Abstract

A space-charge buildup under the blocking mask in a field-assisted Ag+Na+ ion-exchange modeling is assumed. It results in the distortion of electric field lines in the direction under the mask edges. As a result, side diffusion occurs and the numerical model shows the same range of side diffusion as the experimental data. Explicit consideration of the space-charge buildup under the mask and solving the Poisson equation for the electric field determination make it possible to use more realistic boundary conditions in the numerical model, compared to the boundary conditions generally used.

© 2006 Optical Society of America

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References

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  1. C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
    [CrossRef]
  2. A. Belkhir, "A comparative study of silver diffusion in a glass substrate for optical waveguide applications," IEEE J. Quantum Electron. 35, 306-311 (1999).
    [CrossRef]
  3. A. Tervonen, S. Honkanen, and S. Najafi, "Analysis of symmetric directional couplers and asymmetric Mach-Zehnder interferometers as 1.30- and 1.55-µm dual wavelength demultiplexers/multiplexers," Opt. Eng. 32, 2083-2090 (1993).
    [CrossRef]
  4. R. G. Walker, C. D. W. Wilkinson, and J. A. H. Wilkinson, "Integrated optical waveguiding structures made by silver ion-exchange in glass. 1: The propagation characteristics of stripe ion-exchanged waveguides; a theoretical and experimental investigation," Appl. Opt. 22, 1923-1935 (1983).
    [CrossRef] [PubMed]
  5. R. G. Walker and C. D. W. Wilkinson, "Integrated optical waveguiding structures made by silver ion-exchange in glass. 2: Directional coupler and bends," Appl. Opt. 22, 1929-1936 (1983).
    [CrossRef] [PubMed]
  6. B. Pantchev, Z. Nikolov, and E. Voges, "Coupling efficiency in ion-exchanged homogeneous refracting waveguide lenses in glass," Opt. Commun. 135, 247-250 (1997).
    [CrossRef]
  7. V. Francois, T. Ohtsuki, N. Peyghambarian, and S. I. Najafi, "Thermally silver ion exchanged integrated-optic lasers in neodymium-doped silicate glass," Opt. Commun. 119, 104-108 (1995).
    [CrossRef]
  8. E. Mrozek and P. Mrozek, "Thin dielectric electrodiffusion masks for Ag+-Na+ ion exchange applications," J. Tech. Phys. 44, 289-294 (2003).
  9. D. Cheng, J. Saarinen, H. Saarikoski, and A. Tervonen, "Simulation of field-assisted ion exchange for glass channel waveguide fabrication: effect of nonhomogeneous time-dependent electric conductivity," Opt. Commun. 137, 233-238 (1997).
    [CrossRef]
  10. J. Albert and J. W. Y. Lit, "Full modeling of field-assisted ion exchange for graded index buried channel optical waveguides," Appl. Opt. 29, 2798-2804 (1990).
    [CrossRef] [PubMed]
  11. B. R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, "Modeling of ion-exchanged glass waveguide structures," J. Non-Cryst. Solids 347, 18-26 (2004).
    [CrossRef]
  12. A. Tervonen, "A general model for fabrication processes of channel waveguides by ion exchange," J. Appl. Phys. 67, 2746-2752 (1990).
    [CrossRef]
  13. X. Prieto, R. Srivastava, J. Linares, and C. Montero, "Prediction of space-charge density and space-charge field in thermally ion-exchanged planar surface waveguides," Opt. Mater. 5, 145-151 (1996).
    [CrossRef]
  14. G. Wallis, "Direct-current polarization during field-assisted glass-metal sealing," J. Am. Ceram. Soc. 53, 563-567 (1970).
    [CrossRef]
  15. D. E. Carlson, K. W. Hang, and G. F. Stockdale, "Electrode polarization in alkali-containing glasses," J. Am. Ceram. Soc. 55, 337-341 (1972).
    [CrossRef]
  16. P. Sutton, "Space charge and electrode polarization in glass," J. Am. Ceram. Soc. 47, 219-230 (1964).
    [CrossRef]
  17. D. Kapila and J. L. Plawsky, "Diffusion processes for integrated waveguide fabrication in glasses: a solid-state electrochemical approach," Chem. Eng. Sci. 50, 2589-2600 (1995).
    [CrossRef]
  18. R. G. Gossink, "SIMS Analysis of a field-assisted glass-to-metal seal," J. Am. Ceram. Soc. 61, 539-540 (1978).
    [CrossRef]
  19. P. B. DeNee, "Low energy metal-glass bonding," J. Appl. Phys. 40, 5396-5397 (1969).
    [CrossRef]
  20. S. Honkanen and A. Tervonen, "Experimental analysis of Ag+-Na+ exchange in glass with Ag film ion sources for planar optical waveguide fabrication," J. Appl. Phys. 63, 634-639 (1988).
    [CrossRef]

2004 (1)

B. R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, "Modeling of ion-exchanged glass waveguide structures," J. Non-Cryst. Solids 347, 18-26 (2004).
[CrossRef]

2003 (1)

E. Mrozek and P. Mrozek, "Thin dielectric electrodiffusion masks for Ag+-Na+ ion exchange applications," J. Tech. Phys. 44, 289-294 (2003).

2000 (1)

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

1999 (1)

A. Belkhir, "A comparative study of silver diffusion in a glass substrate for optical waveguide applications," IEEE J. Quantum Electron. 35, 306-311 (1999).
[CrossRef]

1997 (2)

B. Pantchev, Z. Nikolov, and E. Voges, "Coupling efficiency in ion-exchanged homogeneous refracting waveguide lenses in glass," Opt. Commun. 135, 247-250 (1997).
[CrossRef]

D. Cheng, J. Saarinen, H. Saarikoski, and A. Tervonen, "Simulation of field-assisted ion exchange for glass channel waveguide fabrication: effect of nonhomogeneous time-dependent electric conductivity," Opt. Commun. 137, 233-238 (1997).
[CrossRef]

1996 (1)

X. Prieto, R. Srivastava, J. Linares, and C. Montero, "Prediction of space-charge density and space-charge field in thermally ion-exchanged planar surface waveguides," Opt. Mater. 5, 145-151 (1996).
[CrossRef]

1995 (2)

D. Kapila and J. L. Plawsky, "Diffusion processes for integrated waveguide fabrication in glasses: a solid-state electrochemical approach," Chem. Eng. Sci. 50, 2589-2600 (1995).
[CrossRef]

V. Francois, T. Ohtsuki, N. Peyghambarian, and S. I. Najafi, "Thermally silver ion exchanged integrated-optic lasers in neodymium-doped silicate glass," Opt. Commun. 119, 104-108 (1995).
[CrossRef]

1993 (1)

A. Tervonen, S. Honkanen, and S. Najafi, "Analysis of symmetric directional couplers and asymmetric Mach-Zehnder interferometers as 1.30- and 1.55-µm dual wavelength demultiplexers/multiplexers," Opt. Eng. 32, 2083-2090 (1993).
[CrossRef]

1990 (2)

A. Tervonen, "A general model for fabrication processes of channel waveguides by ion exchange," J. Appl. Phys. 67, 2746-2752 (1990).
[CrossRef]

J. Albert and J. W. Y. Lit, "Full modeling of field-assisted ion exchange for graded index buried channel optical waveguides," Appl. Opt. 29, 2798-2804 (1990).
[CrossRef] [PubMed]

1988 (1)

S. Honkanen and A. Tervonen, "Experimental analysis of Ag+-Na+ exchange in glass with Ag film ion sources for planar optical waveguide fabrication," J. Appl. Phys. 63, 634-639 (1988).
[CrossRef]

1983 (2)

1978 (1)

R. G. Gossink, "SIMS Analysis of a field-assisted glass-to-metal seal," J. Am. Ceram. Soc. 61, 539-540 (1978).
[CrossRef]

1972 (1)

D. E. Carlson, K. W. Hang, and G. F. Stockdale, "Electrode polarization in alkali-containing glasses," J. Am. Ceram. Soc. 55, 337-341 (1972).
[CrossRef]

1970 (1)

G. Wallis, "Direct-current polarization during field-assisted glass-metal sealing," J. Am. Ceram. Soc. 53, 563-567 (1970).
[CrossRef]

1969 (1)

P. B. DeNee, "Low energy metal-glass bonding," J. Appl. Phys. 40, 5396-5397 (1969).
[CrossRef]

1964 (1)

P. Sutton, "Space charge and electrode polarization in glass," J. Am. Ceram. Soc. 47, 219-230 (1964).
[CrossRef]

Albert, J.

Belkhir, A.

A. Belkhir, "A comparative study of silver diffusion in a glass substrate for optical waveguide applications," IEEE J. Quantum Electron. 35, 306-311 (1999).
[CrossRef]

Carlson, D. E.

D. E. Carlson, K. W. Hang, and G. F. Stockdale, "Electrode polarization in alkali-containing glasses," J. Am. Ceram. Soc. 55, 337-341 (1972).
[CrossRef]

Cheng, D.

D. Cheng, J. Saarinen, H. Saarikoski, and A. Tervonen, "Simulation of field-assisted ion exchange for glass channel waveguide fabrication: effect of nonhomogeneous time-dependent electric conductivity," Opt. Commun. 137, 233-238 (1997).
[CrossRef]

DeNee, P. B.

P. B. DeNee, "Low energy metal-glass bonding," J. Appl. Phys. 40, 5396-5397 (1969).
[CrossRef]

Francois, V.

V. Francois, T. Ohtsuki, N. Peyghambarian, and S. I. Najafi, "Thermally silver ion exchanged integrated-optic lasers in neodymium-doped silicate glass," Opt. Commun. 119, 104-108 (1995).
[CrossRef]

Gossink, R. G.

R. G. Gossink, "SIMS Analysis of a field-assisted glass-to-metal seal," J. Am. Ceram. Soc. 61, 539-540 (1978).
[CrossRef]

Hang, K. W.

D. E. Carlson, K. W. Hang, and G. F. Stockdale, "Electrode polarization in alkali-containing glasses," J. Am. Ceram. Soc. 55, 337-341 (1972).
[CrossRef]

Honkanen, S.

B. R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, "Modeling of ion-exchanged glass waveguide structures," J. Non-Cryst. Solids 347, 18-26 (2004).
[CrossRef]

A. Tervonen, S. Honkanen, and S. Najafi, "Analysis of symmetric directional couplers and asymmetric Mach-Zehnder interferometers as 1.30- and 1.55-µm dual wavelength demultiplexers/multiplexers," Opt. Eng. 32, 2083-2090 (1993).
[CrossRef]

S. Honkanen and A. Tervonen, "Experimental analysis of Ag+-Na+ exchange in glass with Ag film ion sources for planar optical waveguide fabrication," J. Appl. Phys. 63, 634-639 (1988).
[CrossRef]

Itoh, K.

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

Kapila, D.

D. Kapila and J. L. Plawsky, "Diffusion processes for integrated waveguide fabrication in glasses: a solid-state electrochemical approach," Chem. Eng. Sci. 50, 2589-2600 (1995).
[CrossRef]

Lavers, C. R.

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

Linares, J.

X. Prieto, R. Srivastava, J. Linares, and C. Montero, "Prediction of space-charge density and space-charge field in thermally ion-exchanged planar surface waveguides," Opt. Mater. 5, 145-151 (1996).
[CrossRef]

Lit, J. W. Y.

Madasamy, P.

B. R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, "Modeling of ion-exchanged glass waveguide structures," J. Non-Cryst. Solids 347, 18-26 (2004).
[CrossRef]

Mauchline, I.

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

Montero, C.

X. Prieto, R. Srivastava, J. Linares, and C. Montero, "Prediction of space-charge density and space-charge field in thermally ion-exchanged planar surface waveguides," Opt. Mater. 5, 145-151 (1996).
[CrossRef]

Mrozek, E.

E. Mrozek and P. Mrozek, "Thin dielectric electrodiffusion masks for Ag+-Na+ ion exchange applications," J. Tech. Phys. 44, 289-294 (2003).

Mrozek, P.

E. Mrozek and P. Mrozek, "Thin dielectric electrodiffusion masks for Ag+-Na+ ion exchange applications," J. Tech. Phys. 44, 289-294 (2003).

Murabayashi, M.

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

Najafi, S.

A. Tervonen, S. Honkanen, and S. Najafi, "Analysis of symmetric directional couplers and asymmetric Mach-Zehnder interferometers as 1.30- and 1.55-µm dual wavelength demultiplexers/multiplexers," Opt. Eng. 32, 2083-2090 (1993).
[CrossRef]

Najafi, S. I.

V. Francois, T. Ohtsuki, N. Peyghambarian, and S. I. Najafi, "Thermally silver ion exchanged integrated-optic lasers in neodymium-doped silicate glass," Opt. Commun. 119, 104-108 (1995).
[CrossRef]

Nikolov, Z.

B. Pantchev, Z. Nikolov, and E. Voges, "Coupling efficiency in ion-exchanged homogeneous refracting waveguide lenses in glass," Opt. Commun. 135, 247-250 (1997).
[CrossRef]

Ohtsuki, T.

V. Francois, T. Ohtsuki, N. Peyghambarian, and S. I. Najafi, "Thermally silver ion exchanged integrated-optic lasers in neodymium-doped silicate glass," Opt. Commun. 119, 104-108 (1995).
[CrossRef]

Pantchev, B.

B. Pantchev, Z. Nikolov, and E. Voges, "Coupling efficiency in ion-exchanged homogeneous refracting waveguide lenses in glass," Opt. Commun. 135, 247-250 (1997).
[CrossRef]

Peyghambarian, N.

B. R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, "Modeling of ion-exchanged glass waveguide structures," J. Non-Cryst. Solids 347, 18-26 (2004).
[CrossRef]

V. Francois, T. Ohtsuki, N. Peyghambarian, and S. I. Najafi, "Thermally silver ion exchanged integrated-optic lasers in neodymium-doped silicate glass," Opt. Commun. 119, 104-108 (1995).
[CrossRef]

Plawsky, J. L.

D. Kapila and J. L. Plawsky, "Diffusion processes for integrated waveguide fabrication in glasses: a solid-state electrochemical approach," Chem. Eng. Sci. 50, 2589-2600 (1995).
[CrossRef]

Prieto, X.

X. Prieto, R. Srivastava, J. Linares, and C. Montero, "Prediction of space-charge density and space-charge field in thermally ion-exchanged planar surface waveguides," Opt. Mater. 5, 145-151 (1996).
[CrossRef]

Saarikoski, H.

D. Cheng, J. Saarinen, H. Saarikoski, and A. Tervonen, "Simulation of field-assisted ion exchange for glass channel waveguide fabrication: effect of nonhomogeneous time-dependent electric conductivity," Opt. Commun. 137, 233-238 (1997).
[CrossRef]

Saarinen, J.

D. Cheng, J. Saarinen, H. Saarikoski, and A. Tervonen, "Simulation of field-assisted ion exchange for glass channel waveguide fabrication: effect of nonhomogeneous time-dependent electric conductivity," Opt. Commun. 137, 233-238 (1997).
[CrossRef]

Srivastava, R.

X. Prieto, R. Srivastava, J. Linares, and C. Montero, "Prediction of space-charge density and space-charge field in thermally ion-exchanged planar surface waveguides," Opt. Mater. 5, 145-151 (1996).
[CrossRef]

Stewart, G.

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

Stockdale, G. F.

D. E. Carlson, K. W. Hang, and G. F. Stockdale, "Electrode polarization in alkali-containing glasses," J. Am. Ceram. Soc. 55, 337-341 (1972).
[CrossRef]

Stout, T.

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

Sutton, P.

P. Sutton, "Space charge and electrode polarization in glass," J. Am. Ceram. Soc. 47, 219-230 (1964).
[CrossRef]

Tervonen, A.

D. Cheng, J. Saarinen, H. Saarikoski, and A. Tervonen, "Simulation of field-assisted ion exchange for glass channel waveguide fabrication: effect of nonhomogeneous time-dependent electric conductivity," Opt. Commun. 137, 233-238 (1997).
[CrossRef]

A. Tervonen, S. Honkanen, and S. Najafi, "Analysis of symmetric directional couplers and asymmetric Mach-Zehnder interferometers as 1.30- and 1.55-µm dual wavelength demultiplexers/multiplexers," Opt. Eng. 32, 2083-2090 (1993).
[CrossRef]

A. Tervonen, "A general model for fabrication processes of channel waveguides by ion exchange," J. Appl. Phys. 67, 2746-2752 (1990).
[CrossRef]

S. Honkanen and A. Tervonen, "Experimental analysis of Ag+-Na+ exchange in glass with Ag film ion sources for planar optical waveguide fabrication," J. Appl. Phys. 63, 634-639 (1988).
[CrossRef]

Voges, E.

B. Pantchev, Z. Nikolov, and E. Voges, "Coupling efficiency in ion-exchanged homogeneous refracting waveguide lenses in glass," Opt. Commun. 135, 247-250 (1997).
[CrossRef]

Walker, R. G.

Wallis, G.

G. Wallis, "Direct-current polarization during field-assisted glass-metal sealing," J. Am. Ceram. Soc. 53, 563-567 (1970).
[CrossRef]

West, B. R.

B. R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, "Modeling of ion-exchanged glass waveguide structures," J. Non-Cryst. Solids 347, 18-26 (2004).
[CrossRef]

Wilkinson, C. D. W.

Wilkinson, J. A. H.

Wu, S. C.

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

Appl. Opt. (3)

Chem. Eng. Sci. (1)

D. Kapila and J. L. Plawsky, "Diffusion processes for integrated waveguide fabrication in glasses: a solid-state electrochemical approach," Chem. Eng. Sci. 50, 2589-2600 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Belkhir, "A comparative study of silver diffusion in a glass substrate for optical waveguide applications," IEEE J. Quantum Electron. 35, 306-311 (1999).
[CrossRef]

J. Am. Ceram. Soc. (4)

R. G. Gossink, "SIMS Analysis of a field-assisted glass-to-metal seal," J. Am. Ceram. Soc. 61, 539-540 (1978).
[CrossRef]

G. Wallis, "Direct-current polarization during field-assisted glass-metal sealing," J. Am. Ceram. Soc. 53, 563-567 (1970).
[CrossRef]

D. E. Carlson, K. W. Hang, and G. F. Stockdale, "Electrode polarization in alkali-containing glasses," J. Am. Ceram. Soc. 55, 337-341 (1972).
[CrossRef]

P. Sutton, "Space charge and electrode polarization in glass," J. Am. Ceram. Soc. 47, 219-230 (1964).
[CrossRef]

J. Appl. Phys. (3)

A. Tervonen, "A general model for fabrication processes of channel waveguides by ion exchange," J. Appl. Phys. 67, 2746-2752 (1990).
[CrossRef]

P. B. DeNee, "Low energy metal-glass bonding," J. Appl. Phys. 40, 5396-5397 (1969).
[CrossRef]

S. Honkanen and A. Tervonen, "Experimental analysis of Ag+-Na+ exchange in glass with Ag film ion sources for planar optical waveguide fabrication," J. Appl. Phys. 63, 634-639 (1988).
[CrossRef]

J. Non-Cryst. Solids (1)

B. R. West, P. Madasamy, N. Peyghambarian, and S. Honkanen, "Modeling of ion-exchanged glass waveguide structures," J. Non-Cryst. Solids 347, 18-26 (2004).
[CrossRef]

J. Tech. Phys. (1)

E. Mrozek and P. Mrozek, "Thin dielectric electrodiffusion masks for Ag+-Na+ ion exchange applications," J. Tech. Phys. 44, 289-294 (2003).

Opt. Commun. (3)

D. Cheng, J. Saarinen, H. Saarikoski, and A. Tervonen, "Simulation of field-assisted ion exchange for glass channel waveguide fabrication: effect of nonhomogeneous time-dependent electric conductivity," Opt. Commun. 137, 233-238 (1997).
[CrossRef]

B. Pantchev, Z. Nikolov, and E. Voges, "Coupling efficiency in ion-exchanged homogeneous refracting waveguide lenses in glass," Opt. Commun. 135, 247-250 (1997).
[CrossRef]

V. Francois, T. Ohtsuki, N. Peyghambarian, and S. I. Najafi, "Thermally silver ion exchanged integrated-optic lasers in neodymium-doped silicate glass," Opt. Commun. 119, 104-108 (1995).
[CrossRef]

Opt. Eng. (1)

A. Tervonen, S. Honkanen, and S. Najafi, "Analysis of symmetric directional couplers and asymmetric Mach-Zehnder interferometers as 1.30- and 1.55-µm dual wavelength demultiplexers/multiplexers," Opt. Eng. 32, 2083-2090 (1993).
[CrossRef]

Opt. Mater. (1)

X. Prieto, R. Srivastava, J. Linares, and C. Montero, "Prediction of space-charge density and space-charge field in thermally ion-exchanged planar surface waveguides," Opt. Mater. 5, 145-151 (1996).
[CrossRef]

Sens. Actuators B (1)

C. R. Lavers, K. Itoh, S. C. Wu, M. Murabayashi, I. Mauchline, G. Stewart, and T. Stout, "Planar optical waveguides for sensing applications," Sens. Actuators B 69, 85-95 (2000).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Configurations of field-assisted Ag + Na + ion-exchange process: (a) field-assisted ion exchange from Ag thin film source; (b), (c) field-assisted ion exchange from Ag thin-film source with a mask; (d) field-assisted ion exchange from a molten salt; (e) one-step field-assisted ion exchange for fabrication of buried waveguide from Ag thin-film stripe.

Fig. 2
Fig. 2

Electric potential φ boundary conditions usually used in ion-exchange modeling.

Fig. 3
Fig. 3

Electric potential φ boundary conditions used in numerical simulations and the schematic for Ag + and Na + migration during negative space-charge buildup under the mask.

Fig. 4
Fig. 4

E y component of field E ¯ at a time t = 0 s .

Fig. 5
Fig. 5

E x component of field E ¯ at a time t = 0 s .

Fig. 6
Fig. 6

Concentration C Na after t = 10 min of the simulation.

Fig. 7
Fig. 7

Concentration C Ag after t = 10 min of the simulation.

Fig. 8
Fig. 8

E y component of field ̄E after t = 10 min of the simulation.

Fig. 9
Fig. 9

E x component of field E ¯ after t = 10 min of the simulation.

Fig. 10
Fig. 10

Ag concentration contour lines in the direction to the inside of glass: 0.9 C Ag max , 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.0 C Ag max ( C Ag max is the maximum Ag concentration); (a) experimental and (b) numerical simulation results of field-assisted waveguide formation for the process parameters: voltage U = 20 V , time t = 10 min , glass thickness h = 1.5 mm , mask aperture width d = 33 µm , and temperature T = 628 K .

Fig. 11
Fig. 11

Ag concentration contour lines in the direction to the inside of glass: 0.9 C Ag max , 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.0 C Ag max ( C Ag max is the maximum Ag concentration); (a) experimental and (b) numerical simulation results of field-assisted waveguide formation for the process parameters: voltage U = 20 V , time t = 10 min , glass thickness h = 1.5 mm , mask aperture width d = 85 µm , and temperature T = 588 K .

Fig. 12
Fig. 12

Ag concentration contour lines in the direction to the inside of glass: 0.9 C Ag max , 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.0 C Ag max ( C Ag max is the maximum Ag concentration); (a) experimental and (b) numerical simulation results of field-assisted waveguide formation for the process parameters: voltage U = 20 V , time t = 5 min , glass thickness h = 1.5 mm , mask aperture width d = 142 µm , and temperature T = 628 K .

Equations (13)

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

j ¯ = D c 0 c + M j e ¯ c c ( M 1 ) + c 0 ,
c t = · j ¯ ,
c t = D 1 ( 1 M ) c [ 2 c + ( 1 M ) ( c ) 2 1 ( 1 M ) c q E ¯ ext · c k T ] ,
q E ¯ ext k T = j e ¯ c 0 D 0 [ 1 ( 1 M ) c ] .
j e ¯ = σ E ¯
2 φ = 0 ,
σ ( c ) 2 φ + σ ( c ) · φ = 0 ,
2 φ = ρ ε r ε 0 ,
C Ag t = D Ag 2 C Ag μ Ag ( C Ag · E ¯ + E ¯ · C Ag ) ,
C Na t = D Na 2 C Na μ Na ( C Na · E ¯ + E ¯ · C Na ) ,
ρ 0 = ( C Ag + C Na C 0 Na ) F ,
ρ = m ρ 0 .
μ / D = q / ( f k T ) ,

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