We report on the development of a method that records spatially dependent intensity patterns of polarized light that is diffusely backscattered from highly scattering media. It is demonstrated that these intensity patterns can be used to differentiate turbid media, such as polystyrene-sphere and biological-cell suspensions. Our technique employs polarized light from a He-Ne laser (λ = 543 nm), which is focused onto the surface of the scattering medium. A surface area of approximately 4x4 cm centered on the light input point is imaged through polarization-analysis optics onto a CCD camera. One can observe a large variety of intensity patterns by varying the polarization state of the incident laser light and changing the analyzer configuration to detect different polarization components of the backscattered light. Introducing the Mueller-matrix concept for diffusely backscattered light, a framework is provided to select a subset of measurements that comprehensively describe the optical properties of backscattering media.
© Optical Society of America
In recent years there has been an increased interest in the propagation of polarized light in highly scattering media, especially biological materials. For example, Schmitt et al  proposed to use transmitted polarized light for imaging tissue heterogeneities. Anderson et al  Jacques et al  and Demos et al  studied the use of backreflected polarized light for surface and beneath-the-surface imaging. Furthermore, our group previously demonstrated that diffusely backscattered, linearly polarized light can be used to determine the scattering coefficient, μs, anisotropy factor, g, and average particle size of polystyrene-sphere and biological-cell suspensions. [5,6]
In this work, we generalize our studies of diffusely backscattered polarized light from highly scattering media by including measurements of circularly polarized light and introducing the Mueller-matrix concept. While the Mueller matrix is well known to describe many optical elements and materials, [7–9] matrix elements have not been obtained for diffusely backscattered light from turbid media. We will show matrix elements for various polystyrene-sphere suspensions, and for cancerous and noncancerous cell suspensions. In particular it is demonstrated that cancerous rat fibroblast cells, MR1, can best be distinguished from noncancerous cells, M1, by measuring backreflected circularly polarized light.
2.1 Experimental setup
The experimental setup for studying the diffuse backreflectance of polarized light is depicted in Fig. 1. A He-Ne laser with an output power of 5 mW at a wavelength of λ = 543 nm is used as the light source. Polarization optics P1 are placed in front of the HeNe laser. To generate linearly polarized light at various angles a dichroic sheet polarizer (Melles Griot) is mounted onto a rotation stage. Circularly polarized light is generated, by inserting a λ/4 mica retardation plate (Melles Griot) behind the linear polarizer, with the retarder principal plane at ±45° with respect to the electrical field vector of the incident linearly polarized beam. The polarized light is focused to a spot with a diameter d < 500 μm by lens L1 (f = 15 cm) via mirror M1, through a hole in the center of mirror M2, onto the surface of the scattering medium. The beam is perpendicular to the surface of the medium. Lens L2 (f = 15 cm) is used to generate an image of the surface-area around the input point at the image plane IP1. This intermediate image plane allows us to place an optical mask (diameter = 2 mm) in the center of this image. In this way the strong specular reflection from the surface at the laser input point is rejected, and only multiply scattered light is recorded. The image at IP1 including the optical mask was viewed with a 55-mm Nikon camera lens L3 (Micro-Nikkor, 55 mm, f/2.8), which was mounted on a cooled CCD camera (Princeton Instrument, Inc.). The CCD chip has 575x384 pixels with a pixel size of 20x20 μm. The surface-area under observation around the laser entrance point could be varied from 0.5x0.5 cm to 4x4 cm, depending on the position of L2 and L3. A polarizer P2 was placed in front of L3 to serve as an analyzer. The CCD camera was controlled and operated with IPLab software (Signal Analytics Cooperation). The dynamic range of the CCD camera was 14 bits. Two typical examples of images obtained with this set-up are shown in Fig. 2.
We prepared four different types of aqueous, monodisperse polystyrene-sphere suspensions. The spheres had diameters of 204 nm, 497 nm, 890 nm, 2040 nm and were suspended in deionized water with trace amounts of surfactants (Duke Scientific Corporation, 2463 Faber Place, P.O. Box 50005, Palo Alto, California 94303). We chose particle concentrations that resulted in reduced scattering coefficients, μs' = (1-g) μs, in the range of 1.8 to 2.2 cm-1, which is comparable to values found for the cell suspensions used in this work.
To study diffuse backreflection from biological material, we prepared suspensions of cancerous (MR1) and noncancerous (M1) rat fibroblast cells. Fibroblasts are most commonly found in connective tissue and are responsible for the synthesis of fibers and amorphous intercellular substances. Fibroblasts are rich in mitochondria, lipid droplets, Golgi complexes, and rough endoplasmic reticulum.  The cells used in this study came from the same rat-embryo fibroblasts into which a mutated myc-gene is transfected to obtain M1 cells, while an additional mutant ras-gene transfection leads to MR1 cells.  Only MR1 cells form a tumor if injected into a rat. Cell suspensions were prepared from monolayers grown in roller bottles and suspended in ice-cold phosphate-buffered salineat a concentration of 108 cells/cm3. [6,12] Approximately 30 ml of each suspension was used for the backscattering experiments. To obtain the backscattered intensity images, the suspensions were placed in a 4 cm diameter cylindrical beaker, with a suspension depth of approximately 2.5 cm.
2.3 Polarization arithmetic for diffusely backscattered light
A large number of different experiments are possible if one wants to study the polarization-dependent scattering properties of turbid media. The probing light may be linearly polarized at various angles, right-hand circularly, left-hand circularly, or elliptically polarized. Light coming from the scattering medium can be analyzed in the same numerous ways. However, only a few measurements are needed to completely characterize the optical properties of any material. The necessary procedure is elegantly demonstrated by the Stokes-vector Mueller-matrix approach to polarization and light scattering. 
2.3.1 Stokes vector
Each light beam can be represented by its four-component Stokes vector, S = (I,Q,U,V). The components of the Stokes vector are related to the electric field vectors as follows:
where Eh and Ev are electric field components horizontal and vertical with respect to the direction of light travel. The “*“ indicates the complex conjugate, and the brackets represent time averages over time periods longer than T = 1/f, where f is the optical frequency (~1015/s). Some of the Stokes components have an obvious interpretation. For example, I = < Eh + Ev > is the time average of the sum of the products of the E-field amplitudes, which is equal to the total intensity. Examples for the Stokes vector notation are, unpolarized light: [1,0,0,0], horizontal linearly polarized light: [1,1,0,0], vertical linearly polarized light: [1,-1,0,0], right-hand circular polarized light: [1,0,0,1], and left-hand circular polarized light: [1,0,0,-1].
2.3.2 Mueller matrix
In general, the interaction of light with optical elements such as lenses, polarizers, filters, surfaces, scattering media etc., changes the polarization state of the light from S to S'. When light is described by a four-component vector, this interaction with any optical element or material can be described as a multiplication of the Stokes vector with a 4x4 matrix, S' = M S. This sixteen-element matrix is called the Mueller matrix or, if scattering is involved, the scattering matrix:
This matrix completely characterizes any component or material in terms of its optical properties. For example, a linear polarizer is described by M11 = M12 = M21 = M22 = 1 and all other elements equal to zero. A right-hand circular polarizer is represented by M11 = M14 = M41 = M44 = 1, and all other elements equal to zero.
In this work we consider diffusely backscattered light from turbid media. We photograph the surface around the laser input point (see Figs. 1 and 2). Therefore, rather than being just one number, each of the sixteen elements of the scattering matrix is a 2D array of numbers referring to different spatial locations on the surface of the medium. An example is shown in the next section.
2.3.3 Determining the Mueller matrix
If the Mueller matrix is not known, all the elements can be determined experimentally. It can be shown that 49 intensity measurements with various orientations of polarizers and analyzers are necessary to obtain the 16 elements of the Mueller matrix.  Figure 3 lists the necessary measurements for each matrix element. For example, to obtain the M11 element one needs to measure the total reflected intensity from an unpolarized incident beam. The M12 element is obtained by measuring the total reflected intensity for a horizontally, linearly polarized incoming beam and subtracting from this the total reflected intensity for a vertically, linearly polarized incoming beam. Fig. 4 graphically depicts the calculation of the M22 element.
Once all 16 elements of the matrix are obtained, the medium is completely described in terms of its optical characteristics. In our case this means: Given any state of polarization of the incident beam it is possible to calculate the state of polarization (Stokes vector) of the diffusely backscattered light. For example if the incident beam is linearly polarized along the x-axes, the Stokes vector is given by S = [1,1,0,0]. This vector is multiplied by the Mueller matrix, M, to calculate the Stokes vector of the diffusely backscattered light. This light encounters an analyzer consisting of a linear polarizer along the y-axis, with M11 = M22 = 1 and M21 = M12 = -1 and all other elements equal to 0. Putting it all together we obtain:
As pointed out in section (2.3.1) the first component, I, of the Stokes vector is equal to the total intensity. This is the parameter measured with the setup shown in Fig. 1. Therefore the measured intensity will equal the sum of the four Mueller matrix elements M11 + M12 - M21 - M22, of which M22 for a 204-nm-sphere suspension is displayed in Fig. 4.
It should be noted that the M11 element, which is obtained by using no polarizing optics in the setup, equals the measurements done by Jacques et al  and Kienle et al , who call this “video-reflectometry“. They show that one can determine μs' and μa of the medium by analyzing the intensity decay as a function of r, the distance from the laser entry point into the medium. If M11 does not depend on the azimuthal angle, α (see Fig. 2a), it is possible to measure the intensity decay as a function of r in one dimension, rather than taking a two-dimensional image of the surface intensities. This method, often referred to as spatially-resolved diffuse-reflectance measurement, has been used by many authors to determine optical properties of tissue phantoms and biological tissues. [15–24]
The introduction of the Mueller-matrix concept for the diffusely backscattered light provides, in addition to the widely used M11 element, 15 more elements, which can be evaluated to obtain further information about scattering media. Because the description of the optical characteristics with the Mueller matrix is complete, any information about particle size, refractive index, particle shape etc. has to be found in the Mueller matrix by careful analysis of the matrix elements. If some information about the medium cannot be extracted from the various matrix elements, this information cannot be extracted from any other additional backscattering measurements. However, further information may be gained, for example, by measuring the diffuse transmittance, the diffuse backreflectance at different incident and observation angles, or time-dependent polarization effects.
Given the vast literature that deals with M11-type measurements and how these can be used to determine optical properties of turbid media, it is clear that one could devote an entire study to each of the other 15 elements. Such a detailed analysis is beyond the scope of this paper. Instead we have chosen several examples that illustrate the potential of the Mueller matrix approach to characterize highly scattering media.
3.1. Polystyrene-sphere suspensions
Figure 5 shows all 16 Mueller matrix elements obtained for a polystyrene-sphere suspension, with particles of 204-nm diameter. The particle concentration is 0.05% by weight, which results in a reduced scattering coefficient of μs' = 1.9 cm-1. The displayed intensities are relative to the maximum intensity of the M11 element. This element is always positive, however, all other elements can have negative values, because different intensity measurements are subtracted from each other (See Fig. 3). Positive values are displayed in red-yellow colors, while negative values are indicated in blue tones.
We can make several observations: First we notice that Mnm = Mmn, which means that this Mueller matrix is symmetric. The difference between the measurement of Mmn and of Mnm is simply that the polarization optics before and after the medium are exchanged. (see. Fig. 1, optical elements P1 and P2). Therefore, if the scattering medium is not optically active and has no internal structure that results in some preferred optical axis, Mnm should indeed equal Mmn. Furthermore, several pairs of elements exist that differ only in the orientation of the intensity pattern. For example M13 can be obtained by rotating M12 by 45 degrees. The same relation holds for the pairs M31 and M21, and M22 and M33. The only difference between the measurements of these pairs is a rotation of the polarization and analyzer optics (P1 and P2 in Fig. 1) by 45 degrees. Again, if the suspension is not optically active and has no preferential optical direction this result should be expected. Finally, we observe that the elements in the last column (M14, M24, M34, M44) and the last row (M41, M42, M43, M44) are almost zero, while all other elements show strong azimuthal variations in the relative intensity.
The last observation is especially interesting, when the 204-nm-particle suspension of Fig. 5 is compared to the suspension of 2040-nm-diameter spheres (μs' = 1.9 cm-1) in Fig. 6. In this case the last row and last column show azimuthal intensity variations, while the first row and first column (M12, M13, M14, and M21, M31, M41) are almost zero. Other differences are the more pronounced azimuthal variation of the center elements M22, M32, M23, and M33. The matrix symmetries are the same as for the 204-nm sphere suspensions. Also, since the reduced scattering coefficient of the two suspensions are identical, the M11 elements are the same. (The radial intensity decay only depends on (μs' and μa,  and μa is negligible in both cases.) This demonstrates the advantages of the Mueller matrix approach over standard video reflectometry, which is based on unpolarized light measurements. The addition of polarization properties allows to distinguish between 204-nm and 2040-nm sphere suspensions, even when the reduced scattering coefficients are the same.
All sixteen Mueller matrix components together provide a “finger print” of the scattering medium under investigation. As just shown, looking at the entire Mueller matrix often enables one to distinguish qualitatively between two media. To obtain a more quantitative distinction between different media, more detailed analyses of single matrix elements are necessary. As an example we investigated changes in the M44 element as a function of particle diameter in monodisperse, polystyrene-sphere suspensions.
The M44 element is calculated from 4 measurements, which involve only circularly polarized light (see Fig.3: M44 = LL + RR - LR - RL). This element is a measure of how effectively a medium flips the helicity of the backscattered light. For a perfect mirror the element M44 = -1, because perpendicular-incoming left-hand circularly polarized light returns right-hand circularly polarized and vice a versa. (Therefore, LL = RR = 0 and LR = RL = 1).
We found that in the case of polystyrene-sphere suspension, M44 has no azimuthal dependence. Therefore, we extracted the values of this element along the y-axes from the 2-dimensional images of M44. The results for four different sphere suspensions are shown in Fig. 7. It can be seen that M44 decreases with increasing sphere diameter. Therefore, the suspensions with larger spheres flip the helicity more effectively, and most of the backscattered light has a different helicity than the incident light. This behavior can be understood if one considers that with increasing diameter the spheres become more and more like a mirror, which can be thought of as a sphere with infinite radius. If the polystyrene spheres are small (d = 204 nm) compared to the wavelength (λ = 543 nm), the backscattered light is equally left-hand and right-hand polarized and the element is zero. Furthermore, the effect is strongest in the center, near the laser entry point. Here the backscattered light has undergone only a few scattering events and the polarization effects are strongest. With increasing distance from the point of light incident, the number of scattering events increases and eventually the polarization information is lost - the value of the M44 approaches 0.
3.2 Cells suspensions
In addition to monodisperse particle suspensions with known optical properties, we also applied the Mueller matrix approach to biological cell suspensions. Figures 8 and 9 show the results obtained for cancerous (MR1) and noncancerous (M1) rat fibroblast suspensions. Both suspensions contain 108cells/cm3 and the scattering coefficient were μs' = 2.2 cm-1(±10%) and μs' = 2.1 cm-1 (±10%) for the MR1 and M1 cells, respectively. The two matrixes are almost identical. Small differences can be observed for the M22 and M33 elements. In these matrix elements the negative values close to the center appear slightly more pronounced. Larger differences between the tumorigenic MR1 cells and the nontumorigenic M1 cells can be seen in the M44 elements.
The M44 element is radially symmetric for the MR1 cells as well as the M1 cells. Figure 10 shows the radial dependence along the y-axes of M44 for r = 0.4 to 1.6 mfp' (mfp' = transport mean free path = 1/μs'). The laser light enters the medium at r = 0 mfp'. Data for r < 0.4 mfp' is omitted because this range is covered or distorted by the optical mask. For radial distances r >0.5 mfp' the M44 element for the MR1 suspension is smaller. This means that in this range the cancerous cell suspension flips the helicity of the incident light more than the noncancerous cell suspension.
Comparing Fig. 10 with the results found for polystyrene-sphere suspensions in Fig. 7, it is interesting to note that for distances larger than 0.9 mfp' from the light input point, backscattering from the cell suspensions is similar to backscattering from 497-nm-sphere suspension. However, for smaller distances the value of the M44 element for the polystyrene-sphere suspension decreases as the light incident point is approached, while the value of M44 for the cell suspensions increases. The reason for this behavior is unclear and further studies will be necessary. Possible explanations may be related to the nonspherical shapes of the scatterers within the cell, scattering contributions from particles with various sizes, or optical activity of the medium.
4. Summary and conclusion
In this work we present a method and general framework for the study of diffusely backscattered polarized light from highly scattering media. Our method is an extension of the well known techniques of spatially resolved reflectometry and video reflectometry. These methods use unpolarized light in the study of optical properties (μs' and μa) of turbid media. We show that with the use of a polarized incident beam and the analysis of various polarization components in the diffusely backscattered light, complex, 2-dimensional, spatially varying surface-intensity patterns can be observed. These patterns can be used to gain additional information about the scattering medium and to distinguish between media with the same μs' and μa.
An infinite number of different backscattering measurements using polarized light can be performed by varying the polarization state of the incident beam and detecting different polarization components of the diffusely backscattered light. However, by introducing the Stokes-vector, Mueller-matrix concept for diffusely backscattered light, we provide a framework to select subsets of measurements that comprehensively describe the optical properties of backscattering media. With 49 measurements all 16 elements of the diffuse backscattering Mueller matrix can be determined. Knowing all 16 elements of this matrix one can calculate the polarization state of the backscattered light given any polarization state of the incident light. Therefore, knowing all 16 elements completely describes the medium in terms of optical properties.
To illustrate the potential of the Mueller-matrix approach we studied polystyrene-sphere and biological-cell suspensions. Clear differences among different monodisperse polystyrene-sphere suspensions can be found in several matrix elements, even when the suspensions have the same scattering and absorption coefficients. We have shown that the M44 element, which is obtained from measurements employing right and left-hand polarized light, is especially sensitive to particle sizes in the suspension. Furthermore we have shown that suspensions of cancerous and noncancerous rat fibroblast cells can be best differentiated by the M44 element.
Further studies are necessary to fully explore the Mueller-matrix approach to analyzing diffusely backscattered light and it's possible use in biomedical diagnostics. For example, obtaining the full matrix for particle suspensions with various well-known size distributions may help understanding the patterns observed from cell suspensions. In contrast to suspensions, tissues are highly structured and often have preferred optical axes. These structures should affect the elements of the Mueller matrix. Furthermore, tissues have, in general, a higher scattering coefficient than the suspensions described in this study. This means that polarization effects will be limited to a smaller distance from the laser incident point. This may require the use of a microscope or fiber optics to spatially resolve the patterns in the matrix elements. Finally, it is highly desirable to develop an analytical or numerical model to simulate and predict the type of results discussed in this study.
This work was supported in part by grants CA71898 and ES07845 from the National Institutes of Health, a Postdoctoral Fellowship from the Director's Office at Los Alamos National Laboratory (LANL), the Laboratory Directed Research and Development Program at LANL, and funds from the Science and Engineering Research Semester (SERS) program operated by the Department of Energy.
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