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Abstract
The effects of scatterer size and scattering coefficient on backscattered linearly and circularly polarized light are investigated through Stokes polarimetry. High-SNR polarization modulation/synchronous detection measurements are corroborated by polarization-sensitive Monte Carlo simulations. Circular degree of polarization (DOP) is found to be sensitive to scatterer size, but is equivocal at times due to helicity flipping effects; linear DOP appears to be mostly dependent on the medium scattering coefficient. We exploit these trends to generate a DOPC - DOPL response surface which clusters turbid samples based on these medium properties. This work may prove useful in biomedicine, for example in noninvasive assessment of epithelial precancer progression.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Turbid media scatter light in interesting and characteristic ways that can convey information about its properties. When polarized light encounters a heterogeneous medium, the primary modulation mechanism is polarization decoherence (commonly referred to as depolarization) whereby the collectively shared phases and amplitudes of the incident photons’ electric field components are decorrelated and randomized during scattering [1]. Additionally, when polarized light is backscattered from a medium, another modulation mechanism becomes prominent, that of orthogonal flipping of the incident polarization orientation (e.g., right-circular polarization converts to left-circular polarization). Measuring backscattered light is integral for polarized light assessment of tissue in vivo [2] due mainly to the significantly limited polarimetric sampling depth (a few millimeters at most [3]) which is not conducive for transmission through bulk tissue. The mechanisms of polarization decoherence and orientational flipping can be indicative of a medium’s scattering coefficient (essentially its turbidity) and scatterer size distribution, thus enabling backscattered polarized light to probe these two biophysically important sample properties. For example, during epithelial precancer-to-cancer progression, changes occur to both tissue turbidity and average scatterer size as cells undergo morphological changes to their nuclear size (pleomorphism) and concentration (cellular proliferation) [4–7]. Thus, backscattered polarized light may be used to noninvasively detect epithelial precancers in vivo, an improvement to the current invasive and subjective ex-vivo assessment by tissue biopsy [8,9]. Furthermore, polarimetric optical diagnostics of scatterer properties may also benefit photodynamic therapy, a variant of which employs photosensitive nanoparticles to treat cancers [10], where there is a need for assessing photosensitizer concentration and aggregation [11]. Therefore, a noninvasive in-vivo ability to assess scatterer size distributions and scattering coefficient may prove important in biomedicine, and polarized light is promising for this task through its characteristic polarization responses, particularly in the biomedically important backscattering detection geometry.
Prior to in-vivo biomedicine however, many fundamental and practical aspects of the enabling polarimetric methodology must be assessed and optimized. In this study, we thus rigorously examine the polarization responses of backscattered linearly and circularly polarized light in a model system composed of uniform-sized spherical microparticles suspended in a liquid, simulating cell nuclei in tissue and simplifying the light interaction dynamics. Broadly speaking, back-scattered light from such a suspension of particles consists of two subpopulations of photons: type-1 – those that have undergone a reflection-like backscattering event by a single scatterer (large scattering angle, possibly approaching 180°) at some depth in the medium, and type-2 – those that have been forward-scattered at small scattering angles multiple times and eventually redirected into the backwards hemisphere direction (i.e., no single reflection-like scattering event) [12–16]. Importantly, polarized light encodes information about both of these backscattering routes through polarization decoherence and orientational flipping mechanisms. Specifically, circularly polarized light undergoes a helicity flip after a reflection-like event (type-1) whereby its initial and scattered polarization states are related by mirror symmetry (e.g., incident right circularly polarized light emerges as left circularly polarized light). In contrast, circular polarization helicity is preserved after forward scattering (type-2 subpopulation) and possibly measurable due to the resultant modest circular depolarization effects [12,17]. Linear states, however, effectively depolarize with the randomization of their photon’s directions [18]; hence, the multiply scattered redirected backscattering (type-2) sub-population will be greatly depolarized compared to the minimally scattered type-1 photons. In addition to geometric (randomization) processes, so-called “dynamical” depolarization effects have been explored in detail; the reader may refer to [19–21] for largely analytical treatment of these problems. In general, these considerations suggest that one can gain insight into the complexities of the light-medium interactions and by analyzing the behaviour of backscattered linearly and circularly polarized light.
Relating these interactions to medium properties, we note that both scatterer size and scattering coefficient will influence the relative contributions of type-1 and type-2 backscattered photon interactions. Particles large compared to the interrogating light’s wavelength (Mie regime) scatter predominantly in the forward direction, thus increasing the relative contribution of type-2 backscattered photons; this may lead to preferential retention of helicity-maintaining circularly polarized light fraction. Conversely, smaller sized scatterers (Rayleigh regime), with their more symmetric backward-forward scattering patterns, increase the relative probability of directly reflected type-1 photons; this may favour linear polarization preservation over the helicity-flipped circular subpopulation. The influence of scattering coefficient is somewhat less clear, other than the general expectation that degree of polarization (DOP) should overall decrease with increasing turbidity [22]. For example, it has been suggested that low turbidity (such as that of eye) backscatters mainly type-1 photons and high turbidity (such as that of tissue epithelia) backscatters mainly type-2 photons [13]; however, these effects should be further explored. Overall then, a careful rigorous study of backscattered polarization effects in media with well-defined scatterer sizes and scattering coefficients will provide insight into the polarimetric probing capabilities for assessing these two biophysically important parameters.
The detailed polarization responses from turbid media were first observed in a seminal 1989 study by MacKintosh et al. [12] who investigated the effects of scatterer size on linearly and circularly polarized light and found a stronger circular “polarization memory” for Mie (larger) scatterers and stronger linear polarization memory for Rayleigh (smaller) scatterers. This study sparked a strong research interest in linearly and circularly polarized light and their interactions with simple scattering media (particle suspensions) [14–19,23–28], including on the question of which polarization state is better preserved under various experimental conditions (such as medium properties and measurement geometry). These have largely supported the initial observations of linear and circular polarization retention with respect to scatterer size from the MacKintosh study. However, there have also been contrary observations in backscattered light from more complex media such as tissue phantoms with roughened surfaces, polydispersed suspensions of aspherical particles, and actual biological tissues [29–37]; here the optical properties are less understood which make it difficult to determine the underlying reason(s) for these contradictory results. Effects of speckle, absorption, polydispersity [38], or preferential interactions with smaller scatterers have been invoked; however, as noted recently [33], this issue deserves further investigation. There is thus a strong need to better understand backscattered polarized light to gain clarity on the question of linear versus circular polarization preservation in media of different scatterer size distributions and turbidity, particularly to facilitate the development of polarimetric techniques that aim to utilize these polarization states on bulk tissue such as cancer detection and assessment [33,34,39] and optical sampling depth sensitivity [40–42].
Here we attempt to gain insight on these issues by studying backscattered polarized light within the context of the type-1 and type-2 interaction media. This model enables us to better understand the light-medium interactions based on the characteristic polarization modulations discussed above, namely, the helicity response of circularly polarized light (flipped for type-1 and preserved for type-2) and linear DOP (non-zero for type-1 and effectively zero for type-2). Furthermore, since backscattering experiments typically measure differences between polarization intensities of orthogonal states, an ambiguity arises when the resultant circular DOP is low. Is it because the light field is ‘truly’ random and depolarized, or is significant polarization coherence present but masked by the presence of roughly similar amounts of orthogonal states (helicity-preserved and helicity-flipped)? Resolving this interesting ambiguity is also worthy of further investigation and may help provide insight on the much-discussed question of linear versus circular polarization preservation.
In this study, we thus investigate the responses of backscattered linearly and circularly polarized light in controlled turbid media of varying scatterer sizes and scattering coefficients to better understand their relationships with these medium properties, while also providing insight on the issue of ambiguous circular polarization Stokes measurements. The findings enable a novel polarimetric methodology to discriminate turbid media simultaneously by scatterer size and scattering coefficient. To perform the measurements, we employ a practical high-SNR experimental system with dynamic polarization modulation and phase-sensitive synchronous detection, which requires only a single input polarization configuration to measure both linear and circular responses. We also present simulation results from a polarization-sensitive Monte Carlo model to validate the experimental measurements, help resolve the low DOPC ambiguity, and provide additional insights (e.g., scattering interaction numbers within the scattering volume).
2. Methods
2.1 Turbid media samples
The turbid samples were aqueous suspensions of monodispersed polystyrene microspheres (Bangs Laboratories, Inc) of different sizes and scattering coefficients. Three sphere diameters were used – 0.21 μm, 0.42 μm, and 0.96 μm – which at λ = 635 nm yield scattering efficiencies Q and scattering anisotropy factors g as (0.08, 0.35), (0.52, 0.74) and (2.45, 0.92), respectively, as calculated by Mie theory [43]. The spheres had a refractive index of $n = 1.59 $. Concentrations were varied to yield scattering coefficients ranging from ${\mu _s} = $ 5 cm-1 to ${\mu _s} = 200$ cm-1. Aqueous suspensions were placed into a plastic cuvette with a 1 × 1 cm silica optical window.
2.2 Experimental system and signal analysis
Figure 1 shows the schematic for the experimental system. The light source was a continuous-wave diode laser operating at λ = 635 nm. The light was modulated sequentially by a mechanical chopper at 207 Hz, linear polarizer (P1), and photoelastic modulator (PEM) before impinging on the sample. We can describe the polarization states of light in the form of Stokes vectors [44],
Fig. 1. Schematic of the experimental system used to measure the Stokes parameters I, U, and V from scattering samples. P1 is a linear polarizer with its optical axis oriented at 45° counterclockwise from the vertical when viewing from the incident side, the birefringent axis of the PEM is oriented vertically, R is a quarter wave retarder with its fast axis oriented vertically and P2 is a linear polarizer with its optical axis oriented 45° clockwise from the vertical. R was placed in the beam path to form a circular analyzer and removed to expose the linear analyzer. To measure the incident light properties, the sample was removed and the detection axis was swivelled parallel to the beam line.
The state of light after transmitting through P1 is represented as
The retardation of the PEM, $\delta $, varies sinusoidally with time at frequency $f = \omega /2\pi $,
where ${\delta _o}$ is the user-selected peak retardance. The $\textrm{cos}\; \delta $ and $\textrm{sin}\; \delta $ terms can then be expanded in terms of Bessel functions,To simplify analysis, we set the peak PEM retardance ${\delta _o}$ to 2.405 rad where ${J_0}({{\delta_o}} )= 0$ (see Eqn. (5)). With this simplification, the PEM-transmitted light oscillated between right- and left-circularly polarized light at the fundamental frequency $\omega $ (with amplitude 2J1) and between $+ 45$° and $- 45$° linearly polarized light at the harmonic frequency $2\omega $ (with amplitude 2J2).
The light is then scattered in the turbid sample, represented by the product of the Mueller matrix of a depolarizer and ${{\boldsymbol S}_2}$:
The light was measured before scattering and after scattering by swiveling the detection axis parallel to the beam line in the former case and to 25° with respect to the incident beam in the latter case (25° was the closest-to-180° angle permitted by the optics). An off-axis backscatter detection geometry was chosen as an alternative to exact backscattering to avoid the use of a beam splitter which both complicates the Mueller matrix mathematics and reduces the measurable intensity [22]. The details of Stokes measurements using photoelastic modulation and synchronous detection are available in previous rigorous publications [45–49]. Briefly, the Stokes parameters $\textrm{I}$, $\textrm{U}$, and $\textrm{V}$ are measured using a lock-in amplifier [50] by detecting the photodiode signal synchronously with the chopper reference signal (DC intensity, or I), the fundamental ($\mathrm{\omega }$) PEM reference signal, and the harmonic (2$\mathrm{\omega }$) PEM reference signal, respectively. The fractional linear polarization, $\textrm{U}/\textrm{I}$, and fractional circular polarization, $\textrm{V}/\textrm{I}$, of the scattered light are normalized by that of the incident light to obtain DOPL and DOPC,
Notably, the above-described method whereby the scattered polarization fractions are normalized by the incident polarization fractions, enabled the measurement of both linear and circular DOPs using a fixed PEM-enabled incidence. This simplified the measurement process and reduced the chance of experimental error. In this direct experimental approach, the diffusive component of the backscattered light is gated out, leading to high-SNR detection of primarily ballistic (type-1) and snake (type-2) photons which are important in retaining physical information from interrogated media. As shown in several previous studies [14,51,52], such continuous-wave phase modulation polarization gating rather than temporal polarization gating obviates the need for complex and expensive ultrashort-pulsed lasers and rapid-response detectors such as streak cameras [53–55]. To make the measurement system even faster and more robust, the quarter wave retarder on the detection side can be replaced by another PEM to enable fully no-moving-parts measurement of U and V [49]. Such a robust and rapid two-PEM based experimental system without mechanically moving parts would be well suited for the challenges of reflection-mode bulk tissue in-vivo polarimetry, furnishing excellent phase-sensitive detection sensitivity [56] to address severe tissue depolarization and enables rapid signal acquisition times to minimize motion artifacts [3].
2.3 Monte Carlo model
We employed our previously developed, validated and publicly available Monte Carlo model [57] to simulate polarized light propagation in multiply scattering media. The model corroborated the experimental DOP measurements and tracked the average number of scattering events per detected photon (including both polarization-retaining and depolarized populations), thus offering insights into the backscattering mechanisms and facilitating data interpretations.
The relevant aspects of the polarization-sensitive Monte Carlo model are briefly summarized here, with detailed description, validation and usage previously published [58]. The model tracks the position, direction, and polarization via Stokes vectors of photons as they propagate through a scattering medium and sums the photon histories to determine the macroscopic polarization properties of the scattered light. Each photon takes on a new propagation direction after a scattering event by determining the scattering angle through statistical sampling. The Stokes vector of each photon is transformed via the Mueller matrix of each scattering event as calculated by Mie theory. Upon encountering a sample boundary, reflection or transmission probabilities are calculated from the polarization-dependent Fresnel coefficients; reflected photons continue their propagation in the medium with their properties transformed accordingly. A fraction of scattered photons escapes the medium and impinges on a detector element at a specific measurement angle (25° off the retroreflection direction in our study) and surface area (5 cm2 in our study), yielding a summation of the detected photon statistics.
Each simulation was initialized with parameters matching the experimental setup which describe the detection geometry, the medium properties (scatterer size and refractive index, host medium refractive index, scattering coefficient, and medium dimensions), and incident light properties (polarization, number of photons (∼108), and wavelength). The incident state of light was initialized with the same Stokes parameters as the experimentally measured excitation light onto the sample. A typical simulation took ∼20 minutes on a laptop PC with an Intel Core i5-7200U 2.71 GHz processor. Stokes DOPL and DOPC fractions were calculated same as in the experimental methodology, by normalizing the linear and circular polarization fractions of the scattered light by that of the incident light.
3. Results and discussion
We first investigate the effects of scatterer size on DOPL and DOPC. Plotted along the left vertical axis in Fig. 2 are the simulated and experimental DOPL and DOPC values for monodispersed suspensions with sphere diameters ranging from 0.21 µm to 0.96 µm, all with the scattering coefficient kept constant at µs $= 100$ cm-1 (mean free path (MFP) $= 1/{\mu _s} = 0.01$ cm). The scattering anisotropy of each mean sphere diameter is plotted along the top horizontal axis. Experimental measurements were performed for suspensions of 0.21 µm, 0.42 µm, and 0.96 µm diameter spheres; d = 0.6 and 0.8 µm were added in the simulations. The experimentally measured DOP curves are corroborated by the Monte Carlo simulation results, showing strong agreement and lending credence to the observed trends. The red curve on the right vertical axis of Fig. 2 displays the corresponding Monte Carlo calculated average number of scattering events detected in the backscattering direction for the different sphere sizes at ${\mathrm{\mu }_\textrm{s}} = 100$ cm-1. As seen, the average number of scattering events increases with sphere diameter, which can be attributed to the stronger forward scattering of larger spheres, leading to higher relative contributions of the multiply scattered type-2 photons. As such, DOPC also increases with sphere diameter since type-2 photons are helicity-preserved and modestly circularly depolarized with increasing scatterer size [12]. Conversely, the type-2 photons are highly linearly depolarized [18] and thus an overall low and slightly decreasing response is observed for DOPL with particle size, and with far less dynamic range than DOPC. As expected, the DOPL signal is maximum for smallest examined microspheres (d = 0.21 µm), which is likely due to the higher preponderance of minimally scattered type-1 photons (g = 0.35) [12]. Interestingly, DOPL remains non-zero at this high turbidity level for all scatterer sizes; this phenomenon likely arises due to ever-present minimally scattered type-1 photons from shallow layers of the sample [22,59]. Overall we note that DOPC shows greater sensitivity to medium scatterer size compared to DOPL, due to its greater dynamic range and more pronounced variation.
Fig. 2. Simulated and experimentally measured linear DOPL and circular DOPC fractions in monodispersed polystyrene suspensions with microsphere diameters ranging from 0.21 µm to 0.96 µm. Each suspension had a scattering coefficient ${\mathrm{\mu }_\textrm{s}} = 100$ cm-1 (MFP $= 1/{\mathrm{\mu }_\textrm{s}} = 0.01$ cm). The corresponding scattering anisotropy for each microsphere diameter is plotted along the top horizontal axis (not to scale). Experimental results are presented for suspensions of 0.21 µm, 0.42 µm, and 0.96 µm diameter spheres; Monte Carlo simulations also show d = 0.6 µm and 0.8 µm results. The red symbols and the right vertical axis show the average number of scattering events per detected photon for each simulation run. Solid symbols = experimental results; hollow symbols = Monte Carlo simulation results; lines = guide for the eye.
However, we also note that the trendline of DOPC intersects zero at approximately d = 0.3 µm (g = 0.65), which can be interpreted as either (a) circular decoherence or (b) a condition where the helicity-preserved intensity roughly equals the helicity-flipped intensity whereby the corresponding $\textrm{V} = {\textrm{I}_\textrm{R}} - {\textrm{I}_\textrm{L}}$ measurement yields ∼ zero. We posit that the latter is the true mechanism, based on (a) the overall continuity of the DOPC curve, (b) the non-zero value of DOPL for sphere diameter of 0.3 µm which indicates that the minimally scattered and polarization preserving type-1 photons are present, and (c) some degree of backscattered polarization will theoretically always remain due to the aforementioned shallow-layer backscattering [60]; thus, total decoherence of circular polarization is unlikely despite DOPC ∼ 0. Furthermore, this ambiguity was directly observed, although not pointed out, by MacKintosh et al. [12] who measured separate signals for left- and right-circular polarization intensities of backscattered light from monodispersed polystyrene microsphere suspensions and showed that the two signals take on ∼equal non-zero magnitudes at a certain microsphere concentration and average diameter, which would make the Stokes V value = 0 even though considerable circular polarization coherence remained. This ambiguity must be kept in mind when designing / interpreting polarization studies in turbid media, in that circular DOP may appear to be ∼ 0 for certain medium properties (specifically scatterer size), despite there being significant underlying circular polarization preservation via two separate but opposite-helicity subpopulations. One useful means to combat this ambiguity would be to quantify and retain the orthogonal polarization intensities, instead of automatically subtracting them, thus preserving rather than degrading the circular polarization information. We note that some previous studies may have missed this mathematical caveat with circular depolarization. For example, Louie et al. [33] have recently reported stronger depolarization of circularly than linearly polarized light in tissue phantoms and skin lesions in backscattered Stokes DOP measurements. In fact, these observations may or may not be due to strong circular polarization decoherence, but could also stem from there being closely valued helicity-preserved and helicity-flipped intensities (for example, the tissue phantoms used in their study had scattering anisotropies ranging from g = 0.69 to g = 0.82, close to g = 0.65 where DOPC ∼ 0 under our experimental conditions). Similar considerations may need to be invoked in the complex and often conflicting debates of whether linear or circular polarization states are better preserved in particulate phantom media and in tissues.
The increase in the average number of scattering events N with microsphere diameter implies the corresponding increase in the total penetration depth D of the light, estimated here as D ∼ N * MFP * g / 2 for retroreflection geometry. This however represents the average sampling depth of the entire detected light field, and not specifically the polarization-maintaining fraction, as the relatively large values of N reported here include all detected photons; the average sampling depths of the polarization-preserving fractions are likely somewhat lower [61]. It has also been shown that linearly and circularly polarized light fractions exhibit different depth sensitivities, thus suggesting polarization ellipticity as a tuning mechanism for depth selectivity [40–42]. The results in Fig. 2 show that such an approach must consider the scattering anisotropy of the interrogated medium to first determine the difference in polarization retention rates between both polarization states which would in turn affect the depth sensitivities. In fact, there may be no difference at all in some media: for example in Fig. 2 there is an intersection point between DOPL and DOPC at sphere diameter of ∼0.4 µm, which would preclude the use of an ellipticity tuning mechanism for differential depth discrimination. Here is thus another example of the intricacies surrounding the question of linear versus circular polarization retentions, the insights afforded by careful measurements/analysis and associated modeling, and resultant consequences for potential applications.
We next study the effects of scattering coefficient on DOPL and DOPC for suspensions with three different microsphere sizes. Figures 3(a), 3(b), 3(c) show the simulated and experimentally measured DOPL and DOPC values (left vertical axes) and simulation-obtained average number of scattering events per detected photon (right vertical axes) plotted against scattering coefficient for suspensions containing spheres with diameters of $0.21\; $μm, $0.42$ μm, and $0.96$ μm, respectively. Strong agreement is observed between the experimental and Monte Carlo simulated DOP curves, again firming up the observed trends. That said, slight discrepancies are evident at lower scattering coefficients (more pronounced for larger spheres and circular polarization; see Fig. 3(c), triangular symbols). The cause of this discrepancy for the more transparent media $({\mathrm{\mu }_\textrm{s}} < 50$ cm-1) is currently being investigated; but since we are primarily interested in higher scattering media of tissue-like levels of turbidity [62], this lower-scattering discrepancy is not consequential for the current discussion.
Fig. 3. Simulated and experimentally measured DOPL and DOPC signals for monodispersed suspensions with scattering coefficients ranging from 5 cm-1 to 200 cm-1 and spheres of diameter (a) 0.21 μm, (b) 0.42 μm, (c) 0.96 μm. The red symbols and the right vertical axis show the average number of scattering events per detected photon for each simulation run. Solid symbols = experimental results; hollow symbols = Monte Carlo simulation results; lines = guide for the eye.
The number of scattering events increases with scattering coefficient in each plot in Fig. 3. First focusing on the DOPC curves for the two larger sphere sizes (Fig. 3(b) and Fig. 3(c)), we note that the increase in the number of scattering events must arise primarily from increased photon redirection (type-2 backscattering) since the DOPC values increase from negative helicity-flipped signals to positive helicity-preserved signals. In this case, the relative contribution of type-2 light increases with scattering coefficient, whereby correspondingly lower MFPs enhance the mediums’ ability to redirect photons back towards the detector. Eventually, the number of scattering events become sufficiently high such that the depolarization effects begin to dominate, manifesting as slight decreases in DOPC for scattering coefficients > ∼60 cm-1 and ∼100 cm-1 (Fig. 3(b) and Fig. 3(c), respectively). Again, we attribute the DOPC ${\sim} \; 0$ level to the difference in orthogonal circular polarization intensities being nullified, and not to the true polarization loss due to decoherence. The small anisotropy factor of the smallest sphere size likely does not strongly support the multiple-forward-scattered photon redirection mechanism [15] regardless of scattering coefficient, thus DOPC remains helicity-flipped (negative) in Fig. 3(a). In this case, the backscattered intensity primarily consists of type-1 light, and thus we observe an expected simple exponential decay (in magnitude) with the increasing number of scattering events as indicative of polarization decoherence [1] (i.e., not an interplay of helicity-flipped and helicity-preserved intensities as observed in Fig. 3(b) and Fig. 3(c)). Further, since linearly polarized backscattered intensity essentially comprises only type-1 photons, DOPL curves also exhibit exponentially decaying profiles for all three plots in Fig. 3 (with highest values for the lowest diameter (lowest g-value) medium of Fig. 3(a)). We thus note that DOPL is a better predictor of scattering coefficient due to its consistent and simple decaying dependence on it, as opposed to the rise-and-fall behaviour of DOPC in suspensions of larger scatterers. Again we note that both DOPL and DOPC both exhibit non-zero asymptotes with increasing scattering coefficient, thus some polarization information is always preserved in backscattering; this arises from the presence of minimally scattered shallow-layer type-1 photons regardless of increasing medium turbidity.
Since DOPL and DOPC were found to exhibit predictive features of scatterer size and scattering coefficient, we generate a parametric plot where experimentally measured pairs (DOPL, DOPC) are plotted as shown in Fig. 4. The DOPL and DOPC data points are a subset from the results presented in Fig. 2 and Fig. 3, concentrating on the three sphere diameters which have been measured in the lab and on the tissue-relevant range of scattering coefficients, 50 cm-1 to 200 cm-1. On the resultant response surface, there are visible three distinct clusters which correspond to the three different sphere diameters. We again note that DOPC exhibits a far wider dynamic range than DOPL due to its stronger variance with sphere size (see Fig. 2), and thus the size clusters vary mainly along the DOPC axis. The points in each cluster are differentiated by scattering coefficient along both DOPL and DOPC axes, though not as visible on the latter due to its larger scale. However, DOPL is a better predictor of scattering coefficient based on its simple decaying relationship with scattering coefficient, as opposed to the more complex rise-and-fall behaviour of DOPC for the two larger sphere suspensions.
Fig. 4. A DOPL/C response surface for monodispersed microsphere suspensions with a given scattering coefficient (ranging from 50 cm-1 to 200 cm-1) and mean sphere diameter ($0.21\; $μm, $0.42$ μm, and $0.96$ μm). Clustering according to particle size (distinct along the DOPC axis) and scattering coefficient (somewhat along the DOPL axis) is evident; for details, see text.
It is thus evident that the Stokes measurements of DOPL and DOPC are capable of discriminating turbid samples simultaneously by scatterer size and scattering coefficient. The generalizability of these results to a broader parameter space, and their direct applicability to tissue polarimetry, is not yet certain; thus, further investigations are needed in random media with more tissue-like fluctuations in the refractive index and non-uniformly sized/shaped scatterers. Nonetheless, these encouraging early results suggest that once we measure DOPL and DOPC of an unknown sample and plot the resultant pair (DOPL, DOPC) on a response surface similar to Fig. 4, we will learn something about its dominant scatterer size and the magnitude of its scattering coefficient. Such biophysical insights can indeed prove useful in biological tissue characterization applications, for example in the case of epithelial precancer-to-cancer progression that can exhibit simultaneous changes to scatterer size distribution (via pleomorphism) and turbidity (via cell proliferation). Plotting measured DOPL and DOPC values from these tissues on the response surface such as in Fig. 4 may thus prove helpful in detecting and staging cancers based on the average nuclear size and cellular concentration [5–7]. Additionally, a similar response surface approach may furnish useful assessment information on particulate photosensitizer properties (e.g., aggregation) during photodynamic therapy to improve its efficacy [11].
4. Conclusion
Herein, we investigate the linear and circular polarization responses in media with different scatterer sizes and scattering coefficients by performing Stokes measurements to obtain their degrees of polarization. Circular DOP is found to be sensitive to both sub-populations of backscattered photons – directly reflected and forward-scattered-redirected type – whereas linear DOP shows sensitivity primarily to the former type (due to its depolarization for the latter type interactions). Thus, DOPC varies (1) significantly with the scatterer size, and (2) complexly with the scattering coefficient, based on the interplay of both types of photons as well as overall depolarization. DOPC is thus a good marker for scatterer size, but its Stokes backscattering measurements can be ambiguous, arising from either true polarization decoherence or from closely valued orthogonal polarization intensities, or a combination of the two. Linear DOP exhibits a weak dependence on scatterer size in the Mie (larger-scatterers) regime, and a decrease with the scattering coefficient for all scattering diameters. A simple strategy to use linear and circular polarimetry to assess dominant scatterer size and magnitude of the scattering coefficient of an unknown sample is thus presented.
Notably, the backscattered linearly and circularly polarized light signals were measured using a simplified and robust PEM-based polarimetric system which requires only a single incident polarization configuration. The resultant high-SNR ability to detect faint polarization signals amid the large depolarizing background from highly turbid suspensions is an encouraging development for the challenging bulk tissue polarimetry applications currently pursued in our laboratory. Future phantom studies will use model media that better approximate tissue properties, including broader and more biologically relevant scatterer size distributions (polydispersed suspensions), aspherical scatterers, absorptive media, and refractive index effects. To assist these investigations, improved Monte Carlo modelling methods should also be developed to simulate polarized photon propagation in tissue and tissue-like media.
Funding
Canadian Institutes of Health Research (CIHR, PJT-156110); Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-04930); New Frontiers in Research Fund (NFRFE-2019-01049).
Acknowledgment
The foundational contributions of Dr. Daniel Cote in developing the pol-MC code are gratefully acknowledged.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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