Abstract

The imaging and radiometric properties of erect lens arrays made up of small biconvex microlenses are derived from a ray analysis. The lens arrays provide erect, unit magnification images. The relationship between the radii of curvature, the lens thickness, and the one-to-one conjugate distance is derived for both the single-layer case and a double-layer structure, which contains field lenses. Radiometric properties of the microlens and the array are derived for both structures. The results are compared to experimentally measured values obtained from arrays fabricated by a photothermal process.

© 1991 Optical Society of America

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References

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  1. M. Kawazu, Y. Ogura, “Application of gradient-index fiber arrays to copying machines,” Appl. Opt. 19, 1105–1112 (1980).
    [CrossRef] [PubMed]
  2. N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” J. Appl. Phys. 24, 2520–2525 (1985).
  3. N. F. Borrelli, D. L. Morse, “Microlens arrays produced by a photolytic technique,” Appl. Opt. 27, 476–479 (1988).
    [CrossRef] [PubMed]
  4. N. F. Borrelli, D. L. Morse, R. H. Bellman, “Further properties of lens arrays produced by photolysis,” presented at the Second Microoptics Conference, MOC/GRIN 89', 24–26 July 1989, Tokyo.
  5. W. Brower, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
  6. H. Kogelnik, “Propagation of laser beams,” in Applied Optics and Engineering, (Academic, New York, 1979), Vol. 7.
  7. W. L. Lama, “Optical properties of GRIN fiber lens arrays: dependence on fiber length,” Appl. Opt. 21, 2739–2746 (1982).
    [CrossRef] [PubMed]

1988 (1)

1985 (1)

N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” J. Appl. Phys. 24, 2520–2525 (1985).

1982 (1)

1980 (1)

Bellman, R. H.

N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” J. Appl. Phys. 24, 2520–2525 (1985).

N. F. Borrelli, D. L. Morse, R. H. Bellman, “Further properties of lens arrays produced by photolysis,” presented at the Second Microoptics Conference, MOC/GRIN 89', 24–26 July 1989, Tokyo.

Borrelli, N. F.

N. F. Borrelli, D. L. Morse, “Microlens arrays produced by a photolytic technique,” Appl. Opt. 27, 476–479 (1988).
[CrossRef] [PubMed]

N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” J. Appl. Phys. 24, 2520–2525 (1985).

N. F. Borrelli, D. L. Morse, R. H. Bellman, “Further properties of lens arrays produced by photolysis,” presented at the Second Microoptics Conference, MOC/GRIN 89', 24–26 July 1989, Tokyo.

Brower, W.

W. Brower, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

Kawazu, M.

Kogelnik, H.

H. Kogelnik, “Propagation of laser beams,” in Applied Optics and Engineering, (Academic, New York, 1979), Vol. 7.

Lama, W. L.

Morgan, W. L.

N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” J. Appl. Phys. 24, 2520–2525 (1985).

Morse, D. L.

N. F. Borrelli, D. L. Morse, “Microlens arrays produced by a photolytic technique,” Appl. Opt. 27, 476–479 (1988).
[CrossRef] [PubMed]

N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” J. Appl. Phys. 24, 2520–2525 (1985).

N. F. Borrelli, D. L. Morse, R. H. Bellman, “Further properties of lens arrays produced by photolysis,” presented at the Second Microoptics Conference, MOC/GRIN 89', 24–26 July 1989, Tokyo.

Ogura, Y.

Appl. Opt. (3)

J. Appl. Phys. (1)

N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” J. Appl. Phys. 24, 2520–2525 (1985).

Other (3)

N. F. Borrelli, D. L. Morse, R. H. Bellman, “Further properties of lens arrays produced by photolysis,” presented at the Second Microoptics Conference, MOC/GRIN 89', 24–26 July 1989, Tokyo.

W. Brower, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

H. Kogelnik, “Propagation of laser beams,” in Applied Optics and Engineering, (Academic, New York, 1979), Vol. 7.

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Figures (13)

Fig. 1
Fig. 1

Schematic representation of two-dimensional thick lens arrays with single and double layers, respectively.

Fig. 2
Fig. 2

Lens of thick lens array defining parameters: (a) lens radius, thickness, radius of curvature, and sag; (b) lens thickness to radius, approximately to scale; (c) field angle.

Fig. 3
Fig. 3

Diagrams defining (a) ray height, ray slope, and lens power; (b) erect one-to-one imaging condition demonstrated by the a and b rays; (c) maximum field height condition.

Fig. 4
Fig. 4

Diagrams defining (a) ray height, ray slope, element thickness, and lens powers for the stacked assembly; (b) erect one-to-one condition.

Fig. 5
Fig. 5

(a) Schematic ray representation of the method for determining the irradiance profile; (b) overlap of the exit and entrance pupils.

Fig. 6
Fig. 6

Calculated irradiance profiles for the erect imaging condition: A, single element; B, double element with optimum field lens curvature, Eq. (26), and C, double element with all lens powers equal.

Fig. 7
Fig. 7

Hexagonal close-packed array defining spacing parameters.

Fig. 8
Fig. 8

Irradiance overlap produced by 0.16-mm-diameter lenses spaced on 0.175-μm centers: lens thickness, 1.45 mm; total conjugate, 6 mm. The illumination field 2k is 0.75 mm.

Fig. 9
Fig. 9

Computed irradiance modulation of a hexagonal close-packed array of lenses in the one-to-one imaging condition as a function of the generalized parameter, a/b = k/Rb. The modulation was computed as a contrast [I(bright) − I(dark)]/[I(bright) − I(dark)] at the spatial position of its largest value for the given value of a/b. The calculation was done for single-layer and stacked configurations, respectively.

Fig. 10
Fig. 10

Variation of the total conjugate distance with lens power: solid curve, computed from Eq. (16); circles, experimental data; (a) two-element stacked arrays; (b) single-element arrays.

Fig. 11
Fig. 11

Schematic diagram for the method of measurement of the radiometric efficiency.

Fig. 12
Fig. 12

Electron photomicrograph of the interface between the clear glass lens and the crystalline surround; the cross section through the lens shown in both figures shows a higher magnification of the boundary.

Figure 13
Figure 13

Ray trace diagrams comparing the extent of vignetting in the one-to-one imaging condition for A–C, the single-element array with increasing object height; D–F, the stacked double-element array with increasing object height.

Tables (2)

Tables Icon

Table I Radiometric Efficiencya

Tables Icon

Table II Comparison of Percent Contrast versus Spatial Frequency of Single- and Double-Element Arrays

Equations (42)

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δ 2 2 R c δ + R 2 = 0 .
δ = R c { 1 [ 1 ( R / R c ) 2 ] 1 / 2 } .
ϕ = ( n 1 ) / R c = ( n 1 ) c ,
( y 2 u 2 ) = M ( y 1 u 1 ) ,
M = [ 1 ( T 1 ϕ 1 / n ) T 1 / n ( T 1 ϕ 1 ϕ 2 / n ) ϕ 1 ϕ 2 1 ( T 1 ϕ 2 / n ) ] .
M = R 2 T n R 1 = [ 1 0 ϕ 2 n n ] [ 1 T 1 0 1 ] [ 1 0 ϕ 1 / n 1 / n ] .
T = [ 1 t 2 0 1 ] ,
( y i , a , b u i , a , b ) = TM ( y a , b u a , b ) .
t 1 = ( T 1 / n ) ϕ 2 ϕ 1 ϕ 2 ( T / n ) ( ϕ 1 + ϕ 2 ) ,
t 1 / t 2 = ϕ 2 / ϕ 1 .
( y a , b r u a , b r ) = T n R 1 ( y a , b u a , b ) ,
k = R ( ϕ 1 t 1 1 ) .
k = 2 nRt / T 1 .
tan θ = 2 n R / T 1 .
( y 4 u 4 ) = M 2 M 1 ( y 1 u 1 ) .
t = T / ( T ϕ n ) .
k = nRt / T .
H ( r ) = H 0 A ( y ) / π R 2 ,
H 0 = N π R 2 / t 2 2 ,
Δ = | y 2 b | = T y / n t 1 = 2 R ( y / k ) ,
r p = ( 1 / 2 ) | y 2 c y 2 a | = R | ( T / n ) ( ϕ 1 1 / t 1 ) 1 | = R ϕ 1 / ϕ 2 .
H ( r ) = ( 2 / π ) H 0 { cos 1 ( r / k ) ( r / k ) [ 1 ( r / k ) 2 ] 1 / 2 } .
P 1 = H ( r ) r d r d θ ,
P 1 = ( π / 4 ) H 0 k 2 = N ( π n R / T ) 2 .
y 4 = y 1 + 2 ( 1 T ϕ 2 / n ) [ y 1 ( 1 T ϕ 1 / n ) + T u 1 / n ] .
ϕ 2 = n / T .
H ( r ) = H ( 0 ) .
P 1 = π H 0 k 2 ,
ϕ 2 = n / T 1 ; ϕ 3 = n / T 2 .
Δ = 2 R ( r / k ) | ( 1 T ϕ 2 / n ) | ,
H = H 0 ( 2 / π ) { cos 1 ( y q / k ) ( y q / k ) 2 [ 1 ( y q / k ) 2 ] 1 / 2 } ,
q = | ( 1 T ϕ 2 / n ) | .
P 1 = π H 0 k 2 [ 1 ( 8 / 3 π ) q ] .
A = ( 2 b R n x ) [ ( 3 ) 1 / 2 b R n y ]
P = n x n y P 1 ,
= [ π / 2 ( 3 ) 1 / 2 ] ( n R / b T ) 2 .
f / No . = 1 / 2 ( ) 1 / 2 .
NA = ( ) 1 / 2 .
n L = 6 k / 2 b R .
[ I ( bright ) I ( dark ) ] / [ I ( bright ) + I ( dark ) ] ,
d t / d ϕ = [ T / ( T ϕ n ) ] 2 = t 2
= ( A s / z 2 π ) ( S 1 / S 0 ) ,

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