Abstract

Numerical studies of ray-trace aberrations of a curved graded-index medium as a function of launching conditions and refractive-index profile are reported. The variation of meridional and skew ray aberrations with the launching angle for specific point objects shows that the axis of symmetry of the medium shifts on curving. Further, the aberrations of rays emanating from a point object increase or decrease with curvature, depending on the position of the object point with respect to the new axis of symmetry.

© 1983 Optical Society of America

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References

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  1. A. Gupta, K. Thyagarajan, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
    [CrossRef]
  2. K. Thyagarajan, A. Rohra, A. K. Ghatak, Opt. Acta 26, 863 (1979).
    [CrossRef]
  3. D. T. Moore, Appl. Opt. 19, 1035 (1980).
    [CrossRef] [PubMed]
  4. K. Iga, Appl. Opt. 19, 1039 (1980).
    [CrossRef] [PubMed]
  5. E. W. Marchand, Appl. Opt. 19, 1044 (1980).
    [CrossRef] [PubMed]
  6. Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981).
  7. A. Rohra, K. Thyagarajan, A. K. Ghatak, J. Opt. Soc. Am. 69, 300 (1979).
    [CrossRef]
  8. A. Kumar, I. C. Goyal, A. K. Ghatak, Opt. Acta 22, 447 (1975).
  9. Y. Ohtsuka, T. Senga, T. Sugano, Appl. Phys. Lett. 29, 298 (1976).
    [CrossRef]
  10. A. K. Ghatak, I. C. Goyal, A. Gupta, Opt. Acta 25, 1 (1978).
    [CrossRef]
  11. K. S. Kaufman, R. Terras, R. F. Mathis, J. Opt. Soc. Am. 71, 1513 (1981).
    [CrossRef]
  12. See, e.g., A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
    [CrossRef]
  13. Throughout the calculations we have considered rays launched either in the plane of curvature, ζ-ξ plane, or in the plane perpendicular to it, the η-ξ plane. The rays launched in the ζ-ξ plane are referred to as the meridional rays, whereas those launched in the η-ξ plane are referred to as the skew rays. The launching angle γ is therefore related to the initial direction cosine of the ray through the relation [(dζ)/(dξ)]ξ=0 = tanγ or [(dη)/(dξ)]ξ=0 = tanγ.
  14. We have not plotted Δζ for the skew rays as it was very small, ~10−4 cm.
  15. Once again Δζ for the skew rays was too small (~10−5 cm) to be plotted.
  16. For an axial object point, since ζ-ξ and η-ξ would be equivalent planes, one would expect that a ray launched in the η-ξ plane should not have a ζ component of aberration and that its aberration in the η direction should be the same as the aberration along the ζ direction of a ray launched in the ζ-ξ plane. However, in this case, although the axis of symmetry has shifted effectively, the curvature is still in the ζ-ξ plane only, and hence ζ-ξ and η-ξ planes are not completely equivalent, even for a point at the new axis of symmetry.

1981 (1)

1980 (3)

1979 (2)

A. Rohra, K. Thyagarajan, A. K. Ghatak, J. Opt. Soc. Am. 69, 300 (1979).
[CrossRef]

K. Thyagarajan, A. Rohra, A. K. Ghatak, Opt. Acta 26, 863 (1979).
[CrossRef]

1978 (1)

A. K. Ghatak, I. C. Goyal, A. Gupta, Opt. Acta 25, 1 (1978).
[CrossRef]

1976 (2)

1975 (1)

A. Kumar, I. C. Goyal, A. K. Ghatak, Opt. Acta 22, 447 (1975).

Ghatak, A. K.

K. Thyagarajan, A. Rohra, A. K. Ghatak, Opt. Acta 26, 863 (1979).
[CrossRef]

A. Rohra, K. Thyagarajan, A. K. Ghatak, J. Opt. Soc. Am. 69, 300 (1979).
[CrossRef]

A. K. Ghatak, I. C. Goyal, A. Gupta, Opt. Acta 25, 1 (1978).
[CrossRef]

A. Gupta, K. Thyagarajan, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
[CrossRef]

A. Kumar, I. C. Goyal, A. K. Ghatak, Opt. Acta 22, 447 (1975).

See, e.g., A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Goyal, I. C.

A. K. Ghatak, I. C. Goyal, A. Gupta, Opt. Acta 25, 1 (1978).
[CrossRef]

A. Gupta, K. Thyagarajan, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
[CrossRef]

A. Kumar, I. C. Goyal, A. K. Ghatak, Opt. Acta 22, 447 (1975).

Gupta, A.

Iga, K.

Kaufman, K. S.

Kumar, A.

A. Kumar, I. C. Goyal, A. K. Ghatak, Opt. Acta 22, 447 (1975).

Marchand, E. W.

Mathis, R. F.

Moore, D. T.

Ohtsuka, Y.

Y. Ohtsuka, T. Senga, T. Sugano, Appl. Phys. Lett. 29, 298 (1976).
[CrossRef]

Rohra, A.

K. Thyagarajan, A. Rohra, A. K. Ghatak, Opt. Acta 26, 863 (1979).
[CrossRef]

A. Rohra, K. Thyagarajan, A. K. Ghatak, J. Opt. Soc. Am. 69, 300 (1979).
[CrossRef]

Senga, T.

Y. Ohtsuka, T. Senga, T. Sugano, Appl. Phys. Lett. 29, 298 (1976).
[CrossRef]

Sugano, T.

Y. Ohtsuka, T. Senga, T. Sugano, Appl. Phys. Lett. 29, 298 (1976).
[CrossRef]

Terras, R.

Thyagarajan, K.

A. Rohra, K. Thyagarajan, A. K. Ghatak, J. Opt. Soc. Am. 69, 300 (1979).
[CrossRef]

K. Thyagarajan, A. Rohra, A. K. Ghatak, Opt. Acta 26, 863 (1979).
[CrossRef]

A. Gupta, K. Thyagarajan, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
[CrossRef]

See, e.g., A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

Y. Ohtsuka, T. Senga, T. Sugano, Appl. Phys. Lett. 29, 298 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Acta (3)

A. Kumar, I. C. Goyal, A. K. Ghatak, Opt. Acta 22, 447 (1975).

K. Thyagarajan, A. Rohra, A. K. Ghatak, Opt. Acta 26, 863 (1979).
[CrossRef]

A. K. Ghatak, I. C. Goyal, A. Gupta, Opt. Acta 25, 1 (1978).
[CrossRef]

Other (6)

See, e.g., A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Throughout the calculations we have considered rays launched either in the plane of curvature, ζ-ξ plane, or in the plane perpendicular to it, the η-ξ plane. The rays launched in the ζ-ξ plane are referred to as the meridional rays, whereas those launched in the η-ξ plane are referred to as the skew rays. The launching angle γ is therefore related to the initial direction cosine of the ray through the relation [(dζ)/(dξ)]ξ=0 = tanγ or [(dη)/(dξ)]ξ=0 = tanγ.

We have not plotted Δζ for the skew rays as it was very small, ~10−4 cm.

Once again Δζ for the skew rays was too small (~10−5 cm) to be plotted.

For an axial object point, since ζ-ξ and η-ξ would be equivalent planes, one would expect that a ray launched in the η-ξ plane should not have a ζ component of aberration and that its aberration in the η direction should be the same as the aberration along the ζ direction of a ray launched in the ζ-ξ plane. However, in this case, although the axis of symmetry has shifted effectively, the curvature is still in the ζ-ξ plane only, and hence ζ-ξ and η-ξ planes are not completely equivalent, even for a point at the new axis of symmetry.

Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of a graded-index medium bent along the arc of a circle of radius R, with center of curvature at O. The plane of the paper corresponds to the plane of curvature (ζ-ξ plane), and the η axis is perpendicular to it.

Fig. 2
Fig. 2

Variation of the meridional ray aberration Δζ with launching angle γ. The dashed and the solid curves correspond to c = 0 and c = 0.1 cm−1, respectively, for the index profiles indicated against them (SH and HE denoting secant hyperbolic and helically exact profiles, respectively). We have followed the same notation for curves in all the following figures, and the value of α has also been retained constant throughout all the calculations at α = 2 cm−1. The curve for SH and c = 0 coincides with the γ axis. The values of other parameters used are ζi = 0.095 ξ1,ηi = 0.

Fig. 3
Fig. 3

Plots of skew ray aberration in the ζ direction as a function of launching angle γ for the same values of ζi,ηi as in Fig. 2.

Fig. 4
Fig. 4

Plots of skew ray aberration in the η direction as a function of launching angle γ for the same object point as in Fig. 3.

Fig. 5
Fig. 5

Meridional ray aberration Δζ as a function of launching angle γ for an axial object point, i.e., ζi = 0,ηi = 0. The curve for SH and c = 0 coincides with the γ axis. Notice that the curves for c = 0.1 cm−1 are not symmetric in γ.

Fig. 6
Fig. 6

Skew ray aberration in the η direction as a function of launching angle γ for the same object point as in Fig. 5.

Fig. 7
Fig. 7

Variation of meridional ray aberration Δζ with launching angle γ for a point object situated at ζi = 1/(Rα2),ηi = 0. The plots are only for c = 0.1 cm−1, as c = 0 corresponds to an axial object point.

Fig. 8
Fig. 8

Plots of skew ray aberration in the η direction as a function of launching angle γ for the same object point as in Fig. 7.

Fig. 9
Fig. 9

Meridional ray aberration Δζ as a function of β. The curves marked 11 and 12 are for an object point situated at ζi = 0,ηi = 0 with values of launching angle γ as 10° and −10°, respectively. Similarly, the curves denoted by 21 and 22 are for a point object at ζi = 0.095 ξ1,ηi = 0 with γ = 10° and −10°, respectively.

Equations (14)

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d d s ( n d r d s ) = n ,
ζ 2 R ( 1 + ζ / R ) ζ 2 1 R ( 1 + ζ R ) = ( 1 + ζ / R ) 4 A 2 n n ζ ,
η 2 R ( 1 + ζ / R ) ζ η = ( 1 + ζ / R ) 4 A 2 n n η ,
A = n ( 1 + ζ / R ) 2 [ ( 1 + ζ / R ) 2 + ζ 2 + η 2 ] 1 / 2 = n ( 1 + ζ / R ) 2 cos γ ,
Δ ζ = ζ ( ξ 1 ) ζ p ( ξ 1 ) ,
Δ η = η ( ξ 1 ) η p ( ξ 1 ) .
n = n 0 sech α ζ 2 + η 2 ,
n = n 0 [ 1 + α 2 ( ζ 2 + η 2 ) ] 1 / 2 ,
n 2 = n 0 2 [ 1 α 2 ( ζ 2 + η 2 ) ] .
n = n 0 [ 1 α 2 2 ( ζ 2 + η 2 ) + β α 4 ( ζ 2 + η 2 ) 2 + ] ,
ξ 1 = 2 π / α .
ζ p ( ξ ) = 1 cos α ξ R α 2 + ζ i cos α ξ + μ sin α ξ ,
η p ( ξ ) = η i cos α ξ + ν sin α ξ ,
μ = 1 α ( d ζ d ξ ) ξ = 0 , ν = 1 α ( d η d ξ ) ξ = 0 ·

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