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Optical Materials Express

Optical Materials Express

  • Editor: David J. Hagan
  • Vol. 3, Iss. 1 — Jan. 1, 2013
  • pp: 95–104
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Modeling of the induced refractive index kinetics in photo-thermo-refractive glass

Julien Lumeau and Leonid B. Glebov  »View Author Affiliations


Optical Materials Express, Vol. 3, Issue 1, pp. 95-104 (2013)
http://dx.doi.org/10.1364/OME.3.000095


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Abstract

Photo-thermo-refractive (PTR) glass is a photosensitive multi-component silicate glass that exhibits refractive index change after successive UV-exposure and thermal treatment. The refractive index change was demonstrated to be associated with the precipitation of NaF nano-crystals in the glass matrix. This paper presents a systematic study of the dependence of the refractive index change on dosage of UV-exposure and thermal treatment duration and temperature in already UV-exposed and nucleated samples. It is shown using Avrami plots that the refractive index change in PTR glass is determined by a diffusion-controlled growth of particles from pre-existing nuclei while all coefficients have exponential dependence on temperature (Boltzmann law). The developed model, which includes both photo-chemistry and crystallization processes, allow describing the refractive index change dependence on dosage for a wide range of thermal treatment duration and temperature and an accuracy better than 10%.

© 2012 OSA

1. Introduction

Photo-thermo-refractive (PTR) glass is a class of glasses, which undergo photo-thermo-induced crystallization. It was invented by Stookey [1

1. S. D. Stookey, “Photosensitive glass,” Ind. Eng. Chem. 41(4), 856–861 (1949). [CrossRef]

] many years ago and has been studied as a new material for hologram writing in the last 20 years [2

2. L. B. Glebov, N. V. Nikonorov, E. I. Panysheva, G. T. Petrovskii, V. V. Savvin, I. V. Tunimanova, and V. A. Tsekhomskii, “Polychromatic glasses—a new material for recording volume phase holograms,” Sov. Phys. Dokl. 35, 878–880 (1990).

5

5. O. M. Efimov, L. B. Glebov, and V. I. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. 25(23), 1693–1695 (2000). [CrossRef] [PubMed]

]. Currently, PTR glass is sodium-potassium-zinc-aluminum-fluorine-bromine silicate glass doped with antimony, tin, cerium, and silver. It exhibits a localized refractive index decrement after UV-exposure and successive thermal treatment above the glass transition temperature, Tg, which results from the crystallization of about 0.5 wt.% sodium fluoride nano-crystals [6

6. T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of X-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325(1-3), 275–281 (2003). [CrossRef]

]. The possibility of recording phase holograms in this glass has potential for many high-tech applications, such as optical filtering [7

7. L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE 4724, 101–109 (2002). [CrossRef]

] and spectral beam combining of high power lasers [8

8. L. B. Glebov, “Photosensitive holographic glass—new approach to creation of high power lasers,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48(3), 123–128 (2007).

].

A description of the complex photo-thermo-induced crystallization mechanisms in this type of glass is given in reference [9

9. S. D. Stookey, G. H. Beall, and J. E. Pierson, “Full color photosensitive glass,” J. Appl. Phys. 49(10), 5114–5123 (1978). [CrossRef]

]. The evolution of the material’s nanostructure and optical properties after UV-exposure and thermal treatment are reported in several publications e.g [6

6. T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of X-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325(1-3), 275–281 (2003). [CrossRef]

,10

10. O. M. Efimov, L. B. Glebov, and H. P. Andre, “Measurement of the induced refractive index in a photothermorefractive glass by a liquid-cell shearing interferometer,” Appl. Opt. 41(10), 1864–1871 (2002). [CrossRef] [PubMed]

14

14. L. B. Glebov, “Linear and nonlinear photoionization of silicate glasses,” Glass Sci. Technol. 75(C1), 73–90 (2002).

]. Actually, it was shown in [6

6. T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of X-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325(1-3), 275–281 (2003). [CrossRef]

] that the photosensitivity of PTR glass results from the precipitation of nano-sized sodium fluoride crystals within the glass matrix in the UV-exposed regions after heat treatment. A simplified proposal for photo-thermal crystallization is the following: before any thermal development of the glass, sodium, fluorine and all other ions are dissolved in the matrix and the material is totally vitreous. When PTR glass is exposed to long wavelength UV radiation λ > 250 nm (e.g. a He-Cd laser at 325 nm), Ce3+ releases an electron and converts to hole-type Ce3++ center. The released electron is then trapped by intrinsic defects of the glass matrix or dopants in the highest valence state, including antimony and silver ions dispersed in the glass matrix. Then silver ions convert to silver atoms. When a UV-exposed glass is nucleated at temperatures between 450 and 500°C, silver atoms agglomerate and form colloidal silver containing particles. It was also demonstrated that silver bromide clusters form [15

15. J. Lumeau, L. Glebova, and L. B. Glebov, “Evolution of absorption spectra in the process of nucleation in photo-thermo-refractive glass,” Adv. Mater. Res. 39–40, 395–398 (2008). [CrossRef]

]. The second part of the crystallization process consists in the heterogeneous precipitation and growth of sodium fluoride crystals on top of the silver (or silver bromide) clusters. NaF growth is then controlled by diffusion of sodium and fluorine from the glass matrix to the crystals [14

14. L. B. Glebov, “Linear and nonlinear photoionization of silicate glasses,” Glass Sci. Technol. 75(C1), 73–90 (2002).

].

It is seen that the induced refractive index change is obtained only after a very complex series of chemical and physical process. Moreover, refractive index change is controlled by three interconnected parameters: the dosage of UV-exposure, the thermal treatment temperature and duration. Controlling the final refractive index change in a refractive or diffractive optical element is a key point in order to obtain an optimized element. For example, in case of volume Bragg gratings, perfect control of the refractive index modulation would allow achieving an element with expected diffraction efficiency and bandwidth. To achieve the goal, physical models were developed for the description of the photoionization and the crystallization, and were combined to an integrated comprehensive model. This model finally allows describing the refractive index change after exposure to ionizing UV radiation and thermal development within wide ranges of temperature and duration. It is shown that this model predicts the refractive index change with an accuracy of about 10%.

2. Experimental

2.1 PTR glass preparation

2.2 UV-exposure and heat-treatments

UV-exposure of samples was performed by a He-Cd laser (4 mW, 325 nm). A stripe with Gaussian distribution of dosage and maximum dosage about 1 J/cm2 was recorded in each sample by scanning the laser beam over the sample’s surface. Dosage was controlled with the scanning speed [10

10. O. M. Efimov, L. B. Glebov, and H. P. Andre, “Measurement of the induced refractive index in a photothermorefractive glass by a liquid-cell shearing interferometer,” Appl. Opt. 41(10), 1864–1871 (2002). [CrossRef] [PubMed]

]. This maximum dosage was chosen because it provides induced refractive index close to saturation. The samples were then nucleated at 485°C for 100 minutes and heat-treated for different durations at temperatures between 485 and 535°C. For each thermal treatment, samples were heated from room temperature to the development temperature at a rate of about 20 K/min and then, at the end of the development, they were cooled down to room temperature, in the furnace following the natural decrease of the furnace temperature (about 2.5 K/min). Temperature was measured using a calibrated thermocouple.

2.3 Refractive index change measurements

Refractive index changes were measured in each sample using a shearing interferometer setup [10

10. O. M. Efimov, L. B. Glebov, and H. P. Andre, “Measurement of the induced refractive index in a photothermorefractive glass by a liquid-cell shearing interferometer,” Appl. Opt. 41(10), 1864–1871 (2002). [CrossRef] [PubMed]

]. Its basic principle is to create an interferogram that converts the phase change at propagation through the glass to a fringe shift. A liquid cell with an index matching fluid was used to prevent thickness variations of the sample which would contribute to a fringe shift. Therefore the interferometer fringe distortions resulted only from refractive index variations. Precision on each measurement was demonstrated to be better than 10 ppm.

3. Modeling of the refractive index change

3.1 Refractive index change measurements versus time and dosage

PTR glass samples exposed with a Gaussian profile stripe with dosage at maximum (E0) of 0.9 J/cm2 were developed for different durations at different temperatures between 485 and 535°C. From the refractive index change measured on each of these samples, two sets of curves were obtained. First we plotted the dependence of the refractive index change on dosage for different thermal treatment durations and for different temperatures, as shown in Fig. 1
Fig. 1 Dependence of the refractive index change (RIC) on dosage of UV-exposure measured in PTR glass after nucleation and thermal treatment at ~515°C for different durations.
. From these curves, the dependence of the refractive index change at maximum dosage on time of thermal treatment and for different temperatures was extracted (Fig. 2
Fig. 2 Dependence of the maximum refractive index change (RIC) on thermal treatment duration measured in PTR glass UV-exposed with dosage E0 of 0.9 J/cm2 and developed at different temperatures.
). From the curve in Fig. 1, one can see that the longer the thermal treatment duration, the higher the saturation level of the refractive index change.From the curve of Fig. 2, we can justify the range of temperatures that was chosen for this experiment. At development temperatures of 485°C, the refractive index change becomes very slow as 40 + hours are required to reach refractive index change showing some approaching to saturation; hence using temperatures lower than 485°C is out of any reasonable application. In contrary, when developing at 535°C, saturation is obtained after less than one hour, such as the precise control of the refractive index requires extremely precise control on the thermal treatment procedure; therefore using temperatures above than 535°C is also out of reasonable application. Thus, the range of temperatures that provides controllable regimes of thermal development is equal to only 50°C. The curves in Fig. 2 allow extracting a basic parameter defining the refractive index change kinetics, i.e. the slope at t = 0. The evolution of this slope (in ppm/hour) as a function of 1/T (in K−1) is plotted in Fig. 3
Fig. 3 Temperature dependence of the slope of the refractive index change versus thermal treatment duration at t = 0 (extracted from Fig. 2).
.One can see that the slope at t = 0 follows a Boltzmann law and that the induced refractive index kinetics is changed by 2 orders of magnitude within 50°C change of the development temperature. This result demonstrates high sensitivity of the thermal treatment in respect to temperature and justifies the usefulness of a model that can predict the refractive index change for any given dosage of UV-exposure and thermal treatment temperature and duration.

3.2 Modeling of the isothermal refractive index change versus time at constant dosage

3.3 Modeling of the isothermal refractive index change versus dosage

In order to model the refractive index change versus dosage for any thermal treatment duration and temperature as measured in Fig. 1, the results observed in [25

25. L. B. Glebov, “Kinetics modeling in photosensitive glass,” Opt. Mater. 25(4), 413–418 (2004). [CrossRef]

] were used. It was shown in [25

25. L. B. Glebov, “Kinetics modeling in photosensitive glass,” Opt. Mater. 25(4), 413–418 (2004). [CrossRef]

] that by writing balance equations associated with the photoionization of PTR glass and supposing that trapping of electrons by silver and holes have comparable probability, the dependence of the refractive index change (Δn) on dosage (D) follows an hyperbolic function:
Δn(D)=nSDD+ε
(5)
where nS is the refractive change at saturation and nS/ε is the inverse of the slope at D = 0. Using the refractive index change at E0 ( = 0.9 J/cm2) (Δn(E0, t,T)) that can be predicted using JMAK theory (Eq. (1)), the Eq. (5) becomes:
Δn(D,t,T)=Δn(E0,t,T)(E0+ε(t,T))DE0(D+ε(t,T))
(6)
In our case ε(t,T) is a thermodynamic parameter that needs to be determined for each thermal treatment temperature and duration. Combining the data of the section 3.2 with the Eq. (6) allows fitting each of the curves of the Fig. 1 with a precision better than 10% (Fig. 8
Fig. 8 Example of a fit of the dependence on the dosage of UV-exposure of the refractive index change (RIC) measured in PTR glass developed for ~60 minutes at ~515°C.
) and to extract ln(1/ε(t,T)) as a function of the thermal treatment duration t for each used temperature T (Fig. 9
Fig. 9 Dependence of 1/ε(t) on the thermal treatment duration at ~515°C.
). It is seen that ln(1/ε(t,T)) evolves almost linearly for any of the temperature used in our study. Therefore an equation that accurately predicts 1/ε(t,T) is following:
1ε(t,T)=exp(β(T)t)
(7)
where β(T) is a temperature dependent parameter. Despite the fact that physical meaning to this equation still requires additional study, it allows predicting the refractive index versus dosage for any thermal treatment at constant temperature.The evolution of ln(β(T)) as a function of 1/T is shown in Fig. 10
Fig. 10 Dependence of ln(β(T)) on 1/T.
. A linear dependence is observed proving that β(T) is a thermodynamic parameter that follows a Boltzmann law:
β(T)=β0×exp(EβRT)
(8)
Where Eβ is the activation energy and is equal to 393 KJ/mol.

4. Conclusion

The basic physical model that describes the refractive index change in PTR glass includes three independent parameters: the dosage of UV-exposure, the thermal treatment duration and the thermal treatment temperature. The use of the Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory permits to accurately predict the refractive index for a constant dosage at any of the thermal treatment procedure. The Avrami coefficient, describing the type of crystallization process, is equal to 1.5, demonstrating that the refractive index change is based on a diffusion-controlled growth of particles from pre-existing nuclei. The usage of hyperbolic functions permits to accurately predict dependence of refractive index on dosage of UV-radiation for fixed conditions of thermal development. Kinetic coefficients in these models show Boltzmann origin (exponential dependence on temperature).

Acknowledgments

This work is supported in part by DARPA/ADHELS program (contract H0011-06-1-0010).

References and links

1.

S. D. Stookey, “Photosensitive glass,” Ind. Eng. Chem. 41(4), 856–861 (1949). [CrossRef]

2.

L. B. Glebov, N. V. Nikonorov, E. I. Panysheva, G. T. Petrovskii, V. V. Savvin, I. V. Tunimanova, and V. A. Tsekhomskii, “Polychromatic glasses—a new material for recording volume phase holograms,” Sov. Phys. Dokl. 35, 878–880 (1990).

3.

O. M. Efimov, L. B. Glebov, L. N. Glebova, K. C. Richardson, and V. I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass,” Appl. Opt. 38(4), 619–627 (1999). [CrossRef] [PubMed]

4.

O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical-holographic-element writing,” Proc. SPIE 3578, 564–575 (1999). [CrossRef]

5.

O. M. Efimov, L. B. Glebov, and V. I. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. 25(23), 1693–1695 (2000). [CrossRef] [PubMed]

6.

T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of X-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids 325(1-3), 275–281 (2003). [CrossRef]

7.

L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE 4724, 101–109 (2002). [CrossRef]

8.

L. B. Glebov, “Photosensitive holographic glass—new approach to creation of high power lasers,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48(3), 123–128 (2007).

9.

S. D. Stookey, G. H. Beall, and J. E. Pierson, “Full color photosensitive glass,” J. Appl. Phys. 49(10), 5114–5123 (1978). [CrossRef]

10.

O. M. Efimov, L. B. Glebov, and H. P. Andre, “Measurement of the induced refractive index in a photothermorefractive glass by a liquid-cell shearing interferometer,” Appl. Opt. 41(10), 1864–1871 (2002). [CrossRef] [PubMed]

11.

L. B. Glebov and L. Glebova, “Swelling of photo-thermo-refractive glass resulted from thermal development,” Glass Sci. Technol. 75(C2), 294–297 (2002).

12.

L. Glebov, L. Glebova, V. Tsechomskii, and V. Golubkov, “Study of structural transformations in photo-thermo-refractive glass by SAXS and XRD,” Proceedings of XX International Congress on Glass, Kyoto, Japan (September 2004), paper O-07–082.

13.

J. Lumeau, L. Glebova, G. P. Souza, E. D. Zanotto, and L. B. Glebov, “Effect of cooling on the optical properties and crystallization of UV-exposed photo-thermo-refractive glass,” J. Non-Cryst. Solids 354(42-44), 4730–4736 (2008). [CrossRef]

14.

L. B. Glebov, “Linear and nonlinear photoionization of silicate glasses,” Glass Sci. Technol. 75(C1), 73–90 (2002).

15.

J. Lumeau, L. Glebova, and L. B. Glebov, “Evolution of absorption spectra in the process of nucleation in photo-thermo-refractive glass,” Adv. Mater. Res. 39–40, 395–398 (2008). [CrossRef]

16.

J. Lumeau, A. Sinitskii, L. Glebova, L. B. Glebov, and E. D. Zanotto, “Method to assess the homogeneity of partially crystallized glasses: application to a photo-thermo-refractive glass,” J. Non-Cryst. Solids 355(34-36), 1760–1768 (2009). [CrossRef]

17.

A. N. Kolmogorov, “On the statistical theory of crystallization of metals,” Izv. Akad. Nauk. SSSR Ser. Mat. 1, 355–359 (1937).

18.

W. A. Johnson and R. F. Mehl, “Reaction kinetics in processes of nucleation and growth,” Trans. AIME 135, 416–442 (1939).

19.

M. Avrami and J. Chem, “Kinetics of phase change. I. General theory,” J. Chem. Phys. 7(12), 1103–1112 (1939). [CrossRef]

20.

V. Fokin, E. D. Zanotto, N. Yuritsyn, and J. W. Schmelzer, “Homogeneous crystal nucleation in silicate glasses: a 40 years perspective,” J. Non-Cryst. Solids 352(26-27), 2681–2714 (2006). [CrossRef]

21.

K. Chamma, J. Lumeau, L. Glebova, and L. B. Glebov, “Effect of dosage and thermal treatment on photo-thermo-refractive glass crystallization properties,” International Congress on Glass (Salvador, Brazil) (September 2010), paper 326.

22.

G. P. Souza, V. M. Fokin, E. D. Zanotto, J. Lumeau, L. Glebova, and L. B. Glebov, “Spontaneous and photo-induced crystallization of PTR glass,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 50(5), 311–320 (2009).

23.

J. Lumeau, L. Glebova, V. Golubkov, E. D. Zanotto, and L. B. Glebov, “Origin of crystallization-induced refractive index changes in photo-thermo-refractive glass,” Opt. Mater. 32(1), 139–146 (2009). [CrossRef]

24.

G. P. Souza, V. M. Fokin, E. D. Zanotto, J. Lumeau, L. Glebova, and L. B. Glebov, “Diffusion zone in crystallized photo-thermo-refractive glass,” in 8th Pacific Rim Conference on Ceramic and Glass Technology (PACRIM 8) (Vancouver, British Columbia, Canada) (June 2009), paper PACRIM8–S23–P176–2009.

25.

L. B. Glebov, “Kinetics modeling in photosensitive glass,” Opt. Mater. 25(4), 413–418 (2004). [CrossRef]

OCIS Codes
(160.4760) Materials : Optical properties
(350.5130) Other areas of optics : Photochemistry
(160.5335) Materials : Photosensitive materials

ToC Category:
Glass and Other Amorphous Materials

History
Original Manuscript: October 16, 2012
Revised Manuscript: December 17, 2012
Manuscript Accepted: December 17, 2012
Published: December 19, 2012

Citation
Julien Lumeau and Leonid B. Glebov, "Modeling of the induced refractive index kinetics in photo-thermo-refractive glass," Opt. Mater. Express 3, 95-104 (2013)
http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-3-1-95


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References

  1. S. D. Stookey, “Photosensitive glass,” Ind. Eng. Chem.41(4), 856–861 (1949). [CrossRef]
  2. L. B. Glebov, N. V. Nikonorov, E. I. Panysheva, G. T. Petrovskii, V. V. Savvin, I. V. Tunimanova, and V. A. Tsekhomskii, “Polychromatic glasses—a new material for recording volume phase holograms,” Sov. Phys. Dokl.35, 878–880 (1990).
  3. O. M. Efimov, L. B. Glebov, L. N. Glebova, K. C. Richardson, and V. I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass,” Appl. Opt.38(4), 619–627 (1999). [CrossRef] [PubMed]
  4. O. M. Efimov, L. B. Glebov, S. Papernov, and A. W. Schmid, “Laser-induced damage of photo-thermo-refractive glasses for optical-holographic-element writing,” Proc. SPIE3578, 564–575 (1999). [CrossRef]
  5. O. M. Efimov, L. B. Glebov, and V. I. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett.25(23), 1693–1695 (2000). [CrossRef] [PubMed]
  6. T. Cardinal, O. M. Efimov, H. G. Francois-Saint-Cyr, L. B. Glebov, L. N. Glebova, and V. I. Smirnov, “Comparative study of photo-induced variations of X-ray diffraction and refractive index in photo-thermo-refractive glass,” J. Non-Cryst. Solids325(1-3), 275–281 (2003). [CrossRef]
  7. L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE4724, 101–109 (2002). [CrossRef]
  8. L. B. Glebov, “Photosensitive holographic glass—new approach to creation of high power lasers,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B48(3), 123–128 (2007).
  9. S. D. Stookey, G. H. Beall, and J. E. Pierson, “Full color photosensitive glass,” J. Appl. Phys.49(10), 5114–5123 (1978). [CrossRef]
  10. O. M. Efimov, L. B. Glebov, and H. P. Andre, “Measurement of the induced refractive index in a photothermorefractive glass by a liquid-cell shearing interferometer,” Appl. Opt.41(10), 1864–1871 (2002). [CrossRef] [PubMed]
  11. L. B. Glebov and L. Glebova, “Swelling of photo-thermo-refractive glass resulted from thermal development,” Glass Sci. Technol.75(C2), 294–297 (2002).
  12. L. Glebov, L. Glebova, V. Tsechomskii, and V. Golubkov, “Study of structural transformations in photo-thermo-refractive glass by SAXS and XRD,” Proceedings of XX International Congress on Glass, Kyoto, Japan (September 2004), paper O-07–082.
  13. J. Lumeau, L. Glebova, G. P. Souza, E. D. Zanotto, and L. B. Glebov, “Effect of cooling on the optical properties and crystallization of UV-exposed photo-thermo-refractive glass,” J. Non-Cryst. Solids354(42-44), 4730–4736 (2008). [CrossRef]
  14. L. B. Glebov, “Linear and nonlinear photoionization of silicate glasses,” Glass Sci. Technol.75(C1), 73–90 (2002).
  15. J. Lumeau, L. Glebova, and L. B. Glebov, “Evolution of absorption spectra in the process of nucleation in photo-thermo-refractive glass,” Adv. Mater. Res.39–40, 395–398 (2008). [CrossRef]
  16. J. Lumeau, A. Sinitskii, L. Glebova, L. B. Glebov, and E. D. Zanotto, “Method to assess the homogeneity of partially crystallized glasses: application to a photo-thermo-refractive glass,” J. Non-Cryst. Solids355(34-36), 1760–1768 (2009). [CrossRef]
  17. A. N. Kolmogorov, “On the statistical theory of crystallization of metals,” Izv. Akad. Nauk. SSSR Ser. Mat.1, 355–359 (1937).
  18. W. A. Johnson and R. F. Mehl, “Reaction kinetics in processes of nucleation and growth,” Trans. AIME135, 416–442 (1939).
  19. M. Avrami and J. Chem, “Kinetics of phase change. I. General theory,” J. Chem. Phys.7(12), 1103–1112 (1939). [CrossRef]
  20. V. Fokin, E. D. Zanotto, N. Yuritsyn, and J. W. Schmelzer, “Homogeneous crystal nucleation in silicate glasses: a 40 years perspective,” J. Non-Cryst. Solids352(26-27), 2681–2714 (2006). [CrossRef]
  21. K. Chamma, J. Lumeau, L. Glebova, and L. B. Glebov, “Effect of dosage and thermal treatment on photo-thermo-refractive glass crystallization properties,” International Congress on Glass (Salvador, Brazil) (September 2010), paper 326.
  22. G. P. Souza, V. M. Fokin, E. D. Zanotto, J. Lumeau, L. Glebova, and L. B. Glebov, “Spontaneous and photo-induced crystallization of PTR glass,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B50(5), 311–320 (2009).
  23. J. Lumeau, L. Glebova, V. Golubkov, E. D. Zanotto, and L. B. Glebov, “Origin of crystallization-induced refractive index changes in photo-thermo-refractive glass,” Opt. Mater.32(1), 139–146 (2009). [CrossRef]
  24. G. P. Souza, V. M. Fokin, E. D. Zanotto, J. Lumeau, L. Glebova, and L. B. Glebov, “Diffusion zone in crystallized photo-thermo-refractive glass,” in 8th Pacific Rim Conference on Ceramic and Glass Technology (PACRIM 8) (Vancouver, British Columbia, Canada) (June 2009), paper PACRIM8–S23–P176–2009.
  25. L. B. Glebov, “Kinetics modeling in photosensitive glass,” Opt. Mater.25(4), 413–418 (2004). [CrossRef]

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