Abstract

A model is proposed for spherical aberration in a single radial gradient-rod lens. The aberration on a near one-quarter pitch lens is evaluated by employing the model, and the refractive-index coefficients up to sixth order estimated by ray trace analysis.

© 1984 Optical Society of America

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References

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  1. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
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  2. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
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  4. S. D. Fantone, Appl. Opt. 22, 1815 (1983); T. Sakamoto, Jpn. J. Opt. 12, 38 (1983), in Japanese.
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  5. K. B. Paxton, W. Streifer, Appi. Opt. 10, 2090 (1971).
    [CrossRef]
  6. A. Gupta, K. Thyagarajan, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
    [CrossRef]
  7. S. D. Fantone, J. Opt. Soc. Am. 73, 1149, 1162 (1983).
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  8. T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
    [CrossRef]
  9. Y. Kokubun, K. Iga, Appl. Opt. 21, 1030 (1982).
    [CrossRef] [PubMed]
  10. T. Sakamoto, Opt. Eng. 22, 359 (1983).
    [CrossRef]
  11. T. Sakamoto, Appl. Opt. 22, 3064 (1983).
    [CrossRef] [PubMed]
  12. E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
    [CrossRef]
  13. T. W. Cline, R. B. Jander, Appl. Opt. 21, 1035 (1982).
    [CrossRef] [PubMed]
  14. N. Yamamoto, K. Iga, Appl. Opt. 19, 1101 (1980).
    [CrossRef] [PubMed]
  15. M. Toyama, N. Nishi, Jpn. J. Opt. 11, 546 (1982), in Japanese.

1983 (4)

1982 (3)

1980 (1)

1977 (1)

1976 (1)

1973 (2)

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

1971 (2)

K. B. Paxton, W. Streifer, Appi. Opt. 10, 2090 (1971).
[CrossRef]

D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
[CrossRef]

1970 (1)

Cline, T. W.

Fantone, S. D.

Furukawa, M.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Ghatak, A. K.

Goyal, I. C.

Gupta, A.

Iga, K.

Jander, R. B.

Kitano, I.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Kitano, T.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Kokubun, Y.

Matsumura, H.

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Moore, D. T.

Murray, R. G.

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Nishi, N.

M. Toyama, N. Nishi, Jpn. J. Opt. 11, 546 (1982), in Japanese.

Paxton, K. B.

K. B. Paxton, W. Streifer, Appi. Opt. 10, 2090 (1971).
[CrossRef]

Rawson, E. G.

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Sakamoto, T.

Sands, P. J.

Streifer, W.

K. B. Paxton, W. Streifer, Appi. Opt. 10, 2090 (1971).
[CrossRef]

Thyagarajan, K.

Toyama, M.

M. Toyama, N. Nishi, Jpn. J. Opt. 11, 546 (1982), in Japanese.

Yamamoto, N.

Appi. Opt. (1)

K. B. Paxton, W. Streifer, Appi. Opt. 10, 2090 (1971).
[CrossRef]

Appl. Opt. (5)

IEEE J. Quantum Electron. (2)

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

J. Opt. Soc. Am. (5)

Jpn. J. Opt. (1)

M. Toyama, N. Nishi, Jpn. J. Opt. 11, 546 (1982), in Japanese.

Opt. Eng. (1)

T. Sakamoto, Opt. Eng. 22, 359 (1983).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

GRIN lens parameters. The radial index distribution n(r) is assumed to obey n 2 ( r ) = n 0 2 [ 1 - ( g r ) 2 + h 4 ( g r ) 4 + h 6 ( g r ) 6 + ], where r is the distance from the center axis z, n0 is the index on the center axis, and g, h4, and h6 are the second-, fourth-, and sixth-order coefficients, respectively.

Fig. 2
Fig. 2

Accuracy of the formula. The abscissa denotes the normalized lens length gZ. The ordinate in the upper part is the normalized maximum transverse aberration gsT within 0 ≤ S gxi ≤ 0.3. The error shown in the lower part of the figure is the maximum at a given gZ.

Fig. 3
Fig. 3

Axial deviations observed on a SLS 2.0-mm lens.11 The test lens parameters are n0 = 1.552 and Z = 7.0 mm (near one-quarter pitch). The distance l is set to 450 mm, and the light source is a He–Ne laser (6328 Å). The solid line is the least-squares solution with the fifth-order axial deviation polynomial whose root mean deviation is 0.63 mm.

Fig. 4
Fig. 4

Spherical aberration curves of the test lens. The paraxial focal length lf is 28 μm.

Fig. 5
Fig. 5

Reduction of the aberration.

Fig. 6
Fig. 6

Spot diagrams. The input plane was meshed into ~3000, and parallel incident rays were used.

Fig. 7
Fig. 7

MTF; the parameter is the defocus Δf.

Equations (8)

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x l = a ( g ) x i + b ( g , h 4 ) x i 3 + c ( g , h 4 , h 6 ) x i 5 + 0 ( 7 ) ,
a ( g ) = cos g Z - g l n 0 sin g Z , b ( g , h 4 ) = g 2 ( H 3 + g l n 0 K 3 ) , c ( g , h 4 , h 6 ) = g 4 ( H 5 + g l n 0 K 5 ) ,
g l f n 0 = cot g Z .
g s T = S 3 ( g x i ) 3 + S 5 ( g x i ) 5 + 0 ( 7 ) ,
S 3 = 3 4 ( h 4 - 2 3 ) g Z csc g Z + [ 3 4 ( h 4 - 2 3 ) - 1 2 ( n 0 2 - 1 ) ] cos g Z + [ 1 2 ( h 4 + 1 ) + 1 2 ( n 0 2 - 1 ) ] cos 3 g Z , S 5 = [ 11 16 ( h 4 - 2 3 ) - 39 64 ( h 4 - 2 3 ) 2 + 15 16 ( h 6 + 17 45 ) ] g Z csc g Z + [ 21 8 ( h 4 - 2 3 ) + 9 4 ( h 4 - 2 3 ) 2 + 9 8 ( n 0 2 - 1 ) ( h 4 - 2 3 ) ] g Z sin g Z - [ 15 8 ( h 4 - 2 3 ) + 9 8 ( h 4 - 2 3 ) 2 + 9 8 ( n 0 2 - 1 ) ( h 4 - 2 3 ) ] g Z sin 3 g Z + [ 3 4 h 4 - 3 8 + 33 64 h 4 2 + 15 16 h 6 + 3 4 ( n 0 2 - 1 ) ( h 4 - 1 ) - 3 8 ( n 0 2 - 1 ) 2 ] cos g Z + [ 3 8 h 4 + 3 4 + 11 32 h 4 2 + 5 8 h 6 + 3 8 ( n 0 2 - 1 ) ( h 4 + 4 ) + 3 4 ( n 0 2 - 1 ) 2 ] cos 3 g Z - [ 9 8 h 4 + 3 8 + 1 4 h 4 2 - 1 2 h 6 + 9 8 ( n 0 2 - 1 ) ( h 4 + 2 3 ) + 3 8 ( n 0 2 - 1 ) 2 ] cos 5 g Z .
g s L = g l x l = 0 - g l f .
g s L n 0 = S ^ 2 ( g x i ) 2 + S ^ 4 ( g x i ) 4 + 0 ( 6 ) ,
S ^ 2 = 3 4 ( h 4 - 2 3 ) g Z + 5 4 h 4 sin g Z cos g Z - [ 1 2 ( h 4 + 1 ) + 1 2 ( n 0 2 - 1 ) ] sin 3 g Z cos g Z , S ^ 4 = H 5 sin g Z + H 3 K 3 sin 2 g Z + K 3 2 cos g Z sin 3 g Z + K 5 cos g Z sin 2 g Z .

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