Abstract

A planar microlens with N.A. = 0.54 has been obtained (Δn = 0.27) by stacking two lenses. The maximally obtainable N.A. is then expected to be ~0.7 but it is limited by aberration, which is discussed theoretically using a Luneburg lens model.

© 1984 Optical Society of America

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References

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  1. M. Oikawa, K. Iga, T. Sanada, Jpn. J. Appl. Phys. 20, L51 (1981).
    [CrossRef]
  2. M. Oikawa, K. Iga, Appl. Opt. 21, 1052 (1982).
    [CrossRef] [PubMed]
  3. M. Oikawa, K. Iga, S. Misawa, Y. Kokubun, Appl. Opt. 22, 441 (1983).
    [CrossRef] [PubMed]
  4. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1966).
  5. Y. Kokubun, K. Iga, Appl. Opt. 21, 1030 (1982).
    [CrossRef] [PubMed]
  6. S. Misawa, M. Oikawa, K. Iga, Jpn. J. Appl. Phys. 21, L589 (1982).
    [CrossRef]

1983 (1)

1982 (3)

1981 (1)

M. Oikawa, K. Iga, T. Sanada, Jpn. J. Appl. Phys. 20, L51 (1981).
[CrossRef]

Iga, K.

Kokubun, Y.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1966).

Misawa, S.

M. Oikawa, K. Iga, S. Misawa, Y. Kokubun, Appl. Opt. 22, 441 (1983).
[CrossRef] [PubMed]

S. Misawa, M. Oikawa, K. Iga, Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

Oikawa, M.

M. Oikawa, K. Iga, S. Misawa, Y. Kokubun, Appl. Opt. 22, 441 (1983).
[CrossRef] [PubMed]

S. Misawa, M. Oikawa, K. Iga, Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

M. Oikawa, K. Iga, Appl. Opt. 21, 1052 (1982).
[CrossRef] [PubMed]

M. Oikawa, K. Iga, T. Sanada, Jpn. J. Appl. Phys. 20, L51 (1981).
[CrossRef]

Sanada, T.

M. Oikawa, K. Iga, T. Sanada, Jpn. J. Appl. Phys. 20, L51 (1981).
[CrossRef]

Appl. Opt. (3)

Jpn. J. Appl. Phys. (2)

S. Misawa, M. Oikawa, K. Iga, Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

M. Oikawa, K. Iga, T. Sanada, Jpn. J. Appl. Phys. 20, L51 (1981).
[CrossRef]

Other (1)

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1966).

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Figures (3)

Fig. 1
Fig. 1

(a) Hemispherical microlens and (b) stacked spherical microlens.

Fig. 2
Fig. 2

Maximally available N.A. when the index difference is known. The solid line is a single lens and the broken line is a stacked lens. The ao-23-11-1784-i001 indicates the effective N.A. of a single lens obtained by measurement, and ao-23-11-1784-i002 shows the N.A. for a stacked lens.

Fig. 3
Fig. 3

Comparison between the index profiles of the present lens and that of the hemispherical Luneburg lens. Two solid lines express the profile of the fabricated lens which was measured by the fringe pattern in this figure obtained by the Mach-Zehnder interference method. 5 The broken line corresponds to a Luneburg lens.

Equations (12)

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N . A . = n 2 sin θ ,
n ^ = exp [ ω ( ρ , z ¯ 0 ) + ω ( ρ , z ¯ 1 ) ] ,
r ¯ = ρ exp [ - ω ( ρ , z ¯ 0 ) - ω ( ρ , z ¯ 1 ) ] ,
ω ( ρ , z ¯ ) = 1 π ρ 1 sin - 1 ( t / z ¯ ) t 2 - ρ 2 d t ,
n ^ = exp [ 2 ω ( ρ , f ¯ ) ] ,
r ¯ = ρ exp [ - 2 ω ( ρ , f ¯ ) ] .
N . A . = n 2 a / f = n 2 / f ¯ ,
Δ n = n 2 [ n ^ ( 0 ) - 1 ] ,             n ^ ( 0 ) = exp [ 2 π 0 1 sin - 1 ( t / f ¯ ) t d t ] ,
n ^ = exp [ ω ( ρ , f ¯ ) ] ,
r ¯ = ρ exp [ - ω ( ρ , f ¯ ) ] .
N . A . = n 2 a / f = n 2 / f ¯ ,
Δ n = n 2 [ n ^ ( 0 ) - 1 ] ,             n ^ ( 0 ) = exp [ 1 π 0 1 sin - 1 ( t / f ¯ ) t d t ] .

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