Abstract

There has been a growing interest in disordered optical media in recent years due to their potential applications in solar collectors, random lasers, light confinement, and other advanced photonic functions. This paper studies the transport of light for different incidence angles in a strongly disordered optical medium composed of core-shell TiO2@Silica nanoparticles suspended in an ethanol solution. A decrease of optical conductance and an increase of absorption near the input border are reported when the incidence angle increases. The specular reflection, measured for the photons that enter the sample, is lower than the effective internal reflection undergone by the coherently backscattered photons in the exact opposite direction, indicating a nonreciprocal propagation of light. This study represents a novel approach in order to understand the complex physics involved at the phase transition to localization.

© 2018 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
  3. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [Crossref]
  4. S. John, “Localization of light,” Phys. Today 44, 32–40 (1991).
    [Crossref]
  5. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
    [Crossref]
  6. E. Jiménez-Villar, I. F. da Silva, V. Mestre, P. C. de Oliveira, W. M. Faustino, and G. F. de Sá, “Anderson localization of light in a colloidal suspension (TiO2@silica),” Nanoscale 8, 10938–10946 (2016).
    [Crossref]
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  10. J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Phys. Rep. 195, 127–293 (1990).
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  11. P. Tierno, F. Sagués, T. H. Johansen, and I. M. Sokolov, “Antipersistent random walk in a two state flashing magnetic potential,” Phys. Rev. Lett. 109, 070601 (2012).
    [Crossref]
  12. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673–676 (1979).
    [Crossref]
  13. F. Evers and A. D. Mirlin, “Anderson transitions,” Rev. Mod. Phys. 80, 1355–1417 (2008).
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  14. S. E. Skipetrov and J. H. Page, “Red light for Anderson localization,” New J. Phys. 18, 021001 (2016).
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  20. F. Scheffold and D. Wiersma, “Inelastic scattering puts in question recent claims of Anderson localization of light,” Nat. Photonics 7, 934 (2013).
    [Crossref]
  21. T. Van Der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, and A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).
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  24. E. Jimenez-Villar, V. Mestre, W. S. Martins, G. F. Basso, I. F. da Silva, and G. F. de Sá, “Core-shell TiO2@Silica nanoparticles for light confinement,” Mater. Today 4, 11570–11579 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. 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]
  29. E. Jimenez-Villar, V. Mestre, P. C. de Oliveira, W. M. Faustino, D. S. Silva, and G. F. de Sá, “TiO2@Silica nanoparticles in a random laser: strong relationship of silica shell thickness on scattering medium properties and random laser performance,” Appl. Phys. Lett. 104, 081909 (2014).
    [Crossref]
  30. E. Rodriguez, E. Jimenez, G. J. Jacob, A. A. R. Neves, C. L. Cesar, and L. C. Barbosa, “Fabrication and characterization of a PbTe quantum dots multilayer structure,” Physica E 26, 361–365 (2005).
    [Crossref]
  31. E. Rodriguez, G. Kellermann, A. F. Craievich, E. Jimenez, C. L. César, and L. C. Barbosa, “All-optical switching device for infrared based on PbTe quantum dots,” Superlattices Microstruct. 43, 626–634 (2008).
    [Crossref]
  32. E. Jiménez, K. Abderrafi, R. Abargues, J. L. Valdés, and J. P. Martínez-Pastor, “Laser-ablation-induced synthesis of SiO2-capped noble metal nanoparticles in a single step,” Langmuir 26, 7458–7463 (2010).
    [Crossref]
  33. E. Jiménez, K. Abderrafi, J. Martínez-Pastor, R. Abargues, J. Luís Valdés, and R. Ibáñez, “A novel method of nanocrystal fabrication based on laser ablation in liquid environment,” Superlattices Microstruct. 43, 487–493 (2008).
    [Crossref]
  34. J. R. González-Castillo, E. Rodriguez, E. Jimenez-Villar, D. Rodríguez, I. Salomon-García, G. F. de Sá, T. García-Fernández, D. B. Almeida, C. L. Cesar, R. Johnes, and J. C. Ibarra, “Synthesis of Ag@Silica nanoparticles by assisted laser ablation,” Nanosc. Res. Lett. 10, 399 (2015).
    [Crossref]
  35. J. R. González-Castillo, E. Rodríguez-González, E. Jiménez-Villar, C. L. Cesar, and J. A. Andrade-Arvizu, “Assisted laser ablation: silver/gold nanostructures coated with silica,” Appl. Nanosci. 7, 597–605 (2017).
    [Crossref]
  36. G. Fuertes, O. L. Sánchez-Muñoz, E. Pedrueza, K. Abderrafi, J. Salgado, and E. Jiménez, “Switchable bactericidal effects from novel silica-coated silver nanoparticles mediated by light irradiation,” Langmuir 27, 2826–2833 (2011).
    [Crossref]
  37. E. Rodríguez, E. Jimenez, L. A. Padilha, A. A. R. Neves, G. J. Jacob, C. L. César, and L. C. Barbosa, “SiO2/PbTe quantum-dot multilayer production and characterization,” Appl. Phys. Lett. 86, 113117 (2005).
    [Crossref]
  38. G. Kellermann, E. Rodriguez, E. Jimenez, C. L. Cesar, L. C. Barbosa, and A. F. Craievich, “Structure of PbTe(SiO2)/SiO2 multilayers deposited on Si(111),” J. Appl. Crystallogr. 43, 385–393 (2010).
    [Crossref]
  39. A. D. Mirlin, “Spatial structure of anomalously localized states in disordered conductors,” J. Math. Phys. 38, 1888–1917 (1997).
    [Crossref]
  40. A. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259–382 (2000).
    [Crossref]
  41. G. Campagnano and Y. V. Nazarov, “GQ corrections in the circuit theory of quantum transport,” Phys. Rev. B 74, 125307 (2006).
    [Crossref]
  42. A. L. R. Barbosa, D. Bazeia, and J. G. G. S. Ramos, “Universal Braess paradox in open quantum dots,” Phys. Rev. E 90, 042915 (2014).
    [Crossref]
  43. E. Jiménez-Villar, I. F. da Silva, V. Mestre, N. U. Wetter, C. Lopez, P. C. de Oliveira, W. M. Faustino, and G. F. de Sá, “Random lasing at localization transition in a colloidal suspension (TiO2@Silica),” ACS Omega 2, 2415–2421 (2017).
    [Crossref]
  44. N. U. Wetter, J. M. Giehl, F. Butzbach, D. Anacleto, and E. Jiménez-Villar, “Polydispersed powders (Nd3+:YVO4) for ultra efficient random lasers,” Part. Part. Syst. Charact. 35, 1700335 (2017).
    [Crossref]
  45. 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]
  46. B. L. Al’tshuler, I. K. Zharekeshev, S. A. Kotochigova, and V. I. Shklovskiĭ, “Repulsion between energy levels and the metal-insulator transition,” Zhurnal Eksp. i Teor. Fiz. 67, 343–355 (1988).
  47. B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005).
    [Crossref]
  48. B. Maes, M. Soljacic, J. D. Joannopoulos, P. Bienstman, R. Baets, S.-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006).
    [Crossref]
  49. P. Hamel, S. Haddadi, F. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson, and A. M. Yacomotti, “Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers,” Nat. Photonics 9, 311–315 (2015).
    [Crossref]
  50. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [Crossref]
  51. 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]
  52. 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]
  53. O. L. Sánchez-Muñoz, J. Salgado, J. Martínez-Pastor, and E. Jiménez-Villar, “Synthesis and physical stability of novel Au-Ag@SiO2 alloy nanoparticles,” Nanosci. Nanotechnol. 2, 1–7 (2012).
    [Crossref]
  54. E. Jimenez-Villar, V. Mestre, N. U. Wetter, and G. F. de Sá, “Core-shell (TiO2@Silica) nanoparticles for random lasers,” Proc. SPIE 10549, 105490D (2018).
    [Crossref]
  55. M. Büttiker and M. Moskalets, “From Anderson localization to mesoscopic physics,” Int. J. Mod. Phys. B 24, 1555–1576 (2010).
    [Crossref]
  56. S. E. Skipetrov and B. A. Van Tiggelen, “Dynamics of Anderson localization in open 3D media,” Phys. Rev. Lett. 96, 2–5 (2006).
    [Crossref]

2018 (2)

E. Jiménez-Villar, M. C. S. Xavier, J. G. G. S. Ramos, N. U. Wetter, V. Mestre, W. S. Martins, G. F. Basso, V. A. Ermakov, F. C. Marques, and G. F. de Sá, “Localization of light: beginning of a new optics,” Proc. SPIE 10549, 1054905 (2018).
[Crossref]

E. Jimenez-Villar, V. Mestre, N. U. Wetter, and G. F. de Sá, “Core-shell (TiO2@Silica) nanoparticles for random lasers,” Proc. SPIE 10549, 105490D (2018).
[Crossref]

2017 (5)

J. R. González-Castillo, E. Rodríguez-González, E. Jiménez-Villar, C. L. Cesar, and J. A. Andrade-Arvizu, “Assisted laser ablation: silver/gold nanostructures coated with silica,” Appl. Nanosci. 7, 597–605 (2017).
[Crossref]

E. Jiménez-Villar, I. F. da Silva, V. Mestre, N. U. Wetter, C. Lopez, P. C. de Oliveira, W. M. Faustino, and G. F. de Sá, “Random lasing at localization transition in a colloidal suspension (TiO2@Silica),” ACS Omega 2, 2415–2421 (2017).
[Crossref]

N. U. Wetter, J. M. Giehl, F. Butzbach, D. Anacleto, and E. Jiménez-Villar, “Polydispersed powders (Nd3+:YVO4) for ultra efficient random lasers,” Part. Part. Syst. Charact. 35, 1700335 (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]

E. Jimenez-Villar, V. Mestre, W. S. Martins, G. F. Basso, I. F. da Silva, and G. F. de Sá, “Core-shell TiO2@Silica nanoparticles for light confinement,” Mater. Today 4, 11570–11579 (2017).
[Crossref]

2016 (3)

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]

S. E. Skipetrov and J. H. Page, “Red light for Anderson localization,” New J. Phys. 18, 021001 (2016).
[Crossref]

E. Jiménez-Villar, I. F. da Silva, V. Mestre, P. C. de Oliveira, W. M. Faustino, and G. F. de Sá, “Anderson localization of light in a colloidal suspension (TiO2@silica),” Nanoscale 8, 10938–10946 (2016).
[Crossref]

2015 (2)

P. Hamel, S. Haddadi, F. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson, and A. M. Yacomotti, “Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers,” Nat. Photonics 9, 311–315 (2015).
[Crossref]

J. R. González-Castillo, E. Rodriguez, E. Jimenez-Villar, D. Rodríguez, I. Salomon-García, G. F. de Sá, T. García-Fernández, D. B. Almeida, C. L. Cesar, R. Johnes, and J. C. Ibarra, “Synthesis of Ag@Silica nanoparticles by assisted laser ablation,” Nanosc. Res. Lett. 10, 399 (2015).
[Crossref]

2014 (3)

A. L. R. Barbosa, D. Bazeia, and J. G. G. S. Ramos, “Universal Braess paradox in open quantum dots,” Phys. Rev. E 90, 042915 (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]

E. Jimenez-Villar, V. Mestre, P. C. de Oliveira, W. M. Faustino, D. S. Silva, and G. F. de Sá, “TiO2@Silica nanoparticles in a random laser: strong relationship of silica shell thickness on scattering medium properties and random laser performance,” Appl. Phys. Lett. 104, 081909 (2014).
[Crossref]

2013 (4)

E. Jimenez-Villar, V. Mestre, P. C. de Oliveira, and G. F. de Sá, “Novel core-shell (TiO2@Silica) nanoparticles for scattering medium in a random laser: higher efficiency, lower laser threshold and lower photodegradation,” Nanoscale 5, 12512–12517 (2013).
[Crossref]

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Photonics 7, 48–52 (2013).
[Crossref]

F. Scheffold and D. Wiersma, “Inelastic scattering puts in question recent claims of Anderson localization of light,” Nat. Photonics 7, 934 (2013).
[Crossref]

L. Bressel, R. Hass, and O. Reich, “Particle sizing in highly turbid dispersions by photon density wave spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 126, 122–129 (2013).
[Crossref]

2012 (4)

P. Tierno, F. Sagués, T. H. Johansen, and I. M. Sokolov, “Antipersistent random walk in a two state flashing magnetic potential,” Phys. Rev. Lett. 109, 070601 (2012).
[Crossref]

T. Van Der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, and A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).
[Crossref]

K. Abderrafi, E. Jiménez, T. Ben, S. I. Molina, R. Ibáñez, V. Chirvony, and J. P. Martínez-Pastor, “Production of nanometer-size GaAs nanocristals by nanosecond laser ablation in liquid,” J. Nanosci. Nanotechnol. 12, 6774–6778 (2012).
[Crossref]

O. L. Sánchez-Muñoz, J. Salgado, J. Martínez-Pastor, and E. Jiménez-Villar, “Synthesis and physical stability of novel Au-Ag@SiO2 alloy nanoparticles,” Nanosci. Nanotechnol. 2, 1–7 (2012).
[Crossref]

2011 (1)

G. Fuertes, O. L. Sánchez-Muñoz, E. Pedrueza, K. Abderrafi, J. Salgado, and E. Jiménez, “Switchable bactericidal effects from novel silica-coated silver nanoparticles mediated by light irradiation,” Langmuir 27, 2826–2833 (2011).
[Crossref]

2010 (3)

G. Kellermann, E. Rodriguez, E. Jimenez, C. L. Cesar, L. C. Barbosa, and A. F. Craievich, “Structure of PbTe(SiO2)/SiO2 multilayers deposited on Si(111),” J. Appl. Crystallogr. 43, 385–393 (2010).
[Crossref]

M. Büttiker and M. Moskalets, “From Anderson localization to mesoscopic physics,” Int. J. Mod. Phys. B 24, 1555–1576 (2010).
[Crossref]

E. Jiménez, K. Abderrafi, R. Abargues, J. L. Valdés, and J. P. Martínez-Pastor, “Laser-ablation-induced synthesis of SiO2-capped noble metal nanoparticles in a single step,” Langmuir 26, 7458–7463 (2010).
[Crossref]

2009 (2)

A. F. Demirörs, A. van Blaaderen, and A. Imhof, “Synthesis of eccentric titania-silica core-shell and composite particles,” Chem. Mater. 21, 979–984 (2009).
[Crossref]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref]

2008 (3)

F. Evers and A. D. Mirlin, “Anderson transitions,” Rev. Mod. Phys. 80, 1355–1417 (2008).
[Crossref]

E. Jiménez, K. Abderrafi, J. Martínez-Pastor, R. Abargues, J. Luís Valdés, and R. Ibáñez, “A novel method of nanocrystal fabrication based on laser ablation in liquid environment,” Superlattices Microstruct. 43, 487–493 (2008).
[Crossref]

E. Rodriguez, G. Kellermann, A. F. Craievich, E. Jimenez, C. L. César, and L. C. Barbosa, “All-optical switching device for infrared based on PbTe quantum dots,” Superlattices Microstruct. 43, 626–634 (2008).
[Crossref]

2006 (4)

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref]

S. E. Skipetrov and B. A. Van Tiggelen, “Dynamics of Anderson localization in open 3D media,” Phys. Rev. Lett. 96, 2–5 (2006).
[Crossref]

B. Maes, M. Soljacic, J. D. Joannopoulos, P. Bienstman, R. Baets, S.-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006).
[Crossref]

G. Campagnano and Y. V. Nazarov, “GQ corrections in the circuit theory of quantum transport,” Phys. Rev. B 74, 125307 (2006).
[Crossref]

2005 (3)

E. Rodríguez, E. Jimenez, L. A. Padilha, A. A. R. Neves, G. J. Jacob, C. L. César, and L. C. Barbosa, “SiO2/PbTe quantum-dot multilayer production and characterization,” Appl. Phys. Lett. 86, 113117 (2005).
[Crossref]

B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005).
[Crossref]

E. Rodriguez, E. Jimenez, G. J. Jacob, A. A. R. Neves, C. L. Cesar, and L. C. Barbosa, “Fabrication and characterization of a PbTe quantum dots multilayer structure,” Physica E 26, 361–365 (2005).
[Crossref]

2000 (2)

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref]

A. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259–382 (2000).
[Crossref]

1999 (1)

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, “Localization or classical diffusion of light?” Nature 398, 206–207 (1999).
[Crossref]

1997 (2)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

A. D. Mirlin, “Spatial structure of anomalously localized states in disordered conductors,” J. Math. Phys. 38, 1888–1917 (1997).
[Crossref]

1991 (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]

S. John, “Localization of light,” Phys. Today 44, 32–40 (1991).
[Crossref]

1990 (1)

J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Phys. Rep. 195, 127–293 (1990).
[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 (2)

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]

B. L. Al’tshuler, I. K. Zharekeshev, S. A. Kotochigova, and V. I. Shklovskiĭ, “Repulsion between energy levels and the metal-insulator transition,” Zhurnal Eksp. i Teor. Fiz. 67, 343–355 (1988).

1987 (1)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref]

1985 (1)

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[Crossref]

1984 (1)

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[Crossref]

1979 (1)

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673–676 (1979).
[Crossref]

1960 (1)

A. F. Ioffe and A. R. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

Abargues, R.

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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]

S. E. Skipetrov and B. A. Van Tiggelen, “Dynamics of Anderson localization in open 3D media,” Phys. Rev. Lett. 96, 2–5 (2006).
[Crossref]

Sokolov, I. M.

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]

P. Tierno, F. Sagués, T. H. Johansen, and I. M. Sokolov, “Antipersistent random walk in a two state flashing magnetic potential,” Phys. Rev. Lett. 109, 070601 (2012).
[Crossref]

Soljacic, M.

Sperling, T.

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]

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Photonics 7, 48–52 (2013).
[Crossref]

Störzer, M.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref]

Stout, S.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref]

Stoytchev, M.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref]

Suteewong, T.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref]

Tierno, P.

P. Tierno, F. Sagués, T. H. Johansen, and I. M. Sokolov, “Antipersistent random walk in a two state flashing magnetic potential,” Phys. Rev. Lett. 109, 070601 (2012).
[Crossref]

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F. Scheffold, R. Lenke, R. Tweer, and G. Maret, “Localization or classical diffusion of light?” Nature 398, 206–207 (1999).
[Crossref]

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E. Jiménez, K. Abderrafi, R. Abargues, J. L. Valdés, and J. P. Martínez-Pastor, “Laser-ablation-induced synthesis of SiO2-capped noble metal nanoparticles in a single step,” Langmuir 26, 7458–7463 (2010).
[Crossref]

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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).
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A. F. Demirörs, A. van Blaaderen, and A. Imhof, “Synthesis of eccentric titania-silica core-shell and composite particles,” Chem. Mater. 21, 979–984 (2009).
[Crossref]

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T. Van Der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, and A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).
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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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S. E. Skipetrov and B. A. Van Tiggelen, “Dynamics of Anderson localization in open 3D media,” Phys. Rev. Lett. 96, 2–5 (2006).
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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).
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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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Wetter, N. U.

E. Jimenez-Villar, V. Mestre, N. U. Wetter, and G. F. de Sá, “Core-shell (TiO2@Silica) nanoparticles for random lasers,” Proc. SPIE 10549, 105490D (2018).
[Crossref]

E. Jiménez-Villar, M. C. S. Xavier, J. G. G. S. Ramos, N. U. Wetter, V. Mestre, W. S. Martins, G. F. Basso, V. A. Ermakov, F. C. Marques, and G. F. de Sá, “Localization of light: beginning of a new optics,” Proc. SPIE 10549, 1054905 (2018).
[Crossref]

N. U. Wetter, J. M. Giehl, F. Butzbach, D. Anacleto, and E. Jiménez-Villar, “Polydispersed powders (Nd3+:YVO4) for ultra efficient random lasers,” Part. Part. Syst. Charact. 35, 1700335 (2017).
[Crossref]

E. Jiménez-Villar, I. F. da Silva, V. Mestre, N. U. Wetter, C. Lopez, P. C. de Oliveira, W. M. Faustino, and G. F. de Sá, “Random lasing at localization transition in a colloidal suspension (TiO2@Silica),” ACS Omega 2, 2415–2421 (2017).
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Wiersma, D.

F. Scheffold and D. Wiersma, “Inelastic scattering puts in question recent claims of Anderson localization of light,” Nat. Photonics 7, 934 (2013).
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T. Van Der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, and A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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E. Jiménez-Villar, M. C. S. Xavier, J. G. G. S. Ramos, N. U. Wetter, V. Mestre, W. S. Martins, G. F. Basso, V. A. Ermakov, F. C. Marques, and G. F. de Sá, “Localization of light: beginning of a new optics,” Proc. SPIE 10549, 1054905 (2018).
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P. Hamel, S. Haddadi, F. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson, and A. M. Yacomotti, “Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers,” Nat. Photonics 9, 311–315 (2015).
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B. L. Al’tshuler, I. K. Zharekeshev, S. A. Kotochigova, and V. I. Shklovskiĭ, “Repulsion between energy levels and the metal-insulator transition,” Zhurnal Eksp. i Teor. Fiz. 67, 343–355 (1988).

Zhu, G.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[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).
[Crossref]

ACS Omega (1)

E. Jiménez-Villar, I. F. da Silva, V. Mestre, N. U. Wetter, C. Lopez, P. C. de Oliveira, W. M. Faustino, and G. F. de Sá, “Random lasing at localization transition in a colloidal suspension (TiO2@Silica),” ACS Omega 2, 2415–2421 (2017).
[Crossref]

Ann. Phys. (1)

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]

Appl. Nanosci. (1)

J. R. González-Castillo, E. Rodríguez-González, E. Jiménez-Villar, C. L. Cesar, and J. A. Andrade-Arvizu, “Assisted laser ablation: silver/gold nanostructures coated with silica,” Appl. Nanosci. 7, 597–605 (2017).
[Crossref]

Appl. Phys. Lett. (2)

E. Jimenez-Villar, V. Mestre, P. C. de Oliveira, W. M. Faustino, D. S. Silva, and G. F. de Sá, “TiO2@Silica nanoparticles in a random laser: strong relationship of silica shell thickness on scattering medium properties and random laser performance,” Appl. Phys. Lett. 104, 081909 (2014).
[Crossref]

E. Rodríguez, E. Jimenez, L. A. Padilha, A. A. R. Neves, G. J. Jacob, C. L. César, and L. C. Barbosa, “SiO2/PbTe quantum-dot multilayer production and characterization,” Appl. Phys. Lett. 86, 113117 (2005).
[Crossref]

Chem. Mater. (1)

A. F. Demirörs, A. van Blaaderen, and A. Imhof, “Synthesis of eccentric titania-silica core-shell and composite particles,” Chem. Mater. 21, 979–984 (2009).
[Crossref]

Int. J. Mod. Phys. B (1)

M. Büttiker and M. Moskalets, “From Anderson localization to mesoscopic physics,” Int. J. Mod. Phys. B 24, 1555–1576 (2010).
[Crossref]

J. Appl. Crystallogr. (1)

G. Kellermann, E. Rodriguez, E. Jimenez, C. L. Cesar, L. C. Barbosa, and A. F. Craievich, “Structure of PbTe(SiO2)/SiO2 multilayers deposited on Si(111),” J. Appl. Crystallogr. 43, 385–393 (2010).
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J. Math. Phys. (1)

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[Crossref]

J. Nanosci. Nanotechnol. (1)

K. Abderrafi, E. Jiménez, T. Ben, S. I. Molina, R. Ibáñez, V. Chirvony, and J. P. Martínez-Pastor, “Production of nanometer-size GaAs nanocristals by nanosecond laser ablation in liquid,” J. Nanosci. Nanotechnol. 12, 6774–6778 (2012).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

L. Bressel, R. Hass, and O. Reich, “Particle sizing in highly turbid dispersions by photon density wave spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 126, 122–129 (2013).
[Crossref]

Langmuir (2)

E. Jiménez, K. Abderrafi, R. Abargues, J. L. Valdés, and J. P. Martínez-Pastor, “Laser-ablation-induced synthesis of SiO2-capped noble metal nanoparticles in a single step,” Langmuir 26, 7458–7463 (2010).
[Crossref]

G. Fuertes, O. L. Sánchez-Muñoz, E. Pedrueza, K. Abderrafi, J. Salgado, and E. Jiménez, “Switchable bactericidal effects from novel silica-coated silver nanoparticles mediated by light irradiation,” Langmuir 27, 2826–2833 (2011).
[Crossref]

Mater. Today (1)

E. Jimenez-Villar, V. Mestre, W. S. Martins, G. F. Basso, I. F. da Silva, and G. F. de Sá, “Core-shell TiO2@Silica nanoparticles for light confinement,” Mater. Today 4, 11570–11579 (2017).
[Crossref]

Nanosc. Res. Lett. (1)

J. R. González-Castillo, E. Rodriguez, E. Jimenez-Villar, D. Rodríguez, I. Salomon-García, G. F. de Sá, T. García-Fernández, D. B. Almeida, C. L. Cesar, R. Johnes, and J. C. Ibarra, “Synthesis of Ag@Silica nanoparticles by assisted laser ablation,” Nanosc. Res. Lett. 10, 399 (2015).
[Crossref]

Nanoscale (2)

E. Jimenez-Villar, V. Mestre, P. C. de Oliveira, and G. F. de Sá, “Novel core-shell (TiO2@Silica) nanoparticles for scattering medium in a random laser: higher efficiency, lower laser threshold and lower photodegradation,” Nanoscale 5, 12512–12517 (2013).
[Crossref]

E. Jiménez-Villar, I. F. da Silva, V. Mestre, P. C. de Oliveira, W. M. Faustino, and G. F. de Sá, “Anderson localization of light in a colloidal suspension (TiO2@silica),” Nanoscale 8, 10938–10946 (2016).
[Crossref]

Nanosci. Nanotechnol. (1)

O. L. Sánchez-Muñoz, J. Salgado, J. Martínez-Pastor, and E. Jiménez-Villar, “Synthesis and physical stability of novel Au-Ag@SiO2 alloy nanoparticles,” Nanosci. Nanotechnol. 2, 1–7 (2012).
[Crossref]

Nat. Photonics (3)

P. Hamel, S. Haddadi, F. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson, and A. M. Yacomotti, “Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers,” Nat. Photonics 9, 311–315 (2015).
[Crossref]

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Photonics 7, 48–52 (2013).
[Crossref]

F. Scheffold and D. Wiersma, “Inelastic scattering puts in question recent claims of Anderson localization of light,” Nat. Photonics 7, 934 (2013).
[Crossref]

Nature (4)

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, “Localization or classical diffusion of light?” Nature 398, 206–207 (1999).
[Crossref]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[Crossref]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[Crossref]

New J. Phys. (2)

S. E. Skipetrov and J. H. Page, “Red light for Anderson localization,” New J. Phys. 18, 021001 (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]

Opt. Express (1)

Part. Part. Syst. Charact. (1)

N. U. Wetter, J. M. Giehl, F. Butzbach, D. Anacleto, and E. Jiménez-Villar, “Polydispersed powders (Nd3+:YVO4) for ultra efficient random lasers,” Part. Part. Syst. Charact. 35, 1700335 (2017).
[Crossref]

Philos. Mag. B (1)

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[Crossref]

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]

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[Crossref]

Phys. Rev. A (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]

Phys. Rev. B (3)

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).
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T. Van Der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, and A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).
[Crossref]

Phys. Rev. E (1)

A. L. R. Barbosa, D. Bazeia, and J. G. G. S. Ramos, “Universal Braess paradox in open quantum dots,” Phys. Rev. E 90, 042915 (2014).
[Crossref]

Phys. Rev. Lett. (8)

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. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref]

P. Tierno, F. Sagués, T. H. Johansen, and I. M. Sokolov, “Antipersistent random walk in a two state flashing magnetic potential,” Phys. Rev. Lett. 109, 070601 (2012).
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E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
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S. E. Skipetrov and B. A. Van Tiggelen, “Dynamics of Anderson localization in open 3D media,” Phys. Rev. Lett. 96, 2–5 (2006).
[Crossref]

Phys. Today (1)

S. John, “Localization of light,” Phys. Today 44, 32–40 (1991).
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Physica E (1)

E. Rodriguez, E. Jimenez, G. J. Jacob, A. A. R. Neves, C. L. Cesar, and L. C. Barbosa, “Fabrication and characterization of a PbTe quantum dots multilayer structure,” Physica E 26, 361–365 (2005).
[Crossref]

Proc. SPIE (2)

E. Jiménez-Villar, M. C. S. Xavier, J. G. G. S. Ramos, N. U. Wetter, V. Mestre, W. S. Martins, G. F. Basso, V. A. Ermakov, F. C. Marques, and G. F. de Sá, “Localization of light: beginning of a new optics,” Proc. SPIE 10549, 1054905 (2018).
[Crossref]

E. Jimenez-Villar, V. Mestre, N. U. Wetter, and G. F. de Sá, “Core-shell (TiO2@Silica) nanoparticles for random lasers,” Proc. SPIE 10549, 105490D (2018).
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Superlattices Microstruct. (2)

E. Rodriguez, G. Kellermann, A. F. Craievich, E. Jimenez, C. L. César, and L. C. Barbosa, “All-optical switching device for infrared based on PbTe quantum dots,” Superlattices Microstruct. 43, 626–634 (2008).
[Crossref]

E. Jiménez, K. Abderrafi, J. Martínez-Pastor, R. Abargues, J. Luís Valdés, and R. Ibáñez, “A novel method of nanocrystal fabrication based on laser ablation in liquid environment,” Superlattices Microstruct. 43, 487–493 (2008).
[Crossref]

Zhurnal Eksp. i Teor. Fiz. (1)

B. L. Al’tshuler, I. K. Zharekeshev, S. A. Kotochigova, and V. I. Shklovskiĭ, “Repulsion between energy levels and the metal-insulator transition,” Zhurnal Eksp. i Teor. Fiz. 67, 343–355 (1988).

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Figures (7)

Fig. 1.
Fig. 1. For sample [140×1010  NPs·mL1], transmitted total intensity versus incidence angle. (a) Transmission coefficient for incidence angles θ of 0°, 30°, 60°, and 70° as a function of slab thickness (d). The black, red, blue, and green dotted lines represent the fitting β(d0+d)2 with experimental points for 0°, 30°, 60°, and 70°, respectively. (b) Relative conductance G(d;θ) as a function of d; (c) asymptotic values of relative conductance G(;θ) as a function of the incidence angle.
Fig. 2.
Fig. 2. Measurement of intensity profiles at the sample output face. (a) For 140×1010  NPs·mL1 (localization) and 14×1010  NPs·mL1 (diffusive regime), G(;θ)=I(0°)/I(θ) versus incidence angle. (b) For 140×1010  NPs·mL1, (left, red) ωeff and (right, black) the relative effective width (normalized width) versus incidence angle; (c) for 14×1010  NPs·mL1 (diffusive), (left, red) ωeff and (right, black) the relative effective width (normalized width) are also plotted as a function of the incidence angle. The error bars are the statistic standard deviation of relative intensity and effective width (ωeff). For 140×1010  NPs·mL1 (localization), normalized intensity profiles for incidence angles of (d) 0°, (e) 30°, (f) 60°, and (g) 70°. Red arrows point quicker decay for large r. The intensity profiles are fitted to exp(2(|r|/σ))1+ν (red solid lines), where 0<ν<1.
Fig. 3.
Fig. 3. For 140×1010  NPs·mL1 (localization regime), coherent backscattering cones for incidence angles of (a) 0°, (b) 30°, (c) 60°, and (d) 70°. The red solid lines represent the background intensity (incoherently backscattered photons), taking into account the internal reflection at the interface of silica–air (light coming out of the cuvette). The coherent backscattering cones obtained by subtraction of the background intensity are shown below each graph. (e) (Left, red) ICBC and (right, black) IR (%) as a function of the incidence angle; (f) (left, red) half-width of backscattering cone and (right, black) lT0 as a function of IR (%); (g) (left, red) asymptotic values of relative conductance (G(;θ)), extracted from the transmission and propagation experiments, and (right, black) enhancement factor of backscattering cone as a function of IR (%). The black dotted lines in (f) and (g) represent linear fittings with the experimental points. Error bars correspond to the standard deviation of the intensity of the backscattering cone (ICBC) and the calculated IR (%).
Fig. 4.
Fig. 4. Schematic diagram of the experimental setup for determination of transmission coefficient. L1 and L2, lens; PH, pinhole; F + F, cell consisting of two optical flat (fused silica) mounted on a translation stage; IS, integrating sphere is placed in contact with the back cell; OF, optical fiber to collect the light in the spectrometer. An He–Ne laser beam with perpendicular polarization with regard to the incidence plane is introduced at different incidence angles, θ, with regard to the normal incidence (0°, 30°, 60°, 70°), which correspond to incidence angles into the sample of 0° (0 mrad), 19.07° (333 mrad), 34.47° (600 mrad), and 37.89° (661 mrad), respectively.
Fig. 5.
Fig. 5. Schematic diagram of the experimental setup for determination of the intensity profile after propagating through samples. L1 and L2, lens; PH, pinhole; CV, fused silica cuvette of 2.3  mm optical pathlength; CCD, camera; NDF, neutral density filter. At different angles of incidence, θ (0°, 30°, 60°, 70°), an He–Ne laser beam is introduced with perpendicular polarization with regard to the incidence plane.
Fig. 6.
Fig. 6. (a) Schematic diagram of the experimental setup for ITC(d) determination as a function of slab thickness (d) for a very small detection solid angle. L1 and L2, lens; F + F, cell consisting of two optical flats mounted on a translation stage; PH1 and PH2, pinholes; OF, optical fiber to collect the light in the spectrometer. An He–Ne laser beam with perpendicular polarization with regard to the incidence plane is introduced at different incidence angles, θ, with regard to the normal incidence (0°, 30°, 60°, 70°). (b)–(e) Transmission curves for incidence angles of (b) 0°, (c) 30°, (d) 60°, and (e) 70°. The black, red, blue, and green lines represent the fitting with an exponential function exp(d/lMA) for the respective incidence angle. lMA values are displayed in each figure, and are shown to be insensitive to the incidence angle. (f) Experimental setup for FAP measurement as a function of the incidence angle. The He–Ne laser is polarized perpendicular to the incident plane by a polarizer (P) and reflected by a BS onto the sample (CV), which is mounted on a rotation stage (RS). The samples (CV), with and without dye, were rotated horizontally 30°, 60°, and 70°; BD, beam dump; OF, optical fiber to collect the backscattered light in the spectrometer. (g) Left, black and right, red represent leO and FAP values, respectively, measured for 14×1010  NPs·mL1 (dots) and 140×1010  NPs·mL1 (squares), as a function of the incidence angle.
Fig. 7.
Fig. 7. Experimental setup for determination of the coherent backscattering cone. L1, L2, and L3, lens; PH, pinhole; BS, beam splitter; CV, cuvette of 2 mm optical pathlength; CCD, camera; BD, beam dump. The sample (CV) was rotated horizontally 30°, 60°, and 70° with respect to the normal incidence, which correspond to incidence angles into the sample of 0° (0 mrad), 19.07° (333 mrad), 34.47° (600 mrad), and 37.89° (661 mrad), respectively. The backscattered intensity was measured as a function of the horizontal collection angle.

Tables (1)

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Table 1. For Each Incidence Angle θ (0°, 30°, 60°, and 70°), Asymptotic Value of Relative Conductance (G(;θ)), the Enhanced Absorption Factor (γ0(θ)) and the Effective Refractive Index (neff0(θ)) for Depth Near Zero, and the Transport Mean Free Path Corrected by Internal Reflection Considering the Effective Refractive Index neff0(θ)

Equations (9)

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P[I(x,y)2xy]/[I(x,y)xy]2=1/π[I(r)2r]/[I(r)r]2,
T(d)=β*(d0+d)2[20ϑ1f(ϑ)dϑ],
β=[20ϑ1f(ϑ)dϑ]β*.
d090°=d030°×T30°(0),
T30°(0)=0˜30°(T(ϑ)+T(ϑ)2)×cosϑdϑ0.450.540°90°.
lT(d;0°)lT(d;θ)=β(0°)2(d+d0(0°))β(θ)2(d+d0(θ))β(0°)β(θ);I(0°)I(θ)T(d;0°)T(d;θ)β(0°)β(θ).
(RC(ϑ)+RC(ϑ)2)×cosϑcosθ×I0,
neff0(θ)=1+(neff1)γ0(θ),
γ0(θ)=γ0×G(;θ),

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