Abstract

To study the influence of scattering of the distribution parameter in a gradient-index rod lens on ray tracing by statistical linearization, a method of obtaining the mean and variance is presented. According to examples studied, it is found that the ray trajectory is remarkably affected by the amount of scattering of the parameter.

© 1984 Optical Society of America

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References

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  1. M. S. Sodha, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 63, 940 (1973).
    [CrossRef]
  2. K. Iga, S. Hata, Trans. IECE Jpn. 55-C, 84 (1972).
  3. K. Sato, O. Kamada, S. Yamamoto, N. Takatsu, J. Opt. Soc. Am. 73, 855 (1983).
    [CrossRef]
  4. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [CrossRef]
  5. A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
    [CrossRef]
  6. K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber and Integrated Optics (Crane, Trenton, N.J., 1979).
  7. K. Matsushita, M. Toyama, Appl. Opt. 19, 1070 (1980).
    [CrossRef] [PubMed]
  8. M. Toyama, H. Nishi, Kogaku (Jpn. J. Opt.) 11, 6, 546 (1982).
  9. K. Iga, Appl. Opt. 21, 988 (1982).
    [CrossRef] [PubMed]
  10. W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  11. Y. Sawaragi, T. Soeda, S. Tagami, Trans. JSME 34, 261, 864 (1968).
    [CrossRef]
  12. A. A. Pervozvanskii, Random Process in Nonlinear Control Systems (Academic, New York, 1965).

1983 (1)

1982 (2)

M. Toyama, H. Nishi, Kogaku (Jpn. J. Opt.) 11, 6, 546 (1982).

K. Iga, Appl. Opt. 21, 988 (1982).
[CrossRef] [PubMed]

1980 (1)

1973 (1)

1972 (1)

K. Iga, S. Hata, Trans. IECE Jpn. 55-C, 84 (1972).

1971 (1)

1970 (1)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

1969 (1)

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

1968 (1)

Y. Sawaragi, T. Soeda, S. Tagami, Trans. JSME 34, 261, 864 (1968).
[CrossRef]

French, W. G.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Ghatak, A. K.

Goyal, I. C.

Hata, S.

K. Iga, S. Hata, Trans. IECE Jpn. 55-C, 84 (1972).

Iga, K.

K. Iga, Appl. Opt. 21, 988 (1982).
[CrossRef] [PubMed]

K. Iga, S. Hata, Trans. IECE Jpn. 55-C, 84 (1972).

Ishikawa, R.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber and Integrated Optics (Crane, Trenton, N.J., 1979).

Kamada, O.

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Kobayashi, K.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber and Integrated Optics (Crane, Trenton, N.J., 1979).

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Matsushita, K.

Minemura, K.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber and Integrated Optics (Crane, Trenton, N.J., 1979).

Nishi, H.

M. Toyama, H. Nishi, Kogaku (Jpn. J. Opt.) 11, 6, 546 (1982).

Paxton, K. B.

Pearson, A. D.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Pervozvanskii, A. A.

A. A. Pervozvanskii, Random Process in Nonlinear Control Systems (Academic, New York, 1965).

Rawson, E. G.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Sato, K.

Sawaragi, Y.

Y. Sawaragi, T. Soeda, S. Tagami, Trans. JSME 34, 261, 864 (1968).
[CrossRef]

Sodha, M. S.

Soeda, T.

Y. Sawaragi, T. Soeda, S. Tagami, Trans. JSME 34, 261, 864 (1968).
[CrossRef]

Streifer, W.

Sugimoto, S.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber and Integrated Optics (Crane, Trenton, N.J., 1979).

Tagami, S.

Y. Sawaragi, T. Soeda, S. Tagami, Trans. JSME 34, 261, 864 (1968).
[CrossRef]

Takatsu, N.

Toyama, M.

M. Toyama, H. Nishi, Kogaku (Jpn. J. Opt.) 11, 6, 546 (1982).

K. Matsushita, M. Toyama, Appl. Opt. 19, 1070 (1980).
[CrossRef] [PubMed]

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Yamamoto, S.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

J. Opt. Soc. Am. (2)

Kogaku (Jpn. J. Opt.) (1)

M. Toyama, H. Nishi, Kogaku (Jpn. J. Opt.) 11, 6, 546 (1982).

Trans. IECE Jpn. (1)

K. Iga, S. Hata, Trans. IECE Jpn. 55-C, 84 (1972).

Trans. JSME (1)

Y. Sawaragi, T. Soeda, S. Tagami, Trans. JSME 34, 261, 864 (1968).
[CrossRef]

Other (2)

A. A. Pervozvanskii, Random Process in Nonlinear Control Systems (Academic, New York, 1965).

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber and Integrated Optics (Crane, Trenton, N.J., 1979).

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Figures (3)

Fig. 1
Fig. 1

Mean and variance of ray position. Solid and dotted lines denote mean and variance, respectively.

Fig. 2
Fig. 2

Mean and variance of ray slope. Solid and dotted lines denote mean and variance, respectively.

Fig. 3
Fig. 3

Mean and variance of ray position. Solid and dotted lines denote mean and variance, respectively.

Equations (42)

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n 2 ( x ) = n 0 2 ( 1 - g 2 x 2 + h 4 g 4 x 4 + h 6 g 6 x 6 + h 8 g 8 x 8 ) ,
2 x z 2 = 1 2 c 2 n 2 ( x ) x ,
x ¨ + w 0 2 x = w 0 2 ( 2 h 4 g 2 x 3 + 3 h 6 g 4 x 5 + 4 h 8 g 6 x 7 ) ,
x ¨ + g 2 x = g 4 ( 2 h 4 x 3 + 3 h 6 g 2 x 5 + 4 h 8 g 4 x 7 ) .
x ( z 0 ) = x 0 ,             x ˙ ( z 0 ) = x 10 ,
κ = x f ( x ) p ( x , z ) d x / x 2 p ( x , z ) d x ,
x ¨ + ω 1 2 ( g , h 4 , h 6 , h 8 , κ ) x = 0 ,
x ˙ = x 1 ,             x ˙ 1 = - ω 1 2 x ,             x ( z 0 ) = x 0 ,             x ˙ ( z 0 ) = x 10 .
p z = - x 1 p x + ω 1 2 x p x 1 ,
p ( x , x 1 , ω 1 2 ; z 0 ) = p 0 ( x , x 1 , ω 1 2 ) ,
d z 1 = d x x 1 = - d x 1 ω 1 2 x ,
x = x 0 cos ω 1 ( z - z 0 ) + x 10 ω 1 sin ω 1 ( z - z 0 ) , x 1 = - ω 1 x 0 sin ω 1 ( z - z 0 ) + x 10 cos ω 1 ( z - z 0 ) .
x 0 = x cos ω 1 ( z - z 0 ) - x 1 ω 1 sin ω 1 ( z - z 0 ) u ( x , x 1 , ω 1 2 ) , x 10 = x ω 1 sin ω 1 ( z - z 0 ) + x 1 cos ω 1 ( z - z 0 ) v ( x , x 1 , ω 1 2 ) .
p 0 ( u , v , ω 1 2 ) = c .
p ( x , x 1 , ω 1 2 ; z ) = p 0 ( v , v , ω 1 2 ) = p 0 [ x cos ω 1 ( z - z 0 ) - x 1 ω 1 sin ω 1 ( z - z 0 ) , x ω 1 sin ω 1 ( z - z 0 ) + x 1 cos ω 1 ( z - z 0 ) , ω 1 2 ] .
m x ( z ) = x ( z ) = d x d x 1 d ω 1 2 x p ( x , x 1 , ω 1 2 ; z ) F 1 ( κ , z ) ,
ψ x ( z ) = [ x ( z ) ] 2 = d x d x 1 d ω 1 2 x 2 p ( x , x 1 , ω 1 2 ; z ) F 2 ( κ , z ) ,
m x 1 ( z ) = x 1 ( z ) = d x d x 1 d ω 1 2 x 1 p ( x , x 1 , ω 1 2 ; z ) G 1 ( κ , z ) ,
ψ x 1 ( z ) = [ x 1 ( z ) ] 2 = d x d x 1 d ω 1 2 x 1 2 p ( x , x 1 , ω 1 2 ; z ) G 2 ( κ , z ) ,
z j + 1 - z j = Δ j             ( j = 0 , 1 , 2 , , n ) ,
κ ( j ) = H [ m x ( z j ) , ψ x ( z j ) , z j ] .
m x ( z j ) = F 1 [ κ ( j - 1 ) , z j ] , ψ x ( z j ) = F 2 [ κ ( j - 1 ) , z j ] , m x 1 ( z j ) = G 1 [ κ ( j - 1 ) , z j ] , ψ x 1 ( z j ) = G 2 [ κ ( j - 1 ) , z j ] , κ ( j - 1 ) = H [ m x ( z j - 1 ) , ψ x ( z j - 1 ) , z j - 1 ] ,             ( j = 1 , 2 , , n ) .
m x ( z ) z = 0 = a , σ x 2 ( z ) z = 0 = σ x 0 2 , m x 1 ( z ) z = 0 = 0 , σ x 1 2 ( z ) z = 0 = σ x 10 2 .
p 0 ( x , x 1 , ω 1 2 ) = 1 2 π σ x 0 σ x 10 exp [ - ( x - a ) 2 2 σ x 0 2 - x 1 2 2 σ x 10 2 ] × i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 × δ ( g - g i 1 ) δ ( h 4 - h 4 i 2 ) δ ( h 6 - h 6 i 3 ) δ ( h 8 - h 8 i 4 )
i n p i = 1 ,             ( i = i 1 - i 4 , n = n 1 - n 4 ) ,
p ( x , x 1 , ω 1 2 ; z ) = 1 2 π σ x 0 σ x 10 [ exp ( - 1 2 σ x 0 2 x cos ω 1 z - x 1 ω 1 sin ω 1 z - a ) 2 - 1 2 σ x 10 2 × ( ω 1 x sin ω 1 z + x 1 cos ω 1 z ) 2 ) ] × i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 δ ( g - g i 1 ) δ ( h 4 - h 4 i 2 ) × δ ( h 6 - h 6 i 3 ) δ ( h 8 - h 8 i 4 ) .
m x ( z ) = 1 2 π σ x 0 σ x 10 d x d x 1 d g d h 4 d h 6 d h 8 x × { exp [ - 1 2 σ x 0 2 ( x cos ω 1 z - x 1 ω 1 sin ω 1 z - a ) 2 - 1 2 σ x 10 2 × ( ω 1 x sin ω 1 z + x 1 cos ω 1 z ) 2 ] p i 1 p i 2 p i 3 p i 4 × δ ( g - g i 1 ) δ ( h 4 - h 4 i 2 ) δ ( h 6 - h 6 i 3 ) δ ( h 8 - h 8 i 4 ) } = 1 2 π σ x 0 σ x 10 p i 1 p i 2 p i 3 p i 4 d x d x 1 x × { exp [ - 1 2 σ x 0 2 ( x cos ω 1 i z - x 1 ω 1 i sin ω 1 i z - a ) 2 - 1 2 σ x 10 2 ( ω 1 i x sin ω 1 i z + x 1 cos ω 1 i z ) 2 ] } = a i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 cos ω 1 i z ,
ω 1 i = ω 1 ( g i 1 , h 4 i 2 , h 6 i 3 , h 8 i 4 , κ 4 , κ 6 , κ 8 ) = g i 1 2 [ 1 - g i 1 2 ( 2 h 4 i 2 κ 4 + 3 h 6 i 3 κ 6 g i 1 2 + 4 h 8 i 4 κ 8 g i 1 4 ) ] ,
ψ x ( z ) = 1 2 π σ x 0 σ x 10 d x d x 1 d g d h 4 d h 6 d h 8 x 2 × { exp [ - 1 2 σ x 0 2 ( x cos ω 1 z - x 1 ω 1 sin ω 1 z - a ) 2 - 1 2 σ x 10 2 × ( ω 1 x sin ω 1 z + x 1 cos ω 1 z ) 2 ] p i 1 p i 2 p i 3 p i 4 × δ ( g - g i 1 ) δ ( h 4 - h 4 i 2 ) δ ( h 6 - h 6 i 3 ) δ ( h 8 - h 8 i 4 ) } = i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 [ ( σ x 0 2 + a 2 ) × cos 2 ω 1 i z + σ x 10 2 ω 1 i 2 sin 2 ω 1 i z ] .
σ x 2 ( z ) = ψ x ( z ) - [ m x ( z ) ] 2 .
m x 1 ( z ) = - a i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 ω 1 i sin ω 1 i z ,
ψ x 1 ( z ) = i 1 n 1 i 2 n 2 i 3 n 1 i 4 n 4 p i 1 p i 2 p i 3 p i 4 × [ σ x 10 2 cos 2 ω 1 i z + ( σ x 0 2 + a 2 ) ω 1 i 2 sin 2 ω 1 i z ] ,
σ x 1 2 ( z ) = ψ x 1 ( z ) - [ m x 1 ( z ) ] 2 .
x ( z ) z = 0 = a ,             x 1 ( z ) z = 0 = 0.
p 0 ( x , x 1 , ω 1 2 ) = δ ( x - a ) δ ( x 1 ) i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 × δ ( g - g i 1 ) δ ( h 4 - h 4 i 2 ) δ ( h 6 - h 6 i 3 ) δ ( h 8 - h 8 i 4 ) ,
p ( x , x 1 , ω 1 2 , z ) = δ ( x cos ω 1 z - x 1 ω 1 sin ω 1 z - a ) × δ ( ω 1 x sin ω 1 z + x 1 cos ω 1 z ) i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 × p i 1 p i 2 p i 3 p i 4 δ ( g - g i ) δ ( h 4 - h 4 i 2 ) × δ ( h 6 - h 6 i 3 ) δ ( h 8 - h 8 i 4 ) .
m x ( z ) = a i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 cos ω 1 i z ,
ψ x ( z ) = a 2 i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 cos 2 ω 1 i z ,
m x 1 ( z ) = - a i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 ω 1 1 sin ω 1 i z ,
ψ x 1 ( z ) = a 2 i 1 n 1 i 2 n 2 i 3 n 3 i 4 n 4 p i 1 p i 2 p i 3 p i 4 ω 1 i 2 sin 2 ω 1 i z .
κ 4 = ( 3 σ x 4 + 6 m x 2 σ x 2 + m x 4 ) / ψ x ,
ω 1 i = g i 1 2 ( 1 - 2 g i 1 2 h 4 i 2 κ 4 ) ,             ( a = 0.5 mm ) .

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