Abstract

The aberrations of lenses containing spherical index gradients are considered. Coordinate transformations from spherical to rectangular geometry are presented. These transformations enable calculation of the index polynomial in rectangular coordinates and subsequent calculation of aberration coefficients and ray trace data. The ability to alter the sign of the gradient aberration contribution by simply changing the curvature of the gradient is demonstrated.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. P., London, 1970).
  2. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  3. E. W. Marchand, J. Opt. Soc. Am. 60, 1 (1970).
    [CrossRef]
  4. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [CrossRef]
  5. L. von Seidel, Astron. Nach. 43 (1856), No. 1027, 289; No. 1028, 305; No. 1029, 321.
  6. D. T. Moore, J. Opt. Soc. Am. 67, 1137 (1977).
    [CrossRef]
  7. S. D. Fantone, “Design Engineering and Manufacturing Aspects of Gradient Index Optical Components,” Ph.D. Thesis, U. Rochester (1979).

1977

1970

1856

L. von Seidel, Astron. Nach. 43 (1856), No. 1027, 289; No. 1028, 305; No. 1029, 321.

Buchdahl, H. A.

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

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Fantone, S. D.

S. D. Fantone, “Design Engineering and Manufacturing Aspects of Gradient Index Optical Components,” Ph.D. Thesis, U. Rochester (1979).

Marchand, E. W.

Moore, D. T.

Sands, P. J.

von Seidel, L.

L. von Seidel, Astron. Nach. 43 (1856), No. 1027, 289; No. 1028, 305; No. 1029, 321.

Astron. Nach.

L. von Seidel, Astron. Nach. 43 (1856), No. 1027, 289; No. 1028, 305; No. 1029, 321.

J. Opt. Soc. Am.

Other

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

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

S. D. Fantone, “Design Engineering and Manufacturing Aspects of Gradient Index Optical Components,” Ph.D. Thesis, U. Rochester (1979).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Spherical gradient geometry.

Tables (7)

Tables Icon

Table I Rectangular Coordinate Terms

Tables Icon

Table II Rectangular Coordinate Terms Introduced by Nρ1

Tables Icon

Table III Rectangular Coordinate Terms Introduced by Nρ2

Tables Icon

Table IV Rectangular Coordinate Terms Introduced by Nρ3

Tables Icon

Table V Rectangular Coordinate Terms Introduced by Nρ4

Tables Icon

Table VI Axial Singlet Collimator

Tables Icon

Table VII Shallow Radial Singlet Collimator

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

N ( ξ ) = N 00 + N 10 ξ + N 20 ξ 2 + , , N n 0 ξ n ,
N ( x ) = N 00 + N 01 x + N 02 x 2 + , , N 0 n x n ,
N ( x , ξ ) + N 00 + N 01 x + N 02 x 2 + + N 10 ξ + N 11 ξ x + N 12 ξ x 2 + + N 20 ξ 2 + N 21 ξ 2 x + N 22 ξ 2 x 2 .
N ( R - ρ ) = N ρ 0 + N ρ 1 ( R - ρ ) + N ρ 2 ( R - ρ ) 2 + N ρ 3 ( R - ρ ) 3 + N ρ 4 ( R - ρ ) 4 + .
( R - ρ ) = R - [ ( R - x ) 2 + Y 2 + Z 2 ] 1 / 2 , ( R - ρ ) = R - ( R 2 - 2 R x + x 2 + ξ ) 1 / 2 .
R - ρ = R [ 1 - ( 1 + x 2 + ξ - 2 R x R 2 ) 1 / 2 ] , R - ρ = - x 2 + ξ - 2 R x 2 R + ( x 2 + ξ - 2 R x ) 2 8 R 3 + .
N ρ 1 · depth of gradient · ( 17 R 786432 × S 10 ) .
OPD = 0.05 17 R 786432 × S 10 < 5 × 10 - 6 cm , OPD = 1.1 × 10 - 6 R S 10 < 5 × 10 - 6 cm .
OPD error = Δ n 2.2 × 10 - 5 R S 10 ,
N 1 ( ρ ) = N ρ 0 + N ρ 1 ( R - ρ ) ,
N 1 ( x , ξ ) = N ρ 0 + N ρ 1 x - N ρ 1 ξ ( 1 2 R + x 2 R 2 + x 2 2 R 3 + x 3 2 R 4 + x 4 2 R 5 + x 5 2 R 6 + x 6 2 R 7 + x 7 2 R 8 ) + N ρ 1 ξ 2 ( 1 8 R 3 + 3 x 8 R 4 + 3 4 R 5 x 2 + 5 4 R 6 x 3 + 15 8 R 7 x 4 + 21 8 R 8 x 5 ) - N ρ 1 ξ 3 ( 1 16 R 5 + 5 16 R 4 x + 15 16 R 7 x 2 + 35 16 R 8 x 3 ) + N ρ 1 ξ 4 ( 5 128 R 7 + 35 128 R 8 x ) + N ρ 1 0 ( ξ n x 10 - 2 n ) .             n = 0 , 1 , 2 , 3 , 4 , 5
N 2 ( x , ξ ) = N ρ + N ρ 4 x 4 - 2 N ρ 4 ξ ( 1 R x 3 + 1 R 2 x 4 + 1 R 3 x 5 + 1 R 4 x 6 + 1 R 5 x 7 ) + N ρ 4 ξ 2 ( 3 2 R 2 x 2 + 7 2 R 3 x 3 + 6 R 4 x 4 + 9 R 5 x 5 ) + N ρ 4 ξ 3 ( - 1 2 R 3 x + - 9 4 R 4 x 2 - 25 4 R 5 x 3 ) + N ρ 4 ξ 4 ( 1 16 R 4 + 5 8 R 5 x ) + 0 ( ξ n x 10 - 2 n ) .             n = 0 , 12 , 3 , 4 , 5
N 3 ( x , ξ ) = N 1 ( x , ξ ) + N 2 ( x , ξ ) - N 0 ,
N 3 ( x , ξ ) = N ρ 0 + N ρ 1 x + N ρ 4 x 4 - ξ ( N ρ 1 2 R + N ρ 1 2 R 2 x + N ρ 1 2 R 3 x 2 + ( N ρ 1 2 R 4 + 2 N ρ 4 R ) x 3 + ) + ξ 2 ( N ρ 1 8 R 3 + N ρ 1 3 8 R 4 x + ( 3 N ρ 1 4 R 5 + 3 N ρ 4 2 R 2 ) x 2 + ) + ξ 3 ( ) + ξ 4 ( ) .
σ i = σ i h + σ i a + σ i t + σ i s ,             i = 1 , 2 , 3 , 4 , 5
σ i s = μ j a s i j ,             j = surface number
a s 1 = - c Δ ( 2 N 1 + ½ c N ˙ 0 ) y a 4 .
N ˙ 0 = N ρ 1 , N 10 = - ½ c g N ρ 1 ,
a s 1 = - c ( 2 N 1 + ½ c N ˙ 0 ) y a 4 = - c ( - c g N ρ 1 + ½ c N ρ 1 ) y a 4 = ( c c g + - ½ c 2 ) N ρ 1 y a 4 .
a s 1 = + ½ c 2 N ρ 1 y a 4 .
a s 1 = - ½ c 2 N ρ 1 y a 4 .
N ˙ 0 = 0 , N 10 = 0 , N 20 0.
c g = c , N 01 = 0 ; N 02 = N ρ 2 ; N 10 = 0 ; N 11 = - c g N ρ 2 ; N 12 = - c g 2 N ρ 3 ; etc .
N ( x , ξ ) = N 00 + N ρ 2 x 2 - c g N ρ 2 ξ x - c g 2 N ρ 2 ξ x 2 + + ¼ c g 2 ξ 2 N ρ 2 + 3 / 4 N ρ 2 c g 3 ξ 2 x + .
N ( x , ξ ) = N 00 + t 2 N ρ 2 + 2 x t N ρ 2 + x N ρ 2 + c g N ρ 2 ξ x + c g N ρ 2 ξ t + .

Metrics