The polarization properties of light backscattered from a two layer scattering medium are investigated. Linear, circular and elliptical polarization states are considered and it is demonstrated that the degree of polarization of the backscattered light is sensitive to the optical properties of both layers and to layer thickness. Furthermore, it is shown that the polarization memory of circularly polarized light enables deeper layers to be probed whereas linearly polarized light is more sensitive to surface layers. This has applications for characterizing burns and melanoma.
©2000 Optical Society of America
Light based techniques are becoming an increasingly popular method of probing heavily scattering media such as body tissue [1,2]. The techniques are not harmful to the patient, offer additional spectroscopic information and are often low cost and simple to use. The main drawback is that light is heavily scattered by body tissue, resulting in uncertainty in the volume probed. To overcome this problem techniques such as time of flight and diffusive wave imaging have been applied to thick tissue such as breast and brain. For surface tissue or thin tissue samples it is possible to use the coherence [3,4] or polarization properties of the light [5–11]. Polarized light studies are not applicable to thick tissue samples such as breast or brain as the polarization information is destroyed after relatively few scattering events. However, polarization analysis of light backscattered from tissue offers a potential method for probing the skin.
Early work in the field of tissue optics assumed a homogenous scattering medium and measured the bulk scattering and absorption properties. More recently, attempts have been made to characterize the effect of different tissue layers. The problem using light to probe layered scattering media is that the scattered light propagates through many random paths through the different layers. It would be extremely useful to be able to characterize the optical properties and thickness of the different layers from measurements of the properties of the emerging light. For example, in studies of thick tissues models of the brain have been developed to include the effects of skin, bone, cerebrospinal fluid and white and gray matter . Another important application of characterization of layered media is in thickness measurements of the skin. This is important for burn diagnosis as the thickness determines whether a skin graft is necessary . It is also arguably the most important parameter for melanoma diagnosis as the cancer will spread to other parts of the body if it breaks through the epidermis/dermis boundary .
Diffusive wave [15,16], time of flight  and continuous wave  techniques have been applied to layered scattering media. These papers have demonstrated that the temporal and spatial properties of the emerging light are dependent on the thickness, scattering and absorption properties of the media under investigation. Photons that have traveled longer times and distances are more likely to have probed deeper tissue.
Using the polarization properties of the backscattered light potentially offers a very simple method to characterize thin, layered scattering media. This utilizes the property that scattering reduces the degree of polarization. Light emerging from the medium consists of a mixture of photons that have maintained some degree of polarization due to undergoing relatively few scattering events and those that have random polarization as they have been scattered many times. In transmission measurements of scattered light the polarization maintaining light corresponds to that which has traveled a route closer to the optical axis [5–7]. In reflection measurements this corresponds to light that has propagated nearer the surface. The polarization maintaining light can be extracted by a simple subtraction method [7–9] and this has been used in the imaging problem.
An important property of scattered polarized light is that, for large particles (diameter >λ) circularly polarized light maintains its polarization after more scattering events than linearly polarized [5,19,20]. For the interested reader, a detailed account of the physical basis of this effect is provided elsewhere [19,20]. Basically, for large particles the light at each scattering event is forward scattered within a small cone of angles. The degree of polarization of linearly polarized light is dependent on the azimuthal angle whereas for circularly polarized light it is independent of azimuthal angle. It has been shown  that for pathlengths of the order of a transport mean free path linearly polarized light is randomized whereas circularly polarized light has maintained its original polarization. Furthermore polarization effects have been shown to be dependent on the size of the scatterers [7,19–21]. For small particles linear polarization is maintained for longer than circular as scattering is equally likely in all directions. In this case the linear character of linearly polarized light is not affected by backscattering, whereas backscattering reverses the helicity of circularly polarized light and randomizes it more rapidly. Tissue scatters predominantly in the forward direction and under these conditions circular polarization is maintained longer than linear.
Light emerging from the tissue will consist of both polarized and randomly polarized light. The polarized component will have undergone relatively few scattering events and so will have probed nearer the surface. Also, as circularly polarized light maintains its polarization state after more scattering events this will probe further beneath the surface. In this paper the polarization properties of light backscattered from a two layer scattering media are studied. We demonstrate that the degree of polarization of the detected light is dependent on the input polarization, thickness and optical properties of the layers and that in future this could be used as a tool to determine skin layer thickness.
The next section describes the experimental configuration. Section 3 presents the results demonstrating the polarization properties of light backscattered from a two layer scattering medium. Discussion and conclusions follow in section 4.
Fig. 1 shows the experimental system. Linearly polarized light from a HeNe laser (λ=633nm, optical power=10mW) is modulated using an optical chopper to enable detection in a small bandwidth using a lock-in amplifier. The polarization state of the light incident on the medium is set by a λ/4 plate and the scattered light is analyzed using a λ/4 plate and linear polarizer [11,22]. The light is then detected by a PIN photodiode and the signal fed to a lock-in amplifier. The surface scattering medium (, ) consists of a quartz cuvette (thickness d=10mm, lateral dimensions 50×50mm) containing a solution of polystyrene microspheres (diameter=1.4µm, refractive index=1.6, mean cosine of scattering angle, g=0.92). The microspheres are predominantly forward scattering which represents similar scattering properties to tissue. The second scattering medium is either a solid tissue phantom  of fixed optical coefficients (=40mm-1, g=0.95, =0.009mm-1, thickness=20mm) or no medium is present which corresponds to a totally absorbing second medium. The detector is placed at an angle of α=15° from the optical axis so that specular reflections from the surface of the cuvette do not affect the measurements and at a distance of 200mm from the surface of medium 1.
Two experiments are performed; the first involves only the cuvette so that any light that reaches the second surface of the medium is transmitted (totally absorbing second medium). To cover a wide range of scattering properties of the upper layer, an initial µs=13.6 mm-1 concentration of medium 1 is diluted to µs=0.17 mm-1. Diluting the solution increases the mean free path between scattering events and decreases the number of scattering events in medium 1. This can also be thought of as simulating layers with the same scattering properties but different thickness. The major difference between using this simulation and physically changing the layer thickness is that although the light distribution at the surface of the medium will have the same properties (i.e. shape, intensity, polarization) the width of the spatial distribution will be different. For example, halving µs for a fixed layer thickness will simulate a medium with the same µs but half the thickness. In this case the light distribution at the surface of the medium will have the same properties but will be twice as wide. However in the experimental set up considered in fig. 1 the detector is in the far field and detects the total surface distribution integrated over the numerical aperture of the detector. Under these conditions varying the scattering properties of layer 1 can be considered as simulating layers of different thickness as the system cannot discriminate between media where the product of µs and the layer thickness is a constant. Thus one can easily scale between scattering coefficient and layer thickness using the following relationship;
thus by knowing d and µs of a medium a, one can assume a fixed µs of a medium b (similar to that of skin) and calculate the equivalent thickness.
When linearly polarized light is incident on the medium, measurements are made of the co and cross polarized components to record the degree of polarization of the emerging light. When circularly polarized light is incident on the medium the amount of clockwise and anticlockwise light is recorded to obtain the degree of polarization. The degree of polarization is defined as;
where Ico is the intensity of light maintaining its original polarization, Icross is the intensity of light emerging in the opposite polarization state i.e. cross linear or helicity flipped circular.
In the second experiment the first process is repeated but with the solid tissue phantom present as the underlying medium. In this case it is possible for light to propagate within the second medium and then be scattered back to the detector. This medium is more representative of an underlying tissue layer and enables comparison with the totally absorbing second layer. Any differences between the results obtained with a different second medium can then be attributed to light propagating within the second layer. Another possibility is to use a mirror as a second medium to represent an infinite second medium. This is acceptable for studies not analyzing polarization but is not appropriate for polarized light studies as it introduces additional mirror reflected photons, which simulates increased single scattering, not an infinite medium.
Fig. 2 shows the results from the first experiment. The graph shows the degree of polarization of the detected light for different scatterer concentrations, a concentration of unity represents µs=13.6 mm-1 and this solution is diluted for the other measurements. It can be seen that at low scatterer concentration linearly polarized light has a degree of polarization close to unity, which then decreases as the scatterer concentration increases. Circularly polarized light has a degree of polarization close to -1 at low concentration, which then rises to a value of 0.42 and then decreases. Elliptical polarization has properties between the two.
The linearly polarized case is relatively straightforward to explain. For the concentrations examined here there is always a fraction of the light that has been directly backscattered by the medium in a single, or at least relatively few, scattering events (fig. 3a). In addition there is a proportion that have been multiply scattered and contribute equally to each detected polarization state. At a high scatterer concentration there is a high proportion of photons that have been multiply scattered and return to the detector. These are generally depolarized and contribute equally to each polarization state, which results in a relatively low degree of polarization. As the concentration becomes weaker more depolarized light is transmitted by the cuvette and does not contribute to the detected signal. This causes an increase in the degree of polarization until the mirror reflected light dominates and the degree of polarization is close to unity.
Circularly polarized light presents an interesting case as one can define three types of photons emerging from the scattering medium (fig. 3b). The first are those that are directly backscattered by the medium. These obtain a flip in helicity as they undergo a mirror reflection and, when analyzed, contribute to the cross polar component. The second type emerges from the scattering medium via a series of forward scattering events and maintains its original polarization state. When analyzed at the detector these contribute to the co polar component. The third type of photon has been depolarized by many scattering events and contributes equally to both polarization states. It should be noted that the depolarized component is different for circular and linear as it takes more scattering events to depolarize circular.
Fig. 2 also demonstrates the effect of scatterer concentration on the degree of circular polarization detected. At high scatterer concentrations the amount of polarization maintaining photons dominate those detected with helicity flip so the degree of polarization is positive. As there are more polarization maintaining photons than in linear the degree of polarization is higher. As the scatterer concentration decreases the amount of depolarized light transmitted through the cuvette increases and the degree of polarization initially increases. Decreasing the scatterer concentration further demonstrates that polarization maintaining photons begin to be transmitted through the cuvette. The degree of polarization decreases and eventually becomes negative when the helicity flip photons dominate. It is interesting to note that the light backscattered is completely depolarized at a concentration of 0.035 (µs=0.48mm-1) when the helicity flip and polarization maintaining photons cancel each other out.
In the second set of experiments a solid tissue phantom (both absorbing and scattering) is situated directly behind the cuvette. Photons that were transmitted from the first medium in the previous experiment now propagate within the second and can possibly return to the detector. Fig. 4 demonstrates similar trends to the first experiment. The most significant effect of the second medium is to increase the amount of depolarized light returning to the detector, thus reducing the degree of polarization in all cases. Again, for linearly polarized light at high scatterer concentrations there is a small fraction of light detected that maintains its original polarization state. The degree of polarization is reduced as the amount of depolarized light detected has increased. Again, reducing the amount of scattering in the top layer increases the degree of polarization monotonically, although at a reduced rate to that of the previous experiment. This demonstrates that even with a second scattering medium present the degree of polarization detected is dependent on the scattering properties of the upper layer. Similar effects to the first experiment are also seen for circularly polarized light. For the highest scatterer concentrations considered there is no significant difference between the curves in figs 2 and 4, indicating that there is no significant proportion of light that propagates in the second medium that reaches the detector. Circular polarization has a zero degree of polarization at approximately the same concentration as the previous experiment but at lower concentrations it is not as negative due to the presence of the second medium.
4. Discussion and conclusions
The results demonstrate that polarized light backscattered from a two layer scattering medium is sensitive to the state of input polarization and the optical properties and thickness of both layers. Figs 2 and 4 both indicate that as the number of scattering events within the upper layer decreases, simulating layers of decreasing thickness, the degree of polarization of the emerging light changes significantly. The difference between figs. 2 and 4 indicates that the polarization properties of the emerging light are also sensitive to the properties of the underlying medium. Larger differences are observed at low concentration as there is a higher probability of light being scattered from the second medium back to the detector. As the scatterer concentration gets higher the difference in the degree of polarization between figs 2 and 4 becomes negligible as the polarized light no longer probes the second medium. For linearly polarized light there is no significant difference above a scatterer concentration of 0.25 (µs=3.4mm-1) whereas for circularly polarized light there is no significant difference above a scatterer concentration of 0.75 (µs=10.2mm-1). This demonstrates that for the particles considered here circularly polarized light probes a factor of 3 deeper than linearly polarized. The particles (size parameter ka=6.95) considered in this paper are predominantly forward scattering which provide scattering properties similar to those of body tissue. It is interesting to note that for smaller particles (ka < 1) linear polarization is maintained for more scattering events than circular [19,20]. This effect is the opposite to that observed in this paper and under these conditions linear polarization can be used to probe deeper than circular. Assuming µs=40mm-1 for dermal skin  we can use the experimental values of µs to make a coarse approximation for the depth probed in homogeneous tissue. Using equation (2) this corresponds to a maximum depth probed of 0.85mm for linear polarization and 2.55mm for circular polarization. This is a coarse approximation as the effects of absorption (which will attenuate longer pathlength photons and decrease the depths probed), different scattering properties and different refractive indices of the layers have been neglected in this calculation. We are currently using a vector Monte Carlo model  to investigate the forward problem over a range of parameters and quantify the depths probed more accurately. However, the trends observed in these experiments highlight the potential of polarized light for characterizing layered media.
The fundamental problem to be solved is whether there is a unique solution to the inverse problem. If this is the case, then it should be possible to extract both the thickness of the upper layer, and the scattering and absorption properties of both layers. Clearly this is not the case for a single detector and a single input polarization state. For example in figs 2 and 4 the same degree of circular polarization is obtained at a scatterer concentration of 0.05 and 0.42 in the first experiment, and 0.19 in the second, so the solution is far from unique. We are assisted in this problem by the fact that different polarization states maintain their polarization over different numbers of scattering events. Therefore a range of depths are probed by different polarization states, with linear probing nearer the surface, circular probing deepest, and a range of elliptical states in between. These photons are weighted in different ways by the distance they have propagated in either medium e.g. if the underlying layer has greater absorption than the upper layer then circularly polarized light will be more heavily attenuated by this region. Plotting the degree of polarization against polarization state will result in a curve that is weighted by different amounts depending on the thickness and optical properties of the layer. We are currently investigating whether the inversion is well conditioned by modeling the forward problem and obtaining a database of such curves and fitting to these to perform the inversion. Additional parameters can also be incorporated to improve performance such as the spatial distribution [18,21] and additional wavelengths . As has already been described in section 2 the system used cannot discriminate between media where the product of µs and the layer thickness is a constant. However the spatial distribution in these cases will be different and so imaging the surface distribution onto the detector plane will enable these to be discriminated.
The different depths probed by different polarization states offers several interesting possibilities. Polarization subtraction [7–9] offers the possibility of obtaining depth discrimination. In addition to subtracting cross and co polarization states different polarization states can be subtracted from one another to obtain coarse optical sectioning. We are also applying this approach to the imaging problem where sub-surface images could be obtained. However, the spatial resolution degrades when selecting between single scattered and polarization maintaining photons and this is currently being quantified.
In conclusion, the polarization properties of light backscattered from a two layer scattering medium have been investigated. It has been shown that the degree of polarization is sensitive to the optical properties and layer thickness and this offers a potential tool for characterizing burns and melanoma.
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