Abstract

A gradient-index fiber array arranged in a plurality of rows, which forms an erect image with unit magnification, is employed as a novel optical system for use in a copying machine. In this paper, the optical characteristics of the fiber array are discussed. In particular, an equation indicating the quantity of light for slit-exposure scanning is introduced. The f/No. of the N-row array is defined in a form equivalent to the f/No. of a conventional lens.

© 1980 Optical Society of America

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References

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  1. K. Nishizawa, Opt. Eng.Contact Jpn. 6, No. 5, 25 (1978).
  2. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [CrossRef]
  3. M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).
  4. H. Ochi, J. Inst. Image Electron. Eng. Jpn. 4, 13 (1975).
  5. H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
    [CrossRef]

1978 (1)

K. Nishizawa, Opt. Eng.Contact Jpn. 6, No. 5, 25 (1978).

1975 (2)

H. Ochi, J. Inst. Image Electron. Eng. Jpn. 4, 13 (1975).

H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
[CrossRef]

1971 (1)

M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).

1970 (1)

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

Furukawa, M.

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

Fuyukawa, M.

M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).

Kitano, I.

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

Koizumi, K.

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

Matsumura, H.

H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
[CrossRef]

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

Nishizawa, K.

K. Nishizawa, Opt. Eng.Contact Jpn. 6, No. 5, 25 (1978).

Ochi, H.

H. Ochi, J. Inst. Image Electron. Eng. Jpn. 4, 13 (1975).

Sakamoto, M.

M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).

Toyama, M.

M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).

Uchida, T.

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

Uejima, H.

M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).

Yamada, T.

M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).

Electrophotogr. Jpn. (1)

M. Sakamoto, H. Uejima, M. Fuyukawa, M. Toyama, T. Yamada, Electrophotogr. Jpn. 12, 12 (1971).

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. Inst. Image Electron. Eng. Jpn. (1)

H. Ochi, J. Inst. Image Electron. Eng. Jpn. 4, 13 (1975).

Opt. Eng.Contact Jpn. (1)

K. Nishizawa, Opt. Eng.Contact Jpn. 6, No. 5, 25 (1978).

Opt. Quantum Electron. (1)

H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Optical path and index distribution of the fiber.

Fig. 2
Fig. 2

Optical dimensions of the fiber.

Fig. 3
Fig. 3

Erect image formation with unit magnification.

Fig. 4
Fig. 4

Image formation by the fiber array.

Fig. 5
Fig. 5

Projected image of a two-row fiber array.

Fig. 6
Fig. 6

Depth of focus of the fiber array.

Fig. 7
Fig. 7

Projected resolution with defocusing.

Fig. 8
Fig. 8

MTF of fiber array and lens: —, fiber array; - - -, copying lens center; - · -, copying lens image, height 85 mm.

Fig. 9
Fig. 9

Conjugate length of conventional lens and fiber array.

Fig. 10
Fig. 10

Irregularity of pitch of fiber array.

Fig. 11
Fig. 11

Spectral transmittance: – –, cesium-based fiber; – · –, thallium-based fiber; - - -, yellow-coated copying lens.

Fig. 12
Fig. 12

Distribution of light of one GRIN fiber.

Fig. 13
Fig. 13

Distribution E(X) of light of one GRIN fiber.

Fig. 14
Fig. 14

Distribution of light with a one-row fiber array.

Fig. 15
Fig. 15

Illuminance of N-row array.

Fig. 16
Fig. 16

Definition of f/No.

Fig. 17
Fig. 17

PPC system.

Fig. 18
Fig. 18

EF system.

Tables (1)

Tables Icon

Table I Number of Reflections in Optical Systems of Copying Machines

Equations (32)

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n ( r ) = n 0 ( 1 ½ A r 2 ) ,
n 0 ( d 2 r d Z 2 ) = dn ( r ) dr .
f = 1 ( A ) 1 / 2 n 0 sin [ ( A ) 1 / 2 Z 0 ] ,
H = 1 n 0 ( A ) 1 / 2 tan [ ( A ) 1 / 2 2 Z 0 ] ,
l i = sin [ ( A ) 1 / 2 Z 0 ] + n 0 ( A ) 1 / 2 l 0 cos [ ( A ) 1 / 2 Z 0 ] ( A ) 1 / 2 n 0 { n 0 ( A ) 1 / 2 l 0 sin [ ( A ) 1 / 2 Z 0 ] + cos [ ( A ) 1 / 2 Z 0 ] } ,
M = 1 n 0 ( A ) 1 / 2 l 0 sin [ ( A ) 1 / 2 Z 0 ] + cos [ ( A ) 1 / 2 Z 0 ] ,
1 a 1 b = 1 f ,
b a = 1 .
η θ = θ × Δ l η α = α × Δ l = γ l × Δ l η β = β [ Δ l 2 + ( θ × Δ l ) 2 ] 1 / 2 = β × Δ l ( 1 + θ 2 ) 1 / 2 = γ l × Δ l η θ θ > η α , η α = η β } ,
Δ l = ± η θ / θ .
l 0 = 1 η 0 ( A ) 1 / 2 tan [ ( A ) 1 / 2 2 Z 0 ] ,
X 2 X 0 2 + Z 2 Z 0 2 = 1 .
Y 0 = X 0 = l × θ .
Z 0 = τ π B sin 2 α = ¼ τ π B ( D l ) 2 ,
Z 0 = k π 4 ( D l ) 2 .
Z 2 = Z 0 2 ( 1 X 2 X 0 2 ) .
e ( X ) = Z 0 ( 1 X 2 X 0 2 ) 1 / 2 .
e ( X ) = k π 4 ( D l ) 2 [ 1 X 2 ( l θ ) 2 ] 1 / 2 .
E ( X ) = 1 2 π Y 0 Z 0 ( 1 X 2 X 0 2 ) = π 2 l θ k π 4 ( D l ) 2 [ 1 X 2 ( l θ ) 2 ] = k π 2 8 θ D 2 l [ 1 X 2 ( l θ ) 2 ] .
η 0 = 2 × 0 l θ E ( X ) d X = k π 2 6 θ 2 D 2 .
F 1 ( X ) = 1 D E ( X ) = k π 2 8 θ D l [ 1 X 2 ( l θ ) 2 ] .
I 1 = 2 0 l θ F 1 ( X ) dx = k × π 2 6 θ 2 D .
I 1 = k π 4 6 × D 3 n 0 2 P 2 .
F N ( X ) = i = 1 N / 2 F 1 i + i = 1 N / 2 F 2 j F 11 = F 1 [ X ( 3 ) 1 / 2 4 D ] F 21 = F 1 [ X + ( 3 ) 1 / 2 4 D ] F 12 = F 1 [ X 3 ( 3 ) 1 / 2 4 D ] F 22 = F 1 [ X + 3 ( 3 ) 1 / 2 4 D ] F 1 i = F 1 [ X ( 2 i 1 ) × ( 3 ) 1 / 2 4 D ] F 2 j = F 1 [ X + ( 2 i 1 ) × ( 3 ) 1 / 2 4 D ] F 1 2 N = F 1 [ X N 1 4 ( 3 ) 1 / 2 D ] F 2 2 N = F 1 [ X + N 1 4 ( 3 ) 1 / 2 D ] } .
F N ( X ) = F 1 ( X ) + i = 1 N 1 2 F 1 i + j = 1 N 1 2 F 2 j F 11 = F 1 [ X ( 3 ) 1 / 2 4 D ] F 1 i = F 1 [ X i ( 3 ) 1 / 2 2 D ] F 1 2 N 1 = F 1 [ X N 1 2 ( 3 ) 1 / 2 2 D ] F 21 = F 1 [ X + ( 3 ) 1 / 2 4 D ] F 2 j = F 1 [ X + j ( 3 ) 1 / 2 2 D ] F 2 2 N 1 = F 1 [ X + N 1 2 ( 3 ) 1 / 2 2 D ] } .
I N = N × I 1 = k π 6 N θ 2 D = k × π 4 6 N D 3 n 0 2 P 2 .
I = k × π 4 ( 1 F ) 2 1 ( 1 + β ) 2 .
I = k π 16 ( 1 F ) 2 .
I 0 = I N W = k × π 4 6 N W D 3 n 0 2 P 2 .
( 1 F ) 2 = 16 6 π 3 N W D 3 n 0 2 P 2 .
W = 2 × l θ + ( 3 ) 1 / 2 D D ( N 1 ) .
F = ( 8 3 π 3 N W D 3 n 0 2 P 2 ) 1 / 2 = 4.8 .

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