Abstract

A fifth-order algebraic formula is derived for the axial deviation of the ray observed behind the GRIN lens exit face, whose expansion coefficients characterize the coefficients up to sixth-order in the index profile of a GRIN lens with arbitrary length; the index coefficients up to sixth-order of a near one-quarter pitch lens are estimated with the formula. The proposed technique would be useful for the evaluation of a GRIN lens with larger spherical aberrations and a longer GRIN rod.

© 1983 Optical Society of America

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References

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  1. E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
    [CrossRef]
  2. T. W. Cline, R. B. Jander, Appl. Opt. 21, 1035 (1982).
    [CrossRef] [PubMed]
  3. T. Kitano, H. Matsumura, M. Furukawa, I. Kitano, IEEE J. Quantum Electron. QE-9, 967 (1973).
    [CrossRef]
  4. N. Yamamoto, K. Iga, Appl. Opt. 19, 1101 (1980).
    [CrossRef] [PubMed]
  5. T. Sakamoto, Opt. Eng. 22, 359 (1983).
    [CrossRef]
  6. W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  7. T. Sakamoto, “Fifth-Order Spherical Aberration Formula of Gradient-Index Rod Lenses,” in Digest of 4th Topical Meeting on Gradient-Index Optical Imaging Systems (Japan Society of Applied Physics, Optical Society of America and International Commission for Optics, Kobe, 1983), paper B3.

1983 (1)

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

1982 (1)

1980 (1)

1973 (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]

1971 (1)

Cline, T. W.

Furukawa, M.

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

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]

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]

Paxton, K. B.

Rawson, E. G.

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

Sakamoto, T.

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

T. Sakamoto, “Fifth-Order Spherical Aberration Formula of Gradient-Index Rod Lenses,” in Digest of 4th Topical Meeting on Gradient-Index Optical Imaging Systems (Japan Society of Applied Physics, Optical Society of America and International Commission for Optics, Kobe, 1983), paper B3.

Streifer, W.

Yamamoto, N.

Appl. Opt. (3)

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]

Opt. Eng. (1)

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

Other (1)

T. Sakamoto, “Fifth-Order Spherical Aberration Formula of Gradient-Index Rod Lenses,” in Digest of 4th Topical Meeting on Gradient-Index Optical Imaging Systems (Japan Society of Applied Physics, Optical Society of America and International Commission for Optics, Kobe, 1983), paper B3.

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Figures (5)

Fig. 1
Fig. 1

Observed axial deviation on a near one-quarter pitch GRIN lens. Open circles denote an experimental point. The best-fit theoretical curve (â = −155.7, b ̂ = 15.2 , ĉ = −11.1) is shown as a solid line, and the value of ε2 is 16.1.

Fig. 2
Fig. 2

Experimental setup. The parameters of the test lens are n0 = 1.552, Z = 7.00 mm, and 1.0 mm in radius. The distance l is set to 450 mm, and the light source is a He–Ne laser (6328 Å).

Fig. 3
Fig. 3

Nonlinear terms in the theoretical curve.

Fig. 4
Fig. 4

Signal-to-noise ratio of the nonlinear terms.

Fig. 5
Fig. 5

Example of estimated nonlinearities for various h4 and h6 at xi = 1.0 mm. The second-order coefficient g is assumed to be 0.223 mm −1, and the other parameters are the same as those in Fig. 2.

Tables (1)

Tables Icon

Table I Estimation of the Coefficients

Equations (36)

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n 2 ( r ) = n 0 2 [ 1 ( g r ) 2 + h 4 ( g r ) 4 + h 6 ( g r ) 6 + ] ,
g x ( z ) = [ g x i + h 4 2 4 ( g x i ) 3 + 1 2 8 ( 23 h 4 2 + 32 h 6 ) ( g x i ) 5 ] cos Ω z [ h 4 2 4 ( g x i ) 3 + 1 2 8 ( 24 h 4 2 + 30 h 6 ) ( g x i ) 5 ] cos 3 Ω z + 1 2 8 ( h 4 2 2 h 6 ) ( g x i ) 5 cos 5 Ω z ,
Ω g = 1 3 4 ( h 4 2 3 ) ( g x i ) 2 [ 21 16 ( h 4 2 3 ) + 21 64 ( h 4 2 3 ) 2 + 15 16 ( h 6 + 17 45 ) ] ( g x i ) 4 .
g x l = g x z + g l n z d x d z z = Z 1 + ( 1 n z 2 ) ( d x d z z = Z ) 2 ,
g x z = g x i cos g Z + H 3 ( g x i ) 3 + H 5 ( g x i ) 5 + O [ ( g x i ) 7 ] ,
H 3 ( Z ) = 3 4 ( h 4 2 3 ) g Z sin g Z + h 4 4 cos g Z h 4 4 cos 3 g Z , H 5 ( Z ) = [ 17 16 ( h 4 2 3 ) 3 64 ( h 4 2 3 ) 2 + 15 16 ( h 6 + 17 45 ) ] g Z sin g Z + [ 3 8 ( h 4 2 3 ) + 9 16 ( h 4 2 3 ) 2 ] g Z sin 3 g Z + [ 25 64 h 4 2 + 7 16 h 6 9 32 ( h 4 2 3 ) 2 ( g Z ) 2 ] cos g Z ( 29 64 h 4 2 + 5 16 h 6 ) cos 3 g Z + ( 1 16 h 4 2 1 8 h 6 ) cos 5 g Z ,
d x d z z = Z = g x i sin g Z + 1 g d H 3 d Z ( g x i ) 3 + 1 g d H 5 d Z ( g x i ) 5 + O [ ( g x i ) 7 ] .
n z n 0 = 1 ( g x i ) 2 1 2 cos 2 g Z ( g x i ) 4 [ 3 4 ( h 4 2 3 ) g Z sin g Z cos g Z + h 4 4 cos 2 g Z 3 4 ( h 4 1 6 ) cos 4 g Z ] + O [ ( g x i ) 6 ] .
g x l = g x i cos g Z + H 3 ( g x i ) 3 + H 5 ( g x i ) 5 + g l n 0 [ g x i sin g Z + K 3 ( g x i ) 3 + K 5 ( g x i ) 5 ] + O [ ( g x i ) 7 ] ,
K 3 ( Z ) = 3 4 ( h 4 2 3 ) g Z cos g Z + 5 4 h 4 sin g Z [ 3 4 ( h 4 + 2 3 ) + 1 2 ( n 0 2 1 ) ] sin 3 g Z , K 5 ( Z ) = [ 47 16 ( h 4 2 3 ) + 69 64 ( h 4 2 3 ) 2 + 15 16 ( h 6 + 17 45 ) + 9 8 ( n 0 2 1 ) ( h 4 2 3 ) ] g Z cos g Z [ 9 4 ( h 4 2 3 ) + 27 16 ( h 4 2 3 ) 2 + 9 8 ( n 0 2 1 ) ( h 4 2 3 ) ] g Z cos 3 g Z + [ 39 64 h 4 2 + 33 16 h 6 + 9 32 ( h 4 2 3 ) 2 ( g Z ) 2 ] sin g Z + [ 15 8 h 4 11 64 h 4 2 35 16 h 6 + 15 8 ( n 0 2 1 ) h 4 ] sin 3 g Z [ 9 8 h 4 + 3 8 + 5 16 h 4 2 5 8 h 6 + 9 8 ( n 0 2 1 ) ( h 4 + 2 3 ) + 3 8 ( n 0 2 1 ) 2 ] sin 5 g Z .
y = a ( g ) x + b ( g , h 4 ) x 3 + c ( g , h 4 , h 6 ) x 5 .
ε 2 = k = 1 m ( a x k + b x k 3 + c x k 5 y k ) 2 .
ε 2 h 6 = 2 c h 6 ( a x k 6 + b x k 8 + c x k 10 x k 5 y k ) = 0 ,
ε 2 h 4 = 2 b h 4 ( a x k 4 + b x k 6 + c x k 8 x k 3 y k ) + 2 c h 4 ( a x k 6 + b x k 8 + c x k 10 x k 5 y k ) = 0 ,
ε 2 g = 2 a g ( a x k 2 + b x k 4 + c x k 6 x k y k ) + 2 b g ( a x k 4 + b x k 6 + c x k 8 x k 3 y k ) + 2 c g ( a x k 6 + b x k 8 + c x k 10 x k 5 y k ) = 0 ,
a x k 6 + b x k 8 + c x k 10 x k 5 y k = 0 ,
c h 6 = 0 ,
a x k 4 + b x k 6 + c x k 8 x k 3 y k = 0 ,
a x k 2 + b x k 4 + c x k 6 x k y k = 0.
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 ) .
( SNR ) = 10 log ( Δ x l σ N ) 2 ,
x k 2 = 14.35
x k 4 = 9.033
x k 6 = 6.764
x k 8 = 5.511
b ̂ = 15.2 ± 1.2
x k 10 = 4.719
x k y k = 2172
x k 3 y k = 1365
x k 5 y k = 1022
g = a l n 0 ,
h 4 = 8 b / g 2 + 2 π + 4 g l n 0 3 3 π + 8 g l n 0 ,
h 6 = A ( c , h 4 , g , l , n 0 ) B ( g , l , n 0 ) ,
A = c g 4 [ 23 16 ( h 4 2 3 ) + 33 64 ( h 4 2 3 ) 2 + 17 48 ] π 2 g l n 0 [ 9 32 ( π 2 ) 2 ( h 4 2 3 ) 2 + 3 4 h 4 3 8 1 8 h 4 2 + 3 4 ( n 0 2 1 ) ( h 4 1 ) 3 8 ( n 0 2 1 ) 2 ] , B = 15 π 32 + 1 2 g l n 0 ,

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