Characterization of Mean Absorbance of Anisotropic Turbid Media Using Stokes–Mueller Matrix Polarimetry Approach

A Mueller matrix polarimetry was designed to extract the mean absorbance of an anisotropic turbid media using an ink solution diluted with lipoplus as the sample for testing. The experimental results showed that the mean absorbance of ink solution with 2% lipoplus increases linearly with increasing ink concentration, and the other anisotropic optical parameters extracted by two decoupling algorithms were also in good agreement. As a result, the slope of 0.132 (mean absorbance/ink concentration) was found in an ink solution with 2% lipoplus within a 2-mm width of a cuvette. The corresponding average deviation of mean absorbance is 0.77%. Alternatively, a simulation of a Mueller matrix polarimetry was successfully demonstrated using LighTools to extract the mean absorbance based on a new methodology. In addition to extracting anisotropic parameters such as linear birefringence (LB), linear dichroism (LD), optical rotation, circular dichroism (CD), and depolarization index using a typical Mueller matrix polarimetry system, the present study proposes a new methodology for determining the mean absorbance of anisotropic turbid media without scattering effects after subtracting the effective Mueller matrix of depolarization.


I. INTRODUCTION
R ECENT years have seen the development of many noninvasive technologies for human biomonitoring based on natural matrices such as saliva, urine, meconium, nail, hair, skin, and semen [1]. Among the various optical methods, terahertz spectroscopy operates over a frequency range of 2-13.5 THz and demonstrates the capacity to differentiate D-and L-glucoses, which have the same molecular conformation and crystal structure [2], [3]. A white-light time-resolved reflectance spectroscopy [4] is used to monitor absorption changes in a layered diffusive medium. A short-wave infrared LED [5] is based on the detection of sample transmittance or reflectance to study the analytical composition of substances that presents absorption peaks at specific wavelengths. According to the Beer-Lambert law, the attenuation of light as it propagates through a medium is directly proportional to the concentration of the medium and the path length of the light. As a result, many methods have been proposed for determining the unknown concentration of an analyte in solution by measuring the reduction in intensity of a light beam as it passes through the sample. Kim et al. [6] estimated the concentration of glucose solutions by measuring the optical absorbance at four discrete probe wavelengths (1064, 1550, 1685, and 1798 nm) and a single reference wavelength (1310 nm), respectively. Mandal and Manasreh [7] proposed an optical sensor for measuring the percentage of HbA1c in hemoglobin based on the relative change in absorbance at wavelengths of 535 and 593 nm, respectively. Gobrecht et al. [8] presented a method based on polarized light spectroscopy for mitigating the effects of highly scattering samples on the linear Beer-Lambert relationship between the spectral absorbance and the concentration. Piao et al. [9] proposed a noncontact diffuse reflectance method for glucose sensing based on a center illumination area detection approach. Maureen et al. [10] developed an algorithm for determining the reduced scattering coefficient (µ s ′ ) of tissues from a single optical reflectance spectrum measured with a small needle-like probe. Swami et al. [11] examined the effects of absorption on the relationship between the polarization parameters and the morphological features of turbid samples which contain ink as the absorber. In addition, Ninni et al. [12] examined the extinction coefficient and absorption coefficient properties of various India inks used in the preparation of tissue simulation phantoms. It is noted that how to distinguish the scattering and absorption properties in the turbid media is still tricky based on the Beer-Lambert law.
A polarimetry [13], [14], [15], [16], [17] is one of the most commonly used and has found widespread application in many fields including disease diagnosis, glucose monitoring, cancer detection, and so on. In such methods, the polarization properties of the anisotropic optical sample are expressed using the Stokes vectors and are manipulated and derived using a Mueller matrix formalism. Various decoupled analytical techniques based on Mueller matrices have been proposed for extracting the effective anisotropic parameters of turbid media, including the linear birefringence's (LB) orientation angle (α), This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ LB retardance (β), circular birefringence's (CB) optical rotation angle (γ ), linear dichroism (LD) orientation angle (θ d ), linear dichroism (D), circular dichroism's (CD) axis angle (R), and depolarization index ( ) [18], [19], [20], [21], [22], [23]. However, while the literature contains many studies on the use of Mueller matrix methods to decouple the anisotropic parameters of turbid media, the use of Mueller matrix polarimetry to examine the isotropic parameters of optical samples (e.g., the mean absorbance) has attracted only little attention.
In this study, a Mueller matrix polarimetry consisting of polarizers and two liquid-crystal modulators is designed for extracting the mean absorbance of an anisotropic turbid media, and the feasibility of the proposed method is demonstrated both numerically and experimentally using ink samples with and without lipoplus scattering medium, respectively. It is additionally shown that for turbid samples, the accuracy of the concentration estimates determined from the measured mean absorbance values can be improved by subtracting the effective Mueller matrix of depolarization calculated using the differential Mueller matrix and decomposition decoupling methods from the Mueller matrix of the turbid sample.

II. EXTRACTION OF MEAN ABSORBANCE USING BEER-LAMBERT LAW AND MUELLER MATRIX FORMALISM
For a simple medium without scattering, the Beer-Lambert law states that the absorbance (A) is equal to the log 10 of the ratio of the transmission intensity (I ) of the light as it passes through the medium to the transmission intensity (I 0 ) of the same light as it propagates through air. When measuring the absorbance using a Stokes polarimeter, the Beer-Lambert law can be formulated as follows [8]: where S 0sample and S 0air are the first Stokes parameters obtained when the light passes through the sample and through air, respectively, ε (L mol −1 cm −1 ) is the molar absorptivity or molar absorption coefficient of the sample at the measurement wavelength (λ ), l is the optical path length, and X (mol L −1 ) is the molar concentration of the solution [24]. Shindo et al. [25] reported the Mueller matrix approach to investigate the artifacts resulting from the coupling of anisotropic samples and imperfect hardware of a commercially available CD spectropolarimeter and showed that the mean absorbance, A e , of a sample is related as follows: where S is the non-normalized Mueller matrix of the sample, M is the normalized Mueller matrix, I is an identity matrix, and || || is the determinant of the Mueller matrix. Rearranging (2) and (3), the mean absorbance of the sample can be obtained as Strictly speaking, (5) is valid only for nonturbid media without scattering effects. However, in many applications, e.g., biomonitoring, disease detection, and health care, the samples are turbid anisotropic media. Hence, before (5) can be applied, it is necessary to identify and remove the scattering-induced depolarization effect from the Mueller matrix of the sample. Therefore, in the present study, the anisotropic optical parameters of the sample are determined using a simple Mueller matrix polarimetry system consisting of a single polarizer and two liquid crystal variable retarders (LCVRs), and the depolarization effect of the sample is extracted using two different Mueller matrix methods, namely, the decomposition method and the differential Mueller matrix method. The depolarization effects obtained using the two methods are subtracted from the Mueller matrix of the sample, and the mean absorbance values obtained using (5) are then evaluated and compared.

A. Decomposition Method
For a turbid medium with scattering effects, a total of nine effective optical parameters should be extracted, namely, the LB principal axis angle (α), the LB retardance (β), the CB optical rotation angle (γ ), the LD principal axis angle (θ d ), the linear dichroism (D), the CD axis angle (R), the degrees of linear depolarization (e 1 and e 2 ), and the degree of circular depolarization (e 3 ). Many studies have shown that the decomposition method allows for a straightforward interpretation and parameterization of the experimentally determined Mueller matrix in terms of the characteristic parameters (i.e., dichroism, retardance, and depolarization) of the polarization devices and sample, respectively [26], [27], [28], [29], [30]; however, the limitation in the sequential order always exists. In addition, Liao and Lo [20] proposed a hybrid model derived by the decomposition method and the differential calculation for turbid media to resolve the limitation in the sequential order in the decomposition method and extend the measurement in full range. In the conventional Stokes-Mueller matrix formulation, the Stokes vector of the output light [S] is given by the product of the 4 × 4 Mueller matrix [M] of the sample and the Stokes vector of the input light [S ′ ]. That is, where S 0 represents the total intensity of the output light, S 1 is the difference in intensities of the linearly horizontal and vertical polarized components, respectively, S 2 is the difference in intensities of the linearly polarized components oriented at +45 • and −45 • , respectively, and S 3 is the difference in intensities of the right-and left-hand circular polarized components, respectively. In the decomposition method, the 4 × 4 Mueller matrix [M] of the sample is replaced by a sequence , describing the LD, LB, CB, CD, and depolarization properties of the sample, respectively. The detailed derivation of these matrices is described in [18], [19], [21], [31], and [32].

B. Differential Mueller Matrix Method
The optical properties of anisotropic turbid media can also be decoupled using a differential Mueller matrix formalism derived from an eigenanalysis of the corresponding Mueller matrix. It is noted that the subsequent decoupled properties of the corresponding differential matrix are expanded into the complete set of 16 differential matrices which characterize anisotropic turbid media [33]. The derivation process considers a beam propagating along the z-axis in a right-handed Cartesian coordinate system and formulates the differential matrix (m) as follows [33]: where M is the macroscopic Mueller matrix of the sample and is acquired via Mueller matrix polarimetry. The eigenvalues of the macroscopic and differential matrices are derived as λ M and λ m , respectively, and are in full parallelism, while the eigenvectors of the macroscopic and differential matrices, i.e., V M and V m , respectively, are the same. Assuming that m λ is a diagonal matrix with nonzero diagonal elements of λ (m), the differential Mueller matrix can be obtained from an eigenanalysis of M as In the differential Mueller matrix method, the Mueller matrix of the anisotropic sample is partitioned into 16 elements, and the LB, CB, LD, CD, and depolarization properties of the sample are described using macroscopic Mueller matrices M LB , M CB , M LD , M CD , and M respectively. Thus, the differential Mueller matrix describing all the optical properties of the sample can be obtained simply by summing the corresponding differential Mueller matrices (i.e., m lb , m cb , m ld , m cd , and m ) together. The detailed description of the differential Mueller matrix method and its derivation is available in [20], [34], and [35].  of a single polarizer and two linear LCVRs. As shown, the principal axes of the polarizer and second LCVR are set as 45 o , while that of the first retarder is set as 0 • . As described in [14], all 16 elements of the Mueller matrix of the sample can be obtained using just four input lights, namely, three linearly polarized lights (0 o , 45 o , and 90 o ) and one right-hand circular (RHC) polarized light. For the system shown in Fig. 1, the input lights are obtained simply by adjusting the retardance values (δ 1 and δ 2 ) of the two LCVRs [14], [15], [36], as shown in Table I. In calculating the Stokes vectors of the four input lights, the Mueller matrices of the 45 • polarizer and LCVRs are given, respectively, as in (9) and (10), shown at the bottom of the next page, where θ is the principal angle of the LCVR fast axis, and δ is the phase retardance between the fast and slow axes. It is noted that the system shown in Fig. 1 is exactly the same as the one designed in [14] and [15] except that two LCVRs are replaced by two electro-optic (EO) modulators.

B. Simulation Results for Mean Absorbance and Effective Optical Parameters of Turbid Media
The simulations were performed using LightTools software (Synopsis Optical Solutions Group). In constructing the optical model (see Fig. 2), the wavelength of the illumination light was set as 532 nm, the diameters of the polarizer and LCVRs were set as 25 and 10 mm, respectively, and the thicknesses of the polarizer and LCVRs were set as 2 and 5 mm, respectively. In addition, the polarizer, first LCVR, and second LCVR were placed at 65, 75, and 105 mm from the light source, respectively. The simulated tissue phantom samples had a size of 2 × 10 × 43 mm 3 (length × width × height) and were placed at a distance of 165 mm from the light source. Finally, the detector had dimensions of 6.25 × 6.25 mm 2 (width × height) and was placed at 225 mm from the light source. The total number of input rays was set as 10 000 in every case.
To mimic the interstitial fluid of human tissue, the simulated samples were assigned various absorption coefficients (µ a ) in the range of 0-0.1 mm −1 and scattering coefficients (µ s ′ ) of 0, 0.2, 0.5, and 1 mm −1 , respectively. In accordance with [37] and [38], the refractive index and anisotropy factor of the samples were set as 1.4 and 0.9, respectively.   are equal to 3.14. In contrast, for µ s ′ = 0.2 and 1 mm −1 , γ is equal to 0 in both the cases (see Figs. 4(b) and 6(b), respectively). In other words, for values of the CB optical rotation angle equal to 0 and 3.14, the corresponding Mueller matrices are the same [39]. However, for both the decoupling methods, the other LB and LD parameters (i.e., β, D, and R) have values equal to (or very close to) 0, irrespective of the absorbance coefficient (µ a ) or scattering coefficient (µ s ) values. Hence, it is inferred that the corresponding LB and LD matrices are identity matrices.
Observing Figs. 3(b)-6(b) and 3(c)-6(c), it is seen that the depolarization index ( ) has a value close to 0 for all the values of the absorbance coefficient and scattering coefficient, irrespective of the decoupling method applied. A detailed inspection shows that = 0.0003, 0.0004, 0.0003, and 0.0003 for µ s ′ = 0, 0.2, 0.5, and 1 mm −1 , respectively, when extracted using the decomposition method, and = 0.0003, 0.0004, 0.0003, and 0.0004, respectively, when extracted using the differential Mueller matrix method. Overall, the results suggest that the depolarization index is related not only to the scattering coefficient but also to the relative positions of the optical components in the polarimetry system, particularly the distance between the scattering object and the surface detector. The sensing area of the detector is one of the key factors affecting the depolarization index. It can be proven in Fig. 2 that the red rays between the turbid media and the surface detector are almost parallel to each other, which means that the polarized rays are still kept intact.

V. EXPERIMENTAL SETUP AND RESULTS
The practical feasibility of the proposed absorbance measurement method was evaluated by means of experimental trials. In implementing the polarimeter system shown in Fig. 1, the illumination light was provided by a diode-pumped solid-state laser with a wavelength of 532 nm (GPD01005-MH2-P, LASOS) and was passed through a 25-mm-diameter polarizer (Linear Glass Polarizing Filter, Edmund Optics) and two LCVRs (LCC2415-VIS, Thorlabs) to generate the required input polarization lights. The intensity of the light  emerging from the sample was measured using a commercial Stokes polarimeter (PAX1000VIS, Thorlabs). The experiments considered two different types of samples, namely, pure ink samples (commercial black ink, ChungHua) and turbid ink samples containing 2% lipoplus (MCT-10 LCT-8 ω3-2, Braun). For both the samples (i.e., pure and turbid), six different solutions were prepared with ink concentrations ranging from 0% to 0.001% in increments of 0.0002%. For all the experiments, the ink solutions were stored in glass cuvettes with an optical path length of 2 mm. Fig. 7 shows the results obtained for the mean absorbance, A e , of the six pure ink samples. It is seen that the mean absorbance increases linearly with an increasing ink concentration. Furthermore, for the sample with no ink (i.e., the sample consists of only the pure deionized (DI) water), the mean absorbance has a value of around 3.5. Fig. 8 shows the absorbance values obtained from the original Beer-Lambert law [see (1)] for the six pure ink samples when the initial intensity term I 0 in (1) is set as the detected intensity for a pure water sample. It is seen that compared with Fig. 7, the absorbance has a value equal to 0 for the sample with no ink. In other words, it is inferred that the mean absorbance values extracted using (5) (the proposed Mueller matrix method) are relative values rather than absolute values (as defined by the Beer-Lambert law). Accordingly, some form of calibration process is required to eliminate the residual mean absorbance when applying the proposed Mueller matrix polarimetry technique. Thus, in accordance with the results presented in Fig. 8, the mean absorbance of pure DI water without ink (3.5) was subtracted from the results obtained using (5) to obtain the absolute values of the measured absorbance for the different samples. Fig. 9 shows the results obtained for the absorbance of pure ink samples following the calibration process. It is seen that the results are in good agreement with those presented in Fig. 8. For example, the absorbance values in Fig. 8 have a mean standard deviation of 0.1%, while those in Fig. 9 have a mean standard deviation of 0.2%. Furthermore, the rate of change in the absorbance with the ink concentration is equal to 0.114 (A/ink concentration) in Fig. 8 and 0.112 (A e /ink concentration) in Fig. 9. In other words, the effectiveness of the proposed Mueller matrix method (with calibration) in extracting the mean absorbance values of the pure ink samples is confirmed. Fig. 10(a) shows the values of the effective anisotropic properties of the pure ink samples obtained by the decomposition method. It is seen that the LB orientation angle and LD angle have values of around α = 1.54 and θ d = 1.16 radians, respectively. The circular birefringence (γ ), linear dichroism (D), and circular dichroism (R) all have values close to 0. However, the LB retardance (β) has a value of around 1.52 radians. Fig. 10(b) shows the effective anisotropic parameters extracted by the differential Mueller matrix method. The LB orientation angle and LD angle have values of around α = 0.18 and θ d = 0.13 radians, respectively. However, the linear birefringence (β), circular birefringence (γ ), linear dichroism (D), and circular dichroism (R) all have values close to 0. For both the methods the depolarization index ( ) has a value of around 0.005 and is thus close to the simulation results presented in Figs. 3-6.  see Fig. 9). However, the standard deviation of the mean absorbance values (0.77%) is significantly higher than that of the pure ink samples (0.2%) due to the higher scattering effect and hence greater variation in the optical path length. Fig. 12(a) shows the extraction results obtained by the decomposition method for the effective optical parameters of the turbid ink solutions with 2% lipoplus. As shown, the LB orientation angle has a value of α = 1.58 radians, while the LD angle varies in the range of 1.57-3 radians. Meanwhile, the linear birefringence (β) and linear dichroism (D) have values of approximately 0.7 and 0 radians, respectively. The circular birefringence (γ ) and circular dichroism (R) also have values close to 0. Fig. 12(b) shows the extraction results obtained

C. Enhanced Concentration Estimation Performance of Ink Solutions With Scattering Effects
For a turbid medium, the normalized Mueller matrix without scattering effects can be obtained simply by removing the normalized Mueller matrix of depolarization from the original normalized matrix of the medium. The corresponding nonnormalized Mueller matrix can be obtained in a similar manner. Thus, the potential exists to improve the ability of the proposed method to predict the concentration of turbid samples by removing the depolarization content of the sample matrices prior to processing by (5). Fig. 13 shows the mean absorbance measurements obtained for the turbid ink samples given three different treatments of the measured Mueller matrix data. In the first case (denoted as A e ), the Mueller matrix is processed directly by (5) However, in the second and third cases, the depolarization matrices obtained by the decomposition and differential Mueller matrix methods, respectively, are subtracted from the sample Mueller matrix before being processed by (5). It is seen that the measurement results obtained using the full Mueller matrix (A e ) and Mueller matrix minus the depolarization matrix determined using the differential Mueller matrix method, respectively, are almost identical. In other words, it is inferred that the Mueller matrix of depolarization (M ) extracted by the differential Mueller matrix method is close to an identity matrix, and thus has no significant effect on the measurement performance of the proposed method. However, the measurement results obtained when subtracting the depolarization matrix determined using the decomposition method are virtually superimposed on those obtained for the pure ink sample with no scattering. In other words, the depolarization matrix extracted by the decomposition method provides an effective means of calibrating the sample Mueller matrix prior to processing by (5) and improves the concentration prediction performance of the proposed method when applied to turbid media accordingly.

VI. CONCLUSION
This study has presented a method for determining the mean absorbance of turbid media using a Mueller matrix polarimeter and the Beer-Lambert model. The validity of the proposed approach has been demonstrated numerically for turbid samples with various absorption coefficients and scattering coefficients. Experimental trials have also been performed using both pure ink samples and turbid ink samples with 2% lipoplus. The results have shown that for both the samples, the measured value of the mean absorbance increases linearly with an increasing ink concentration. As a result, the mean absorbance measurements obtained using the proposed method provide a viable means of predicting the analyte concentration. For the pure ink samples, the mean concentration measurements have a mean standard deviation of 0.2%. However, for the turbid samples, the mean standard deviation increases to 0.77% as a result of the greater nonuniformity in the optical path length.
Finally, it has been shown that the concentration estimation performance of the proposed method for turbid media can be further improved by subtracting the depolarization matrix calculated using the decomposition decoupling method from the sample matrix. In particular, by removing the depolarization matrix, the mean absorbance measurements enable the sample concentration to be more reliably predicted. It is noted that for higher scattering effects, the method proposed in this study will be modified in getting more accurate mean absorbance in the near future. Also, it is found that more parameter such as A e can be extracted and it will be better for the future Big Data analysis.