Abstract

Abstract: Based on the theory of Goos-Hänchen shift and its continuity near the critical angle, we introduce the concept of penetration depth below the critical angle, and obtain the general formula of reflectance using the gradient complex refractive index multilayered model. Compared with the fitting curve with Fresnel's Formula, our calculated results are more consistent with experimental results of Intralipid solution and the suspension of rutile TiO 2 powder. Combining the change of penetration depth near the critical angle with our model, we also reveal the essence of a simple method used to obtain the non-surface complex refractive index of turbid media.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hanchen effect I,” Optik (Stuttg.) 32(2), 116–137 (1970).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2015 (1)

2014 (1)

K. Yasumoto, Y. Ōishi, K. Yasumoto, and Y. Oishi, “A new evaluation of the Goos – Hänchen shift and associated time delay,” J. Appl. Phys. 2170(1983), 1–8 (2014).

2013 (1)

2012 (1)

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]

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]

2008 (1)

2007 (1)

2005 (1)

1995 (2)

1986 (1)

1970 (1)

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hanchen effect I,” Optik (Stuttg.) 32(2), 116–137 (1970).

1964 (1)

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

Bali, L. M.

Bali, S.

Barrera, R.

Birge, R. R.

Calhoun, W. R.

Cheng, F. C.

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]

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.

Fan, J.

García-Valenzuela, A.

Goyal, K. G.

Gross, R. B.

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]

Judge, P. T.

Köhler, W.

Ku, C. Y.

Lai, H. M.

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]

Liu, Y.

Lotsch, H. K. V.

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hanchen effect I,” Optik (Stuttg.) 32(2), 116–137 (1970).

Lu, Z. 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]

Mei, J.

Méndez, E.

Michalak, R.

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]

Oishi, Y.

K. Yasumoto, Y. Ōishi, K. Yasumoto, and Y. Oishi, “A new evaluation of the Goos – Hänchen shift and associated time delay,” J. Appl. Phys. 2170(1983), 1–8 (2014).

K. Yasumoto, Y. Ōishi, K. Yasumoto, and Y. Oishi, “A new evaluation of the Goos – Hänchen shift and associated time delay,” J. Appl. Phys. 2170(1983), 1–8 (2014).

Peiponen, K.-E.

Räty, J.

Renard, R. H.

Reyes-Coronado, A.

Sánchez-Pérez, C.

Schwefel, H. G. L.

Song, Q. W.

Sun, J.

Tang, W. K.

Tian, J.

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]

Wang, J.

J. Sun, J. Wang, Y. Liu, Q. Ye, H. Zeng, W. Zhou, J. Mei, C. Zhang, and J. Tian, “Effect of the gradient of complex refractive index at boundary of turbid media on total internal reflection,” Opt. Express 23(6), 7320–7332 (2015).
[Crossref] [PubMed]

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, L. J.

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]

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]

Yasumoto, K.

K. Yasumoto, Y. Ōishi, K. Yasumoto, and Y. Oishi, “A new evaluation of the Goos – Hänchen shift and associated time delay,” J. Appl. Phys. 2170(1983), 1–8 (2014).

K. Yasumoto, Y. Ōishi, K. Yasumoto, and Y. Oishi, “A new evaluation of the Goos – Hänchen shift and associated time delay,” J. Appl. Phys. 2170(1983), 1–8 (2014).

Ye, Q.

J. Sun, J. Wang, Y. Liu, Q. Ye, H. Zeng, W. Zhou, J. Mei, C. Zhang, and J. Tian, “Effect of the gradient of complex refractive index at boundary of turbid media on total internal reflection,” Opt. Express 23(6), 7320–7332 (2015).
[Crossref] [PubMed]

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]

Zeng, H.

Zhang, C.

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.

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]

Ann. Phys. (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

J. Appl. Phys. (1)

K. Yasumoto, Y. Ōishi, K. Yasumoto, and Y. Oishi, “A new evaluation of the Goos – Hänchen shift and associated time delay,” J. Appl. Phys. 2170(1983), 1–8 (2014).

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. Opt. Soc. Am. (1)

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

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

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 (2)

Opt. Lett. (3)

Optik (Stuttg.) (1)

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hanchen effect I,” Optik (Stuttg.) 32(2), 116–137 (1970).

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Figures (7)

Fig. 1
Fig. 1 Illustration of the triangular relationship between Goos-Hänchen shift D s (TE wave) and penetration depth d p
Fig. 2
Fig. 2 (a)Illustration of the GCRIMM. The boundary part of turbid media is divided into M layers and energy flows in different layers attenuate in terms of CRI in corresponding layer. (b) Illustration of the simplified three-layer model. The boundary part of turbid media is divided into 3 layers.
Fig. 3
Fig. 3 Schematic diagram of experimental setup
Fig. 4
Fig. 4 Goos-Hänchen shift (red curve) and the penetration depth (blue curve) near the critical angle for (a) 20% Intralipid solution and (b) 30% Intralipid solution
Fig. 5
Fig. 5 The comparisons of calculated results with adjusted fitting parameters based on our three-layer model and fitting curves with FF for Intralipid solutions with different concentrations in different incident ranges (a) for 20% Intralipid solution below the critical angle (b) for 20% Intralipid solution above the criticle angle (c) for 30% Intralipid solution below the critical angle (d) for 30% Intralipid solution above the criticle angle
Fig. 6
Fig. 6 The comparison of calculated results with adjusted fitting parameters based on our three-layer model and fitting curves with FF for the suspension of rutile T i O 2 powder (volume fraction of about 1.8%) (a) below the critical angle (b) below the critical angle
Fig. 7
Fig. 7 The proportions of incident energy flow in three different layers to total incident energy flow versus incident angle for (a) 20% Intralipid solution and (b) 30% Intralipid solution

Tables (3)

Tables Icon

Table 1 Original fitting results of 20% and 30% Intralipid solutions with our three-layer model

Tables Icon

Table 2 Adjusted fitting results of the suspension of rutile T i O 2 powder (volume fraction of about 1.8%) with our three-layer model

Tables Icon

Table 3 Comparison of CRIs fitted with all data and partial data for Intralipid solutions and the suspension of rutile T i O 2 powder

Equations (4)

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D s = λ π sin θ i ( n 1 2 sin 2 θ i n 2 2 ) 1 / 2 = 2 Z cos θ i
d p = λ 2 π ( n 1 2 sin 2 θ i n 2 2 ) 1 / 2 = D s 2 sin θ i = Z tan θ i
R = ( 1 e 2 d 1 d p ) e 4 π λ k 1 l 1 + ( e 2 d 1 d p e 2 d 1 + d 2 d p ) e 4 π λ k 2 l 2 + ( e 2 d 1 + d 2 d p e 2 ) e 4 π λ k 3 l 3 1 e 2 ,
R = R f { ( 1 e 2 d 1 d p ) e 4 π λ k 1 l 1 + ( e 2 d 1 d p e 2 d 1 + d 2 d p ) e 4 π λ k 2 l 2 + ( e 2 d 1 + d 2 d p e 2 ) e 4 π λ k l 3 1 e 2 , d p > d 1 + d 2 , ( 1 e 2 d 1 d p ) e 4 π λ k 1 l 1 + ( e 2 d 1 d p e 2 ) e 4 π λ k 2 l 2 1 e 2 , d 1 < d p < d 1 + d 2 , R f = { 1 , θ i θ c , ( n a cos θ i n 0 cos θ t n a cos θ i + n 0 cos θ t ) 2 , θ i < θ c .

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