Abstract

This paper presents a new technique for evaluating a gradient-index rod lens that has a radial gradient index of refraction and can be used as an imaging or light-focusing device. The index profile is analytically related to imaging in terms of simple expressions for the aberrations. It is shown that the index profile can be easily checked by observing the distortion or field curvature of the image. A plastic gradient-index rod lens is evaluated by means of a proposed aberration-testing method, and the index profile is estimated.

© 1980 Optical Society of America

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References

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  1. K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. K. Iga, K. Yokomori, T. Sakayori, Appl. Phys. Lett. 26, 578 (1975).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

1978

1977

1975

K. Iga, K. Yokomori, T. Sakayori, Appl. Phys. Lett. 26, 578 (1975).
[CrossRef]

1974

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

1973

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]

1970

Fukuyo, H.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Furukawa, M.

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

Hata, S.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Iga, K.

K. Iga, Y. Kokubun, Appl. Opt. 17, 1972 (1978).
[CrossRef] [PubMed]

K. Iga, N. Yamamoto, Appl. Opt. 16, 1305 (1977).
[CrossRef] [PubMed]

K. Iga, K. Yokomori, T. Sakayori, Appl. Phys. Lett. 26, 578 (1975).
[CrossRef]

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Kapron, F. P.

Kato, Y.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

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]

Murray, R. G.

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

Rawson, E. G.

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

Sakayori, T.

K. Iga, K. Yokomori, T. Sakayori, Appl. Phys. Lett. 26, 578 (1975).
[CrossRef]

Yamamoto, N.

Yokomori, K.

K. Iga, K. Yokomori, T. Sakayori, Appl. Phys. Lett. 26, 578 (1975).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

K. Iga, K. Yokomori, T. Sakayori, Appl. Phys. Lett. 26, 578 (1975).
[CrossRef]

IEEE J. Quantum Electron.

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]

J. Opt. Soc. Am.

Jpn. J. Appl. Phys.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Comparison of normalized sinusoidal wave number Ω/g obtained from Eq. (2) (dashed curve and broken curve) with that calculated numerically using the Runge-Kutta method (solid curve) for initial conditions, gxi = i = 0.1. Dashed curve shows the result obtained by taking only the first and second terms of Eq. (2) into consideration.

Fig. 2
Fig. 2

Image formation by GRIN rod lens.

Fig. 3
Fig. 3

Changes of aberration coefficients K1, K2, and K3, normalized image distance gcn(0), and ideal lateral magnification m as a function of normalized length gb of GRIN lens for normalized object distance gan(0) = 4.

Fig. 4
Fig. 4

Distortion and curvature of field (to fourth-order coefficient).

Fig. 5
Fig. 5

Complex distortion (to sixth-order coefficient). When H4K2 < 0 and H 4 K 2 > 0, for example, distortion becomes barrel near-axis and pincushion far off-axis. Aberration coefficient of higher-order K 2 has the same sign as K2.

Fig. 6
Fig. 6

Aberration-testing system used for evaluating aberrations.

Fig. 7
Fig. 7

Mesh image observed by placing a slit 0.3 mm in width in front of plastic GRIN lens 8 mm in length and 4 mm in diam. Distance between the mesh and the lens is 50 mm.

Fig. 8
Fig. 8

Observed barrel distortion for a quarter-pitch lens with larger aberration. Flat observation field is placed (a) on the center axis, and (b) far off-axis.

Fig. 9
Fig. 9

Observed pincushion distortion for a quarter-pitch lens.

Fig. 10
Fig. 10

Observed image of a mesh through plastic GRIN lens 7 mm in length and 2 mm in diam. Object distance between the mesh and the lens is 65 mm, and line interval of the mesh is 2 mm.

Tables (1)

Tables Icon

Table I Expected Image Defects for a Quarter-Pitch Lens

Equations (11)

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n 2 ( r ) = n 2 ( 0 ) [ 1 ( gr ) 2 + h 4 ( gr ) 4 + h 6 ( gr ) 6 + . . . ] ,
Ω / g = 1 ( 3 / 4 ) ( h 4 2 / 3 ) [ ( g x i ) 2 + x ˙ i 2 ] ( 3 / 4 ) ( h 4 2 / 3 ) [ 21 ( g x i ) 4 + 46 ( g x i ) 2 x ˙ i 2 + 17 x ˙ i 4 ] / 12 [ ( 3 / 4 ) ( h 4 2 / 3 ) ] 2 [ 7 ( g x i ) 4 + 46 ( g x i ) 2 x ˙ i 2 + 23 x ˙ i 4 ] / 12 ( 15 / 16 ) ( h 6 + 17 / 45 ) [ ( g x i ) 2 + x ˙ i 2 ] 2 .
g ( x p m x 0 ) = m H 4 { K 1 [ ( g x i ) γ ( g x 0 ) ] 3 + K 2 ( g x 0 ) 3 } ,
c δ = c H 4 K 3 ( g x 0 ) 2 ,
H 4 = 3 ( h 4 2 / 3 ) / 4 .
( x ¯ p m x 0 ) / m x 0 = H 4 K 2 ( g x 0 ) 2 ,
h 4 > 2 3
h 6 > 17 45
h 6 < 17 45
h 4 < 2 3
h 6 > 17 45

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