Abstract

We study the diffuse transport of light through polymer slabs containing TiO2 scattering particles. The slabs are diffuser plates typical of a commercial white light-emitting diode (LED) module (Fortimo). We have measured the diffuse transmission and reflection properties over a broad wavelength range (470–840 nm) from which we derive the transport mean free path using the theory of light diffusion. With increasing scatterer density, the mean free path becomes shorter. The mean free path increases with wavelength; hence, blue light is scattered more strongly than red light. To interpret the results, we propose an ab initio model without adjustable parameters for the mean free path by using Mie theory. We include inhomogeneous broadening as a result of the size distribution of the scattering particles as measured by dynamic light scattering. Surprisingly, the calculated mean free path decreases with wavelength, at variance with our experiments, which is caused by particles with radii R in excess of 0.25 μm. Close inspection of the scatterers by electron microscopy reveals that large particles (R>0.4μm) consist of clusters of small particles (R<0.13μm). Therefore, we have improved our model by only taking into account the individual scatterers within the clusters. This model predicts mean free paths in good agreement with our experimental results. We discuss consequences of our results to white LED lighting modules.

© 2013 Optical Society of America

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  3. 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]
  4. 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. Quantum Electron. 15, 1181–1188 (2009).
    [CrossRef]
  5. C. Gilray and I. Lewin, “Monte Carlo techniques for the design of illumination optics,” in Illuminating Engineering Soc North America (IESNA) Annual Conference Technical Papers (IESNA, 1996), Paper no. 85, pp. 65–80.
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  7. 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, 247–257 (2010).
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  12. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).
  13. J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
    [CrossRef]
  14. P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
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  15. O. L. Muskens and A. Lagendijk, “Broadband enhanced backscattering spectroscopy of strongly scattering media,” Opt. Express 16, 1222–1231 (2008).
    [CrossRef]
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  21. R. J. Hunter, Foundations of Colloid Science (Clarendon, 1993).
  22. From the change and prominence of the 676 nm feature at constant TiO2 density and sample thickness, we conclude that it is neither related to TiO2 nor to the polymer matrix itself, but probably to unknown additives.
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    [CrossRef]
  26. A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
    [CrossRef]
  27. J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef]
  28. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).
  29. C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  30. See: www.philiplaven.com/mieplot.htm , retrieved multiple instances since 2010. As mentioned on the website: “MiePlot was originally designed to provide a simple interface (for PCs using Microsoft Windows) to the classic BHMIE algorithm for Mie scattering from a sphere—as published by Bohren and Huffmann [29]”.
  31. K. Ishida, I. Mituishi, Y. Hattori, and S. Nunoue, “A revised Kubelka-Munk theory for spectral simulation of phosphor-based white light-emitting diodes,” Appl. Phys. Lett. 93, 241910 (2008).
    [CrossRef]
  32. F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
    [CrossRef]
  33. M. I. Mishchenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4666(1993).
    [CrossRef]
  34. While performing computations of Eqs. (8)–(10) with a size distribution cutoff at R=Rmax, we have carefully applied proper normalization. Thus the denominator of the equations was also integrated from R=0 to R=Rmax to obtain the proper total number density of scatterers n≡(N/V).
  35. A. Lagendijk, in Ultrashort Processes in Condensed Matter, W. E. Bron, ed. (Plenum, 1993), pp. 197–238.
  36. X. H. Yan, J. W. Ding, and Q. B. Yang, “Size and randomness effects on the temperature-dependent hopping conductivity of nanocrystalline chains,” Eur. Phys. J. B 20, 157–163 (2001).
    [CrossRef]
  37. M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random photonic media,” Phys. Rev. Lett. 105, 013904 (2010).
    [CrossRef]
  38. A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
    [CrossRef]

2010 (2)

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, 247–257 (2010).
[CrossRef]

M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random photonic media,” Phys. Rev. Lett. 105, 013904 (2010).
[CrossRef]

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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

2008 (4)

K. Ishida, I. Mituishi, Y. Hattori, and S. Nunoue, “A revised Kubelka-Munk theory for spectral simulation of phosphor-based white light-emitting diodes,” Appl. Phys. Lett. 93, 241910 (2008).
[CrossRef]

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]

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

O. L. Muskens and A. Lagendijk, “Broadband enhanced backscattering spectroscopy of strongly scattering media,” Opt. Express 16, 1222–1231 (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,” J. Display Technology 3, 160–175 (2007).
[CrossRef]

2005 (1)

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

2002 (1)

L. Bechger, A. F. Koenderink, and W. L. Vos, “Emission spectra and lifetimes of R6G dye on silica-coated titania powder,” Langmuir 18, 2444–2447 (2002).
[CrossRef]

2001 (1)

X. H. Yan, J. W. Ding, and Q. B. Yang, “Size and randomness effects on the temperature-dependent hopping conductivity of nanocrystalline chains,” Eur. Phys. J. B 20, 157–163 (2001).
[CrossRef]

1999 (2)

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]

J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

1996 (1)

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

1994 (1)

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

1993 (1)

1992 (1)

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

1991 (1)

J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef]

1989 (1)

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

1988 (1)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

1965 (1)

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[CrossRef]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Altube, A.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Bechger, L.

L. Bechger, A. F. Koenderink, and W. L. Vos, “Emission spectra and lifetimes of R6G dye on silica-coated titania powder,” Langmuir 18, 2444–2447 (2002).
[CrossRef]

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]

Bertolotti, J.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Birowosuto, M. D.

M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random photonic media,” Phys. Rev. Lett. 105, 013904 (2010).
[CrossRef]

Blanco, A.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Bohren, C. F.

C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bond, W. L.

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[CrossRef]

Bret, B. P. J.

B. P. J. Bret, Multiple light scattering in porous gallium phosphide, Ph.D. thesis (University of Twente, 2005).

Busch, K.

J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

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]

Cassarly, W.

W. Cassarly, “Nonimaging optics: concentration and illumination,” in Handbook of Optics, M. Bass, J. M. Enoch, E. W. van Stryland, and W. L. Wolfe, eds. 2nd ed. (McGraw-Hill, 2001), Vol. 3, p. 20.

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,” J. Display Technology 3, 160–175 (2007).
[CrossRef]

de Haan, J. F.

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

de Vries, P.

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Ding, J. W.

X. H. Yan, J. W. Ding, and Q. B. Yang, “Size and randomness effects on the temperature-dependent hopping conductivity of nanocrystalline chains,” Eur. Phys. J. B 20, 157–163 (2001).
[CrossRef]

Durian, D. J.

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

García, P. D.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Gilray, C.

C. Gilray and I. Lewin, “Monte Carlo techniques for the design of illumination optics,” in Illuminating Engineering Soc North America (IESNA) Annual Conference Technical Papers (IESNA, 1996), Paper no. 85, pp. 65–80.

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,” J. Display Technology 3, 160–175 (2007).
[CrossRef]

Hartmann, 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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Hattori, Y.

K. Ishida, I. Mituishi, Y. Hattori, and S. Nunoue, “A revised Kubelka-Munk theory for spectral simulation of phosphor-based white light-emitting diodes,” Appl. Phys. Lett. 93, 241910 (2008).
[CrossRef]

Hovenier, J. W.

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

Huffmann, D. R.

C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Hunter, R. J.

R. J. Hunter, Foundations of Colloid Science (Clarendon, 1993).

Ishida, K.

K. Ishida, I. Mituishi, Y. Hattori, and S. Nunoue, “A revised Kubelka-Munk theory for spectral simulation of phosphor-based white light-emitting diodes,” Appl. Phys. Lett. 93, 241910 (2008).
[CrossRef]

Koenderink, A. F.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

L. Bechger, A. F. Koenderink, and W. L. Vos, “Emission spectra and lifetimes of R6G dye on silica-coated titania powder,” Langmuir 18, 2444–2447 (2002).
[CrossRef]

Krames, M. 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,” J. Display Technology 3, 160–175 (2007).
[CrossRef]

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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Kuik, F.

F. Kuik, J. F. de Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

Lagendijk, A.

O. L. Muskens and A. Lagendijk, “Broadband enhanced backscattering spectroscopy of strongly scattering media,” Opt. Express 16, 1222–1231 (2008).
[CrossRef]

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

A. Lagendijk, in Ultrashort Processes in Condensed Matter, W. E. Bron, ed. (Plenum, 1993), pp. 197–238.

Lewin, I.

C. Gilray and I. Lewin, “Monte Carlo techniques for the design of illumination optics,” in Illuminating Engineering Soc North America (IESNA) Annual Conference Technical Papers (IESNA, 1996), Paper no. 85, pp. 65–80.

Liu, S.

Liu, Z.

López, C.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Luo, X.

Martín, M. D.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Mishchenko, M. I.

Mituishi, I.

K. Ishida, I. Mituishi, Y. Hattori, and S. Nunoue, “A revised Kubelka-Munk theory for spectral simulation of phosphor-based white light-emitting diodes,” Appl. Phys. Lett. 93, 241910 (2008).
[CrossRef]

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Mosk, A. P.

M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random photonic media,” Phys. Rev. Lett. 105, 013904 (2010).
[CrossRef]

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,” J. Display Technology 3, 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,” J. Display Technology 3, 160–175 (2007).
[CrossRef]

Muskens, O. L.

Nieuwenhuizen, T. M.

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]

Nunoue, S.

K. Ishida, I. Mituishi, Y. Hattori, and S. Nunoue, “A revised Kubelka-Munk theory for spectral simulation of phosphor-based white light-emitting diodes,” Appl. Phys. Lett. 93, 241910 (2008).
[CrossRef]

Pachler, 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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Pine, D. J.

J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef]

Rivas, J. G.

J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

Sapienza, R.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Schmidt, P.

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]

Schreinemacher, B. S.

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]

Schubert, E. F.

E. F. Schubert, Light Emitting Diodes (Cambridge University, 2006).

Schweighart, M.

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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

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,” J. Display Technology 3, 160–175 (2007).
[CrossRef]

Skipetrov, S. E.

M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random photonic media,” Phys. Rev. Lett. 105, 013904 (2010).
[CrossRef]

Sommer, C.

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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Soukoulis, C. M.

J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

Sprik, R.

J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Tukker, T. W.

T. W. Tukker, “Fluorescence modeling in remote and close LED illumination devices,” in SPIE International Optical Design Conference 2010 (SPIE, 2010), Paper no. ITuE2.

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M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

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H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

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M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

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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]

van Tiggelen, B. A.

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Viña, L.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Vos, W. L.

M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random photonic media,” Phys. Rev. Lett. 105, 013904 (2010).
[CrossRef]

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

L. Bechger, A. F. Koenderink, and W. L. Vos, “Emission spectra and lifetimes of R6G dye on silica-coated titania powder,” Langmuir 18, 2444–2447 (2002).
[CrossRef]

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A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

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Weitz, D. A.

J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef]

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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. Quantum Electron. 15, 1181–1188 (2009).
[CrossRef]

Wiersma, D. S.

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
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[CrossRef]

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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,” J. Display Technology 3, 160–175 (2007).
[CrossRef]

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J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
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Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. Ishida, I. Mituishi, Y. Hattori, and S. Nunoue, “A revised Kubelka-Munk theory for spectral simulation of phosphor-based white light-emitting diodes,” Appl. Phys. Lett. 93, 241910 (2008).
[CrossRef]

Eur. Phys. J. B (1)

X. H. Yan, J. W. Ding, and Q. B. Yang, “Size and randomness effects on the temperature-dependent hopping conductivity of nanocrystalline chains,” Eur. Phys. J. B 20, 157–163 (2001).
[CrossRef]

Europhys. Lett. (1)

J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Optical transmission through strong scattering and highly polydisperse media,” Europhys. Lett. 48, 22–28 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (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. Quantum Electron. 15, 1181–1188 (2009).
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[CrossRef]

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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,” J. Display Technology 3, 160–175 (2007).
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[CrossRef]

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L. Bechger, A. F. Koenderink, and W. L. Vos, “Emission spectra and lifetimes of R6G dye on silica-coated titania powder,” Langmuir 18, 2444–2447 (2002).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

A. Lagendijk, R. Vreeker, and P. de Vries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Phys. Rep. (1)

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Phys. Rev. A (2)

J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef]

P. D. García, R. Sapienza, J. Bertolotti, M. D. Martín, A. Blanco, A. Altube, L. Viña, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Phys. Rev. B (2)

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random photonic media,” Phys. Rev. Lett. 105, 013904 (2010).
[CrossRef]

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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]

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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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Other (17)

B. P. J. Bret, Multiple light scattering in porous gallium phosphide, Ph.D. thesis (University of Twente, 2005).

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

J. Bartels, R. Bechmann, A. Eucken, A. M. Hellwege, and K. H. Hellwege, eds., Landolt-Börnstein, Zahlenwerte und funktionen, Optische Konstanten (Springer-Verlag, 1962), Vol. 2, Part 8.

T. W. Tukker, “Fluorescence modeling in remote and close LED illumination devices,” in SPIE International Optical Design Conference 2010 (SPIE, 2010), Paper no. ITuE2.

From: www.theplasticshop.co.uk/plastic_technical_data_sheets/lexan_polycarbonate_9030_technical_properties_data_sheet.pdf , retrieved May, 2011.

From: www.azom.com/article.aspx?ArticleID=1179 , retrieved May, 2011.

R. J. Hunter, Foundations of Colloid Science (Clarendon, 1993).

From the change and prominence of the 676 nm feature at constant TiO2 density and sample thickness, we conclude that it is neither related to TiO2 nor to the polymer matrix itself, but probably to unknown additives.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

See: www.philiplaven.com/mieplot.htm , retrieved multiple instances since 2010. As mentioned on the website: “MiePlot was originally designed to provide a simple interface (for PCs using Microsoft Windows) to the classic BHMIE algorithm for Mie scattering from a sphere—as published by Bohren and Huffmann [29]”.

E. F. Schubert, Light Emitting Diodes (Cambridge University, 2006).

C. Gilray and I. Lewin, “Monte Carlo techniques for the design of illumination optics,” in Illuminating Engineering Soc North America (IESNA) Annual Conference Technical Papers (IESNA, 1996), Paper no. 85, pp. 65–80.

W. Cassarly, “Nonimaging optics: concentration and illumination,” in Handbook of Optics, M. Bass, J. M. Enoch, E. W. van Stryland, and W. L. Wolfe, eds. 2nd ed. (McGraw-Hill, 2001), Vol. 3, p. 20.

While performing computations of Eqs. (8)–(10) with a size distribution cutoff at R=Rmax, we have carefully applied proper normalization. Thus the denominator of the equations was also integrated from R=0 to R=Rmax to obtain the proper total number density of scatterers n≡(N/V).

A. Lagendijk, in Ultrashort Processes in Condensed Matter, W. E. Bron, ed. (Plenum, 1993), pp. 197–238.

See catalog at: www.lighting.philips.co.uk/pwc_li/gb_en/subsites/oem/fortimo-led-catalogue/files/assets/downloads/files/Fortimo-LED-catalogue.pdf , retrieved April, 2012.

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Figures (7)

Fig. 1.
Fig. 1.

SEM of TiO2 particles as used in the experiments. The scale bar is 10 μm long. The image shows a large variation in sizes as illustrated in Fig. 2. Inset: zoom-in of a large particle that consists of a cluster of smaller particles (R<0.13μm); scale bar is 0.2 μm long.

Fig. 2.
Fig. 2.

Distribution of the radii n(R) of the TiO2 light scattering particles used in the experiments. Since the distribution is obtained from dynamic light scattering, the radii are hydrodynamic radii [21]. The particles are approximated to be spherical, and the distribution is normalized to 1.

Fig. 3.
Fig. 3.

Total transmission T versus wavelength of polymer slabs with increasing TiO2 particle density (indicated), from top to bottom: f=0vol.% (black and red symbols), 0.003 vol. % (blue and magenta symbols), 0.015 vol. % (dark yellow and navy symbols), 0.03 vol. % (purple and wine symbols), 0.3 vol. % (olive and dark cyan symbols). For each sample two different measurements are shown, indicating the good reproducibility.

Fig. 4.
Fig. 4.

Reflectivity R versus wavelength of the polymer slabs with increasing TiO2 particle density: f=0vol.% (black squares), 0.003 vol. % (red circles), 0.015 vol. % (blue triangles), 0.03 vol. % (magenta inverted triangles), and 0.3 vol. % (dark yellow diamonds).

Fig. 5.
Fig. 5.

Transport mean free path versus wavelength for polymer slabs with increasing TiO2 content, from bottom to top: f=0.3vol.% (dark blue), 0.03 vol. % (magenta), and 0.015 vol. % (red). The horizontal dashed–dotted line illustrates the sample thickness L.

Fig. 6.
Fig. 6.

Inverse of the transport mean free path versus density of TiO2 scatterers in vol. % in a log–log representation. Data are shown for three different wavelengths: λ=500nm in black, 600 nm in red, 700 nm in blue. For each data set, a linear dependence with zero offset is plotted [cf. Eq. (9)] with slope 57.7, 39.4, and 29.4 from top to bottom.

Fig. 7.
Fig. 7.

Transport mean free path versus wavelength. Symbols are our measurements for a Fortimo slab with f=0.3vol.% TiO2 scatterers. The spurious λ=676nm absorption band has been omitted [22]. The black dashed–dotted curve is calculated for a polydisperse assembly of spherical scatters with a hydrodynamic size distribution (see Fig. 2). The blue drawn curve is calculated for a polydisperse assembly of spherical scatters with a size distribution cutoff at R=130nm based on the SEM results.

Equations (10)

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

T(λ)=(1Rs(λ))(λ)+ze(λ)L+2ze(λ).
ze(λ)=2(λ)31+R¯(λ)1R¯(λ).
(λ)=Lt(λ)(1+23r(λ)43r(λ)t(λ)),
t(λ)[1Rs(λ)]T(λ)=L+2ze(λ)(λ)+ze(λ),
r(λ)1+R¯(λ)1R¯(λ).
lim(L,Rs1)T(λ)=(λ)+ze(λ)2ze(λ)=1+23r43r.
1lscat=NVCavg=NVN1C1+N2C2++NmCmN1+N2++Nm,
1lscat=nCavg=n0RmaxdRn(R)C(R)0RmaxdRn(R),
1=nCavg(1cos(θ)avg)=
=n0RmaxdRn(R)C(R)(1cos(θ)(R))0RmaxdRn(R),

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