Abstract

A matrix method is developed that allows a new set of Zernike coefficients that describe a surface or wave front appropriate for a new aperture size to be found from an original set of Zernike coefficients that describe the same surface or wave front but use a different aperture size. The new set of coefficients, arranged as elements of a vector, is formed by multiplying the original set of coefficients, also arranged as elements of a vector, by a conversion matrix formed from powers of the ratio of the new to the original aperture and elements of a matrix that forms the weighting coefficients of the radial Zernike polynomial functions. In developing the method, a new matrix method for expressing Zernike polynomial functions is introduced and used. An algorithm is given for creating the conversion matrix along with computer code to implement the algorithm.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. K. Goldberg, K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
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  5. L. N. Thibos, R. A. Applegate, J. T. Schweigerling, R. Webb and VISA Standards Taskforce members, Standards for Reporting the Optical Aberration of Eyes, Vol. 35 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

2002

2001

1994

1993

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schweigerling, R. Webb and VISA Standards Taskforce members, Standards for Reporting the Optical Aberration of Eyes, Vol. 35 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Bille, J.

Fischer, D. J.

Geary, K.

Goelz, S.

Goldberg, K.

Grimm, B.

Liang, J.

Lopez, R.

O’Bryan, J. T.

Schweigerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schweigerling, R. Webb and VISA Standards Taskforce members, Standards for Reporting the Optical Aberration of Eyes, Vol. 35 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Schwiegerling, J. T.

Stahl, H. P.

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schweigerling, R. Webb and VISA Standards Taskforce members, Standards for Reporting the Optical Aberration of Eyes, Vol. 35 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schweigerling, R. Webb and VISA Standards Taskforce members, Standards for Reporting the Optical Aberration of Eyes, Vol. 35 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

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Equations (37)

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Znm=NnmRn|m|(ρ)M(mθ),
Nnm=[(2-δ0m)(n+1)]1/2,δ0m=1ifm=0,δ0m=0ifmisnot0;
Rn|m|(ρ)=s=00.5(n-|m|)×(-1)s(n-s)!s![0.5(n+|m|)-s]![0.5(n-|m|)-s]! ρn-2s,
M(n, m)=cos(mθ)ifm0,mnn-|m|even,M(n, m)=sin(mθ)ifm<0.
S(r, θ)=cnmZnm=Snm.
ρ=r/a.
Z|=Zn max-n maxZn max-1-(n max-1)Zn max-2-(n max-2)Zn max-(n max-2)Zn max(n max-2)Zn max-1n max-1Zn maxn max
Z|=|Z3-3Z2-2Z1-1Z3-1Z00Z20Z11Z31Z22Z33|.
Z|=R|[M][N].
R|=ρ|[R].
ρ|=|ρn maxρn max-1ρn max-2ρn maxρn maxρn max-1ρn max|,
[R]=[R(-n max)]0000[R(-(n max-1))]0000[R(n max-1)]0000[R(n max)].
Rn|m|(ρ)=p=|m|,inc=2n(-1)0.5(n-p)[0.5(n+p)]![0.5(n-p)]![0.5(p+|m|)]![0.5(p-|m|)]! ρp.
[M]=[M(-n max)]0000{M[-(n max-1)]}0000[M(n max-1)]0000[M(n max)].
[N]=Nn max-n max000000Nnm000000Nn maxn max.
|c=cnmax-nmaxcnmax-1-(nmax-1)cnmax-2-(nmax-2)cnmax-(nmax-2)cnmaxnmax-2cnmax-1nmax-1cnmaxnmax.
S=Z|c=ρ|[R][M][N]|c.
ρ=ra;ρ=ra=aara=ηρ;ρ=ηρ,
[η]=ηnmax000000ηnmax-1000000ηnmax-2000000ηnmax00000ηnmax000ηnmax-100000ηnmax.
ρ|=ρ|[η].
S=Z|c=ρ|[R][M][N]|c
ρ|[R][M][N]|c=ρ|[R][M][N]|c.
ρ|[η][R][M][N]|c=ρ|[R][M][N]|c.
[η][R][M][N]|c=[R][M][N]|c.
|c=[C]|c,
[η][R][M][N]|c=[R][M][N][C]|c.
[η][R][M][N]=[R][M][N][C].
[R]-1[η][R][M][N]=[M][N][C].
[M]-1[R]-1[η][R][M][N]=[N][C].
[M]-1[R]-1[η][R][M][N]=[R]-1[η][R][N];
[R]-1[η][R][N]=[N][C].
[N]-1[R]-1[η][R][N]=[C],
[C]=[P]T[N]-1[R]-1[η][R][N][P].
j=n(n+2)-m2.
p=|m|-2(r-1),
R(counter+r, counter+c)=(-1)0.5(n-p)[0.5(n+p)]![0.5(n-p)]![0.5(p+|m|)]!(0.5(p-|m|))!,
S(r, θ)=cnmZnm+,

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