Abstract

We report the fabrication of planar microlenses with numerical apertures (N.A.’s) of 0.2 by field-assisted Ag–Na ion exchange in glass. To measure the N.A. of microlenses, different definitions can be used. We discuss the issue of measuring the N.A. and suggest an additional definition based on diffraction-limited performance. According to a simple model, the N.A. of a spherical lens is limited by the maximum index difference. Owing to this model, the N.A. for Ag–Na ion exchange is limited to a value of ~0.1. From measurements of microlenses, fabricated by field-assisted ion exchange, we obtained N.A.’s as high as 0.2, providing for diffraction-limited performance within the whole aperture.

© 1996 Optical Society of America

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References

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  1. B. Messerschmidt, T. Possner, R. Göhring, “Colorless gradient-index cylindrical lenses with high numerical apertures produced by silver-ion exchange,” Appl. Opt. 34, 7825–7830 (1995).
    [CrossRef] [PubMed]
  2. K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and multiple Fourier transformation using planar microlens array,” Appl. Opt. 25, 4064–4070 (1990).
    [CrossRef]
  3. M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
    [CrossRef]
  4. R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).
  5. M. Oikawa, K. Iga, “Distributed-index planar microlens,” Appl. Opt. 21, 1052–1056 (1982).
    [CrossRef] [PubMed]
  6. J. Bähr, K.-H. Brenner, S. Sinzinger, W. Singer, T. Spick, 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. 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]
  8. W. Singer, M. Testorf, K. H. Brenner, “Gradient-index microlenses: numerical investigation of different spherical index profiles with the wave propagation method,” Appl. Opt. 34, 2165–2171 (1995).
    [CrossRef] [PubMed]
  9. J. R. Flores, J. Sochnacki, “Design of gradient-index microlenses for stacked planar optics,” Appl. Opt. 33, 3409–3414 (1994).
    [CrossRef] [PubMed]
  10. D. Intani, T. Baba, K. Iga, “Planar microlens relay optics utilizing lateral focusing,” Appl. Opt. 31, 5255–5258 (1992).
    [CrossRef] [PubMed]
  11. G. Chartier, P. Collier, A. Guez, P. Jaussaud, Y. Won, “Graded-index surface or buried waveguides by ion exchange in glass,” Appl. Opt. 19, 1092–1095 (1980).
    [CrossRef] [PubMed]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  13. 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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).
  14. J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.
  15. M. Oikawa, H. Nemoto, K. Hamanaka, E. Okuda, “High numerical aperture planar microlens with swelled structure,” Appl. Opt. 29, 4077–4080 (1990).
    [CrossRef] [PubMed]

1995

1994

1992

1990

M. Oikawa, H. Nemoto, K. Hamanaka, E. Okuda, “High numerical aperture planar microlens with swelled structure,” Appl. Opt. 29, 4077–4080 (1990).
[CrossRef] [PubMed]

K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and multiple Fourier transformation using planar microlens array,” Appl. Opt. 25, 4064–4070 (1990).
[CrossRef]

M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
[CrossRef]

1982

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]

M. Oikawa, K. Iga, “Distributed-index planar microlens,” Appl. Opt. 21, 1052–1056 (1982).
[CrossRef] [PubMed]

1980

Baba, T.

Bähr, J.

J. Bähr, K.-H. Brenner, S. Sinzinger, W. Singer, T. Spick, 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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

Born, M.

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

Brenner, K. H.

W. Singer, M. Testorf, K. H. Brenner, “Gradient-index microlenses: numerical investigation of different spherical index profiles with the wave propagation method,” Appl. Opt. 34, 2165–2171 (1995).
[CrossRef] [PubMed]

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

Brenner, K.-H.

Chartier, G.

Collier, P.

Dändliker, R.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Eisner, M.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Flores, J. R.

Göhring, R.

Guez, A.

Hamanaka, K.

M. Oikawa, H. Nemoto, K. Hamanaka, E. Okuda, “High numerical aperture planar microlens with swelled structure,” Appl. Opt. 29, 4077–4080 (1990).
[CrossRef] [PubMed]

K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and multiple Fourier transformation using planar microlens array,” Appl. Opt. 25, 4064–4070 (1990).
[CrossRef]

Haselbeck, S.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Herzig, H. P.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Hutley, M. C.

M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
[CrossRef]

Iga, K.

Intani, D.

Jaussaud, P.

Kishimoto, T.

K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and multiple Fourier transformation using planar microlens array,” Appl. Opt. 25, 4064–4070 (1990).
[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]

Messerschmidt, B.

Moisel, J.

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

Nemoto, H.

K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and multiple Fourier transformation using planar microlens array,” Appl. Opt. 25, 4064–4070 (1990).
[CrossRef]

M. Oikawa, H. Nemoto, K. Hamanaka, E. Okuda, “High numerical aperture planar microlens with swelled structure,” Appl. Opt. 29, 4077–4080 (1990).
[CrossRef] [PubMed]

Nussbaum, Ph.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Oikawa, M.

Okuda, E.

K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and multiple Fourier transformation using planar microlens array,” Appl. Opt. 25, 4064–4070 (1990).
[CrossRef]

M. Oikawa, H. Nemoto, K. Hamanaka, E. Okuda, “High numerical aperture planar microlens with swelled structure,” Appl. Opt. 29, 4077–4080 (1990).
[CrossRef] [PubMed]

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]

Possner, T.

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]

Schwider, J.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Singer, W.

W. Singer, M. Testorf, K. H. Brenner, “Gradient-index microlenses: numerical investigation of different spherical index profiles with the wave propagation method,” Appl. Opt. 34, 2165–2171 (1995).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, S. Sinzinger, W. Singer, T. Spick, 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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

Sinzinger, S.

J. Bähr, K.-H. Brenner, S. Sinzinger, W. Singer, T. Spick, 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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

Sochnacki, J.

Spick, T.

J. Bähr, K.-H. Brenner, S. Sinzinger, W. Singer, T. Spick, 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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

Testorf, M.

W. Singer, M. Testorf, K. H. Brenner, “Gradient-index microlenses: numerical investigation of different spherical index profiles with the wave propagation method,” Appl. Opt. 34, 2165–2171 (1995).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, S. Sinzinger, W. Singer, T. Spick, 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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

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]

Völkel, R.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Weible, K. J.

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

Wolf, E.

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

Won, Y.

Appl. Opt.

G. Chartier, P. Collier, A. Guez, P. Jaussaud, Y. Won, “Graded-index surface or buried waveguides by ion exchange in glass,” Appl. Opt. 19, 1092–1095 (1980).
[CrossRef] [PubMed]

M. Oikawa, K. Iga, “Distributed-index planar microlens,” Appl. Opt. 21, 1052–1056 (1982).
[CrossRef] [PubMed]

M. Oikawa, H. Nemoto, K. Hamanaka, E. Okuda, “High numerical aperture planar microlens with swelled structure,” Appl. Opt. 29, 4077–4080 (1990).
[CrossRef] [PubMed]

D. Intani, T. Baba, K. Iga, “Planar microlens relay optics utilizing lateral focusing,” Appl. Opt. 31, 5255–5258 (1992).
[CrossRef] [PubMed]

J. R. Flores, J. Sochnacki, “Design of gradient-index microlenses for stacked planar optics,” Appl. Opt. 33, 3409–3414 (1994).
[CrossRef] [PubMed]

J. Bähr, K.-H. Brenner, S. Sinzinger, W. Singer, T. Spick, 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]

W. Singer, M. Testorf, K. H. Brenner, “Gradient-index microlenses: numerical investigation of different spherical index profiles with the wave propagation method,” Appl. Opt. 34, 2165–2171 (1995).
[CrossRef] [PubMed]

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

K. Hamanaka, H. Nemoto, M. Oikawa, E. Okuda, T. Kishimoto, “Multiple imaging and multiple Fourier transformation using planar microlens array,” Appl. Opt. 25, 4064–4070 (1990).
[CrossRef]

IEEE J. Quantum Electron.

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]

J. Mod. Opt.

M. C. Hutley, “Optical techniques for the generation of microlens arrays,” J. Mod. Opt. 37, 253–265 (1990).
[CrossRef]

Other

R. Völkel, Ph. Nussbaum, K. J. Weible, H. P. Herzig, R. Dändliker, S. Haselbeck, M. Eisner, J. Schwider, “Fabrication of nonconventional microlens arrays,” in Proceedings of European Optical Society, Topical Meeting on Microlens Arrays (European Optical Society, 1995).

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

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 glass types,” in Topical Meeting on Optical Computing, Z. I. Alferov, J. W. Goodman, A. L. Mikaelian, eds., Proc. SPIE1806, 234–242 (1992).

J. Bähr, K. H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, M. Testorf, “Modification of the imaging properties of ion-exchange microlenses by mask shaping,” in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems, (European Optical Society, 1992), p. 187.

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Figures (10)

Fig. 1
Fig. 1

Longitudinal index distribution for field-assisted and thermal exchange.

Fig. 2
Fig. 2

Microscope image of a cut through the plane of the lens axis.

Fig. 3
Fig. 3

Measurement of a planar gradient-index microlens in a transmission interferometer.

Fig. 4
Fig. 4

Radial index distribution for different postheating times at 306 °C.

Fig. 5
Fig. 5

Interferogram of a microlens optimized by postheating.

Fig. 6
Fig. 6

Phase profile of a microlens at different postheating times (continuous curve) at 306 °C, best-fit hyperbola according to the Maréchal criterion (heavy curve), fit hyperbola in the region of wave-front deviations beyond the Maréchal criterion (dashed curve): (a) 0 h, (b) 28 h, (c) 52 h.

Fig. 7
Fig. 7

Lateral aberrations for different postheating times at 306 °C.

Fig. 8
Fig. 8

Working distance for different postheating times.

Fig. 9
Fig. 9

Lens diameter for different postheating times.

Fig. 10
Fig. 10

N.A.DL for different postheating times.

Equations (12)

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

φ ( x , y ) = k 0 ( R 2 - x 2 - y 2 ) 1 / 2 Δ n .
φ ( x , y ) k 0 x 2 + y 2 2 f p ,
f p = R Δ n
N . A . = n 0 sin ( ϑ max ) = n 0 R ( R 2 + f p 2 ) 1 / 2 = n 0 ( 1 + 1 Δ n 2 ) 1 / 2 n 0 Δ n .
exp [ i φ ( x , y , z = 0 ) ] = exp [ 2 π i λ ( x 2 + y 2 + f w 2 ) 1 / 2 ] .
W = φ lens - k 0 ( x 2 + y 2 + f w 2 ) 1 / 2 .
N . A . DL = n 0 r max ( r max 2 + f w 2 ) 1 / 2 .
φ ( x , y ) = k 0 ( r lens 2 + f w 2 ) 1 / 2 - k 0 ( x 2 + y 2 + f w 2 ) 1 / 2 .
φ ( 0 ) = k 0 ( r lens 2 + f w 2 ) 1 / 2 - k 0 f w = k 0 0 d m [ n ( 0 , z ) - n 0 ] d z ,
0 d m [ n ( 0 , z ) - n 0 ] d z = α Δ n d m ,
N . A . max = r lens α Δ n r lens + f w .
N . A . max = 2 α Δ n 1 + ( α Δ n ) 2 .

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