Abstract

In this paper we report studies on imaging through a defocusing gradient-index rod. We have obtained analytic expressions for the third-order aberration of this system and compared it with the total aberration obtained by numerical ray tracing. Our studies show that the rod length can be critically optimized for low aberrations depending on the numerical aperture requirement. Further, a value of the fourth-order grading parameter exists for which the aberration can be reduced to a minimum for fixed values of other parameters.

© 1983 Optical Society of America

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  1. J. D. Forer, S. N. Houde-Walter, J. J. Miceli, D. T. Moore, M. J. Nadeau, D. P. Ryan, J. M. Stagaman, N. J. Sullo, Appl. Opt. 22, 407 (1983).
    [CrossRef] [PubMed]
  2. K. Iga, M. Oikawa, S. Misawa, J. Banno, Y. Kokubun, Appl. Opt. 21, 3456 (1982).
    [CrossRef] [PubMed]
  3. K. Thyagarajan, A. Rohra, A. K. Ghatak, Appl. Opt. 19, 1061 (1980).
    [CrossRef] [PubMed]
  4. M. Kawazu, Y. Ogura, Appl. Opt. 19, 1105 (1980).
    [CrossRef] [PubMed]
  5. W. J. Tomlinson, Appl. Opt. 19, 1127 (1980).
    [CrossRef] [PubMed]
  6. Y. Ohtsuka, K. Maeda, Appl. Opt. 20, 3562 (1981).
    [CrossRef] [PubMed]
  7. K. Thyagarajan, A. K. Ghatak, Optik 44, 329 (1976).
  8. A. Gupta, K. Thyagarajan, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
    [CrossRef]
  9. See, e.g., A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
    [CrossRef]
  10. We have given these equations here for the sake of completeness; a detailed discussion of the derivation of third-order aberration coefficients using this approach can be found in Ref. 9 or in R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).
  11. Since the system is rotationally symmetric we assume y0 = 0 without any loss of generality.
  12. Since the direction cosines of the ray at the object plane are given by (dx/ds)z=0 = sinγ cosψ, (dy/ds)z=0 = sinγ sinψ, and (dz/ds)z=0 = cosγ, where γ is the angle which the ray makes with the z axis and ψ is the angle which the projection of the ray on the x-y plane makes with the x axis, meridional rays correspond to ψ = 0 and skew rays correspond to ψ = π/2.
  13. We have considered rays launched within γ = ±10°.

1983 (1)

1982 (1)

1981 (1)

1980 (3)

1976 (2)

Banno, J.

Forer, J. D.

Ghatak, A. K.

K. Thyagarajan, A. Rohra, A. K. Ghatak, Appl. Opt. 19, 1061 (1980).
[CrossRef] [PubMed]

K. Thyagarajan, A. K. Ghatak, Optik 44, 329 (1976).

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]

Goyal, I. C.

Gupta, A.

Houde-Walter, S. N.

Iga, K.

Kawazu, M.

Kokubun, Y.

Luneburg, R. K.

We have given these equations here for the sake of completeness; a detailed discussion of the derivation of third-order aberration coefficients using this approach can be found in Ref. 9 or in R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

Maeda, K.

Miceli, J. J.

Misawa, S.

Moore, D. T.

Nadeau, M. J.

Ogura, Y.

Ohtsuka, Y.

Oikawa, M.

Rohra, A.

Ryan, D. P.

Stagaman, J. M.

Sullo, N. J.

Thyagarajan, K.

K. Thyagarajan, A. Rohra, A. K. Ghatak, Appl. Opt. 19, 1061 (1980).
[CrossRef] [PubMed]

K. Thyagarajan, A. K. Ghatak, Optik 44, 329 (1976).

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]

Tomlinson, W. J.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

Optik (1)

K. Thyagarajan, A. K. Ghatak, Optik 44, 329 (1976).

Other (5)

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

We have given these equations here for the sake of completeness; a detailed discussion of the derivation of third-order aberration coefficients using this approach can be found in Ref. 9 or in R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

Since the system is rotationally symmetric we assume y0 = 0 without any loss of generality.

Since the direction cosines of the ray at the object plane are given by (dx/ds)z=0 = sinγ cosψ, (dy/ds)z=0 = sinγ sinψ, and (dz/ds)z=0 = cosγ, where γ is the angle which the ray makes with the z axis and ψ is the angle which the projection of the ray on the x-y plane makes with the x axis, meridional rays correspond to ψ = 0 and skew rays correspond to ψ = π/2.

We have considered rays launched within γ = ±10°.

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Figures (6)

Fig. 1
Fig. 1

Aberration of rays emanating from an axial object point as a function of the launching angle γ. The dashed and solid curves correspond to the third-order and the total aberration, respectively. The values of parameters used are L = 11.0 mm, δ2 = 5.54 × 10−3 mm−2, and β = 0.

Fig. 2
Fig. 2

Variation of the aberration of meridional rays emanating from a point object situated at x0 = 0.6 mm, y0 = 0 with the launching angle γ. The notation of the curves and the values of parameters used are the same as in Fig. 1.

Fig. 3
Fig. 3

(a) Skew ray aberration along the y direction as a function of the launching angle γ for the same object point as considered for Fig. 2. The notation of the curves and the values of parameters used are the same as in Fig. 1. (b) Skew ray aberration along the x direction as a function of the launching angle γ for the same conditions as in (a).

Fig. 4
Fig. 4

Plot of Δr as a function of the rod length L for a point object situated at x0 = 0.6 mm, y0 = 0. The values of grading parameters are δ2 = 5.54 × 10−3 mm−2 and β = 0.

Fig. 5
Fig. 5

Variation of meridional ray aberration as a function of the grading parameter δ2 for the point object situated at x0 = 0.6 mm, y0 = 0. The values of γ are as shown against the curves, L = 10.0 mm and β = 0.

Fig. 6
Fig. 6

Plot of meridional ray aberration as a function of β for the values of γ shown against the curves. The off-axial object point is the same as considered in Fig. 5, L = 11.0 mm and δ2 = 5.54 × 10−3 mm−2.

Equations (64)

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n = n 0 ( 1 + δ 2 2 r 2 + β δ 4 r 4 )
H = n 2 ( u , z ) υ ,
dx dz = H p ; dy dz = H q ,
dp dz = δ H x ; dq dz = H y .
x = x 1 + x 3 + x 5 + ,
p = p 1 + p 3 + p 5 + ,
H = H 0 + H 1 u + H 2 υ + 1 2 ( H 11 u 2 + 2 H 12 u υ + H 22 υ 2 ) + ,
dx 1 dz = 2 H 2 p 1 ,
dp 1 dz = 2 H 1 x 1 ,
dx 3 dz = 2 H 2 p 3 + 2 p 1 [ H 12 ( x 1 2 + y 1 2 ) + H 22 ( p 1 2 + q 1 2 ) ] ,
dp 3 dz = 2 H 1 x 3 2 x 1 [ H 11 ( x 1 2 + y 1 2 ) + H 12 ( p 1 2 + q 1 2 ) ] ,
g ( z 0 ) = 0 , g ( z = ζ ) = 1 ,
G ( z 0 ) = 1 , G ( z = ζ ) = 0 , respectively ,
g ( z 0 ) = 0 , ġ ( z 0 ) = constant 0 ,
G ( z 0 ) = 1 , Ġ ( z 0 ) = 0 ,
x ( z 0 ) = x 0 , y ( z 0 ) = y 0 ,
dx dz | z = z 0 = ξ ġ ( z 0 ) , dy dz | z = z 0 = η ġ ( z 0 ) ,
x 1 = x 0 G ( z ) + ξ g ( z ) ; y 1 = y 0 G ( z ) + η g ( z ) ,
p 1 = x 0 Θ ( z ) + ξ θ ( z ) ; q 1 = y 0 Θ ( z ) + η θ ( z ) ,
x 3 ( z 0 ) = 0 = y 3 ( z 0 ) = x 5 ( z 0 ) = y 5 ( z 0 ) = ,
3 ( z 0 ) = 0 = 3 ( z 0 ) = 5 ( z 0 ) = 5 ( z 0 ) = .
x 3 ( z ) = ξ [ tI A + 2 sI B 1 + r ( I C + I D ) ] + x 0 ( tI B 2 + 2 sI C + rI E ) ,
y 3 ( z ) = η [ tI A + 2 s I B 1 + r ( I C + I D ) ] + y 0 ( tI B 2 + 2 sI C + rI E ) ,
x p = x 1 ( z = z 1 ) = x 1 L ( L z 1 ) dx 1 dz | z = L ,
y p = y 1 ( z = z 1 ) = y 1 L ( L z 1 ) dy 1 dz | z = L ,
Δ x 3 = x 3 L ( L z 1 ) dx 3 dz | z = L ,
Δ y 3 = y 3 L ( L z 1 ) dy 3 dz | z = L .
H 1 = n 0 δ 2 2 , H 2 = 1 2 n 0 ,
H 11 = 2 n 0 β δ 4 , H 12 = δ 2 4 n 0 , H 22 = 1 4 n 0 3 .
g ( z ) = sinh δ z , θ ( z ) = n 0 δ cosh δ z ,
G ( z ) = cosh δ z , Θ ( z ) = n 0 δ sinh δ z ,
x 1 ( z ) = x 0 cosh δ z + ξ sinh δ z ,
y 1 ( z ) = y 0 cosh δ z + η sinh δ z ,
p 1 ( z ) = n 0 x 0 δ sinh δ z + n 0 ξ δ cosh δ z ,
q 1 ( z ) = n 0 y 0 δ sinh δ z + n 0 η δ cosh δ z .
x 1 L = ξ sinh δ L , dx 1 dz | z = L = ξ δ cosh δ L .
O = x 1 L ( L z 1 ) dx 1 dz | z = L ,
z 1 = L 1 δ tanh δ L .
x p = x 0 sech δ L .
d ds ( n d r ds ) = n ,
d 2 x dz 2 = n 0 n c 2 δ 2 x [ 1 + 4 β δ 2 ( x 2 + y 2 ) ]
d 2 y dz 2 = n 0 n c 2 δ 2 y [ 1 + 4 β δ 2 ( x 2 + y 2 ) ] ,
x ( z 1 ) = x ( L ) ( L z 1 ) ( L ) ,
y ( z 1 ) = y ( L ) ( L z 1 ) ( L ) .
Δ x = x ( z 1 ) x p ,
Δ y = y ( z 1 ) ,
I A = 2 g ( z ) [ z 0 z dz ( H 12 g θ + H 22 θ 3 g ) z 0 z ġ g 2 θ I a dz ] ,
I B 1 = 2 g ( z ) [ z 0 z dz ( H 12 θ G + H 22 Θ θ 2 g ) z 0 z ġ g 2 θ I b dz ] ,
I B 2 = 2 g ( z ) [ z 0 z dz ( H 12 g Θ + H 22 Θ θ 2 g ) z 0 z ġ g 2 θ I b dz ] ,
I C = 2 g ( z ) [ z 0 z dz ( H 12 G Θ + H 22 Θ 2 θ g ) z 0 z ġ g 2 θ I c dz ] ,
I D = 2 g ( z ) [ z 0 z dz ( H 12 G g Γ ) z 0 z ġ g 2 θ I d dz ] ,
I E = 2 g ( z ) [ z 0 z dz ( H 12 G 2 Θ g + H 22 Θ g ) z 0 z ġ g 2 θ I e dz ] ,
I a = z 0 z dz ( H 11 g 4 + 2 H 12 g 2 θ 2 + H 22 θ 4 ) ,
I b = z 0 z dz [ H 11 g 3 G + H 12 g θ ( g Θ + θ G ) + H 22 θ 3 Θ ] ,
I c = z 0 z dz ( H 11 g 2 G 2 + 2 H 12 g G θ Θ + H 22 θ 2 Θ 2 ) ,
I d = z 0 z dz H 12 Γ 2 ,
I e = z 0 z dz [ H 11 g G 3 + H 12 G Θ ( g Θ + θ G ) + H 22 θ Θ 3 ] ,
Γ = θ G g Θ = constant .
I A = δ 2 32 [ ( 8 β + 1 ) sinh δ z cosh δ z + 2 ( 5 24 β ) δ z cosh δ z ( 11 40 β ) sinh δ z ] ,
I B 1 = δ 2 16 ( 8 β + 1 ) [ sinh 2 δ z cosh δ z δ z sinh δ z ) ,
I B 2 = δ 2 16 [ ( 7 8 β ) δ z sinh δ z + ( 8 β + 1 ) sinh 2 δ z cosh δ z ] ,
I C = δ 2 32 ( 8 β + 1 ) ( sinh δ z cosh 2 δ z + 2 δ z cosh δ z 3 sinh δ z ) ,
I D = δ 2 2 ( sinh δ z δ z cosh δ z ) ,
I E = δ 2 16 [ ( 8 β + 1 ) sinh 2 δ z cosh δ z ( 5 24 β ) dz sinh δ z ] .

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