Abstract

An analytical derivation of Zernike aberration coefficients is presented, starting from the Seidel and higher-order power-series aberration coefficients. An azimuth symmetrical aberration function has been used. The first 155 nonvanishing terms of the conversion matrix are also given in rational form.

© 1983 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.
  2. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  3. For the inverse problems, see R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 6, 262 (1982).
    [CrossRef]

1982 (1)

For the inverse problems, see R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 6, 262 (1982).
[CrossRef]

1976 (1)

Born, M.

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

Noll, R. J.

Tyson, R. K.

For the inverse problems, see R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 6, 262 (1982).
[CrossRef]

Wolf, E.

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

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

For the inverse problems, see R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 6, 262 (1982).
[CrossRef]

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 Matrix Elements Tnmkl in Rational Form

Equations (9)

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Φ ( ρ , θ ) = k l S k l ρ k cos l θ ,
Φ ( ρ , θ ) = n = 0 m = 0 n A n m R n m ( ρ ) cos m θ ,
R n m ( ρ ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! ( n + m 2 s ) ! ( n m 2 s ) ! ρ n 2 s .
V n m ( ρ , θ ) = R n m ( ρ ) cos m θ
0 2 π d θ 0 1 V n m ( ρ , θ ) V m n ( ρ , θ ) ρ d ρ = 1 + δ m o 2 ( n + 1 ) π δ n n δ m m ,
A n m = l = m k = 0 T n m k l S k l ,
T nmkl = 2 ( n + 1 ) ( 1 + δ m o ) π × 0 2 π d θ 0 1 R n m ( ρ ) ρ k + 1 cos m θ cos l θ d ρ .
cos l θ = 1 2 l t = 0 l l ! t ! ( l t ) ! cos ( l 2 t ) θ ,
T n m k l = ( n + s ) l ! 2 2 l ( 1 + δ m o ) ( l + m 2 ) ! ( l m 2 ) ! × s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! ( n + m 2 s ) ! ( n m 2 s ) ! ( n 2 s + k + 2 )

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