Abstract

We have investigated the transport of light through slabs that both scatter and strongly absorb, a situation that occurs in diverse application fields ranging from biomedical optics, powder technology, to solid-state lighting. In particular, we study the transport of light in the visible wavelength range between 420 and 700 nm through silicone plates filled with YAG:Ce3+ phosphor particles, that even re-emit absorbed light at different wavelengths. We measure the total transmission, the total reflection, and the ballistic transmission of light through these plates. We obtain average single particle properties namely the scattering cross-section σs, the absorption cross-section σa, and the anisotropy factor µ using an analytical approach, namely the P3 approximation to the radiative transfer equation. We verify the extracted transport parameters using Monte-Carlo simulations of the light transport. Our approach fully describes the light propagation in phosphor diffuser plates that are used in white LEDs and that reveal a strong absorption (L/la > 1) up to L/la = 4, where L is the slab thickness, la is the absorption mean free path. In contrast, the widely used diffusion theory fails to describe this parameter range. Our approach is a suitable analytical tool for industry, since it provides a fast yet accurate determination of key transport parameters, and since it introduces predictive power into the design process of white light emitting diodes.

© 2017 Optical Society of America

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References

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2017 (1)

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

2016 (2)

M. L. Meretska, A. Lagendijk, H. Thyrrestrup, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “How to distinguish elastically scattered light from Stokes shifted light for solid-state lighting?” J. Appl. Phys. 119(9), 093102 (2016).
[Crossref]

X. Li, J. M. Zhao, C. C. Wang, and L. H. Liu, “Improved transmission method for measuring the optical extinction coefficient of micro/nano particle suspensions,” Appl. Opt. 55(29), 8171 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (3)

2013 (3)

R. Uppu and S. Mujumdar, “Dependence of the Gaussian-Lévy transition on the disorder strength in random lasers,” Phys. Rev. A 87, 013822 (2013).
[Crossref]

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white LED lighting,” Appl. Opt. 52(12), 2602–2609 (2013).
[Crossref] [PubMed]

D. S. Wiersma, “Disordered photonics,” Nature Photon. 7, 188 (2013).
[Crossref]

2012 (1)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photon. 6, 283 (2012).
[Crossref]

2011 (1)

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011)
[Crossref] [PubMed]

2010 (2)

S. Mujumdar, R. Torre, H. Ramachandran, and D. Wiersma, “Monte Carlo calculations of spectral features in random lasing,” J. Nanophotonic 4(1), 041550 (2010).
[Crossref]

Z. Liu, S. Liu, K. Wang, and X. Luo, “Measurement and numerical studies of optical properties of YAG:Ce phosphor for white light-emitting diode packaging,” Appl. Opt. 49(2), 247–257 (2010).
[Crossref] [PubMed]

2009 (1)

C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
[Crossref]

2008 (1)

H. Bechtel, P. Schmidt, W. Busselt, and B. S. Schreinemacher, “Lumiramic new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[Crossref]

2007 (1)

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
[Crossref]

2006 (1)

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441 (2006).
[Crossref]

2005 (1)

2004 (1)

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Progr. Oceanogr. 61(1), 27 (2004).
[Crossref]

2002 (1)

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Intern J. Med. Phys. 29(9), 2013–2023 (2002).

2001 (2)

F. Fell and J. Fischer, “Numerical simulation of the light field in the atmosphere-ocean system using the matrix-operator method r,” J. Quant. Spectros. Radiat. Transfer 69(3), 351 (2001).
[Crossref]

D. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46, 2359 (2001).
[Crossref] [PubMed]

1999 (2)

R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick-Muraca, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity,” J. Pharm. Sci. 88(10), 959 (1999).
[Crossref] [PubMed]

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

1998 (1)

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modeling using the P3 approximation,” Phys. Med. Biol. 43(12), 3559 (1998).
[Crossref] [PubMed]

1997 (2)

S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

T. Burger, J. Kuhn, R. Caps, and J. Fricke, “Quantitative determination of the scattering and absorption coefficients from diffuse reflectance and transmittance measurements: application to pharmaceutical powders,” Appl. Spectrosc. 51(3), 309 (1997).
[Crossref]

1996 (1)

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270(3), 143–216 (1996).
[Crossref]

1995 (1)

D. Boas and H. Liu, “Photon migration within the P3 approximation,” Proc. SPIE 2386, 240 (1995).
[Crossref]

1994 (2)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE. 2258, 193 (1994).

D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E. 50, 857–866 (1994).
[Crossref]

1993 (2)

1989 (1)

W. M. Star, “Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Instit. Ser. IS5, 146 (1989).

1988 (1)

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Bio. 34(4), 437 (1988).
[Crossref]

1987 (1)

K. A. O’Donnell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 106(7), 1194–1205 (1987)
[Crossref]

1973 (1)

Akbulut, D.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011)
[Crossref] [PubMed]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic physics of electrons and photons (Cambridge University, 2007).
[Crossref]

Aldridge, P. K.

S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

Aydin, E. D.

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Intern J. Med. Phys. 29(9), 2013–2023 (2002).

Balgi, G. V.

R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick-Muraca, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity,” J. Pharm. Sci. 88(10), 959 (1999).
[Crossref] [PubMed]

Barajas, O.

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modeling using the P3 approximation,” Phys. Med. Biol. 43(12), 3559 (1998).
[Crossref] [PubMed]

Bechtel, H.

H. Bechtel, P. Schmidt, W. Busselt, and B. S. Schreinemacher, “Lumiramic new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[Crossref]

Beek, J. F.

Bertolotti, J.

H. Yılmaz, E. G. van Putten, J. Bertolotti, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica 2(5), 424–429 (2015)
[Crossref]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011)
[Crossref] [PubMed]

Boas, D.

D. Boas and H. Liu, “Photon migration within the P3 approximation,” Proc. SPIE 2386, 240 (1995).
[Crossref]

Bogucki, D.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Progr. Oceanogr. 61(1), 27 (2004).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffmann, Absorption and scattering of light by small particles (Wiley, 1983).

Boss, E.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Progr. Oceanogr. 61(1), 27 (2004).
[Crossref]

Brannegan, D. R.

S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

Brown, K.

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modeling using the P3 approximation,” Phys. Med. Biol. 43(12), 3559 (1998).
[Crossref] [PubMed]

Burger, T.

Busselt, W.

H. Bechtel, P. Schmidt, W. Busselt, and B. S. Schreinemacher, “Lumiramic new phosphor technology for high performance solid state light sources,” Proc. SPIE 7058, 70580E (2008).
[Crossref]

Caps, R.

Cassarly, W.

W. Cassarly, “Nonimaging optics: concentration and illumination,” in Handbook of Optics Volume III, 2nd ed., M. Bass, J.M. Enoch, E.W. van Stryland, and W.L. Wolfe, eds. (McGraw-Hill, 2001).

Chandrasekhar, S.

S. Chandrasekhar, Radiative transfer (Dover, 1960).

Craford, M. G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
[Crossref]

de Oliveira, C. R. E.

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Intern J. Med. Phys. 29(9), 2013–2023 (2002).

Dickey, D.

D. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46, 2359 (2001).
[Crossref] [PubMed]

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modeling using the P3 approximation,” Phys. Med. Biol. 43(12), 3559 (1998).
[Crossref] [PubMed]

Durian, D. J.

D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E. 50, 857–866 (1994).
[Crossref]

Evans, C. L.

S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

Faris, G. W.

Fell, F.

F. Fell and J. Fischer, “Numerical simulation of the light field in the atmosphere-ocean system using the matrix-operator method r,” J. Quant. Spectros. Radiat. Transfer 69(3), 351 (2001).
[Crossref]

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photon. 6, 283 (2012).
[Crossref]

Fischer, J.

F. Fell and J. Fischer, “Numerical simulation of the light field in the atmosphere-ocean system using the matrix-operator method r,” J. Quant. Spectros. Radiat. Transfer 69(3), 351 (2001).
[Crossref]

Forand, J. L.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE. 2258, 193 (1994).

Fournier, G. R.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE. 2258, 193 (1994).

Fricke, J.

Funk, C. J.

Gaonkar, H. A.

Gigan, S.

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Gilray, C.

C. Gilray and I. Lewin, “Monte Carlo techniques for the design of illumination optics,” Illuminating Engineering Society North America (IESNA) Annual Conference Technical Papers (July 1996), Paper 85, pp. 65–80.

Goddard, A. J. H.

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Intern J. Med. Phys. 29(9), 2013–2023 (2002).

Hailey, P. A.

S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

Harbers, G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
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C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
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A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441 (2006).
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M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
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Krenn, J.R.

C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
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Kumar, D.

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M. L. Meretska, A. Lagendijk, H. Thyrrestrup, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “How to distinguish elastically scattered light from Stokes shifted light for solid-state lighting?” J. Appl. Phys. 119(9), 093102 (2016).
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H. Yılmaz, E. G. van Putten, J. Bertolotti, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica 2(5), 424–429 (2015)
[Crossref]

V. Y. F. Leung, A. Lagendijk, T. W. Tukker, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “Interplay between multiple scattering, emission, and absorption of light in the phosphor of a white light-emitting diode,” Opt. Express 22(7), 8190–8204 (2014).
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W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white LED lighting,” Appl. Opt. 52(12), 2602–2609 (2013).
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A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photon. 6, 283 (2012).
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E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011)
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A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441 (2006).
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Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photon. 6, 283 (2012).
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W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Bio. 34(4), 437 (1988).
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Mendez, E. R.

K. A. O’Donnell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 106(7), 1194–1205 (1987)
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Meretska, M. L.

M. L. Meretska, A. Lagendijk, H. Thyrrestrup, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “How to distinguish elastically scattered light from Stokes shifted light for solid-state lighting?” J. Appl. Phys. 119(9), 093102 (2016).
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D. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46, 2359 (2001).
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D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modeling using the P3 approximation,” Phys. Med. Biol. 43(12), 3559 (1998).
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Mosk, A. P.

M. L. Meretska, A. Lagendijk, H. Thyrrestrup, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “How to distinguish elastically scattered light from Stokes shifted light for solid-state lighting?” J. Appl. Phys. 119(9), 093102 (2016).
[Crossref]

H. Yılmaz, E. G. van Putten, J. Bertolotti, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica 2(5), 424–429 (2015)
[Crossref]

V. Y. F. Leung, A. Lagendijk, T. W. Tukker, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “Interplay between multiple scattering, emission, and absorption of light in the phosphor of a white light-emitting diode,” Opt. Express 22(7), 8190–8204 (2014).
[Crossref] [PubMed]

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white LED lighting,” Appl. Opt. 52(12), 2602–2609 (2013).
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A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photon. 6, 283 (2012).
[Crossref]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011)
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Mueller, G. O.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
[Crossref]

Mueller-Mach, R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
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Mujumdar, S.

R. Uppu and S. Mujumdar, “Dependence of the Gaussian-Lévy transition on the disorder strength in random lasers,” Phys. Rev. A 87, 013822 (2013).
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S. Mujumdar, R. Torre, H. Ramachandran, and D. Wiersma, “Monte Carlo calculations of spectral features in random lasing,” J. Nanophotonic 4(1), 041550 (2010).
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R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick-Muraca, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity,” J. Pharm. Sci. 88(10), 959 (1999).
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K. A. O’Donnell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 106(7), 1194–1205 (1987)
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C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
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Rayner, D. C.

D. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46, 2359 (2001).
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S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

Sekulic, S. S.

S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

Sevick-Muraca, E. M.

R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick-Muraca, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity,” J. Pharm. Sci. 88(10), 959 (1999).
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Shchekin, O. B.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
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Shinde, R. R.

R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick-Muraca, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity,” J. Pharm. Sci. 88(10), 959 (1999).
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C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
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S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

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Stramski, D.

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Tasch, S.

C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
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Thyrrestrup, H.

M. L. Meretska, A. Lagendijk, H. Thyrrestrup, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “How to distinguish elastically scattered light from Stokes shifted light for solid-state lighting?” J. Appl. Phys. 119(9), 093102 (2016).
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Torre, R.

S. Mujumdar, R. Torre, H. Ramachandran, and D. Wiersma, “Monte Carlo calculations of spectral features in random lasing,” J. Nanophotonic 4(1), 041550 (2010).
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Tukker, T. W.

Tulip, J.

D. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46, 2359 (2001).
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D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modeling using the P3 approximation,” Phys. Med. Biol. 43(12), 3559 (1998).
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Uppu, R.

R. Uppu and S. Mujumdar, “Dependence of the Gaussian-Lévy transition on the disorder strength in random lasers,” Phys. Rev. A 87, 013822 (2013).
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J. W. Pickering, S. A. Prahl, N. Wieringen, J. F. Beek, H. J. C. M. Sterenborg, and M. J. C. van Gemert, “Double-integrating-sphere system for measuring the optical properties of tissue,” Appl. Opt. 32(4), 399 (1993).
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van Putten, E. G.

H. Yılmaz, E. G. van Putten, J. Bertolotti, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica 2(5), 424–429 (2015)
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E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011)
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van Rossum, M. C. W.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
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van Tiggelen, B. A.

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270(3), 143–216 (1996).
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Vos, W. L.

Voss, K. J.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Progr. Oceanogr. 61(1), 27 (2004).
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Wang, C. C.

Wang, K.

Ward, H. W.

S. S. Sekulic, H. W. Ward, D. R. Brannegan, E. D. Stanley, C. L. Evans, S. T. Sciavolino, P. A. Hailey, and P. K. Aldridge, “On-line monitoring of powder blend homogeneity by near-infrared spectroscopy,” Appl. Spectrosc. 51(3), 509–513 (1997).

Wenzl, F. P.

C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
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Wieringen, N.

Wiersma, D.

S. Mujumdar, R. Torre, H. Ramachandran, and D. Wiersma, “Monte Carlo calculations of spectral features in random lasing,” J. Nanophotonic 4(1), 041550 (2010).
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M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
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Appl. Spectrosc. (2)

T. Burger, J. Kuhn, R. Caps, and J. Fricke, “Quantitative determination of the scattering and absorption coefficients from diffuse reflectance and transmittance measurements: application to pharmaceutical powders,” Appl. Spectrosc. 51(3), 309 (1997).
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IEEE J. Disp. Tech. (1)

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Tech. 3(2), 160–175 (2007).
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IEEE J. Sel. Top. Quant. Elec. (1)

C. Sommer, J.R. Krenn, P. Hartmann, P. Pachler, M. Schweighart, S. Tasch, and F. P. Wenzl, “Effect of phosphor particle sizes on the angular homogeneity of phosphor-converted high-power white LED light sources,” IEEE J. Sel. Top. Quant. Elec. 15, 1181–1188 (2009).
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Intern J. Med. Phys. (1)

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J. Appl. Phys. (1)

M. L. Meretska, A. Lagendijk, H. Thyrrestrup, A. P. Mosk, W. L. IJzerman, and W. L. Vos, “How to distinguish elastically scattered light from Stokes shifted light for solid-state lighting?” J. Appl. Phys. 119(9), 093102 (2016).
[Crossref]

J. Comput. Phys. (1)

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441 (2006).
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J. Nanophotonic (1)

S. Mujumdar, R. Torre, H. Ramachandran, and D. Wiersma, “Monte Carlo calculations of spectral features in random lasing,” J. Nanophotonic 4(1), 041550 (2010).
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J. Opt. Soc. Am. A (1)

K. A. O’Donnell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 106(7), 1194–1205 (1987)
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J. Pharm. Sci. (1)

R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick-Muraca, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity,” J. Pharm. Sci. 88(10), 959 (1999).
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J. Quant. Spectros. Radiat. Transfer (1)

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Med. Phys. (1)

A. Liemert and A. Kienle, “Explicit solutions of the radiative transport equation in the P3 approximation,” Med. Phys. 41(11), 111916 (2014).
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Nat. Photon. (1)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photon. 6, 283 (2012).
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Nature Photon. (1)

D. S. Wiersma, “Disordered photonics,” Nature Photon. 7, 188 (2013).
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Opt. Express (1)

Optica (1)

Phys. Med. Bio. (1)

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Bio. 34(4), 437 (1988).
[Crossref]

Phys. Med. Biol. (2)

D. Dickey, O. Barajas, K. Brown, J. Tulip, and R. B. Moore, “Radiance modeling using the P3 approximation,” Phys. Med. Biol. 43(12), 3559 (1998).
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Figures (8)

Fig. 1
Fig. 1

Transport parameters in the plane spanned by absorption and diffusion. The absorption on the abscissa is gauged by the ratio of sample thickness L and absorption mean free path la and the diffusion strength on the ordinate by the ratio of sample thickness L and transport mean free path ltr. The shaded yellow range indicates the area where the diffusion theory performs well. The red range represents the transport parameter range accessed in the current manuscript. Symbols represent presently measured absorption and scattering as a function of wavelength for phosphor diffuser plates with particle concentrations 1 – 10 wt%. The gradient on the figure shows artistic representation of the diffusion theory applicability.

Fig. 2
Fig. 2

Scheme of light incident on and exiting from a slab. Plane waves with intensity Ii(λ, 0, ρ) are incident in the z−direction on a slab of scattering material that contains phosphor particles (represented by yellow circles). Id(λ, 0, ρ) is the diffuse reflected intensity, Id(λ, L, ρ) is the diffuse transmitted intensity, and Ii(λ, L, ρ) is the transmitted ballistic intensity. T and R are the total transmission and the total reflection, respectively, and Tb is the ballistic transmission.

Fig. 3
Fig. 3

Experimental setup. (a) The light source used in the experiment. (b) The configuration of the integrating sphere used for the total reflection measurements. (c) The configuration of the integrating sphere used for the total transmission measurements. (d) The configuration for ballistic Lamber-Beer measurements. F: Supercontinuum white light source, NDF: Neutral density filter, DM: Dichroic mirror, L1: Achromatic doublet (AC080-010-A-ML, f=10 mm), L2: Achromatic doublet (f=50 mm), L3: Achromatic doublet (AC254-050-A-ML, f=50mm), I: Integrating sphere, S: Spectrometer, P: Prism monochromator (f# = 4.6), p1: port1, p2: port 2, p3: port 3.

Fig. 4
Fig. 4

Ballistic transmission of the silicone plate with 1 wt% of phosphor particles measured at different spatial positions. The beam position during the measurements is shown in the inset.

Fig. 5
Fig. 5

Measured scattering and absorption cross section. The sum of the scattering cross section σs and absorption cross section σa measured as a function of wavelength for two different phosphor concentrations. Green triangles represent data extracted from the diffused light measurements.

Fig. 6
Fig. 6

Transmission and total reflection spectra of the silicone plates (a) Broadband transmission T as a function of concentration. (b) Broadband reflection as a function of concentration. Concentration of the phosphor particles is indicated on the figures. Arrows indicate the wavelengths for the Fig. 7

Fig. 7
Fig. 7

Transmission and reflection as a function of concentration for three chosen wavelength. (a) Transmission as a function of concentration shown for three different wavelengths. (b) Reflection as a function of concentration shown for the same three different wavelengths. In both figures symbols represent measured values. Solid lines are analytical results obtained using the P3 approximation to the radiative transfer equation.

Fig. 8
Fig. 8

Transport parameters of YAG:Ce3+ phosphor particles extracted with analytical model and Monte-Carlo simulations. Open symbols are the transport parameters values extracted using analytical approach. The solid, dashed, dot-dashed lines are the Monte-Carlo simulation results with respective error bars. (a) Scattering cross section σs as a function of wavelength. (b) Anisotropy factor µ as a function of wavelength. (c) Absorption cross section σa as a function of wavelength calculated using diffusion theory (P1), P3 approximation, and Monte-Carlo simulations.

Equations (14)

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T ( λ , L , ρ ) I d ( λ , L , ρ ) + I i ( λ , L , ρ ) I i ( λ , 0 , ρ ) ,
R ( λ , 0 , ρ ) I d ( λ , 0 , ρ ) I i ( λ , 0 , ρ ) ,
T b ( λ , L ) I i ( λ , L , ρ ) I i ( λ , 0 , ρ ) ,
T m ( λ , L , ρ ; σ s , σ a , μ ) F ( L ) F 0 ,
R m ( λ , L , ρ ; σ s , σ a , μ ) F ( 0 ) F 0 ,
F ( z ) = i = 1 4 B li exp ( μ i z ) + G 1 exp ( ρ σ t z ) ,
T b m ( λ , L , ρ ; σ a , σ s ) F 0 e ( ρ σ t L ) F 0 exp ( ρ σ t L ) .
l s = 1 ρ σ s ,
l a = 1 ρ σ a ,
1 l tr = ( 1 μ ) l s + 1 l a ,
η I d ( z , η ) z = ρ σ t I d ( z , η ) + ρ σ t 1 1 p ( η , η ) I d ( z , η ) d η + ρ σ t 4 π 4 π p ( s ^ , s ^ ) I b ( r , s ^ ) d ω ,
I d ( r , s ^ ) = l = 0 N m = l l ψ lm ( r ) Y lm ( s ^ ) ,
( 2 m + 1 ) ρ σ t exp ( ρ σ t z ) W m = ( m + 1 ) d ψ m + 1 ( z ) d z + m d ψ m 1 ( z ) d z + ρ σ t ( 1 W m ) ( 2 m + 1 ) ψ m ( z ) ,
ψ m = i = 1 m H mi C i exp ( μ i z ) + G m exp ( ρ σ t z ) ,

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