S}}_{45^{0}} = {[\begin{matrix} 1, & 0, & 1, & 0 \end{matrix}]}^{T}

, and

{\hat{S}}_{90^{0}} = {[\begin{matrix} 1, & - 1, & 0, & 0 \end{matrix}]}^{T}

) and one right-hand circular polarization light (i.e.,

{\hat{S}}_{R H C} = {[\begin{matrix} 1, & 0, & 0, & 1 \end{matrix}]}^{T}

). From Eq. (1), the elements of M_sample (i.e., M₁₁ ~M₄₄) are then obtained as

M_{s a m p l e} = [\begin{matrix} M_{11} & M_{12} & M_{13} & M_{14} \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ M_{41} & M_{42} & M_{43} & M_{44} \end{matrix}] = S_{o u t p u t [0 °, 45 °, 90 °, R H C]} S_{i n p u t [0 °, 45 °, 90 °, R H C]}^{- 1}

3. Differential Mueller matrix

In characterizing depolarizing anisotropic media using the differential Mueller matrix method, an eigenanalysis of the Mueller matrix is performed and the subsequent decoupled properties of the resulting differential matrix are then expanded into a set of 16 differential matrices which collectively describe the optical properties of the sample [26]. In calculating the differential matrix, it is assumed that the illuminating beam propagates along the z-axis of a right-handed Cartesian coordinate system. The differential matrix is thus given by [25]

m = (\frac{d M}{d z}) M^{- 1}

where M is the macroscopic Mueller matrix of the sample. The eigenvalues of the macroscopic and differential matrices are denoted as λ_M and λ_m, respectively, and are in full parallelism with Eq. (3). Meanwhile, the eigenvectors of the macroscopic and differential matrices are denoted as V_M and V_m, respectively, and are also in full parallelism with Eq. (3) [29]. Assume that the initial condition matrix M_z _{= 0} has the form of an identity matrix, λ_M_{(z = 0)} = 1. Thus, the eigenvalues of M and m are related as [25]

λ_{m} = \frac{\ln (λ_{M})}{z}

Assuming that m_λ is a diagonal matrix with non-zero diagonal elements of λ_m, the differential Mueller matrix can be obtained from an eigen analysis of M as follows [25]:

m = V_{M} m_{λ} V_{M}^{- 1} = [\begin{matrix} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{21} & m_{22} & m_{23} & m_{24} \\ m_{31} & m_{32} & m_{33} & m_{34} \\ m_{41} & m_{42} & m_{43} & m_{44} \end{matrix}]

In accordance with the differential Mueller matrix method, the Mueller matrix of a general anisotropic sample can be partitioned into 16 different elements, with each element describing a different aspect of the basic optical behavior. Let M_LB, M_CB, M_LD and M_CD be the macroscopic Mueller matrices describing the LB, CB, LD and CD properties of the sample, respectively [24, 30, 31]. The corresponding differential matrix for each property can be derived using Eq. (5). The differential matrix m_BD of a composite sample with LB, CB, LD and CD optical properties can be expressed as

m_{B D} = \frac{1}{d} [\begin{matrix} \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] & - \ln \sqrt{\frac{1 - D}{1 + D}} \cos (2 θ_{d}) & - \ln \sqrt{\frac{1 - D}{1 + D}} \sin (2 θ_{d}) & \ln (\frac{1 + R}{1 - R}) \\ - \ln \sqrt{\frac{1 - D}{1 + D}} \cos (2 θ_{d}) & \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] & 2 γ & - β \sin (2 α) \\ - \ln \sqrt{\frac{1 - D}{1 + D}} \sin (2 θ_{d}) & - 2 γ & \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] & β \cos (2 α) \\ \ln (\frac{1 + R}{1 - R}) & β \sin (2 α) & - β \cos (2 α) & \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] \end{matrix}]

where d is the sample thickness; α and β are the orientation angle and phase retardation of the LB property, respectively; γ is the optical rotation angle of the CB property; θ_d and D are the orientation angle and linear dichroism of the LD property, respectively; and R is the circular amplitude anisotropy of the CD property (i.e., (r_R-r_L)/(r_R + r_L), −1 ≤ R ≤ + 1, where r_R and r_L are the absorptions of right-hand circular polarized light and left-hand circular polarized light, respectively).

The differential matrix describing the depolarization effect in depolarizing anisotropic media has the form [26]

m_{Δ} = \frac{1}{d} [\begin{matrix} 0 & κ_{q}^{'} & κ_{u}^{'} & κ_{v}^{'} \\ - κ_{q}^{'} & - κ_{i q}^{'} & η_{v}^{'} & η_{u}^{'} \\ - κ_{u}^{'} & η_{v}^{'} & - κ_{i u}^{'} & η_{q}^{'} \\ - κ_{v}^{'} & η_{u}^{'} & η_{q}^{'} & - κ_{i v}^{'} \end{matrix}]

where the diagonal depolarization is characterized by the differential parameters

κ_{i q, i u, i v}^{'}

, while the anomalous dichroism and anomalous depolarization are characterized by the differential parameters

κ_{q, u, v}^{'}

and

η_{q, u, v}^{'}

, respectively.

The differential matrix describing all of the LB, CB, LD, CD and Dep properties of an anisotropic optical sample can be obtained by summing Eqs. (6) and (7) to give

\begin{array}{l} m_{B D Δ} = m_{B D} + m_{Δ} = \\ \frac{1}{d} [\begin{matrix} \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] & - \ln \sqrt{\frac{1 - D}{1 + D}} \cos (2 θ_{d}) + κ_{q}^{'} & - \ln \sqrt{\frac{1 - D}{1 + D}} \sin (2 θ_{d}) + κ_{u}^{'} & \ln (\frac{1 + R}{1 - R}) + κ_{v}^{'} \\ - \ln \sqrt{\frac{1 - D}{1 + D}} \cos (2 θ_{d}) - κ_{q}^{'} & \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] - κ_{i q}^{'} & 2 γ + η_{v}^{'} & - β \sin (2 α) + η_{u}^{'} \\ - \ln \sqrt{\frac{1 - D}{1 + D}} \sin (2 θ_{d}) - κ_{u}^{'} & - 2 γ + η_{v}^{'} & \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] - κ_{i u}^{'} & β \cos (2 α) + η_{q}^{'} \\ \ln (\frac{1 + R}{1 - R}) - κ_{v}^{'} & β \sin (2 α) + η_{u}^{'} & - β \cos (2 α) + η_{q}^{'} & \ln [(1 - R^{2}) \sqrt{\frac{1 - D}{1 + D}}] - κ_{i v}^{'} \end{matrix}] \end{array}

In practice, the macroscopic Mueller matrix of the sample can be determined from Eq. (2), and the differential Mueller matrix of the sample calculated using Eq. (5). The optical parameters describing the anisotropic behavior of the sample can then be obtained by equating the differential Mueller matrix obtained from Eq. (5) with that given in Eq. (8). By doing so, the orientation angle α, phase retardation β, optical rotation angle γ, orientation angle θ_d, linear dichroism D, and circular dichroism R can be obtained respectively as

α = \frac{1}{2} \tan^{- 1} (\frac{m_{42} - m_{24}}{m_{34} - m_{43}})

β = \sqrt{{[\frac{(m_{42} - m_{24})}{2}]}^{2} + {[\frac{(m_{34} - m_{43})}{2}]}^{2}}

γ = \frac{(m_{23} - m_{32})}{4}

θ_{d} = \frac{1}{2} \tan^{- 1} (\frac{m_{13} + m_{31}}{m_{12} + m_{21}})

D = \frac{1 - e^{- 2 \sqrt{{(m_{12} + m_{21})}^{2} + {(m_{13} + m_{31})}^{2}}}}{1 + e^{- 2 \sqrt{{(m_{12} + m_{21})}^{2} + {(m_{13} + m_{31})}^{2}}}}

R = \frac{e^{(\frac{m_{14} + m_{41}}{2})} - 1}{e^{(\frac{m_{14} + m_{41}}{2})} + 1}

Similarly, the differential Mueller matrix describing the depolarization effect can be obtained as

m_{Δ} = [\begin{matrix} 0 & \frac{(m_{12} - m_{21})}{2} & \frac{(m_{13} - m_{31})}{2} & \frac{(m_{14} - m_{41})}{2} \\ \frac{(m_{21} - m_{12})}{2} & m_{22} - m_{11} & \frac{(m_{23} + m_{32})}{2} & \frac{(m_{24} + m_{42})}{2} \\ \frac{(m_{31} - m_{13})}{2} & \frac{(m_{23} + m_{32})}{2} & m_{33} - m_{11} & \frac{(m_{34} + m_{43})}{2} \\ \frac{(m_{41} - m_{14})}{2} & \frac{(m_{24} + m_{42})}{2} & \frac{(m_{34} + m_{43})}{2} & m_{44} - m_{11} \end{matrix}]

Performing inverse differential calculation [25], the macroscopic Mueller matrix describing the depolarization effect is obtained as

Μ_{Δ} = [\begin{matrix} 1 & K_{12} & K_{13} & K_{14} \\ - K_{12} & K_{22} & K_{23} & K_{24} \\ - K_{13} & K_{23} & K_{33} & K_{34} \\ - K_{14} & K_{24} & K_{34} & K_{44} \end{matrix}]

where K₂₂ and K₃₃ are degrees of linear depolarization and K₄₄ is the degree of circular depolarization. In general, the degree of depolarization is quantified by the depolarization index, Δ, which has a value of 0 for a non-depolarizing sample and 1 for an ideal depolarizer [32]. Having obtained K₂₂, K₃₃ and K₄₄, the depolarization index can be obtained as [33, 34]

Δ = 1 - \sqrt{\frac{K_{22}^{2} + K_{33}^{2} + K_{44}^{2}}{3}}, 0 \leq Δ \leq 1

In the differential Mueller matrix formalism described above, the measurement ranges of the phase retardation (β) and rotation angle (γ) given in Eqs. (10) and (11), respectively, are restricted due to an ambiguity in the value of the angle in the corresponding trigonometric functions. Thus, as described in the following subsection, the present study proposes a hybrid model in which the differential Mueller matrix formalism is combined with the Mueller matrix decomposition method.

4. Hybrid Mueller matrix model

In the proposed hybrid model, the macroscopic Mueller matrix describing the depolarization effect [Eq. (16)] and the macroscopic Mueller matrix describing the other anisotropic properties of the sample [derived from Eq. (6)] are combined by means of the Mueller matrix decomposition method. In practice, it is extremely difficult to derive a single macroscopic Mueller matrix containing all of the anisotropic optical parameters (i.e., LB, CB, LD and CD) from Eq. (6). Thus, in the following subsections, hybrid models are derived for composite samples with three specific combinations of anisotropic parameters, namely LB/CB/Dep, CB/CD/Dep and LB/LD/Dep, respectively. However, instead of directly deriving a single macroscopic Mueller matrix from Eq. (6), a modified algorithm to extract all anisotropic parameters, namely LB/CB/LD/CD/Dep, from a turbid media is studied.

4.1 LB/CB/Dep composite sample

In developing the optically equivalent model of the composite sample, it is assumed that the LB, CB and Dep components are arranged as shown in Fig. 2.

Fig. 2 Schematic illustration of composite sample with LB, CB and Dep properties.

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The equivalent Mueller matrix of the composite sample has the form

M_{B Δ} = M_{B} M_{Δ} or M_{Δ} M_{B}

where M_B is the macroscopic matrix describing the LB and CB properties of the sample and M_Δ is the macroscopic Mueller matrix describing the depolarization properties. It is noted in Fig. 2 that two different combinations of the Dep and LB/CB properties are considered in order to check which solution of Eq. (18) is closer to that obtained from the differential model given in Eq. (8). In Eq. (18), the macroscopic depolarization matrix, M_Δ, can be obtained directly from Eq. (16). Thus, matrix M_B describing the LB and CB properties of the sample can be calculated from the measured matrix M_BΔ as follows:

M_{B} = M_{B Δ} M_{Δ}^{- 1} or M_{Δ}^{- 1} M_{B Δ} = [\begin{matrix} M_{11} & M_{12} & M_{13} & M_{14} \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ M_{41} & M_{42} & M_{43} & M_{44} \end{matrix}]

Alternatively, M_B can be derived from Eq. (6) via a process of inverse differential calculation and formulated as

M_{B} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & B_{22} & B_{23} & B_{24} \\ 0 & B_{32} & B_{33} & B_{34} \\ 0 & B_{42} & B_{43} & B_{44} \end{matrix}]

\begin{array}{l} where B_{22} = \cos (\sqrt{β^{2} + 4 γ^{2}}) + \frac{β^{2} \cos^{2} (2 α)}{β^{2} + 4 γ^{2}} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{33} = \cos (\sqrt{β^{2} + 4 γ^{2}}) + \frac{β^{2} \sin^{2} (2 α)}{β^{2} + 4 γ^{2}} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{44} = \cos (\sqrt{β^{2} + 4 γ^{2}}) + \frac{4 γ^{2}}{β^{2} + 4 γ^{2}} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{23} = \frac{2 γ}{\sqrt{β^{2} + 4 γ^{2}}} \sin (\sqrt{β^{2} + 4 γ^{2}}) + \frac{β^{2} \sin (4 α)}{2 (β^{2} + 4 γ^{2})} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{32} = \frac{- 2 γ}{\sqrt{β^{2} + 4 γ^{2}}} \sin (\sqrt{β^{2} + 4 γ^{2}}) + \frac{β^{2} \sin (4 α)}{2 (β^{2} + 4 γ^{2})} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{24} = \frac{- β \sin (2 α)}{\sqrt{β^{2} + 4 γ^{2}}} \sin (\sqrt{β^{2} + 4 γ^{2}}) + \frac{2 γ β \cos (2 α)}{β^{2} + 4 γ^{2}} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{42} = \frac{β \sin (2 α)}{\sqrt{β^{2} + 4 γ^{2}}} \sin (\sqrt{β^{2} + 4 γ^{2}}) + \frac{2 γ β \cos (2 α)}{β^{2} + 4 γ^{2}} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{34} = \frac{β \cos (2 α)}{\sqrt{β^{2} + 4 γ^{2}}} \sin (\sqrt{β^{2} + 4 γ^{2}}) + \frac{2 γ β \sin (2 α)}{β^{2} + 4 γ^{2}} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})], \\ B_{43} = \frac{- β \cos (2 α)}{\sqrt{β^{2} + 4 γ^{2}}} \sin (\sqrt{β^{2} + 4 γ^{2}}) + \frac{2 γ β \sin (2 α)}{β^{2} + 4 γ^{2}} [1 - \cos (\sqrt{β^{2} + 4 γ^{2}})] . \end{array}

Equation (20) in comparison with Eq. (19), γ, βsin(2α) and βcos(2α) can be obtained respectively as

γ = \frac{\cos^{- 1} (\frac{M_{22} + M_{33} + M_{44} - 1}{2}) (\frac{M_{23} - M_{32}}{4})}{\sin [\cos^{- 1} (\frac{M_{22} + M_{33} + M_{44} - 1}{2})]}

β \sin (2 α) = \frac{\cos^{- 1} (\frac{M_{22} + M_{33} + M_{44} - 1}{2}) (M_{42} - M_{24})}{2 \sin [\cos^{- 1} (\frac{M_{22} + M_{33} + M_{44} - 1}{2})]} = P

β \cos (2 α) = \frac{\cos^{- 1} (\frac{M_{22} + M_{33} + M_{44} - 1}{2}) (M_{34} - M_{43})}{2 \sin [\cos^{- 1} (\frac{M_{22} + M_{33} + M_{44} - 1}{2})]} = Q

From Eqs. (22) and (23), the orientation angle (α) and phase retardation (β) of the composite sample can be obtained as

α = \frac{1}{2} \tan^{- 1} \frac{P}{Q}

β = \sqrt{P^{2} + Q^{2}}

It is noted that in the decomposition method used within the proposed hybrid model, the parameters given in Eqs. (21), (24) and (25) are extracted for both cases shown in Fig. 2 (i.e.,

M_{B Δ} = M_{B} M_{Δ} or M_{Δ} M_{B}

). The parameters extracted in each case are then used to establish the corresponding differential Mueller matrices given in Eq. (8) in order to determine which particular sequence of optical parameters yields a solution closer to that of the measured differential Mueller matrix in Eq. (5). It is noted that the measurement ranges of α, β and γ in Eqs. (21), (24) and (25) are equal to 0<α<180°, 0<β<360° and 0<γ<180°, respectively. In other words, the proposed hybrid model enables parameters α, β and γ to be measured over the full range.

4.2 CB/CD/Dep composite sample

In developing the hybrid model of the CB/CD/Dep composite sample, two different arrangements of the CB/CD and Dep parameters are considered, as shown in Fig. 3

Fig. 3 Schematic illustration of composite sample with CB, CD and Dep properties.

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The equivalent Mueller matrix of the composite sample shown in Fig. 3 has the form

M_{C Δ} = M_{C} M_{Δ} or M_{Δ} M_{C}

where M_C is the macroscopic matrix describing the CB and CD properties of the sample and M_Δ is the macroscopic Mueller matrix describing the depolarization properties. In practice, M_CΔ can be determined experimentally from the Stokes polarimeter measurements, while M_Δ can be calculated from Eq. (16). Matrix M_C can thus be obtained as

M_{C} = M_{C Δ} M_{Δ}^{- 1} or M_{Δ}^{- 1} M_{C Δ} = [\begin{matrix} N_{11} & N_{12} & N_{13} & N_{14} \\ N_{21} & N_{22} & N_{23} & N_{24} \\ N_{31} & N_{32} & N_{33} & N_{34} \\ N_{41} & N_{42} & N_{43} & N_{44} \end{matrix}]

Alternatively, matrix M_C can be derived from the differential matrix given in Eq. (6) as follows:

M_{C} = [\begin{matrix} 1 + R^{2} & 0 & 0 & 2 R \\ 0 & (1 - R^{2}) \cos (2 γ) & (1 - R^{2}) \sin (2 γ) & 0 \\ 0 & - (1 - R^{2}) \sin (2 γ) & (1 - R^{2}) \cos (2 γ) & 0 \\ 2 R & 0 & 0 & 1 + R^{2} \end{matrix}]

Equating Eqs. (27) and (28), the circular birefringence (γ) and circular dichroism (R) can be obtained respectively as

γ = \frac{1}{2} \tan^{- 1} (\frac{N_{23}}{N_{22}})

R = \frac{N_{11} - (N_{12} / \cos 2 γ)}{N_{14}}

As for the previous sample, the values of γ and R are calculated for each configuration shown in Fig. 3. The extracted parameters are then used to establish the corresponding differential Mueller matrix given in Eq. (8) in order to check which solutions are closer to those determined from the measured differential Mueller matrix in Eq. (5). It is noted that Eqs. (29) and (30) yield values of γ and R in the ranges 0<γ<180° and −1<R<1, respectively. In other words, the proposed hybrid model provides full range measurements of both the CB properties and the CD properties of the composite sample.

4.3 LB/LD/Dep composite sample

In constructing the hybrid model of the composite sample with LB, LD and Dep properties, two different arrangements of the LB/LD and Dep properties are considered, as shown in Fig. 4. The equivalent Mueller matrix of the composite sample has the form

M_{L Δ} = M_{L} M_{Δ} or M_{Δ} M_{L}

where M_L is the macroscopic Mueller matrix describing the LB and LD properties of the sample, while M_Δ is the macroscopic Mueller matrix describing the depolarization properties.

Fig. 4 Schematic illustration of composite sample with LB/LD and Dep properties.

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The macroscopic depolarization matrix M_Δ can be obtained directly from Eq. (16). Matrix M_L can then be determined from the measured matrix M_LΔ as follows:

M_{L} = M_{L Δ} M_{Δ}^{- 1} or M_{Δ}^{- 1} Μ_{L Δ} = [\begin{matrix} L_{11} & L_{12} & L_{13} & L_{14} \\ L_{21} & L_{22} & L_{23} & L_{24} \\ L_{31} & L_{32} & L_{33} & L_{34} \\ L_{41} & L_{42} & L_{43} & L_{44} \end{matrix}]

Alternatively, M_L can be inversely derived from Eq. (6) as

m_{L} = \frac{1}{d} [\begin{matrix} \ln \sqrt{\frac{1 - D}{1 + D}} & - \ln \sqrt{\frac{1 - D}{1 + D}} \cos (2 θ_{d}) & - \ln \sqrt{\frac{1 - D}{1 + D}} \sin (2 θ_{d}) & 0 \\ - \ln \sqrt{\frac{1 - D}{1 + D}} \cos (2 θ_{d}) & \ln \sqrt{\frac{1 - D}{1 + D}} & 0 & - β \sin (2 α) \\ - \ln \sqrt{\frac{1 - D}{1 + D}} \sin (2 θ_{d}) & 0 & \ln \sqrt{\frac{1 - D}{1 + D}} & β \cos (2 α) \\ 0 & β \sin (2 α) & - β \cos (2 α) & \ln \sqrt{\frac{1 - D}{1 + D}} \end{matrix}] \overset{\begin{array}{l} Inverse \\ differential \\ calculation \end{array}}{\to} M_{L}

It is noted that the macroscopic Mueller matrix M_L is highly complicated and is therefore presented in the appendix to this study. In the present study, the unknown parameters in Eq. (33) are extracted by means of a Genetic Algorithm [35–37] using the measured Mueller matrix M_L given in Eq. (32) as the target function. It is again noted that the GA yields full range measurements of all of the LB and LD properties.

4.4 LB/CB/LD/CD/Dep composite sample

Instead of directly deriving a single macroscopic Mueller matrix from Eq. (6) for the cases in Subsections 4.1, 4.2, and 4.3, a modified algorithm based on using the differential Mueller matrix formalism and the hybrid model is introduced for a composite sample containing LB, CB, LD, CD, and depolarization. The flow chart of a modified algorithm is illustrated in Fig. 5. Firstly, from the differential Mueller matrix formalism shown in Eqs. (9)-(17), the total parameters of α, β, γ, θ_d, D, R and Δ can be obtained. It is found that α, θ_d, D, and R are measured over the full range, but β and γ are measured over the restricted range. Subsequently, in order to improve the restricted ranges of β and γ, the hybrid model is employed. The extracted β and γ obtained by the differential Mueller matrix formalism are used to re-build a macroscopic LB/CB Mueller matrix by the inverse differential calculation. Finally, according to the macroscopic LB/CB Mueller matrix, β and γ then can be extracted by employing the hybrid model for the LB/CB case and using Eqs. (21), (22), (23), and (25). The problem in the restricted ranges of β and γ thus can be improved.

Fig. 5 Flow chart of a modified algorithm based on using the differential calculation method and the hybrid model

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Due to all parameters are simultaneously extracted by the differential Mueller matrix formalism, errors are first induced while the extracted values of β and γ are out of range. Thus, it affects the correction in deriving the macroscopic LB/CB Mueller matrix by the inverse differential calculation. Therefore, the modified algorithm proposed here has some limitation but still provides a good extension of the measurement range than just only using the differential Mueller matrix formalism. The robustness of this modified algorithm can be confirmed in the simulation.

5. Simulation results

In this section, simulations are performed to compare the performance of the hybrid model in extracting the optical parameters of the four composite samples described in Section 4 with that of the differential Mueller matrix method and the Mueller matrix decomposition method, respectively.

In performing the simulations, the measured Mueller matrix is obtained from Eq. (8) via a process of inverse differentiation. The theoretical input values of the sample parameters are inserted into the measured Mueller matrix, and are then extracted using the hybrid model, the differential Muller matrix method and the Mueller matrix decomposition method, respectively. Finally, the extracted values of the optical parameters are compared with the theoretical values in order to quantify the accuracy of the corresponding extraction method. Note that for each of the extracted parameters, the theoretical input value is varied over the full range, i.e., α: 0~180°, θ_d: 0~180°; γ: 0~180°; β: 0~360°; D: 0~1; and R: −1~1).

5.1 LB/CB/Dep composite sample

The ability of the proposed hybrid model to determine the optical parameters of LB/CB/Dep composite samples was evaluated by extracting the values of α, β and γ for an anisotropic sample with known LB, CB and Dep properties and then comparing the extracted values with the theoretical input values. Figure 6(a) compares the extracted value of the principal axis of phase retardation (α) with the input value of α over the full measurement range, i.e., α: 0~180°. Note that the remaining sample parameters have values of β = 60°, γ = 15° and Δ = 0.4. Figures 6(b) and 6(c) present the equivalent results for the phase retardation (β) and orientation angle (γ), respectively. Note that parameters α, β and γ are extracted using Eqs. (21), (24) and (25) in the hybrid model, Eqs. (9)-(11) in the differential Mueller matrix method, and the method proposed in [24] in the decomposition method.

Fig. 6 Extracted values of α, β and γ for LB/CB/Dep sample using hybrid model, differential calculation method and decomposition method. Note that theoretical input parameters are as follows: (a) α: 0~180°, β = 60°, γ = 15°, Δ = 0.4; (b) β: 0~360°, α = 30°, γ = 15°, Δ = 0.4; and (c) γ: 0~180°, α = 30°, β = 60°, Δ = 0.4.

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The results presented in Fig. 6(a) show that the hybrid method and the differential Muller matrix method both enable the LB orientation angle α to be measured over the full range. However, it is seen in Figs. 6(b) and 6(c) that the differential Mueller matrix method is unable to obtain full-range measurements of β and γ. By contrast, the proposed hybrid model enables both parameters to be measured over the full range. In general, it is noted that the parameter values extracted by the decomposition method all deviate from the input values for the composite sample. Finally, it is noted that the depolarization is correctly extracted as Δ = 0.4 in every case.

5.2 CB/CD/Dep composite sample

The ability of the proposed hybrid model to determine the optical parameters of CB/CD/Dep composite samples was investigated by extracting the values of the optical rotation angle (γ) and circular dichroism (R), respectively. Figure 7(a) compares the extracted value of γ with the input value over the full range of γ: 0~180° given R = 0.2 and Δ = 0.4. Similarly, Fig. 7(b) compares the extracted value of R with the input value of R over the full range of R: −1 ~1 given γ = 15° and Δ = 0.4. Note that in the hybrid model, γ and R are extracted using Eqs. (29) and (30), respectively, while in the differential Mueller matrix method, γ and R are extracted using Eqs. (11) and (14), respectively.

Fig. 7 Extracted values of γ and R for CB/CD/Dep sample using hybrid model, differential calculation method and decomposition method. Note that theoretical input parameters are as follows: (a) γ: 0~180°, R = 0.2 and Δ = 0.4; and (b) R = −1~1, γ = 15° and Δ = 0.4.

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The results presented in Fig. 7(a) show that the hybrid model and the decomposition method both provide full-range measurements of the optical rotation angle. However, the differential calculation method enables γ to be measured only over the range of γ: 0~90°. Figure 7(b) shows that the hybrid model and the differential calculation method both enable the circular dichroism to be measured over the full range. However, the values of R extracted by the decomposition method deviate from the theoretical input values since the composite sample is assumed.

5.3 LB/LD/Dep composite sample

The ability of the proposed hybrid model to determine the optical parameters of LB/LD/Dep composite samples was evaluated by extracting the values of α, β, θ_d, and D and then comparing the extracted values with the known input values. Figure 8(a) presents the extracted values of α for input values of α over the full range (i.e., α: 0~180°) given parameter settings of β = 60°, θ_d = 35°, D = 0.4, and Δ = 0.4. Figures 8(b)-8(d) present the equivalent results for parameters β, θ_d and D, respectively. Note that in the hybrid model, parameters α, β, θ_d, and D are extracted using a GA, while in the differential calculation method, α, β, θ_d, and D are extracted using Eqs. (9)-(10) and (12)-(13).

Fig. 8 Extracted values of α, β, θ_d, and D for LB/LD/Dep composite sample using hybrid model (GA), differential calculation method and decomposition method. Note that theoretical input parameters are as follows: (a) α: 0~180°, β = 60°, θ_d = 35°, D = 0.5, Δ = 0.4; (b) β: 0~360°, α = 30°, θ_d = 35°, D = 0.5, Δ = 0.4; (c) θ_d: 0~180°, α = 30°, β = 60°, D = 0.5, Δ = 0.4; and (d) D: 0~1, α = 30°, β = 60°, θ_d = 35°, Δ = 0.4.

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In general, the results presented in Fig. 8 show that the proposed hybrid model enables all four properties of the LB/LD/Dep composite sample to be measured over the full range. The differential Mueller matrix method also enables α, θ_d, and D to be measured over the full range. However, β can only be measured over the half-range (i.e., β: 0~180°). It is noted that the decomposition method enables all four parameters to be measured over the full range. However, as for the two previous composite samples, the extracted parameter values deviate from the input values.

5.4 LB/CB/LD/CD/Dep composite sample

Based upon the modified algorithm introduced in Subsection 4.4, a simulation with a composite sample containing LB, CB, LD, CD and depolarization is introduced. Figure 9(a) presents the extracted value of α for input value of α over the full range (i.e., α: 0~180°) given parameter settings of β = 60°, θ_d = 35°, D = 0.4, and Δ = 0.4. Figures 9(b)-9(f) present the equivalent results for parameters θ_d, D, R, β, and γ, respectively. Note that α, θ_d, D, and R are extracted using differential Mueller matrix formalism in Eqs. (9), (12), (13), and (14), and β and γ are extracted by the modified algorithm as described in Subsection 4.4.

Fig. 9 Extracted values of α, β, γ, θ_d, D, and R for LB/CB/LD/CD/Dep sample using differential calculation method, and exactly β and γ are modified by the hybrid model. Note that theoretical input parameters are as follows: (a) α: 0~180°, β = 60°, γ = 15°, θ_d = 35°, D = 0.5, R = 0.2, Δ = 0.4; (b) θ_d: 0~180°, α = 30°, β = 60°, γ = 15°, D = 0.5, R = 0.2, Δ = 0.4; (c) D: 0~1, α = 30°, β = 60°, γ = 15°, θ_d = 35°, R = 0.2, Δ = 0.4; (d) R = −1~1, α = 30°, β = 60°, γ = 15°, θ_d = 35°, D = 0.5, Δ = 0.4; (e) β: 0~360°, α = 30°, γ = 15°, θ_d = 35°, D = 0.5, R = 0.2, Δ = 0.4; and (f) γ: 0~180°, α = 30°, β = 60°, θ_d = 35°, D = 0.5, R = 0.2, Δ = 0.4.

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The results presented in Fig. 9 show that the differential Mueller matrix formalism enables α, θ_d, D, and R to be measured over the full range. However, β and γ can only be measured over the half-range by the differential Mueller matrix formalism. The limitation is then improved by the hybrid model, and it enables β and γ to be measured over the near full-range (i.e., β: 0~350° and γ: 0~170°)). It can be seen in Figs. 9(e) and 9(f) in black dots.

Again, another two cases ((1) α = 60°, β = 30°, γ = 30°, θ_d = 70°, D = 0.2, R = 0.6, Δ = 0.3; (2) α = 120°, β = 15°, γ = 45°, θ_d = 105°, D = 0.6, R = 0.4, Δ = 0.5) are simulated and the similar results are observed. While β and γ are out of range (i.e., β > 350° and γ >170°), the deviations in the extracted values of β and γ occur. In the proposed modified algorithm, the hybrid model can provide the exact quadrant determination but cannot eliminate the existing errors induced by using the differential Mueller matrix formalism.

6. Experimental measurement of anisotropic parameters of turbid media

The validity of the proposed hybrid model was further evaluated by extracting the anisotropic parameters of three real-world turbid samples. Figure 10 presents a schematic illustration of the experimental setup used to perform the measurement process. The input light was provided by a frequency-stable He-Ne laser (SL 02/2, SIOS Co.) with a central wavelength of 632.8 nm. The laser beam was passed through a polarizer (GTH5M, Thorlabs Co.) and a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.) in order to produce three linear polarization lights (0°, 45° and 90°) and a single right-hand circular polarization light. A neutral density filter (NDC-100-2, ONET Co.) and power meter detector (8842A, OPHIT Co.) were used to ensure that each of the polarization lights incident on the sample had an identical intensity. The output Stokes parameters were computed from the detected intensity measurements using a commercial Stokes polarimeter (PAX5710, Thorlabs Co.) at a sampling rate of 30 samples per second. A minimum of 1024 data measurement points were obtained for each sample. Of these 1024 data points, 100 points were chosen in order to calculate the mean value of each parameter.

Fig. 10 Schematic illustration of experimental measurement system (NDF: Neutral Density Filter).

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The validity of the proposed hybrid method was evaluated using three optical turbid samples, namely a composite LB/CB/Dep sample comprising chitosan, glucose and suspended polystyrene microspheres and two ferrofluidic samples with CB/CD/Dep and LB/LD/Dep properties, respectively.

6.1 Composite chitosan-glucose-microsphere sample

Chitosan hydrogel suspension solutions were prepared by dissolving chitosan powder (Aldrich 417963, DD ≥ 75%) in 0.1 M acetic acid (Fluka Ltd) under moderate magnetic stirring for 3 hours at room temperature. (Note that suspension solutions with chitosan concentrations of 0, 0.01, 0.02, 0.03, 0.04 and 0.05 g chitosan/100 ml were prepared.) The chitosan solution was then mixed with 0.01 M sodium hydroxide (Merck Ltd) at room temperature in order to form a homogenous suspension with no phase separation or precipitation. The suspension was mixed with D-glucose (Merck Ltd) in concentrations of 0, 0.5, 1.0, 1.5, 2.0 and 2.5 M, respectively, Suspended polystyrene microspheres (Thermo Scientific Ltd.) with a diameter of 5 μm and a concentration of 0.2 g/ml were then added to each mixture. Finally, the chitosan hydrogel suspension, glucose solutions and suspended particles were mixed thoroughly and placed in square quartz containers with an external side-length of 12.5 mm and an internal side-length of 10 mm. The LB, CB and Dep properties induced by the chitosan, glucose and suspended particles, respectively, were then extracted using the hybrid model proposed in this study and the decomposition method described in [24]. As shown in the previous section, the parameters extracted using the differential calculation method are very close to those obtained using the hybrid model (albeit not over the full range in every case). Thus, in performing the experimental investigation, the differential calculation method was omitted for reasons of convenience.

Figure 11 illustrates the experimental results obtained for the optical parameters of chitosan / glucose solutions with and without suspended particles, respectively. Note that the results relate to samples with chitosan concentrations ranging from 0~0.05 g/100ml in increments of 0.01 g/100ml and a constant glucose concentration of 2.5 M. It is seen that the measured values of the phase retardation (β) obtained using the hybrid method vary approximately linearly with the chitosan concentration. Moreover, the optical rotation angle (γ) and depolarization have approximately constant values since the different samples contain equal amounts of glucose and microparticles. In addition, it is seen that the measured values and tendencies of β and γ are similar in both types of chitosan / glucose sample (i.e., with and without suspended particles, respectively. From inspection, the standard deviations of the extracted values of α, β and γ are found to be just 0.43°, 0.04° and 0.01°, respectively. By contrast, the standard deviations of α, β and γ extracted using the decomposition method are 0.85°, 0.05° and 0.08°, respectively. In other words, the measurement performance of the hybrid model is more stable than that of the decomposition method.

Fig. 11 Experimental results for LB/CB/Dep properties of chitosan/glucose solutions with various chitosan concentrations. Note that figures (a)~(d) relate to chitosan / glucose samples containing suspended particles, while figures (e)~(h) relate to chitosan / glucose samples with no suspended particles.

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Figure 12 presents the average measured values of the linear birefringence, circular birefringence and depolarization parameters of samples containing glucose concentrations ranging from 0~2.5 M in increments of 0.5 M and a constant chitosan concentration of 0.05 g/100 ml. It is seen that a good correlation exists between the measured values of the optical rotation angle (γ) and the glucose concentration over the considered range. Moreover, it is seen that the phase retardation (β) and depolarization have approximately constant values since all of the samples contain the same amount of chitosan and microspheres. In addition, it is observed that for both types of sample, the measurement results obtained using the hybrid model are more stable than those obtained using the decomposition method.

Fig. 12 Experimental results for LB/CB/Dep properties of chitosan/glucose solutions with various glucose concentrations. Note that figures (a)~(d) relate to chitosan / glucose samples containing suspended particles, while figures (e)~(h) relate to chitosan / glucose samples with no suspended particles.

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6.2 Ferrofluidic samples with CB/CD/Dep and LB/LD/Dep properties

Ferrofluids (or magnetic liquids) are stable colloidal suspensions of nanometric magnetic particles within a suitable carrier liquid. In the present study, ferrofluidic samples containing monodispersed magnetite (Fe₃O₄) nanoparticles were synthesized using the thermal decomposition method described in [38]. Briefly, iron (III) acetylacetonate (Strem Chemicals, Inc) was mixed with 1, 2-hexadecanediol in the presence of oleic acid, oleylamine and high boiling point solvents. The reaction mixture was then heated to 200°C under continuous stirring for 30 minutes, and was then heated to the reflux temperature for a further 30 minutes. The resultant product was collected by centrifugation and washed three times in pure ethanol. Finally, the Fe₃O₄ nanoparticles were dispersed in hexane to create stock solutions with concentrations of 0.01 M and 0.025 M, respectively. Figure 13 shows the image of Fe₃O₄ nanoparticles obtained from the transmission electron microscopy (JEM-2010 Electron Microscope, JEOL Co. 200 KV) by our group. From inspection, the average diameter of the Fe₃O₄ nanoparticles is 6 nm. The ferrofluidic suspensions were placed in square quartz containers with external side-lengths of 12.5 mm and internal side-lengths of 10 mm.

Fig. 13 Transmission electron microscopy image of the Fe₃O₄ nanoparticles

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As shown in Fig. 14(a), ferrofluidic samples with CB and CD properties were induced by applying an external magnetic field in a direction parallel to the illuminating beam. Similarly, ferrofluidic samples with LB and LD properties were induced by applying an external magnetic field in a direction perpendicular to the illuminating beam [Fig. 14(b)]. In performing the experiments, the magnetic field was produced by a magnetic generator (PHYWE SYSTEM, U-shaped core, 0651.01, GMBH, Germany) and measured by a Gauss/Tesla meter (Model 5180, Pacific Scientific-OECO).

Fig. 14 Ferrofluidic samples with (a) CB and CD properties, and (b) LB and LD properties

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6.2.1 Ferrofluidic sample with CB/CD/Dep properties

Figure 15 presents the experimental results obtained by the hybrid model and the decomposition model, respectively, for the circular birefringence, circular dichroism and depolarization of the ferrofluidic sample shown in Fig. 14(a) given magnetic field intensities ranging from 0 to 450 G in increments of 50 G. Note that the nanoparticle concentration is equal to 0.01 M. For both methods, a good correlation exists between the measured values of the optical rotation angle (γ) and the external magnetic field. Moreover, as expected, the circular dichroism (R) increases as the intensity of the external magnetic field increases. However, the depolarization of the ferrofluidic sample remains approximately constant for all values of the magnetic field intensity since the sample contains a fixed concentration of Fe₃O₄ nanoparticles. From inspection, the standard deviations of the values of γ and R obtained using the hybrid model are found to be 0.02° and 5.45 × 10⁻⁴, respectively. Thus, the experimental stability of the proposed hybrid model is confirmed. Also, from inspection the experimental results of the measured R and Dep obtained by the decomposition method and the hybrid model, the hybrid model is more stable than that of the decomposition method.

Fig. 15 Experimental results for CB/CD/Dep properties of ferrofluidic sample with Fe₃O₄ concentration of 0.01 M.

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6.2.2 Ferrofluidic sample with LB/LD/Dep properties

Figure 16 presents the experimental results obtained by the hybrid model and the decomposition method, respectively, for the linear birefringence, linear dichroism and depolarization of the ferrofluidic sample with an Fe₃O₄ nanoparticle concentration of 0.025 M given external magnetic field intensities ranging from 0 to 450 G. For both methods, a good correlation exists between the measured values of the phase retardation (β) and the intensity of the external magnetic field. Moreover, the linear dichroism (D) increases with an increasing magnetic field intensity. The standard deviations of α, β, θ_d, and D extracted using the hybrid model are 0.05°, 0.04°, 2.03° and 9.5 × 10⁻⁴, respectively. As expected, the extracted value of the depolarization is around twice that of the sample with a Fe₃O₄ concentration of 0.01 M. It is seen that the orientation angle of the phase retardation (α) is more closely aligned with the direction of the magnetic field as the intensity of the magnetic field is increased. Notably, the orientation angle of the linear dichroism (θ_d) extracted by the hybrid model also converges toward the direction of the magnetic field as the intensity of the magnetic field increases. However, such a tendency is not apparent in the results extracted using the decomposition method. In other words, the superior measurement performance of the proposed hybrid model is once again confirmed.

Fig. 16 Experimental results for LB/LD/Dep properties of ferrofluidic sample with Fe₃O₄ concentration of 0.025 M.

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7. Conclusions and discussions

This study has developed a hybrid model based on the differential Mueller matrix formalism and the Mueller matrix decomposition method for analyzing and measuring the anisotropic properties of LB/CB/Dep, CB/CD/Dep and LB/LD/Dep composite optical samples. Also, the case for simultaneously extracting all LB, CB, LD, CD, and Dep properties of turbid media is studied. The validity of the hybrid model is confirmed by testing the chitosan-glucose compound with nanoparticle and the ferrofluid material. Experimental results have shown that the hybrid model can provide a good agreement with the corresponding tendency. It is also confirmed that the hybrid model is suitable for turbid media containing depolarization effect. It has been shown that unlike the differential Mueller matrix method, the proposed model enables all of the anisotropic parameters of a turbid optical sample to be measured over the full range. Moreover, it has been shown that the hybrid method is both more stable and more precise than the Mueller matrix decomposition method since the hybrid model overcomes the sequence limitation of the decomposition method. In addition, the proposed hybrid method more accurately captures the tendency of the orientation angle of the linear birefringence (α) and linear dichroism (θ_d) of the ferrofluidic sample under high magnetic field intensities along the direction of the magnetic field.

The proposed method has significant potential for the analysis of real-world anisotropic samples with scattering effects. For example, it can be applied for collagen and muscle structure characterization (based on LB measurements only), glucose detection in skin (based on CB measurement only), protein structure characterization (based on CB/CD measurements), and ferrofluid material analysis (based on CB/CD or LB/LD measurements).

Appendix

The hybrid Mueller matrix containing LB and LD properties can be derived via a process of inverse differential calculation as follows:

M_{L} = [\begin{matrix} L_{11} & L_{12} & L_{13} & L_{14} \\ L_{21} & L_{22} & L_{23} & L_{24} \\ L_{31} & L_{32} & L_{33} & L_{34} \\ L_{41} & L_{42} & L_{43} & L_{44} \end{matrix}]

A = {(\ln P)}^{4} + β^{4} + 2 {(\ln P)}^{2} β^{2} \cos [4 (α - θ_{d})]

B = {(\ln P)}^{2} - β^{2} + \sqrt{A}

C = {(\ln P)}^{2} - β^{2} - \sqrt{A}

Therefore,

L_{11} = \frac{P {[{(\ln P)}^{2} + β^{2}] (\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}) + \sqrt{A} (\cosh \sqrt{\frac{B}{2}} + \cosh \sqrt{\frac{C}{2}})}}{2 \sqrt{A}}

L_{12} = \frac{- P \ln P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{C}{2}} [- β^{2} \cos (4 α - 2 θ_{d}) + (\sqrt{A} - {(\ln P)}^{2}) \cos (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{B}{2}} [β^{2} \cos (4 α - 2 θ_{d}) + (\sqrt{A} + {(\ln P)}^{2}) \cos (2 θ_{d})]}}{\sqrt{A B C}}

L_{13} = \frac{- P \ln P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{C}{2}} [- β^{2} \sin (4 α - 2 θ_{d}) + (\sqrt{A} - {(\ln P)}^{2}) \sin (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{B}{2}} [β^{2} \sin (4 α - 2 θ_{d}) + (\sqrt{A} + {(\ln P)}^{2}) \sin (2 θ_{d})]}}{\sqrt{A B C}}

L_{14} = \frac{P \ln P β \sin [2 (α - θ_{d})] {\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}}}{\sqrt{A}}

L_{21} = \frac{- P \ln P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{C}{2}} [- β^{2} \cos (4 α - 2 θ_{d}) + (\sqrt{A} - {(\ln P)}^{2}) \cos (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{B}{2}} [β^{2} \cos (4 α - 2 θ_{d}) + (\sqrt{A} + {(\ln P)}^{2}) \cos (2 θ_{d})]}}{\sqrt{A B C}}

L_{22} = \frac{P {[β^{2} \cos 4 α + {(\ln P)}^{2} \cos 4 θ_{d}] (\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}) + \sqrt{A} (\cosh \sqrt{\frac{B}{2}} + \cosh \sqrt{\frac{C}{2}})}}{2 \sqrt{A}}

L_{23} = \frac{P [β^{2} \sin (4 α) + {(\ln P)}^{2} \sin (4 θ_{d})] [\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}]}{2 \sqrt{A}}

L_{24} = \frac{P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{B}{2}} [- β^{2} \sin (4 α - 2 θ_{d}) + [\sqrt{A} - {(\ln P)}^{2}] \sin (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{C}{2}} [β^{2} \sin (4 α - 2 θ_{d}) + [\sqrt{A} + {(\ln P)}^{2}] \sin (2 θ_{d})]}}{2 β \cos [2 (α - θ_{d})] \sqrt{A}}

L_{31} = \frac{- P \ln P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{C}{2}} [- β^{2} \sin (4 α - 2 θ_{d}) + (\sqrt{A} - {(\ln P)}^{2}) \sin (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{B}{2}} [β^{2} \sin (4 α - 2 θ_{d}) + (\sqrt{A} + {(\ln P)}^{2}) \sin (2 θ_{d})]}}{\sqrt{A B C}}

L_{32} = \frac{P [β^{2} \sin (4 α) + {(\ln P)}^{2} \sin (4 θ_{d})] [\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}]}{2 \sqrt{A}}

L_{33} = \frac{P {- [β^{2} \cos 4 α + {(\ln P)}^{2} \cos 4 θ_{d}] (\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}) + \sqrt{A} (\cosh \sqrt{\frac{B}{2}} + \cosh \sqrt{\frac{C}{2}})}}{2 \sqrt{A}}

L_{34} = \frac{- P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{B}{2}} [- β^{2} \cos (4 α - 2 θ_{d}) + [\sqrt{A} - {(\ln P)}^{2}] \cos (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{C}{2}} [β^{2} \cos (4 α - 2 θ_{d}) + [\sqrt{A} + {(\ln P)}^{2}] \cos (2 θ_{d})]}}{2 β \cos [2 (α - θ_{d})] \sqrt{A}}

L_{41} = \frac{- P \ln P β \sin [2 (α - θ_{d})] {\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}}}{\sqrt{A}}

L_{42} = \frac{- P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{B}{2}} [- β^{2} \sin (4 α - 2 θ_{d}) + [\sqrt{A} - {(\ln P)}^{2}] \sin (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{C}{2}} [β^{2} \sin (4 α - 2 θ_{d}) + [\sqrt{A} + {(\ln P)}^{2}] \sin (2 θ_{d})]}}{2 β \cos [2 (α - θ_{d})] \sqrt{A}}

L_{43} = \frac{- P {\sqrt{\frac{B}{2}} \sinh \sqrt{\frac{B}{2}} [- β^{2} \cos (4 α - 2 θ_{d}) + [\sqrt{A} - {(\ln P)}^{2}] \cos (2 θ_{d})] + \sqrt{\frac{C}{2}} \sinh \sqrt{\frac{C}{2}} [β^{2} \cos (4 α - 2 θ_{d}) + [\sqrt{A} + {(\ln P)}^{2}] \cos (2 θ_{d})]}}{2 β \cos [2 (α - θ_{d})] \sqrt{A}}

L_{44} = \frac{P {- [{(\ln P)}^{2} + β^{2}] (\cosh \sqrt{\frac{B}{2}} - \cosh \sqrt{\frac{C}{2}}) + \sqrt{A} (\cosh \sqrt{\frac{B}{2}} + \cosh \sqrt{\frac{C}{2}})}}{2 \sqrt{A}}

Acknowledgments

The authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under Grant No. NSC101-2221-E-006-028-MY3.

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Extraction of anisotropic parameters of turbid media using hybrid model comprising differential- and decomposition-based Mueller matrices

Abstract

1. Introduction

2. Mueller-stokes parameter measurement method

3. Differential Mueller matrix

4. Hybrid Mueller matrix model

4.1 LB/CB/Dep composite sample

4.2 CB/CD/Dep composite sample

4.3 LB/LD/Dep composite sample

4.4 LB/CB/LD/CD/Dep composite sample

5. Simulation results

5.1 LB/CB/Dep composite sample

5.2 CB/CD/Dep composite sample

5.3 LB/LD/Dep composite sample

5.4 LB/CB/LD/CD/Dep composite sample

6. Experimental measurement of anisotropic parameters of turbid media

6.1 Composite chitosan-glucose-microsphere sample

6.2 Ferrofluidic samples with CB/CD/Dep and LB/LD/Dep properties

6.2.1 Ferrofluidic sample with CB/CD/Dep properties

6.2.2 Ferrofluidic sample with LB/LD/Dep properties

7. Conclusions and discussions

Appendix

Acknowledgments

References and links

Cited By

Figures (16)

Equations (54)

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