Abstract

A method of ray tracing through curved gradient-index imaging rods is developed. A study of the imaging ability of a radial gradient-index relay rod, with end-to-end imaging of an object in the meridional plane, bent along some simple curves is performed. The study includes an analysis of changes in the first order of properties and the transverse aberrations in the image plane. Approximations of the first-, second-, and third-order transverse aberration coefficients are made.

© 1986 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  2. R. W. Wood, Physical Optics (Macmillan, New York, 1905).
  3. Selfoc is a registered trade name of the Nippon Sheet Glass Co., Ltd., Osaka, Japan.
  4. K. Iga, N. Yamamoto, “Plastic Focusing Fiber for Imaging Applications,” Appl. Opt. 16, 1305 (1977).
    [CrossRef] [PubMed]
  5. A. Rohra, K. Thyagarajan, “Aberrations in Curved Graded-Index Fibers,” J. Opt. Soc. Am. 69, 300 (1979).
    [CrossRef]
  6. M. J. Nadeau, “Image Analysis of Curved Gradient Index Rods,” M.S. Thesis, U. Rochester (1984).
  7. A. Sharma, D. Kumar, G. Vizia, A. K. Ghatak, “Tracing Rays Through Graded-Index Media-A New Method,” Appl. Opt. 21, 984 (1982).
    [CrossRef] [PubMed]
  8. D. T. Moore, J.M. Stagaman, “Ray Tracing in Anamorphic Gradient-Index Media,” Appl. Opt. 21, 999 (1982).
    [CrossRef] [PubMed]
  9. P. J. Sands, “Off-Axis Aberration Coefficients,” Ph.D. Thesis, Australian National U. (1967).
  10. L. G. Atkinson, D. T. Moore, N. J. Sullo, “Imaging Capabilities of a Long Gradient-Index Rod,” Appl. Opt. 21, 1004 (1982).
    [CrossRef] [PubMed]

1982 (3)

1979 (1)

1977 (1)

Atkinson, L. G.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Ghatak, A. K.

Iga, K.

Kumar, D.

Moore, D. T.

Nadeau, M. J.

M. J. Nadeau, “Image Analysis of Curved Gradient Index Rods,” M.S. Thesis, U. Rochester (1984).

Rohra, A.

Sands, P. J.

P. J. Sands, “Off-Axis Aberration Coefficients,” Ph.D. Thesis, Australian National U. (1967).

Sharma, A.

Stagaman, J.M.

Sullo, N. J.

Thyagarajan, K.

Vizia, G.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1905).

Yamamoto, N.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Other (5)

M. J. Nadeau, “Image Analysis of Curved Gradient Index Rods,” M.S. Thesis, U. Rochester (1984).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

R. W. Wood, Physical Optics (Macmillan, New York, 1905).

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

P. J. Sands, “Off-Axis Aberration Coefficients,” Ph.D. Thesis, Australian National U. (1967).

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Figures (15)

Fig. 1
Fig. 1

Imaging GRIN rod; +1 magnification.

Fig. 2
Fig. 2

Axis representation of the curved GRIN rod.

Fig. 3
Fig. 3

Translation and rotation of a segment of an equivalent straight rod to the curved rod.

Fig. 4
Fig. 4

Rotation and translation of the index expansion for equivalent planes.

Fig. 5
Fig. 5

Definition of the paraxial image plane.

Fig. 6
Fig. 6

Definition of the paraxial magnification.

Fig. 7
Fig. 7

Transverse aberration plots of the straight half-period GD3 rod.

Fig. 8
Fig. 8

Imaging of the axial object point in (a) straight and (b) curved rods.

Fig. 9
Fig. 9

Transverse aberration plots for a GRIN rod bent along a series of cubic curves.

Fig. 10
Fig. 10

Displacement and rotation of the image plane.

Fig. 11
Fig. 11

Variation of the focal shift and displacement of the image plane with greater bendings of the rod.

Fig. 12
Fig. 12

Imaging of the axial object point in a parabolic rod.

Fig. 13
Fig. 13

Transverse aberration plots at the half-period image plane. (a) fob = 0.0, θ = 0, (b) fob = 0.0, θ = 90, (c) fob = 0.6, (d) fob = −0.6, (e) fob = −0.9, and (f) distortion.

Fig. 14
Fig. 14

Transverse aberration plots at the one period image plane. (a) fob = 0.0, θ = 0, (b) fob = 0.0, θ = 90, (c) fob = 0.6, (d) fob = −0.6, and (e) fob = −0.9.

Fig. 15
Fig. 15

Imaging of the axial object point in the antisymmetric rod.

Tables (4)

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Table I Transverse Aberration Polynomial up to Third Order

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Table II Physical Parameters and First-Order Properties of the GD3 Rod

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Table III Aberration Polynomial Coefficients for a GRIN Rod Curved Along a Parabolic Curve

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Table IV Aberration Polynomial Coefficients for a GRIN Rod Curved Along an Antisymmetric Curve

Equations (7)

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Y ( x ) = a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 , Z ( x ) = b 1 x + b 2 x 2 + b 3 x 3 + b 4 x 4 + b 5 x 5 .
n ( x , ζ ) = n = 0 N m = 0 M N n m x m ζ n ,
ζ = a Y 2 + b Z 2 + c x 2 + 2 dYZ + 2 eYx + 2 fZx + gY + hZ + ix + j .
M = h / h ,
d = 2.44 ( f / No . ) λ .
Y = 0.00048 x 2 .
Y = 0.219 x 2 0.987 x 3 + 0.16 x 4 0.865 x 5 .

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