1.

Introduction

Polarization is a fundamental property of light and has many practical applications in industry and engineering science. Many studies have shown that the polarization state of a light beam (including depolarization effects) can be fully described by four Stokes parameters ($S_0$, $S_1$, $S_2$, and $S_3$).^1–3 Meanwhile, the change in optical properties of a sample given different polarization states of the incident light can be completely described by the Mueller matrix formalism.⁴ Consequently, Stokes–Mueller matrix polarimetry provides a powerful technique for characterizing a wide range of materials, biomaterials, and turbid media.^5,6 Lu and Chipman⁷ proposed a pioneering method for dealing with the strong multiple scattering effects of turbid media by decomposing the Mueller matrix into a sequence of three matrix factors corresponding to the diattenuation, retardation, and depolarization properties, respectively. Kumar et al.⁸ and Ghosh and Vitkin⁹ utilized polar decomposition and Mueller matrix decomposition methods to extract/quantify the linear retardance, optical rotation, and depolarization parameters of complex tissue-like turbid media with simultaneous scattering and depolarization effects. Pham and Lo^10,11 proposed a Stokes–Mueller decomposition matrix-based method for extracting all the effective parameters of turbid media, including the linear birefringence, linear dichroism, circular birefringence (CB), circular dichroism (CD), linear depolarization, and circular depolarization properties. However, the methods in Refs. 89.10.–11 require a strict decoupling of the optical properties of the sample. As a result, they are unsuitable for samples containing multiple properties. Several studies have shown that this problem can be resolved by a differential Mueller matrix formalism. Quijano et al.¹² proposed a differential Mueller matrix decomposition method for extracting the polarimetric properties of general depolarizing anisotropic media. Liao and Lo¹³ developed a hybrid model comprising the Mueller matrix decomposition method and the differential Mueller matrix formalism for obtaining full-range measurements of the anisotropic optical properties of turbid media. However, the methods in Refs. 12 and 13 require light to be transmitted through the sample and are thus unsuitable for noninvasive practical medical diagnoses.

Surface plasmon resonance (SPR) is a charge density oscillation at the interface of two media with dielectric constants of opposite signs and has significant potential for sensing applications.^14–16 Generally speaking, existing SPR sensors are based on either prism couplers or diffraction gratings. Of the two types of sensor, those based on prism couplers tend to have a better sensitivity and resolution.¹⁷ One of the most common uses of prism coupler-based SPR sensors is that of total internal reflection ellipsometry (TIRE).¹⁸ Le et al.¹⁹ used TIRE to visualize the behavior of giant lipid vesicles interacting with an adhesive surface coated with poly-l-lysine. Olender et al.²⁰ used TIRE to perform the real-time monitoring of the absorption of lipopolysaccharide molecules and whole gram-negative bacteria cells. Balevicius et al.^21,22 utilized TIRE to study the interaction of biomolecule layers and the optical anisotropy of biorecognition molecule layers.

In a previous study,²³ the present group proposed an SPR-based technique for enhanced CD/degree of polarization (DOP) sensing. In the present study, the proposed method is extended to the detection of the CB/CD/DOP properties of complex turbid media. The validity of the proposed method is demonstrated both numerically and by means of experimental investigations using glucose–chlorophyllin compound samples containing polystyrene microspheres.

2.

Total Internal Reflection Ellipsometry Surface Plasmon Resonance Prism Coupler

Figure 1(a) presents a schematic illustration of the SPR prism coupler used in the present study. As described in Ref. 23, the coupler comprises a half-ball glass lens, a Cr–Au isotropic thin-film layer, and a ${\mathrm{Ta}}_2{\mathrm{O}}_5$ anisotropic layer. The half-ball lens couples the incident polarized light into the isotropic and anisotropic films and provides total internal reflection. Meanwhile, the isotropic and anisotropic layers enhance the sensing performance by manipulating the incident polarized light and inducing SPR at the sensed interface. The half-ball lens in the SPR prism coupler was fabricated of BK7 glass with a refractive index of $n_0=1.517$. The refractive indices of the isotropic and anisotropic layers were the same as those in Ref. 23, i.e., $n_1=0.36$–$2.9i$, $n_{21}=1.637$; and $n_{22}=1.449$, $n_{23}=1.589$. However, the thicknesses of the two layers were increased to $d_1=30\mathrm{nm}$ and $d_2=10\mathrm{nm}$, respectively, to improve the measurement sensitivity of the CB/CD/DOP properties. The resonance angle of the prism coupler was found to be around 76 deg at a wavelength of 632.8 nm and resulted in a reflectance coefficient $R_{\mathrm{pp}}$ of $<0.2$, as shown in Fig. 1(b).

Fig. 1

(a) Schematic illustration of SPR prism coupler and (b) resonance angle determination.

3.

Analytical Model for Extracting Circular Birefringence/Circular Dichroism/Degree of Polarization Properties

An optical sample can be described by the matrix formulation $S=M{S}^{\prime }$, where $S$ is the Stokes vector of the output light, $M$ is the $4\times 4$ Mueller matrix of the sample, and $S^{\prime }$ is the Stokes vector of the input light. The general form of this relation is given as follows:

Compared to the model presented in Ref. 23 for CD/DOP measurement, the Mueller matrix in the present study contains an additional term ($M_{\mathrm{CB}}$) to describe the CB properties of the sample. In other words, the Mueller matrix is given by

Thus, Eq. (1) can be expressed as follows:

Note that $R$ is the CD property of the sample; $m_{11}$, $m_{12}$, $m_{33}$ and $m_{34}$ are the elements of the $M_{\mathrm{R}}$ Mueller matrix; and $p_1$, $p_2$, $p_3$, $e_1$, $e_2$, and $e_3$ are the elements of the $M_{\mathrm{D}}$ Mueller matrix.

As described in the following, the use of four input lights [namely, three linear polarization lights (0, 45, and 90 deg) and one right-hand circular polarization light] yields a sufficient number of equations to determine parameters $\gamma$, $R$, and $\mathrm{\Delta }$ (DOP) of the optical sample. The Stokes vectors of the four input lights are given as follows: $S_{0\mathrm{deg}}^{\prime }={[\mathrm{1,1},\mathrm{0,0}]}^{\mathrm{T}}$, $S_{45\mathrm{deg}}^{\prime }={[\mathrm{1,0},\mathrm{1,0}]}^{\mathrm{T}}$, $S_{90\mathrm{deg}}^{\prime }={[1,-\mathrm{1,0},0]}^{\mathrm{T}}$, and $S_R^{\prime }={[\mathrm{1,0},\mathrm{0,1}]}^{\mathrm{T}}$. Thus, $R$ and $\gamma$ can be obtained directly as follows:

In addition, the elements of the depolarization Mueller matrix are obtained as follows:

Thus, the $\mathrm{\Delta }$ can then be obtained as follows:

It is noted that $e_1$ and $e_2$ in Eqs. (11) and (12) are functions of $\gamma$ and represent the only difference from the expressions given in Ref. 23 for the depolarization Mueller matrix.

4.

Validity of Analytical Model

The validity of the analytical model derived above was investigated by comparing the values obtained for parameters $\mathrm{\Delta }$, $\gamma$, and $R$ of a hypothetical sample with the known values inserted into the sample matrix in Eq. (2). In performing the simulations, the refractive index of the CB/CD/DOP sample was set as 1.33 and the incident angle ${\theta }_i$ was set equal to the SPR angle of 76 deg. As shown in Fig. 2, a good agreement was obtained between the two sets of values in every case. In other words, the ability of the proposed model to extract the values of $\mathrm{\Delta }$, $R$, and $\gamma$ over the full range is confirmed.

Fig. 2

Comparison of extracted values and input values of: (a) $\gamma$, (b) $R$, and (c) $\mathrm{\Delta }$.

5.

Sensitivity of Circular Dichroism Measurements to Chlorophyllin Concentration

Chlorophyllin is a common photosynthetic pigment and can be obtained from spinach leaves or grass. The molecular structure of chlorophyllin results in a high optical absorbance and a strong CD effect.^24,25 In the present study, simulations were performed to investigate the sensitivity of the extracted $R$ value to the change in concentration of chlorophyllin sodium copper salt samples (referred to hereafter simply as chlorophyllin samples) in aqueous solution. In performing the simulations, the change in concentration of the chlorophyllin sample was modeled by a change in the refractive index $N$.²⁶ Based on this assumption, the $R$ values of samples with refractive indices ranging from 1.3 to 1.4 (with a step size of 0.01) were computed using Eq. (9) for scanning angles $\theta$ in the range of 0 to 180 deg. The corresponding results are presented in Fig. 3. It is noted that scanning angle $\theta$ is the angle between the original $X-Y$ coordinate system and the measured $P^{\prime }-S^{\prime }$ coordinate system of the polarization scanning ellipsometry technique developed by the current group in Ref. 27. As described in Ref. 27, in applying Eq. (9) to derive the $R$ parameter, the traditional algorithm must be modified from the laboratory ($X-Y$) coordinate system to the measured $P^{\prime }-S^{\prime }$ coordinate system. Thus, the axes of the four polarization input light beams (i.e., 0 deg, 45 deg, 90 deg, and $R-$) in the $P^{\prime }-S^{\prime }$ coordinate system must be rotated through an additional scanning angle of $\theta$ to convert them to the $X-Y$ coordinate frame.

Fig. 3

(a) Variation of extracted $R$ value with polarization scanning angle for chlorophyllin samples with refractive indices ranging from 1.3 to 1.4 (step size of 0.01). Note that the arrows show the direction of increasing refractive index. (b) Variation of extracted $R$ value with refractive index given scanning angle of $\theta =135\mathrm{deg}$. Estimated resolution of CD measurement is equal to ${10}^{-4}\mathrm{RIU}$.

As shown in Fig. 3(a), the extracted $R$ values are particularly sensitive to changes in the refractive index (i.e., the chlorophyllin concentration) at scanning angles of 45 and 135 deg. Furthermore, Fig. 3(b) shows that the $R$ value increases approximately linearly over the refractive index range of 1.3 to 1.4 given a fixed scanning angle of 135 deg. Assuming the output Stokes vectors are obtained using a commercial Stokes polarimeter (PAX5710, Thorlabs Co.) with a deviation of $\pm {10}^{-3}$, the estimated resolution of the extracted $R$ values is of the order of ${10}^{-4}\mathrm{RIU}$. Note that the results presented in Fig. 3(a) show that the sensitivity of the extraction results is the same given a scanning angle of 45 deg as that for a scanning angle of 135 deg. Thus, only the data obtained for a scanning angle of 135 deg are considered in deriving Fig. 3(b).

6.

Sensitivity of Circular Birefringence Measurements to Glucose Concentration

Simulations were performed to investigate the sensitivity of the extracted $\gamma$ value to the change in concentration of glucose aqueous solutions. In performing the simulations, it was assumed that the refractive index increased linearly from 1.3 to 1.4 as the glucose concentration increased from 0 to $100\mathrm{mg}/\mathrm{mL}$.^28,29 Based on this assumption, the $\gamma$ values of samples with refractive indices ranging from 1.3 to 1.4 (with a step size of 0.01) were computed using Eq. (10) for scanning angle $\theta$ in the range of 0 to 180 deg. It is noted that scanning angle $\theta$ is the angle between the original $X–Y$ coordinate system and the measured $P^{\prime }-S^{\prime }$ coordinate system.²⁷

As shown in Fig. 4(a), the extracted optical rotation angle $\gamma$ is particularly sensitive to changes in the refractive index (i.e., the glucose concentration) at polarization scanning angles of 40, 60, 120, and 135 deg. Furthermore, Fig. 4(b) shows that $\gamma$ decreases with the increase of refractive index over a range of 1.3 to 1.4 given a fixed scanning angle of 135 deg. Assuming that the output Stokes vectors are again obtained using a commercial Stokes polarimeter (PAX5710, Thorlabs Co.) with a deviation of $\pm {10}^{-3}$, the estimated resolution of the extracted CB values is of the order of $5\times {10}^{-5}\mathrm{RIU}$. Note that the results presented in Fig. 4(a) show that a scanning angle of 135 deg results in the highest measurement sensitivity. Thus, only the data obtained for a scanning angle of 135 deg are considered in deriving the results presented in Fig. 4(b).

Fig. 4

(a) Variation of extracted $\gamma$ value with polarization scanning angle for glucose samples with refractive indices ranging from 1.3 to 1.4 (step size of 0.01). Note that the arrows show the direction of increasing refractive index. (b) Variation of extracted $\gamma$ value with refractive index given scanning angle of $\theta =135\mathrm{deg}$. Estimated resolution of $\gamma$ measurement is equal to ${10}^{-5}\mathrm{RIU}$.

Figure 5 shows the simulation results for the sensitivity of the DOP ($\mathrm{\Delta }$) to changes in the refractive index (1.3 to 1.4) for glucose samples with small [Fig. 5(a)] and high [Fig. 5(b)] degrees of depolarization of $\mathrm{\Delta }=0.4$ and 0.85, respectively. Note that in performing the simulation, the value of $\mathrm{\Delta }$ was calculated using Eq. (19). As shown, the extracted values of $\mathrm{\Delta }$ are insensitive to the refractive index over the full range of the polarization scanning angle for both samples. This finding is reasonable since, although $\mathrm{\Delta }$ is a function of both $R$ and $\gamma$, when the refractive index increases, $R$ increases but $\gamma$ decreases (or $R$ decreases but $\gamma$ increases, depending on the scanning angle). As a result, the value of $\mathrm{\Delta }$ remains approximately unchanged. The thickness of the isotropic/anisotropic layers of the prism coupler affects both the working range and the sensitivity of the sensor. Thus, the measurement sensitivity of $\mathrm{\Delta }$ can be enhanced through an appropriate design of the prism coupler structure. In the present study, however, the prism coupler is designed only to maximize the measurement sensitivity of $\gamma$ and $R$ over the considered measurement range.

Fig. 5

Variation of extracted $\mathrm{\Delta }$ value with polarization scanning angle for glucose samples with refractive indices ranging from 1.3 to 1.4 (step size of 0.01): (a) small degree of depolarization and (b) high degree of depolarization.

7.

Experimental Setup and Results

Figure 6 presents a schematic illustration of the PSR-based scanning polarization ellipsometry system proposed in the present study. As shown, the major items of equipment include a He–Ne laser (SL 02/2, SIOS Co., central wavelength 632.8 nm), a polarizer (GTH5M, Thorlabs Co.) to produce linear polarized light, a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.) to convert the linear polarized light into circular polarized light, a second polarizer (GTH5M, Thorlabs Co.) set to a scanning angle in the range of $\theta =0$ to 180 deg, and a neutral density filter (NDC-100C-2, Oneset Co.) and power detector (8842A, OPHIR Co.) to calibrate the intensity of the input polarization light. Following the calibration process, the power detector was removed from the experimental setup, and the light emerging from the neutral density filter was reflected on the SPR sensor and detected by a commercial Stokes polarimeter (PAX5710, Thorlabs Co.). To achieve a precise alignment of the optical components in the experimental setup, a pin hole was placed in front of the polarizers and quarter-wave plate and the components were then adjusted such that the reflected laser beam passed though the pin hole in turn.

Fig. 6

Schematic illustration of experimental setup.

To enhance the SPR effect at the sensed interface, the Stokes polarimeter was placed on a mechanical stage (SGSP-60YAW, Sigma Koki Co.) controlled by a step motor (Mark 204-MS, Sigma Koki Co.) and rotated through an angle of 28 deg such that the incident light was incident on the prism coupler at the resonance angle of 76 deg (see Fig. 6). In performing the experiments, the linear scanning polarization lights were produced by manually rotating the second polarizer from 0 to 180 deg in steps of 15 deg using a mechanical stage (SGSP-60YAW, Sigma Koki Co.). In addition, the right-hand circular polarization light ($R-$) was produced by removing the second polarizer from the system. The samples were stored in plastic cuvettes with dimensions of $10\times 10\times 1\mathrm{mm}$. The prism coupler was attached to the cuvettes by means of industrial glue and a layer of silicon around the border edge. Prior to mounting the coupler, the cuvette was drilled with a small hole with a diameter of 6 mm such that the sample made direct contact with the half-ball lens (thereby avoiding optical interference by the cuvette material).

The CD samples were prepared using 10-mL chlorophyllin solutions (C6003, Sigma-Aldrich Co. LLC) with concentrations ranging from 0 to $100\mu \mathrm{g}/\mathrm{mL}$ in $20\mu \mathrm{g}/\mathrm{mL}$ increments mixed with 2 mL d-glucose solution ($100\mathrm{mg}/\mathrm{mL}$ Merck Ltd.) and 0.5-mL polystyrene microspheres ($0.2\mathrm{mg}/\mathrm{mL}$ Duke Standard™). Figure 7(a) shows the experimental and simulated $R$ values of the chlorophyllin samples given a scanning angle of 135 deg in every case. As shown, $R$ increases linearly with an increasing chlorophyllin concentration over the considered concentration range. Figures 7(b) and 7(c) show the experimental and simulated values of $\gamma$ and $\mathrm{\Delta }$, respectively, for the various samples. It is seen that $\gamma$ decreases linearly with an increasing chlorophyllin concentration, while $\mathrm{\Delta }$ remains approximately constant. The standard deviations of the measured values of $R$, $\gamma$, and $\mathrm{\Delta }$ obtained on four repeated tests over the measured concentration range of chlorophyllin samples are shown in Table 1. Moreover, for all of the samples, a good qualitative agreement exists between the experimental and simulation results for all three properties. Hence, the basic validity of the proposed approach is confirmed. The slight discrepancy between the two sets of results is most likely due to alignment errors in the optical system or imperfections in the optical components themselves.

Fig. 7

Experimental and simulated results for: (a) $R$, (b) $\gamma$, and (c) $\mathrm{\Delta }$ properties of CD samples with chlorophyllin concentrations ranging from 0 to $100\mu \mathrm{g}/\mathrm{mL}$. Note that the scanning angle is $\theta =135\mathrm{deg}$ and the incident angle is 76 deg in every case.

Table 1

The standard deviations of the measured values of R, γ, and Δ obtained on four repeated tests over the measured concentration range of chlorophyllin samples.

The CB samples were prepared using 10-mL glucose solution samples ($100\mathrm{mg}/\mathrm{mL}$ Merck Ltd) with concentrations ranging from 0 to $100\mathrm{mg}/\mathrm{mL}$ in $20\mathrm{mg}/\mathrm{mL}$ increments mixed with 2-mL chlorophyllin (C6003, Sigma-Aldrich Co. LLC) and 0.5-mL polystyrene microspheres ($0.2\mathrm{mg}/\mathrm{mL}$ Duke Standard™). Figure 8(a) shows the experimental and simulation results for the $\gamma$ values of the various samples given a scanning angle of 135 deg in every case. As shown, $\gamma$ decreases linearly with an increasing glucose concentration. Figures 8(b) and 8(c) show the experimental and simulation results for the $R$ and $\mathrm{\Delta }$ values of the various samples. It is seen that the $R$ value increases linearly with an increasing glucose concentration, while the $\mathrm{\Delta }$ value remains approximately constant. As for the CD samples, a good qualitative agreement exists between the experimental and simulated values in every case. The standard deviations of the measured values of $\gamma$, $R$, and $\mathrm{\Delta }$ obtained on four repeated tests over the measured concentration range of glucose samples are shown in Table 2.

Fig. 8

Experimental and simulated results for: (a) $\gamma$; (b) $R$; and (c) $\mathrm{\Delta }$ of CB samples with glucose concentrations ranging from 0 to $100\mu \mathrm{g}/\mathrm{mL}$. Note that the scanning angle is $\theta =135\mathrm{deg}$ and the incident angle is 76 deg in every case.

Table 2

The standard deviations of the measured values of γ, R, and Δ obtained on four repeated tests over the measured concentration range of glucose samples.

Overall, the results presented in Figs. 7 and 8 show that the extracted values of $R$ and $\gamma$ are linearly correlated with the chlorophyllin and glucose concentration over the considered range. As a result, the feasibility of the proposed method for extracting the optical properties of turbid media is confirmed. In addition, the results show that the extracted $\mathrm{\Delta }$ value is insensitive to changes in the chlorophyllin and glucose concentration over the range of 0 to $100\mu \mathrm{g}/\mathrm{mL}$ and 0 to $100\mathrm{mg}/\mathrm{mL}$.

8.

Conclusions and Suggestions

This study presents a CB/CD/DOP measurement technique based on an SPR prism coupler and Stokes–Mueller matrix polarimetry. The validity of the proposed method is demonstrated both numerically and experimentally. The simulation results show that the proposed method enables the CB and CD properties to be measured with resolutions of ${10}^{-5}$ and ${10}^{-4}\mathrm{RIU}$, respectively, for refractive indices in the range of 1.3 to 1.4. The simulation results also show that the measured DOP is insensitive to changes in the refractive index of the sample over the range of 1.3 to 1.4. However, in practice, the sensitivity of the $\mathrm{\Delta }$ measurements can be enhanced through an appropriate design of the prism coupler structure. The experimental results show that the measured CB and CD values are linearly related to the chlorophyllin and glucose concentrations, respectively, over the measured range. Furthermore, the average deviations of the CD, CB, and DOP measurements over four repeated tests are approximately $\pm 1.59\times {10}^{-4}$, $\pm 4.1\times {10}^{-2}$, and $\pm 2.01\times {10}^{-3}$, respectively. Thus, the feasibility of the SPR-enhanced Stokes–Mueller matrix polarimetry technique proposed in this study for practical CB/CD/DOP sensing applications is confirmed.