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Diffuse backscattering Mueller matrices of highly scattering media

Andreas H. Hielscher, Angelia A. Eick, Judith R. Mourant, Dan Shen, James P. Freyer, and Irving J. Bigio

Open Access Open Access

Abstract

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.

©1997 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. Experimental setup for measuring diffusely backscattered polarized light.

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Fig. 2.
Fig. 2. Two examples of images taken with the experimental setup depicted in Fig.1. The scattering medium was a suspension of polystyrene-spheres with a diameter of 204 nm at a concentration of 0.05% by weight. Using Mie-theory8 the scattering coefficient of the suspension was calculated to be μs' = 1.9 cm-1. In Fig. 2a (left) the incident beam was linearly polarized at +45° with respect to the x-axis and the analyzer consisted of a linear polarizer oriented along the y-axis. Fig. 2b (right) was obtained with a right-hand circularly polarized incident beam and the linear polarization analyzer oriented along x-axis. The line that enters the images from the right is caused by the needle that holds the optical mask, which obscures the center of the images. To better illustrate the azimuthal dependence of the intensity decay the images are shown in a false-color depiction. The yellow areas in the corner of the images are outside the cylindrical beaker that contains the particle suspensions.

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Fig. 3.
Fig. 3. This chart shows how to measure the various Mueller matrix elements. A two letter combination stands for one measurement. For example the combination (HV) means that the incoming light is linearly polarized along the horizontal axis (x-axis, see Fig. 2) and the analyzer is set to transmit light that is linearly polarized along the vertical axis (y-axis). For example, to calculate M22 four measurements are necessary: (HH), (VV), (HV) and (VH).

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Fig. 4.
Fig. 4. Calculation of the Matrix element M22 from images obtained for the same polystyrene-sphere suspensions as shown in Fig. 2. Note that the color codes are different for the two sides of the equation, to allow for negative values of the matrix elementt.

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Fig. 5.
Fig. 5. Diffuse backscattering Mueller matrix elements of polystyrene-sphere suspension withmonodisperse particle diameter of d = 2040nm (μs' = 1.9 cm-1). The upper left corner is M11, the lower right corner is M44 (see Eq. 2 and Fig. 3). The color scale is normalized so that the maximum intensity of the M11 element equals 1. All images displayed are 3.5cm x 3.5cm.

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Fig. 6.
Fig. 6. Diffuse backscattering Mueller matrix elements of a polystyrene-sphere suspension with monodisperse particle diameter of d = 2040nm (with μs' = 1.9 cm-1). All images displayed are 3.5cm x 3.5cm.

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Fig. 7.
Fig. 7. Radial dependence of M44 obtained from 4 different polystyrene-sphere suspensions. The numbers indicate the diameter, d, of the spheres in the suspension. The radial distance is given in units of transport mean free path (mfp').

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Fig.8.
Fig.8. Diffuse-backscattering Mueller matrix of M1 cell suspension with 108 cells/cm-3.

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Fig.9.
Fig.9. Diffuse-backscattering Mueller matrix of cancerous MR1 cell suspension with 108 cells/cm3.

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Fig. 10.
Fig. 10. Radial dependence of M44 obtained from cell suspensions . As a reference the result for a 497-nm-polystyrene-sphere suspension is also shown (see Fig.7).

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

S = [ I Q U V ] = [ 〈 E h E h * + E v E v * 〉 〈 E h E h * − E v E v * 〉 〈 E h E v * + E v E h * 〉 〈 i ( E h E v * − E v E h * ) 〉 ]
M = [ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ]
[ I Q U V ] = [ 1 − 1 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 ] • [ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] • [ 1 1 0 0 ]
[ I Q U V ] = [ M 11 + M 12 − M 21 − M 22 − M 11 − M 12 + M 21 + M 22 0 0 ]
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