Abstract

Ion-exchange microlenses are available with different gradient-index profiles. We investigate the dependence of the imaging properties on the steepness of the index profiles. Therefore we model the index distribution by the Fermi function as radial distribution with spherical symmetry. The results are compared to index profiles according to the Doremus model.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Shimada, O. Ohguchi, R. Sawada, “Efficient coupling device between a laser diode integrated with a microlens and an optical fiber,” presented at the European Conference on Integrated Optics 1993, Neuchâtel, Switzerland (1993), pp. 11-6–11-7.
  2. K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and Fourier transformation using planar microlens arrays,” Appl. Opt. 29, 4064–4070 (1990).
    [CrossRef] [PubMed]
  3. A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
    [CrossRef]
  4. A. Tervonen, “A general model for the fabrication processes of channel waveguides by ion exchange,” J. Appl. Phys. 67, 2746–2752 (1990).
    [CrossRef]
  5. J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).
  6. J. Bähr, K.-H. Brenner, T. Spick, S. Sinzinger, M. Testorf, “Index-distributed planar microlenses for three-dimensional micro-optics fabricated by silver–sodium ion exchange in BGG35 substrates,” Appl. Opt. 33, 5919–5924 (1994).
    [CrossRef] [PubMed]
  7. K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
    [CrossRef] [PubMed]
  8. R. H. Doremus, “Exchange and diffusion of ions in glass,” J. Phys. Chem. 68, 2212–2218 (1964).
    [CrossRef]
  9. J. Albert, G. L. Yip, “Refractive-index profiles of planar waveguides made by ion-exchange in glass,” Appl. Opt. 24, 3692–3693 (1985).
    [CrossRef] [PubMed]

1994

1993

1990

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

K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and Fourier transformation using planar microlens arrays,” Appl. Opt. 29, 4064–4070 (1990).
[CrossRef] [PubMed]

1985

1969

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

1964

R. H. Doremus, “Exchange and diffusion of ions in glass,” J. Phys. Chem. 68, 2212–2218 (1964).
[CrossRef]

Albert, J.

Bähr, J.

J. Bähr, K.-H. Brenner, T. Spick, S. Sinzinger, M. Testorf, “Index-distributed planar microlenses for three-dimensional micro-optics fabricated by silver–sodium ion exchange in BGG35 substrates,” Appl. Opt. 33, 5919–5924 (1994).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

Brenner, K.-H.

J. Bähr, K.-H. Brenner, T. Spick, S. Sinzinger, M. Testorf, “Index-distributed planar microlenses for three-dimensional micro-optics fabricated by silver–sodium ion exchange in BGG35 substrates,” Appl. Opt. 33, 5919–5924 (1994).
[CrossRef] [PubMed]

K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

Doremus, R. H.

R. H. Doremus, “Exchange and diffusion of ions in glass,” J. Phys. Chem. 68, 2212–2218 (1964).
[CrossRef]

French, W. G.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

Hamanaka, K.

Kishimoto, T.

Moisel, J.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

Nemoto, H.

Ohguchi, O.

J. Shimada, O. Ohguchi, R. Sawada, “Efficient coupling device between a laser diode integrated with a microlens and an optical fiber,” presented at the European Conference on Integrated Optics 1993, Neuchâtel, Switzerland (1993), pp. 11-6–11-7.

Oikawa, M.

Okuda, E.

Pearson, A. D.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

Rawson, E. G.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

Sawada, R.

J. Shimada, O. Ohguchi, R. Sawada, “Efficient coupling device between a laser diode integrated with a microlens and an optical fiber,” presented at the European Conference on Integrated Optics 1993, Neuchâtel, Switzerland (1993), pp. 11-6–11-7.

Shimada, J.

J. Shimada, O. Ohguchi, R. Sawada, “Efficient coupling device between a laser diode integrated with a microlens and an optical fiber,” presented at the European Conference on Integrated Optics 1993, Neuchâtel, Switzerland (1993), pp. 11-6–11-7.

Singer, W.

K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

Sinzinger, S.

J. Bähr, K.-H. Brenner, T. Spick, S. Sinzinger, M. Testorf, “Index-distributed planar microlenses for three-dimensional micro-optics fabricated by silver–sodium ion exchange in BGG35 substrates,” Appl. Opt. 33, 5919–5924 (1994).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

Spick, T.

J. Bähr, K.-H. Brenner, T. Spick, S. Sinzinger, M. Testorf, “Index-distributed planar microlenses for three-dimensional micro-optics fabricated by silver–sodium ion exchange in BGG35 substrates,” Appl. Opt. 33, 5919–5924 (1994).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

Tervonen, A.

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

Testorf, M.

J. Bähr, K.-H. Brenner, T. Spick, S. Sinzinger, M. Testorf, “Index-distributed planar microlenses for three-dimensional micro-optics fabricated by silver–sodium ion exchange in BGG35 substrates,” Appl. Opt. 33, 5919–5924 (1994).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

Yip, G. L.

Appl. Opt.

Appl. Phys. Lett.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

J. Appl. Phys.

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

J. Phys. Chem.

R. H. Doremus, “Exchange and diffusion of ions in glass,” J. Phys. Chem. 68, 2212–2218 (1964).
[CrossRef]

Other

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Diffusion elements in glass: comparison and optimization of the diffusion response in different substrates,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1806, 234–242 (1993).

J. Shimada, O. Ohguchi, R. Sawada, “Efficient coupling device between a laser diode integrated with a microlens and an optical fiber,” presented at the European Conference on Integrated Optics 1993, Neuchâtel, Switzerland (1993), pp. 11-6–11-7.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Model index distribution: the Fermi function for different decay parameters.

Fig. 2
Fig. 2

Propagation of a single plane wave parallel to the optical axis through index distributions. A comparison of intensity cuts through the xz plane behind index distributions with a equal to (a) 10 μm, (b) 20 μm, (c) 30 μm, (d) 60 μm.

Fig. 3
Fig. 3

Same index distributions as in Fig. 2 but now with a tilted plane wave having an incident angle of 3°.

Fig. 4
Fig. 4

Propagation of several plane waves through an index distribution with a = 30 μm.

Fig. 5
Fig. 5

Phase difference relative to an ideal sphere for different decay parameters.

Fig. 6
Fig. 6

Numerical aperture, geometrical aperture, and focal length for different decay parameters.

Fig. 7
Fig. 7

MTF for different decay parameters a.

Fig. 8
Fig. 8

Field curvature for different decay parameters.

Fig. 9
Fig. 9

Comparison of a Fermi profile with a = 30 μm and a simulated diffusion response that was obtained by a least-squares fit with the Doremus model.

Fig. 10
Fig. 10

Relation between nonlinearity α and decay parameter a.

Equations (9)

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

d 2 d z 2 E ˜ ( v x , z ) + k 2 [ n 2 ¯ ( v x , z ) - ( 2 π v x ) 2 ] E ˜ ( v x , z ) = 0.
E ˜ ( v x , z + δ z ) = E ˜ ( v x , z ) exp [ i 2 π λ δ z ( n 2 - λ 2 v x 2 ) 1 / 2 ] .
E ( x , z + δ z ) = - E ˜ ( v x , z ) exp ( 2 π i v x x ) × exp { - i 2 π λ δ z [ n 2 ( x , z ) - λ 2 v x 2 ] 1 / 2 } d v x .
δ z ( λ / δ n ) ,
n ( r ) = n 0 + δ n 1 + exp [ ( r - r 0 ) / a ] .
r - r 0 a = r 0 a ( r r 0 - 1 ) .
f = ( δ x / tan φ ) .
t C ( r , t ) = [ D ( C ) C ( r , t ) ] ,
D ( C ) = D 0 1 - α C .

Metrics