Abstract

The best focus for a GRIN fiber array is shown to occur when the object and image plane vertex distances are equal, even when neither plane is the proper Gaussian unit magnification plane. This analytical result agrees with previous experimental data. With non-Gaussian imaging, the image plane of best focus for the GRIN array is not coincident with the plane of best focus for the individual gradient-index fibers.

© 1982 Optical Society of America

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References

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  1. Selfoc is a registered trade name of the Nippon Sheet Glass Co., Ltd., Osaka, Japan.
  2. K. Matsushita, N. Akazawa, M. Toyama, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC2.
  3. J. D. Rees, W. Lama, Appl. Opt. 19, 1065 (1980).
    [CrossRef] [PubMed]
  4. K. Matsushita, M. Toyama, Appl. Opt. 19, 1070 (1980).
    [CrossRef] [PubMed]
  5. M. Kawazu, Y. Ogura, Appl. Opt. 19, 1105 (1980).
    [CrossRef] [PubMed]
  6. M. Toyama, N. Akazawa, K. Matsushita, in Selfoc ’81 (Nippon Sheet Glass Co., Ltd., Osaka, Japan, 1981).
  7. F. P. Kapron, J. Opt. Soc. Am. 60, 1433 (1970).
    [CrossRef]
  8. I. Kitano, K. Koizumi, H. Matsumura, K. Ikeda, T. Uchida, “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (25Apr.1972).
  9. S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

1980 (3)

1970 (1)

1965 (1)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

Akazawa, N.

K. Matsushita, N. Akazawa, M. Toyama, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC2.

M. Toyama, N. Akazawa, K. Matsushita, in Selfoc ’81 (Nippon Sheet Glass Co., Ltd., Osaka, Japan, 1981).

Ikeda, K.

I. Kitano, K. Koizumi, H. Matsumura, K. Ikeda, T. Uchida, “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (25Apr.1972).

Kapron, F. P.

Kawazu, M.

Kitano, I.

I. Kitano, K. Koizumi, H. Matsumura, K. Ikeda, T. Uchida, “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (25Apr.1972).

Koizumi, K.

I. Kitano, K. Koizumi, H. Matsumura, K. Ikeda, T. Uchida, “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (25Apr.1972).

Lama, W.

Matsumura, H.

I. Kitano, K. Koizumi, H. Matsumura, K. Ikeda, T. Uchida, “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (25Apr.1972).

Matsushita, K.

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

K. Matsushita, N. Akazawa, M. Toyama, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC2.

M. Toyama, N. Akazawa, K. Matsushita, in Selfoc ’81 (Nippon Sheet Glass Co., Ltd., Osaka, Japan, 1981).

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

Ogura, Y.

Rees, J. D.

Toyama, M.

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

K. Matsushita, N. Akazawa, M. Toyama, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC2.

M. Toyama, N. Akazawa, K. Matsushita, in Selfoc ’81 (Nippon Sheet Glass Co., Ltd., Osaka, Japan, 1981).

Uchida, T.

I. Kitano, K. Koizumi, H. Matsumura, K. Ikeda, T. Uchida, “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (25Apr.1972).

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

J. Opt. Soc. Am. (1)

Other (4)

Selfoc is a registered trade name of the Nippon Sheet Glass Co., Ltd., Osaka, Japan.

K. Matsushita, N. Akazawa, M. Toyama, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC2.

M. Toyama, N. Akazawa, K. Matsushita, in Selfoc ’81 (Nippon Sheet Glass Co., Ltd., Osaka, Japan, 1981).

I. Kitano, K. Koizumi, H. Matsumura, K. Ikeda, T. Uchida, “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (25Apr.1972).

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Figures (7)

Fig. 1
Fig. 1

Side view of a single radial GRIN fiber lens. The fiber is producing an erect reduced image. A limiting axial ray is shown which just misses the fiber periphery. This ray defines the longitudinal position of the aperture stop. The diameter of the aperture stop is equal to the diameter of the fiber. Object and image vertex distances l0 and l1 are given by Eqs. (2) and (3).

Fig. 2
Fig. 2

Side view of a single Selfoc fiber lens. An axial and chief ray are traced. The entrance pupil is located at a distance ENP ¯ from the object-side fiber face. The distance ENP ¯ is given by Eq. (12).

Fig. 3
Fig. 3

Gaussian unit magnification imaging with a Selfoc lens array. Object and image vertex distances l are equal and given by Eq. (7). Images P 1 and P 2 of the object point P are formed by fiber lenses 1 and 2, respectively. Images P 1 and P 2 are coincident.

Fig. 4
Fig. 4

Non-Gaussian imaging with a Selfoc lens array. The object plane has been moved a distance Δ to the left of the Gaussian unit magnification plane. Fibers 1 and 2 image the object point P as noncoincident images P 1 and P 2 in the paraxial image plane. Best array imaging occurs in the plane located by the chief ray intersection point P″.

Fig. 5
Fig. 5

Non-Gaussian imaging with a Selfoc lens array. Symbols for the entrance and exit ray heights and angles are shown for the chief ray passing through fiber 2. The plane of best array focus is located at an array image vertex distance x.

Fig. 6
Fig. 6

Difference between the exact [Eq. (17)] and approximate [Eq. (18)] values for the array image vertex distance x as a function of the object plane defocus Δ. The difference is very small for absolute values of Δ less than 2.0 mm. In this figure n0 = 1.538, A = 0.1269 mm - 1, and L = 29.0 mm.

Fig. 7
Fig. 7

Non-Gaussian imaging with a Selfoc lens array. Chief rays are shown imaging an object of lateral extent h in the plane of best array focus. The lateral magnification in the plane of best array focus is one for all values of the object plane defocus Δ.

Equations (19)

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n ( r ) = n 0 ( 1 - A r 2 / 2 ) ,             A r 2 1 ,
l 0 = 1 / m - cos ( A L ) - n 0 A sin ( A L ) ,             π < A L < 2 π ,
l 1 = m - cos ( A L ) - n 0 a sin ( A L ) ,             π < A L < 2 π .
l 1 = l 0 + tan ( A L ) n 0 A n 0 A l 0 tan ( A L ) - 1 .
d l 1 = - 1 [ n 0 A l 0 sin ( A L ) - cos ( A L ) ] 2 d l 0 .
d m = ( - m 2 ) d l 0 .
l 0 ( m = 1 ) = l 1 ( m = 1 ) = l = 1 - cos ( A L ) - n 0 A sin ( A L ) .
d l 1 = - d l 0 ,             m = 1 ,
d m = - d l 0 ,             m = 1.
y = y cos ( A L ) + β sin ( A L ) n 0 A ,
β = - y n 0 A sin ( A L ) + β cos ( A L ) ,
ENP ¯ = 1 / ( n 0 2 A l 0 ) .
N . A . = - n 0 A R cos ( A L / 2 ) .
y = - β ENP ¯ ;
h = - β ( l + Δ + ENP ¯ ) ;
x = ( h - y ) / β .
x = l + Δ + ENP ¯ [ 1 - cos ( A L ) ] + [ sin ( A L ) ] / n 0 A - ENP ¯ n 0 A sin ( A L ) - cos ( A L ) ,
ENP ¯ = 1 n 0 2 A ( l + Δ ) ,             l = 1 - cos ( A L ) - n 0 A sin ( A L ) .
x l + Δ ,             Δ / l 1.

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