1.

Introduction

Diabetes mellitus is a medical condition in which the body cannot adequately produce or effectively use insulin that is needed to maintain a normal blood glucose level. It is reported that diabetes afflicts over 15 million people in North America and 100 million people worldwide.¹ To avoid or to delay the complication of diabetes including blindness, heart disease, and kidney failure, diabetics must regulate their insulin injections and caloric intake by monitoring their blood glucose levels. Most current methods for glucose monitoring require drawing blood (typically via a finger prick), which is painful, inconvenient, potentially infectious, and otherwise suboptimal. Significant efforts have centered on the search for noninvasive glucose monitoring methods. Optical methods are potentially well-suited to tackle this currently unsolved problem due to their noninvasive, nonharmful nature and sensitivity to glucose-induced optical changes. ^{2, 3, 4, 5, 6, 7, 8, 9, 10}

Recently, polarized light has been used to extract quantitative information from the optically thick tissues with which it interacts. ^{3, 10, 11, 12, 13, 14, 15} Because the optics of skin is dominated by scattering effects in the tissue optical window range of 600 to $1200\mathrm{nm}$ , the understanding of polarization properties of light that has escaped from highly scattering media is of great importance. In addition to the forward direction, some research has examined the detection of scattered light in other directions,^{11, 12, 13} to investigate the detection geometry where the experimental observables such as optical rotation, degree of polarization, and intensity respond most to changes in glucose concentration. These attempts were challenged by the complexity of light-media interactions and experimental difficulties. ^{12, 13, 14, 15}

The approach presented in this paper employs a refined linear Stokes polarimeter to detect the emerging light polarization states in a variety of detection angles around a scattering sample and examines the sensitivity of the derived changes in optical rotation and surviving linear polarization fraction to changes in the glucose concentration. This polarimeter design is similar to the one presented by Hunt and Huffman in the 1970s and subsequently used by many researchers. ^{16, 17, 18, 19} The main innovation in the present embodiment lies in the measurement of various harmonics of the signal, which are directly related to the linear Stokes vector components of the sample-scattered light. In other words, the current arrangement with the polarization modulation in the analyzer arm after the sample, permits completely assumption-independent determination of sample-induced light polarization parameters. This is a significant improvement over many previous attempts at Mueller matrix or Stokes vector polarimetry, which had to assume specific forms of the sample matrix or of the scattered light components for signal analysis. These assumptions become questionable as one encounters complicated media, such as biological tissues, that may exhibit several optically active effects at once (linear and circular birefringence or dichroism) while also causing light depolarization due to multiple scattering. Also, the electronic synchronous detection at three different modulation frequencies provides sufficient information to determine the Stokes parameters of interest, without the need for mechanical moving parts (rotating linear polarizers and/or wave plates, swapping in and out of polarizing optics). This reduces noise in the polarization measurements. Utilizing the linear Stokes polarimeter, the relative contributions of glucose concentration, sample turbidity, and detection angle geometry to the polarization properties of the emerging light are systematically investigated. The role of optical activity of glucose, the foundation for glucometry in clear and low turbid media, in the observed optical rotation in highly turbid media is also investigated.

2.

Theory and Signal Analysis

2.1.

Theory

Figure 1 illustrates the simplified linear Stokes polarimeter setup. The approach is centered on polarization modulation accomplished by the photoelastic modulator (PEM) and synchronous detection via a lock-in amplifier, allowing a sensitive means of detecting low levels of polarization signals in the presence of largely depolarized background. Specifically, the laser beam passes through a linear polarizer $P$ (pass axis horizontal, parallel to the optical table) and impinges on the sample. The scattered light from the sample passes through a quarter wave plate, whose fast-axis orientation is $-45°$ with respect to optical table, and goes through the PEM block orientated horizontally. The PEM is a resonant device operating at $50\mathrm{kHz}$ whose user-defined peak retardance ${\delta }_0$ is selected to optimize polarization signal strength (see below). The light then passes through an analyzer $A$ set at $45°$ with respect to the optical table and finally reaches the detector. Using standard Stokes vector notation for light polarization and Mueller matrices to represent the optical elements that transform these polarization states, we now analyze the system in Fig. 1. ${\mathsf{S}}_i$ represents the Stokes vector of the incident beam

Eq. 1

$${\mathsf{S}}_i={\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^T,$$

where horizontal notation is used for convenience, with $T$ standing for the transpose of the vector. ${\mathsf{S}}_e$ is the Stokes vector of the emerging beam after sample interactions,

Eq. 2

$${\mathsf{S}}_e={\left[\begin{array}{cccc}\hfill I\hfill & \hfill Q\hfill & \hfill U\hfill & \hfill V\hfill \end{array}\right]}^T.$$

If the elements of the Stokes vector in Eq. 2 are normalized by intensity $I$ , Eq. 2 becomes

Eq. 3

$${\mathsf{S}}_e=I{\left[\begin{array}{cccc}\hfill 1\hfill & \hfill q\hfill & \hfill u\hfill & \hfill v\hfill \end{array}\right]}^T,$$

where $q=\frac{Q}{I}$ , $u=\frac{U}{I}$ , and $v=\frac{V}{I}$ . ${\mathsf{S}}_f$ is the Stokes vector of the beam reaching the detector

Eq. 4

$${\mathsf{S}}_f={\left[\begin{array}{cccc}\hfill I_f\hfill & \hfill Q_f\hfill & \hfill U_f\hfill & \hfill V_f\hfill \end{array}\right]}^T.$$

Relating the detected signal to the sample-emerging beam via Mueller calculus,

Eq. 5

$${\mathsf{S}}_f={\mathsf{M}}_A{\mathsf{M}}_{PEM}{\mathsf{M}}_{WP}{\mathsf{S}}_e,$$

where ${\mathsf{M}}_{WP}$ is the Mueller matrix of the quarter wave plate, ${\mathsf{M}}_{PEM}$ is the Mueller matrix of the PEM, and ${\mathsf{M}}_A$ is the Mueller matrix of the analyzer. For the selected orientations, the Mueller matrices are

Eq. 6

$${\mathsf{M}}_{WP}(-45°)=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$$

Eq. 7

$${\mathsf{M}}_{PEM}(0°)=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cos \delta \hfill & \hfill \sin \delta \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\sin \delta \hfill & \hfill \cos \delta \hfill \end{array}\right],$$

Eq. 8

$${\mathsf{M}}_A(45°)=\frac{1}{2}\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right].$$

Putting Eqs. 6, 7, 8, 3 into Eq. 5, we obtain

Eq. 9

$${\mathsf{S}}_f=\frac{1}{2}\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cos \delta \hfill & \hfill \sin \delta \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\sin \delta \hfill & \hfill \cos \delta \hfill \end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ \hfill q\hfill \\ \hfill u\hfill \\ \hfill v\hfill \end{array}\right].$$

Performing the multiplications, the measurable part of the final Stokes vector reaching the detector is the time-dependent intensity expressed as

Eq. 10

$$I_f\left(t\right)=\frac{1}{2}[1-q\sin \delta +u\cos \delta ]=\frac{I}{2}[1-q\sin \left({\delta }_0\sin \omega t\right)+u\cos \left({\delta }_0\sin \omega t\right)],$$

where the time-dependent PEM retardation follows $\delta \left(t\right)={\delta }_0\sin \left(\omega t\right)$ . Applying Fourier expansion in terms of Bessel functions

Eq. 11

$$\sin \delta =\sin \left({\delta }_0\sin \omega t\right)=2\sum _{n=1}^{\infty }{J}_{2n-1}\left({\delta }_0\right)\sin (2n-1)\omega t,$$

Eq. 12

$$\cos \delta =\cos \left({\delta }_0\sin \omega t\right)={\mathrm{J}}_0\left({\delta }_0\right)+2\sum _{n=1}^{\infty }{J}_{2n}\left({\delta }_0\right)\cos 2n\omega t,$$

to sine and cosine terms in Eq. 10, there results

Eq. 13

$$I_f\left(t\right)=\frac{1}{2}\{1-2J_1\left({\delta }_0\right)q\sin \omega t+u[J_0\left({\delta }_0\right)+J_2\left({\delta }_0\right)\cos 2\omega t]+\cdots \}.$$

If the peak retardance of PEM is set to ${\delta }_0=2.405$ radians so that $J_0\left({\delta }_0\right)=0$ , then

Eq. 14

$$I_f\left(t\right)=\frac{1}{2}[1-2J_1\left({\delta }_0\right)q\sin \omega t+2J_2\left({\delta }_0\right)u\cos 2\omega t+\cdots ].$$

The elements of the Stokes vector of the sample-scattered light, $I$ , $q$ , and $u$ [Eq. 3] can be extracted from the dc component, first harmonic component, and second harmonic component of Eq. 14, respectively. Equation 14 is valid for all scattering directions because no angle-dependent assumptions are used in its derivation. Further, this formulation is based on the general form of sample polarization in Eq. 3, thus avoiding the assumption of specific (and generally unknown) forms of the sample Mueller matrix that may simultaneously exhibit optical rotation, depolarization, and linear anisotropy (birefringence).^{12, 14} As such, the relationship of the derived polarization parameters to the intrinsic properties of the interrogated sample is further strengthened.

Fig. 1

Simplified linear Stokes polarimeter set-up. P, polarizer; A, analyzer; WP, quarter wave plate; PEM, photoelastic modulator; ${\mathrm{S}}_{\mathrm{i}}$ , Stokes vector of the incident beam; ${\mathrm{S}}_{\mathrm{e}}$ , Stokes vector of the emerging beam; ${\mathrm{S}}_{\mathrm{f}}$ , Stokes vector of the beam reaching detector. All angles are with respect to the optical table.

3.

Materials and Experiments

3.1.

Experiment Setup

Figure 2 shows the diagram of linear Stokes polarimeter, demonstrating the specific embodiment of the general set-up of Fig. 1. The $632.8\text{-}\mathrm{nm}$ light from a $\mathrm{HeNe}$ laser (Research Electro-Optics, LHRR-120M), chopped by a mechanical chopper at a frequency of $500\mathrm{Hz}$ , passes through a polarizer (pass axis horizontal) and an iris of $1\mathrm{mm}$ diameter, and strikes a round glass cuvette of $1\mathrm{cm}$ inner diameter that contains the sample. For the reported experiments, the effective sampling volume is a function of detection geometry and sample scattering properties. Its approximate range varies from a few cubic millimeters (highly scattering, large detection angles) to tens of cubic millimeters. Further quantification of this important but difficult parameter, and a related measure we call the average photon pathlength, will be reported in a forthcoming publication. The incident power on the sample is $\sim 1\mathrm{mW}$ . Part of the emerging light enters an analyzing block consisting of a pair of plano-convex lenses; a quarter wave plate (fast axis at $-45°$ ); a PEM (Hinds Instruments IS-90; horizontal retardation axis, oscillating at $50\mathrm{kHz}$ ), a linear analyzer (pass axis at $+45°$ ), and a polarization-insensitive avalanche photodiode detector (Hamamatsu, C5460), which has a flat frequency response range from dc to $1\mathrm{MHz}$ . The analyzing block is installed on a horizontally rotatable platform forming a detection angle $\theta$ with respect to the incident light, with the center of the cylindrical cuvette positioned at the rotation axis of the block. The use of the pair of plano-convex lenses increases measurement sensitivity. The first lens picks up emerging light from an area of the turbid sample that is small enough to avoid significant blurring of the polarization states engendered by spatial averaging. The collected light is then converted into a collimated beam by the lens, an important step as the PEM functions optimally with collimated light. The second lens converges the collimated light, focusing it onto the sensing area of the detector. The signal from the analyzing block is sent to a lock-in amplifier (Standford Research Systems, SR830). The reference input of the lock-in amplifier is toggled between the chopper and PEM controllers. Data acquisition is carried out under computer control, obtaining within $2\min$ 10-measurement averages at $500\mathrm{Hz}$ , $50\mathrm{kHz}$ , and $100\mathrm{kHz}$ as $V_{chopper}$ , $V_{1f}$ , and $V_{2f}$ for further processing. The detection angle scanning is achieved by rotating the analyzing block platform as it isocentrically pivots around the round cuvette. The angles in the experiment vary from 0 to $155°$ at $5°$ intervals. Because of the PEM’s size $(15\times 2.5\times 5.5{\mathrm{cm}}^3)$ , the measurements at angles larger than $\theta >155°$ are hindered due to the obstruction of incident beam by the analyzing block.

Fig. 2

Schematic diagram of linear Stokes polarimeter. C, optical chopper; P, polarizer; $\theta$ , detection angle; ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ , plano-convex lenses; WP, quarter wave plate; PEM, photoelastic modulator; A, analyzer; D, detector.

4.

Results and Discussion

Figure 3 shows how optical rotation $\alpha$ , surviving linear polarization ${\beta }_{linear}$ , and intensity $I$ of the light emerging from turbid media (no glucose) change as a function of detection angle $\theta$ and turbidity of sample, indicated by scattering coefficient ${\mu }_s$ . The results of about 32 measurements are summarized in the figure. Somewhat surprisingly, Fig. 3a shows large optical rotations even in the absence of glucose, as suggested by previous initial observations.^{3, 15, 18} As seen, the magnitude of $\alpha$ is a strong and complex function of both sample turbidity and detection angle. For lower turbidity ( ${\mu }_s=20$ to $40{\mathrm{cm}}^{-1}$ ), $\alpha$ increases with $\theta$ from about $1°$ to about $4°$ and then decreases again for $\theta >130°$ . For turbid media with ${\mu }_s=50$ to $80{\mathrm{cm}}^{-1}$ , $\alpha$ displays a monotonic increase with $\theta$ from $\sim 1$ to $12°$ . For highly turbid media ( ${\mu }_s=90$ to $100{\mathrm{cm}}^{-1}$ ), $\alpha$ exhibits oscillations at smaller detection angles and then increases monotonically from ∼2 to $12°$ when $\theta >90°$ . It is noticed that in these highly turbid samples, $\alpha$ is slightly nonzero even in the forward direction, possibly arising from minor detector misalignment about $\theta =0°$ . Clearly then, multiple scattering significantly changes the orientation of the linear polarization plane, thus resulting in an apparent optical rotation in most directions. The magnitude of this change in linear polarization orientation depends strongly on both sample turbidity and detector orientation angle. Larger $\theta$ generally means larger change of the polarization plane orientation. Further studies are underway to interpret these trends. Regardless of the mechanism, the data unequivocally shows that optical rotation results in a turbid polarimetry setting must be interpreted with caution, in that $\alpha$ is not only caused by chirality of glucose molecules as is the case for clear-media glucometry.

Fig. 3

Measured angular and turbidity dependence of polarization states of emerging light from turbid media in the absence of glucose (a) Optical rotation; (b) surviving linear polarization fraction; (c) normalized intensity. The orientation of the horizontal axes is different for the three plots to enable better visualization.

Figure 3b shows that ${\beta }_{linear}$ decreases significantly with increasing turbidity and detection angle for low and moderately turbid media. In the forward direction $(\theta =0°)$ , ${\beta }_{linear}$ drops from 0.9 at ${\mu }_s=20{\mathrm{cm}}^{-1}$ to 0.35 at ${\mu }_s=60{\mathrm{cm}}^{-1}$ . For a turbid medium with ${\mu }_s=20{\mathrm{cm}}^{-1}$ , ${\beta }_{linear}$ drops from 0.9 at $\theta =0°$ to 0.2 at $\theta =155°$ . With further increases of turbidity of the sample, ${\beta }_{linear}$ still decreases but becomes less dependent on detection angle as its overall levels drop. The data suggests that for low and moderately turbid media, photons exiting at smaller angles may be scattered fewer times than the photons exiting at larger detection angles, yielding the smaller depolarization effects. With the increase of sample turbidity, photons encounter more scattering events on route to reaching the detector, resulting in greater depolarization. With further increase in sample turbidity, photons exiting in any direction encounter almost the same amount of scattering events, as suggested by the decreasing angular dependence of ${\beta }_{linear}$ .

Figure 3c shows that the intensity $I$ is not strongly dependent on the detection angle $\theta$ except for low turbidity media $({\mu }_s\sim 25{\mathrm{cm}}^{-1})$ , where $I$ drops from $\sim 0.9$ to 0.1 when $\theta$ increases from 0 to $155°$ . For samples with ${\mu }_s>40{\mathrm{cm}}^{-1}$ , the intensity is low and largely independent of detection angle, although exhibiting a minor increase for $\theta >90°$ .

Selected subsets of the above experimental results were compared with polarization-sensitive Monte Carlo (MC) simulations developed in our group,^{3, 25} a validated model that expands upon the prior work of Kaplan ²⁶ and Jaillon ²⁷ In these simulations, the photons are propagated between scattering events, as determined by pseudorandom sampling of the scattering mean free path similar to conventional (intensity-only) MC models²⁸ of light propagation in tissue; the polarization information is tracked in the form of individual Stokes vectors, which are summed over a large number of tracked photon histories, to yield the experimentally observable macroscopic properties of interest. Optical activity due to the chiral nature of dissolved glucose is handled by imposing the rotation of the plane of linear polarization between the scattering events, using a Mueller matrix for a standard optical rotator.^{3, 25} The aim of the comparison is to validate the measurement results and predict the trends of polarization states of scattered light at $\theta >155°$ , where measurements are not currently possible. Typical experiment-simulation comparison curves for glucose-free turbid suspensions are shown in Fig. 4 . The MC simulations of the surviving linear polarization fraction as a function of detection angle [for ${\mu }_s=20{\mathrm{cm}}^{-1}$ , Fig. 4a] and as a function of scattering coefficient [for $\theta =0°$ , Fig. 4b] show similar trends to the experimental measurements, although further refinement may be necessary to improve agreement. The simulation in Fig. 4a predicts a significant increase of ${\beta }_{linear}$ beyond $160°$ , a region currently unconfirmed with experimental measurements. This suggests the potential benefit of backward detection for highly turbid media, where surviving linear polarization fraction and signal intensity are relatively high. System modifications are underway to enable experimental validations at larger detection angles approaching the exact backscattering direction.

Fig. 4

Experimental data (squares) and MC simulations (line) for glucose-free turbid media in a cylindrical cuvette of $1\mathrm{cm}$ diameter. (a) Surviving linear polarization fraction as a function of detection angle for ${\mu }_s=20{\mathrm{cm}}^{-1}$ ; (b) surviving linear polarization fraction as a function of scattering coefficient at $\theta =0°$ .

Figure 5 shows the effects of increasing glucose (20 to $900\mathrm{mM}$ ) in moderately turbid media $({\mu }_s=30{\mathrm{cm}}^{-1}$ in glucose-free suspension) on polarization states of the emerging light detected at five different detection angles of $\theta =0$ , 45, 90, 135, and $155°$ . Figure 5a shows that optical rotation decreases with glucose concentration at all detection angles except in the forward direction. The observed decrease in $\alpha$ may indicate that the optical rotation due to scattering and the optical rotation due to glucose chirality are in the opposite directions. If former is greater than the latter, addition of glucose can cause a decrease in the net apparent optical rotation. The optical rotation change detected at $\theta =155°$ is most sensitive to glucose level change, yielding $\mathrm{\Delta }\alpha \sim -4.5°$ for examined concentration change. Selecting a slightly different backward detection angle of $\theta =135°$ yields a $\mathrm{\Delta }\alpha \sim -2.0°$ for the same glucose range. For comparison, the conventional detection geometry in the forward direction results in a much smaller $\mathrm{\Delta }\alpha$ of $\sim +0.8°$ . The data trends suggest that in the forward direction, glucose chirality is the main contributor to the observed optical activity; in other detection directions, the measured optical activity is dominated by scattering effects (as modulated by the addition of glucose, which (1) reduces scattering (2) is optically active). It is expected that there exists a detection angle at which the measured optical rotation has equal contributions from these reactive index matching and optical activity effects (as borne by the experimental results summarized in Fig. 7 below).

Fig. 5

Effects of glucose concentration (20 to $900\mathrm{mM}$ ) in moderately turbid media ( ${\mu }_s=30{\mathrm{cm}}^{-1}$ at $0\mathrm{M}$ glucose) on polarization states of emerging light detected at different detection geometries. (a) Optical rotation; (b) surviving linear polarization fraction; (c) normalized intensity. The symbols are data points; the lines are a guide for the eye.

Fig. 7

Glucose effects on optical rotation in turbid media, in order to examine the chirality and refractive index matching contributions. (a) Moderately turbid media $({\mu }_s\sim 30{\mathrm{cm}}^{-1})$ ; (b) highly turbid media $({\mu }_s\sim 100{\mathrm{cm}}^{-1})$ . The symbols are data points; the lines are a guide for the eye.

Figure 5b shows the change of corresponding surviving linear polarization fraction due to glucose for the same samples as in Fig. 5a. ${\beta }_{linear}$ increases with glucose concentration at all detection angles with $\mathrm{\Delta }\beta \sim +0.15$ in forward direction and $\mathrm{\Delta }\beta \sim +0.1$ at $\theta =155°$ . This change is most likely due to the glucose refractive index matching, which effectively reduces the sample turbidity, thus decreasing the depolarizing effects of multiple scattering. Even through the backward detection displays a slightly reduced $\mathrm{\Delta }{\beta }_{linear}$ and overall lower levels of surviving linear polarization, it is still an attractive detection geometry if both $\alpha$ and ${\beta }_{linear}$ are used as metrics of glucose concentration.

Figure 5c shows the change of normalized intensity $I$ in the presence of varying glucose amounts. As seen, the intensity is higher in the forward hemisphere, and glucose-induced changes are very much directions-dependent, being negligible for $\theta >90°$ and most pronounced in the forward direction. These polarization-independent changes in detected light intensity with varying glucose levels most likely originate from the index-matching effect of glucose, which reduces sample scattering properties as previously outlined.

Figure 6 is analogous to Fig. 5 but displays the results for the case of higher scattering ( ${\mu }_s=100{\mathrm{cm}}^{-1}$ in glucose-free suspension). Opposite to Fig. 5a, the optical rotation $\alpha$ in Fig. 6a decreases with glucose concentration at a forward detection direction and increases with glucose concentration at larger detection angles (135 and $155°$ ). The monotonically decreasing $\alpha$ in the forward direction indicates an overall dominance of the glucose refractive index matching effect that reduces scattering and the apparent optical rotation it causes in this high-scattering regime. The increase in $\alpha$ at larger angles implies that more complicated mechanisms are contributing to optical rotation. For example, the reduction in scattering due to glucose-induced refractive index matching effect can cause the observed increase in $\alpha$ at large scattering angles, by lengthening the effective pathlength over which the detected photons can accrue additional optical rotation. Figure 6b shows that ${\beta }_{linear}$ levels and their sensitivity to glucose changes are different in their dependence on $\theta$ as compared with optical rotation. The magnitude and increase of ${\beta }_{linear}$ caused by increasing glucose levels are both largest in backward directions. The intensity dependence of Fig. 6c shows that more photons escape the highly turbid medium in the backward hemisphere and the glucose-induced intensity reduction is the largest at $\theta =155°$ .

Fig. 6

Effects of glucose concentration (30 to $900\mathrm{mM}$ ) in highly turbid media ( ${\mu }_s=100{\mathrm{cm}}^{-1}$ at $0\mathrm{M}$ glucose) on polarization states of emerging light detected at different detection geometries. (a) Optical rotation; (b) surviving linear polarization fraction; (c) normalized intensity. The symbols are data points; the lines are a guide for the eye.

Figure 7a displays the relative importance of scattering-induced and optical activity-induced glucose contributions to the observed optical rotation in moderately turbid samples. As $\theta$ is increased from 5 to $30°$ in samples with ${\mu }_s\sim 30{\mathrm{cm}}^{-1}$ , two distinct regimes of optical rotation contribution are evident. Near the forward direction ( $\theta =5$ and $10°$ ), increasing the glucose concentration leads to nearly monotonic increase in measured optical rotation. Conversely, for detection angles of 25 and $30°$ , a decrease in optical rotation is seen. We interpret the former behavior as being dominated by optical activity, whereas the latter is indicative of scattering effect dominance as engendered by glucose refractive index matching. The $\theta =15°$ detection geometry represents an intermediate case of roughly equal contributions, at least for this sample geometry and this scattering coefficient $({\mu }_s\sim 30{\mathrm{cm}}^{-1})$ . An analogous graph for higher scattering $({\mu }_s\sim 100{\mathrm{cm}}^{-1})$ seen in Fig. 7b shows that the transition region occurs at much higher detection angles and also indicates overall higher optical rotation levels. It appears that $\alpha$ detected at $\theta =125°$ has roughly equal contributions from chirality and from index matching for the examined range of glucose concentration. Given the larger magnitude of the drop, we conclude that the refractive index matching effect is the stronger one in highly scattering media.

Glycerol comparison measurement results shown in Fig. 8 reveal that the same solution refractive index change (from 1.337 to 1.355) due to presence of optically active glucose (243 to $900\mathrm{mM}$ glucose concentration) and optically inactive glycerol (2.49 to 17.5%) yields similar amounts of optical rotation detected at $\theta =0$ , 45, and $135°$ in media with ${\mu }_s\sim 100{\mathrm{cm}}^{-1}$ . This implies that the refractive index matching effect dominates over optical activity in its contribution to optical rotation in highly turbid media, as is also suggested by Fig. 7b. A similar conclusion was reached by Hadley for media with lower scattering.²⁹

Fig. 8

Comparison measurements in highly turbid suspensions $({\mu }_s\sim 100{\mathrm{cm}}^{-1})$ containing glycerol or glucose. For the same amount of refractive index change caused by both optically active glucose and optically inactive glycerol, similar amounts of optical rotation in both forward and backward detection geometries are seen. For the range of index matching values shown, the scattering coefficient changes from $98.3{\mathrm{cm}}^{-1}$ $(n_{medium}=1.337)$ to $92.7{\mathrm{cm}}^{-1}$ $(n_{medium}=1.355)$ . The symbols are data points; the lines are a guide for the eye.

5.

Summary

We have demonstrated that the polarization parameters α, ${\beta }_{linear}$ and overall scattered intensity $I$ , simultaneously obtained from polarimetric measurements in different detection directions around a scattering sample, depend in a varied and complex way on a number of factors, including sample glucose concentration. For example, in scattering media with ${\mu }_s$ in the 30 to $100{\mathrm{cm}}^{-1}$ range, large changes in $\alpha$ and ${\beta }_{linear}$ were detected in the backward direction as glucose levels increased to $\sim 1\mathrm{M}$ . The three parameters ( $\alpha$ , ${\beta }_{linear}$ , and $I$ ) form a potentially attractive three-channel probe for glucose level monitoring in turbid media, although their dependence on sample properties and experimental geometry is complicated. The weak dependence of the derived optical rotation on the chiral nature of glucose in multiply scattering samples, while surprising and disappointing initially, may actually remove some of the difficulties of polarimetry caused by other chiral confounders in biological tissues. The range of validity of the observed trends is currently being explored in other chiral scattering systems. Further research will focus on the optimal detection geometry for maximizing sensitivity to glucose concentration levels closer to the physiological level. Other alternative detection geometries, such as out-of-plane detection, are beginning to be explored.³⁰ Expanding upon the presented methodology, the effects of media absorption, linear birefringence, and the possibility of multiwavelength (spectroscopic) polarimetry coupled with chemometric analysis are also under investigation.

6.

Conclusions

We have demonstrated the feasibility of studying the polarization states of scattered light from turbid media in different directions using a refined linear Stokes polarimeter. An extensive experimental study was performed, concentrating on sample (turbidity, chirality) and system (detection angle) variables. Selected validation with polarization-sensitive MC simulation model was performed. It is found that in turbid media, the scattering process itself can cause a large optical rotation, even in the absence of glucose. In general, the magnitude of the optical rotation is strongly dependent on sample turbidity, detection angle, and glucose levels. For example, in highly turbid media $({\mu }_s=100{\mathrm{cm}}^{-1})$ , optical rotation can reach up to $20°$ levels and can exhibit large changes ( $\mathrm{\Delta }\alpha$ up to 10 to $15°$ ) over the $\sim 1\mathrm{M}$ glucose concentration range. The surviving linear polarization fraction decreases with increasing turbidity and also with the detection angle for low and moderately turbid media. Its angular dependence becomes less pronounced with increasing turbidity and, in fact, reverses for highly turbid samples. It is demonstrated that the three derived polarization parameters ( $\alpha ,{\beta }_{\mathit{linear}}$ , and $I$ ) depend on glucose levels, albeit in a complicated way. The sensitivity to glucose is strongly detection-angle–dependent, suggesting that the backward detection angles may be preferable for glucose sensing in highly turbid media, such as biological tissue, although the complicated angular and turbidity dependence of optical rotation $\alpha$ may make this metric less glucose-specific. Comparison measurements in turbid media containing (optically inactive) glycerol indicate that the optical rotation dependence on (optically active) glucose concentration is not due to its chiral nature, but is mainly caused by the refractive index matching effect.