Abstract

We demonstrate, to the best of our knowledge, a first accurate empirical model for reflectance measurements from highly turbid media over the full range of incident angles, i.e., for reflectivity values going from unity in the total internal reflection regime to nearly zero when almost all the light is transmitted. Evidence that our model is accurate is provided by extraction of the particle size, followed by independent verification with dynamic light scattering. Our methodology is in direct contrast with the prevalent approach in turbid media of focusing on only the critical angle region, which is just a small subset of the entire reflectance data.

© 2013 Optical Society of America

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  1. S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
    [CrossRef]
  2. W. Calhoun, H. Maeta, A. Combs, L. Bali, and S. Bali, Opt. Lett. 35, 1224 (2010).
    [CrossRef]
  3. W. Calhoun, H. Maeta, A. Combs, L. Bali, and S. Bali, Opt. Lett. 36, 3172 (2011).
  4. G. Meeten and A. North, Meas. Sci. Technol. 6, 214 (1995).
    [CrossRef]
  5. I. Niskanen, J. Räty, and K. Peiponen, Opt. Lett. 32, 862 (2007).
    [CrossRef]
  6. G. Meeten, Meas. Sci. Technol. 8, 728 (1997).
    [CrossRef]
  7. A. Garcia-Valenzuela, R. Barrera, C. Sanchez-Perez, A. Reyes-Coronado, and E. Mendez, Opt. Express 13, 6723 (2005).
    [CrossRef]
  8. W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
    [CrossRef]
  9. W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
    [CrossRef]
  10. S. A. Prahl, http://omlc.ogi.edu/calc/mie_calc.html .
  11. N. Ren, J. Liang, X. Qu, J. Li, B. Lu, and J. Tian, Opt. Express 18, 6811 (2010).
    [CrossRef]
  12. K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
    [CrossRef]
  13. M. McClimans, C. LaPlante, D. Bonner, and S. Bali, Appl. Opt. 45, 6477 (2006).
    [CrossRef]
  14. C. A. Risset and J. M. Vigoureaux, Opt. Commun. 91, 155 (1992).
    [CrossRef]
  15. K. H. Lan, N. Ostrowsky, and D. Sornette, Phys. Rev. Lett. 57, 17 (1986).
    [CrossRef]
  16. H. Matsuoka, Macromol. Rapid Commun. 22, 51 (2001).
    [CrossRef]
  17. V. Kontturi, P. Turunen, J. Uozumi, and K. Peiponen, Opt. Lett. 34, 3743 (2009).
    [CrossRef]
  18. M. A. C. Potenza, D. Brogioli, and M. Giglio, Appl. Phys. Lett. 85, 2730 (2004).
    [CrossRef]
  19. D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
    [CrossRef]
  20. Alternatively one may use manufacturer provided values based on dry microscopy. Our conclusions are unaltered.
  21. X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
    [CrossRef]

2012 (2)

K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
[CrossRef]

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

2011 (2)

W. Calhoun, H. Maeta, A. Combs, L. Bali, and S. Bali, Opt. Lett. 36, 3172 (2011).

W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
[CrossRef]

2010 (2)

2009 (2)

V. Kontturi, P. Turunen, J. Uozumi, and K. Peiponen, Opt. Lett. 34, 3743 (2009).
[CrossRef]

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef]

2007 (2)

I. Niskanen, J. Räty, and K. Peiponen, Opt. Lett. 32, 862 (2007).
[CrossRef]

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

2006 (1)

2005 (1)

2004 (1)

M. A. C. Potenza, D. Brogioli, and M. Giglio, Appl. Phys. Lett. 85, 2730 (2004).
[CrossRef]

2003 (1)

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

2001 (1)

H. Matsuoka, Macromol. Rapid Commun. 22, 51 (2001).
[CrossRef]

1997 (1)

G. Meeten, Meas. Sci. Technol. 8, 728 (1997).
[CrossRef]

1995 (1)

G. Meeten and A. North, Meas. Sci. Technol. 6, 214 (1995).
[CrossRef]

1992 (1)

C. A. Risset and J. M. Vigoureaux, Opt. Commun. 91, 155 (1992).
[CrossRef]

1986 (1)

K. H. Lan, N. Ostrowsky, and D. Sornette, Phys. Rev. Lett. 57, 17 (1986).
[CrossRef]

Bahng, J.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Bali, L.

Bali, S.

Barrera, R.

Bonner, D.

Brock, R.

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Brogioli, D.

M. A. C. Potenza, D. Brogioli, and M. Giglio, Appl. Phys. Lett. 85, 2730 (2004).
[CrossRef]

Calhoun, W.

Combs, A.

Dai, J.

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
[CrossRef]

Dong, M.

K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
[CrossRef]

Faez, S.

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef]

Fowlkes, J.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Garcia-Valenzuela, A.

Giglio, M.

M. A. C. Potenza, D. Brogioli, and M. Giglio, Appl. Phys. Lett. 85, 2730 (2004).
[CrossRef]

Goyal, K.

K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
[CrossRef]

Grotberg, J.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Guo, W.

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
[CrossRef]

Hu, X.

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Huh, D.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Jacobs, K.

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Johnson, P. M.

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef]

Kane, D.

K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
[CrossRef]

Kontturi, V.

Kripfgans, O.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Lagendijk, A.

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef]

Lan, K. H.

K. H. Lan, N. Ostrowsky, and D. Sornette, Phys. Rev. Lett. 57, 17 (1986).
[CrossRef]

LaPlante, C.

Lei, W.

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

Li, J.

Li, W.

W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
[CrossRef]

Liang, J.

Ling, Y.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Lu, B.

Lu, J.

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Ma, X.

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Maeta, H.

Makkar, S.

K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
[CrossRef]

Matsuoka, H.

H. Matsuoka, Macromol. Rapid Commun. 22, 51 (2001).
[CrossRef]

McClimans, M.

Meeten, G.

G. Meeten, Meas. Sci. Technol. 8, 728 (1997).
[CrossRef]

G. Meeten and A. North, Meas. Sci. Technol. 6, 214 (1995).
[CrossRef]

Mendez, E.

Niskanen, I.

North, A.

G. Meeten and A. North, Meas. Sci. Technol. 6, 214 (1995).
[CrossRef]

Ostrowsky, N.

K. H. Lan, N. Ostrowsky, and D. Sornette, Phys. Rev. Lett. 57, 17 (1986).
[CrossRef]

Peiponen, K.

Potenza, M. A. C.

M. A. C. Potenza, D. Brogioli, and M. Giglio, Appl. Phys. Lett. 85, 2730 (2004).
[CrossRef]

Qu, X.

Räty, J.

Ren, N.

Reyes-Coronado, A.

Risset, C. A.

C. A. Risset and J. M. Vigoureaux, Opt. Commun. 91, 155 (1992).
[CrossRef]

Sanchez-Perez, C.

Sornette, D.

K. H. Lan, N. Ostrowsky, and D. Sornette, Phys. Rev. Lett. 57, 17 (1986).
[CrossRef]

Takayama, S.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Tian, J.

Turunen, P.

Uozumi, J.

Vigoureaux, J. M.

C. A. Risset and J. M. Vigoureaux, Opt. Commun. 91, 155 (1992).
[CrossRef]

Wei, H.

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Worth, B.

K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
[CrossRef]

Xia, M.

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
[CrossRef]

Yang, K.

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
[CrossRef]

Yang, P.

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Zhang, X.

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

Anal. Chem. (1)

D. Huh, J. Bahng, Y. Ling, H. Wei, O. Kripfgans, J. Fowlkes, J. Grotberg, and S. Takayama, Anal. Chem. 79, 1369 (2007).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. A. C. Potenza, D. Brogioli, and M. Giglio, Appl. Phys. Lett. 85, 2730 (2004).
[CrossRef]

Macromol. Rapid Commun. (1)

H. Matsuoka, Macromol. Rapid Commun. 22, 51 (2001).
[CrossRef]

Meas. Sci. Technol. (3)

G. Meeten and A. North, Meas. Sci. Technol. 6, 214 (1995).
[CrossRef]

G. Meeten, Meas. Sci. Technol. 8, 728 (1997).
[CrossRef]

W. Guo, M. Xia, W. Lei, J. Dai, X. Zhang, and K. Yang, Meas. Sci. Technol. 23, 047001 (2012).
[CrossRef]

Opt. Commun. (1)

C. A. Risset and J. M. Vigoureaux, Opt. Commun. 91, 155 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Med. Biol. (1)

X. Ma, J. Lu, R. Brock, K. Jacobs, P. Yang, and X. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

S. Faez, P. M. Johnson, and A. Lagendijk, Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef]

K. H. Lan, N. Ostrowsky, and D. Sornette, Phys. Rev. Lett. 57, 17 (1986).
[CrossRef]

Rev. Sci. Instrum. (2)

K. Goyal, M. Dong, D. Kane, S. Makkar, B. Worth, L. Bali, and S. Bali, Rev. Sci. Instrum. 83, 086107 (2012).
[CrossRef]

W. Guo, M. Xia, W. Li, J. Dai, and K. Yang, Rev. Sci. Instrum. 82, 053108 (2011).
[CrossRef]

Other (2)

S. A. Prahl, http://omlc.ogi.edu/calc/mie_calc.html .

Alternatively one may use manufacturer provided values based on dry microscopy. Our conclusions are unaltered.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Prism-sample interface for measuring Ir/Ii(θi) with a divergent incident beam. For low values of θi, light mostly refracts through (It), but TIR occurs for higher values. (b) and (c) depict angle-dependent penetration in TIR for transparent samples. The penetration is maximum at the critical angle. In a turbid medium, the picture is similar, but θc is not defined. G-H, Goos–Hänchen.

Fig. 2.
Fig. 2.

Reflectance data (1000 datapoints) for aqueous solution of polystyrene spheres (dia 0.356±0.014μm). Solid lines are theoretical fits; dots are data. Both the angle-dependent model [dark blue; see text for explanation of spike (bold)] and traditional Fresnel theory (light orange) fit data closely, yet only one is correct.

Tables (2)

Tables Icon

Table 1. AM and F Models Yield Different Particle Sizes

Tables Icon

Table 2. Comparison of Particle-Sizing by AM and F Models for Different Particle Sizes and Concentrations (Size Errors in Columns 6 and 7 are Calculated Relative to the DLS Sizes in Column 3)

Equations (1)

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κ(θi)=(4πnprism(ML)/2)1,

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