Abstract

The numerical modeling of field-assisted ion exchange in glass through a finite aperture is carried out. The effects of unequal ion mobilities and thermal diffusion are included we believe for the first time in the 2-D case. This allows for the modeling of optical channel waveguides with graded index profiles. It is demonstrated that annealing of backdiffused channel guides is far superior to backdiffusion alone in improving their circular symmetry for better coupling to optical fibers.

© 1990 Optical Society of America

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References

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  1. R. V. Ramaswamy, R. Srivastava, “Ion-Exchanged Glass Waveguides: A Review,” IEEE/OSA J. Lightwave Technol. LT-6, 984–1002 (1988).
    [CrossRef]
  2. H. J. Lilienhof, E. Voges, D. Ritter, B. Pantschew, “Field-Induced Index Profiles of Multimode Ion-Exchanged Strip Waveguides,” IEEE J. Quantum Electron. QE-18, 1877–1883 (1982).
    [CrossRef]
  3. A. Tervonen, S. Honkanen, M. Leppihalme, “Control of Ion-Exchanged Waveguide Profiles with Ag Thin-Film Sources,” J. Appl. Phys. 62, 759–763 (1987).
    [CrossRef]
  4. C. Nissim, A. Beguin, R. Jansen, P. Laborde, “Fabrication and Characterization of Buried Single-Mode Waveguides and Couplers made by Ion Exchange in Glass,” in Technical Digest, Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1989), paper WM2.
  5. R. H. Doremus, “Ion Exchange in Glasses,” in Ion-Exchange, Vol. 2, J. A. Marinsky, Ed. (Marcal Dekker, New York, 1969).
  6. S. D. Fantone, “Refractive Index and Spectral Models for Gradient-Index Materials,” Appl. Opt. 22, 432–440 (1983).
    [CrossRef] [PubMed]
  7. R. G. Walker, C. D. W. Wilkinson, J. A. H. Wilkinson, “Integrated Optical Waveguiding Structures Made by Ion-Exchange in Glass. 1: The Propagation Characteristics of Stripe Ion-Exchanged Waveguides; a Theoretical and Experimental Investigation,” Appl. Opt. 22, 1923–1928 (1983).
    [CrossRef] [PubMed]
  8. M. Abou-El-Leil, A. R. Cooper, “Analysis of Field-Assisted Binary Ion-Exchange,” J. Am. Ceram. Soc. 62, 390–395 (1979).
    [CrossRef]
  9. S. N. Houde-Walter, D. T. Moore, “Gradient-Index Profile Control by Field-Assisted Ion Exchange in Glass,” Appl. Opt. 24, 4326–4333 (1985).
    [CrossRef] [PubMed]
  10. E. Voges, “Planar Tees and Star Couplers,” in Integrated Optics, S. Martellucci, A. N. Chester, Eds. (Plenum, New York, 1983). Note that the expression for the electric field cited here is corrected from that of Ref. 2 with a square root on the right-hand side.
  11. H. Yoshida, T. Kataoka, “Migration of Two Ions During Electrolysis of Glass Waveguide,” J. Appl. Phys. 58, 1739–1743 (1985).
    [CrossRef]
  12. S. Honkanen, Nokia Research Center Espoo, Finland; private communication.
  13. R. H. Doremus, “Mixed-Alkali Effect and Interdiffusion of Na and K Ions in Glass,” J. Am. Ceram. Soc. 57, 478–480 (1974).
    [CrossRef]
  14. A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
    [CrossRef]

1988 (2)

R. V. Ramaswamy, R. Srivastava, “Ion-Exchanged Glass Waveguides: A Review,” IEEE/OSA J. Lightwave Technol. LT-6, 984–1002 (1988).
[CrossRef]

A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
[CrossRef]

1987 (1)

A. Tervonen, S. Honkanen, M. Leppihalme, “Control of Ion-Exchanged Waveguide Profiles with Ag Thin-Film Sources,” J. Appl. Phys. 62, 759–763 (1987).
[CrossRef]

1985 (2)

S. N. Houde-Walter, D. T. Moore, “Gradient-Index Profile Control by Field-Assisted Ion Exchange in Glass,” Appl. Opt. 24, 4326–4333 (1985).
[CrossRef] [PubMed]

H. Yoshida, T. Kataoka, “Migration of Two Ions During Electrolysis of Glass Waveguide,” J. Appl. Phys. 58, 1739–1743 (1985).
[CrossRef]

1983 (2)

1982 (1)

H. J. Lilienhof, E. Voges, D. Ritter, B. Pantschew, “Field-Induced Index Profiles of Multimode Ion-Exchanged Strip Waveguides,” IEEE J. Quantum Electron. QE-18, 1877–1883 (1982).
[CrossRef]

1979 (1)

M. Abou-El-Leil, A. R. Cooper, “Analysis of Field-Assisted Binary Ion-Exchange,” J. Am. Ceram. Soc. 62, 390–395 (1979).
[CrossRef]

1974 (1)

R. H. Doremus, “Mixed-Alkali Effect and Interdiffusion of Na and K Ions in Glass,” J. Am. Ceram. Soc. 57, 478–480 (1974).
[CrossRef]

Abou-El-Leil, M.

M. Abou-El-Leil, A. R. Cooper, “Analysis of Field-Assisted Binary Ion-Exchange,” J. Am. Ceram. Soc. 62, 390–395 (1979).
[CrossRef]

Beguin, A.

A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
[CrossRef]

C. Nissim, A. Beguin, R. Jansen, P. Laborde, “Fabrication and Characterization of Buried Single-Mode Waveguides and Couplers made by Ion Exchange in Glass,” in Technical Digest, Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1989), paper WM2.

Cooper, A. R.

M. Abou-El-Leil, A. R. Cooper, “Analysis of Field-Assisted Binary Ion-Exchange,” J. Am. Ceram. Soc. 62, 390–395 (1979).
[CrossRef]

Doremus, R. H.

R. H. Doremus, “Mixed-Alkali Effect and Interdiffusion of Na and K Ions in Glass,” J. Am. Ceram. Soc. 57, 478–480 (1974).
[CrossRef]

R. H. Doremus, “Ion Exchange in Glasses,” in Ion-Exchange, Vol. 2, J. A. Marinsky, Ed. (Marcal Dekker, New York, 1969).

Dumas, T.

A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
[CrossRef]

Fantone, S. D.

Hackert, M. J.

A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
[CrossRef]

Honkanen, S.

A. Tervonen, S. Honkanen, M. Leppihalme, “Control of Ion-Exchanged Waveguide Profiles with Ag Thin-Film Sources,” J. Appl. Phys. 62, 759–763 (1987).
[CrossRef]

S. Honkanen, Nokia Research Center Espoo, Finland; private communication.

Houde-Walter, S. N.

Jansen, R.

A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
[CrossRef]

C. Nissim, A. Beguin, R. Jansen, P. Laborde, “Fabrication and Characterization of Buried Single-Mode Waveguides and Couplers made by Ion Exchange in Glass,” in Technical Digest, Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1989), paper WM2.

Kataoka, T.

H. Yoshida, T. Kataoka, “Migration of Two Ions During Electrolysis of Glass Waveguide,” J. Appl. Phys. 58, 1739–1743 (1985).
[CrossRef]

Laborde, P.

C. Nissim, A. Beguin, R. Jansen, P. Laborde, “Fabrication and Characterization of Buried Single-Mode Waveguides and Couplers made by Ion Exchange in Glass,” in Technical Digest, Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1989), paper WM2.

Leppihalme, M.

A. Tervonen, S. Honkanen, M. Leppihalme, “Control of Ion-Exchanged Waveguide Profiles with Ag Thin-Film Sources,” J. Appl. Phys. 62, 759–763 (1987).
[CrossRef]

Lilienhof, H. J.

H. J. Lilienhof, E. Voges, D. Ritter, B. Pantschew, “Field-Induced Index Profiles of Multimode Ion-Exchanged Strip Waveguides,” IEEE J. Quantum Electron. QE-18, 1877–1883 (1982).
[CrossRef]

Moore, D. T.

Nissim, C.

A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
[CrossRef]

C. Nissim, A. Beguin, R. Jansen, P. Laborde, “Fabrication and Characterization of Buried Single-Mode Waveguides and Couplers made by Ion Exchange in Glass,” in Technical Digest, Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1989), paper WM2.

Pantschew, B.

H. J. Lilienhof, E. Voges, D. Ritter, B. Pantschew, “Field-Induced Index Profiles of Multimode Ion-Exchanged Strip Waveguides,” IEEE J. Quantum Electron. QE-18, 1877–1883 (1982).
[CrossRef]

Ramaswamy, R. V.

R. V. Ramaswamy, R. Srivastava, “Ion-Exchanged Glass Waveguides: A Review,” IEEE/OSA J. Lightwave Technol. LT-6, 984–1002 (1988).
[CrossRef]

Ritter, D.

H. J. Lilienhof, E. Voges, D. Ritter, B. Pantschew, “Field-Induced Index Profiles of Multimode Ion-Exchanged Strip Waveguides,” IEEE J. Quantum Electron. QE-18, 1877–1883 (1982).
[CrossRef]

Srivastava, R.

R. V. Ramaswamy, R. Srivastava, “Ion-Exchanged Glass Waveguides: A Review,” IEEE/OSA J. Lightwave Technol. LT-6, 984–1002 (1988).
[CrossRef]

Tervonen, A.

A. Tervonen, S. Honkanen, M. Leppihalme, “Control of Ion-Exchanged Waveguide Profiles with Ag Thin-Film Sources,” J. Appl. Phys. 62, 759–763 (1987).
[CrossRef]

Voges, E.

H. J. Lilienhof, E. Voges, D. Ritter, B. Pantschew, “Field-Induced Index Profiles of Multimode Ion-Exchanged Strip Waveguides,” IEEE J. Quantum Electron. QE-18, 1877–1883 (1982).
[CrossRef]

E. Voges, “Planar Tees and Star Couplers,” in Integrated Optics, S. Martellucci, A. N. Chester, Eds. (Plenum, New York, 1983). Note that the expression for the electric field cited here is corrected from that of Ref. 2 with a square root on the right-hand side.

Walker, R. G.

Wilkinson, C. D. W.

Wilkinson, J. A. H.

Yoshida, H.

H. Yoshida, T. Kataoka, “Migration of Two Ions During Electrolysis of Glass Waveguide,” J. Appl. Phys. 58, 1739–1743 (1985).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

H. J. Lilienhof, E. Voges, D. Ritter, B. Pantschew, “Field-Induced Index Profiles of Multimode Ion-Exchanged Strip Waveguides,” IEEE J. Quantum Electron. QE-18, 1877–1883 (1982).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (2)

R. V. Ramaswamy, R. Srivastava, “Ion-Exchanged Glass Waveguides: A Review,” IEEE/OSA J. Lightwave Technol. LT-6, 984–1002 (1988).
[CrossRef]

A. Beguin, T. Dumas, M. J. Hackert, R. Jansen, C. Nissim, “Fabrication and Performance of Low Loss Optical Components Made by Ion Exchange in Glass,” IEEE/OSA J. Lightwave Technol. LT-6, 1483–1487 (1988).
[CrossRef]

J. Am. Ceram. Soc. (2)

R. H. Doremus, “Mixed-Alkali Effect and Interdiffusion of Na and K Ions in Glass,” J. Am. Ceram. Soc. 57, 478–480 (1974).
[CrossRef]

M. Abou-El-Leil, A. R. Cooper, “Analysis of Field-Assisted Binary Ion-Exchange,” J. Am. Ceram. Soc. 62, 390–395 (1979).
[CrossRef]

J. Appl. Phys. (2)

A. Tervonen, S. Honkanen, M. Leppihalme, “Control of Ion-Exchanged Waveguide Profiles with Ag Thin-Film Sources,” J. Appl. Phys. 62, 759–763 (1987).
[CrossRef]

H. Yoshida, T. Kataoka, “Migration of Two Ions During Electrolysis of Glass Waveguide,” J. Appl. Phys. 58, 1739–1743 (1985).
[CrossRef]

Other (4)

S. Honkanen, Nokia Research Center Espoo, Finland; private communication.

C. Nissim, A. Beguin, R. Jansen, P. Laborde, “Fabrication and Characterization of Buried Single-Mode Waveguides and Couplers made by Ion Exchange in Glass,” in Technical Digest, Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1989), paper WM2.

R. H. Doremus, “Ion Exchange in Glasses,” in Ion-Exchange, Vol. 2, J. A. Marinsky, Ed. (Marcal Dekker, New York, 1969).

E. Voges, “Planar Tees and Star Couplers,” in Integrated Optics, S. Martellucci, A. N. Chester, Eds. (Plenum, New York, 1983). Note that the expression for the electric field cited here is corrected from that of Ref. 2 with a square root on the right-hand side.

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Figures (8)

Fig. 1
Fig. 1

Geometry of the exchange domain and flux lines of the ionic current. D is the width of the mask opening, and d is the substrate depth.

Fig. 2
Fig. 2

Comparison between the 1-D result of Ref. 11 (——) and our result (- - -) for D = 100 μm, d = 50 μm, T = 616 K, Da = 2.6 × 10−15 m2/s, α = 0.9, t = 96″, and V = 1.6 V.

Fig. 3
Fig. 3

Comparison between the partial 2-D results of Ref. 2 (dotted contour line marking the boundary of the exchanged area) and our results (set of continuous contour lines): (a) D = 106 μm, d = 300 μm, α = 0.5, t = 30′, and V = 30 V (other parameters as in Fig. 2); (b) D = 36 μm, α = 0.2 [others as in (a)].

Fig. 4
Fig. 4

Evolution of the exchanged profile for α = 0.7. In this figure and all the following, the contour lines go from 0.1 to 0.9 as in Fig. 3 with the results normalized to the maximum value: a, D = 36 μm, t = 15′, V = 30 V; b, t = 30′, V =30 V; c, t = 30′, V = 60 V. Other parameters as in Fig. 3.

Fig. 5
Fig. 5

Backdiffusion [the contours remain normalized to the maximum of the original profile 5(a)]: (a) original exchange: D = 4 μm, α = 0.5, t = 30′, V = 30 V; (b) backdiffusion (no mask), t = 30′, V = 30 V; (c) additional backdiffusion, cumulative t = 90′, V = 30 V.

Fig. 6
Fig. 6

Annealing of the profile in Fig. 5(c): (a) no field, no mask, no sources, t = 120′; (b) cumulative t = 240′; (c) cumulative t = 600′.

Fig. 7
Fig. 7

Ellipticity e and asymmetry a of the profiles shown in Figs. 5 and 6 (parameters defined in the text).

Fig. 8
Fig. 8

Maximum index change of the profiles shown in Figs. 5 and 6 [normalized to the maximum occurring in Fig. 5(a)].

Equations (28)

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J D = - D c .
J E = c μ E .
μ = e D k T ,
J a = - D a ( c a - e E k T c a ) ,
J b = - D b ( c b - e E k T c b ) .
c a t = - · J a
= D a [ 2 c a - e k T ( c a · E + E · c a ) ] .
c a + c b = c b ( t = 0 ) = c 0 ,
· ( J a + J b ) = 0.
2 c a = - 2 c b ,
( D b - D a ) 2 c a - ( D b - D a ) e k T · ( E c a ) + D b e k T · ( E c 0 ) = 0.
· E = α E · c - k T e 2 c 1 - α c ,
c = c a c 0 ;             α = 1 - D a D b ,
c t = D a 1 - α c ( 2 c - e k T E · c ) .
c t = D a · ( 1 1 - α c c )
= D a 1 - α c [ 2 c + α ( c ) 2 1 - α c ] .
J 0 c 0 D b = e E k T [ D a c D b + ( 1 - c ) ] + ( 1 - D a D b ) c .
e E k T = J 0 c 0 D b - α c 1 - α c .
E = E ext + E diff ,
e E ext k T = J 0 c 0 D b ( 1 - α c ) ,
e E diff k T = - α c ( 1 - α c ) .
c t = D a 1 - α c [ 2 c + α ( c ) 2 1 - α c - e E ext k T c ] .
E y + i E x = i π 2 d V K ( β ) ( tanh 2 u - 1 tanh 2 u - β 2 ) 1 / 2
u = π ( y - i x ) 2 d and β = tanh ( π D / 4 d ) ,
c Φ = 1 1 - α c [ 2 c + α ( c ) 2 1 - α c - e E ext k T c ] ,
Φ = S 2 D a t ,             x = x / S ,             y = y / S ,             E ext = E ext S
c ( 0 , y ) = 1             ( y D 2 ) ,
c x = 0             ( y > D 2 ) ,

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