Abstract

Discrete random media have been investigated extensively over the past century due to their ability to scatter light. Even so, the link between the three-dimensional (3D) spatial distribution of the scattering elements and the resulting opacity is still lively debated to date due to different experimental conditions, range of parameters explored, or sample formulations. On the other hand, a unified numerical survey with controlled parameters has been impractical up to date due to the sheer computational power required to address samples with representative size. In this work, we exploit a graphics processing unit implementation of the T-matrix method to investigate the complete range of particle volume concentration and packing-induced spatial correlations, allowing us to reveal and elucidate a twofold role played by spatial correlations in either enhancing or suppressing opacity. By applying these findings to the illustrative case of white paint, we determine the optimal combination of density and spatial correlations corresponding to the highest opacity.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (4)

S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective radiative cooling by paint-format microsphere-based photonic random media,” ACS Photon. 5, 1181–1187 (2018).
[Crossref]

B. X. Wang and C. Y. Zhao, “Analysis of dependent scattering mechanism in hard-sphere Yukawa random media,” J. Appl. Phys. 123, 223101 (2018).
[Crossref]

M. A. Klatt and S. Torquato, “Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions,” Phys. Rev. E 97, 012118 (2018).
[Crossref]

G. Shang, L. Maiwald, H. Renner, D. Jalas, M. Dosta, S. Heinrich, A. Petrov, and M. Eich, “Photonic glass for high contrast structural color,” Sci. Rep. 8, 7804 (2018).
[Crossref]

2017 (8)

A. Egel, L. Pattelli, G. Mazzamuto, D. S. Wiersma, and U. Lemmer, “CELES: CUDA-accelerated simulation of electromagnetic scattering by large ensembles of spheres,” J. Quant. Spectrosc. Radiat. Transfer 199, 103–110 (2017).
[Crossref]

J. M. Escalante and S. E. Skipetrov, “Longitudinal optical fields in light scattering from dielectric spheres and Anderson localization of light,” Ann. Phys. 529, 1700039 (2017).
[Crossref]

G. J. Aubry, L. Schertel, M. Chen, H. Weyer, C. M. Aegerter, S. Polarz, H. Cölfen, and G. Maret, “Resonant transport and near-field effects in photonic glasses,” Phys. Rev. A 96, 043871 (2017).
[Crossref]

Y. Zhai, Y. Ma, S. N. David, D. Zhao, R. Lou, G. Tan, R. Yang, and X. Yin, “Scalable-manufactured randomized glass-polymer hybrid metamaterial for daytime radiative cooling,” Science 355, 1062–1066 (2017).
[Crossref]

L. S. Froufe-Pérez, M. Engel, J. J. Sáenz, and F. Scheffold, “Band gap formation and Anderson localization in disordered photonic materials with structural correlations,” Proc. Natl. Acad. Sci. USA 114, 9570–9574 (2017).
[Crossref]

F. Riboli, F. Uccheddu, G. Monaco, N. Caselli, F. Intonti, M. Gurioli, and S. Skipetrov, “Tailoring correlations of the local density of states in disordered photonic materials,” Phys. Rev. Lett. 119, 043902 (2017).
[Crossref]

P. Garcia and P. Lodahl, “Physics of quantum light emitters in disordered photonic nanostructures,” Ann. Phys. 529, 1600351 (2017).
[Crossref]

B. D. Wilts, X. Sheng, M. Holler, A. Diaz, M. Guizar-Sicairos, J. Raabe, R. Hoppe, S.-H. Liu, R. Langford, O. D. Onelli, D. Chen, S. Torquato, U. Steiner, C. G. Schroer, S. Vignolini, and A. Sepe, “Evolutionary-optimized photonic network structure in white beetle wing scales,” Adv. Mater. 30, 1702057 (2017).
[Crossref]

2016 (6)

O. Leseur, R. Pierrat, and R. Carminati, “High-density hyperuniform materials can be transparent,” Optica 3, 763–767 (2016).
[Crossref]

R. R. Naraghi, S. Sukhov, and A. Dogariu, “Disorder fingerprint: intensity distributions in the near field of random media,” Phys. Rev. B 94, 174205 (2016).
[Crossref]

J. M. Rieser, C. P. Goodrich, A. J. Liu, and D. J. Durian, “Divergence of Voronoi cell anisotropy vector: a threshold-free characterization of local structure in amorphous materials,” Phys. Rev. Lett. 116, 088001 (2016).
[Crossref]

G. Mazzamuto, L. Pattelli, C. Toninelli, and D. Wiersma, “Deducing effective light transport parameters in optically thin systems,” New J. Phys. 18, 023036 (2016).
[Crossref]

L. Pattelli, R. Savo, M. Burresi, and D. S. Wiersma, “Spatio-temporal visualization of light transport in complex photonic structures,” Light Sci. Appl. 5, e16090 (2016).
[Crossref]

T. Sperling, L. Schertel, M. Ackermann, G. J. Aubry, C. M. Aegerter, and G. Maret, “Can 3D light localization be reached in “white paint”?” New J. Phys. 18, 013039 (2016).
[Crossref]

2015 (2)

R. R. Naraghi, S. Sukhov, J. Sáenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

M. Burresi, F. Pratesi, F. Riboli, and D. S. Wiersma, “Complex photonic structures for light harvesting,” Adv. Opt. Mater. 3, 722–743 (2015).
[Crossref]

2014 (6)

L. Bressel and O. Reich, “Theoretical and experimental study of the diffuse transmission of light through highly concentrated absorbing and scattering materials: part I: Monte-Carlo simulations,” J. Quant. Spectrosc. Radiat. Transfer 146, 190–198 (2014).
[Crossref]

M. Burresi, L. Cortese, L. Pattelli, M. Kolle, P. Vukusic, D. S. Wiersma, U. Steiner, and S. Vignolini, “Bright-white beetle scales optimise multiple scattering of light,” Sci. Rep. 4, 6075 (2014).
[Crossref]

N. Elton and A. Legrix, “Spatial point statistics for quantifying TiO2distribution in paint,” J. Coat. Technol. Res. 11, 443–454 (2014).

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112, 143901 (2014).
[Crossref]

S. E. Skipetrov and I. M. Sokolov, “Absence of Anderson localization of light in a random ensemble of point scatterers,” Phys. Rev. Lett. 112, 023905 (2014).
[Crossref]

M. A. Klatt and S. Torquato, “Characterization of maximally random jammed sphere packings: Voronoi correlation functions,” Phys. Rev. E 90, 052120 (2014).
[Crossref]

2013 (4)

D. Mackowski and M. I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 103–112 (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 light-emitting diode lighting,” Appl. Opt. 52, 2602–2609 (2013).
[Crossref]

D. S. Wiersma, “Disordered photonics,” Nat. Photonics 7, 188–196 (2013).
[Crossref]

L. Shi, Y. Zhang, B. Dong, T. Zhan, X. Liu, and J. Zi, “Amorphous photonic crystals with only short-range order,” Adv. Mater. 25, 5314–5320 (2013).
[Crossref]

2012 (3)

D. Molinari and A. Fratalocchi, “Route to strong localization of light: the role of disorder,” Opt. Express 20, 18156–18164 (2012).
[Crossref]

J.-C. Auger and B. Stout, “Dependent light scattering in white paint films: clarification and application of the theoretical concepts,” J. Coat. Technol. Res. 9, 287–295 (2012).

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
[Crossref]

2011 (5)

M. P. Diebold, “A Monte Carlo determination of the effectiveness of nanoparticles as spacers for optimizing TiO2 opacity,” J. Coat. Technol. Res. 8, 541–552 (2011).

S. F. Liew, J. Forster, H. Noh, C. F. Schreck, V. Saranathan, X. Lu, L. Yang, R. O. Prum, C. S. O’Hern, E. R. Dufresne, and H. Cao, “Short-range order and near-field effects on optical scattering and structural coloration,” Opt. Express 19, 8208–8217 (2011).
[Crossref]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transfer 112, 671–692 (2011).
[Crossref]

J. M. Dlugach, M. I. Mishchenko, L. Liu, and D. W. Mackowski, “Numerically exact computer simulations of light scattering by densely packed, random particulate media,” J. Quant. Spectrosc. Radiat. Transfer 112, 2068–2078 (2011).
[Crossref]

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
[Crossref]

2010 (1)

J. Bertolotti, K. Vynck, L. Pattelli, P. Barthelemy, S. Lepri, and D. S. Wiersma, “Engineering disorder in superdiffusive levy glasses,” Adv. Funct. Mater. 20, 965–968 (2010).
[Crossref]

2009 (4)

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. 34, 130–132 (2009).
[Crossref]

C. H. Rycroft, “Voro++: a three-dimensional Voronoi cell library in C++,” Chaos 19, 041111 (2009).
[Crossref]

J.-C. Auger, V. A. Martinez, and B. Stout, “Theoretical study of the scattering efficiency of rutile titanium dioxide pigments as a function of their spatial dispersion,” J. Coat. Technol. Res. 6, 89–97 (2009).

Y. Okada and A. Kokhanovsky, “Light scattering and absorption by densely packed groups of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 902–917 (2009).
[Crossref]

2008 (1)

S. H. Tseng, “Optical characteristics of a cluster of closely-packed dielectric spheres,” Opt. Commun. 281, 1986–1990 (2008).
[Crossref]

2007 (2)

O. Berger, D. Inns, and A. G. Aberle, “Commercial white paint as back surface reflector for thin-film solar cells,” Sol. Energy Mater. Sol. Cells 91, 1215–1221 (2007).
[Crossref]

X. Peng and A. Dinsmore, “Light propagation in strongly scattering, random colloidal films: the role of the packing geometry,” Phys. Rev. Lett. 99, 143902 (2007).
[Crossref]

2006 (1)

M. Skoge, A. Donev, F. H. Stillinger, and S. Torquato, “Packing hyperspheres in high-dimensional Euclidean spaces,” Phys. Rev. E 74, 041127 (2006).
[Crossref]

2005 (1)

A. Donev, F. H. Stillinger, and S. Torquato, “Unexpected density fluctuations in jammed disordered sphere packings,” Phys. Rev. Lett. 95, 090604 (2005).
[Crossref]

2004 (1)

L. F. Rojas-Ochoa, J. Mendez-Alcaraz, J. Sáenz, P. Schurtenberger, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. 93, 073903 (2004).
[Crossref]

2003 (1)

J.-C. Auger, R. G. Barrera, and B. Stout, “Scattering efficiency of clusters composed by aggregated spheres,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 521–531 (2003).
[Crossref]

2000 (2)

L. McNeil and R. French, “Multiple scattering from rutile TiO2 particles,” Acta Mater. 48, 4571–4576 (2000).
[Crossref]

S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is random close packing of spheres well defined?” Phys. Rev. Lett. 84, 2064–2067 (2000).
[Crossref]

1998 (1)

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[Crossref]

1997 (2)

1995 (2)

1994 (3)

R. West, D. Gibbs, L. Tsang, and A. Fung, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
[Crossref]

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

C. Soukoulis, S. Datta, and E. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810 (1994).
[Crossref]

1992 (1)

1990 (1)

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[Crossref]

1989 (1)

B. Palmer, P. Stamatakis, C. Bohren, and G. Salzman, “Multiple-scattering model for opacifying particles in polymer films,” J. Coat. Technol. 61, 41–47 (1989).

1988 (2)

J. Braun, “Crowding and spacing of titanium dioxide pigments,” J. Coat. Technol. 60, 67–71 (1988).

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]

1987 (1)

B. Drolen and C. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. Heat Transfer 1, 63–68 (1987).
[Crossref]

1985 (1)

S. Fitzwater and J. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39–47 (1985).

1980 (1)

1974 (1)

D. Tunstall and M. Hird, “Effect of particle crowding on scattering power of TiO2 pigments,” J. Paint Technol. 46, 33–40 (1974).

1971 (1)

H. Hottel, A. Sarofim, W. Dalzell, and I. Vasalos, “Optical properties of coatings. Effect of pigment concentration,” AIAA J. 9, 1895–1898 (1971).
[Crossref]

1961 (1)

R. Bruehlman, L. Thomas, and E. Gonick, “Effect of particle size and pigment volume concentration on hiding power of titanium dioxide,” Off. Dig. 33, 252–267 (1961).

Aberle, A. G.

O. Berger, D. Inns, and A. G. Aberle, “Commercial white paint as back surface reflector for thin-film solar cells,” Sol. Energy Mater. Sol. Cells 91, 1215–1221 (2007).
[Crossref]

Ackermann, M.

T. Sperling, L. Schertel, M. Ackermann, G. J. Aubry, C. M. Aegerter, and G. Maret, “Can 3D light localization be reached in “white paint”?” New J. Phys. 18, 013039 (2016).
[Crossref]

Aegerter, C. M.

G. J. Aubry, L. Schertel, M. Chen, H. Weyer, C. M. Aegerter, S. Polarz, H. Cölfen, and G. Maret, “Resonant transport and near-field effects in photonic glasses,” Phys. Rev. A 96, 043871 (2017).
[Crossref]

T. Sperling, L. Schertel, M. Ackermann, G. J. Aubry, C. M. Aegerter, and G. Maret, “Can 3D light localization be reached in “white paint”?” New J. Phys. 18, 013039 (2016).
[Crossref]

Angelani, L.

Asano, S.

Atiganyanun, S.

S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective radiative cooling by paint-format microsphere-based photonic random media,” ACS Photon. 5, 1181–1187 (2018).
[Crossref]

Aubry, G. J.

G. J. Aubry, L. Schertel, M. Chen, H. Weyer, C. M. Aegerter, S. Polarz, H. Cölfen, and G. Maret, “Resonant transport and near-field effects in photonic glasses,” Phys. Rev. A 96, 043871 (2017).
[Crossref]

T. Sperling, L. Schertel, M. Ackermann, G. J. Aubry, C. M. Aegerter, and G. Maret, “Can 3D light localization be reached in “white paint”?” New J. Phys. 18, 013039 (2016).
[Crossref]

Auger, J.-C.

J.-C. Auger and B. Stout, “Dependent light scattering in white paint films: clarification and application of the theoretical concepts,” J. Coat. Technol. Res. 9, 287–295 (2012).

J.-C. Auger, V. A. Martinez, and B. Stout, “Theoretical study of the scattering efficiency of rutile titanium dioxide pigments as a function of their spatial dispersion,” J. Coat. Technol. Res. 6, 89–97 (2009).

J.-C. Auger, R. G. Barrera, and B. Stout, “Scattering efficiency of clusters composed by aggregated spheres,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 521–531 (2003).
[Crossref]

Barrera, R. G.

J.-C. Auger, R. G. Barrera, and B. Stout, “Scattering efficiency of clusters composed by aggregated spheres,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 521–531 (2003).
[Crossref]

Barthelemy, P.

J. Bertolotti, K. Vynck, L. Pattelli, P. Barthelemy, S. Lepri, and D. S. Wiersma, “Engineering disorder in superdiffusive levy glasses,” Adv. Funct. Mater. 20, 965–968 (2010).
[Crossref]

Berger, O.

O. Berger, D. Inns, and A. G. Aberle, “Commercial white paint as back surface reflector for thin-film solar cells,” Sol. Energy Mater. Sol. Cells 91, 1215–1221 (2007).
[Crossref]

Bertolotti, J.

J. Bertolotti, K. Vynck, L. Pattelli, P. Barthelemy, S. Lepri, and D. S. Wiersma, “Engineering disorder in superdiffusive levy glasses,” Adv. Funct. Mater. 20, 965–968 (2010).
[Crossref]

Bohren, C.

B. Palmer, P. Stamatakis, C. Bohren, and G. Salzman, “Multiple-scattering model for opacifying particles in polymer films,” J. Coat. Technol. 61, 41–47 (1989).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley Science Paperback Series (Wiley, 1983).

Boriskina, S. V.

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
[Crossref]

Braun, J.

J. Braun, “Crowding and spacing of titanium dioxide pigments,” J. Coat. Technol. 60, 67–71 (1988).

Bressel, L.

L. Bressel and O. Reich, “Theoretical and experimental study of the diffuse transmission of light through highly concentrated absorbing and scattering materials: part I: Monte-Carlo simulations,” J. Quant. Spectrosc. Radiat. Transfer 146, 190–198 (2014).
[Crossref]

Bruehlman, R.

R. Bruehlman, L. Thomas, and E. Gonick, “Effect of particle size and pigment volume concentration on hiding power of titanium dioxide,” Off. Dig. 33, 252–267 (1961).

Burresi, M.

L. Pattelli, R. Savo, M. Burresi, and D. S. Wiersma, “Spatio-temporal visualization of light transport in complex photonic structures,” Light Sci. Appl. 5, e16090 (2016).
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M. Burresi, F. Pratesi, F. Riboli, and D. S. Wiersma, “Complex photonic structures for light harvesting,” Adv. Opt. Mater. 3, 722–743 (2015).
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M. Burresi, L. Cortese, L. Pattelli, M. Kolle, P. Vukusic, D. S. Wiersma, U. Steiner, and S. Vignolini, “Bright-white beetle scales optimise multiple scattering of light,” Sci. Rep. 4, 6075 (2014).
[Crossref]

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112, 143901 (2014).
[Crossref]

Cairns, B.

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transfer 112, 671–692 (2011).
[Crossref]

Cao, H.

Carminati, R.

Caselli, N.

F. Riboli, F. Uccheddu, G. Monaco, N. Caselli, F. Intonti, M. Gurioli, and S. Skipetrov, “Tailoring correlations of the local density of states in disordered photonic materials,” Phys. Rev. Lett. 119, 043902 (2017).
[Crossref]

Chen, D.

B. D. Wilts, X. Sheng, M. Holler, A. Diaz, M. Guizar-Sicairos, J. Raabe, R. Hoppe, S.-H. Liu, R. Langford, O. D. Onelli, D. Chen, S. Torquato, U. Steiner, C. G. Schroer, S. Vignolini, and A. Sepe, “Evolutionary-optimized photonic network structure in white beetle wing scales,” Adv. Mater. 30, 1702057 (2017).
[Crossref]

Chen, M.

G. J. Aubry, L. Schertel, M. Chen, H. Weyer, C. M. Aegerter, S. Polarz, H. Cölfen, and G. Maret, “Resonant transport and near-field effects in photonic glasses,” Phys. Rev. A 96, 043871 (2017).
[Crossref]

Cölfen, H.

G. J. Aubry, L. Schertel, M. Chen, H. Weyer, C. M. Aegerter, S. Polarz, H. Cölfen, and G. Maret, “Resonant transport and near-field effects in photonic glasses,” Phys. Rev. A 96, 043871 (2017).
[Crossref]

Conley, G. M.

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112, 143901 (2014).
[Crossref]

Conti, C.

Contini, D.

Cortese, L.

M. Burresi, L. Cortese, L. Pattelli, M. Kolle, P. Vukusic, D. S. Wiersma, U. Steiner, and S. Vignolini, “Bright-white beetle scales optimise multiple scattering of light,” Sci. Rep. 4, 6075 (2014).
[Crossref]

Cytrynbaum, J.

S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective radiative cooling by paint-format microsphere-based photonic random media,” ACS Photon. 5, 1181–1187 (2018).
[Crossref]

Dal Negro, L.

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
[Crossref]

Dalzell, W.

H. Hottel, A. Sarofim, W. Dalzell, and I. Vasalos, “Optical properties of coatings. Effect of pigment concentration,” AIAA J. 9, 1895–1898 (1971).
[Crossref]

Datta, S.

C. Soukoulis, S. Datta, and E. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810 (1994).
[Crossref]

David, S. N.

Y. Zhai, Y. Ma, S. N. David, D. Zhao, R. Lou, G. Tan, R. Yang, and X. Yin, “Scalable-manufactured randomized glass-polymer hybrid metamaterial for daytime radiative cooling,” Science 355, 1062–1066 (2017).
[Crossref]

Debenedetti, P. G.

S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is random close packing of spheres well defined?” Phys. Rev. Lett. 84, 2064–2067 (2000).
[Crossref]

Diaz, A.

B. D. Wilts, X. Sheng, M. Holler, A. Diaz, M. Guizar-Sicairos, J. Raabe, R. Hoppe, S.-H. Liu, R. Langford, O. D. Onelli, D. Chen, S. Torquato, U. Steiner, C. G. Schroer, S. Vignolini, and A. Sepe, “Evolutionary-optimized photonic network structure in white beetle wing scales,” Adv. Mater. 30, 1702057 (2017).
[Crossref]

Diebold, M. P.

M. P. Diebold, “A Monte Carlo determination of the effectiveness of nanoparticles as spacers for optimizing TiO2 opacity,” J. Coat. Technol. Res. 8, 541–552 (2011).

Ding, K.

Ding, K. H.

Dinsmore, A.

X. Peng and A. Dinsmore, “Light propagation in strongly scattering, random colloidal films: the role of the packing geometry,” Phys. Rev. Lett. 99, 143902 (2007).
[Crossref]

Dlugach, J. M.

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transfer 112, 671–692 (2011).
[Crossref]

J. M. Dlugach, M. I. Mishchenko, L. Liu, and D. W. Mackowski, “Numerically exact computer simulations of light scattering by densely packed, random particulate media,” J. Quant. Spectrosc. Radiat. Transfer 112, 2068–2078 (2011).
[Crossref]

Dogariu, A.

R. R. Naraghi, S. Sukhov, and A. Dogariu, “Disorder fingerprint: intensity distributions in the near field of random media,” Phys. Rev. B 94, 174205 (2016).
[Crossref]

R. R. Naraghi, S. Sukhov, J. Sáenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

Donev, A.

M. Skoge, A. Donev, F. H. Stillinger, and S. Torquato, “Packing hyperspheres in high-dimensional Euclidean spaces,” Phys. Rev. E 74, 041127 (2006).
[Crossref]

A. Donev, F. H. Stillinger, and S. Torquato, “Unexpected density fluctuations in jammed disordered sphere packings,” Phys. Rev. Lett. 95, 090604 (2005).
[Crossref]

Dong, B.

L. Shi, Y. Zhang, B. Dong, T. Zhan, X. Liu, and J. Zi, “Amorphous photonic crystals with only short-range order,” Adv. Mater. 25, 5314–5320 (2013).
[Crossref]

Dosta, M.

G. Shang, L. Maiwald, H. Renner, D. Jalas, M. Dosta, S. Heinrich, A. Petrov, and M. Eich, “Photonic glass for high contrast structural color,” Sci. Rep. 8, 7804 (2018).
[Crossref]

Dreisow, F.

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
[Crossref]

Drolen, B.

B. Drolen and C. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. Heat Transfer 1, 63–68 (1987).
[Crossref]

Dufresne, E. R.

Durian, D. J.

J. M. Rieser, C. P. Goodrich, A. J. Liu, and D. J. Durian, “Divergence of Voronoi cell anisotropy vector: a threshold-free characterization of local structure in amorphous materials,” Phys. Rev. Lett. 116, 088001 (2016).
[Crossref]

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

Economou, E.

C. Soukoulis, S. Datta, and E. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810 (1994).
[Crossref]

Egel, A.

A. Egel, L. Pattelli, G. Mazzamuto, D. S. Wiersma, and U. Lemmer, “CELES: CUDA-accelerated simulation of electromagnetic scattering by large ensembles of spheres,” J. Quant. Spectrosc. Radiat. Transfer 199, 103–110 (2017).
[Crossref]

Eich, M.

G. Shang, L. Maiwald, H. Renner, D. Jalas, M. Dosta, S. Heinrich, A. Petrov, and M. Eich, “Photonic glass for high contrast structural color,” Sci. Rep. 8, 7804 (2018).
[Crossref]

Elton, N.

N. Elton and A. Legrix, “Spatial point statistics for quantifying TiO2distribution in paint,” J. Coat. Technol. Res. 11, 443–454 (2014).

Engel, M.

L. S. Froufe-Pérez, M. Engel, J. J. Sáenz, and F. Scheffold, “Band gap formation and Anderson localization in disordered photonic materials with structural correlations,” Proc. Natl. Acad. Sci. USA 114, 9570–9574 (2017).
[Crossref]

Escalante, J. M.

J. M. Escalante and S. E. Skipetrov, “Longitudinal optical fields in light scattering from dielectric spheres and Anderson localization of light,” Ann. Phys. 529, 1700039 (2017).
[Crossref]

Fitzwater, S.

S. Fitzwater and J. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39–47 (1985).

Forster, J.

Fraden, S.

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[Crossref]

Fratalocchi, A.

French, R.

L. McNeil and R. French, “Multiple scattering from rutile TiO2 particles,” Acta Mater. 48, 4571–4576 (2000).
[Crossref]

French, R. H.

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[Crossref]

Froufe-Pérez, L. S.

L. S. Froufe-Pérez, M. Engel, J. J. Sáenz, and F. Scheffold, “Band gap formation and Anderson localization in disordered photonic materials with structural correlations,” Proc. Natl. Acad. Sci. USA 114, 9570–9574 (2017).
[Crossref]

Fung, A.

Garcia, P.

P. Garcia and P. Lodahl, “Physics of quantum light emitters in disordered photonic nanostructures,” Ann. Phys. 529, 1600351 (2017).
[Crossref]

Gentilini, S.

Gibbs, D.

Gonick, E.

R. Bruehlman, L. Thomas, and E. Gonick, “Effect of particle size and pigment volume concentration on hiding power of titanium dioxide,” Off. Dig. 33, 252–267 (1961).

Goodrich, C. P.

J. M. Rieser, C. P. Goodrich, A. J. Liu, and D. J. Durian, “Divergence of Voronoi cell anisotropy vector: a threshold-free characterization of local structure in amorphous materials,” Phys. Rev. Lett. 116, 088001 (2016).
[Crossref]

Guizar-Sicairos, M.

B. D. Wilts, X. Sheng, M. Holler, A. Diaz, M. Guizar-Sicairos, J. Raabe, R. Hoppe, S.-H. Liu, R. Langford, O. D. Onelli, D. Chen, S. Torquato, U. Steiner, C. G. Schroer, S. Vignolini, and A. Sepe, “Evolutionary-optimized photonic network structure in white beetle wing scales,” Adv. Mater. 30, 1702057 (2017).
[Crossref]

Gurioli, M.

F. Riboli, F. Uccheddu, G. Monaco, N. Caselli, F. Intonti, M. Gurioli, and S. Skipetrov, “Tailoring correlations of the local density of states in disordered photonic materials,” Phys. Rev. Lett. 119, 043902 (2017).
[Crossref]

Han, S. E.

S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective radiative cooling by paint-format microsphere-based photonic random media,” ACS Photon. 5, 1181–1187 (2018).
[Crossref]

Han, S. J.

S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective radiative cooling by paint-format microsphere-based photonic random media,” ACS Photon. 5, 1181–1187 (2018).
[Crossref]

Han, S. M.

S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective radiative cooling by paint-format microsphere-based photonic random media,” ACS Photon. 5, 1181–1187 (2018).
[Crossref]

Heinrich, M.

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
[Crossref]

Heinrich, S.

G. Shang, L. Maiwald, H. Renner, D. Jalas, M. Dosta, S. Heinrich, A. Petrov, and M. Eich, “Photonic glass for high contrast structural color,” Sci. Rep. 8, 7804 (2018).
[Crossref]

Hird, M.

D. Tunstall and M. Hird, “Effect of particle crowding on scattering power of TiO2 pigments,” J. Paint Technol. 46, 33–40 (1974).

Holler, M.

B. D. Wilts, X. Sheng, M. Holler, A. Diaz, M. Guizar-Sicairos, J. Raabe, R. Hoppe, S.-H. Liu, R. Langford, O. D. Onelli, D. Chen, S. Torquato, U. Steiner, C. G. Schroer, S. Vignolini, and A. Sepe, “Evolutionary-optimized photonic network structure in white beetle wing scales,” Adv. Mater. 30, 1702057 (2017).
[Crossref]

Hook, J.

S. Fitzwater and J. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39–47 (1985).

Hoppe, R.

B. D. Wilts, X. Sheng, M. Holler, A. Diaz, M. Guizar-Sicairos, J. Raabe, R. Hoppe, S.-H. Liu, R. Langford, O. D. Onelli, D. Chen, S. Torquato, U. Steiner, C. G. Schroer, S. Vignolini, and A. Sepe, “Evolutionary-optimized photonic network structure in white beetle wing scales,” Adv. Mater. 30, 1702057 (2017).
[Crossref]

Hottel, H.

H. Hottel, A. Sarofim, W. Dalzell, and I. Vasalos, “Optical properties of coatings. Effect of pigment concentration,” AIAA J. 9, 1895–1898 (1971).
[Crossref]

Hsu, K.

S. Atiganyanun, J. B. Plumley, S. J. Han, K. Hsu, J. Cytrynbaum, T. L. Peng, S. M. Han, and S. E. Han, “Effective radiative cooling by paint-format microsphere-based photonic random media,” ACS Photon. 5, 1181–1187 (2018).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley Science Paperback Series (Wiley, 1983).

IJzerman, W. L.

Inns, D.

O. Berger, D. Inns, and A. G. Aberle, “Commercial white paint as back surface reflector for thin-film solar cells,” Sol. Energy Mater. Sol. Cells 91, 1215–1221 (2007).
[Crossref]

Intonti, F.

F. Riboli, F. Uccheddu, G. Monaco, N. Caselli, F. Intonti, M. Gurioli, and S. Skipetrov, “Tailoring correlations of the local density of states in disordered photonic materials,” Phys. Rev. Lett. 119, 043902 (2017).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Wiley, 1999), Vol. 12.

Jalas, D.

G. Shang, L. Maiwald, H. Renner, D. Jalas, M. Dosta, S. Heinrich, A. Petrov, and M. Eich, “Photonic glass for high contrast structural color,” Sci. Rep. 8, 7804 (2018).
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M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer, 2013), Vol. 16.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

V. P. Tishkovets and E. V. Petrova, “Light scattering by densely packed systems of particles: near-field effects,” in Light Scattering Reviews 7 (Springer, 2013), pp. 3–36.

Supplementary Material (1)

NameDescription
» Supplement 1       Brief discussion regarding the wavelength dependence and the estimation of the effective refractive index relative to the optimally scattering configurations.

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Figures (6)

Fig. 1.
Fig. 1. Rendering of illustrative configurations relative to values of f v and f p equal to 0.01, 0.04, 0.16, and 0.64. Panel (a) shows configurations with an equal particle/area ratio ( N / A = 1.125 · 10 2 μm 2 ) and variable thickness ( 0.75 μm L 48 μm ) comprising up to 3.6 · 10 6 spheres. Panel (b) shows configurations with equal thickness ( L = 3 μm ) and variable density ( 7    μm 2 N / A 450 μm 2 ) comprising up to 8.0 · 10 4 particles per sample. For simplicity, particles beyond a diameter/height ratio of 8 are discarded in the actual T -matrix calculations.
Fig. 2.
Fig. 2. Numerical results and cubic spline adaptation of integrated reflectivity calculations. Panels show data relative to (a) fixed particle area density and (b) fixed-thickness configurations. Edges of the phase space corresponding to f p = f v and f p = 0.64 , representing, respectively, the least and maximum amount of spatial correlations, are plotted in the inset. Pairs of configurations most impacted by spatial correlations are highlighted in panel (b).
Fig. 3.
Fig. 3. Maximum integrated reflectance for equal-thickness samples. Panel (a) shows a cross-cut of Fig. 2(b), showing that the maximum opacity is reached for an intermediate degree of spatial correlations. Panel (b) shows the corresponding estimated transport mean free path l * obtained by a Monte Carlo fit of the total reflectance. Error bars correspond to a thickness uncertainty between L L + d / 2 L + d . The absolute value of the total electric field in the x z plane for the configuration with f v = 0.25 and f p = 0.49 is depicted in panel (c).
Fig. 4.
Fig. 4. Illustration of structural descriptors and their typical probability density functions as derived from auxiliary Voronoi tessellations for f v = f p = { 0.01 , 0.04 , 0.16 , 0.64 } (solid lines) and for f v = 0.01 , f p = { 0.04 , 0.16 , 0.64 } (dotted lines). Panel (a) refers to the normalized surface-to-surface distance between nearest-neighbors particles δ nn 1 . Panel (b) shows the distribution for the normalized pore diameter 2 δ p estimated using the trial sphere method with 10 6 trials. Insets show simplified illustrations of the structural descriptors in a 2D geometry. All distributions have been evaluated on independent configurations containing 8 · 10 6 particles.
Fig. 5.
Fig. 5. Structural descriptors for the ensemble of particles over the space of configurations. Panel (a) shows the average surface-to-surface distance between nearest-neighbor spheres. Panel (b) shows the average size (diameter) of empty regions in the particle ensemble. All quantities are normalized by the particle diameter d = 200 nm .
Fig. 6.
Fig. 6. Phase diagram for the role of spatial correlations in terms of the pore size and the nearest-neighbor distance based on the numerical results of Fig. 2(b). Structural parameters are shown normalized either by the diameter d of the scattering spheres (lower and left axes) or by the wavelength in the host medium λ = λ 0 / 1.5 (upper and right axes). Circles indicate the location of the simulated samples. Dotted lines highlight the combination of structural parameters resulting in the highest turbidity.

Equations (2)

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δ nn = 1 e 3 · 2 3 f v 1 r y 2 G ( y ) d y d r ,
δ p = 0 δ p p ( δ p ) d δ p

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