Abstract

The use of a continuously varying index of refraction provides new degrees of freedom in the design of lens systems. By use of the third-order aberration theory developed by Sands, the effectiveness of these new degrees of freedom in correcting third-order aberrations is determined. After the theory is cast into a form suitable for a computer, two major designs are done. The first is a singlet with an axial gradient that is corrected for third-order spherical and chromatic aberrations and has no third-order distortion. A radial gradient is used in a singlet to correct all third-order monochromatic aberrations except Petzval curvature of field. The results of the study indicate that axial gradients have the same effect as aspheric surfaces for third-order aberrations. The study also shows that radial gradients are more effective in aberration correction than axial gradients and have a greater potential in lens design.

© 1971 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1968), pp. 269–302.
  2. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), pp. 305–310.
  3. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. P., Cambridge, England, 1970).
  4. L. Montagnino, J. Opt. Soc. Am. 58, 1667 (1968).
    [CrossRef]
  5. E. W. Marchand, J. Opt. Soc. Am. 60, 1 (1970).
    [CrossRef]
  6. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [CrossRef]
  7. The optical axis is assumed to be in the positive x direction.
  8. E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 753 (1970).
    [CrossRef] [PubMed]
  9. The S and T are normalized coordinates in their respective planes. For a discussion of the S and T,see Ref. 6 and Ref. 2.
  10. R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 71–77.
  11. D. T. Moore, thesis, University of Rochester, 1970.
  12. The ∇X is the difference of X between one surface and the next. ∇X= Xj+1− Xj.

1970 (3)

1968 (1)

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), pp. 305–310.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. P., Cambridge, England, 1970).

Herriott, D. R.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1968), pp. 269–302.

Marchand, E. W.

McKenna, J.

Montagnino, L.

Moore, D. T.

D. T. Moore, thesis, University of Rochester, 1970.

Rawson, E. G.

Sands, P. J.

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 71–77.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Other (8)

The optical axis is assumed to be in the positive x direction.

The S and T are normalized coordinates in their respective planes. For a discussion of the S and T,see Ref. 6 and Ref. 2.

R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 71–77.

D. T. Moore, thesis, University of Rochester, 1970.

The ∇X is the difference of X between one surface and the next. ∇X= Xj+1− Xj.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1968), pp. 269–302.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), pp. 305–310.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. P., Cambridge, England, 1970).

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Figures (10)

F. 1
F. 1

Double graphing for aplanatic singlet with axial index gradient and focal length= 10.0. Spherical aberration is plotted at the x coordinate; coma, at the y. The coordinates are (C1,N01). The predicted solution is (0.16, −0.05).

F. 2
F. 2

Double graphing for radial-gradient singlet with focal length 10.0. Coma is the x coordinate, astigmatism the y. For each point (C1,N10), the spherical aberration has been corrected to zero by properly choosing N20. The predicted solution is (0.24, −0.089).

F. 3
F. 3

Solutions for a radial gradient with zero third-order spherical aberration, coma, and astigmatism with backwardcurving field. Focal length is 10.0. The curvature of the first surface is the y coordinate; the thickness, the x.

F. 4
F. 4

Solutions for a radial gradient with zero third-order spherical aberration, coma, and astigmatism with backwardcurving field. Focal length is 10.0. The value of N10 is the y coordinate; the thickness, the x.

F. 5
F. 5

Solutions for a radial gradient with zero third-order spherical aberration, coma, and astigmatism with backwardcurving field. Focal length is 10.0. The resulting field curvature of each of the solutions is the y coordinate; thickness, the x coordinate.

F. 6
F. 6

Solutions for a radial gradient with zero third-order spherical aberration, coma, and astigmatism with inward-curving field. Focal length is 10.0. The thickness for each solution is the y coordinate; curvature of the first surface, the x coordinate.

F. 7
F. 7

Solutions for a radial gradient with zero third-order spherical aberration, coma, and astigmatism with inward-curving field. Focal length is 10.0. The value of N10 is the y coordinate; the first curvature, the x coordinate.

F. 8
F. 8

Solutions for a radial gradient with zero third-order spherical aberration, coma, and astigmatism with inward-curving field. Focal length is 10.0. The resulting field curvature of each solution is the y coordinate; first curvature, the x.

F. 9
F. 9

Spherical aberration of a grin iod vs value of N20. The value of N20 predicted by Rawson is 5N12/6N0=0.003 375.

F. 10
F. 10

Sagittal and tangential fields of a grin rod. The value of N2 predicted by Rawson to vield the best compromise for off-axis rays is 3N12/2N0 = 0.006 075.

Tables (4)

Tables Icon

Table I. Design parameters for a singlet with an axial-gradient stop placed for zero distortion.

Tables Icon

Table II Singlet with axial gradient moved for zero-distortion third-order aberration coefficients.

Tables Icon

Table III Design parameters for a singlet with a radial gradient.

Tables Icon

Table IV. Singlet with a radial-gradient third-order aberration coefficient.

Equations (32)

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V = a b N d s ,
δ N d s = 0 .
δ N ( x , ξ ) ( 1 + 2 + Ż 2 ) 1 2 d x ,
L = N ( x , ξ ) ( 1 + 2 + Ż 2 ) 1 2 .
d d x ( L ) L Y = 0
d d x ( L Ż ) L Z = 0
N ( x , ξ ) x ( 1 + 2 ) + N ( x , ξ ) Ϋ ( 1 + 2 ) N ( x , ξ ) Y = 0 .
[ N ( x ) / x ] ( 1 + 2 ) + N ( x ) Ϋ = 0 .
N ( Y ) Ϋ ( 1 + 2 ) [ N ( Y ) / Y ] = 0 .
Y ( x ) = y a ( x ) S + y b ( x ) T .
N ( x , Y ) = N 0 ( x ) + N 1 ( x ) Y 2 + N 2 ( x ) Y 4 + .
N ( Y ) = N 0 + N 1 Y 2 + N 2 Y 4 + ,
N 0 ( x ) = N 00 + N 01 x + N 02 x 2 + , N 1 ( x ) = N 10 + N 11 x + N 12 x 2 + , N 2 ( x ) = N 20 + N 21 x + N 22 x 2 + .
N ( x ) = N 00 + N 01 x + N 02 x 2 + .
N 0 [ ÿ a ( x ) S + ÿ b ( x ) T ] 2 N 1 [ y a ( x ) S + y b ( x ) T ] = 0 .
N 0 ÿ a ( x ) 2 N 1 y a ( x ) = 0 , N 0 ÿ b ( x ) 2 N 1 y b ( x ) = 0 .
y a ( x ) = C 1 e 2 α x + C 2 e 2 α x , y b ( x ) = C 3 e 2 α x + C 4 e 2 α x ,
y = Y m y , z = Z m z .
y = 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 , z = σ 1 ρ 3 sin θ + σ 2 ρ 2 h sin 2 θ + ( σ 3 + σ 4 ) ρ h 2 sin θ ,
σ i = μ j = 1 k a i j + μ j = 1 k a i j * ,
a 1 = a + K y a 4 , a 2 = q a + K y a 3 y b , a 3 = q 2 a + K y a 2 y b 2 , a 4 = ( 1 2 ) λ 2 C Δ ( 1 / N 0 ) , a 5 = q 3 a + a 4 + K y a y b 3 ,
a 1 * = 1 2 N 0 y a υ a 3 + ( 4 N 2 y a 4 + 2 N 1 y a 2 υ a 2 1 2 N 0 υ a 4 ) d x , a 2 * = 1 2 N 0 y a υ a 2 υ b + [ 4 N 2 y a 3 y b + N 1 y a υ a ( y a υ b + y b υ a ) 1 2 N 0 υ a 3 υ b ] d x , a 3 * = 1 2 N 0 y a υ a υ b 2 + ( 4 N 2 y a 2 y b 2 + 2 N 1 y a y b υ a υ b 1 2 N 0 υ a 2 υ b 2 d x , a 4 * = λ 2 ( N 1 N 0 2 ) d x , a 5 * = 1 2 N 0 y a υ b 3 + [ 4 N 2 y a y b 3 + N 1 y b υ b ( y a υ b + y b υ a ) 1 2 N 0 υ a υ b 2 ] d x .
Y ( x ) = y a ( x ) S + y b ( x ) T .
( x ) = a ( x ) S + b ( k ) T , Ϋ ( x ) = ÿ a ( x ) S + ÿ b ( x ) T .
ÿ a ( x ) ( N 00 + N 01 x + N 02 x 2 + ) + ÿ a ( x ) ( N 01 + 2 N 02 x + ) 2 y a ( x ) ( N 10 + N 11 x + N 12 x 2 + ) = 0 .
y a ( x ) = n = 0 A n x n .
a ( x ) = n = 0 ( n + 1 ) A n + 1 x n , ÿ a ( x ) = n = 0 ( n + 2 ) ( n + 1 ) A n + 2 x n .
[ n = 0 [ n + 2 ) ( n + 1 ) A n + 2 x n ] [ N 00 + N 01 x + N 02 x 2 ] + [ n = 0 ( n + 1 ) A n + 1 x n ] [ N 01 + 2 N 02 x + 3 N 03 x 2 ] 2 [ n = 0 A n x n ] [ N 10 + N 11 x + N 12 x 2 + ] = 0 .
A m = { n = 2 m 1 [ 2 A n 2 N 1 , m n n ( n 1 ) A n N 0 , m n ( m n + 1 ) ( n 1 ) A n 1 N 0 , m n + 1 ] ( m 1 ) A m 1 N 01 + 2 A m 2 N 10 } / ( m ) ( m 1 ) N 00 ,
y a ( x ) = A 0 + A 1 x + A 2 x 2 + , a ( x ) = A 1 + 2 A 2 x + 3 A 3 x 2 .
λ = N ( x ) [ y a ( x ) b ( x ) y b ( x ) a ( x ) ] .
N ( Y ) = N 0 + N 1 Y 2 + N 2 Y 4 .