2.1.

Monte Carlo Simulation

The polarization Monte Carlo model simulates illumination with a pencil beam of polarized light perpendicular to its surface and models individual photons (or light packets) propagating through a layered scattering medium composed of Mie scattering particles.²¹ The details of this model have been discussed previously^{22 23} and is, therefore, addressed only briefly here. Photons are individually tracked through the medium and at each photon-particle collision the direction and polarization are modified, via adjustment of the directional cosines and Stokes parameters. The characteristics of photons backscattered from the medium are recorded. Spatial intensity distributions, I(r), are obtained by measuring the frequency of photons emerging within annuli centered on the source and normalized by the annular area. All results presented are for a bin size of 0.25 mean free paths (MFP=1/μ_s).

2.2.

Samples and Sample Geometries

Figure 1 shows the general form of the sample under investigation. The model is capable of simulating multiple layers each infinite in the x–y plane with a semi-infinite lower layer. However, for simplicity we consider a two layer medium in this study, with an upper planar layer and a semi-infinite lower layer. There is a mismatch of 1.4 in the refractive index at the air-tissue interface and at the tissue–tissue interface the layers are considered to be index matched. The medium is comprised of a monodispersion of Mie scatterers with the mean cosine of the scattering angle, g=0.92 and size parameter ka=13.9, and are consistent with typical tissue scatterers.¹ Each layer has different scattering (μ_s) and absorption (μ_a) coefficients and absorption is added postsimulation using Lambert–Beer’s law, depending on the propagation distance within each layer. The absorption and scattering properties of the media analyzed are stated in MFPs so that the results can be easily scaled for a wide range of media.

Figure 1

General sample geometry.

2.3.

Polarization Analysis

Light of different initial polarization states can be used to obtain sensitivity to different depths within the layered medium.¹⁹ The technique relies on the property that different illuminating polarization states depolarize at different rates on propagation through the medium. The ability of a photon to maintain its polarization is dependent on the shape, size and material properties of scattering particles encountered and this has been explored extensively elsewhere.^{9 18 21 24} For the particles considered in this investigation, light that is initially linearly polarized maintains its polarization for fewer scattering events than for circularly polarized illumination.

To utilize these properties, a four channel detection scheme is used with different illumination and detection arrangements (Table 1). These channels can be easily measured experimentally using a simple detection scheme.^{15 19} The different channels allow various categories of backscattered photons to be detected (Figure 2). Considering the linear case [Figure 2(a)], there are two categories of backscattered light. The first is light that has undergone relatively few scattering events and maintains its initial polarization, and has thus propagated close to the surface of the medium. The second is light that has traveled a greater distance and emerges with random polarization, thus contributing equally to channels 1 and 2.

Figure 2

Photon categories emerging from the surface of a scattering a medium. (a) Linearly polarized illumination showing the visitation region of polarization maintaining light (A) and random polarization (B). (b) Circularly polarized illumination showing the regions probed by mirror reflected light (C), polarization maintaining light (D) and randomly polarized light (E).

Table 1

For circularly polarized light, three categories of emerging light can be defined. The first is that which is almost immediately backscattered upon entering the scattering medium. This mirror reflection results in a flip in helicity of the incoming circular polarization and contributes to the orthogonal circularly polarized channel 4. The second category is that which has undergone a series of forward scattering events and emerges from the medium having maintained its initial polarization state. As the particles considered here are heavily forward scattering, there is a significant contribution from this second category to channel 3. Finally, photons that have random polarization through multiple scattering contribute equally to channels 3 and 4. The different categories of polarized light have different penetration depths (Figure 2) and so can potentially be used to characterize layered media. The detected channels can be used to define the degree of polarization (DOP) of the backscattered light with respect to its initial polarization. In the case of the linear illumination, this is defined as

2.4.

Moments of Distributions

Moment analysis is a widely used technique for curve analysis and provides a useful method of characterizing the spatial intensity distributions obtained in this study. The first order moment, M₁, and normalized second order moment, N₂, are defined as

3.1.

Semi-Infinite Homogeneous Medium

Figure 3 shows the spatial intensity and spatial DOP distributions for a semi-infinite homogeneous medium with unity MFP and zero absorption. Considering first the linear channels, it can be seen that, for radial distances up to 15 MFPs from the source, channel 1 is greater in intensity than channel 2. As discussed in Sec. 2.3, both channels contain equal amounts of multiple scattered light but channel 1 contains an additional component that has maintained the original polarization. At high radial distances there is no polarization maintaining light and so the DOP tends to zero [Figure 3(b)]. It is interesting to note that the peak in the linear DOP is positioned away from the source (r=0.5 MFPs). At positions close to the source there is a high proportion of singly scattered photons that exhibit Mie scattering properties, e.g., in the bin nearest the source 40 of the photons are singly scattered. The detailed structure of the Mie scattering footprint contributes to a variation in intensity in the co- and cross channels with detector position which, when averaged over the detection bins, results in a higher DOP away from the source. This effect arises as we have used a single particle size in the simulation. In practice for tissue scattering, where there is a range of particle sizes, it is unlikely that this feature will be observed. A simple subtraction of these two channels retrieves light that has probed only a shallow depth. This technique has been used previously to remove multiple scattered light and improve image quality of structures within scattering media^{11 13 14 16} Figure 4(a) demonstrates how subtraction of linear polarization channels enables lateral localization to be obtained. The subtraction reduces the tail of the distribution due to elimination of scattered light. However, this has negligible effect on the 1/e width of the distributions due to the high intensity close to the source.

Figure 3

(a) Backscattered intensity spatial distributions in the four illumination and detection regimes and (b) the resulting spatial DOP distribution for linearly and circularly polarized illuminations.

Figure 4

Lateral localization obtained by extracting (a) linear polarization maintaining light (ch1-ch2), (b) circular polarization maintaining light (ch3-ch2).

Figure 3(a) shows a difference between channels 3 and 4 for all radial positions considered, resulting in a significantly higher circular DOP compared to linear [Figure 3(b)]. This confirms the greater ability for circularly polarized light to maintain its polarization. The peak in the DOP is away from the source (r=2.75) as there is a contribution from the mirror reflected light to the cross-polar channel 4 at the source. Circular polarization maintaining photons can also be extracted using subtraction techniques as shown in Figure 4(b), again reducing the tail of the distribution. The circular polarization maintaining light emerges at a greater radial distance from the source than linear polarization maintaining light suggesting that it has probed a deeper level. This is confirmed by plotting the maximum photon visitation depth (normalized to peak intensity in channel) for linear and circular polarization maintaining light extracted by calculating I₁−I₂ and I₃−I₂, respectively (Figure 5). ( I₂−I₃ assumes that the amount of depolarized light is the same in both linear and circular channels. This is not strictly true for pathlengths ∼1 transport MFP due to the different polarization memory. However, it successfully enables the elimination of heavily scattered light as shown in Figure 5 and so provides a reasonable approximation.) Also plotted for reference is the maximum photon visitation depth for multiple scattered light (channel 2). This clearly shows that the extraction of different maintained polarizations provides interrogation of different depths, with the peak in the linear maintaining distribution occurring at 2 MFPs into the sample and 7 MFPs for the circularly polarized maintaining illumination, compared with a peak at 10 MFPs for the multiple scattered light. This property suggests that different polarizations can be used to characterize different layers within the medium.

Figure 5

Visitation depth of polarization maintaining illumination for linearly maintaining, circularly maintaining and multiple scattered light.

3.2.

Layered Media

Before considering medium absorption and layer thickness it is useful to demonstrate the general properties of the intensity distributions of layered media. Initially a two layer medium with top and bottom MFPs of 1.0 and 2.0, respectively, is considered. Both layers have zero absorption, the top layer is 10 MFPs thick (all measurements of absorption and layer thickness are relative to the top layer MFP) and the bottom layer is semi-infinite. Figure 6 shows the spatial intensity distributions of the four polarization channels, superimposed with the response of a semi-infinite homogeneous medium with the same properties as the individual layers. All channels show that at low radial distances (typically r<4 MFPs ) the spatial response of the two layer medium follows that of a semi-infinite homogeneous medium with the properties of the top layer. Conversely, further from the source (typically r>30 MFPs ), the intensity distribution tends towards that of a semi-infinite homogeneous medium with the properties of the second layer, confirming the interdependence of depth and radial emergence distance. Another observation that can be made from these plots is that channel 4 tends toward the response of the second layer at a slower rate than channel 3. This subtle difference is due to channel 4 containing a light component that has propagated a shorter distance and is therefore more representative of the top layer. Having demonstrated the dependence of the distributions on the two layers, the absorption and layer thickness is considered.

Figure 6

Spatial intensity variations for a two layer medium (solid line) with a top layer MFP of unity, bottom layer MFP of 2.0, top layer thickness of 10.0 MFPs and zero absorption in (a) channel 1, (b) channel 2, (c) channel 3, (d) channel 4. Superimposed on the curves are measurements for semi-infinite media with optical properties of the top layer (-▵- line) and lower layer (-□- line).

3.2.1.

Layer Thickness

Using an approach described previously¹⁹ we have modeled layer thickness using a single layer medium. This is equivalent to an upper layer with varying thickness above a second layer that is totally absorbing. The advantage of this approach is that a single Monte Carlo simulation can be used to model different layer thickness by recording the maximum visitation depth of a photon and discarding those absorbed in the second medium. This improves the efficiency of the simulations and enables sufficient photons to be obtained to calculate the moments of subtracted distributions. However, it should be noted that this represents the extreme contrast for a two layer medium.

Figures 7(a) and 7(b) show the first and normalized second order moments for the four polarization channels. All channels demonstrate a sensitivity to layer thickness and some differences are observable between the channels containing polarization maintaining light (1 and 3) and channels 2 and 4. These differences are relatively small as the channels contain a common multiple scattered background. Figures 7(c) and 7(d) show the first and normalized second order moments for extracted linearly and circularly polarized light and those of multiple scattered light (channel 2) for reference. As demonstrated in Figure 5 the majority of linearly and circularly polarized light is maintained within the top 15 and 30 MFPs, respectively, and so the moment calculations are insensitive to thicker layers. The elimination of multiple scattered light enhances the differences between the moments of these cases and potentially may be better conditioned for inversion.

Figure 7

Top layer thickness variation (top layer MFP=1.0, absorption=0, bottom layer absorption=∞ ). (a) First order moment of channels 1–4, (b) normalized second order moment of channels 1–4, (c) first order moment of extracted linear (ch1-ch2) and circular (ch3-ch2) polarization states, (d) normalized second order moment of extracted polarization states.

When the two layer model is used to simulate layer thickness the differences between the polarization channels reduces due to a high contribution of multiple scattered light from the lower layer. It would be useful to measure the moments of the distributions from the polarization maintaining light over the range of thickness for a two layer model. Under these conditions the elimination of multiple scattered light should provide a greater sensitivity to layer thickness and the results should have comparable sensitivity to Figures 7(c) and 7(d). At present this is impractical due to long computation time and an accelerated Monte Carlo²³ is a more viable option.

3.2.2.

Layer Absorption

Absorption is also an important consideration due to factors such as the melanin content of the upper layer and blood concentration of underlying layers. In this section the MFP of both layers was set to unity to remove any influence due to a scattering mismatch between layers. The top layer thickness is fixed at 10 MFPs and the top and bottom layer absorption is varied. The first and normalized second order moments of the different polarization channels are presented in Figure 8 using contour plots to show variation with absorption of both layers. Featured on these plots are lines of constant first order (solid) and normalized second order moments (dashed) with variation of absorption in the top and bottom layers. It can be seen from these that if both the first and normalized second order moments are known the absorption of the two separate layers can be determined. Practical application of such a technique indicates that the more perpendicular the two sets of contours, the lower the sensitivity to noise on either value.²⁵

Figure 8

Contour plots of the first (solid line) and normalized second order (dotted line) moments for different top and bottom layer absorptions in (a) channel 1, (b) channel 2, (c) channel 3 and (d) channel 4.

The first observation is that the first order moment is heavily dependent on the characteristics of the top layer. This can be seen from the contours being nearly vertical, hence a variation in bottom layer absorption results in a small change in first order moment, but a top layer alteration produces a significant change in this parameter. Another feature that confirms this dependence is that the first order contours are steeper in channels 1 and 4 than in 2 and 3 as these channels are more dependent on the top layer. The second order moment is heavily influenced by light emerging further from the source [Eq. (3)] and so is more dependent on the characteristics of the second layer, resulting in more horizontal contours.