Abstract

A novel concept of a gradient-index lens array for reduction and enlargement is described. The array is composed of radial gradient-index rod lenses arrayed in a fanlike manner. Design and radiometric equations are presented. Some image aberrations are discussed in a qualitative fashion. The imaging performance and radiometric properties of a sample lens array are given.

© 1984 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 Company, Ltd., Osaka, Japan.
  2. F. P. Kapron, “Parabolic Index-Gradient Cyclindrical Lens,” J. Opt. Soc. Am 60, 1433 (1970).
    [CrossRef]
  3. Selfoc Handbook, (NSG America, Inc., Clark, N.J., 1981).
  4. T. Scheimpflug, “Der Photoperspektograph und Seine Anwendung,” Photogr. Korresp. 43, 516 (1906).
  5. J. D. Rees, W. Lama, “Some Radiometric Properties of Gradient-Index Fiber-Lenses,” Appl. Opt. 19, 1065 (1980).
    [CrossRef] [PubMed]
  6. K. Matsushita, M. Toyama, “Uneveness of Illuminance Caused by Gradient-Index Fiber Arrays,” Appl. Opt. 19, 1070 (1980).
    [CrossRef] [PubMed]
  7. W. Lama, J. Durbin, in Technical Digest, Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.

1980 (2)

1970 (1)

F. P. Kapron, “Parabolic Index-Gradient Cyclindrical Lens,” J. Opt. Soc. Am 60, 1433 (1970).
[CrossRef]

1906 (1)

T. Scheimpflug, “Der Photoperspektograph und Seine Anwendung,” Photogr. Korresp. 43, 516 (1906).

Durbin, J.

W. Lama, J. Durbin, in Technical Digest, Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.

Kapron, F. P.

F. P. Kapron, “Parabolic Index-Gradient Cyclindrical Lens,” J. Opt. Soc. Am 60, 1433 (1970).
[CrossRef]

Lama, W.

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

W. Lama, J. Durbin, in Technical Digest, Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.

Matsushita, K.

Rees, J. D.

Scheimpflug, T.

T. Scheimpflug, “Der Photoperspektograph und Seine Anwendung,” Photogr. Korresp. 43, 516 (1906).

Toyama, M.

Appl. Opt. (2)

J. Opt. Soc. Am (1)

F. P. Kapron, “Parabolic Index-Gradient Cyclindrical Lens,” J. Opt. Soc. Am 60, 1433 (1970).
[CrossRef]

Photogr. Korresp. (1)

T. Scheimpflug, “Der Photoperspektograph und Seine Anwendung,” Photogr. Korresp. 43, 516 (1906).

Other (3)

W. Lama, J. Durbin, in Technical Digest, Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-B3.

Selfoc Handbook, (NSG America, Inc., Clark, N.J., 1981).

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

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Figures (16)

Fig. 1
Fig. 1

Schematic side view of a unit magnification Selfoc lens array. The inverted intermediate images are jumbled. However, by forming erect images, a usuable composite image is produced by the array.

Fig. 2
Fig. 2

Schematic side view of a single radial gradient-index lens forming a reduced erect image. A limiting chief ray and an axial ray which just miss the edges of the GRIN lens are traced.

Fig. 3
Fig. 3

Schematic representations of two two-lens gradient-index lens arrays making reduced images. (a) When the two lenses are parallel, the two reduced images do not overlap properly. (b) If the lenses are tilted slightly, the images can be made to overlap properly.

Fig. 4
Fig. 4

Multilens reduction/enlargement lens array shown in the reduction mode. The individual GRIN lenses are positioned in a fanlike manner. The top and bottom surfaces of the array are weakly convex. Central axial rays which coincide with the physical axes of the GRIN rods lenses are traced.

Fig. 5
Fig. 5

Three lens systems imaging a tilted object at unit magnification. (a) A nongradient-index inverting system forms an image that is tilted an equal amount but in the direction opposite to that of the object. The nongradient-index (b) and gradient-index (c) erecting systems form images that are tilted an equal amount in the same direction as the object, a desirable result.

Fig. 6
Fig. 6

Schematic diagram of a tilted GRIN lens showing the geometrical design variables. A central axial ray is traced from an object a distance X0 from the center of the array. The image is formed at X1 = m0X0.

Fig. 7
Fig. 7

Single GRIN lens forming a reduced image of a tilted object. The image plane is also tilted and is parallel to the tilted object plane. In the tilted image plane, magnification varies with field position; i.e., distortion is present.

Fig. 8
Fig. 8

Schematic diagram of three GRIN lenses imaging an object point P in a tilted object plane. Due to distortion resulting from the tilted object and image planes, rays from the object point P are not all imaged at the desired image point P′. Rays through lenses (1) and (3), respectively, undergo an image plane magnification less than and greater than the desired magnification m0,2.

Fig. 9
Fig. 9

Schematic diagram of a single tilted GRIN lens with endfaces that are not perpendicular to the lens axis. A white light ray is incident on the object side face and then is dispersed into red, yellow, and blue components. Note the large angle of incidence resulting from the nonsquare endface. This leads to dispersion and finally lateral chromatic aberration in the image.

Fig. 10
Fig. 10

Exposure modulation as a function of the ratio a1/b.

Fig. 11
Fig. 11

Example one-row reduction/enlargement gradient-index lens array. The array is shown in the reduction mode forming an image at an array magnification of 0.707. The design parameters of lenses 0 and 103 are given in Table I.

Fig. 12
Fig. 12

GRIN rod lens imaging an axial object point at nonunit magnification. A sinusoidal ray path is shown.

Fig. 13
Fig. 13

GRIN lens imaging an object of height d0 at nonunit magnification.

Fig. 14
Fig. 14

Geometry used to determine the relationship between the lens endface angle and the radius of curvature of a surface of the array.

Fig. 15
Fig. 15

Geometry used to determine the relationship between the lateral magnification m and the array magnification m0.

Fig. 16
Fig. 16

GRIN lens making a reduced image of an elemental object of radiance N used in the radiometric analysis.

Tables (1)

Tables Icon

Table I Design Parameters for a Central and End Lens in an Example Reduction GRIN Arraya

Equations (73)

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n ( r ) = n 0 ( 1 - A r 2 / 2 ) ,             0 r R ,
m = k 1 / k 0 = θ 0 / θ 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 .
m = m 0 cos ( ϕ 1 - β ) cos ( ϕ 0 + α )
tan α = 2 X 2 [ l 0 cos ( ϕ 0 + α ) - l 0 ] X 2 2 - [ l 0 cos ( ϕ 0 + α ) - l 0 ] 2 ,
X 2 = X 0 - l 0 sin ( ϕ 0 + α ) ,
tan β = 2 X 3 [ l 1 cos ( ϕ 1 - β ) - l 1 ] X 3 2 - [ l 1 cos ( ϕ 1 - β ) - l 1 ] 2 ,
X 3 = X 0 - l 0 sin ( ϕ 0 + α ) - L sin ( ϕ 0 + α ) ,
TC = l 0 cos ( ϕ 0 + α ) + L cos ( ϕ 0 + α ) + l 1 cos ( ϕ 1 - β ) .
X 0 ( 1 - m 0 ) = l 0 sin ( ϕ 0 + α ) + L sin ( ϕ 0 + α ) + l 1 sin ( ϕ 1 - β ) .
sin ϕ 0 = n 0 sin ϕ 0 ¢ ;
sin ϕ 1 = n 0 sin ϕ 1 ;
ϕ 1 - β = ϕ 0 + α .
R 0 = X 0 - l 0 sin ( ϕ 0 + α ) sin α ,
R 1 = X 0 - l 0 sin ( ϕ 0 + α ) - L sin ( ϕ 0 + α ) sin β .
m 0 , R < m 0 , C < m 0 , L .
h 0 = π N T n 0 2 A R 2 / a 1 2 ,
a 1 = m [ sin 2 ( A L ) + [ 1 / m - cos ( A L ) ] 2 sin 2 ( A L ) ] 1 / 2
h ( r ) = h 0 [ 1 - ( r / k 1 ) 2 ] 1 / 2 ,
k 1 = a 1 R
E ( x ) = 1 v H ( x , y ) d y ,
E ¯ = h ( x , y ) d x d y 2 b R v ,
E ¯ = π h 0 k 1 2 3 b R v = π 2 N T n 0 2 A R 3 3 b v .
E ¯ E / E ¯ 1 × = b 1 × / b E = 1 / m E ,
h ( x , y ) = N T Ω cos ψ ,
h ( x , y ) = h 0 ( 1 - x 2 / k x 2 - y 2 / k y 2 ) 1 / 2 ,
E ¯ = π h 0 k x k y 3 b R v .
E R = 6.28 × 10 - 3 ( N / v ) ,
n ( r ) = n 0 ( 1 - A r 2 / 2 ) ,
x = B 1 sin ( A z ) + B 2 cos ( A z ) ,
tan θ = d x d z = B 1 A cos ( A z ) - B 2 A sin ( A z ) .
x = θ 0 n 0 A sin ( A z ) + x 0 cos ( A z ) ,
θ = - x 0 A sin ( A z ) + θ 0 n 0 cos ( A z ) .
x 1 = θ 0 n 0 A sin ( A L ) + x 0 cos ( A L ) ,
θ 1 = - x 0 A n 0 sin ( A L ) + θ 0 cos ( A L ) .
l 1 = sin ( A L ) / n 0 A + l 0 cos ( A L ) l 0 n 0 A sin ( A L ) - cos ( A L ) .
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 ) .
x 0 = d 0 + l 0 θ 0 .
R = θ 0 n 0 A sin ( A z ) + x 0 cos ( A z ) ,
θ = - x 0 A sin ( A z ) + θ 0 n 0 cos ( A z ) .
θ 0 = - n 0 2 A l 0 d 0 ± n 0 A [ ( 1 + n 0 2 A l 0 2 ) R 2 - d 0 2 ] 1 / 2 1 + n 0 2 A l 0 2 .
θ 0 , m = ± n 0 A R ( 1 + n 0 2 A l 0 2 ) 1 / 2 .
k 0 = R ( 1 + n 0 2 A l 0 2 ) 1 / 2 .
k 0 = R { sin 2 ( A L ) + [ 1 / m - cos ( A L ) ] 2 sin 2 ( A L ) } 1 / 2 .
S = R 0 - y = R 0 - ( R 0 2 - x 2 ) 1 / 2 .
tan α = d S d x = x ( R 0 2 - x 2 ) 1 / 2 = x R 0 - s .
y = ( R 0 2 - x 2 ) 1 / 2 = R 0 - s .
R 0 = ( S 2 + x 2 ) / 2 S ,
tan α = 2 x S x 2 - S 2 .
x = X 0 - l 0 sin ( θ 0 + α ) ,
S = l 0 cos ( ϕ 0 + α ) - l 0 ,
sin α = x / R 0 ,
R 0 = X 0 - l 0 sin ( ϕ 0 + α ) sin α .
X 0 , l = k 0 sec ( ϕ 0 + α ) 1 + ( k 0 / l 0 ) tan ( ϕ 0 + α ) ,
X 0 , r = k 0 sec ( ϕ 0 + α ) 1 - ( k 0 / l 0 ) tan ( ϕ 0 + α ) ,
X 1 , l = k 1 sec ( ϕ 1 - β ) 1 - ( k 1 / l 1 ) tan ( ϕ 1 - β ) ,
X 1 , r = k 1 sec ( ϕ 1 - β ) 1 + ( k 1 / l 1 ) tan ( ϕ 1 - β ) .
m 0 , l = X 1 , l / X 0 , l ;
m 0 , r = X 1 , r / X 0 , r .
m 0 = m cos ( ϕ 0 + α ) cos ( ϕ 1 - β ) ,
P = 2 π N Δ a 0 θ 0 , m sin θ 0 cos θ 0 d θ 0 , P = π N Δ a sin 2 θ 0 , m .
h 0 = T P m 2 Δ a = π N T sin 2 θ 0 , m m 2 .
h 0 = π N T sin 2 θ 1 , m ,
sin θ 0 θ 0 = n 0 A R 2 / k 0 .
h 0 = π N T n 0 2 A R 2 / a 1 2 ,
a 1 = m { sin 2 ( A L ) + [ 1 / m - cos ( A L ) ] 2 sin 2 ( A L ) } 1 / 2 ,
h ( x , y ) = h 0 [ 1 - ( x 2 + y 2 ) / k 1 2 ] 1 / 2 ,
H ( x ¯ , y ) = N h ( x , y ) d x N ( 2 b R ) ,
H ( x ¯ , y ) = π h 0 ( k 1 2 - y 2 ) 4 b R k 1 .
E ¯ = 1 v H ( x ¯ , y ) d y ,
E ¯ = π h 0 k 1 2 3 b R v .

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