Abstract

We measured angular-resolved diffuse reflectance in tissue samples of different anisotropic characteristics. Experimental measurements were compared with theoretical results based on the diffusion approximation. The results indicated that the angular distribution in isotropic tissue was the same as in isotropic phantoms. Under normal incidence, the measured angular profiles of diffuse reflectance approached the Lambertian distribution when the evaluation location was far away from the incident point. The skewed angular profiles observed under oblique incidence could be explained using the diffuse model. The anisotropic tissue structures in muscle showed clear effects on the measurements especially at locations close to the light incidence. However, when measuring across the muscle fiber orientations, the results were in good agreement with those obtained in isotropic samples.

© 2007 Optical Society of America

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2007

2006

J. Ranasinghesagara, F. Hsieh, and G. Yao, "A photon migration method for quantifying fiber formation in meat analogs," J. Food Sci. 71, E227-231 (2006).
[CrossRef]

A. Kienle and R. Hibst, "Light guiding in biological tissue due to scattering," Phys. Rev. Lett. 97, 018104 (2006).
[CrossRef] [PubMed]

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, "Monitoring sarcomere structure changes in whole muscle using diffuse light reflectance," J. Biomed. Opt. 11, 040504 (2006).
[CrossRef] [PubMed]

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

2005

2004

A. Garcia-Uribe, N. Kehtarnavaz, G. Marquez, V. Prieto, M. Duvic, and L. H. Wang, "Skin cancer detection by spectroscopic oblique-incidence reflectometry: classification and physiological origins," Appl. Opt. 43, 2643-2650 (2004).
[CrossRef] [PubMed]

I. J. Bigio and S. G. Bown, "Spectroscopic sensing of cancer and cancer therapy: current status of translational research," Cancer Biother. 3, 259-267 (2004).

M. A. ElHelw, B. P. Lo, A. J. Chung, A. Darzi, and G. Z. Yang, "Photorealistic rendering of large tissue deformation for surgical simulation," Lecture Notes Comput. Sci. 3217, 355-362 (2004).
[CrossRef]

2002

K. Sokolov, M. Follen, and R. Richards-Kortum, "Optical spectroscopy for detection of neoplasia," Curr. Opin. Chem. Biol. 6, 651-658 (2002).
[CrossRef] [PubMed]

2000

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Kramer, and M. S. Patterson, "Anisotropy of light propagation in human skin," Phys. Med. Biol. 45, 2873-2886 (2000).
[CrossRef] [PubMed]

J. Y. Qu, Z. Huang, and J. Hua, "Excitation-and-collection geometry insensitive fluorescence imaging of tissue-simulating turbid media," Appl. Opt. 39, 3344-3356 (2000).
[CrossRef]

1998

1997

1996

1995

1994

K. M. Hebeda, T. Menovsky, J. F. Beek, J. G. Wolbers, and M. J. C. van Gemert, "Light propagation in the brain depends on nerve fiber orientation," Neurosurgery 35, 720-724 (1994).
[CrossRef] [PubMed]

G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, and W. Zinth, "Time-gated transillumination of biological tissues and tissue-like phantoms," Appl. Opt. 33, 6699-6710 (1994).
[CrossRef] [PubMed]

1992

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

1989

A. M. Pearson and R. B. Young, Muscle and Meat Biochemistry (Academic, 1989).

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, and M. S. Patterson, "Monte-Carlo modeling of light propagation in highly scattering tissues-II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

1978

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Chaps. 7 and 9.

Alfano, R. R.

B. Chance and R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, Proc. SPIE 2979 (1997).

R. R. Alfano and J. G. Fujimoto, eds., Advances in Optical Imaging and Photon Imaging, Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1996).

Beek, J. F.

K. M. Hebeda, T. Menovsky, J. F. Beek, J. G. Wolbers, and M. J. C. van Gemert, "Light propagation in the brain depends on nerve fiber orientation," Neurosurgery 35, 720-724 (1994).
[CrossRef] [PubMed]

Bevilacqua, F.

Bigio, I. J.

I. J. Bigio and S. G. Bown, "Spectroscopic sensing of cancer and cancer therapy: current status of translational research," Cancer Biother. 3, 259-267 (2004).

Binzoni, T.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Bown, S. G.

I. J. Bigio and S. G. Bown, "Spectroscopic sensing of cancer and cancer therapy: current status of translational research," Cancer Biother. 3, 259-267 (2004).

Chance, B.

B. Chance and R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, Proc. SPIE 2979 (1997).

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

Chung, A. J.

M. A. ElHelw, B. P. Lo, A. J. Chung, A. Darzi, and G. Z. Yang, "Photorealistic rendering of large tissue deformation for surgical simulation," Lecture Notes Comput. Sci. 3217, 355-362 (2004).
[CrossRef]

Coquoz, O.

Courvoisier, C.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Darzi, A.

M. A. ElHelw, B. P. Lo, A. J. Chung, A. Darzi, and G. Z. Yang, "Photorealistic rendering of large tissue deformation for surgical simulation," Lecture Notes Comput. Sci. 3217, 355-362 (2004).
[CrossRef]

Delpy, D. T.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Depeursinge, C.

Drezek, R.

Duvic, M.

ElHelw, M. A.

M. A. ElHelw, B. P. Lo, A. J. Chung, A. Darzi, and G. Z. Yang, "Photorealistic rendering of large tissue deformation for surgical simulation," Lecture Notes Comput. Sci. 3217, 355-362 (2004).
[CrossRef]

Essenpreis, M.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Kramer, and M. S. Patterson, "Anisotropy of light propagation in human skin," Phys. Med. Biol. 45, 2873-2886 (2000).
[CrossRef] [PubMed]

Farrell, T. J.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Kramer, and M. S. Patterson, "Anisotropy of light propagation in human skin," Phys. Med. Biol. 45, 2873-2886 (2000).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

Flock, S. T.

S. T. Flock, B. C. Wilson, and M. S. Patterson, "Monte-Carlo modeling of light propagation in highly scattering tissues-II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

Follen, M.

K. Sokolov, M. Follen, and R. Richards-Kortum, "Optical spectroscopy for detection of neoplasia," Curr. Opin. Chem. Biol. 6, 651-658 (2002).
[CrossRef] [PubMed]

Fu, K.

Fujimoto, J. G.

R. R. Alfano and J. G. Fujimoto, eds., Advances in Optical Imaging and Photon Imaging, Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1996).

Garcia-Uribe, A.

Gebhart, S. C.

Gerrard, D. E.

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, "Monitoring sarcomere structure changes in whole muscle using diffuse light reflectance," J. Biomed. Opt. 11, 040504 (2006).
[CrossRef] [PubMed]

Gharbi, T.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Giust, R.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Hebden, J. C.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Hebeda, K. M.

K. M. Hebeda, T. Menovsky, J. F. Beek, J. G. Wolbers, and M. J. C. van Gemert, "Light propagation in the brain depends on nerve fiber orientation," Neurosurgery 35, 720-724 (1994).
[CrossRef] [PubMed]

Hermann, M.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Kramer, and M. S. Patterson, "Anisotropy of light propagation in human skin," Phys. Med. Biol. 45, 2873-2886 (2000).
[CrossRef] [PubMed]

Hibst, R.

Hsieh, F.

J. Ranasinghesagara, F. Hsieh, and G. Yao, "A photon migration method for quantifying fiber formation in meat analogs," J. Food Sci. 71, E227-231 (2006).
[CrossRef]

Hua, J.

Huang, Z.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Chaps. 7 and 9.

Jacques, S. L.

Jiang, H. B.

Kehtarnavaz, N.

Kienle, A.

Kolzer, J.

Kramer, U.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Kramer, and M. S. Patterson, "Anisotropy of light propagation in human skin," Phys. Med. Biol. 45, 2873-2886 (2000).
[CrossRef] [PubMed]

Leung, T. S.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Lilge, L.

Lin, A. W. H.

Lin, S. P.

Lin, S.-P.

Lin, W. C.

Lo, B. P.

M. A. ElHelw, B. P. Lo, A. J. Chung, A. Darzi, and G. Z. Yang, "Photorealistic rendering of large tissue deformation for surgical simulation," Lecture Notes Comput. Sci. 3217, 355-362 (2004).
[CrossRef]

Mahadevan-Jansen, A.

Marquet, P.

Marquez, G.

Menovsky, T.

K. M. Hebeda, T. Menovsky, J. F. Beek, J. G. Wolbers, and M. J. C. van Gemert, "Light propagation in the brain depends on nerve fiber orientation," Neurosurgery 35, 720-724 (1994).
[CrossRef] [PubMed]

Mitic, G.

Nickell, S.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Kramer, and M. S. Patterson, "Anisotropy of light propagation in human skin," Phys. Med. Biol. 45, 2873-2886 (2000).
[CrossRef] [PubMed]

Otto, J.

Patterson, M. S.

S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U. Kramer, and M. S. Patterson, "Anisotropy of light propagation in human skin," Phys. Med. Biol. 45, 2873-2886 (2000).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, "Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue," Appl. Opt. 35, 2304-2314 (1996).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, and M. S. Patterson, "Monte-Carlo modeling of light propagation in highly scattering tissues-II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

Pearson, A. M.

A. M. Pearson and R. B. Young, Muscle and Meat Biochemistry (Academic, 1989).

Plies, E.

Prieto, V.

Qu, J. Y.

Ranasinghesagara, J.

J. Ranasinghesagara and G. Yao, "Imaging 2D diffuse reflectance in skeleton muscle," Opt. Express 15, 3998-4007 (2007).
[CrossRef] [PubMed]

J. Ranasinghesagara, F. Hsieh, and G. Yao, "A photon migration method for quantifying fiber formation in meat analogs," J. Food Sci. 71, E227-231 (2006).
[CrossRef]

Richards-Kortum, R.

K. Sokolov, M. Follen, and R. Richards-Kortum, "Optical spectroscopy for detection of neoplasia," Curr. Opin. Chem. Biol. 6, 651-658 (2002).
[CrossRef] [PubMed]

Roux, J.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Schwartz, J. A.

Sokolov, K.

K. Sokolov, M. Follen, and R. Richards-Kortum, "Optical spectroscopy for detection of neoplasia," Curr. Opin. Chem. Biol. 6, 651-658 (2002).
[CrossRef] [PubMed]

Solkner, G.

Steiner, R.

Sun, J.

Thomsen, S. L.

Tittel, F. K.

Tribillion, G.

T. Binzoni, C. Courvoisier, R. Giust, G. Tribillion, T. Gharbi, J. C. Hebden, T. S. Leung, J. Roux, and D. T. Delpy, "Anisotropic photon migration in human skeletal muscle," Phys. Med. Biol. 51, N79-N90 (2006).
[CrossRef] [PubMed]

Utzinger, U.

Van, M. J. C.

A. J. Welch and M. J. C. Van, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum Press, 1995).

van Gemert, M. J. C.

K. M. Hebeda, T. Menovsky, J. F. Beek, J. G. Wolbers, and M. J. C. van Gemert, "Light propagation in the brain depends on nerve fiber orientation," Neurosurgery 35, 720-724 (1994).
[CrossRef] [PubMed]

Wang, A.

Wang, L.

Wang, L. H.

Wang, L. V.

Wang, L.-H.

Weaver, A.

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, "Monitoring sarcomere structure changes in whole muscle using diffuse light reflectance," J. Biomed. Opt. 11, 040504 (2006).
[CrossRef] [PubMed]

Welch, A. J.

A. J. Welch and M. J. C. Van, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum Press, 1995).

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

Wilson, B. C.

Wolbers, J. G.

K. M. Hebeda, T. Menovsky, J. F. Beek, J. G. Wolbers, and M. J. C. van Gemert, "Light propagation in the brain depends on nerve fiber orientation," Neurosurgery 35, 720-724 (1994).
[CrossRef] [PubMed]

Xia, J.

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, "Monitoring sarcomere structure changes in whole muscle using diffuse light reflectance," J. Biomed. Opt. 11, 040504 (2006).
[CrossRef] [PubMed]

Yang, G. Z.

M. A. ElHelw, B. P. Lo, A. J. Chung, A. Darzi, and G. Z. Yang, "Photorealistic rendering of large tissue deformation for surgical simulation," Lecture Notes Comput. Sci. 3217, 355-362 (2004).
[CrossRef]

Yao, G.

J. Ranasinghesagara and G. Yao, "Imaging 2D diffuse reflectance in skeleton muscle," Opt. Express 15, 3998-4007 (2007).
[CrossRef] [PubMed]

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, "Monitoring sarcomere structure changes in whole muscle using diffuse light reflectance," J. Biomed. Opt. 11, 040504 (2006).
[CrossRef] [PubMed]

J. Ranasinghesagara, F. Hsieh, and G. Yao, "A photon migration method for quantifying fiber formation in meat analogs," J. Food Sci. 71, E227-231 (2006).
[CrossRef]

Young, R. B.

A. M. Pearson and R. B. Young, Muscle and Meat Biochemistry (Academic, 1989).

Zinth, W.

Appl. Opt.

G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, and W. Zinth, "Time-gated transillumination of biological tissues and tissue-like phantoms," Appl. Opt. 33, 6699-6710 (1994).
[CrossRef] [PubMed]

L.-H. Wang and S. L. Jacques, "Use of a laser beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium," Appl. Opt. 34, 2362-2366 (1995).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, "Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue," Appl. Opt. 35, 2304-2314 (1996).
[CrossRef] [PubMed]

S.-P. Lin, L.-H. Wang, S. L. Jacques, and F. K. Tittel, "Measurement of tissue optical properties by the use of oblique-incidence optical fiber reflectometry," Appl. Opt. 36, 136-143 (1997).
[CrossRef] [PubMed]

F. Bevilacqua, P. Marquet, O. Coquoz, and C. Depeursinge, "Role of tissue structure in photon migration through breast tissues," Appl. Opt. 36, 44-51 (1997).
[CrossRef] [PubMed]

J. Y. Qu, Z. Huang, and J. Hua, "Excitation-and-collection geometry insensitive fluorescence imaging of tissue-simulating turbid media," Appl. Opt. 39, 3344-3356 (2000).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram for measuring the angular resolved diffuse reflectance. In the experiments, the incident fiber delivered light at 0° or 40° within the incident plane; the detection fiber collected reflectance while rotating within the detection plane.

Fig. 2
Fig. 2

Normalized-angular distribution of diffuse reflectance for three phantoms at (a) Δ y = 0.8   mfp and (b) Δ y = 2.0   mfp . Optical properties of the phantoms were: μ s = 3.3 cm 1 and μ a = 2.0 × 10 4 cm 1 , μ s = 5.8 cm 1 and μ a = 1.0 × 10 4 cm 1 , and μ s = 12.4 cm 1 and μ a = 1.0 × 10 4 cm 1 . The comparison between experiments and the Lambertian distribution was shown in (c), where the phantom that we used had μ s = 5.8 cm 1 and μ a = 1.0 × 10 4 cm 1 .

Fig. 3
Fig. 3

Normalized angular distribution of diffuse reflectance measured in chicken liver tissue at Δ y = 2.36 and Δ y = 3.64   mfp from the incident location. (Sample optical properties were μ s = 8.7 cm 1 and μ a = 0.5 cm 1 .)

Fig. 4
Fig. 4

Normalized angular distribution of diffuse reflectance in muscle measured with (a) across, (b) parallel, and (c) end arrangements. Optical properties of muscle tissue were: μ s = 3.6 cm 1 and μ a = 2.0 × 10 2 cm 1 measured with the cross arrangement, μ s = 6.0 cm 1 and μ a = 7.5 × 10 2 cm 1 measured with the parallel arrangement, and μ s = 4.6 cm 1 , μ a = 4.2 × 10 3 cm 1 measured with end arrangement.

Fig. 5
Fig. 5

Normalized angular distribution of diffuse reflectance of intralipid phantom under 40° incidence measured around an axis passing through (a) incident point, and (b) diffuse center. The optical properties of the phantom were the same as in Fig. 2(c).

Fig. 6
Fig. 6

The normalized angular distribution of diffuse reflectance centered at the incident line in liver tissue under 40° incidence. The distance between the incident and detection planes was Δ y = 2.36 and Δ y = 3.64   mfp (Sample optical properties were μ s = 8.7 cm 1 and μ a = 0.5 cm 1 .).

Fig. 7
Fig. 7

Normalized angular distribution of diffuse reflectance measured at the incident line in skeletal muscle under 40° incidence, and with different geometrical arrangements. Optical properties of muscle tissue were: μ s = 3.6 cm 1 and μ a = 1.5 × 10 2 c m 1 with the cross arrangement, μ s = 7.1 cm 1 and μ a = 4.0 × 10 4 cm 1 with the end arrangement, and μ s = 4.5 cm 1 and μ a = 6.6 × 10 2 cm 1 with the parallel arrangement.

Fig. 8
Fig. 8

Normalized angular distribution of diffuse reflectance measured at the diffuse line in skeletal muscle under 40° incidence, and with different geometrical arrangements. Optical properties of muscle tissue were: μ s = 3.6 cm 1 and μ a = 1.5 × 10 2 cm 1 with the cross arrangement, μ s = 7.1 cm 1 and μ a = 4.0 × 10 4 cm 1 with the end arrangement, and μ s = 4.5 cm 1 and μ a = 6.6 × 10 2 cm 1 with the parallel arrangement.

Equations (119)

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μ s
400   μm
( 692.4   nm )
( Δ y )
40 °
( n = 1.52 )
0.13 0.16   mm
1 .0   mm
0.63   mm
0 .85   mm
m f p
1  mfp = 1 / μ t
μ t = μ s + μ a
1 / μ t + 4 A D
R ( r )
R ( r ) = D ϕ z | z = 0 ,
z = 0
R ( r , u ) = D ( ϕ x u x + ϕ y u y + ϕ z u z ) | z = 0 ,
u = ( u x , u y , u z )
u x 2 + u y 2 + u z 2 = 1
Δ x = 0
u y = 0
R ( r , θ z ) = D ϕ z | z , x = 0 cos ( θ z ) cos ( θ z ) ,
θ z
θ z
Δ x
Δ x = sin θ t μ t ,
θ t
( Δ y < 1   mfp )
( Δ y > 2   mfp )
( μ a )
( μ s )
Δ x = 0
Δ y = 0.8
2.0   mfp
Δ y
3.64   mfp
μ s
5.8 cm 1
Δ y = 0.8
3.64   mfp
Δ y = 2.36
3.64   mfp
μ s = 8.7 cm 1
μ a = 0.5 cm 1
Δ y = 2.36   mfp
Δ y = 2.36
3.64   mfp
Δ y = 0.8 3.64   mfp
Δ y = 0.8
3.64   mfp
Δ y = 0.8   mfp
3 .64   mfp
Δ y = 0.8
3.64   mfp
( Δ y )
2   mfp
( Δ y > 1   mfp )
( Δ x = 0 )
Δ y = 0.8
3.64   mfp
Δ y = 0.8
3.64   mfp
Δ y = 2
3.64   mfp
( Δ y = 0.8   mfp )
Δ y = 0.8   mfp
Δ x = 0
Δ y = 2.36
3.64   mfp
Δ y = 3.64   mfp
Δ y = 2.36
3.64   mfp
Δ y = 2.36
3.64   mfp
Δ y = 0.8
3.64   mfp
Δ y = 3.64   mfp
Δ y = 0.8   mfp
( Δ y = 0.8   mfp )
Δ y = 3.64   mfp
Δ y = 3.64   mfp
( Δ y = 3.64   mfp )
Δ y = 0.8   mfp
Δ y = 2.0   mfp
μ s = 3.3 cm 1
μ a = 2.0 × 10 4 cm 1
μ s = 5.8 cm 1
μ a = 1.0 × 10 4 cm 1
μ s = 12.4 cm 1
μ a = 1.0 × 10 4 cm 1
μ s = 5.8 cm 1
μ a = 1.0 × 10 4 cm 1
Δ y = 2.36
Δ y = 3.64   mfp
μ s = 8.7 cm 1
μ a = 0.5 cm 1
μ s = 3.6 cm 1
μ a = 2.0 × 10 2 cm 1
μ s = 6.0 cm 1
μ a = 7.5 × 10 2 cm 1
μ s = 4.6 cm 1
μ a = 4.2 × 10 3 cm 1
Δ y = 2.36
Δ y = 3.64   mfp
μ s = 8.7 cm 1
μ a = 0.5 cm 1
μ s = 3.6 cm 1
μ a = 1.5 × 10 2 c m 1
μ s = 7.1 cm 1
μ a = 4.0 × 10 4 cm 1
μ s = 4.5 cm 1
μ a = 6.6 × 10 2 cm 1
μ s = 3.6 cm 1
μ a = 1.5 × 10 2 cm 1
μ s = 7.1 cm 1
μ a = 4.0 × 10 4 cm 1
μ s = 4.5 cm 1
μ a = 6.6 × 10 2 cm 1

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