Abstract

In a recent paper [J. Opt. Soc. Am. A 19, 1937 (2002) ] a recursive analytical formula was derived to calculate a set of new Zernike polynomial expansion coefficients from an original set when the size of the aperture is reduced. In the current paper I describe a more intuitive derivation of a simpler, nonrecursive formula, which is used to calculate the instantaneous refractive power.

© 2006 Optical Society of America

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References

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  1. J. Liang, B. Grimm, S. Goelz, and J. Bille, 'Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor,' J. Opt. Soc. Am. A 11, 1949-1957 (1994).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965).
  3. R. J. Noll, 'Zernike polynomials and atmospheric turbulence,' J. Opt. Soc. Am. 66, 203-211 (1976).
    [CrossRef]
  4. G.-m. Dai, 'Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions,' J. Opt. Soc. Am. A 12, 2182-2193 (1995).
    [CrossRef]
  5. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' in Vision Science and Its Applications, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.
  6. K. A. Goldberg and K. Geary, 'Wave-front measurement errors from restricted concentric subdomains,' J. Opt. Soc. Am. A 18, 2146-2152 (2001).
    [CrossRef]
  7. J. Schwiegerling, 'Scaling Zernike expansion coefficients to different pupil sizes,' J. Opt. Soc. Am. A 19, 1937-1945 (2002).
    [CrossRef]
  8. C. E. Campbell, 'Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,' J. Opt. Soc. Am. A 20, 209-217 (2003).
    [CrossRef]
  9. The last term in Eq. (A8) of Ref. should read ×r1∣m∣+2i/r2∣m∣+2i.
  10. G.-m. Dai, 'Optical surface optimization for the correction of presbyopia,' Appl. Opt. (to be published).
  11. G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power-series coefficients,' Opt. Lett. 8, 390-391 (1983).
    [CrossRef]
  12. M. Koomen, R. Tousey, and R. Scolnik, 'The spherical aberration of the eye,' J. Opt. Soc. Am. 39, 370-376 (1949).
    [CrossRef] [PubMed]

2003 (1)

2002 (1)

2001 (1)

1995 (1)

1994 (1)

1983 (1)

G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power-series coefficients,' Opt. Lett. 8, 390-391 (1983).
[CrossRef]

1976 (1)

R. J. Noll, 'Zernike polynomials and atmospheric turbulence,' J. Opt. Soc. Am. 66, 203-211 (1976).
[CrossRef]

1949 (1)

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' in Vision Science and Its Applications, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Bille, J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965).

Campbell, C. E.

Conforti, G.

G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power-series coefficients,' Opt. Lett. 8, 390-391 (1983).
[CrossRef]

Dai, G.-m.

Geary, K.

Goelz, S.

Goldberg, K. A.

Grimm, B.

Koomen, M.

Liang, J.

Noll, R. J.

R. J. Noll, 'Zernike polynomials and atmospheric turbulence,' J. Opt. Soc. Am. 66, 203-211 (1976).
[CrossRef]

Schwiegerling, J.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' in Vision Science and Its Applications, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Scolnik, R.

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' in Vision Science and Its Applications, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Tousey, R.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' in Vision Science and Its Applications, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965).

J. Opt. Soc. Am. (2)

R. J. Noll, 'Zernike polynomials and atmospheric turbulence,' J. Opt. Soc. Am. 66, 203-211 (1976).
[CrossRef]

M. Koomen, R. Tousey, and R. Scolnik, 'The spherical aberration of the eye,' J. Opt. Soc. Am. 39, 370-376 (1949).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power-series coefficients,' Opt. Lett. 8, 390-391 (1983).
[CrossRef]

Other (4)

The last term in Eq. (A8) of Ref. should read ×r1∣m∣+2i/r2∣m∣+2i.

G.-m. Dai, 'Optical surface optimization for the correction of presbyopia,' Appl. Opt. (to be published).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965).

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' in Vision Science and Its Applications, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

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Figures (1)

Fig. 1
Fig. 1

Contour plots of a wavefront map with pupil radius R 1 (left panel) and the wavefront map when the pupil size constricts to pupil radius R 2 (right panel). The two maps are in the same scale. Units are in micrometers of optical path difference. Note that the portion of the wavefront defined by R 2 on the left panel is exactly the same as the plot on the right panel.

Equations (34)

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W ( R r , θ ) = i = 0 a i F i ( r , θ ) ,
Z i ( r , θ ) = R n m ( r ) ϴ m ( θ ) ,
R n m ( r ) = s = 0 ( n m ) 2 ( 1 ) s n + 1 ( n s ) ! s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! r n 2 s
ϴ m ( θ ) = { 2 cos m θ ( m > 0 ) 1 ( m = 0 ) . 2 sin m θ ( m < 0 ) }
W 1 ( R 1 r , θ ) = i = 0 a i Z i ( r , θ ) ,
W 2 ( R 2 r , θ ) = i = 0 b i Z i ( r , θ ) ,
W 1 ( R 1 r , θ ) = i = 0 a i Z i ( ϵ r , θ ) .
W 1 ( R 1 r , θ ) = W 2 ( R 2 r , θ ) .
i = 0 b i Z i ( r , θ ) = i = 0 a i Z i ( ϵ r , θ ) .
n m b n m R n m ( r ) = n m a n m R n m ( ϵ r ) .
n = 0 N m b n m s = 0 ( n m ) 2 ( 1 ) s n + 1 ( n s ) ! r n 2 s s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! = n = 0 N m a n m s = 0 ( n m ) 2 ( 1 ) s n + 1 ( n s ) ! ϵ n 2 s r n 2 s s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! ,
b N m = ϵ N a N m .
b N 1 m = ϵ N 1 a N 1 m .
i = 0 ( N n ) 2 b n + 2 i m ( 1 ) i n + 2 i + 1 ( n + i ) ! r n i ! [ ( n + 2 i + m ) 2 i ] ! [ ( n + 2 i m ) 2 i ] ! = i = 0 ( N n ) 2 a n + 2 i m ( 1 ) i n + 2 i + 1 ( n + i ) ! ϵ n r n i ! [ ( n + 2 i + m ) 2 i ] ! [ ( n + 2 i m ) 2 i ] ! .
i = 0 ( N n ) 2 b n + 2 i m i ! ( i ) i n + 2 i + 1 ( n + i ) ! = i = 0 ( N n ) 2 ϵ n a n + 2 i m i ! ( 1 ) i n + 2 i + 1 ( n + i ) ! .
b n m = ϵ n a n m + i = 1 ( N n ) 2 ( 1 ) i i ! n ! n + 2 i + 1 n + 1 ( n + i ) ! ( ϵ n a n + 2 i m b n + 2 i m ) = ϵ n a n m + i = 1 ( N n ) 2 ( 1 ) i i ! ( n + 2 i + 1 ) ( n + 1 ) × ( n + i ) ( n + i 1 ) ( n + 2 ) ( ϵ n a n + 2 i m b n + 2 i m ) = ϵ n a n m + i = 1 ( N n ) 2 ( 1 ) i ( n + i ) ! ( n + 1 ) ! i ! ( n + 2 i + 1 ) ( n + 1 ) ( ϵ n a n + 2 i m b n + 2 i m ) .
b n + 2 m = ϵ n + 2 a n + 2 m + i = 1 ( N n ) 2 1 ( 1 ) i ( n + i + 2 ) ! ( n + 3 ) ! i ! × ( n + 2 i + 3 ) ( n + 3 ) ( ϵ n + 2 a n + 2 i + 2 m b n + 2 i + 2 m ) .
b n m = ϵ n [ a n m + i = 1 ( N n ) 2 a n + 2 i m ( n + 2 i + 1 ) ( n + 1 ) × j = 0 i ( 1 ) i + j ( n + i + j ) ! ( n + j + 1 ) ! ( i j ) ! j ! ϵ 2 j ] .
b n m = ϵ n [ a n m + ( n + 1 ) i = 1 ( N n ) 2 a n + 2 i m j = 0 i ( 1 ) i + j ( n + i + j ) ! ( n + j + 1 ) ! ( i j ) ! j ! ϵ 2 j ] .
P eff = 4 3 a 2 0 R 2 ,
b 2 0 = ϵ 2 [ a 2 0 + i = 1 N 2 1 a 2 ( i + 1 ) 0 3 ( 3 + 2 i ) j = 0 i ( 1 ) i + j ( i + j + 2 ) ! ( i j ) ! ( j + 3 ) ! j ! ϵ 2 j ] .
P eff ( ϵ ) = 4 3 R 2 [ a 2 0 + i = 1 N 2 1 a 2 ( i + 1 ) 0 3 ( 3 + 2 i ) × j = 0 i ( 1 ) i + j ( i + j + 2 ) ! ( i j ) ! ( j + 3 ) ! j ! ϵ 2 j ] .
W ( R r ) = n = 0 N 2 a 2 n r 2 n ,
P eff ( ϵ ) = 12 R 2 n = 1 N 2 n ϵ 2 ( n 1 ) ( n + 1 ) ( n + 2 ) a 2 n .
b n + 2 m = ϵ n + 2 a n + 2 m ( n + 5 ) ( n + 3 ) ( ϵ n + 2 a n + 4 m b n + 4 m ) + 1 2 ( n + 4 ) ( n + 7 ) ( n + 3 ) ( ϵ n + 2 a n + 6 m b n + 6 m ) 1 6 ( n + 5 ) ( n + 4 ) ( n + 9 ) ( n + 3 ) ( ϵ n + 2 a n + 8 m b n + 8 m ) + + ( 1 ) ( N n ) 2 1 ( N + 1 ) ( n + 3 ) [ ( N n ) 2 1 ] ! [ ( N + n ) 2 + 1 ] × [ ( N + n ) 2 ] ( n + 5 ) ( n + 4 ) ( ϵ n + 2 a N m b N m ) ,
b n + 2 m = ϵ n + 2 a n + 2 m ( n + 5 ) ( n + 3 ) ( ϵ n + 2 a n + 4 m b n + 4 m ) + 1 2 ( n + 4 ) ( n + 7 ) ( n + 3 ) ( ϵ n + 2 a n + 6 m b n + 6 m ) 1 6 ( n + 5 ) ( n + 4 ) ( n + 9 ) ( n + 3 ) ( ϵ n + 2 a n + 8 m b n + 8 m ) + + ( 1 ) ( N n 3 ) 2 N ( n + 3 ) [ ( N n 3 ) 2 ] ! [ ( N + n + 1 ) 2 ] [ ( N + n 1 ) 2 ] ( n + 5 ) ( n + 4 ) ( ϵ n + 2 a N 1 m b N 1 m ) .
b n + 4 m = ϵ n + 4 a n + 4 m ( n + 7 ) ( n + 5 ) ( ϵ n + 4 a n + 6 m b n + 6 m ) + 1 2 ( n + 6 ) ( n + 9 ) ( n + 5 ) ( ϵ n + 4 a n + 8 m b n + 8 m ) 1 6 ( n + 7 ) ( n + 6 ) ( n + 11 ) ( n + 5 ) ( ϵ n + 4 a n + 10 m b n + 10 m ) + + ( 1 ) ( N n ) 2 2 ( N + 1 ) ( n + 5 ) [ ( N n ) 2 2 ] ! [ ( N + n ) 2 + 2 ] × [ ( N + n ) 2 + 1 ] ( n + 7 ) ( n + 6 ) ( ϵ n + 4 a N m b N m ) .
b n + 6 m = ϵ n + 6 a n + 6 m ( n + 9 ) ( n + 7 ) ( ϵ n + 6 a n + 8 m b n + 8 m ) + 1 2 ( n + 8 ) ( n + 11 ) ( n + 7 ) ( ϵ n + 6 a n + 10 m b n + 10 m ) 1 6 ( n + 9 ) ( n + 8 ) ( n + 13 ) ( n + 7 ) ( ϵ n + 6 a n + 12 m b n + 12 m ) + + ( 1 ) ( N n ) 2 3 ( N + 1 ) ( n + 7 ) [ ( N n ) 2 3 ] ! [ ( N + n ) 2 + 3 ] × [ ( N + n ) 2 + 2 ] ( n + 9 ) ( n + 8 ) ( ϵ n + 6 a N m b N m ) .
b n m = ϵ n a n m ( n + 3 ) ( n + 1 ) ϵ n ( 1 ϵ 2 ) a n + 2 m + 1 2 [ ( n + 2 ) 2 ( n + 3 ) ϵ 2 + ( n + 4 ) ϵ 4 ] ( n + 5 ) ( n + 1 ) ϵ n a n + 4 m 1 6 [ ( n + 2 ) ( n + 3 ) 3 ( n + 3 ) ( n + 4 ) ϵ 2 + 3 ( n + 4 ) ( n + 5 ) ϵ 4 ( n + 5 ) ( n + 6 ) ϵ 6 ] ( n + 7 ) ( n + 1 ) ϵ n a n + 6 m + .
b n m = ϵ n [ a n m + i = 1 ( N n ) 2 a n + 2 i m ( n + 2 i + 1 ) ( n + 1 ) × j = 0 i ( 1 ) i + j ( n + i + j ) ! ( n + j + 1 ) ! ( i j ) ! j ! ϵ 2 j ] .
Z i ( r , θ ) = R n m ( r ) ϴ m ( θ ) ,
R n { m } ( r ) s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! r n 2 s .
b n m = ϵ n a n m + i = 1 ( N n ) 2 ( 1 ) i ( n + i ) ! n ! i ! ( ϵ n a n + 2 i m b n + 2 i m ) .
b n m = ϵ n [ a n m + ( n + 1 ) i = 1 ( N n ) 2 a n + 2 i m × j = 0 i ( 1 ) i + j ( n + i + j ) ! ( n + j + 1 ) ! ( i j ) ! j ! ϵ 2 j ] .

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