Abstract

We combine a Monte Carlo technique with Mie theory to develop a method for simulating optical coherence tomography (OCT) imaging through homogeneous turbid media. In our model the propagating light is represented by a plane wavelet; its line propagation direction and path length in the turbid medium are determined by the Monte Carlo technique, and the process of scattering by small particles is computed according to Mie theory. Incorporated into the model is the numerical phase function obtained with Mie theory. The effect of phase function on simulation is also illustrated. Based on this improved Monte Carlo technique, OCT imaging is directly simulated and phase information is recorded. Speckles, resolution, and coherence gating are discussed. The simulation results show that axial and transversal resolutions decrease as probing depth increases. Adapting a light source with a low coherence improves the resolution. The selection of an appropriate coherence length involves a trade-off between intensity and resolution.

© 2004 Optical Society of America

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2002

2001

2000

1999

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

1998

D. J. Smithies, T. Lindmo, Z. P. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

1997

1995

Y. Pan, R. Birngruber, J. Rosperich, R. Engelhardt, “Low-coherence optical tomography in turbid tissue: theoretical analysis,” Appl. Opt. 34, 6564–6574 (1995).
[CrossRef] [PubMed]

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. M. Schmit, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).
[CrossRef]

1994

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Andersen, P. E.

Anderson, P. E.

Bartel, S.

Bashkansky, M.

Bigio, I. J.

Birngruber, R.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bonner, R. F.

J. M. Schmit, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, Z.

Chen, Z. P.

D. J. Smithies, T. Lindmo, Z. P. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Daria, V. R.

De Mul, F. F. M.

Ding, Z.

Duker, J. S.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

Dunn, A.

A. Dunn, “Light scattering properties of cells,” Ph.D. dissertation (University of Texas at Austin, 1997).

Eick, A. A.

Engelhardt, R.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Freyer, J. P.

Fujimoto, J. G.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gan, X.

X. Gan, S. Schilders, M. Gu, “Fluorescence microscopic imaging through tissue-like turbid media,” J. Appl. Phys. 87, 3214–3221 (2000).
[CrossRef]

X. Gan, M. Gu, “Spatial distribution of single-photon and two-photon fluorescence light in scattering media: Monte Carlo simulation,” Appl. Opt. 39, 1575–1579 (2000).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gu, M.

X. Gan, M. Gu, “Spatial distribution of single-photon and two-photon fluorescence light in scattering media: Monte Carlo simulation,” Appl. Opt. 39, 1575–1579 (2000).
[CrossRef]

X. Gan, S. Schilders, M. Gu, “Fluorescence microscopic imaging through tissue-like turbid media,” J. Appl. Phys. 87, 3214–3221 (2000).
[CrossRef]

Hee, M. R.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hielscher, A. H.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Izatt, J. A.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

Jacques, S. L.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Jorgensen, T. M.

Kawata, S.

Knuttel, A.

Koblinger, L.

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

Lin, C. P.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lindmo, T.

D. J. Smithies, T. Lindmo, Z. P. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Lux, I.

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

Milner, T. E.

D. J. Smithies, T. Lindmo, Z. P. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Mourant, J. R.

Nelson, J. S.

Y. Zhao, Z. Chen, Z. Ding, H. Ren, J. S. Nelson, “Real-time phase resolved functional optical coherence tomography by use of optical Hilbert transformation,” Opt. Lett. 27, 98–100 (2002).
[CrossRef]

D. J. Smithies, T. Lindmo, Z. P. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Owen, G. M.

Pan, Y.

Petoukhova, A. L.

Puliafito, C. A.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Reichel, E.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

Reintjes, J.

Ren, H.

Rosperich, J.

Saloma, C.

Schilders, S.

X. Gan, S. Schilders, M. Gu, “Fluorescence microscopic imaging through tissue-like turbid media,” J. Appl. Phys. 87, 3214–3221 (2000).
[CrossRef]

Schmit, J. M.

J. M. Schmit, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).
[CrossRef]

Schmitt, J. M.

J. M. Schmitt, “Array detection for speckle reduction in optical coherence microscopy,” Phys. Med. Biol. 42, 1427–1439 (1997).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knuttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
[CrossRef]

S. H. Xiang, L. Zhou, J. M. Schmitt, “Speckle noise reduction for optical coherence tomography,” in Optical and Imaging Techniques for Biomonitoring III, H.-J. Foth, R. Marchesini, H. Podbielska, eds., Proc. SPIE3196, 79–88 (1998).
[CrossRef]

Schuman, J. S.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Shen, D.

Smithies, D. J.

D. J. Smithies, T. Lindmo, Z. P. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Steenbergen, W.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Thrane, L.

Tycho, A.

Wang, L. H.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Wang, L. V.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Xiang, S. H.

S. H. Xiang, L. Zhou, J. M. Schmitt, “Speckle noise reduction for optical coherence tomography,” in Optical and Imaging Techniques for Biomonitoring III, H.-J. Foth, R. Marchesini, H. Podbielska, eds., Proc. SPIE3196, 79–88 (1998).
[CrossRef]

Yadlowsky, M.

J. M. Schmit, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).
[CrossRef]

Yao, G.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Yura, H. T.

Zhao, Y.

Zheng, L. Q.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zhou, L.

S. H. Xiang, L. Zhou, J. M. Schmitt, “Speckle noise reduction for optical coherence tomography,” in Optical and Imaging Techniques for Biomonitoring III, H.-J. Foth, R. Marchesini, H. Podbielska, eds., Proc. SPIE3196, 79–88 (1998).
[CrossRef]

Appl. Opt.

Comput. Methods Programs Biomed.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Dermatology

J. M. Schmit, M. Yadlowsky, R. F. Bonner, “Subsurface imaging of living skin with optical coherence microscopy,” Dermatology 191, 93–98 (1995).
[CrossRef]

J. Appl. Phys.

X. Gan, S. Schilders, M. Gu, “Fluorescence microscopic imaging through tissue-like turbid media,” J. Appl. Phys. 87, 3214–3221 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Opthalmology

C. A. Puliafito, M. R. Hee, C. P. Lin, E. Reichel, J. S. Schuman, J. S. Duker, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, “Imaging of macular disease with optical coherence tomography,” Opthalmology 120, 217–229 (1995).

Phys. Med. Biol.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

D. J. Smithies, T. Lindmo, Z. P. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

J. M. Schmitt, “Array detection for speckle reduction in optical coherence microscopy,” Phys. Med. Biol. 42, 1427–1439 (1997).
[CrossRef] [PubMed]

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other

A. Dunn, “Light scattering properties of cells,” Ph.D. dissertation (University of Texas at Austin, 1997).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

S. H. Xiang, L. Zhou, J. M. Schmitt, “Speckle noise reduction for optical coherence tomography,” in Optical and Imaging Techniques for Biomonitoring III, H.-J. Foth, R. Marchesini, H. Podbielska, eds., Proc. SPIE3196, 79–88 (1998).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Geometrical schematic of the transform of the coordinate systems. ê denotes the unit vector for a given spatial direction. ê sn and ê pn , n = 1 … m denote unit vectors of perpendicular and parallel components, respectively. r n , n = 1 … m, denotes the free path with length l n and direction ê rn . θ n and φ n , n = 1 … m, denote scattering and azimuthal angles, respectively. The relation among these vectors can be denoted by ê s ê p , ê s × ê p = ê r , ê p = ê θ, ê s = -ê φ.

Fig. 2
Fig. 2

Contours of g values for microspheres with different radii and different refractive indices. The refractive index of the medium is 1.33; wavelength, 0.6328 μm. The g value is 0.9 in point A (1.3603, 0.3577) and in point B (1.6303, 7.2737).

Fig. 3
Fig. 3

Mie and H–G phase functions with the same g value of 0.9 for points A and B, which correspond to those in Fig. 2.

Fig. 4
Fig. 4

Contours of the calculated two-dimensional phase function p(θ, φ) with a microsphere refractive index of 1.55, medium refractive index of 1.33, microsphere radius of 0.2 μm, and wavelength of 0.6328 μm.

Fig. 5
Fig. 5

Comparison of our CMC program with that reported in Ref. 18. Plot A is the m 12 value from Ref. 18, and plot B is that from our CMC program.

Fig. 6
Fig. 6

Intensities of transmitted light for different phase functions. Points A and B have the same g value of 0.9 and correspond to those in Fig. 2. Number densities of microspheres in the solution are selected to equal the mfp value of 100 μm. The turbid phantom has a thickness of 300 μm.

Fig. 7
Fig. 7

Patterns of the DOP for different phase functions: (a) point A with Mie phase function, (b) point A with H–G phase function, (c) point B with Mie phase function, (d) point B with H–G phase function. Pattern size is 1 mm × 1 mm.

Fig. 8
Fig. 8

Simulated one-dimensional axial image with the following parameters: microsphere of size 1.2 μm and refractive index of 1.565 in an aqueous suspension (refractive index 1.329) with a number density of 0.006 per cubic micrometer. The sharp-edged mirror is embedded at a depth of 1–6 mfp’s under the surface (1 mfp = 103.15 μm). The wavelength is 0.8 μm. Plot A is the simulated OCT signal, and plot B is the exponential fit curve of data at 1–3 mfp’s. The coherence length of the light source is 7.5 μm in free space. The detector has a radius of 100 μm.

Fig. 9
Fig. 9

Axial images of a sharp-edged mirror embedded at different depths. The parameters are the same as those in Fig. 8. Plots A, B, and C denote the images at depths of 1, 3, and 5 mfp’s, respectively. Plot D is the image without a low-pass filter. Plot C is the envelope of the raw image demodulated after the Hilbert transform.

Fig. 10
Fig. 10

Axial resolution as a function of the depth at which a sharp-edged mirror is embedded. Results with coherence lengths of 30 μm (plot A), 15 μm (plot B), and 7.5 μm (plot C) are presented.

Fig. 11
Fig. 11

Transverse resolution as a function of the depth at which a sharp-edged mirror is embedded. Results with coherence lengths of 30 μm (plot A), 15 μm (plot B), and 7.5 μm (plot C) are presented.

Fig. 12
Fig. 12

Axial resolution as a function of coherence length for different image depths. Light with wavelength of 0.6328 μm incidents on the turbid medium, which is composed of microspheres with a radius of 1.0 μm, a refractive index of 1.573, and a number density of 0.006 per cubic micrometers suspended in water, which has a refractive index of 1.332. Results for image depths of 5 mfp (plot A; 1 mfp = 155 μm) and 3 mfp (plot B) are given.

Fig. 13
Fig. 13

Signal intensity as a function of coherence length for different image depths. The parameters are the same as those used in Fig. 12.

Equations (14)

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En=Tln-1En-1=En-1 expjkln-1êrn-1,
EpnEsn=SθnEpnEsn=expjπ/2s2θns1θnEpnEsn,
EpnEsn=RφnEpnEsn=cos φn-sin φnsin φncos φnEpnEsn,
EpnEsn=SθnRφnTrn-1Epn-1Esn-1.
Em=TlmSθmRφmTl1Sθ1Rφ1Tr0E0.
Emean=i=1NEi,m=i=1NM n=1mTli,nSθi,nRφi,nTl0Ei,0,
êsm, êpm, êrm=êx, êy, êzM,
pθ, φ  Isθ, φ=cos2 φ|s2θEp|2+|s1θEs|2+sin2 φ|s2θEs|2+|s1θEp|2-sin 2φ|s2θ|2-|s1θ|2ReEpEs*,
pθθ=2π pθ, φdφ=|s1θ|2+|s2θ|2|Ep|2+|Es|2=|s1θ|2+|s2θ|2Ii,
pφ|θnφ=A cos2 φ+B sin2 φ+C sin 2φ,
A|s2θnEp|2+|s1θnEs|2,B|s2θnEs|2+|s1θnEp|2,C|s1θn|2-|s2θn|2ReEpEs*.
Ihet=ReER* n EnstΓLnt,
ΓLn=1, if |Ln|<Lc/2 =0, else,
DOP=Q2+U2+V21/2/I.

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