Abstract

Using a computer, we generated a set of filters to aid in the retrieval of aberration functions from Hartmanngrams. These filters consist of discrete two-dimensional data points, like the Hartmanngrams themselves, and are orthogonalized by the Gram–Schmidt procedure. The aberration coefficients are obtained by calculation of the scalar product of the Hartmanngram and each orthogonal filter.

© 1999 Optical Society of America

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References

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  1. Y. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 367–396.
  2. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  3. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  4. D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
    [CrossRef]
  5. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  6. W. H. Swanter, W. H. Lowrey, “Zernike–Tatian polynomials for interferogram reduction,” Appl. Opt. 19, 161–163 (1980).
    [CrossRef]
  7. J. L. Rayces, “Least-squares fitting of orthogonal polynomials for the wave-aberration function,” Appl. Opt. 31, 2223–2228 (1992).
    [CrossRef] [PubMed]
  8. F. T. S. Yu, X. Yang, Introduction to Optical Engineering (Cambridge U. Press, 1997), Chap. 11.
  9. M. Taniguchi, K. Matsuoka, Y. Ichioka, “Computer-generated multiple-object discriminant correlation filters: design by simulated annealing,” Appl. Opt. 34, 1379–1385 (1885).
    [CrossRef]
  10. J. L. Rayces, “Exact relations between wave aberrations and ray aberration,” Opt. Act. 11, 85–88 (1964).
    [CrossRef]

Carpio-Valadéz, J. M.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Cubalchini, R.

Ghozeil, Y.

Y. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 367–396.

Ichioka, Y.

Lowrey, W. H.

Malacara, D.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Matsuoka, K.

Rayces, J. L.

J. L. Rayces, “Least-squares fitting of orthogonal polynomials for the wave-aberration function,” Appl. Opt. 31, 2223–2228 (1992).
[CrossRef] [PubMed]

J. L. Rayces, “Exact relations between wave aberrations and ray aberration,” Opt. Act. 11, 85–88 (1964).
[CrossRef]

Sánchez-Mondragón, J. J.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Silva, D. E.

Southwell, W. H.

Swanter, W. H.

Taniguchi, M.

Wang, J. Y.

Yang, X.

F. T. S. Yu, X. Yang, Introduction to Optical Engineering (Cambridge U. Press, 1997), Chap. 11.

Yu, F. T. S.

F. T. S. Yu, X. Yang, Introduction to Optical Engineering (Cambridge U. Press, 1997), Chap. 11.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Opt. Act. (1)

J. L. Rayces, “Exact relations between wave aberrations and ray aberration,” Opt. Act. 11, 85–88 (1964).
[CrossRef]

Opt. Eng. (1)

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Other (2)

F. T. S. Yu, X. Yang, Introduction to Optical Engineering (Cambridge U. Press, 1997), Chap. 11.

Y. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 367–396.

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Figures (6)

Fig. 1
Fig. 1

The ray through point P is directed to P o according to the gradient of the wavefront f(x, y, z).

Fig. 2
Fig. 2

Nonorthogonal filters F j .

Fig. 3
Fig. 3

Hartmann screen.

Fig. 4
Fig. 4

Orthogonal filters V j .

Fig. 5
Fig. 5

Centers of the experimental Hartmanngram.

Fig. 6
Fig. 6

Experimental setup for testing a mirror.

Tables (3)

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Table 1 Seidel Polynomials in Cartersian Coordinates

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Table 2 Results of the Evaluation of a Simulated Hartmanngram

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Table 3 Results for an Experimental Hartmanngram

Equations (29)

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Po=P-tf,
xo=x-tfx,
yo=y-tfy,
zo=y-tfz.
xo=x+zo-zfxfz,
yo=y+zo-zfyfz,
fxfz=x+R-WW/xz-R,
fyfz=y+R-WW/yz-R.
xo=x1-zoR-zoWx,
yo=y1-zoR-zoWy,
xo=x1-zoR-zoj=1k wjUjx,
yo=y1-zoR-zoj=1k wjUjy,
Fj=Ujx, Ujy.
H=j=1k wjFj.
i=1NHi·Fki=j=1k wji=1NFji·Fki,
H=j=1k vjVj.
vj=i=1NHi·Vji.
V0=F0,V1=F1+D10V0,V2=F2+D20V0+D21V1,V3=F3+D30V0+D31V1+D32V2,Vj=Fj+Dj0V0+Dj1V1++Dj,j-1Vj-1,
Vj=Fj+i=1j-1 DjiVi,
i=1NVji·Vpi=i=1NFji·Vpi+Djpi=1NVpi2=0.
Djp=-i=1NFji·Vpii=1NVpi2,
V0=F0,V1=F1+C10F0,V2=F2+C20F0+C21F1,V3=F3+C30F0+C31F1+C32F2,Vj=Fj+Cj0F0+Cj1F1++Cj,j-1Fj-1,
Vj=Fj+i=1j-1 CjiFi,
C10=D10,C20=D21C10+D20,C21=D21,C30=D32C20+D31C10+D30,C31=D32C21+D31,C32=D32,C40=D43C30+D42C20+D41C10+D40,C41=D43C31+D42C21+D41,C42=D43C32+D42,C43=D43,
Cji=t=0j-i Dj,j-tCj-t,i,
H=voFo+j=1L vjFj+i=0j-1 CjiFi.
wj=vj+i=j+1L viCij,  wL=vL.
Wx, y=s5x2+y22+s4yx2+y2+s3x2+3y2+s2x2+y2+s1y+s0x.
rms=i=1Nxis-xie2+yis-yie2/N1/2,

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