Abstract

The required concentration-dependent diffusion coefficients for both ideal one-dimensional and ideal radial gradient-index profiles are determined. The modified quasi-chemical diffusion model is used to relate the diffusion coefficient to optimum glass composition. Adding aluminum to sodium silicate glasses facilitates the approach to the desired concentration dependence of the diffusion coefficient for silver–sodium ion exchange. A parabolic one-dimensional index profile is fabricated in one of the glasses. It deviates from ideal values by less than 2%.

© 1996 Optical Society of America

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    [CrossRef] [PubMed]
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  4. J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
    [CrossRef]
  5. J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 209–215 (1995).
    [CrossRef]
  6. Y. Suematsu, K. Iga, S. Ito, “A light beam waveguide using hyperbolic type gas lenses,” IEEE Trans. Microwave Theory Tech. MTT-14, 657–665 (1966).
    [CrossRef]
  7. E. W. Marchand, Gradient-Index Optics (Academic, New York, 1978), Chap. 5, p. 61.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  12. L. E. Murr, Electron and Ion Microscopy and Microanalysis (Marcel and Dekker, New York, 1991), Chap. 4, p. 191.
  13. R. Goering, M. Rothhardt, “Application of the refracted near-field technique to multimode planar and channel waveguides in glass,” Opt. Commun. 7, 82–85 (1986).
  14. T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).
  15. B. Messerschmidt, T. Possner, R. Goering, “Colorless gradient-index cylindrical lenses with high numerical apertures produced by silver-ion exchange,” Appl. Opt. 34, 7825–7830 (1995).
    [CrossRef] [PubMed]
  16. S. N. Houde-Walter, J. M. Inman, A. J. Dent, G. N. Greaves, “Sodium and silver environments and ion-exchange processes in silicate and aluminosilicate glasses,” J. Phys. Chem. 97, 9330–9336 (1993).
    [CrossRef]

1996

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

1995

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 209–215 (1995).
[CrossRef]

B. Messerschmidt, T. Possner, R. Goering, “Colorless gradient-index cylindrical lenses with high numerical apertures produced by silver-ion exchange,” Appl. Opt. 34, 7825–7830 (1995).
[CrossRef] [PubMed]

1993

S. N. Houde-Walter, J. M. Inman, A. J. Dent, G. N. Greaves, “Sodium and silver environments and ion-exchange processes in silicate and aluminosilicate glasses,” J. Phys. Chem. 97, 9330–9336 (1993).
[CrossRef]

1992

1991

T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).

C. Kaps, W. Fliegel, “Sodium/silver ion exchange between a non-bridging oxygen-free boroaluminosilicate glass and nitrate melts,” Glastech. Ber. 64, 199–204 (1991).

1988

1986

R. Goering, M. Rothhardt, “Application of the refracted near-field technique to multimode planar and channel waveguides in glass,” Opt. Commun. 7, 82–85 (1986).

1985

T. Findakly, “Glass waveguides by ion exchange: a review,” Opt. Eng. 24, 244–252 (1985).

1983

1966

Y. Suematsu, K. Iga, S. Ito, “A light beam waveguide using hyperbolic type gas lenses,” IEEE Trans. Microwave Theory Tech. MTT-14, 657–665 (1966).
[CrossRef]

Araujo, R.

Asahara, Y.

Bentley, J. L.

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 209–215 (1995).
[CrossRef]

Crank, J.

J. Crank, The Mathematics of Diffusion (Clarendon, Oxford, 1975), Chap. 10, p. 231.

Dent, A. J.

S. N. Houde-Walter, J. M. Inman, A. J. Dent, G. N. Greaves, “Sodium and silver environments and ion-exchange processes in silicate and aluminosilicate glasses,” J. Phys. Chem. 97, 9330–9336 (1993).
[CrossRef]

Fantone, S. D.

Findakly, T.

T. Findakly, “Glass waveguides by ion exchange: a review,” Opt. Eng. 24, 244–252 (1985).

Fliegel, W.

C. Kaps, W. Fliegel, “Sodium/silver ion exchange between a non-bridging oxygen-free boroaluminosilicate glass and nitrate melts,” Glastech. Ber. 64, 199–204 (1991).

Goering, R.

B. Messerschmidt, T. Possner, R. Goering, “Colorless gradient-index cylindrical lenses with high numerical apertures produced by silver-ion exchange,” Appl. Opt. 34, 7825–7830 (1995).
[CrossRef] [PubMed]

R. Goering, M. Rothhardt, “Application of the refracted near-field technique to multimode planar and channel waveguides in glass,” Opt. Commun. 7, 82–85 (1986).

Greaves, G. N.

S. N. Houde-Walter, J. M. Inman, A. J. Dent, G. N. Greaves, “Sodium and silver environments and ion-exchange processes in silicate and aluminosilicate glasses,” J. Phys. Chem. 97, 9330–9336 (1993).
[CrossRef]

Houde-Walter, S. N.

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 209–215 (1995).
[CrossRef]

S. N. Houde-Walter, J. M. Inman, A. J. Dent, G. N. Greaves, “Sodium and silver environments and ion-exchange processes in silicate and aluminosilicate glasses,” J. Phys. Chem. 97, 9330–9336 (1993).
[CrossRef]

Iga, K.

Y. Suematsu, K. Iga, S. Ito, “A light beam waveguide using hyperbolic type gas lenses,” IEEE Trans. Microwave Theory Tech. MTT-14, 657–665 (1966).
[CrossRef]

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens and Stacked Planar Optics (Academic, Tokyo, 1984), Chap. 7, p. 129.

Inman, J. M.

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 209–215 (1995).
[CrossRef]

S. N. Houde-Walter, J. M. Inman, A. J. Dent, G. N. Greaves, “Sodium and silver environments and ion-exchange processes in silicate and aluminosilicate glasses,” J. Phys. Chem. 97, 9330–9336 (1993).
[CrossRef]

Ito, S.

Y. Suematsu, K. Iga, S. Ito, “A light beam waveguide using hyperbolic type gas lenses,” IEEE Trans. Microwave Theory Tech. MTT-14, 657–665 (1966).
[CrossRef]

Izumitani, T.

Kahnt, H.

T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).

Kaps, C.

C. Kaps, W. Fliegel, “Sodium/silver ion exchange between a non-bridging oxygen-free boroaluminosilicate glass and nitrate melts,” Glastech. Ber. 64, 199–204 (1991).

T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).

Kokubun, Y.

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens and Stacked Planar Optics (Academic, Tokyo, 1984), Chap. 7, p. 129.

Liao, Z. M.

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

Marchand, E. W.

E. W. Marchand, Gradient-Index Optics (Academic, New York, 1978), Chap. 5, p. 61.

McIntyre, B. L.

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

Messerschmidt, B.

Mueller, R.

T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).

Murr, L. E.

L. E. Murr, Electron and Ion Microscopy and Microanalysis (Marcel and Dekker, New York, 1991), Chap. 4, p. 191.

Nakayama, S.

Ohmi, S.

Oikawa, M.

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens and Stacked Planar Optics (Academic, Tokyo, 1984), Chap. 7, p. 129.

Parker, R. S.

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

Possner, T.

B. Messerschmidt, T. Possner, R. Goering, “Colorless gradient-index cylindrical lenses with high numerical apertures produced by silver-ion exchange,” Appl. Opt. 34, 7825–7830 (1995).
[CrossRef] [PubMed]

T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).

Rothhardt, M.

R. Goering, M. Rothhardt, “Application of the refracted near-field technique to multimode planar and channel waveguides in glass,” Opt. Commun. 7, 82–85 (1986).

Sakai, H.

Schreiter, G.

T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).

Simmons, V.

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

Suematsu, Y.

Y. Suematsu, K. Iga, S. Ito, “A light beam waveguide using hyperbolic type gas lenses,” IEEE Trans. Microwave Theory Tech. MTT-14, 657–665 (1966).
[CrossRef]

Yoneda, Y.

Appl. Opt.

Glastech. Ber.

T. Possner, G. Schreiter, R. Mueller, C. Kaps, H. Kahnt, “Special glass for integrated and microoptics,” Glastech. Ber. 64, 185–190 (1991).

C. Kaps, W. Fliegel, “Sodium/silver ion exchange between a non-bridging oxygen-free boroaluminosilicate glass and nitrate melts,” Glastech. Ber. 64, 199–204 (1991).

IEEE Trans. Microwave Theory Tech.

Y. Suematsu, K. Iga, S. Ito, “A light beam waveguide using hyperbolic type gas lenses,” IEEE Trans. Microwave Theory Tech. MTT-14, 657–665 (1966).
[CrossRef]

J. Non-Cryst. Solids

J. M. Inman, S. N. Houde-Walter, B. L. McIntyre, Z. M. Liao, R. S. Parker, V. Simmons, “Chemical structure and the mixed mobile ion effect in silver-for-sodium ion exchange in aluminosilicate glasses,” J. Non-Cryst. Solids 194, 85–92 (1996).
[CrossRef]

J. M. Inman, J. L. Bentley, S. N. Houde-Walter, “Modeling ion-exchanged glass photonics: the modified quasi-chemical diffusion coefficient,” J. Non-Cryst. Solids 191, 209–215 (1995).
[CrossRef]

J. Phys. Chem.

S. N. Houde-Walter, J. M. Inman, A. J. Dent, G. N. Greaves, “Sodium and silver environments and ion-exchange processes in silicate and aluminosilicate glasses,” J. Phys. Chem. 97, 9330–9336 (1993).
[CrossRef]

Opt. Commun.

R. Goering, M. Rothhardt, “Application of the refracted near-field technique to multimode planar and channel waveguides in glass,” Opt. Commun. 7, 82–85 (1986).

Opt. Eng.

T. Findakly, “Glass waveguides by ion exchange: a review,” Opt. Eng. 24, 244–252 (1985).

Other

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens and Stacked Planar Optics (Academic, Tokyo, 1984), Chap. 7, p. 129.

E. W. Marchand, Gradient-Index Optics (Academic, New York, 1978), Chap. 5, p. 61.

J. Crank, The Mathematics of Diffusion (Clarendon, Oxford, 1975), Chap. 10, p. 231.

L. E. Murr, Electron and Ion Microscopy and Microanalysis (Marcel and Dekker, New York, 1991), Chap. 4, p. 191.

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Figures (9)

Fig. 1
Fig. 1

Typical case of a diffusion profile in a slab (left) for using the Boltzmann-Matano method. The validity of the method can be extended up to the limit just before the diffusion flanks meet (right).

Fig. 2
Fig. 2

Diffusion coefficients D(χ), normalized by diffusion time t and half of the slab thickness d or radius a = d, respectively, which are required for reproducing parabolic or hyperbolic secant profiles in slabs and rods.

Fig. 3
Fig. 3

Comparison of a normalized hyperbolic secant profile, Eqs. (1), and a parabola as function of x. The example shown corresponds to an index change from nR = 1.5099 at the edge to n0 = 1.6591 at the center.

Fig. 4
Fig. 4

Deviation Δχ of a normalized concentration profile in a rod reproduced by diffusion coefficient (13) from the desired parabola as function of r; a is the radius of the rod.

Fig. 5
Fig. 5

MQC parameter fit performed on the required diffusion coefficient D(χ), Eq. (9), for a parabolic profile in a slab. The resulting MQC parameters are DB/d2 = 0.03692, DB/DA = 0.443, c = 10.55, χ0 = −0.45, and ɛint = −0.01653 eV (633 K).

Fig. 6
Fig. 6

Deviation Δχ(x) of the concentration profile in a slab reproduced by a MQC diffusion coefficient corresponding from desired parabola. Note that maximum deviation is less than 0.7%.

Fig. 7
Fig. 7

Normalized diffusion coefficients D(χ) experimentally obtained for the glasses characterized by the R ratio (solid curves). The corresponding MQC parameters (dashed curves) are presented in Table 4.

Fig. 8
Fig. 8

Normalized MQC diffusion coefficients D(χ) for the binary silicate (BS) and the aluminosilicate (AS) glasses discussed compared with that required for a parabolic profile in a slab.

Fig. 9
Fig. 9

One-dimensional index profile measured by the reflected near-field method13 in the R = 1 aluminosilicate glass (T = 673 K, t = 4 h). The profile deviates by 2% maximum of the obtained index change from a parabola (dashed curve).

Tables (4)

Tables Icon

Table 1 Analyzed Glass Compositions (mole %)

Tables Icon

Table 2 Ion Exchange Conditions

Tables Icon

Table 3 MQC Fit Parameters for Required Diffusion Coefficients for Parabolic and Sech Index Profiles, Resulting Maximal Error and Standard Deviation of the Reproduced Profilea

Tables Icon

Table 4 MQC Fit Parameters on Experimental Diffusion Coefficients in Investigated Glasses

Equations (16)

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n 2 ( x ) = n 0 2 sech 2 ( g x ) = n 0 2 [ 1 - ( g x ) 2 + h 4 ( g x ) 4 + h 6 ( g x ) 6 + ]             h 4 = 2 3 ,             h 6 = - 17 45 ,
n 2 ( x ) = n 0 2 [ 1 - ( g x ) 2 ] ,
n 2 ( x ) = n 0 2 [ 1 + ( g x ) 2 ] ,
n 2 ( x ) = a 1 C ( x ) + a 0 ,
χ t = x ( D ( χ ) χ x ) ,             χ = C C 0 ,
D ( χ ) = - 1 2 t d x d χ 0 χ x d χ .
χ ( x ) = ( 1 - x d ) 2 ,
x ( χ ) = d ( 1 - χ ) .
D ( χ ) = d 2 t ( χ 4 - χ 6 ) .
D ( χ ) = d 2 t 1 4 g ( χ - 1 g { χ b - ( b - χ ) arcsech [ 1 - ( χ / b ) ] } χ b - χ χ / b ) ,
g = g d = arcsech ( n R n 0 ) and b = n 0 2 n 0 2 - n R 2 .
D ( χ ) = f 1 χ - f 2 χ ,
D ( χ ) a 2 t 9 7 ( χ 6 - χ 8 )
D ( χ ) = { c 2 ( χ β ( χ = 1 ) + ( 1 - χ ) β ( χ = 0 ) β - 1 ) + 1 } × [ D B 1 - χ α ] , β = { 1 - 4 ( χ - χ 0 ) [ 1 - ( χ - χ 0 ) ] × [ 1 - exp ( 2 ɛ int / k T ) ] } 1 / 2 , α = 1 - D B D A ,
ɛ int = ɛ A B - ɛ A A + ɛ B B 2 ,
R = Al 2 O 3 mole % Na 2 O mole % .

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