Abstract

A method to evaluate wave-front aberrations in optical telescopes that is based on the method of curvature sensing but that solves the irradiance transport equation by variable separation is presented. This technique is simpler for processing than are previously released techniques and can perform more efficiently, as is required by active and adaptive optics. Testing for consistency of the method by evaluation of several sets of out-of-focus images obtained with the 2-m telescope at the Universidad Nacional Autónoma de México was carried out, and a stability of 10% for the derived values of Zernike coefficients was found.

© 1996 Optical Society of America

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References

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  1. C. Roddier, F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993).
    [CrossRef]
  2. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]
  3. K. Ichikawa, A. W. Lohmann, M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [CrossRef] [PubMed]
  4. I. Han, “New method for estimating the wave front from the curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
    [CrossRef]
  5. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, U.K., 1975), Chap. 9, p. 464.
  7. A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” Tech. rep. 95-02 (Instituto de Astronomía, Universidad Autónoma de México, San Pedro Mártir, Baja California, México, 1995).
  8. W. H. Press, B. P. Flannery, S. Tevkolsky, W. T. Vetterling, Numerical Recipes in C, 1st ed. (Cambridge U. Press, Cambridge, 1988), Chap. 10, p. 343.

1995 (1)

I. Han, “New method for estimating the wave front from the curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

1993 (1)

1991 (1)

1988 (1)

1976 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, U.K., 1975), Chap. 9, p. 464.

Cordero, A.

A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” Tech. rep. 95-02 (Instituto de Astronomía, Universidad Autónoma de México, San Pedro Mártir, Baja California, México, 1995).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. Tevkolsky, W. T. Vetterling, Numerical Recipes in C, 1st ed. (Cambridge U. Press, Cambridge, 1988), Chap. 10, p. 343.

Han, I.

I. Han, “New method for estimating the wave front from the curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

Harris, O.

A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” Tech. rep. 95-02 (Instituto de Astronomía, Universidad Autónoma de México, San Pedro Mártir, Baja California, México, 1995).

Ichikawa, K.

Lohmann, A. W.

Luna, E.

A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” Tech. rep. 95-02 (Instituto de Astronomía, Universidad Autónoma de México, San Pedro Mártir, Baja California, México, 1995).

Noll, R. J.

Press, W. H.

W. H. Press, B. P. Flannery, S. Tevkolsky, W. T. Vetterling, Numerical Recipes in C, 1st ed. (Cambridge U. Press, Cambridge, 1988), Chap. 10, p. 343.

Roddier, C.

Roddier, F.

Takeda, M.

Tevkolsky, S.

W. H. Press, B. P. Flannery, S. Tevkolsky, W. T. Vetterling, Numerical Recipes in C, 1st ed. (Cambridge U. Press, Cambridge, 1988), Chap. 10, p. 343.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. Tevkolsky, W. T. Vetterling, Numerical Recipes in C, 1st ed. (Cambridge U. Press, Cambridge, 1988), Chap. 10, p. 343.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, U.K., 1975), Chap. 9, p. 464.

Zárate, S.

A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” Tech. rep. 95-02 (Instituto de Astronomía, Universidad Autónoma de México, San Pedro Mártir, Baja California, México, 1995).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

I. Han, “New method for estimating the wave front from the curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, U.K., 1975), Chap. 9, p. 464.

A. Cordero, E. Luna, S. Zárate, O. Harris, “Evaluación de la calidad de la imagen del telescopio de 2.1 m,” Tech. rep. 95-02 (Instituto de Astronomía, Universidad Autónoma de México, San Pedro Mártir, Baja California, México, 1995).

W. H. Press, B. P. Flannery, S. Tevkolsky, W. T. Vetterling, Numerical Recipes in C, 1st ed. (Cambridge U. Press, Cambridge, 1988), Chap. 10, p. 343.

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Figures (4)

Fig. 1.
Fig. 1.

Azimuthal plots of the curvature function (1/I)(∂I/∂z) at 16 different radii ρ i that are equally spaced from the inner hole to the outer border. Arbitrary constants have been added for display purposes. The crosses indicate the values of the azimuthal angles used.

Fig. 2.
Fig. 2.

Fourier transforms of each plot shown in Fig. 1: The plots permit the functions β m (ρ) to be obtained according to Eq. (6). Plotted are the logarithm of the modulus plus an arbitrary constant. The crosses indicate the values of the angular frequencies m = 0–8.

Fig. 3.
Fig. 3.

Fitted data of the β m i ) points (indicated by the filled circles) for seven values of m, determined with expansions from Eq. (7). The fitted plots were obtained by the use of an amoeba-minimization routine with simulated annealing.

Fig. 4.
Fig. 4.

Histograms for the largest aberrations found in nine different experiments: Coma, for a coefficient value of A 3,1 = 0.27 ± 0.01 μm and for an angle of θ = −17° ± 7°, trifoil, for A 3,3 = 0.20 ± 0.03 μm and for an angle of θ = 152° ± 4°, astigmatism, for A 2,2 = 1.26 ± 0.12 μm and for an angle of θ = −108° ± 7°. Sphericity (A 4,0) is seen to be a function of the position of the secondary mirror, measured here versus the radii of the nine out-of-focus images (indicated by the filled circles).

Equations (8)

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I z = ( I W + I 2 W ) ,
W ( ρ , θ ) = n = 1 N    m = 0 ( n m ) even n    A n , m Z n m ( ρ , θ ) ,
Z n m ( ρ , θ ) = ( 2 ) n + 1 R n m ( ρ ) { cos   m θ sin   m θ } ,
1 I I z = n = 1 N    m = 0 ( n m ) even n A n , m Φ n m ( ρ ) { cos   m θ sin   m θ } ,
Φ n m ( ρ ) = ( 2 ) n + 1 [ 1 I ( ρ ) I ( ρ ) ∂ρ R n m ∂ρ + 2 R n m ρ 2    + 1 ρ R n m ∂ρ m 2 ρ 2 R n m ∂ρ ]
1 I I z = m = 0 N   β m ( ρ ) { cos   m θ sin   m θ } ,
β m ( ρ ) = n m ( n m ) even N    A n , m Φ n m ( ρ ) .
S ( ρ , θ ) = 1 I I z f ( f l ) l

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