Abstract

This paper studies the effects on Zernike coefficients of aperture scaling, translation, and rotation, when a given aberrated wavefront is described on the Zernike polynomial basis. It proposes an analytical method for computing the matrix that enables the building of transformed Zernike coefficients from the original ones. The technique is based on the properties of Zernike polynomials and Fourier transform, and, in the case of a full aperture without central obstruction, the coefficients of the matrix are given in terms of integrals of Bessel functions. The integral formulas are exact and do not depend on any specific ordering of the polynomials.

© 2013 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd ed. (Pergamon, 1965).
  2. L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
    [CrossRef]
  3. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1431 (1965).
    [CrossRef]
  4. K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
    [CrossRef]
  5. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945(2002).
    [CrossRef]
  6. 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]
  7. H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1966 (2006).
    [CrossRef]
  8. A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
    [CrossRef]
  9. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
    [CrossRef]
  10. S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
    [CrossRef]
  11. R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
    [CrossRef]
  12. Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, 1962).
  13. I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).
  14. R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56(1999).
  15. G. Molodij, “Wavefront propagation in turbulence: an unified approach to the derivation of angular correlation functions,” J. Opt. Soc. Am. A 28, 1732–1740 (2011).
    [CrossRef]
  16. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).

2011 (1)

2007 (1)

2006 (3)

2003 (1)

2002 (2)

2001 (1)

1999 (1)

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56(1999).

1993 (1)

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

1965 (1)

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1431 (1965).
[CrossRef]

Ares, J.

Arines, J.

Bará, S.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd ed. (Pergamon, 1965).

Bradley, A.

Campbell, C. E.

Cheng, X.

Coatrieux, J.-L.

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
[CrossRef]

Fried, D. L.

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1431 (1965).
[CrossRef]

Geary, K.

Goldberg, K. A.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Han, G.

Hong, X.

Janssen, A. J. E. M.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
[CrossRef]

Jeffrey, A.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Luke, Y. L.

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, 1962).

Lundström, L.

Luo, L.

Marchetti, E.

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56(1999).

Molodij, G.

Prado, P.

Ragazzoni, R.

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56(1999).

Rigaut, F.

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56(1999).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Sasiela, R. J.

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

Schwiegerling, J.

Shelton, J. D.

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

Shu, H.

Thibos, L. N.

Unsbo, P.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd ed. (Pergamon, 1965).

Zwillinger, D.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Astron. Astrophys. (1)

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56(1999).

J. Math. Phys. (1)

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

J. Microlithogr. Microfabr. Microsyst. (1)

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
[CrossRef]

J. Opt. Soc. Am. A (9)

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1431 (1965).
[CrossRef]

K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
[CrossRef]

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945(2002).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

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]

H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1966 (2006).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
[CrossRef]

G. Molodij, “Wavefront propagation in turbulence: an unified approach to the derivation of angular correlation functions,” J. Opt. Soc. Am. A 28, 1732–1740 (2011).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd ed. (Pergamon, 1965).

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, 1962).

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).

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Figures (3)

Fig. 1.
Fig. 1.

Sketch of the reference systems used in Section 3. In the Oxy^ reference axis (solid lines), the phase is defined over the aperture of radius R and the unitary coordinate of a point P in this system is ρ. In the Oxy^ reference axis (dashed lines), the phase is defined over the aperture of radius R, and the unitary coordinate of a point P in this system is ρ. The reference change involves a translation of vector Rt and a rotation of angle θr.

Fig. 2.
Fig. 2.

Center: initial phase (top) and associated Zernike coefficients (bottom), with j=1 to 28; that is, the phase is defined with n=6 radial degree modes. Left: reconstructed phase over the scaled aperture (β=1.3) computed from the transformed Zernike coefficients. These coefficients are calculated from the original ones using both the transformation matrix of Eq. (20) (circles) and Shu et al. method [7] (solid line). Right: same as previously for a scaled and translated aperture (β=1.5, |t|=0.3, and αt=π/3). The transformed coefficients derived from Eq. (23) (circles) are compared with those of the Lundström and Unsbo technique [9] (solid line).

Fig. 3.
Fig. 3.

Left: original wavefront. Center: reconstructed wavefront for a pupil rotation of θr=π/3. Right: transformed Zernike coefficients derived from the original ones using Eq. (28) (circles) and Bará et al. method [10] (solid lines).

Equations (58)

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ϕ(r)=ϕ(Rρ)=j=1ajZj(ρ),
Πp(ρ)={1/πif|ρ|10elsewhere.
aj=Πp(ρ)Zj(ρ)ϕ(Rρ)d2ρ.
Znm(ρ,θ)=Zj(ρ,θ)=n+1Rnm(ρ){2cos(|m|θ)ifm>02sin(|m|θ)ifm<01ifm=0,
Rnm(ρ)=s=0(n|m|)/2(1)s(ns)!s![(n+|m|)/2s]![(n|m|)/2s]!ρn2s.
Πp(ρ)Zi(ρ)Zj(ρ)d2ρ=δij,
Qj(κ,α)=(1)nn+1Jn+1(2πκ)πκ{(1)(n|m|)/2i|m|2cos(|m|α)ifm>0(1)(n|m|)/2i|m|2sin(|m|α)ifm<0(1)n/2ifm=0.
Rρ=Rθr[RρRt]ρ=Rθr[β(ρt)]
R|t|+RRβ|t|+1β.
ϕ(Rρ)=j=1NzajZj(ρ).
ϕ(Rρ)=i=1NzbiZi(ρ)
bi=Πp(ρ)Zi(ρ)ϕ(Rρ)d2ρ.
bi=Πp(ρ)Zi(ρ)ϕ(Rρ)d2ρ=j=1NzajΠp(ρ)Zi(ρ)Zj(ρ)d2ρ,
bi=πj=1NzajΠp(ρ)Zi(ρ)Πp(ρ)Zj(ρ)d2ρ,
bi=πj=1NzajΠp(ρ)Zi(ρ)Πp(Rθr[ρ]β+t)Zj(Rθr[ρ]β+t)d2ρ.
bi=β2πj=1NzajQi*(K)QjRθr(βK)exp[2iπβt·K]d2K,
b=M[β,t,θr]·a,
Mij[β,t,θr]=πβ2002πκQi*(κ,α)Qj(βκ,α+θr)exp[2iπβ|t|κcos(ααt)]dαdκ
Mij[β]=βπ(1)3(ni+nj)2(1)(|mi|+|mj|)2i(|mj||mi|)ni+1nj+1×0κ1Jni+1(2πκ)Jnj+1(2πβκ)dκ×02π[2cos(|mi|α)2sin(|mi|α)1][2cos(|mj|α)2sin(|mj|α)1]dα,
Mij[β]={2βni+1nj+10κ1Jni+1(2πκ)Jnj+1(2πβκ)dκifmi=mj0ifmimj.
Mij[β,t]=βπ(1)3(ni+nj)|mi||mj|2i(|mj||mi|)ni+1nj+10κ1Jni+1(2πκ)Jnj+1(2πβκ)×02π[2cos(|mi|α)2sin(|mi|α)1][2cos(|mj|α)2sin(|mj|α)1]exp[2iπβ|t|κcos(ααt)]dαdκ.
Jn(κ)=12πππexp[i(nακsinα)]dα(withninteger).
Mij[β,t]=2δmi,0.δmj,0(1)3(ni+nj)|mi||mj|2i(|mj||mi|)βni+1nj+1×[Aij,αt0κ1Jni+1(2πκ)Jnj+1(2πβκ)J||mi||mj||(2πβ|t|κ)dκ+Aij,αt+0κ1Jni+1(2πκ)Jnj+1(2πβκ)J|mi|+|mj|(2πβ|t|κ)]dκ]
Aij,αt=(1)||mi||mj||2{cos([|mi||mj|]αt)ifsgn(mi)=sgn(mj)sgn(mi)sin([|mi||mj|]αt)ifsgn(mi)sgn(mj)
Aij,αt+=(1)||mi|+|mj||2{sgn(mi)cos([|mi|+|mj|]αt)ifsgn(mi)=sgn(mj)sin([|mi|+|mj|]αt)ifsgn(mi)sgn(mj).
M[β,t,θr]=M[β,t]·M[θr]=M[θr]·M[β,t].
Mij[θr]=1π(1)3(ni+nj)2(1)(|mi|+|mj|)2i(|mj||mi|)ni+1nj+1×0κ1Jni+1(2πκ)Jnj+1(2πκ)dκ×02π[2cos(|mi|α)2sin(|mi|α)1][2cos(|mj|[α+θr])2sin(|mj|[α+θr])1]dα.
Mij[θr]=2ni+1nj+10κ1Jni+1(2πκ)Jnj+1(2πκ)dκ×{cos(mjθr)ifmi=mjsin(mjθr)ifmi=mj(andmi0)0if|mi||mj|.
2ni+1nj+10κ1Jni+1(2πκ)Jnj+1(2πκ)dκ=δni,nj
Mij[θr]=δni,nj×{cos(mjθr)ifmi=mjsin(mjθr)ifmi=mj(andmi0)0if|mi||mj|.
F(κ)=02π[2cos(|mi|α)2sin(|mi|α)1][2cos(|mj|α)2sin(|mj|α)1]exp[2iπβ|t|κcos(ααt)]dα.
02πcos(mα)exp(iycos(ααt))dα={2π(1)|m|2cos(mαt)J|m|(y)ifmeven2iπ(1)|m|12cos(mαt)J|m|(y)ifmodd,
02πsin(mα)exp(iycos(ααt))dα={2π(1)|m|2sin(mαt)J|m|(y)ifmeven2iπ(1)|m|12sin(mαt)J|m|(y)ifmodd.
F(κ)=02πcos([|mi||mj|]α)exp[2iπβ|t|κcos(ααt)]dα+02πcos([|mi|+|mj|]α)exp[2iπβ|t|κcos(ααt)]dα,
F(κ)=2π(1)||mi||mj||2cos([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)+2π(1)||mi|+|mj||2cos([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ);
F(κ)=2iπ(1)||mi||mj||12cos([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)+2iπ(1)||mi|+|mj||12cos([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ).
F(κ)=02πcos([|mi||mj|]α)exp[2iπβ|t|κcos(ααt)]dα02πcos([|mi|+|mj|]α)exp[2iπβ|t|κcos(ααt)]dα,
F(κ)=2π(1)||mi||mj||2cos([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)2π(1)||mi|+|mj||2cos([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ);
F(κ)=2iπ(1)||mi||mj||12cos([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)2iπ(1)||mi|+|mj||12cos([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ).
F(κ)=02πsin([|mi||mj|]α)exp[2iπβ|t|κcos(ααt)]dα+02πsin([|mi|+|mj|]α)exp[2iπβ|t|κcos(ααt)]dα,
F(κ)=2π(1)||mi||mj||2sin([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)+2π(1)||mi|+|mj||2sin([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ);
F(κ)=2iπ(1)||mi||mj||12sin([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)+2iπ(1)||mi|+|mj||12sin([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ).
F(κ)=02πsin([|mi||mj|]α)exp[2iπβ|t|κcos(ααt)]dα+02πsin([|mi|+|mj|]α)exp[2iπβ|t|κcos(ααt)]dα;
F(κ)=2π(1)||mi||mj||2sin([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)+2π(1)||mi|+|mj||2sin([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ);
F(κ)=2iπ(1)||mi||mj||12sin([|mi||mj|]αt)J||mi||mj||(2πβ|t|κ)+2iπ(1)||mi|+|mj||12sin([|mi|+|mj|]αt)J||mi|+|mj||(2πβ|t|κ).
F(κ)=202πcos(|mi|α)exp[2iπβ|t|κcos(ααt)]dα;
F(κ)=2π2(1)|mi|2cos(|mi|αt)J|mi|(2πβ|t|κ);
F(κ)=2iπ2(1)|mi|12cos(|mi|αt)J|mi|(2πβ|t|κ).
F(κ)=202πsin(|mi|α)exp[2iπβ|t|κcos(ααt)]dα;
F(κ)=2π2(1)|mi|2sin(|mi|αt)J|mi|(2πβ|t|κ);
F(κ)=2iπ2(1)|mi|12sin(|mi|αt)J|mi|(2πβ|t|κ).
F(κ)=202πcos(|mj|α)exp[2iπβ|t|κcos(ααt)]dα;
F(κ)=2π2(1)|mj|2cos(|mj|αt)J|mj|(2πβ|t|κ);
F(κ)=2iπ2(1)|mj|12cos(|mj|αt)J|mj|(2πβ|t|κ).
F(κ)=202πsin(|mj|α)exp[2iπβ|t|κcos(ααt)]dα;
F(κ)=2π2(1)|mj|2sin(|mj|αt)J|mj|(2πβ|t|κ);
F(κ)=2iπ2(1)|mj|12sin(|mj|αt)J|mj|(2πβ|t|κ).
F(κ)=02πexp[2iπβ|t|κcos(ααt)]dα=2πJ0(2πβ|t|κ).

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