Abstract

This study is devoted to the development and application of a Monte Carlo ray-tracing model to simulate light scattering when a colloid suspension droplet passes through a highly focused Gaussian laser sheet. Within this study, a colloidal suspension droplet refers to a spherical droplet containing multiple spherical inclusions. Such scattering scenarios arise when using the time-shift measurement technique for particle sizing. The incident laser sheet is treated as a large number of polarized light rays: the Stokes vector of each light ray is tracked, achieved by multiplication of the rotation matrix and the Mueller matrix after each scattering event. For the Monte Carlo simulation of light scattering, a very important issue is to generate the deflection angle and azimuthal angle after each scattering event. The scattering from embedded inclusions is computed using the Lorenz-Mie theory and by employing the rejection sampling technique to update the new propagation direction. Multi-reflection and refraction within the droplet is accounted for, as is total reflection at the drop interface. For this, the Mueller matrix formulation is invoked at the drop surface to update the Stokes vector. To validate this simulation code, the scattering diagram from a nanoparticle is computed with this Monte Carlo method and compared with the scattering diagram computed with the Lorenz-Mie theory, the agreement is excellent. This Monte Carlo code is then applied to simulate signals arising from a time-shift device, when a colloid suspension droplet passes through a focused polarized laser sheet, with the objective of measuring the concentration of colloidal particles in the droplet. Measurements verify the ability of the code to properly simulate this light scattering scenario.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

P. G. Stegmann, B. Sun, J. Ding, P. Yang, and X. Zhang, “Study of the effects of phytoplankton morphology and vertical profile on lidar attenuated backscatter and depolarization ratio,” J. Quant. Spectrosc. Radiat. Transfer 225, 1–15 (2019).
[Crossref]

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Light scattering from a drop with an embedded particle and its exploitation in the time-shift technique,” J. Quant. Spectrosc. Radiat. Transfer 227, 20–31 (2019).
[Crossref]

2018 (2)

M. I. Mishchenko and P. Yang, “Far-field Lorenz–Mie scattering in an absorbing host medium: theoretical formalism and FORTRAN program,” J. Quant. Spectrosc. Radiat. Transfer 205, 241–252 (2018).
[Crossref]

C. Li, X. C. Wu, J. Z. Cao, L. H. Chen, G. Gréhan, and K. F. Cen, “Application of rainbow refractometry for measurement of droplets with solid inclusions,” Opt. Laser Technol. 98, 354–362 (2018).
[Crossref]

2016 (3)

M. P. Sentis, F. Onofri, L. Méès, and S. Radev, “Scattering of light by large bubbles: Coupling of geometrical and physical optics approximations,” J. Quant. Spectrosc. Radiat. Transfer 170, 8–18 (2016).
[Crossref]

P. G. Stegmann, C. Tropea, E. Järvinen, and M. Schnaiter, “Comparison of measured and computed phase functions of individual tropospheric ice crystals,” J. Quant. Spectrosc. Radiat. Transfer 178, 379–389 (2016).
[Crossref]

S. Rosenkranz, W. Schäfer, C. Tropea, and A. M. Zoubir, “Modeling photon transport in turbid media for measuring colloidal concentration in drops using the time-shift technique,” Appl. Opt. 55(34), 9703–9711 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (2)

H. Yu, F. Xu, and C. Tropea, “Simulation of optical caustics associated with the secondary rainbow of oblate droplets,” Opt. Lett. 38(21), 4469–4472 (2013).
[Crossref]

D. Jakubczyk, G. Derkachov, M. Kolwas, and K. Kolwas, “Combining weighting and scatterometry: Application to a levitated droplet of suspension,” J. Quant. Spectrosc. Radiat. Transfer 126, 99–104 (2013).
[Crossref]

2011 (2)

2008 (1)

H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation of light scattering by large air bubbles,” Particuology 6(5), 340–346 (2008).
[Crossref]

2005 (1)

2004 (2)

2003 (2)

2002 (1)

T. Wriedt and R. Schuh, “The inclusion-concentration measurement of suspension droplets based on Monte Carlo ray tracing,” Meas. Sci. Technol. 13(3), 276–279 (2002).
[Crossref]

2001 (2)

Y. X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: A multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 569–579 (2001).
[Crossref]

B. Kaplan, G. Ledanois, and B. Drévillon, “Mueller matrix of dense polystyrene latex sphere suspensions: measurements and Monte Carlo simulation,” Appl. Opt. 40(16), 2769–2777 (2001).
[Crossref]

1999 (2)

1992 (1)

J. P. Briton, B. Maheu, G. Gréhan, and G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9(1-4), 52–58 (1992).
[Crossref]

1971 (1)

E. Collett, “Mueller-Stokes matrix formulation of Fresnel's equations,” Am. J. Phys. 39(5), 517–528 (1971).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Albrecht, H. E.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques. (Springer Science & Business Media, 2013).

Azzam, R. M.

Baum, B.

Y. X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: A multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 569–579 (2001).
[Crossref]

Bergougnoux, L.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. (John Wiley & Sons, 2008).

Borys, M.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques. (Springer Science & Business Media, 2013).

Briton, J. P.

J. P. Briton, B. Maheu, G. Gréhan, and G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9(1-4), 52–58 (1992).
[Crossref]

Cameron, B. D.

Cao, J. Z.

C. Li, X. C. Wu, J. Z. Cao, L. H. Chen, G. Gréhan, and K. F. Cen, “Application of rainbow refractometry for measurement of droplets with solid inclusions,” Opt. Laser Technol. 98, 354–362 (2018).
[Crossref]

Cen, K. F.

C. Li, X. C. Wu, J. Z. Cao, L. H. Chen, G. Gréhan, and K. F. Cen, “Application of rainbow refractometry for measurement of droplets with solid inclusions,” Opt. Laser Technol. 98, 354–362 (2018).
[Crossref]

Chauchard, F.

Chen, L. H.

C. Li, X. C. Wu, J. Z. Cao, L. H. Chen, G. Gréhan, and K. F. Cen, “Application of rainbow refractometry for measurement of droplets with solid inclusions,” Opt. Laser Technol. 98, 354–362 (2018).
[Crossref]

Collett, E.

E. Collett, “Mueller-Stokes matrix formulation of Fresnel's equations,” Am. J. Phys. 39(5), 517–528 (1971).
[Crossref]

Coté, G. L.

Damaschke, N.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques. (Springer Science & Business Media, 2013).

Derkachov, G.

D. Jakubczyk, G. Derkachov, M. Kolwas, and K. Kolwas, “Combining weighting and scatterometry: Application to a levitated droplet of suspension,” J. Quant. Spectrosc. Radiat. Transfer 126, 99–104 (2013).
[Crossref]

Dhez, O.

Ding, J.

P. G. Stegmann, B. Sun, J. Ding, P. Yang, and X. Zhang, “Study of the effects of phytoplankton morphology and vertical profile on lidar attenuated backscatter and depolarization ratio,” J. Quant. Spectrosc. Radiat. Transfer 225, 1–15 (2019).
[Crossref]

Drévillon, B.

Firpo, J. L.

Girasole, T.

Gouesbet, G.

J. P. Briton, B. Maheu, G. Gréhan, and G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9(1-4), 52–58 (1992).
[Crossref]

Gréhan, G.

C. Li, X. C. Wu, J. Z. Cao, L. H. Chen, G. Gréhan, and K. F. Cen, “Application of rainbow refractometry for measurement of droplets with solid inclusions,” Opt. Laser Technol. 98, 354–362 (2018).
[Crossref]

J. P. Briton, B. Maheu, G. Gréhan, and G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9(1-4), 52–58 (1992).
[Crossref]

Hecht, E.

E. Hecht, Optics. (Addison-Wesley, 2002).

Hu, Y. X.

Y. X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: A multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 569–579 (2001).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. (John Wiley & Sons, 2008).

Hulst, H. C.

H. C. Hulst, Light Scattering by Small Particles. (Wiley, 1981).

Jacques, S. L.

Jaillon, F.

Jakubczyk, D.

D. Jakubczyk, G. Derkachov, M. Kolwas, and K. Kolwas, “Combining weighting and scatterometry: Application to a levitated droplet of suspension,” J. Quant. Spectrosc. Radiat. Transfer 126, 99–104 (2013).
[Crossref]

Järvinen, E.

P. G. Stegmann, C. Tropea, E. Järvinen, and M. Schnaiter, “Comparison of measured and computed phase functions of individual tropospheric ice crystals,” J. Quant. Spectrosc. Radiat. Transfer 178, 379–389 (2016).
[Crossref]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics. (Tata McGraw-Hill Education, 1976).

Kaplan, B.

Kattawar, G. W.

Kolwas, K.

D. Jakubczyk, G. Derkachov, M. Kolwas, and K. Kolwas, “Combining weighting and scatterometry: Application to a levitated droplet of suspension,” J. Quant. Spectrosc. Radiat. Transfer 126, 99–104 (2013).
[Crossref]

Kolwas, M.

D. Jakubczyk, G. Derkachov, M. Kolwas, and K. Kolwas, “Combining weighting and scatterometry: Application to a levitated droplet of suspension,” J. Quant. Spectrosc. Radiat. Transfer 126, 99–104 (2013).
[Crossref]

Laurent, J. Y.

Laven, P.

Ledanois, G.

Li, C.

C. Li, X. C. Wu, J. Z. Cao, L. H. Chen, G. Gréhan, and K. F. Cen, “Application of rainbow refractometry for measurement of droplets with solid inclusions,” Opt. Laser Technol. 98, 354–362 (2018).
[Crossref]

Li, L.

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Light scattering from a drop with an embedded particle and its exploitation in the time-shift technique,” J. Quant. Spectrosc. Radiat. Transfer 227, 20–31 (2019).
[Crossref]

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Sensitivity of the time-shift technique in characterizing non-spherical drops,” In: Proceedings of the nineteenth International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics, Lisbon; 16-19 July. 2018.

Lux, I.

I. Lux, Monte Carlo Particle Transport Methods. (CRC, 1991).

Maheu, B.

J. P. Briton, B. Maheu, G. Gréhan, and G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9(1-4), 52–58 (1992).
[Crossref]

Méès, L.

M. P. Sentis, F. Onofri, L. Méès, and S. Radev, “Scattering of light by large bubbles: Coupling of geometrical and physical optics approximations,” J. Quant. Spectrosc. Radiat. Transfer 170, 8–18 (2016).
[Crossref]

Mehrubeoglu, M.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Misguich-Ripault, J.

Mishchenko, M. I.

M. I. Mishchenko and P. Yang, “Far-field Lorenz–Mie scattering in an absorbing host medium: theoretical formalism and FORTRAN program,” J. Quant. Spectrosc. Radiat. Transfer 205, 241–252 (2018).
[Crossref]

Modest, M. F.

M. F. Modest, Radiative Heat Transfer. (Academic, 2013).

Onofri, F.

Poole, L.

Y. X. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, “Identification of cloud phase from PICASSO-CENA lidar depolarization: A multiple scattering sensitivity study,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 569–579 (2001).
[Crossref]

Prahl, S. A.

Radev, S.

M. P. Sentis, F. Onofri, L. Méès, and S. Radev, “Scattering of light by large bubbles: Coupling of geometrical and physical optics approximations,” J. Quant. Spectrosc. Radiat. Transfer 170, 8–18 (2016).
[Crossref]

Rakovic, M. J.

Ramella-Roman, J. C.

Rastegar, S.

Ren, K. F.

Rosenkranz, S.

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Light scattering from a drop with an embedded particle and its exploitation in the time-shift technique,” J. Quant. Spectrosc. Radiat. Transfer 227, 20–31 (2019).
[Crossref]

S. Rosenkranz, W. Schäfer, C. Tropea, and A. M. Zoubir, “Modeling photon transport in turbid media for measuring colloidal concentration in drops using the time-shift technique,” Appl. Opt. 55(34), 9703–9711 (2016).
[Crossref]

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Sensitivity of the time-shift technique in characterizing non-spherical drops,” In: Proceedings of the nineteenth International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics, Lisbon; 16-19 July. 2018.

Rozé, C.

Saint-Jalmes, H.

Schäfer, W.

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Light scattering from a drop with an embedded particle and its exploitation in the time-shift technique,” J. Quant. Spectrosc. Radiat. Transfer 227, 20–31 (2019).
[Crossref]

S. Rosenkranz, W. Schäfer, C. Tropea, and A. M. Zoubir, “Modeling photon transport in turbid media for measuring colloidal concentration in drops using the time-shift technique,” Appl. Opt. 55(34), 9703–9711 (2016).
[Crossref]

W. Schäfer and C. Tropea, “Time-shift technique for simultaneous measurement of size, velocity and relative refractive index of transparent droplets or particles in a flow,” Appl. Opt. 53(4), 588–596 (2014).
[Crossref]

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Sensitivity of the time-shift technique in characterizing non-spherical drops,” In: Proceedings of the nineteenth International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics, Lisbon; 16-19 July. 2018.

Schnaiter, M.

P. G. Stegmann, C. Tropea, E. Järvinen, and M. Schnaiter, “Comparison of measured and computed phase functions of individual tropospheric ice crystals,” J. Quant. Spectrosc. Radiat. Transfer 178, 379–389 (2016).
[Crossref]

Schuh, R.

T. Wriedt and R. Schuh, “The inclusion-concentration measurement of suspension droplets based on Monte Carlo ray tracing,” Meas. Sci. Technol. 13(3), 276–279 (2002).
[Crossref]

Sentis, M. P.

M. P. Sentis, F. Onofri, L. Méès, and S. Radev, “Scattering of light by large bubbles: Coupling of geometrical and physical optics approximations,” J. Quant. Spectrosc. Radiat. Transfer 170, 8–18 (2016).
[Crossref]

M. P. Sentis, F. Onofri, O. Dhez, J. Y. Laurent, and F. Chauchard, “Organic photo sensors for multi-angle light scattering characterization of particle systems,” Opt. Express 23(21), 27536–27541 (2015).
[Crossref]

Shen, J.

H. Yu, J. Shen, and Y. Wei, “Geometrical optics approximation of light scattering by large air bubbles,” Particuology 6(5), 340–346 (2008).
[Crossref]

Stegmann, P. G.

P. G. Stegmann, B. Sun, J. Ding, P. Yang, and X. Zhang, “Study of the effects of phytoplankton morphology and vertical profile on lidar attenuated backscatter and depolarization ratio,” J. Quant. Spectrosc. Radiat. Transfer 225, 1–15 (2019).
[Crossref]

P. G. Stegmann, C. Tropea, E. Järvinen, and M. Schnaiter, “Comparison of measured and computed phase functions of individual tropospheric ice crystals,” J. Quant. Spectrosc. Radiat. Transfer 178, 379–389 (2016).
[Crossref]

Sun, B.

P. G. Stegmann, B. Sun, J. Ding, P. Yang, and X. Zhang, “Study of the effects of phytoplankton morphology and vertical profile on lidar attenuated backscatter and depolarization ratio,” J. Quant. Spectrosc. Radiat. Transfer 225, 1–15 (2019).
[Crossref]

Tropea, C.

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Light scattering from a drop with an embedded particle and its exploitation in the time-shift technique,” J. Quant. Spectrosc. Radiat. Transfer 227, 20–31 (2019).
[Crossref]

P. G. Stegmann, C. Tropea, E. Järvinen, and M. Schnaiter, “Comparison of measured and computed phase functions of individual tropospheric ice crystals,” J. Quant. Spectrosc. Radiat. Transfer 178, 379–389 (2016).
[Crossref]

S. Rosenkranz, W. Schäfer, C. Tropea, and A. M. Zoubir, “Modeling photon transport in turbid media for measuring colloidal concentration in drops using the time-shift technique,” Appl. Opt. 55(34), 9703–9711 (2016).
[Crossref]

W. Schäfer and C. Tropea, “Time-shift technique for simultaneous measurement of size, velocity and relative refractive index of transparent droplets or particles in a flow,” Appl. Opt. 53(4), 588–596 (2014).
[Crossref]

H. Yu, F. Xu, and C. Tropea, “Simulation of optical caustics associated with the secondary rainbow of oblate droplets,” Opt. Lett. 38(21), 4469–4472 (2013).
[Crossref]

C. Tropea, “Optical particle characterization in flows,” Annu. Rev. Fluid Mech. 43(1), 399–426 (2011).
[Crossref]

L. Li, S. Rosenkranz, W. Schäfer, and C. Tropea, “Sensitivity of the time-shift technique in characterizing non-spherical drops,” In: Proceedings of the nineteenth International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics, Lisbon; 16-19 July. 2018.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques. (Springer Science & Business Media, 2013).

Vann, L.

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Figures (12)

Fig. 1.
Fig. 1. Droplet with diameter D containing colloidal particles with diameter of d. The refractive indices of medium, drop and colloidal particles are n1, n2 and n3 respectively.
Fig. 2.
Fig. 2. Local reference coordinate system and rotation that depicts the new propagation direction of the light ray
Fig. 3.
Fig. 3. Comparison of the scattering phase functions of a colloidal particle illuminated by a plane wave for different polarizations. The scattered intensity is expressed by color on a logarithmic scale. a): the incident plane wave is linearly polarized with the Stokes vector [1 -1 0 0]T; b): the incident plane wave is circularly polarized with the Stokes vector [1 0 0 1]T. (Refractive index of medium 1.3431, refractive index of particle 1.6268, size of particle 320 nm, wavelength of the plane wave 405 nm)
Fig. 4.
Fig. 4. Bivariate scattering diagram calculated with a) the Mueller matrix computed from the MiePlot Software [32] and b) the rejection sampling method using 30 billion light rays. The intensity in Z-axis is plotted on a logarithmic scale. The incident plane wave is linearly polarized with the Stokes vector [1 -1 0 0] T. (refractive index of surrounding medium 1.3431, refractive index of colloidal particle 1.6268, size of nanoparticle 320 nm, wavelength of the plane wave 405 nm)
Fig. 5.
Fig. 5. Relative deviation between the scattering diagrams computed from the Lorenz-Mie theory and the rejection sampling method using 30 billion light rays.
Fig. 6.
Fig. 6. Principle of time-shift measurement technique [16]. Θs expresses the angle of the incident point for each scattering order.
Fig. 7.
Fig. 7. Comparison between the simulated and measured time-shift signal when the volume concentration is a) 0.05% and b) 0.14%. (Droplet size = 100 µm; size of colloidal particle = 320 ± 30 nm; droplet refractive index = 1.3431; colloidal particle refractive index = 1.6268; wavelength of laser beam = 405 nm; polarization state of the laser beam is s polarization; Beam waist = 14 µm)
Fig. 8.
Fig. 8. Comparison between the simulated time-shift signals under different polarizations. Droplet size = 130 µm; size of colloidal particle = 320 ± 30 nm; droplet refractive index = 1.3431; colloidal particle refractive index = 1.6268; wavelength of laser beam = 405 nm; volume concentration of the colloidal particle is 0.14%.
Fig. 9.
Fig. 9. Influence of beam waist on time-shift signal. (Droplet size = 100 µm; size of colloidal particle = 320 ± 30 nm; droplet refractive index = 1.3431; colloidal particle refractive index = 1.6268; wavelength of laser beam = 405 nm; polarization state of the laser beam is s polarization.)
Fig. 10.
Fig. 10. Simulated time-shift signal from pure droplet, colloidal droplet and the internal scattering. (Size of colloidal drop: 100 µm, optical mean path length E(L) is 166.7 µm, which corresponds to the volume concentration 0.14%, size of colloidal particle:320 nm, beam waist 14 µm)
Fig. 11.
Fig. 11. Illustration for the signal generation of different scattering orders. Detector scattering angle θ=165°; solid angle of detector is 0.032 sr. The incident and glare points regions corresponding to each scattering order and for this solid angle of detector are shown as thickened lines on the circumference of the droplet.
Fig. 12.
Fig. 12. Simulated time-shift signal as a function of volume concentration. (Droplet size = 100 µm; size of colloidal particle = 320 ± 30 nm; droplet refractive index = 1.3431; colloidal particle refractive index = 1.6268; wavelength of laser beam = 405 nm; polarization state of the laser beam is s polarization, beam waist 4 µm). The times at which the different scattering modes are expected on the detector are marked on the x-axis.

Equations (37)

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E ( L ) = V p C v σ e
S = ( I Q U V ) = ( E | | E | | + E E E | | E E E E | | E + E | | E i ( E | | E E | | E ) )
S s c a ( θ , φ ) = M ( θ ) R ( φ ) S i
R ( φ ) = ( 1 0 0 0 0 c o s 2 φ s i n 2 φ 0 0 s i n 2 φ c o s 2 φ 0 0 0 0 1 )
( E s | | E s ) = exp [ i k ( R z ) ] i k R S ( E i E i )
S = ( A 11 A 12 A 21 A 22 )
( E r | | E r ) = exp [ i k ( R z ) ] i k R ( r p 0 0 r s ) ( E i E i )
( E t | | E t ) = exp [ i k ( R z ) ] i k R ( t p 0 0 t s ) ( E i E i )
M = ( S 11 S 12 0 0 S 12 S 22 0 0 0 0 S 33 S 34 0 0 S 34 S 44 )
S 11 = 1 2 ( | A 11 | 2 + | A 22 | 2 + | A 21 | 2 + | A 12 | 2 )
S 12 = 1 2 ( | A 11 | 2 | A 22 | 2 + | A 21 | 2 | A 12 | 2 )
S 22 = 1 2 ( | A 11 | 2 + | A 22 | 2 | A 21 | 2 | A 12 | 2 )
S 33 = R e ( A 11 A 22 + A 12 A 21 )
S 34 = I m ( A 11 A 22 + A 21 A 12 )
S 44 = R e ( A 11 A 22 A 12 A 21 )
R = ( r p 2 + r s 2 2 r p 2 r s 2 2 0 0 r p 2 r s 2 2 r p 2 + r s 2 2 0 0 0 0 R e ( r p r s ) I m ( r p r s ) 0 0 I m ( r p r s ) R e ( r p r s ) )
T = ( t p 2 + t s 2 2 t p 2 t s 2 2 0 0 t p 2 t s 2 2 t p 2 + t s 2 2 0 0 0 0 R e ( t p t s ) I m ( t p t s ) 0 0 I m ( t p t s ) R e ( t p t s ) )
r p = tan ( θ i θ t ) tan ( θ i + θ t )
r s = sin ( θ i θ t ) sin ( θ i + θ t )
t p = 2 sin θ t cos θ i sin ( θ i + θ t ) cos ( θ i θ t )
t s = 2 s i n θ t c o s θ i s i n ( θ i + θ t )
r p = c o s θ i i m m 2 s i n 2 θ i 1 c o s θ i + i m m 2 s i n 2 θ i 1
r s = m c o s θ i i m 2 s i n 2 θ i 1 m c o s θ i + i m 2 s i n 2 θ i 1
I s c a ( θ , φ ) = S 11 I i + S 12 ( c o s ( 2 φ ) Q i s i n ( 2 φ ) U i ) .
I s c a = θ = 0 π φ = 0 2 π I s c a ( θ , φ ) s i n θ d θ d φ
f ( θ , φ ) = I s c a ( θ , φ ) I s c a
g ( θ , φ ) = I s c a ( θ = 0 , φ ) I s c a
θ 0 = π ξ 1
φ 0 = 2 π ξ 2
F = S 11 ( θ = 0 ) I i + S 12 ( θ = 0 ) ( c o s ( 2 φ 0 ) Q i s i n ( 2 φ 0 ) U i ) .
I s c a = S 11 ( θ 0 ) I i + S 12 ( θ 0 ) ( c o s ( 2 φ 0 ) Q i s i n ( 2 φ 0 ) U i ) .
I c o m = ξ 3 F
( I s c a Q s c a U s c a V s c a ) = ( 1 S 12 / S 11 0 0 S 12 / S 11 S 22 / S 11 0 0 0 0 S 33 / S 11 S 34 / S 11 0 0 S 34 / S 11 S 44 / S 11 ) R ( φ ) ( I i Q i U i V i )
[ e x 1 e y 1 e z 1 ] = [ cos θ 0 0 sin θ 0 0 1 0 sin θ 0 0 cos θ 0 ] [ cos φ 0 sin φ 0 0 sin φ 0 cos φ 0 0 0 0 1 ] [ e x 0 e y 0 e z 0 ]
O L = 4 R c o s ( a r c s i n ( s i n θ i m ) )
I = I 0 e x p ( O L / E ( L ) ) .
η = I S R S   R S :Reflection Scattering;  I S : Internal Scattering .