1.

Introduction

In recent years, there has been an increased interest in using nonharmful light for studying and diagnosing human pathologies because of the noninvasive nature of the light–tissue interaction. One medical condition of particular clinical importance is diabetes, with its associated need for monitoring blood glucose. In order to overcome the discomfort to the patient caused by current commercial devices that require small blood samples, noninvasive glucose monitoring techniques have been investigated by several research groups.¹ Polarimetric measurements of optical activity that is due to glucose in the aqueous humor of the eye, for instance, have shown great promise recently since the eye is the only transparent tissue in the body. However, birefringence of the cornea and multiple reflections in its layered geometry make measurements and their interpretation difficult.² For this and other reasons, measurements performed at other sites on the body should be investigated, including areas that are optically thick.

However, while the propagation of light in a transparent medium such as the eye is well described by Maxwell’s equation, the propagation of light in tissue is more complicated and is better modeled as a combination of scattering and bulk dielectric propagation. Because of the high number of scattering events in tissues, it is natural at first to disregard the polarization of light in models since the scatterers randomize—and eventually completely scramble—the polarization of light.³ Light propagation in tissue is commonly modeled within the framework of light transport theory or diffusion theory, assuming that the particle nature of light is sufficient to describe all the relevant properties of the measurements.³ Monte Carlo calculations, together with some aspects of Mie theory, have become standard for predicting and modeling experiments.⁴

Recently, however, the polarization of light has been used to extract quantitative information from the optically thick tissues with which it interacts. Indeed, although the degree of polarization is significantly lowered after propagating through thick turbid media, often it is not completely destroyed.⁵ Polarization is well suited for measuring birefringence and optical activity, which affect, respectively, the ellipticity and the orientation of the polarization of the incident beam. For instance, it has been used in tissue birefringence measurement,⁶ polarization-sensitive optical coherence tomography (OCT) measurements,^{7 8} and imaging of polarization-sensitive skin pathology⁹ to yield more information about the tissue than is provided by intensity data alone. In the case of noninvasive glucose monitoring, it has been shown that the presence of glucose can be detected in turbid media via index-matching effects that lower the tissue scattering coefficient.^{10 11} However, it is difficult to measure absolute concentrations of the glucose when relying on a change in the scattering coefficient since other factors can affect the scattering (tissue inhomogeneity, for example). Other methods based on optical activity¹² effects have also been used to quantify the presence of glucose in turbid media, and offer a more direct method for glucose quantification since the optical rotation is linear with the glucose concentration. However, the small physiological blood glucose concentration (typically 10 mM) yields a very small optical rotation, on the order of millidegrees through 1 cm of tissue. Since many polarimetric measurements rely on taking the difference of two intensity readings at two orthogonal polarizations, measurements of small polarimetric signals in the presence of a large depolarized background are difficult to perform, especially in the presence of significant low-frequency laser noise.

In the past, we have used a technique that relies on taking multiple measurements at different analyzer optic orientations and performing an analytical fitting procedure based on the expected dependence of the signal on the orientation to obtain precise values of the induced optical rotation and depolarizing parameters.^{12 13} In this paper we demonstrate how to perform sensitive polarimetric measurements in highly scattering media (μ_s≈30 cm ⁻¹) without the need for multiple measurements or user adjustments. The key to the method is to perform two measurements simultaneously and electronically subtract them before doing lock-in detection, thereby canceling any common-mode noise at the receivers (e.g., laser intensity or any polarization-independent noise). The laser used in the experiments is not stabilized in intensity, but the methodology developed here is general and would cancel any remaining intensity noise in a setup that makes use of an intensity-stabilized laser (these lasers typically have a 0.01 noise root mean square in intensity).

There are experimental techniques^{14 15} that allow the complete characterization of the scattering matrix of a sample without any moving parts. The technique presented here, on the other hand, does not attempt to fully characterize the scattering matrix. Instead, it concentrates on a few elements that are important for characterizing optical activity. The strength of this methodology lies in its ability to measure small elements of the scattering matrix. To compare our measurements with predictions, we use a Monte Carlo model that includes polarization effects based on publications by other groups.^{16 17 18}

This paper proceeds as follows. First, the formalism used to model the experiments is reviewed, together with the definitions of the relevant parameters. This is used to introduce the rationale for the balanced detection method. The experimental setup for the measurements of glucose concentrations at physiologically relevant levels in turbid media is described, and the methodology for measurements and analysis is presented. Finally, the effects of turbidity and glucose on light polarization are measured and shown to be in agreement with published experimental values and polarization-sensitive Monte Carlo predictions.

2.

Formalism

It is common to describe polarized light using Stokes parameters. The state of polarization of the light, with respect to a chosen set of orthonormal axes e^_∥ and e^_⊥, is given by a Stokes vector S of the form:

Figure 1

The complex electric field is decomposed into two perpendicular components E_∥ and E_⊥. The reference axis e^_∥ (e^_⊥) refers to a direction parallel (perpendicular) to the scattering plane, which in the situation of interest here is horizontal (vertical) in the laboratory frame. The ellipsometric parameter γ (the orientation of the major axis of the polarization ellipse with respect to e^_∥) is shown in the figure.

To streamline the mathematics, it is convenient to define “detector” operators. When one applies the detector operator to a Stokes vector S, the scalar value of that particular Stokes parameter is obtained. Mathematically, detector operators for the first three Stokes parameters are defined as

Upon propagation through optical elements or scattering with an object, the Stokes vector of a beam is transformed. This can be described with the multiplication of the Stokes vector by Mueller matrices representing the interactions. Mueller matrices for the propagation through standard optical elements [polarizer P(θ), wave plate, etc.] are known and will not be repeated here.¹⁹ When a beam encounters a scatterer, the incident field will be redistributed in all directions, depending on the properties of the scatterer, the surrounding medium, and the illumination wavelength. A scattering direction is completely determined with two angles. The scattering angle θ is the angle between the incident propagation direction z^ and the new direction of propagation of the field z^^′. The plane spanned by z^ and z^^′ is called the scattering plane, and the azimuthal angle ϕ is the orientation of e^_∥ with respect to that scattering plane. In general, one represents a multiply scattering medium by an arbitrary matrix M:

In order to determine the optical activity of a turbid medium, we use two important ratios: m₃₂/m₂₂ and m₂₂/m₁₁. The ratio m₃₂/m₂₂ is a direct measurement of optical rotation α, which can be understood from the following argument. Using Eqs. (7) and (8), the orientation for a horizontally polarized beam with the incident Stokes vector S ₀ is

The object of polarimetry is to experimentally determine one or more elements of the scattering matrix. This is accomplished by measuring the scattered Stokes vector for a known incident Stokes vector to calculate the matrix elements involved in the transformation.^{5 23} A difficult task in polarimetry is the measurement of very small elements of the Stokes vector in the presence of large intensity noise in the light beam. When the noise is arising from the interaction with a medium, one must resort to statistical averaging of several measurements. However, when the noise originates from the laser beam itself, one can suppress it with an appropriate experimental design. For instance, let the intensity of the laser source be

3.

Materials and Methods

3.2.

Experimental Setup

The experimental setup shown schematically in Fig. 2 consists of an unpolarized laser producing approximately 10 mW of power at 633 nm, or about 100 mW at 635 nm. The beam goes through a mechanical chopper, operating at a frequency f_c=(2π)⁻¹ω_c=200 Hz, then through a polarizer P₁(θ₁) oriented with its axis making an angle θ₁ with the horizontal, and a photoelastic modulator (PEM, Hinds, PEM-90) modulating its retardation δ(t) at frequency f_p=(2π)⁻¹ω_p=50 kHz according to a sinusoidal function δ(t)=δ_o sin ω_pt with its axis horizontal. The amplitude of the retardation modulation δ_o is user specified. The beam then goes through a liquid sample contained in a 1-cm-thick spectroscopic glass cuvette. The light scattered in the forward direction is then apertured with two irises with a 4-mm diameter and separated by 10 cm, and sent to a 2.5-cm-wide polarization splitting cube with a 10⁴:1 polarization extinction ratio.

Figure 2

Schematic diagram of the experimental setup used for the measurements. Chop., mechanical chopper; P₁, polarizer; PEM, photoelastic modulator; A₁, A₂, apertures; PC, polarizing beamsplitter cube; L _x , L _y , lenses; D _x , D _y , photodetectors. f_c and f_p are the modulation frequencies for the mechanical chopper and PEM, respectively.

The two beams, linearly polarized along the laboratory vertical and horizontal, are sent to two separate mirrors and focused onto the two photodiodes of a balanced detector (New Focus, Nirvana 2017). The output of the balanced detector is the real-time difference of the measured intensities at both photodiodes, with a 3-dB bandwidth of approximately 100 kHz. A lock-in amplifier (Stanford Research, SR-530), which measures the root mean square (r.m.s.) amplitude of a given harmonic, is used for measuring the 200-Hz signal at the first harmonic of the frequency f_c (while the PEM is turned off) and the 100-kHz signal at the second harmonic of the frequency f_p (while the mechanical chopper is turned off). The combined frequency response function of the photodetector electronics and the lock-in amplifier is identified as F(f ).

4.

Results and Discussion

From the measurements of Q_fc and I_fc , and with the use of Eq. (31), the surviving linear polarization fraction as a function of glucose concentration is obtained and plotted in Fig. 3, with values ranging from 0.77 to 0.89 for the turbid suspensions (μ_s≈30 cm ⁻¹). As previously reported,¹³ the change in the surviving linear polarization fraction with glucose concentration is due to the increase in the refractive index induced by the dissolved glucose in solution. The index of refraction of water increases by 0.027 per molar of glucose.^{10 11} This increase in background index reduces the scattering coefficient μ_s of the suspension^{10 11} and therefore increases polarization retention. A Monte Carlo calculation of the surviving linear polarization fraction as a function of scattering coefficient μ_s is shown in Fig. 4. The predicted β_L that matches the measured β_L of Fig. 3 corresponds to a scattering coefficient in the range of 45 to 60 cm ⁻¹, which is somewhat higher than, but in general qualitative agreement with the conditions of the experiments (where μ_s is estimated to be approximately 27 cm ⁻¹) .

Figure 3

Surviving linear polarization fraction measured in the forward direction as a function of glucose concentration after propagating through a 1-cm cuvette containing an optically active clear solution (round symbols) and an optically active turbid solution with 0.076 microspheres (square symbols). The latter correspond to scattering coefficients ranging from 28 to 26 cm ⁻¹. The error bars, estimated from repeated measurements, are due to the interference of the multiple reflections off the cuvette walls and the slightly different cuvette position at each measurement.

Figure 4

Calculated surviving linear polarization fraction (β_L) in the forward direction as a function of scattering coefficient from Monte Carlo calculations for a 1-cm cuvette, microspheres of radius 0.7 μm in water, λ=633 nm, and no glucose.

Measurements of Q at 2 f_p and f_c are performed and Eqs. (16) and (30) are used to extract the optical rotation of clear and turbid solutions. The results are shown in Fig. 5, along with a linear fit to the data. First, in the case of a clear solution in a 1-cm cuvette and with a laser wavelength of 633 nm, the optical rotation is determined to be 0.82 °M ⁻¹ cm ⁻¹, or a specific rotation of [α]₆₃₃ ^25°=(45.5±0.1) deg ml g ⁻¹ dm ⁻¹, at a temperature of 25 °C. This is in agreement with previously estimated values and the expected dependence of the specific rotation on wavelength,²⁵ and is to our knowledge the first measurement made at that wavelength.

Figure 5

Linear (a) and logarithmic (b) plots of optical rotation α as a function of D-glucose concentration in a clear (open circles) and turbid (solid square) optically active solution of glucose with 0.076 microspheres. The solid lines are linear fits to the data. For the turbid suspensions, the scattering coefficient ranges from 28 to 26 cm ⁻¹. The error bars are smaller than the size of the symbols except for the smallest concentrations.

Second, in the case of turbid suspensions, reliable values and linearity of the rotation as a function of glucose concentration are also observed for concentrations around a physiological range as low as 10 mM, which is a notable improvement over our previous results.¹² We note that although the scattering coefficient μ_s and the surviving polarization fraction change slightly with glucose concentration, the linearity of the optical rotation with respect to the glucose concentration is not affected. However, the measured optical rotation is smaller than in the case of a clear solution by approximately 13, with a slope of 0.72 deg M ⁻¹ cm ⁻¹. This appears to contradict the intuitive argument that in a highly scattering medium photons travel a longer distance than they would in a clear solution before exiting the sample, and therefore they should contribute a larger amount to macroscopic rotation.^{12 26} However, a Monte Carlo calculation suggests that this argument is partially right. Although the photons do travel a longer distance, their polarization is also scrambled more and their contribution to the final macroscopic rotation is smaller if μ_s is significant and if the intensity is integrated over a significant area at the exit face.

Figure 6 shows the calculated optical rotation α of a 0.9 M solution of glucose (an optical rotation of 0.74 deg cm ⁻¹ in a clear medium) using the definition of Eq. (8), as a function of the β_L for a detected area 5 mm in diameter at the exit face of the cuvette and an acceptance angle of 100 mrad, corresponding to our experiments. It can be seen that when the surviving linear polarization fraction is large β_L>0.9 (corresponding to small scattering coefficients), there is indeed an increase in the measured rotation compared with a clear solution. However, for a smaller β_L (and larger scattering coefficients), the measured rotation is smaller. In the conditions of our experiments, where β_L≈0.80 to 0.85, the expected rotation is smaller than in a clear solution by about 10 to 20, as was measured in our experiments. This is a consequence of the fact that although the average path length of the photons increases with μ_s, their contribution to the rotation decreases faster with μ_s. When all the photons are integrated over an area 5 mm in diameter and with a broad acceptance angle, and Eq. (8) is used, the rotation appears smaller. On the other hand, when a small acceptance angle and a small area are used (i.e., approaching the ideal forward-scattering direction) the range over which the rotation is larger than in clear solutions extends to larger scattering coefficients (Monte Carlo results not shown). We note that although β_L changes slightly with glucose concentration, the change is small enough that the linearity of optical rotation as a function of glucose concentration is not affected enough for the present setup to resolve it. The current limitation on the measurements is the low sensitivity of the silicon photodiodes used in the balanced detector and the low electronic signal obtained.

Figure 6

Monte Carlo calculation of the expected measured rotation as calculated from Eq. (8) in a 1-cm cuvette for a solution with an optical activity of 0.74 deg cm ⁻¹ as a function of β_L. The solid line is a guide for the eye. The horizontal dashed line is the expected rotation in a clear solution; the Monte Carlo results predict that measurements in a turbid solution of β_L≈0.8 yield a smaller rotation angle by about 15.

5.

Conclusion

We have demonstrated high-precision, low-noise polarimetric measurements in turbid media using balanced detection. The technique allows the rejection of the noise common to the two polarized intensity measurements necessary to determine Stokes parameters, which increases the precision of the measurements. The specific rotation of D-glucose in a clear solution at 633 nm was determined to be [α]₆₃₃ ^25°=45.5 deg ml g ⁻¹ dm ⁻¹ at room temperature. The surviving linear polarization fractions in chiral (with glucose) and achiral (no glucose) turbid suspensions have been measured in the forward direction and are in reasonable agreement with Monte Carlo calculations. The measurements of optical rotation in turbid suspensions of microspheres demonstrate the possibility of determining glucose concentrations down to levels near 10 mM in a highly scattering and depolarizing medium made of polystyrene spheres. Measurements of optical rotation that is due to glucose in multiply-scattering phantom solutions when the depolarization is significant have been shown to yield slightly smaller values than in clear solutions when intensities are integrated over large areas; this is supported by Monte Carlo calculations.