Abstract

A numerical model explicitly considering the space-charge density evolved both under the mask and in the region of optical structure formation was used to predict the profiles of Ag concentration during field-assisted Ag+Na+ ion exchange channel waveguide fabrication. The influence of the unequal values of diffusion constants and mobilities of incoming and outgoing ions, the value of a correlation factor (Haven ratio), and particularly space-charge density induced during the ion exchange, on the resulting profiles of Ag concentration was analyzed and discussed. It was shown that the incorporation into the numerical model of a small quantity of highly mobile ions other than exclusively Ag+ and Na+ may considerably affect the range and shape of calculated Ag profiles in the multicomponent glass. The Poisson equation was used to predict the electric field spread evolution in the glass substrate. The results of the numerical analysis were verified by the experimental data of Ag concentration in a channel waveguide fabricated using a field-assisted process.

© 2011 Optical Society of America

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  1. S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).
  2. B. West, “Ion-exchanged glass waveguides,” in The Handbook of Photonics, 2nd ed., M.C.Gupta and J.Ballato, eds. (CRC Press, 2007), pp. 13.1–35.
  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]
  4. 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]
  5. 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]
  6. A. Tervonen, “A general model for fabrication processes of channel waveguides by ion exchange,” J. Appl. Phys. 67, 2746–2752 (1990).
    [CrossRef]
  7. 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]
  8. P. Mrozek, E. Mrozek, and T. Lukaszewicz, “Side diffusion modeling by the explicit consideration of a space charge buildup under the mask during a strip waveguide formation in Ag+–Na+ field-assisted ion exchange process,” Appl. Opt. 45, 619–625 (2006).
    [CrossRef] [PubMed]
  9. K. M. Knowles and A. T. J. van Helvoort, “Anodic bonding,” Int. Mater. Rev. 51, 273–311 (2006).
    [CrossRef]
  10. U. K. Krieger and W. A. Lanford, “Field assisted transport of Na+ ions, Ca+ ions and electrons in commercial soda-lime glass I: experimental,” J. Non-Cryst. Solids 102, 50–61(1988).
    [CrossRef]
  11. C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
    [CrossRef]
  12. 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]
  13. G. Wallis, “Direct-current polarization during field-assisted glass-metal sealing,” J. Am. Ceram. Soc. 53, 563–67 (1970).
    [CrossRef]
  14. 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]
  15. P. Mrozek, E. Mrozek, and T. Lukaszewicz, “Determination of refractive index profiles of Ag+–Na+ ion-exchange multimode strip waveguides by variable wavefront shear double-refracting interferometry microinterferometry,” Appl. Opt. 45, 756–763 (2006).
    [CrossRef] [PubMed]

2007 (1)

B. West, “Ion-exchanged glass waveguides,” in The Handbook of Photonics, 2nd ed., M.C.Gupta and J.Ballato, eds. (CRC Press, 2007), pp. 13.1–35.

2006 (4)

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

K. M. Knowles and A. T. J. van Helvoort, “Anodic bonding,” Int. Mater. Rev. 51, 273–311 (2006).
[CrossRef]

P. Mrozek, E. Mrozek, and T. Lukaszewicz, “Side diffusion modeling by the explicit consideration of a space charge buildup under the mask during a strip waveguide formation in Ag+–Na+ field-assisted ion exchange process,” Appl. Opt. 45, 619–625 (2006).
[CrossRef] [PubMed]

P. Mrozek, E. Mrozek, and T. Lukaszewicz, “Determination of refractive index profiles of Ag+–Na+ ion-exchange multimode strip waveguides by variable wavefront shear double-refracting interferometry microinterferometry,” Appl. Opt. 45, 756–763 (2006).
[CrossRef] [PubMed]

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]

1997 (1)

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 (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]

1993 (1)

C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
[CrossRef]

1990 (2)

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]

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

1988 (1)

U. K. Krieger and W. A. Lanford, “Field assisted transport of Na+ ions, Ca+ ions and electrons in commercial soda-lime glass I: experimental,” J. Non-Cryst. Solids 102, 50–61(1988).
[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, 563–67 (1970).
[CrossRef]

Achete, C. A.

C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
[CrossRef]

Albert, J.

Auxier, J.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

Carriere, J.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

Castro, J.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

Frantz, J.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

Freire, F. L.

C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
[CrossRef]

Geraghty, D.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

Giacometti, J. A.

C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
[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.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

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]

Knowles, K. M.

K. M. Knowles and A. T. J. van Helvoort, “Anodic bonding,” Int. Mater. Rev. 51, 273–311 (2006).
[CrossRef]

Kostuk, R.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

Krieger, U. K.

U. K. Krieger and W. A. Lanford, “Field assisted transport of Na+ ions, Ca+ ions and electrons in commercial soda-lime glass I: experimental,” J. Non-Cryst. Solids 102, 50–61(1988).
[CrossRef]

Lanford, W. A.

U. K. Krieger and W. A. Lanford, “Field assisted transport of Na+ ions, Ca+ ions and electrons in commercial soda-lime glass I: experimental,” J. Non-Cryst. Solids 102, 50–61(1988).
[CrossRef]

Leal Ferreira, G. F.

C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
[CrossRef]

Lepienski, C. M.

C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
[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.

Lukaszewicz, T.

Madasamy, P.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

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]

Morrell, M.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

Mrozek, E.

Mrozek, P.

Peyghambarian, N.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

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]

Schu¨lzgen, A.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

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]

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

van Helvoort, A. T. J.

K. M. Knowles and A. T. J. van Helvoort, “Anodic bonding,” Int. Mater. Rev. 51, 273–311 (2006).
[CrossRef]

Wallis, G.

G. Wallis, “Direct-current polarization during field-assisted glass-metal sealing,” J. Am. Ceram. Soc. 53, 563–67 (1970).
[CrossRef]

West, B.

B. West, “Ion-exchanged glass waveguides,” in The Handbook of Photonics, 2nd ed., M.C.Gupta and J.Ballato, eds. (CRC Press, 2007), pp. 13.1–35.

West, B. R.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

Yliniemi, S.

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

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]

Int. Mater. Rev. (1)

K. M. Knowles and A. T. J. van Helvoort, “Anodic bonding,” Int. Mater. Rev. 51, 273–311 (2006).
[CrossRef]

J. Am. Ceram. Soc. (2)

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]

G. Wallis, “Direct-current polarization during field-assisted glass-metal sealing,” J. Am. Ceram. Soc. 53, 563–67 (1970).
[CrossRef]

J. Appl. Phys. (1)

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

J. Non-Cryst. Solids (3)

U. K. Krieger and W. A. Lanford, “Field assisted transport of Na+ ions, Ca+ ions and electrons in commercial soda-lime glass I: experimental,” J. Non-Cryst. Solids 102, 50–61(1988).
[CrossRef]

C. M. Lepienski, J. A. Giacometti, G. F. Leal Ferreira, F. L. Freire Jr., and C. A. Achete, “Electric field distribution and near-surface modifications in soda-lime glass submitted to a dc potential,” J. Non-Cryst. Solids 159, 204–212 (1993).
[CrossRef]

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]

Opt. Commun. (1)

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]

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]

Phys. Chem. Glasses (1)

S. Honkanen, B. R. West, S. Yliniemi, P. Madasamy, M. Morrell, J. Auxier, A. Schu¨lzgen, N. Peyghambarian, J. Carriere, J. Frantz, R. Kostuk, J. Castro, and D. Geraghty, “Recent advances in ion exchanged glass waveguides and devices,” Phys. Chem. Glasses 47, 110–120 (2006).

Other (1)

B. West, “Ion-exchanged glass waveguides,” in The Handbook of Photonics, 2nd ed., M.C.Gupta and J.Ballato, eds. (CRC Press, 2007), pp. 13.1–35.

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Figures (13)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Electric potential φ boundary conditions used in numerical simulations.

Fig. 3
Fig. 3

Lines of constant electric potential (solid lines) decreased by 0.1 V step, in the direction from the anode (top) to the cathode; Ag concentration contour lines (dotted lines), in the direction to the inside of glass: 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.05 C Ag max ( C Ag max is the maximum Ag concentration); the influence of a decrease of M parameter on numerical simulation results of field-assisted waveguide formation: (a), (b) for parameters of item No. 1, 2 in Table 1, respectively (for clarity’s sake only a part of each contour is shown in the vicinity of the mask edge).

Fig. 4
Fig. 4

Ag concentration contour lines (solid lines), in the direction to the inside of glass: 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.05 C Ag max ( C Ag max is the maximum Ag concentration); the influence of a decrease of M parameter on numerical simulation results of field-assisted waveguide formation: (a), (b), (c), (d) for parameters of item No. 1, 2, 3, 4 in Table 1, respectively (only a half of each symmetrical contour is shown for clarity’s sake). In the lower-left corner of each subplot a close-up view of the concentration contours in 3 μm deep region under the mask edge is shown.

Fig. 5
Fig. 5

Ag concentration contour lines, in the direction to the inside of glass: 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.05 C Ag max ( C Ag max is the maximum Ag concentration); the influence of an increase of f parameter on numerical simulation results of field-assisted waveguide formation: (a), (b), (c) for parameters of item No. 5, 6, 7 in Table 1, respectively (only a half of each symmetrical contour is shown for clarity’s sake). In the lower-left corner of each subplot a close-up view of the concentration contours in 3 μm deep region under the mask edge is shown.

Fig. 6
Fig. 6

Ag concentration contour lines, in the direction to the inside of glass: 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.05 C Ag max ( C Ag max is the maximum Ag concentration); the influence of a decrease of D Na parameter on numerical simulation results of field-assisted waveguide formation: (a), (b), (c) for param eters of item No. 8, 9, 10 in Table 1, respectively (only a half of each symmetrical contour is shown for clarity’s sake). In the lower-left corner of each subplot a close-up view of the concentration contours in 3 μm deep region under the mask edge is shown.

Fig. 7
Fig. 7

Ag concentration contour lines, in the direction to the inside of glass: 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.05 C Ag max ( C Ag max is the maximum Ag concentration); the influence of a decrease of m parameter on numerical simulation results of field-assisted waveguide formation: (a), (b), (c) for parameters of item No. 11, 12, 13 in Table 1, respectively (only a half of each symmetrical contour is shown for clarity’s sake). In the lower-left corner of each subplot a close-up view of the concentration contours in 3 μm deep region under the mask edge is shown.

Fig. 8
Fig. 8

Space-charge density ρ distribution in the center of the mask opening ( x = 0 ); the influence of an increase of t parameter on numerical simulation results of field-assisted waveguide formation. Solid line, dashed line, and dotted line for parameters of item No. 14, 15, 16 in Table 1, respectively.

Fig. 9
Fig. 9

Space-charge density ρ distribution in the center of the mask opening ( x = 0 ); the influence of a decrease of M parameter on numerical simulation results of field-assisted waveguide formation. Solid line, dashed line, and dotted line for parameters of item No. 17, 18, 19 in Table 1, respectively.

Fig. 10
Fig. 10

Space-charge density ρ distribution under the mask ( x = 60 μm ); numerical simulation results of field-assisted waveguide formation for parameters of item No. 20 in Table 1.

Fig. 11
Fig. 11

Relative deviation in the initial ionic concentration in glass ( C Ag + C Na ) / C 0 _ Na in the center of the mask opening ( x = 0 ); the influence of a decrease of M parameter on numerical simulation results of field-assisted waveguide formation. Solid line, dashed line, and dotted line for parameters of item No. 21, 22 and 23 in Table 1, respectively.

Fig. 12
Fig. 12

Uniform field interference split image of a cross section of a channel waveguide in white light illumination.

Fig. 13
Fig. 13

Ag concentration contour lines, in the direction to the inside of glass: 0.8 C Ag max , 0.6 C Ag max , 0.4 C Ag max , 0.2 C Ag max , 0.05 C Ag max ( C Ag max is the maximum Ag concentration); (a) numerical simulation for parameters of item No. 24 in Table 1 and (b) experimental results of field-assisted waveguide formation for process parameters. Voltage U = 20 V , time t = 5 min , glass thickness h = 1.5 mm , mask aperture width d = 85 μm , and temperature T = 628 K .

Tables (1)

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Table 1 Values of Process Parameters Used in Numerical Simulations (Time t, Ratio M = D Ag / D Na , Haven Ratio f, Self-diffusion Coefficient D Na , Coefficient Decreasing the Density of a Space Charge m)

Equations (7)

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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 ]
2 φ = ρ ε r ε 0 ,
C Ag ( Na ) t = D Ag ( Na ) 2 C Ag ( Na ) μ Ag ( Na ) ( C Ag ( Na ) · E ¯ + E ¯ · C Ag ( Na ) ) ,
μ / D = q / ( f k T ) ,
ρ 0 = ( C Ag + C Na C 0 _ Na ) F ,
ρ = m ρ 0 .

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