Abstract

The Hartmann test is a well-known technique for testing large telescope mirrors. The Hartmann technique samples the wave front under analysis by use of a screen of uniformly spaced array of holes located at the pupil plane. The traditional technique used to gather quantitative data requires the measurement of the centroid of these holes as imaged near the paraxial focus. The deviation from its unaberrated uniform position is proportional to the slope of the wave-front asphericity. The centroid estimation is normally done manually with the aid of a microscope or a densitometer; however, newer automatic fringe-processing techniques that use the synchronous detection technique or the Fourier phase-estimation method may also be used. Here we propose a new technique based on a regularized phase-tracking (RPT) system to detect the transverse aberration in Hartmanngrams in a direct way. That is, it takes the dotted pattern of the Hartmanngram as input, and as output the RPT system gives the unwrapped transverse ray aberration in just one step. Our RPT is compared with the synchronous and the Fourier methods, which may be regarded as its closest competitors.

© 1999 Optical Society of America

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References

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  1. J. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 367–396.
  2. M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
    [CrossRef]
  3. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  4. M. Takeda, H. Ina, S. Kobayashi, “Fourier transform methods of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  5. M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
    [CrossRef] [PubMed]
  6. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 4, 107–117 (1994).
    [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction problem of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  8. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  9. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  10. A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

1997

1996

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

1994

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 4, 107–117 (1994).
[CrossRef]

1984

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1982

1979

1978

1977

1963

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Cuevas, F. J.

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 4, 107–117 (1994).
[CrossRef]

Ghozeil, J.

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

Hudgin, R. H.

Hunt, B. R.

Ina, H.

Kobayashi, S.

Malacara, D.

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

Marroquin, J. L.

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

Noll, R. J.

Romero, L. A.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 4, 107–117 (1994).
[CrossRef]

Servin, M.

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

Takeda, M.

Tikhonov, A. N.

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 4, 107–117 (1994).
[CrossRef]

Opt. Eng.

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Sov. Math. Dokl.

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Other

J. 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 (10)

Fig. 1
Fig. 1

Simplified experimental setup used to obtain an off-axis Hartmanngram.

Fig. 2
Fig. 2

Framegrabbed Hartmanngram obtained from a primary mirror that has a curvature radius of 280.6 cm, a conic constant of -1.0133, and a diameter of 64 cm.

Fig. 3
Fig. 3

Fringe pattern used as prior phase model for the Hartmanngram aberration. The RPT technique herein presented will then estimate the phase difference between this fringe pattern and the digitized Hartmanngram shown in Fig. 2.

Fig. 4
Fig. 4

Estimated phase difference between the prior phase model shown in Fig. 3 and the actual phase of the digitized Hartmanngram shown in Fig. 2 along the x direction.

Fig. 5
Fig. 5

(a) Fringe pattern modulated by the expected phase model (spherical ray aberration). We have superimposed the digitized Hartmanngram on this fringe pattern to see the initial phase mismatch. (b) Fringe pattern modulated by the expected phase model plus the estimated phase error shown in Fig. 4. We have superimposed the digitized Hartmanngram on this fringe pattern to see clearly the spatial phase match of the Hartmanngram dots along the x direction.

Fig. 6
Fig. 6

Estimated phase difference between the prior phase model and the actual phase of the digitized Hartmanngram shown in Fig. 2 along the y direction.

Fig. 7
Fig. 7

(a) Fringe pattern modulated by the expected phase model (spherical ray aberration). We have superimposed the digitized Hartmanngram on this fringe pattern to see the initial phase mismatch. (b) Fringe pattern modulated by the expected phase model plus the estimated phase error shown in Fig. 6. We have superimposed the digitized Hartmanngram on this fringe pattern to compare its spatial phase with the phase of the Hartmanngram dots along the y direction.

Fig. 8
Fig. 8

Fourier (frequency) spectrum of the Hartmanngram shown in Fig. 2. We may see that the diffraction orders are still separable, so linear phase-detection techniques may be used.

Fig. 9
Fig. 9

Cosine of the estimated modulating phase obtained from the Hartmanngram in Fig. 2 with the synchronous technique.

Fig. 10
Fig. 10

Cosine of the estimated modulating phase obtained from the Hartmanngram in Fig. 2 with the Fourier technique.

Equations (16)

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Hsx1, y1=n=-N/2N/2m=-N/2N/2 hx1-dn, y1-dm,
x=r-Lr x1+L Wx1, y1x1, y=r-Lr y1+L Wx1, y1y1,
Hx, y=n=-N/2N/2m=-N/2N/2 hir-Lr x1-L Wx1, y1x1-dn, r-Lr y1-L Wx1, y1y1-dmPx, y,
Wx1, y1=Bx12+y122,
x=r-Lr x1+4LBx12+y122x1, y=r-Lr y1+4LBx12+y122y1.
Fxx, y=cosω0x+Kx2+y2x+ϕxx, yPx, y,
Ux, y=,ηNx, yPx, yH, η-cos pxx, y2+λϕx, η-ϕxx, y2m, η,
pxx, y=ω0x+Kx2+y2x+ϕxx, y,
pyx, y=ω0y+Kx2+y2y+ϕyx, y,
ϕxk+1x, y=ϕxkx, y-τ Ux, yϕxx, y,
Txx, y=Kx2+y2x+ϕxx, yPx, y, Tyx, y=Kx2+y2y+ϕyx, yPx, y.
I1x, y=1+cosω0x+Kx2+y2xPx, y.
I2x, y=1+cosω0x+Kx2+y2x+ϕxx, yPx, y.
IFx, y=1+cosTFxx, y,
TSxx, y=arctanLPFHx, ysinω0xLPFHx, ycosω0x,
ISx, y=1+cosω0x+TSxx, y,

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