Abstract

Gradient-index lens arrays that make reduced or enlarged images have been invented at Xerox Corp. An important optical property of these lens arrays is their radiometric speed. Thus a general procedure for the radiometric analysis of arbitrary magnification gradient-index lens arrays has been developed. The analysis has been applied to a particular array which was fabricated by Nippon Sheet Glass Co. and tested at Xerox.

© 1985 Optical Society of America

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References

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  1. J. D. Rees, D. Kay, W. L. Lama, “Gradient Index Lens Array Having Reduction Properties,” U.S. Patent4,331,380 (25May1982).
  2. J. D. Rees, W. L. Lama, “Reduction/Enlargement Gradient-Index Lens Arrays,” Appl. Opt. 23, 1715 (1984).
    [CrossRef] [PubMed]
  3. W. L. Lama, J. Durbin, “Radiometric Analysis of Gradient-Index Lens Arrays for Reduction and Enlargement,” Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.
  4. F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368 (1963).
    [CrossRef]
  5. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 186.
  6. K. Matsushita, M. Toyama, “Uneveness of illuminance Caused by Gradient-Index Fiber Arrays,” Appl. Opt. 19, 1070 (1980).
    [CrossRef] [PubMed]
  7. W. L. Lama, “Optical Properties of GRIN Fiber Lens Arrays: Dependence on Fiber Length,” Appl. Opt. 21, 2739 (1982).
    [CrossRef] [PubMed]
  8. J. D. Rees, W. L. Lama, “Some Radiometric Properties of Gradient-Index Fiber Lenses,” Appl. Opt. 19, 1065 (1980).
    [CrossRef] [PubMed]

1984 (1)

1982 (1)

1980 (2)

1963 (1)

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368 (1963).
[CrossRef]

Durbin, J.

W. L. Lama, J. Durbin, “Radiometric Analysis of Gradient-Index Lens Arrays for Reduction and Enlargement,” Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.

Kay, D.

J. D. Rees, D. Kay, W. L. Lama, “Gradient Index Lens Array Having Reduction Properties,” U.S. Patent4,331,380 (25May1982).

Lama, W. L.

J. D. Rees, W. L. Lama, “Reduction/Enlargement Gradient-Index Lens Arrays,” Appl. Opt. 23, 1715 (1984).
[CrossRef] [PubMed]

W. L. Lama, “Optical Properties of GRIN Fiber Lens Arrays: Dependence on Fiber Length,” Appl. Opt. 21, 2739 (1982).
[CrossRef] [PubMed]

J. D. Rees, W. L. Lama, “Some Radiometric Properties of Gradient-Index Fiber Lenses,” Appl. Opt. 19, 1065 (1980).
[CrossRef] [PubMed]

W. L. Lama, J. Durbin, “Radiometric Analysis of Gradient-Index Lens Arrays for Reduction and Enlargement,” Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.

J. D. Rees, D. Kay, W. L. Lama, “Gradient Index Lens Array Having Reduction Properties,” U.S. Patent4,331,380 (25May1982).

Matsushita, K.

Nicodemus, F. E.

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368 (1963).
[CrossRef]

Rees, J. D.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 186.

Toyama, M.

Am. J. Phys. (1)

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368 (1963).
[CrossRef]

Appl. Opt. (4)

Other (3)

J. D. Rees, D. Kay, W. L. Lama, “Gradient Index Lens Array Having Reduction Properties,” U.S. Patent4,331,380 (25May1982).

W. L. Lama, J. Durbin, “Radiometric Analysis of Gradient-Index Lens Arrays for Reduction and Enlargement,” Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 186.

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Figures (12)

Fig. 1
Fig. 1

Schematic view of an R/E GRIN lens array.

Fig. 2
Fig. 2

Schematic diagram of tilted GRIN rod lens showing the geometric design variables. A central axial ray is traced from an object point a distance x0 from the center of the array. Here N is the normal to the rod endface.

Fig. 3
Fig. 3

Schematic side view of a single gradient-index lens forming a reduced erect image. Rays are drawn which show the maximum field height and the numerical aperture of the lens.

Fig. 4
Fig. 4

Diagram of a GRIN lens making a reduced image with irradiance h1 of an elemental object of radiance N.

Fig. 5
Fig. 5

Ellipsoidal image-plane irradiance profile produced by a single GRIN lens at the center of the array.

Fig. 6
Fig. 6

Diagram of three GRIN lenses at the center of the array. The centers of the overlapping irradiance profiles are separated by (2bR).

Fig. 7
Fig. 7

Two-dimensional image plane irradiance profile. At the center of the lens array the width of the profile is (2k1).

Fig. 8
Fig. 8

Diagram showing a cone of rays from the exit pupil of a tilted lens. The rays converge to an image point on the y axis a distance Δy from the CAR intercept.

Fig. 9
Fig. 9

Emitting area on the endface of a tilted rod for an image point displaced by Δy = 1.8 mm from the CAR intercept. In this case S3 = 7.89 mm and V = 14.95 mm.

Fig. 10
Fig. 10

Single-lens irradiance profiles for scans along the x and y axes. S3 = 7.89 mm and V = 14.95 mm. The width of the x profile is 2kx = 4.5 mm, and the width of the y profile is 2ky = 3.8 mm. The lens transmission T = 0.9.

Fig. 11
Fig. 11

Diagram of a single tilted GRIN lens and the 2-D irradiance profile it produces.

Fig. 12
Fig. 12

Diagram of the experimental setup used to measure the irradiance profile H ( x ¯ , y ) at various points x along the array.

Tables (3)

Tables Icon

Table I Typical Lens Array Parameters

Tables Icon

Table II Lens Parameters at Array Center

Tables Icon

Table III Radiometric Results

Equations (40)

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n ( r ) = n 0 ( 1 A r 2 / 2 )
l 0 = 1 / ( m ) cos ( A L ) n 0 A sin ( A L ) ,
l 1 = m cos ( A L ) n 0 A sin ( A L ) ,
k 0 = R [ sin 2 ( A L ) + [ ( 1 / m ) cos ( A L ) ] 2 sin 2 ( A L ) ] 1 / 2 .
m = k 1 k 0 = θ 0 θ 1 ,
θ 0 = n 0 A R 2 / k 0 .
h 1 = π N T sin 2 θ 1 .
h 1 = π N T n 0 2 A R 4 / k 1 2 .
h ( x , y ) = h 1 [ 1 ( x 2 + y 2 ) / k 1 2 ] 1 / 2 ,
H ( x ¯ , y ) = Q h ( x , y ) d x Q ( 2 b R ) = h ( x , y ) d x 2 b R ,
H ( x ¯ , y ) = π h 1 ( k 1 2 y 2 ) 4 b R k 1 .
H ( x ¯ , o ) = π h 1 k 1 4 b R = π 2 N T n 0 2 A R 3 4 b k 1 ,
= π T n 0 2 A R 3 / 4 b k 1 .
E = H ( x ¯ , y ) d t ,
E = 1 υ k 1 k 1 H ( x ¯ , y ) d y ,
E = π h 1 k 1 2 3 b R υ .
E = π 2 N T n 0 2 A R 3 / 3 b υ .
b E = m E b R ,
k 1 E = m E k 1 R ,
E R = b R k 1 R b E k 1 E = 1 m E 2 = m R 2 .
E E E R = b R υ R b E υ E = m R υ R υ E .
Ω = a cos ( θ c ) / l 1 2 ,
l 1 = V / cos θ c ,
cos θ c = V [ ( S 3 Δ x ) 2 + ( Δ y ) 2 + V 2 ] 1 / 2 ,
h ( x , y ) = N T Ω cos θ c .
h ( x , y ) = N T a cos 4 ( θ c ) / V 2 .
h ( x , y ) = N T π r x r y cos 4 ( θ ) c / V 2 .
H ( x ¯ , o ) = 1 2 b R h ( x , o ) d x .
H ( x ¯ , o ) = 5.40 × 10 3 N .
= H π N = 0.172 % .
E = h ( x , y ) d x d y 2 b R υ .
E = 4.14 × 10 3 π N υ .
h ( x , y ) = h p ( 1 x 2 / k x 2 y 2 / k y 2 ) 1 / 2 ,
H ( x ¯ , y ) = Q h ( x , y ) d x Q ( 2 b R ) .
H ( x ¯ , y ) = π h p k x ( k y 2 y 2 ) 4 b R k y 2 .
H ( x ¯ , o ) = π h p k x / 4 b R .
= H ( x ¯ , 0 ) π N = h p k x 4 b R N .
E = 1 υ H ( x ¯ , y ) d y .
E = π h p k x k y 3 b R υ .
E = h p k x k y 3 b R N .

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