Abstract

Allowable deviations in index profiles, dopant distributions, and concentration dependence of diffusion coefficients are determined for collimating microlenses. Examples of high (and low) numerical aperture lenses are given for silver/sodium (and lithium/sodium) ion exchanges. Using the full lens aperture, one can ensure diffraction-limited performance only when the index is measured to within 1.0 × 10-4 (4.3 × 10-5) of the optimum values. Fabrication tolerances for diffraction-limited performance over 80% of the numerical aperture are expressed in terms of the concentration-dependent diffusion coefficient, which typically must be held to within ±4.7% (±11.5%) of ideal values.

© 1997 Optical Society of America

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References

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  1. K. Fujii, S. Ogi, N. Akazawa, “Gradient-index rod lens with a high acceptance angle for color use by Na+ for Li+ exchange,” Appl. Opt. 33, 8087–8093 (1994).
    [Crossref] [PubMed]
  2. B. Messerschmidt, T. Possner, R. Goering, “Colorless gradient-index cylindrical lenses with high numerical aperture pruduced by silver-ion exchange,” Appl. Opt. 34, 7825–7830 (1995).
    [Crossref] [PubMed]
  3. S. Ohmi, H. Sakai, Y. Asahara, S. Nakayama, Y. Yoneda, T. Izumitani, “Gradient-index rod lens made by double ion-exchange process,” Appl. Opt. 27, 496–499 (1988).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  6. R. H. Doremus, Glass Science (Wiley, New York, 1973), Chap. 9.
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    [Crossref] [PubMed]
  8. J. Crank, The Mathematics of Diffusion (Clarendon, Oxford, UK, 1975), Chap. 10, p. 231.
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    [Crossref] [PubMed]
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    [Crossref]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 9.
  13. R. Goering, M. Rothhardt, “Application of the refracted near-field technique to multimode planar waveguides and channel waveguides in glass,” Opt. Commun. 7, 82–85 (1986).
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    [Crossref] [PubMed]
  15. L. E. Murr, Electron and Ion Microscopy and Microanalysis (Marcel Dekker, New York, 1991), Chap. 4, p. 191.
  16. B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
    [Crossref]

1997 (1)

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
[Crossref]

1996 (1)

1995 (1)

1994 (1)

1992 (1)

1991 (1)

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 (2)

1986 (1)

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

1983 (1)

1970 (1)

Akazawa, N.

Andre, R. R.

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
[Crossref]

Araujo, R.

Asahara, Y.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 9.

Crank, J.

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

Doremus, R. H.

R. H. Doremus, Glass Science (Wiley, New York, 1973), Chap. 9.

Fantone, S. D.

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).

Fujii, K.

Goering, R.

B. Messerschmidt, T. Possner, R. Goering, “Colorless gradient-index cylindrical lenses with high numerical aperture pruduced 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 waveguides and channel waveguides in glass,” Opt. Commun. 7, 82–85 (1986).

Houde-Walter, S. N.

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
[Crossref]

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, “Desired concentration-dependent ion exchange for micro-optic lenses,” Appl. Opt. 35, 5670–5676 (1996).
[Crossref] [PubMed]

S. N. Houde-Walter, D. T. Moore, “Real-time index profile measurement during GRIN glass fabrication,” Appl. Opt. 27, 508–515 (1988).
[Crossref] [PubMed]

B. Messerschmidt, C. H. Hsieh, B. L. McIntyre, S. N. Houde-Walter, “Ionic mobility in an ion exchanged silver-sodium boroaluminosilicate glass for micro-optics,” to be published in J. Non-Cryst. Solids (1997).

Hsieh, C. H.

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
[Crossref]

B. Messerschmidt, C. H. Hsieh, B. L. McIntyre, S. N. Houde-Walter, “Ionic mobility in an ion exchanged silver-sodium boroaluminosilicate glass for micro-optics,” to be published in J. Non-Cryst. Solids (1997).

Iga, K.

Izumitani, T.

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).

McIntyre, B. L.

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
[Crossref]

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, “Desired concentration-dependent ion exchange for micro-optic lenses,” Appl. Opt. 35, 5670–5676 (1996).
[Crossref] [PubMed]

B. Messerschmidt, C. H. Hsieh, B. L. McIntyre, S. N. Houde-Walter, “Ionic mobility in an ion exchanged silver-sodium boroaluminosilicate glass for micro-optics,” to be published in J. Non-Cryst. Solids (1997).

Messerschmidt, B.

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
[Crossref]

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, “Desired concentration-dependent ion exchange for micro-optic lenses,” Appl. Opt. 35, 5670–5676 (1996).
[Crossref] [PubMed]

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

B. Messerschmidt, C. H. Hsieh, B. L. McIntyre, S. N. Houde-Walter, “Ionic mobility in an ion exchanged silver-sodium boroaluminosilicate glass for micro-optics,” to be published in J. Non-Cryst. Solids (1997).

Moore, D. T.

Murr, L. E.

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

Nakayama, S.

Ogi, S.

Ohmi, S.

Possner, T.

Rothhardt, M.

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

Sakai, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 9.

Yoneda, Y.

Appl. Opt. (8)

Glastech. Ber. (1)

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).

J. Opt. Mater. (1)

B. Messerschmidt, B. L. McIntyre, S. N. Houde-Walter, R. R. Andre, C. H. Hsieh, “Temperature dependence of silver-sodium interdiffusion in micro-optic glasses,” J. Opt. Mater. 7, 165–171 (1997).
[Crossref]

Opt. Commun. (1)

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

Other (5)

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

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 9.

B. Messerschmidt, C. H. Hsieh, B. L. McIntyre, S. N. Houde-Walter, “Ionic mobility in an ion exchanged silver-sodium boroaluminosilicate glass for micro-optics,” to be published in J. Non-Cryst. Solids (1997).

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

R. H. Doremus, Glass Science (Wiley, New York, 1973), Chap. 9.

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Figures (10)

Fig. 1
Fig. 1

Optical configuration for the calculation of the OPD. A parallel entrance ray at yi is focused near the exit surface of the lens with a quarter-pitch length; M, marginal focus; B, balanced focus; P, paraxial focus.

Fig. 2
Fig. 2

Tolerance range of the higher-order profile parameters h4 and h6, resulting in diffraction-limited performance in 100% of the N.A. of a 1-mm rod or slab lens (quarter-pitch length): top, high-N.A. example (Ag exchange); bottom, low-N.A. example (Li exchange).

Fig. 3
Fig. 3

Possible error range of the diffusion coefficient D(χ), ensuring diffraction-limited performance of a 1-mm slab lens with quarter-pitch length: Ag, high-N.A. example; Li, low-N.A. example.

Fig. 4
Fig. 4

Diffraction-limited N.A. of the high-N.A. 1-mm slab lens (Ag exchange) as a function of the higher-order profile parameters h4 and h6: top, small deviation range; bottom, large deviation range.

Fig. 5
Fig. 5

Shape variation of the normalized concentration profile χ(y) used for all simulations: ideal Ag, free of spherical aberration; S, small deviation range; L, large deviation range.

Fig. 6
Fig. 6

Diffraction-limited N.A. of the low-N.A. 1-mm slab lens (Li-exchange) as a function of the higher-order profile parameters h4 and h6: top, small deviation range; bottom, large deviation range.

Fig. 7
Fig. 7

Possible deviation range of the normalized index or concentration profile c(y) of a 1-mm slab lens for diffraction-limited performance in 80% of the N.A. (0 ≤ χ ≤ 0.64): Li, low-N.A. example; Ag, high-N.A. example.

Fig. 8
Fig. 8

Possible deviation range of the concentration-dependent diffusion coefficient D(χ) (times the diffusion time t to obtain a 1-mm-thick slab lens) for diffraction-limited performance in 80% of the maximum N.A. (χ = 0.64): top, Ag exchange; bottom, Li exchange.

Fig. 9
Fig. 9

Diffraction-limited N.A. of the high-N.A. 1-mm rod lens (Ag exchange) as a function of the higher-order profile parameters h4 and h6: top, small deviation range; bottom, large deviation range.

Fig. 10
Fig. 10

Diffraction-limited N.A. of the low-N.A. 1-mm rod lens (Li exchange) as a function of the higher-order profile parameters h4 and h6: top, small deviation range; bottom, large deviation range.

Tables (3)

Tables Icon

Table 1 Center and Marginal Index n0 and nm of Low-N.A and High-N.A. Gradient-Index Profile Examples

Tables Icon

Table 2 Standard Deviation of Square of Index n, of the Index, of Normalized Concentration χ, and of Concentration-Dependent Diffusion Coefficient D Normalized by Its Maximum Dmaxa

Tables Icon

Table 3 Rule-of-Thumb Fabrication Tolerance on Diffusion Coefficient at 64% of Maximum Dopant Concentration Relative to Its Maximum Valuea

Equations (12)

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

Dχ=-12tdydχ0χydχ,
n2y=a-bχy,
n2y=n021-gy2+h4gy4+h6gy6+,
n2y=n02 sech2gy,
N.A.=n02-nm21/2.
zl=π2g,
hP=1n0g singzl.
OPD=WP-WR.
σ2=OPD2
σ45.2 nm.
σ2=k=1mOPDk2m,
σ2=k=1mOPDk2yikk=1nyik.

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