Abstract

We perform Monte Carlo light scattering simulations to study the angular distribution of the fluorescence emission from turbid media and compare the results to measured angular distributions from fluorescing white paper samples. The angular distribution of fluorescence emission is significantly depending on the concentration of fluorophores. The simulations show also a dependence on the angle of incidence that is however not as evident in the measurements. A detailed analysis of the factors affecting this angular distribution indicates that it is strongly correlated to the mean depth of the fluorescence process. The findings can find applications in fluorescence spectroscopy and are of particular interest when optimizing the impact of fluorescence on e.g. the appearance of paper as the measured values are angle dependent.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Determination of quantum efficiency in fluorescing turbid media

Ludovic Gustafsson Coppel, Mattias Andersson, and Per Edström
Appl. Opt. 50(17) 2784-2792 (2011)

Experimental and simulated angular profiles of fluorescence and diffuse reflectance emission from turbid media

Steven C. Gebhart, Anita Mahadevan-Jansen, and Wei-Chiang Lin
Appl. Opt. 44(23) 4884-4901 (2005)

Angle resolved color of bulk scattering media

Magnus Neuman, Ludovic G. Coppel, and Per Edström
Appl. Opt. 50(36) 6555-6563 (2011)

References

  • View by:
  • |
  • |
  • |

  1. L. G. Coppel, M. Andersson, O. Norberg, and S. Lindberg, “Impact of illumination spectral power distribution on radiance factor of fluorescing materials,” in Proceedings of IEEE Colour and Visual Computing Symposium (IEEE, 2013), pp. 1–4.
  2. R. Donaldson, “Spectrophotometry of fluorescent pigments,” Br. J. Appl. Phys. 5, 210–214 (1954).
    [Crossref]
  3. J. C. Zwinkels, D. S. Gignac, M. Nevins, I. Powell, and A. Bewsher, “Design and testing of a two-monochromator reference spectrofluorimeter for high accuracy total radiance factor measurements,” Appl. Opt. 36, 892–902 (1997).
    [Crossref] [PubMed]
  4. ASTM-E2153-01, “Standard practice for obtaining bispectral photometric data for evaluation of fluorescent color,” (2011).
  5. CIE, “CIE 15: Technical report: Colorimetry, 3rd edition,” Tech. Rep. (2004).
  6. S. Holopainen, F. Manoocheri, and E. Ikonen, “Non-Lambertian behaviour of fluorescence emission from solid amorphous material,” Metrologia 46, S197 (2009).
    [Crossref]
  7. N. Johansson and M. Andersson, “Angular variations of reflectance and fluorescence from paper - the influence of fluorescent whitening agents and fillers,” in Proceedings of 20th Color and Imaging Conference (IS&T, 2012), pp. 236–241.
  8. S. Tominaga, K. Hirai, and T. Horiuchi, “Measurement and modeling of bidirectional characteristics of fluorescence objects,” in Lecture Notes in Computer Science, A. Elmoataz, O. Lezoray, F. Nouboud, and D. Mammass, eds. (Springer International Publishing, 2014), pp. 35–42.
    [Crossref]
  9. M. Neuman, L. G. Coppel, and P. Edström, “Angle resolved color of bulk scattering media,” Appl. Opt. 50, 6555–6563 (2011).
    [Crossref] [PubMed]
  10. S. C. Gebhart, A. Mahadevan-Jansen, and W.-C. Lin, “Experimental and simulated angular profiles of fluorescence and diffuse reflectance emission from turbid media,” Appl. Opt. 44, 4884–4901 (2005).
    [Crossref] [PubMed]
  11. L. G. Coppel, M. Andersson, and P. Edström, “Determination of quantum efficiency in fluorescing turbid media,” Appl. Opt. 50, 2784–2792 (2011).
    [Crossref] [PubMed]
  12. P. Turunen, J. Kinnunen, and J. Mutanen, “Modeling of fluorescent color mixing by regression analysis,” in Proceedings of Fourth European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2010), pp. 94–100.
  13. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, 1977).
  14. B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).
  15. L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [Crossref]
  16. L. Wang, S. L. Jacques, and L. Zheng, “MCML–Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
    [Crossref]
  17. L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proceedings of paper making research symposium, E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009).
  18. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
    [Crossref]
  19. M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express 19, 1915–1920 (2011).
    [Crossref] [PubMed]
  20. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A 27, 1032–1039 (2010).
    [Crossref]
  21. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A 27, 1040–1045 (2010).
    [Crossref]
  22. M. Neuman, L. G. Coppel, and P. Edström, “A partial explanation of the dependence between light scattering and light absorption in the Kubelka-Munk model,” Nord. Pulp Pap. Res. J. 27, 426–430 (2012).
    [Crossref]
  23. N. Johansson, M. Neuman, M. Andersson, and P. Edström, “Influence of finite-sized detection solid angle on bidirectional reflectance distribution function measurements,” Appl. Opt. 53, 1212–1220 (2014).
    [Crossref] [PubMed]

2014 (1)

2012 (1)

M. Neuman, L. G. Coppel, and P. Edström, “A partial explanation of the dependence between light scattering and light absorption in the Kubelka-Munk model,” Nord. Pulp Pap. Res. J. 27, 426–430 (2012).
[Crossref]

2011 (3)

2010 (2)

2009 (1)

S. Holopainen, F. Manoocheri, and E. Ikonen, “Non-Lambertian behaviour of fluorescence emission from solid amorphous material,” Metrologia 46, S197 (2009).
[Crossref]

2005 (3)

B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[Crossref]

S. C. Gebhart, A. Mahadevan-Jansen, and W.-C. Lin, “Experimental and simulated angular profiles of fluorescence and diffuse reflectance emission from turbid media,” Appl. Opt. 44, 4884–4901 (2005).
[Crossref] [PubMed]

1997 (1)

1995 (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML–Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

1954 (1)

R. Donaldson, “Spectrophotometry of fluorescent pigments,” Br. J. Appl. Phys. 5, 210–214 (1954).
[Crossref]

1941 (1)

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Andersson, M.

N. Johansson, M. Neuman, M. Andersson, and P. Edström, “Influence of finite-sized detection solid angle on bidirectional reflectance distribution function measurements,” Appl. Opt. 53, 1212–1220 (2014).
[Crossref] [PubMed]

L. G. Coppel, M. Andersson, and P. Edström, “Determination of quantum efficiency in fluorescing turbid media,” Appl. Opt. 50, 2784–2792 (2011).
[Crossref] [PubMed]

L. G. Coppel, M. Andersson, O. Norberg, and S. Lindberg, “Impact of illumination spectral power distribution on radiance factor of fluorescing materials,” in Proceedings of IEEE Colour and Visual Computing Symposium (IEEE, 2013), pp. 1–4.

N. Johansson and M. Andersson, “Angular variations of reflectance and fluorescence from paper - the influence of fluorescent whitening agents and fillers,” in Proceedings of 20th Color and Imaging Conference (IS&T, 2012), pp. 236–241.

Bernad, B.

B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).

Bewsher, A.

Campos, J.

B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).

Coppel, L. G.

M. Neuman, L. G. Coppel, and P. Edström, “A partial explanation of the dependence between light scattering and light absorption in the Kubelka-Munk model,” Nord. Pulp Pap. Res. J. 27, 426–430 (2012).
[Crossref]

M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express 19, 1915–1920 (2011).
[Crossref] [PubMed]

L. G. Coppel, M. Andersson, and P. Edström, “Determination of quantum efficiency in fluorescing turbid media,” Appl. Opt. 50, 2784–2792 (2011).
[Crossref] [PubMed]

M. Neuman, L. G. Coppel, and P. Edström, “Angle resolved color of bulk scattering media,” Appl. Opt. 50, 6555–6563 (2011).
[Crossref] [PubMed]

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proceedings of paper making research symposium, E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009).

L. G. Coppel, M. Andersson, O. Norberg, and S. Lindberg, “Impact of illumination spectral power distribution on radiance factor of fluorescing materials,” in Proceedings of IEEE Colour and Visual Computing Symposium (IEEE, 2013), pp. 1–4.

Donaldson, R.

R. Donaldson, “Spectrophotometry of fluorescent pigments,” Br. J. Appl. Phys. 5, 210–214 (1954).
[Crossref]

Edström, P.

N. Johansson, M. Neuman, M. Andersson, and P. Edström, “Influence of finite-sized detection solid angle on bidirectional reflectance distribution function measurements,” Appl. Opt. 53, 1212–1220 (2014).
[Crossref] [PubMed]

M. Neuman, L. G. Coppel, and P. Edström, “A partial explanation of the dependence between light scattering and light absorption in the Kubelka-Munk model,” Nord. Pulp Pap. Res. J. 27, 426–430 (2012).
[Crossref]

M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express 19, 1915–1920 (2011).
[Crossref] [PubMed]

L. G. Coppel, M. Andersson, and P. Edström, “Determination of quantum efficiency in fluorescing turbid media,” Appl. Opt. 50, 2784–2792 (2011).
[Crossref] [PubMed]

M. Neuman, L. G. Coppel, and P. Edström, “Angle resolved color of bulk scattering media,” Appl. Opt. 50, 6555–6563 (2011).
[Crossref] [PubMed]

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A 27, 1032–1039 (2010).
[Crossref]

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A 27, 1040–1045 (2010).
[Crossref]

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[Crossref]

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proceedings of paper making research symposium, E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009).

Ferrero, A.

B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).

Gebhart, S. C.

Gignac, D. S.

Ginsberg, I.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, 1977).

Greenstein, J.

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Henyey, L.

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Hernanz, M. L.

B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).

Hirai, K.

S. Tominaga, K. Hirai, and T. Horiuchi, “Measurement and modeling of bidirectional characteristics of fluorescence objects,” in Lecture Notes in Computer Science, A. Elmoataz, O. Lezoray, F. Nouboud, and D. Mammass, eds. (Springer International Publishing, 2014), pp. 35–42.
[Crossref]

Holopainen, S.

S. Holopainen, F. Manoocheri, and E. Ikonen, “Non-Lambertian behaviour of fluorescence emission from solid amorphous material,” Metrologia 46, S197 (2009).
[Crossref]

Horiuchi, T.

S. Tominaga, K. Hirai, and T. Horiuchi, “Measurement and modeling of bidirectional characteristics of fluorescence objects,” in Lecture Notes in Computer Science, A. Elmoataz, O. Lezoray, F. Nouboud, and D. Mammass, eds. (Springer International Publishing, 2014), pp. 35–42.
[Crossref]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, 1977).

Ikonen, E.

S. Holopainen, F. Manoocheri, and E. Ikonen, “Non-Lambertian behaviour of fluorescence emission from solid amorphous material,” Metrologia 46, S197 (2009).
[Crossref]

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML–Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

Johansson, N.

N. Johansson, M. Neuman, M. Andersson, and P. Edström, “Influence of finite-sized detection solid angle on bidirectional reflectance distribution function measurements,” Appl. Opt. 53, 1212–1220 (2014).
[Crossref] [PubMed]

N. Johansson and M. Andersson, “Angular variations of reflectance and fluorescence from paper - the influence of fluorescent whitening agents and fillers,” in Proceedings of 20th Color and Imaging Conference (IS&T, 2012), pp. 236–241.

Kinnunen, J.

P. Turunen, J. Kinnunen, and J. Mutanen, “Modeling of fluorescent color mixing by regression analysis,” in Proceedings of Fourth European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2010), pp. 94–100.

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, 1977).

Lin, W.-C.

Lindberg, S.

L. G. Coppel, M. Andersson, O. Norberg, and S. Lindberg, “Impact of illumination spectral power distribution on radiance factor of fluorescing materials,” in Proceedings of IEEE Colour and Visual Computing Symposium (IEEE, 2013), pp. 1–4.

Lindquister, M.

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proceedings of paper making research symposium, E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009).

Mahadevan-Jansen, A.

Manoocheri, F.

S. Holopainen, F. Manoocheri, and E. Ikonen, “Non-Lambertian behaviour of fluorescence emission from solid amorphous material,” Metrologia 46, S197 (2009).
[Crossref]

Mutanen, J.

P. Turunen, J. Kinnunen, and J. Mutanen, “Modeling of fluorescent color mixing by regression analysis,” in Proceedings of Fourth European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2010), pp. 94–100.

Neuman, M.

Nevins, M.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, 1977).

Norberg, O.

L. G. Coppel, M. Andersson, O. Norberg, and S. Lindberg, “Impact of illumination spectral power distribution on radiance factor of fluorescing materials,” in Proceedings of IEEE Colour and Visual Computing Symposium (IEEE, 2013), pp. 1–4.

Pons, A.

B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).

Powell, I.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, 1977).

Tominaga, S.

S. Tominaga, K. Hirai, and T. Horiuchi, “Measurement and modeling of bidirectional characteristics of fluorescence objects,” in Lecture Notes in Computer Science, A. Elmoataz, O. Lezoray, F. Nouboud, and D. Mammass, eds. (Springer International Publishing, 2014), pp. 35–42.
[Crossref]

Turunen, P.

P. Turunen, J. Kinnunen, and J. Mutanen, “Modeling of fluorescent color mixing by regression analysis,” in Proceedings of Fourth European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2010), pp. 94–100.

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML–Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML–Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

Zwinkels, J. C.

Appl. Opt. (5)

Astrophys. J. (1)

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Br. J. Appl. Phys. (1)

R. Donaldson, “Spectrophotometry of fluorescent pigments,” Br. J. Appl. Phys. 5, 210–214 (1954).
[Crossref]

Comput. Meth. Prog. Bio. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML–Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. 47, 131–146 (1995).
[Crossref]

J. Opt. Soc. Am. A (2)

Metrologia (1)

S. Holopainen, F. Manoocheri, and E. Ikonen, “Non-Lambertian behaviour of fluorescence emission from solid amorphous material,” Metrologia 46, S197 (2009).
[Crossref]

Nord. Pulp Pap. Res. J. (1)

M. Neuman, L. G. Coppel, and P. Edström, “A partial explanation of the dependence between light scattering and light absorption in the Kubelka-Munk model,” Nord. Pulp Pap. Res. J. 27, 426–430 (2012).
[Crossref]

Opt. Express (1)

Proc. SPIE (1)

B. Bernad, A. Ferrero, A. Pons, M. L. Hernanz, and J. Campos, “Upgrade of goniospectrophotometer GEFE for near-field scattering and fluorescence radiance measurements,” Proc. SPIE 9398, 93980E (2005).

SIAM Rev. (1)

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005).
[Crossref]

Other (8)

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proceedings of paper making research symposium, E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009).

N. Johansson and M. Andersson, “Angular variations of reflectance and fluorescence from paper - the influence of fluorescent whitening agents and fillers,” in Proceedings of 20th Color and Imaging Conference (IS&T, 2012), pp. 236–241.

S. Tominaga, K. Hirai, and T. Horiuchi, “Measurement and modeling of bidirectional characteristics of fluorescence objects,” in Lecture Notes in Computer Science, A. Elmoataz, O. Lezoray, F. Nouboud, and D. Mammass, eds. (Springer International Publishing, 2014), pp. 35–42.
[Crossref]

L. G. Coppel, M. Andersson, O. Norberg, and S. Lindberg, “Impact of illumination spectral power distribution on radiance factor of fluorescing materials,” in Proceedings of IEEE Colour and Visual Computing Symposium (IEEE, 2013), pp. 1–4.

P. Turunen, J. Kinnunen, and J. Mutanen, “Modeling of fluorescent color mixing by regression analysis,” in Proceedings of Fourth European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2010), pp. 94–100.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, 1977).

ASTM-E2153-01, “Standard practice for obtaining bispectral photometric data for evaluation of fluorescent color,” (2011).

CIE, “CIE 15: Technical report: Colorimetry, 3rd edition,” Tech. Rep. (2004).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Defining angles for bidirectional reflectance. Light incident from (θi, ϕi) and contained within the differential solid angle i is reflected from a surface element dAi along the direction (θr, ϕr) and into the differential solid angle r.

Fig. 2
Fig. 2

The BRDF measurements are carried out with a setup where the entire illuminated area Ai of the sample is within the detectors field of view Af, for all angles of incidence and all viewing angles.

Fig. 3
Fig. 3

Difference between simulated and measured bispectral radiance factor ΔD versus mean free path le for an opaque pad of samples with 9 kg/T FWA, with g = 0.8 and Q optimized to minimize ΔD0. The value at zero difference gives the optimum le(360) and Q pair at constant single scattering albedo a(360), from which the scattering and absorption coefficient at the excitation wavelength can be estimated. Similar curves are obtained for cases with other g and FWA concentrations.

Fig. 4
Fig. 4

Simulated scaled BLDF at 440 nm for the opaque pad of 18 kg/T samples for different mean free paths le(360) and asymmetry factors g. The single scattering albedo a(360) is kept constant for each g value but differs for different g. The BLDF is less Lambertian for large le(360) and large g. The constant scaled BLDF (dashed curve) shows the case of a Lambertian luminescence for comparison purposes.

Fig. 5
Fig. 5

Simulated BRDF at 360 nm and 0° angle of incidence for the opaque pad of 18 kg/T samples for different mean free paths le(360) and asymmetry factors g. All curves cross each other at 45° since a(360) is optimised to fit the measured D(360, 360) in 0/45 geometry. As opposed to fluorescence in Fig. (4), le does not impact on the BRDF and g only affects the BRDF at large scattering angles.

Fig. 6
Fig. 6

Simulated scaled BLDF of the 3 kg/T FWA sample at 45° angle of incidence at different emission wavelengths λ2 between 400 nm and 540 nm with 20 nm interval. The constant scaled BLDF (dashed curve) shows the case of a Lambertian luminescence for comparison purposes.

Fig. 7
Fig. 7

Estimated absorption coefficient k and relative quantum efficiency Q(360, λ2) of the 3 kg/T FWA sample. At λ2 = 400 nm, the absorption is large and the quantum efficiency low. Hence light fluoresced at that wavelength does not contribute significantly to the measured BLDF.

Fig. 8
Fig. 8

Scaled measured (a–b) and simulated (c–d) BLDF at 0° (a,c) and 45° (b,d) angles of incidence. Only the simulations performed with g = 0.8 are shown. A Lambertian luminescence would have a constant scaled BLDF at 1.

Tables (1)

Tables Icon

Table 1 Estimated scattering (s) and absorption (k) coefficients at 360 nm excitation wavelength and 440 nm emission wavelength, and quantum efficiency (Q) for samples with different amounts of FWA and three different asymmetry factors (g). For the 36 kg/T FWA sample, the values in italic are extrapolated.

Equations (13)

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

f r ( θ i , ϕ i , θ r , ϕ r ) = d L r ( θ i , ϕ i , θ r , ϕ r ) d E i ( θ i , ϕ i ) ,
L r = L d = Φ r A d Ω f ,
L r = Φ r A d Ω f = Φ r A d ω = Φ r r 2 A d A i cos θ r .
E i = Φ i / A i = Φ i cos θ i / A s ,
f r = L r E i = Φ r Φ i r 2 A s cos θ i A d A s cos θ i cos θ r Φ r Φ i r 2 A d cos θ r .
f f ( θ i , ϕ i , θ r , ϕ r , λ 1 , λ 2 ) = L r ( λ 2 ) E i ( λ 1 ) = Φ r ( λ 2 ) Φ i ( λ 1 ) r 2 A d cos θ r ,
V ( θ r ) = λ Φ r ( θ r , λ 2 ) ( λ 2 ) Ψ ( λ 2 ) d λ 2 ,
Φ r ( θ r , λ 2 ) = f 1 ( θ r ) f 2 ( λ 2 ) .
V ( θ r ) = f 1 ( θ r ) × C ,
P ( t ) = ( s + k ) e ( s + k ) t .
a = s s + k ,
l e = 1 / ( s + k ) ,
min Q ( 360 , 440 ) [ D 0 ( 360 , 440 ) | s ( 360 ) , k ( 360 ) D 0 ( 360 , 440 ) ] 2 ,

Metrics