Abstract

Third-order aberration theory is extended to analyze gradient-index arrays. This theory is used to design a reduction array for photocopy applications. An array with untilted endfaces is designed and shown to have better image quality than previously designed arrays.

© 1985 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., Osaka, Japan.
  2. J. D. Rees, D. B. Kay, W. L. Lama, “Gradient-Index Array Having Reduction Properties,” U.S. patent4,331,380 (25May1982).
  3. J. D. Forer, Third Order Aberration Analysis of Gradient Index Arrays, M.S. Thesis, U. Rochester, Rochester, N.Y. (1983).
  4. P. J. Sands, “Third Order Aberrations of Inhomogeneous Lenses,” J. Opt. Soc. Am. 60, 1436 (1970).
    [CrossRef]
  5. Military Standardization Handbook 141, Optical Design, Defense Supply Agency, Washington, D.C. (1962), pp. 8–14.
  6. Selfoc Handbook (Nippon Sheetglass, America, Inc. Location, year).
  7. J. D. Rees, “Reduction/Enlargement Gradient-Index Lens Array,” Appl. Opt. 23, 1715 (1984).
    [CrossRef] [PubMed]
  8. J. D. Rees, “Non-Gaussian Imaging Properties of GRIN Fiber Lens Arrays,” Appl. Opt. 21, 1009 (1982).
    [CrossRef] [PubMed]

1984 (1)

1982 (1)

1970 (1)

P. J. Sands, “Third Order Aberrations of Inhomogeneous Lenses,” J. Opt. Soc. Am. 60, 1436 (1970).
[CrossRef]

Forer, J. D.

J. D. Forer, Third Order Aberration Analysis of Gradient Index Arrays, M.S. Thesis, U. Rochester, Rochester, N.Y. (1983).

Kay, D. B.

J. D. Rees, D. B. Kay, W. L. Lama, “Gradient-Index Array Having Reduction Properties,” U.S. patent4,331,380 (25May1982).

Lama, W. L.

J. D. Rees, D. B. Kay, W. L. Lama, “Gradient-Index Array Having Reduction Properties,” U.S. patent4,331,380 (25May1982).

Rees, J. D.

Sands, P. J.

P. J. Sands, “Third Order Aberrations of Inhomogeneous Lenses,” J. Opt. Soc. Am. 60, 1436 (1970).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

P. J. Sands, “Third Order Aberrations of Inhomogeneous Lenses,” J. Opt. Soc. Am. 60, 1436 (1970).
[CrossRef]

Other (5)

Military Standardization Handbook 141, Optical Design, Defense Supply Agency, Washington, D.C. (1962), pp. 8–14.

Selfoc Handbook (Nippon Sheetglass, America, Inc. Location, year).

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

J. D. Rees, D. B. Kay, W. L. Lama, “Gradient-Index Array Having Reduction Properties,” U.S. patent4,331,380 (25May1982).

J. D. Forer, Third Order Aberration Analysis of Gradient Index Arrays, M.S. Thesis, U. Rochester, Rochester, N.Y. (1983).

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Figures (12)

Fig. 1
Fig. 1

Gradient-index reduction/enlargement array.

Fig. 2
Fig. 2

First-order properties of gradient-index rod.

Fig. 3
Fig. 3

Geometry of plus one magnification array.

Fig. 4
Fig. 4

Rotation of local coordinate system so that the new meridional axis is parallel to the object.

Fig. 5
Fig. 5

Maximum field angle.

Fig. 6
Fig. 6

Geometry used to calculate coordinates of a ray at first surface.

Fig. 7
Fig. 7

Geometry used to calculate the azimuthal angle of a ray at first surface.

Fig. 8
Fig. 8

(a) Gradient-index array with parallel rods (reduced magnification); (b) gradient-index array with tilted rods (reduced magnification).

Fig. 9
Fig. 9

Spot diagram of axial object point for reduction array with tilted surfaces (generated by ray tracing).

Fig. 10
Fig. 10

Geometry of reduction array (untilted endfaces).

Fig. 11
Fig. 11

Detail of geometry of one rod in the reduction array (tilted endfaces).

Fig. 12
Fig. 12

Spot diagram for the reduction array with untilted surfaces. Spots (a), (b), (c), and (d) are from rods (3,0), (2,0), (4,0), and (1,0), respectively.

Tables (7)

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Table I Notation of First-Order Properties

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Table II Physical Parameters of Array (Tilted Endfaces)

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Table III Imaging Conditions in Local Coordinates (Tilted Endfaces)

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Table IV Paraxial Image Points in Universal Coordinates (Tilted Endfaces

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Table V Physical Parameters of Reduction Array (Untilted Endfaces)

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Table VI Paraxial Image Points in Universal Coordinates (Untilted Endfaces)

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Table VII Third-Order Aberration Coefficients for Reduction Array (Untilted Surfaces)

Equations (42)

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n ( r ) = N 00 + N 10 r 2 + N 20 r 4 + ,
y ( Z ) = y 0 cos ( 2 N 10 N 00 Z ) + u 0 2 N 10 N 00 × sin ( 2 N 10 N 00 Z ) ,
u ( Z ) = y 0 2 N 10 N 00 sin ( 2 N 10 N 00 Z ) + u 0 cos ( 2 N 10 N 00 Z ) ,
x = σ 1 ρ ¯ 3 cos ( θ ) + σ 2 ρ ¯ 2 h [ ( 2 + cos 2 θ ) ] + ( 3 σ 3 + σ 4 ) ρ ¯ h 2 cos ( θ ) + σ 5 h 3 + 0 ( 5 ) ,
y = σ 1 ρ ¯ 3 sin ( θ ) + σ 2 ρ ¯ 2 h sin ( 2 θ ) + ( σ 3 + σ 4 ) ρ ¯ h 2 sin ( θ ) + 0 ( 5 )
σ i = j = 1 k ( σ i homogeneous surface + σ i inhomogeneous surface + σ i inhomogeneous transfer ) ,
i = 1,2,3,4,5 , k = number of surfaces .
h ( J , K ) = { [ X LOC ( J , K ) X OB ] 2 + [ Y LOC ( J , K ) Y OB ] 2 } 1 / 2 .
[ X LOC ( J , K ) R ] < x 1 < [ X LOC ( J , K ) + R ] ,
[ Y LOC ( J , K ) R ] < y 1 < [ Y LOC ( J , K ) + R ] ,
ϕ ( J , K ) = tan 1 [ X OB X LOC ( J , K ) Y OB Y LOC ( J , K ) ] .
x 1 = x 1 · cos ϕ y 1 · sin ϕ ,
y 1 = y 1 · cos ϕ + x 1 · sin ϕ .
ρ = ( x 1 2 + y 1 2 ) 1 / 2 ,
ρ ¯ = ρ / R ,
θ = tan 1 ( y 1 x 1 ) .
γ max = 2 N 10 N 00 N 00 R [ 1 ( r R ) 2 ] 1 / 2 [ 1 ( r R ) 2 sin 2 ( β ) ] 1 / 2 ,
x 2 = x 1 · d ENP 2 R ,
y 2 = y 1 · d ENP 2 R .
x 3 = x 2 · t ENP 2 ( l 0 + t ENP ) + x 2 ,
y 3 = + t ENP · [ h ( J , K ) y 2 ] 2 ( l 0 + t ENP ) + y 2 .
r 0 = ( x 32 + y 32 ) 1 / 2 .
a = { [ R cos ( α ) ] 2 + [ h ( J , K ) R sin ( α ) ] 2 } 1 / 2 ,
c = R r 0 ,
b = { [ h ( J , K ) y 3 ] 2 + x 32 } 1 / 2 ,
a = tan 1 ( y 3 x 3 ) ,
β = cos 1 ( b 2 + c 2 a 2 2 b c ) .
γ = tan 1 ( { [ h ( J , K ) y 3 ] 2 + x 32 } 1 / 2 l 0 ) .
x = y sin [ ϕ ( J , K ) ] + x cos [ ϕ ( J , K ) ] ,
y = y cos [ ϕ ( J , K ) ] x sin [ ϕ ( J , K ) ] .
x IM = X OB ,
y IM = Y OB .
X IM = x IM + x ,
y IM = y IM + y .
t 1 = t 0 + tan ( 2 N 10 N 00 L ) 2 N 10 N 00 2 N 10 N 00 t 0 + tan ( 2 N 10 N 00 L ) 1 ,
h = ( X X 0 ) cos ( α ) ,
t 0 = ( X X 0 ) sin ( α ) + l 0 .
X im = X 0 [ l 0 + L + t 1 + h tan ( α ) ] sin ( α ) + h / cos ( α ) ,
Z im = [ l 0 + L + ( t 1 + h tan ( α ) ] cos ( α ) .
T = ( l 0 + L + l 1 ) cos ( α ) ,
X 1 = M X 0 = X 0 ( l 0 + L + l 1 ) sin ( α ) ,
α = tan 1 [ X 0 ( 1 M ) T ] .

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