Abstract

The optical imaging and radiometric properties of arrays of GRIN fiber lenses (Selfoc lens arrays) have been studied as a function of the length of the fibers. Experiments measured the object-to-image distance, image quality (MTF), depth of focus, image irradiance distribution, and exposure for a set of Selfoc lenses with differing fiber length. Simple models were used to explain the observed dependences, and analytical formulas were developed for the prediction of these optical properties.

© 1982 Optical Society of America

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References

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  1. E. G. Rawson, D. R. Herriott, J. McKenna, Appl. Opt. 9, 753 (1970).
    [CrossRef] [PubMed]
  2. F. P. Kapron, J. Opt. Soc. Am. 60, 1433 (1970).
    [CrossRef]
  3. K. Matsushita, K. Ideda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).
  4. I. Kitano et al., “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (1972).
  5. J. D. Rees, W. Lama, Appl. Opt. 19, 1065 (1980).
    [CrossRef] [PubMed]
  6. Selfoc Linear Lens Array, Nippon Sheet Glass Co., Ltd., brochure dated 1Sept.1979.
  7. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 11. For a diffraction-limited lens with a circular aperture the diameter of the diffraction spot is given by 1.22 λ/sinα, where λ is the wavelength, and sinα is the numerical aperture, Eq. (10c).
  8. K. Nishizawa, Appl. Opt. 19, 1052 (1980).
    [CrossRef] [PubMed]
  9. J. Rees, Appl. Opt. 21, 1009 (1982).
    [CrossRef] [PubMed]
  10. M. Kawazu, Y. Ogura, Appl. Opt. 19, 1105 (1980).
    [CrossRef] [PubMed]
  11. K. Matsushita, M. Toyama, Appl. Opt. 19, 1070 (1980).
    [CrossRef] [PubMed]

1982

1980

1972

K. Matsushita, K. Ideda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

1970

Herriott, D. R.

Ideda, K.

K. Matsushita, K. Ideda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

Kapron, F. P.

Kawazu, M.

Kitano, I.

I. Kitano et al., “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (1972).

Lama, W.

Matsushita, K.

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

K. Matsushita, K. Ideda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

McKenna, J.

Nishizawa, K.

Ogura, Y.

Rawson, E. G.

Rees, J.

Rees, J. D.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 11. For a diffraction-limited lens with a circular aperture the diameter of the diffraction spot is given by 1.22 λ/sinα, where λ is the wavelength, and sinα is the numerical aperture, Eq. (10c).

Toyama, M.

Appl. Opt.

J. Opt. Soc. Am.

Proc. Soc. Photo-Opt. Instrum. Eng.

K. Matsushita, K. Ideda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

Other

I. Kitano et al., “Image Transmitter Formed of a Plurality of Graded Index Fibers in Bundled Configuration,” U.S. Patent3,658,407 (1972).

Selfoc Linear Lens Array, Nippon Sheet Glass Co., Ltd., brochure dated 1Sept.1979.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 11. For a diffraction-limited lens with a circular aperture the diameter of the diffraction spot is given by 1.22 λ/sinα, where λ is the wavelength, and sinα is the numerical aperture, Eq. (10c).

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Figures (17)

Fig. 1
Fig. 1

Side view of a cylindrical gradient-index optical fiber. The index of refraction varies radially with distance (r) from the optical axis. The case of unity magnification is illustrated where (k) is the maximum field height for image-forming rays. Note the sinusoidal ray path inside the fiber. Also note the inverted image midway through the fiber.

Fig. 2
Fig. 2

Plot of measured vertex distance (l), total conjugate (TC), and maximum field height (k) vs fiber length (L) for several Selfoc arrays (dots). The solid curves were calculated from the theoretical expressions, Eqs. (3)(5), with A = 0.1256, n0 = 1.538, and R = 0.525 mm. Measured k was determined from irradiance profiles (see Fig. 13).

Fig. 3
Fig. 3

Schematic diagram of the apparatus used to measure the Selfoc (square wave) MTF. The Ronchi ruling shows lines running perpendicular to the array direction. MTF was also measured for lines parallel to the array.

Fig. 4
Fig. 4

Measured (square wave) MTF for perpendicular (x) and parallel (0) lines for two different Selfoc lens arrays. Note that the MTF is greater for the array with longer fibers (shorter total conjugate). The spatial frequency was measured in lp/mm.

Fig. 5
Fig. 5

Measured (square wave) modulation at 6.2 lp/mm for Selfoc arrays with different fiber lengths and total conjugates. Plotted values are the averages for perpendicular and parallel object lines. Measurements were carried out with a broadband (tungsten) light source and with a narrowband (sodium) light source. The upper dashed line shows the calculated modulation due to diffraction alone. For a wavelength λ = 0.589 μm and a spatial frequency f = 6.2 lp/mm, the sine wave modulation is given by7 M(f) = (2/π)[ϕ − sinϕ cosϕ], where ϕ = cos−1f/2 sinα), and sinα is given by Eq. (10c). The plotted square wave modulation is7 S(f) = (4/π)[M(f) − ⅓M(3f) + 1/5M(5f) − …].

Fig. 6
Fig. 6

Measured dependence of vertex distance l on wavelength of the light source for Selfoc arrays with different fiber lengths (L). Note that the dispersion in l is greater for shorter L, i.e., for longer TC. The dispersion in l can be used to compute the dispersion in A for given n0 from Eq. (3).

Fig. 7
Fig. 7

Measured image modulation at 6.2 lp/mm for a Selfoc array with L = 29.29 mm and TC = 67.14 mm vs displacement from the plane of best focus for object lines perpendicular (x) to the array and parallel (0) to the array. Note that the depth of focus is much greater for the parallel lines.

Fig. 8
Fig. 8

Measured DOF for perpendicular (x) and parallel (0) lines vs total conjugate. The theoretical curves (dashed lines) were calculated from Eqs. (8) and (9), with D = 0.109 mm.

Fig. 9
Fig. 9

(a) Simple geometrical model used for calculation of DOF, Eq. (7); (b) model showing a blurred image at best focus leading to a shorter DOF.

Fig. 10
Fig. 10

Side view of a gradient-index fiber. For an object point (Lambertian radiance N) on the fiber axis, all image-forming rays are within the cone defined by the angle (α). The numerical aperture is given by sinα, Eq. (10c).

Fig. 11
Fig. 11

The image-plane irradiance profile for a single GRIN fiber lens. The profile is ellipsoidal as given by Eq. (11). The axial irradiance h0 is given by Eq. (10).

Fig. 12
Fig. 12

End view of a two-row Selfoc array. The center-to-center distance between adjacent fibers is (2bR). The irradiance boundary of an individual fiber is illustrated.

Fig. 13
Fig. 13

Image-plane irradiance profile for a two-row Selfoc lens array. A cross section of the profile is approximately triangular, with a base width of ( 2 k + 3 b R) and a height H ¯ 0 given by Eq. (16). Note the modulation in the y direction. The exposure is proportional to the integral of H over the x dimension, Eq. (21).

Fig. 14
Fig. 14

Normalized peak irradiance ( H ¯ 0 /πN = efficiency) for a two-row Selfoc array vs fiber length and total conjugate. The theoretical curve was calculated from Eq. (18b).

Fig. 15
Fig. 15

Measured relative exposure Ē/(N/v) vs total conjugate for a two-row Selfoc array. The theoretical exposure (solid line) is given by Eq. (22), with p = 2, (N/v) = 1.

Fig. 16
Fig. 16

Calculated irradiance modulation MH and exposure modulation ME vs fiber length.

Fig. 17
Fig. 17

(a) Schematic diagram of a single-row Selfoc array. The average irradiance along the line a distance x from the y axis is given by Eq. (A3); (b) diagram of a two-row array; average irradiance along the y axis given by Eq. (A5); (c) three-row array; average irradiance on y axis given by Eq. (A6).

Tables (1)

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Table I Fiber and Fiber Array Parameters

Equations (41)

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n ( r ) = n 0 ( 1 - A r 2 / 2 ) ,
π < A L < 2 π .
l = - tan ( A L / 2 ) / n 0 A ,
T C = 2 l + L .
k = - R sec ( A L / 2 ) .
D = X l Δ l .
DOF = 2 D ( l / X ) ,
DOF = - 2 D tan ( A L / 2 ) 3 n 0 A b R .
DOF = D sin ( A L / 2 ) n 0 A R .
h 0 = π N T sin 2 ( α )
h 0 = π N T n 0 2 A R 2 cos 2 ( A L / 2 ) ,
sin α - n 0 A R cos ( A L / 2 ) .
h ( x , y ) = h 0 [ 1 - ( x 2 + y 2 ) / k 2 ] 1 / 2 ,
H ( x , y ) = n h n ( x , y )
H ( x , y ) = h 0 n [ 1 - ( x - x n ) 2 / k 2 - ( y - y n ) 2 / k 2 ] 1 / 2 ,
H ¯ 0 = p h 0 ( π a 4 b ) [ 1 - ( p 2 - 1 ) b 2 / 4 a 2 ] ,
a = k R = - sec ( A L / 2 ) ,
b = ( fiber separation ) / ( 2 R ) .
H ¯ 0 = ( π 2 2 a b ) N T n 0 2 A R 2 [ 1 - 3 b 2 4 a 2 ] .
w = 2 k + 3 b R
ɛ p = p ( h 0 π N ) ( π a 4 b ) [ 1 - ( p 2 - 1 ) b 2 / 4 a 2 ] ,
ɛ 2 = ( π 2 a b ) T n 0 2 A R 2 [ 1 - 3 b 2 / a 2 ] .
H 0 = π N T 16 ( f / No . ) 2             ( conventional lens ) .
f / No . = [ a b 4 π p n 0 2 A R 2 [ 1 - ( p 2 - 1 ) b 2 / 4 a 2 ] 1 / 2 .
E ( y ) = 1 v H ( x , y ) d x .
E ¯ = p π 2 N T n 0 2 ( A R 3 / 3 b v )
M H = H 0 , max - H 0 , min H 0 , max + H 0 , min × 100 % ,
M E = E max - E min E max + E min × 100 % .
h ( x , y ) d y = h 0 - k 2 - x 2 + k 2 - x 2 [ 1 - ( x 2 + y 2 ) / k 2 ] 1 / 2 d x = π h 0 2 k ( k 2 - x 2 ) ,
H 1 ( x , y ¯ ) = m π h 0 ( k 2 - x 2 ) / 2 k m ( 2 b R ) + 2 k .
H 1 ( x , y ¯ ) = π h 0 4 k b R ( k 2 - x 2 )
H 1 ( x , y ¯ ) = π a h 0 4 b ( 1 - x 2 a 2 R 2 ) ,
H 1 ( 0 , y ¯ ) = π a h 0 4 b ,
H 2 ( 0 , y ¯ ) = H 1 ( 3 b R / 2 , y ¯ ) + H 1 ( - 3 b R / 2 , y ¯ ) = 2 ( π a h 0 4 b ) [ 1 - ( 3 b R / 2 ) 2 a 2 R 2 ] = ( 2 ) ( π a h 0 4 b ) ( 1 - 3 b 2 4 a 2 )
H 3 ( 0 , y ¯ ) = H 1 ( 0 , y ¯ ) + H 1 ( - 3 b R , y ¯ ) + H 1 ( + 3 b R , y ¯ ) , = π a h 0 4 b + ( 2 ) ( π a h 0 4 b ) [ 1 - ( 3 b R ) 2 a 2 R 2 ] ,
H 3 ( 0 , y ¯ ) = ( 3 ) ( π a h 0 4 b ) ( 1 - 8 b 2 4 a 2 )
H p ( 0 , y ¯ ) = p ( π a h 0 4 b ) [ 1 - ( p 2 - 1 ) b 2 / 4 a 2 ] ,
p 1 + 2 k / 3 b R .
E ¯ 1 = 1 v - k k H 1 ( x , y ¯ ) d x , = 1 v - k k ( π a h 0 4 b ) ( 1 - x 2 a 2 R 2 ) d x , = π a 2 R h 0 3 b v ,
E ¯ 1 = π 2 N T n 0 2 A R 3 / 3 b v ,
E ¯ p = p π 2 N T n 0 2 A R 3 / 3 b v ,

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