Abstract

We report on photoinduced modifications of the refractive index of nanoparticulate TiO2-pHEMA organic-inorganic hybrids. The samples with titania concentration ranging from 0.88·1020 to 17.6·1020 cm−3 were irradiated and analyzed with UV light at 375 nm. A reduction of the refractive index is observed in all samples. Although the photoinduced refraction was stronger in samples with higher titania concentration, its normalized value per Ti3+ center a0 = −2.4·10−23 cm3 remained constant. The change of refractive index correlates with the material photochromic response due to the accumulation of polaronic Ti3+ centers in the material.

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References

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    [Crossref]
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    [Crossref]
  24. B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26(1), 113–122 (1990).
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  25. N. A. Deskins and M. Dupuis, “Electron transport via polaron hopping in bulk TiO2: a density functional theory characterization,” Phys. Rev. B 75(19), 195212 (2007).
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  26. R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
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2012 (1)

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

2011 (3)

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

C. Sanchez, P. Belleville, M. Popall, and L. Nicole, “Applications of advanced hybrid organic-inorganic nanomaterials: from laboratory to market,” Chem. Soc. Rev. 40(2), 696–753 (2011).
[Crossref] [PubMed]

R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
[Crossref]

2010 (2)

L. Nicole, L. Rozes, and C. Sanchez, “Integrative approaches to hybrid multifunctional materials: from multidisciplinary research to applied technologies,” Adv. Mater. 22(29), 3208–3214 (2010).
[Crossref] [PubMed]

R. A. S. Ferreira, P. S. André, and L. D. Carlos, “Organic–inorganic hybrid materials towards passive and active architectures for the next generation of optical networks,” Opt. Mater. 32(11), 1397–1409 (2010).
[Crossref]

2009 (3)

K. Tanaka and K. Shimakawa, “Chalcogenide glasses in Japan: a review on photoinduced phenomena,” Phys. Status Solidi B 246(8), 1744–1757 (2009).
[Crossref]

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

K. Fedus and G. Boudebs, “Determination of photo-induced changes in linear optical coefficients by the Z-scan technique,” J. Opt. Soc. Am. B 26(11), 2171–2175 (2009).
[Crossref]

2007 (2)

N. A. Deskins and M. Dupuis, “Electron transport via polaron hopping in bulk TiO2: a density functional theory characterization,” Phys. Rev. B 75(19), 195212 (2007).
[Crossref]

D. J. Kang and B.-S. Bae, “Photo-imageable sol-gel hybrid materials for simple fabrication of micro-optical elements,” Acc. Chem. Res. 40(9), 903–912 (2007).
[Crossref] [PubMed]

2006 (3)

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

N. Mehta, “Applications of chalcogenide glasses in electronics and optoelectronics: a review,” J. Sci. Ind. Res. (India) 65(10), 777–786 (2006).

T. Keiji, “Photo-induced phenomena in chalcogenide glass: Comparison with those in oxide glass and polymer,” J. Non-Cryst. Solids 352(23–25), 2580–2584 (2006).

2005 (3)

V. Gayvoronsky, S. Yakunin, V. Nazarenko, V. Starkov, and M. Brodyn, “Techniques to characterize the nonlinear optical response of doped nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 426(1), 231–241 (2005).
[Crossref]

N. Bityurin, A. I. Kuznetsov, and A. Kanaev, “Kinetics of UV-induced darkening of titanium-oxide gels,” Appl. Surf. Sci. 248(1–4), 86–90 (2005).
[Crossref]

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

2004 (1)

R. Reisfeld, A. Weiss, T. Saraidarov, E. Yariv, and A. A. Ishchenko, “Solid-state lasers based on inorganic–organic hybrid materials obtained by combined sol–gel polymer technology,” Polym. Adv. Technol. 15(6), 291–301 (2004).
[Crossref]

2003 (3)

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330(1–3), 1–12 (2003).
[Crossref]

J. U. Park, W. S. Kim, and B. S. Bae, “Photoinduced low refractive index in a photosensitive organic–inorganic hybrid material,” J. Mater. Chem. 13(4), 738–741 (2003).
[Crossref]

2002 (2)

1990 (3)

S. Ducharme, J. Hautala, and P. C. Taylor, “Photodarkening profiles and kinetics in chalcogenide glasses,” Phys. Rev. B Condens. Matter 41(17), 12250–12259 (1990).
[Crossref] [PubMed]

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26(1), 113–122 (1990).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Alexandrov, A.

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

André, P. S.

R. A. S. Ferreira, P. S. André, and L. D. Carlos, “Organic–inorganic hybrid materials towards passive and active architectures for the next generation of optical networks,” Opt. Mater. 32(11), 1397–1409 (2010).
[Crossref]

Bae, B. S.

J. U. Park, W. S. Kim, and B. S. Bae, “Photoinduced low refractive index in a photosensitive organic–inorganic hybrid material,” J. Mater. Chem. 13(4), 738–741 (2003).
[Crossref]

Bae, B.-S.

D. J. Kang and B.-S. Bae, “Photo-imageable sol-gel hybrid materials for simple fabrication of micro-optical elements,” Acc. Chem. Res. 40(9), 903–912 (2007).
[Crossref] [PubMed]

Belleville, P.

C. Sanchez, P. Belleville, M. Popall, and L. Nicole, “Applications of advanced hybrid organic-inorganic nanomaterials: from laboratory to market,” Chem. Soc. Rev. 40(2), 696–753 (2011).
[Crossref] [PubMed]

Bennett, B. R.

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26(1), 113–122 (1990).
[Crossref]

Bityurin, N.

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

N. Bityurin, A. I. Kuznetsov, and A. Kanaev, “Kinetics of UV-induced darkening of titanium-oxide gels,” Appl. Surf. Sci. 248(1–4), 86–90 (2005).
[Crossref]

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Bonn, M.

R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
[Crossref]

Boudebs, G.

Bräuer, A.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Brinza, O.

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

Brodyn, M.

V. Gayvoronsky, S. Yakunin, V. Nazarenko, V. Starkov, and M. Brodyn, “Techniques to characterize the nonlinear optical response of doped nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 426(1), 231–241 (2005).
[Crossref]

Buestrich, R.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Carlos, L. D.

R. A. S. Ferreira, P. S. André, and L. D. Carlos, “Organic–inorganic hybrid materials towards passive and active architectures for the next generation of optical networks,” Opt. Mater. 32(11), 1397–1409 (2010).
[Crossref]

Chhor, K.

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Chichkov, B.

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

Cronauer, C.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Dannberg, P.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Del Alamo, J. A.

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26(1), 113–122 (1990).
[Crossref]

Deskins, N. A.

N. A. Deskins and M. Dupuis, “Electron transport via polaron hopping in bulk TiO2: a density functional theory characterization,” Phys. Rev. B 75(19), 195212 (2007).
[Crossref]

Domann, G.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Ducharme, S.

S. Ducharme, J. Hautala, and P. C. Taylor, “Photodarkening profiles and kinetics in chalcogenide glasses,” Phys. Rev. B Condens. Matter 41(17), 12250–12259 (1990).
[Crossref] [PubMed]

Dupuis, M.

N. A. Deskins and M. Dupuis, “Electron transport via polaron hopping in bulk TiO2: a density functional theory characterization,” Phys. Rev. B 75(19), 195212 (2007).
[Crossref]

Elliott, S. R.

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330(1–3), 1–12 (2003).
[Crossref]

Fadeeva, E.

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

Fedus, K.

Ferreira, R. A. S.

R. A. S. Ferreira, P. S. André, and L. D. Carlos, “Organic–inorganic hybrid materials towards passive and active architectures for the next generation of optical networks,” Opt. Mater. 32(11), 1397–1409 (2010).
[Crossref]

Fröhlich, L.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Gayvoronsky, V.

V. Gayvoronsky, S. Yakunin, V. Nazarenko, V. Starkov, and M. Brodyn, “Techniques to characterize the nonlinear optical response of doped nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 426(1), 231–241 (2005).
[Crossref]

Gorbovyi, P.

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Hautala, J.

S. Ducharme, J. Hautala, and P. C. Taylor, “Photodarkening profiles and kinetics in chalcogenide glasses,” Phys. Rev. B Condens. Matter 41(17), 12250–12259 (1990).
[Crossref] [PubMed]

Heinz, T. F.

R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
[Crossref]

Hendry, E.

R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
[Crossref]

Houbertz, R.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Ishchenko, A. A.

R. Reisfeld, A. Weiss, T. Saraidarov, E. Yariv, and A. A. Ishchenko, “Solid-state lasers based on inorganic–organic hybrid materials obtained by combined sol–gel polymer technology,” Polym. Adv. Technol. 15(6), 291–301 (2004).
[Crossref]

Kameneva, O.

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Kanaev, A.

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

N. Bityurin, A. I. Kuznetsov, and A. Kanaev, “Kinetics of UV-induced darkening of titanium-oxide gels,” Appl. Surf. Sci. 248(1–4), 86–90 (2005).
[Crossref]

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Kang, D. J.

D. J. Kang and B.-S. Bae, “Photo-imageable sol-gel hybrid materials for simple fabrication of micro-optical elements,” Acc. Chem. Res. 40(9), 903–912 (2007).
[Crossref] [PubMed]

Keiji, T.

T. Keiji, “Photo-induced phenomena in chalcogenide glass: Comparison with those in oxide glass and polymer,” J. Non-Cryst. Solids 352(23–25), 2580–2584 (2006).

Kim, W. S.

J. U. Park, W. S. Kim, and B. S. Bae, “Photoinduced low refractive index in a photosensitive organic–inorganic hybrid material,” J. Mater. Chem. 13(4), 738–741 (2003).
[Crossref]

Koch, J.

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

Kuznestov, A. I.

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Kuznetsov, A.

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

Kuznetsov, A. I.

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

N. Bityurin, A. I. Kuznetsov, and A. Kanaev, “Kinetics of UV-induced darkening of titanium-oxide gels,” Appl. Surf. Sci. 248(1–4), 86–90 (2005).
[Crossref]

Leger, J. R.

Ljungstrom, A.

Martin, H.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Mehta, N.

N. Mehta, “Applications of chalcogenide glasses in electronics and optoelectronics: a review,” J. Sci. Ind. Res. (India) 65(10), 777–786 (2006).

Monro, T.

Museur, L.

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

Nazarenko, V.

V. Gayvoronsky, S. Yakunin, V. Nazarenko, V. Starkov, and M. Brodyn, “Techniques to characterize the nonlinear optical response of doped nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 426(1), 231–241 (2005).
[Crossref]

Nicole, L.

C. Sanchez, P. Belleville, M. Popall, and L. Nicole, “Applications of advanced hybrid organic-inorganic nanomaterials: from laboratory to market,” Chem. Soc. Rev. 40(2), 696–753 (2011).
[Crossref] [PubMed]

L. Nicole, L. Rozes, and C. Sanchez, “Integrative approaches to hybrid multifunctional materials: from multidisciplinary research to applied technologies,” Adv. Mater. 22(29), 3208–3214 (2010).
[Crossref] [PubMed]

Park, J. U.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

J. U. Park, W. S. Kim, and B. S. Bae, “Photoinduced low refractive index in a photosensitive organic–inorganic hybrid material,” J. Mater. Chem. 13(4), 738–741 (2003).
[Crossref]

Popall, M.

C. Sanchez, P. Belleville, M. Popall, and L. Nicole, “Applications of advanced hybrid organic-inorganic nanomaterials: from laboratory to market,” Chem. Soc. Rev. 40(2), 696–753 (2011).
[Crossref] [PubMed]

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Reisfeld, R.

R. Reisfeld, A. Weiss, T. Saraidarov, E. Yariv, and A. A. Ishchenko, “Solid-state lasers based on inorganic–organic hybrid materials obtained by combined sol–gel polymer technology,” Polym. Adv. Technol. 15(6), 291–301 (2004).
[Crossref]

Rozes, L.

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

L. Nicole, L. Rozes, and C. Sanchez, “Integrative approaches to hybrid multifunctional materials: from multidisciplinary research to applied technologies,” Adv. Mater. 22(29), 3208–3214 (2010).
[Crossref] [PubMed]

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Sanchez, C.

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

C. Sanchez, P. Belleville, M. Popall, and L. Nicole, “Applications of advanced hybrid organic-inorganic nanomaterials: from laboratory to market,” Chem. Soc. Rev. 40(2), 696–753 (2011).
[Crossref] [PubMed]

L. Nicole, L. Rozes, and C. Sanchez, “Integrative approaches to hybrid multifunctional materials: from multidisciplinary research to applied technologies,” Adv. Mater. 22(29), 3208–3214 (2010).
[Crossref] [PubMed]

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Saraidarov, T.

R. Reisfeld, A. Weiss, T. Saraidarov, E. Yariv, and A. A. Ishchenko, “Solid-state lasers based on inorganic–organic hybrid materials obtained by combined sol–gel polymer technology,” Polym. Adv. Technol. 15(6), 291–301 (2004).
[Crossref]

Schmitt, A.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Shan, J.

R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
[Crossref]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Shimakawa, K.

K. Tanaka and K. Shimakawa, “Chalcogenide glasses in Japan: a review on photoinduced phenomena,” Phys. Status Solidi B 246(8), 1744–1757 (2009).
[Crossref]

Smirnova, L. A.

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

Soref, R. A.

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26(1), 113–122 (1990).
[Crossref]

Starkov, V.

V. Gayvoronsky, S. Yakunin, V. Nazarenko, V. Starkov, and M. Brodyn, “Techniques to characterize the nonlinear optical response of doped nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 426(1), 231–241 (2005).
[Crossref]

Streppel, U.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Tanaka, K.

K. Tanaka and K. Shimakawa, “Chalcogenide glasses in Japan: a review on photoinduced phenomena,” Phys. Status Solidi B 246(8), 1744–1757 (2009).
[Crossref]

Taylor, P. C.

S. Ducharme, J. Hautala, and P. C. Taylor, “Photodarkening profiles and kinetics in chalcogenide glasses,” Phys. Rev. B Condens. Matter 41(17), 12250–12259 (1990).
[Crossref] [PubMed]

Tieng, S.

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

Traore, M.

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

Uklein, A.

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

Ulbricht, R.

R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
[Crossref]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Wächter, C.

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Weiss, A.

R. Reisfeld, A. Weiss, T. Saraidarov, E. Yariv, and A. A. Ishchenko, “Solid-state lasers based on inorganic–organic hybrid materials obtained by combined sol–gel polymer technology,” Polym. Adv. Technol. 15(6), 291–301 (2004).
[Crossref]

Yakunin, S.

V. Gayvoronsky, S. Yakunin, V. Nazarenko, V. Starkov, and M. Brodyn, “Techniques to characterize the nonlinear optical response of doped nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 426(1), 231–241 (2005).
[Crossref]

Yariv, E.

R. Reisfeld, A. Weiss, T. Saraidarov, E. Yariv, and A. A. Ishchenko, “Solid-state lasers based on inorganic–organic hybrid materials obtained by combined sol–gel polymer technology,” Polym. Adv. Technol. 15(6), 291–301 (2004).
[Crossref]

Zakery, A.

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330(1–3), 1–12 (2003).
[Crossref]

Zhan, Q.

Acc. Chem. Res. (1)

D. J. Kang and B.-S. Bae, “Photo-imageable sol-gel hybrid materials for simple fabrication of micro-optical elements,” Acc. Chem. Res. 40(9), 903–912 (2007).
[Crossref] [PubMed]

Adv. Mater. (1)

L. Nicole, L. Rozes, and C. Sanchez, “Integrative approaches to hybrid multifunctional materials: from multidisciplinary research to applied technologies,” Adv. Mater. 22(29), 3208–3214 (2010).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys., A Mater. Sci. Process. (1)

E. Fadeeva, J. Koch, B. Chichkov, A. Kuznetsov, O. Kameneva, N. Bityurin, C. Sanchez, and A. Kanaev, “Laser imprinting of 3D structures in gel-based titanium oxide organic-inorganic hybrids,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 27–30 (2006).
[Crossref]

Appl. Surf. Sci. (1)

N. Bityurin, A. I. Kuznetsov, and A. Kanaev, “Kinetics of UV-induced darkening of titanium-oxide gels,” Appl. Surf. Sci. 248(1–4), 86–90 (2005).
[Crossref]

Chem. Soc. Rev. (1)

C. Sanchez, P. Belleville, M. Popall, and L. Nicole, “Applications of advanced hybrid organic-inorganic nanomaterials: from laboratory to market,” Chem. Soc. Rev. 40(2), 696–753 (2011).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (2)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced change in refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Electron. 26(1), 113–122 (1990).
[Crossref]

J. Lumin. (1)

L. Museur, P. Gorbovyi, M. Traore, A. Kanaev, L. Rozes, and C. Sanchez, “Luminescence properties of pHEMA-TiO2 gels based hybrids materials,” J. Lumin. 132(5), 1192–1199 (2012).
[Crossref]

J. Mater. Chem. (2)

O. Kameneva, A. I. Kuznestov, L. A. Smirnova, L. Rozes, C. Sanchez, A. Alexandrov, N. Bityurin, K. Chhor, and A. Kanaev, “New photoactive hybrid organic-inorganic materials based on titanium-oxo-PHEMA nanocomposites exhibiting mixed valence properties,” J. Mater. Chem. 15(33), 3380–3383 (2005).
[Crossref]

J. U. Park, W. S. Kim, and B. S. Bae, “Photoinduced low refractive index in a photosensitive organic–inorganic hybrid material,” J. Mater. Chem. 13(4), 738–741 (2003).
[Crossref]

J. Non-Cryst. Solids (2)

T. Keiji, “Photo-induced phenomena in chalcogenide glass: Comparison with those in oxide glass and polymer,” J. Non-Cryst. Solids 352(23–25), 2580–2584 (2006).

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330(1–3), 1–12 (2003).
[Crossref]

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

J. Sci. Ind. Res. (India) (1)

N. Mehta, “Applications of chalcogenide glasses in electronics and optoelectronics: a review,” J. Sci. Ind. Res. (India) 65(10), 777–786 (2006).

Mol. Cryst. Liq. Cryst. (1)

V. Gayvoronsky, S. Yakunin, V. Nazarenko, V. Starkov, and M. Brodyn, “Techniques to characterize the nonlinear optical response of doped nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 426(1), 231–241 (2005).
[Crossref]

Nanoscale (1)

P. Gorbovyi, A. Uklein, S. Tieng, O. Brinza, M. Traore, K. Chhor, L. Museur, and A. Kanaev, “Novel nanostructured pHEMA-TiO2 hybrid materials with efficient light-induced charge separation,” Nanoscale 3(4), 1807–1812 (2011).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Mater. (1)

R. A. S. Ferreira, P. S. André, and L. D. Carlos, “Organic–inorganic hybrid materials towards passive and active architectures for the next generation of optical networks,” Opt. Mater. 32(11), 1397–1409 (2010).
[Crossref]

Phys. Chem. Chem. Phys. (1)

A. I. Kuznetsov, O. Kameneva, N. Bityurin, L. Rozes, C. Sanchez, and A. Kanaev, “Laser-induced photopatterning of organic-inorganic TiO2-based hybrid materials with tunable interfacial electron transfer,” Phys. Chem. Chem. Phys. 11(8), 1248–1257 (2009).
[Crossref] [PubMed]

Phys. Rev. B (1)

N. A. Deskins and M. Dupuis, “Electron transport via polaron hopping in bulk TiO2: a density functional theory characterization,” Phys. Rev. B 75(19), 195212 (2007).
[Crossref]

Phys. Rev. B Condens. Matter (1)

S. Ducharme, J. Hautala, and P. C. Taylor, “Photodarkening profiles and kinetics in chalcogenide glasses,” Phys. Rev. B Condens. Matter 41(17), 12250–12259 (1990).
[Crossref] [PubMed]

Phys. Status Solidi B (1)

K. Tanaka and K. Shimakawa, “Chalcogenide glasses in Japan: a review on photoinduced phenomena,” Phys. Status Solidi B 246(8), 1744–1757 (2009).
[Crossref]

Polym. Adv. Technol. (1)

R. Reisfeld, A. Weiss, T. Saraidarov, E. Yariv, and A. A. Ishchenko, “Solid-state lasers based on inorganic–organic hybrid materials obtained by combined sol–gel polymer technology,” Polym. Adv. Technol. 15(6), 291–301 (2004).
[Crossref]

Rev. Mod. Phys. (1)

R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83(2), 543–586 (2011).
[Crossref]

Thin Solid Films (1)

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J. U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter, and A. Bräuer, “Inorganic–organic hybrid materials for application in optical devices,” Thin Solid Films 442(1–2), 194–200 (2003).
[Crossref]

Other (4)

P. Gómez-Romero and C. Sanchez, eds., Functional Hybrid Materials (Wiley-VCH Verlag GmbH & Co., 2003).

R. G. Hunsperger, Integrated Optics (Springer, 2009).

A. Afanasiev, A. Alexandrov, N. Agareva, N. Sapogova, L. Smirnova, and N. Bityurin, “UV induced of linear and non-linear IR optical properties of dielectrics for photonic applications,” presented in FLAMN-10, St. Petersburg, Russia (2010).

L. Merhari, Hybrid Nanocomposites for Nanotechnology: Electronic, Optical, Magnetic And Biomedical Applications (Springer, 2009).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup: SFS – spatial filtering system, Sp is a beam splitter, S – sample, D1 and D2 are photodiodes. The beam waist position is taken as the origin of the axis z.

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Fig. 2
Fig. 2

a) Evolution of square beam radius (half width at 1 / e 2 ) along the propagation (Z axis). The full line is a fit with the usual expression of w ( z ) for Gaussian beams (see text). b) Image of the darkened area of sample x10 and spatial variation of the sample transmission.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3

Variation of total transmittance (a) and on-axis transmittance (b) in the hybrid samples as a function of UV irradiation dose. The dark solid line represents the fit of the experimental transmittance of sample X10 by Eq. (9) with parameters η a = 16.5%, η b = 2.5%.

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4

Scheme of the relevant processes involved in the photodarkenning of pHEMA-TiO2 hybrids materials.

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Fig. 5
Fig. 5

Variations of refractive index Δ n 0 (a) and normalized on-axis transmittance (b) of hybrid samples X1 and X10 versus the UV irradiation dose. On (b) the dashed lines represent the fits of on axis transmittance by Eq. (5).

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Fig. 6
Fig. 6

Photoinduced refractive index modification per Ti3+ center a 0 = Δ n 0 / [ T i 3 + ] .

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Fig. 7
Fig. 7

Charges distribution on -OEMA groups.

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Tables (1)

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Table 1 Hybrid samples characteristics: [Ti4+] concentration, thickness (L), quantum yield ηa, photoinduced refractive index change ∆n0, density of [Ti3+] centers and refractive index modification per Ti3+ center a0 a

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Equations (10)

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E e ( r , z ) = E 0 w 0 w ( z ) exp [ − r 2 w 2 ( z ) − i k r 2 2 R ( z ) ] exp [ − i ϕ ( z ) ]
Δ n ( r , D ) = Δ n 0 ( D ) exp [ − 2 r 2 w 2 ( z S ) ] Δ α ( r , D ) = Δ α 0 ( D ) exp [ − 2 r 2 w 2 ( z S ) ]
E s ( r , z , D ) = E e ( r , z ) exp [ − α L / 2 ] exp [ − ( i Δ ϕ 0 + Δ α 0 L 2 ) exp ( − 2 r 2 w 2 ( z S ) ) ]
E a ( r , z ′ = z + d , D ) = E e ( r = 0 , z ) e −   α L / 2 ∑ m = 0 ∞ ( − 1 ) m ( i Δ ϕ 0 + Δ α 0 L 2 ) m m !   w m 0 w m     exp ( − r 2 w m 2 )     exp i ( − k r 2 2 R m   +   θ m )
w m 0 2 = w 2 ( z ) / ( 2 m + 1 )   ,   d m = k w m 0 2 / 2   ,   R m 2 =   d   [ 1   –   g / ( g 2 +   d 2 / d m 2 )   ] − 1 θ m =   tan − 1 [ ( d / d m ) / g ]   ,   w m 2 = w m 0 2 ( g + d 2 / d m 2 )     and   g   =   1   +   d / R ( z )
T ( D ) = P o n   a x i s P t o t a l = ∫ 0 r a | E a ( r , z , D ) | 2 r   d r ∫ 0 ∞ | E a ( r , z , D ) | 2 r   d r
d   [ T i 3 + ] d t     =     −     ( η a   σ a + η b   σ b ) . [ T i 3 + ] . I h ν     +     η a   σ a   [     T i 4 +   ]   0 . I h ν
[ T i 3 + ] = [ T i 4 + ]   0 1 + η b σ b η a σ a . ( 1 − exp ( − ( η a σ a + η b σ b ) h ν . D ) )
T   = P t o t a l ( D ) P t o t a l ( D = 0 ) ∝ exp ( − ( σ b − σ a ) [ T i 3 + ] L )
n 2 − 1 n 2 + 2   =   4   π 3   ρ   β

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