Abstract

We analyze the existence of non-uniformity at the boundary of turbid media, and develop a gradient complex refractive index multilayered model in terms of this fact. Our model reveals the physics mechanism of the discrepancies between experimental data above the critical angle and the fitting curve with Fresnel’s Formula. Also, from the perspective of the energy flow, reflectance R is obtained by the simplified models. We get complex refractive indexes and reflectance curves by fitting experimental data of 20% and 30% Intralipid solutions and rutile TiO2 powder suspension with two different methods. Compared with Fresnel’s Formula, our model can fit experimental data better.

© 2015 Optical Society of America

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References

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  1. Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006).
    [Crossref] [PubMed]
  2. M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004).
    [Crossref]
  3. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995).
    [Crossref] [PubMed]
  4. A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. 84(1), 38–43 (2001).
    [Crossref] [PubMed]
  5. G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6(2), 214–221 (1995).
    [Crossref]
  6. W. R. Calhoun, H. Maeta, A. Combs, L. M. Bali, and S. Bali, “Measurement of the refractive index of highly turbid media,” Opt. Lett. 35(8), 1224–1226 (2010).
    [Crossref] [PubMed]
  7. K. G. Goyal, M. L. Dong, V. M. Nguemaha, B. W. Worth, P. T. Judge, W. R. Calhoun, L. M. Bali, and S. Bali, “Empirical model of total internal reflection from highly turbid media,” Opt. Lett. 38(22), 4888–4891 (2013).
    [Crossref] [PubMed]
  8. W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
    [Crossref]
  9. R. G. Barrera and A. García-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach,” J. Opt. Soc. Am. A 20(2), 296–311 (2003).
    [Crossref] [PubMed]
  10. A. García-Valenzuela, R. Barrera, C. Sánchez-Pérez, A. Reyes-Coronado, and E. Méndez, “Coherent reflection of light from a turbid suspension of particles in an internal-reflection configuration: Theory versus experiment,” Opt. Express 13(18), 6723–6737 (2005).
    [Crossref] [PubMed]
  11. I. Niskanen, J. Räty, and K.-E. Peiponen, “Complex refractive index of turbid liquids,” Opt. Lett. 32(7), 862–864 (2007).
    [Crossref] [PubMed]
  12. I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989).
    [Crossref]
  13. H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991).
    [Crossref] [PubMed]
  14. G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999).
    [Crossref] [PubMed]
  15. L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012).
    [Crossref] [PubMed]
  16. G. A. Parks and A. Parks, “The isoelectric points of solid oxides, solid hydroxides, and aqueous hydroxo complex systems,” Chem. Rev. 65(2), 177–198 (1965).
    [Crossref]
  17. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999)
  18. F. M. Mirabella, Internal Reflection Spectroscopy: Theory and Applications (Chemical Rubber Company, 1992)
  19. G.-Y. Leng, “Study of Goos-Hänchen shift of reflected beam from absorbing medium,” Acta Opt. Sin. 7(5), 464–467 (1987).
  20. Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
    [Crossref] [PubMed]

2013 (1)

2012 (2)

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012).
[Crossref] [PubMed]

2011 (1)

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

2010 (1)

2007 (1)

2006 (1)

Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (1)

M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004).
[Crossref]

2003 (1)

2001 (1)

A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. 84(1), 38–43 (2001).
[Crossref] [PubMed]

1999 (1)

G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999).
[Crossref] [PubMed]

1995 (2)

G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6(2), 214–221 (1995).
[Crossref]

X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995).
[Crossref] [PubMed]

1991 (1)

1989 (1)

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989).
[Crossref]

1987 (1)

G.-Y. Leng, “Study of Goos-Hänchen shift of reflected beam from absorbing medium,” Acta Opt. Sin. 7(5), 464–467 (1987).

1965 (1)

G. A. Parks and A. Parks, “The isoelectric points of solid oxides, solid hydroxides, and aqueous hydroxo complex systems,” Chem. Rev. 65(2), 177–198 (1965).
[Crossref]

Bali, L. M.

Bali, S.

Barrera, R.

Barrera, R. G.

Calhoun, W. R.

Chansiri, G.

G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999).
[Crossref] [PubMed]

Chen, J. Y.

Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006).
[Crossref] [PubMed]

Combs, A.

Dai, J.

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

Dawson, J. B.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989).
[Crossref]

Deng, Z.-C.

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

Dong, M. L.

Driver, I.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989).
[Crossref]

Ebert, M.

M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004).
[Crossref]

Feather, J. W.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989).
[Crossref]

Fry, E. S.

García-Valenzuela, A.

Goyal, K. G.

Guo, W.

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

He, L.

L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012).
[Crossref] [PubMed]

Hem, S. L.

G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999).
[Crossref] [PubMed]

Hoffmann, P.

M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004).
[Crossref]

Hu, Y.

L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012).
[Crossref] [PubMed]

Jääskeläinen, A. J.

A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. 84(1), 38–43 (2001).
[Crossref] [PubMed]

Jin, Y. L.

Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006).
[Crossref] [PubMed]

Judge, P. T.

King, P. R.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989).
[Crossref]

Leng, G.-Y.

G.-Y. Leng, “Study of Goos-Hänchen shift of reflected beam from absorbing medium,” Acta Opt. Sin. 7(5), 464–467 (1987).

Li, W.

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

Lyons, R. T.

G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999).
[Crossref] [PubMed]

Maeta, H.

Meeten, G. H.

G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6(2), 214–221 (1995).
[Crossref]

Méndez, E.

Moes, C. J. M.

Nguemaha, V. M.

Niskanen, I.

North, A. N.

G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6(2), 214–221 (1995).
[Crossref]

Ortner, H. M.

M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004).
[Crossref]

Parks, A.

G. A. Parks and A. Parks, “The isoelectric points of solid oxides, solid hydroxides, and aqueous hydroxo complex systems,” Chem. Rev. 65(2), 177–198 (1965).
[Crossref]

Parks, G. A.

G. A. Parks and A. Parks, “The isoelectric points of solid oxides, solid hydroxides, and aqueous hydroxo complex systems,” Chem. Rev. 65(2), 177–198 (1965).
[Crossref]

Patel, M. V.

G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999).
[Crossref] [PubMed]

Peiponen, K. E.

A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. 84(1), 38–43 (2001).
[Crossref] [PubMed]

Peiponen, K.-E.

Prahl, S. A.

Quan, X.

Räty, J.

Räty, J. A.

A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. 84(1), 38–43 (2001).
[Crossref] [PubMed]

Reyes-Coronado, A.

Sánchez-Pérez, C.

Tian, J.-G.

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

van Gemert, M. J. C.

van Marie, J.

van Staveren, H. J.

Wang, J.

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

Wang, M.

L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012).
[Crossref] [PubMed]

Wang, P. N.

Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006).
[Crossref] [PubMed]

Weinbruch, S.

M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004).
[Crossref]

Worth, B. W.

Xia, M.

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

Xu, L.

Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006).
[Crossref] [PubMed]

Yang, K.

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

Ye, Q.

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

Yin, Y.

L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012).
[Crossref] [PubMed]

Zhang, C.-P.

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

Zhang, X.

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

Zhou, W.-Y.

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

ACS Nano (1)

L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012).
[Crossref] [PubMed]

Acta Opt. Sin. (1)

G.-Y. Leng, “Study of Goos-Hänchen shift of reflected beam from absorbing medium,” Acta Opt. Sin. 7(5), 464–467 (1987).

Appl. Opt. (2)

Atmos. Environ. (1)

M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004).
[Crossref]

Chem. Rev. (1)

G. A. Parks and A. Parks, “The isoelectric points of solid oxides, solid hydroxides, and aqueous hydroxo complex systems,” Chem. Rev. 65(2), 177–198 (1965).
[Crossref]

J. Biomed. Opt. (1)

Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011).
[Crossref] [PubMed]

J. Dairy Sci. (1)

A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. 84(1), 38–43 (2001).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

J. Pharm. Sci. (1)

G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999).
[Crossref] [PubMed]

Meas. Sci. Technol. (2)

G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6(2), 214–221 (1995).
[Crossref]

W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Med. Biol. (2)

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989).
[Crossref]

Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006).
[Crossref] [PubMed]

Other (2)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999)

F. M. Mirabella, Internal Reflection Spectroscopy: Theory and Applications (Chemical Rubber Company, 1992)

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Figures (6)

Fig. 1
Fig. 1 Illustrations of (a) particle distribution near the prism surface (b) the GCRIMM (c) the simplified three-layer model.
Fig. 2
Fig. 2 Schematic diagram of experimental installation.
Fig. 3
Fig. 3 Experimental data above the critical angle (blue dots) and fitting reflectance curves with Fresnel’s Formula (red solid line) and our simplified three-layer model (black solid line) for (a) 20% intralipid solution (b) 30% intralipid solution.
Fig. 4
Fig. 4 (a) penetration depth versus incident angle for 30% Intralipid solution (b) ratios of incident energy flows in different layers to the total incident energy flow versus incident angle for 30% Intralipid solution.
Fig. 5
Fig. 5 The comparison between the three-layer model, the two-layer model and Fresnel’s Formula for 30% Intralipid solution.
Fig. 6
Fig. 6 The comparison between the three-layer model, the two-layer model and Fresnel’s Formula for rutile Ti O 2 powder suspension.

Tables (2)

Tables Icon

Table 1 Fitting results of 20% and 30% Intralipid solutions with Fresnel’s Formula and the simplified three-layer model

Tables Icon

Table 2 Fitting results of rutile Ti O 2 powder suspension with Fresnel’s Formula, the simplified three-layer model and the simplified two-layer model

Equations (9)

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

{ R s = ( n 0 cos θ 1 u 2 ) 2 + v 2 2 ( n 0 cos θ 1 + u 2 ) 2 + v 2 2 R p = [ ( n s 2 k s 2 ) 2 cos θ 1 n 0 u 2 ] 2 + ( 2 n s k s cos θ 1 n 0 v 2 ) 2 [ ( n s 2 k s 2 ) 2 cos θ 1 + n 0 u 2 ] 2 + ( 2 n s k s cos θ 1 n 0 v 2 ) 2 , u 2 2 v 2 2 = n s 2 k s 2 n 1 2 sin 2 θ 1 , u 2 v 2 = n s k s ,
d p = λ 2 π n 0 1 ( υ 2 2 + μ 2 2 + υ 2 ) 1 2 , υ 2 = sin 2 θ 1 ( n s 2 k s 2 )/ n 0 2 , μ 2 =2 n s k s / n 0 2 ,
w= d z d 1 d 2 I e 2 x d p dx= d z I d p 2 ( e 2 d 1 d p e 2 d 2 d p ).
w m = d z I d p 2 [ e 2 ( m1 )d d p e 2 md d p ].
w ' m = w m e 4π λ k m l m .
R= m=1 N w ' m m=1 N w m = m=1 n ( e 2 ( m1 )d d p e 2 md d p ) e 4π λ k m l m 1 e 2 .
l m = D p cos θ 1 +2d,
D p = λ 2π { ( C+D )×[ sin θ 1 ( g 2 q g 1 τ ) n 0 2 A 2 + B 2 sin θ 1 cos 2 θ 1 ( g 2 q+ g 1 τ ) ]+( CD )× 2 n 0 2 qτ A 2 + B 2 sin θ 1 cos θ 1 }, C= 1 ( g 1 cos θ 1 q ) 2 + ( g 2 cos θ 1 τ ) 2 ,D= 1 ( g 1 cos θ 1 +q ) 2 + ( g 2 cos θ 1 +τ ) 2 , g 1 = n s 2 ( 1 k s 2 ) n 0 , g 2 = 2 n s 2 k s n 0 ,A= n s 2 ( 1 k 2 2 ) n s 2 sin 2 θ 1 = q 2 τ 2 ,B=2 n s 2 k s =2qτ.
R= ( 1 e 2 d 1 d p ) e 4π λ k 1 l 1 +( e 2 d 1 d p e 2 d 1 + d 2 d p ) e 4π λ k 2 l 2 +( e 2 d 1 + d 2 d p e 2 ) e 4π λ k 3 l 3 1 e 2 .

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