Abstract

This Letter describes the derivation of a matrix equation that can be used to determine the Seidel and higher-order power-series aberration coefficients from an aberration function expressed in terms of Zernike coefficients. The elements of the conversion matrix are given in analytic form, and the first 195 nonzero elements are given in a table. Two examples of the use of the conversion formula are presented.

© 1982 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 464–473.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 464–473.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 464–473.

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 464–473.

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Tables (1)

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Table 1 Elements of the Conversion Matrix

Equations (15)

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Φ ( r , θ ) = S k l ( r / R ) k cos l θ ,
Φ ( r , θ ) = A 00 + n = 1 m = 0 n A n m ( 1 + δ m 0 ) 1 / 2 R n m ( r R ) cos m θ , n m , n m = even .
R n m ( r R ) = s = 0 n m 2 ( 1 ) s ( n s ) ! s ! ( n + m 2 s ) ! ( n m 2 s ) ! ( r R ) n 2 s ,
s k l = f ( A n m )
cos m θ = ( 1 + δ m 0 ) 2 m 1 cos m θ + m t = 1 m 1 ( 1 ) t ( m t 1 ) ! 2 m 2 t 1 t ( m 2 t ) ! ( t 1 ) ! cos m 2 t θ
S kl = n = 0 m = 0 n a klnm A nm ,
a 0000 = 1 , a klnm = [ ( 1 ) n k 2 ( n + k 2 ) ! ( 1 + δ m 0 ) 1 / 2 ( n k 2 ) ! ( k + m 2 ) ! ( k m 2 ) ! ] × ( 1 + δ m 0 ) 2 l 1
a klnm = [ ( 1 ) n k 2 ( n + k 2 ) ! ( 1 + δ m 0 ) 1 / 2 ( n k 2 ) ! ( k + m 2 ) ! ( k m 2 ) ! ] × [ ( 1 ) m l 2 2 l m ( m + l 2 1 ) ! ( m l ) ( m l 2 1 ) ! l ! ]
Φ ( r , θ ) = ( r / R ) cos θ + 2 ( r / R ) 2 cos 2 θ ,
A 00 = 1 2 , A 20 = 1 / 2 , A 11 = 1 , A 22 = 1 .
S 00 = a 0000 A 00 + a 0020 A 20 + a 0040 A 40 + , S 11 = a 1111 A 11 + a 1131 A 31 + a 1151 A 51 + , S 20 = a 2020 A 20 + a 2022 A 22 + a 2040 A 40 + , S 22 = a 2222 A 22 + a 2242 A 42 + a 2262 A 62 + ,
S 00 = 1 ( 1 2 ) 1 2 ( 1 2 ) + 0 + = 0 , S 11 = 1 ( 1 ) 2 ( 0 ) + 0 + = 1 , S 20 = 2 2 ( 1 2 ) 1 ( 1 ) + 0 + = 0 , S 22 = 2 ( 1 ) 6 ( 0 ) + 0 + = 2 .
A 00 = 1 2 , A 20 = 1 / 2 .
S 00 = a 0000 A 00 + a 0020 A 20 = 1 ( 1 2 ) + ( 1 2 ) 1 2 = 0 , S 11 = a 1111 A 11 + a 1121 A 31 + = 0 , S 20 = a 2020 A 20 + a 2022 A 22 + = 2 2 ( 1 2 ) + 0 + = 1 , S 22 = a 2222 A 22 + a 2242 A 42 + = 0 , S 31 = a 3131 A 31 + a 3133 A 33 + = 0 , S 40 = a 4040 A 40 + a 4042 A 42 + = 0 .
Φ ( r , θ ) = ( r / R ) 2 .

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