Abstract

Instead of measuring the wavefront deformations directly, Hartmann and Shack–Hartmann tests measure the wavefront slopes, which are equivalent to the ray transverse aberrations. Numerous different integration methods have been described in the literature to obtain the wavefront deformations from these measurements. Basically, they can be classified in two different categories, i.e., modal and zonal. In this work we describe a modal method to integrate Hartmann and Shack–Hartmann patterns using orthogonal wavefront slope aberration polynomials, instead of the commonly used Zernike polynomials for the wavefront deformations.

© 2014 Optical Society of America

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References

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  1. J. Hartmann, “Bemerkungen uber den bau und die justirung von spektrographen,” Z. Instrumentenkd 20, 2 (1900).
  2. B. Platt and R. V. Shack, “Lenticular Hartmann screen,” Newsletter 5, 15 (1971).
  3. D. Malacara, Optical Shop Testing (Wiley, 2007), Vol. 59.
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    [CrossRef]
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    [CrossRef]
  6. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
    [CrossRef]
  10. D. R. Neal, J. Copland, and D. A. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
    [CrossRef]
  11. V. P. Aksenov and Y. N. Isaev, “Analytical representation of the phase and its mode components reconstructed according to the wave-front slopes,” Opt. Lett. 17, 1180–1182 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. C. Canovas and E. N. Ribak, “Comparison of Hartmann analysis methods,” Appl. Opt. 46, 1830–1835 (2007).
    [CrossRef]
  16. S. Ríos, E. Acosta, and S. Bará, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
    [CrossRef]
  17. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).
  18. Y. Mejía-Barbosa and D. Malacara-Hernández, “Object surface for applying a modified Hartmann test to measure corneal topography,” Appl. Opt. 40, 5778–5786 (2001).
    [CrossRef]
  19. S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
    [CrossRef]
  20. S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
    [CrossRef]
  21. Y. Mejía-Barbosa, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001).
    [CrossRef]
  22. D. Malacara and S. DeVore, in Optical Shop Testing, 2nd ed. (Wiley-Intersicence, 1992), Chap. 13.

2007

2002

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

D. R. Neal, J. Copland, and D. A. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

2001

2000

1997

S. Ríos, E. Acosta, and S. Bará, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
[CrossRef]

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
[CrossRef]

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef]

1995

1992

1986

1982

1980

1979

1974

1971

B. Platt and R. V. Shack, “Lenticular Hartmann screen,” Newsletter 5, 15 (1971).

1900

J. Hartmann, “Bemerkungen uber den bau und die justirung von spektrographen,” Z. Instrumentenkd 20, 2 (1900).

Acosta, E.

Aksenov, V. P.

Artal, P.

Bará, S.

Bradley, A.

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

Canovas, C.

Copland, J.

D. R. Neal, J. Copland, and D. A. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Cubalchini, R.

DeVore, S.

D. Malacara and S. DeVore, in Optical Shop Testing, 2nd ed. (Wiley-Intersicence, 1992), Chap. 13.

Freischlad, K.

Gavrielides, A.

Ghozeil, I.

Goelz, S.

Hartmann, J.

J. Hartmann, “Bemerkungen uber den bau und die justirung von spektrographen,” Z. Instrumentenkd 20, 2 (1900).

Hong, X.

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).

Isaev, Y. N.

Klein, S. A.

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef]

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
[CrossRef]

Koliopoulos, C. L.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013), Vol. PM221.

Malacara, D.

D. Malacara and S. DeVore, in Optical Shop Testing, 2nd ed. (Wiley-Intersicence, 1992), Chap. 13.

D. Malacara, Optical Shop Testing (Wiley, 2007), Vol. 59.

Malacara-Hernández, D.

Mejía-Barbosa, Y.

Neal, D. A.

D. R. Neal, J. Copland, and D. A. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Neal, D. R.

D. R. Neal, J. Copland, and D. A. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Platt, B.

B. Platt and R. V. Shack, “Lenticular Hartmann screen,” Newsletter 5, 15 (1971).

Prieto, P. M.

Rama, M. A.

Ribak, E. N.

Ríos, S.

Shack, R. V.

B. Platt and R. V. Shack, “Lenticular Hartmann screen,” Newsletter 5, 15 (1971).

Simmons, J. E.

Southwell, W. H.

Thibos, L. N.

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

Vargas-Martín, F.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Newsletter

B. Platt and R. V. Shack, “Lenticular Hartmann screen,” Newsletter 5, 15 (1971).

Ophthalmic Physiolog. Opt.

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

Opt. Commun.

S. Ríos, E. Acosta, and S. Bará, “Hartmann sensing with Albrecht grids,” Opt. Commun. 133, 443–453 (1997).
[CrossRef]

Opt. Lett.

Optom. Vis. Sci.

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 945–962 (1997).
[CrossRef]

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef]

Y. Mejía-Barbosa, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001).
[CrossRef]

Proc. SPIE

D. R. Neal, J. Copland, and D. A. Neal, “Shack–Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Z. Instrumentenkd

J. Hartmann, “Bemerkungen uber den bau und die justirung von spektrographen,” Z. Instrumentenkd 20, 2 (1900).

Other

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013), Vol. PM221.

D. Malacara, Optical Shop Testing (Wiley, 2007), Vol. 59.

D. Malacara and S. DeVore, in Optical Shop Testing, 2nd ed. (Wiley-Intersicence, 1992), Chap. 13.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).

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Figures (1)

Fig. 1.
Fig. 1.

Hartmann screen arrays: (a) square, (b) hexagonal, (c) equally spaced radius-uniform density, and (d) uneven radial distribution. The total number of spots in figures (a) and (b) and also in (c) and (d) is about the same.

Tables (3)

Tables Icon

Table 1. Wavefront Deformations Monomial Terms up to Power 6

Tables Icon

Table 2. Wavefront Radial Slopes Monomial Terms up to Power 6

Tables Icon

Table 3. Polynomials for a Square Hartmann Pattern of Fig. 1(a)

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

W(x,y)=j=1JajZj(x,y),
aj=Zj(x,y)W(x,y)dxdy,
aj=W(x,y)·V⃗(x,y)dxdy,
W(ρ,θ)=n=0km=0nAnlρn{sincos}(|n2m|)θ,
r=n(n+1)2+m+1.
L=(k+1)(k+2)2.
W(ρ,θ)ρ=n=1km=0nAnmnρn1{sincos}(|n2m|)θ,
W(ρ,θ)θ=n=1km=0nAnm(n2m)ρn{sincos}(|n2m|)θ.
r=n(n+1)2+m,
W(ρ,θ)ρ=1rwTAρ(ρ,θ),
W(ρ,θ)θ=ρrwTAθ(ρ,θ),
W(ρ,θ)=W(ρ,θ)ρdρ+a00=n=1km=0nAnmρn{sincos}(|n2m|)θ+a00,
W(ρ,θ)=W(ρ,θ)θdθ+n=0kan0ρn=n=1km=0nAnmρn{sincos}(|n2m|)θ+n=0kan0ρn.
W(ρ,θ)=W(ρ,θ)θdρ,
lmax=int(J12).
TA(ρ,θ)=rwn=0km=0nAnmnρn1{sincos}(|n2m|)θ=r=1LArHr(ρ,θ),
i=1NHr(ρi,θi)Hp(ρi,θi)=δrp,
H1(ρ,θ)=P1(ρ,θ)H2(ρ,θ)=P2(ρ,θ)+D21H1(ρ,θ)H3(ρ,θ)=P3(ρ,θ)+D31H1(ρ,θ)+D32H2(ρ,θ)Hj(ρ,θ)=Pj(ρ,θ)+Dj1H1(ρ,θ)+Dj2H2(ρ,θ)++Dj,j1Hj1(ρ,θ),
Hr(ρ,θ)=Pr(ρ,θ)+s=1r1DrsHs(ρ,θ),
Hr(ρ,θ)Hp(ρ,θ)=Pr(ρ,θ)Hp(ρ,θ)+Hp(ρ,θ)s=1r1DrsHs(ρ,θ).
i=1NHr(ρi,θi)Hp(ρi,θi)=i=1NPr(ρi,θi)Hp(ρi,θi)+Drpi=1NHp2(ρi,θi)=0.
Dpr=i=1NPp(ρi,θi)Hr(ρi,θi)i=1NHr2(ρi,θi),
Hp(ρ,θ)=Pp(ρ,θ)+r=1p1DprHr(ρ,θ).
H1(ρ,θ)=P1(ρ,θ),

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