Abstract

We conduct computer simulations of the reconstruction of a wave front at a telescope pupil with the phase-diversity method. An instantaneous wave front is reconstructed from focused and defocused specklegrams of a point star. In the wave-front reconstruction we do not fit the wave front to Zernike polynomials but retrieve the phase with a phase-unwrapping procedure. Averaging over many atmospherically perturbed wave fronts leads to the residual phase error, namely, the aberration of the telescope. The scintillation effect, nonuniformity of amplitude on a telescope pupil, is also discussed.

© 2001 Optical Society of America

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References

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  1. R. K. Tyson, Principles of Adaptive Optics (Academic, New York, 1991).
  2. L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
    [Crossref]
  3. 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]
  4. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
  5. M. G. Loefdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).
  6. R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
    [Crossref]
  7. M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
    [Crossref]
  8. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  9. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [Crossref]
  10. N. Baba, H. Tomita, N. Miura, “Iterative reconstruction method in phase-diversity imaging,” Appl. Opt. 33, 4428–4433 (1994).
    [Crossref] [PubMed]
  11. J. Strand, T. Taxt, “Performance evaluation of two-dimensional phase unwrapping algorithms,” Appl. Opt. 38, 4333–4344 (1999).
    [Crossref]
  12. 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 11, 107–117 (1994).
    [Crossref]
  13. D. C. Ghiglia, M. D. Pratt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  14. W. H. Press, B. D. Flanmery, S. A. Tenkolsky, W. T. Vallerling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).
  15. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
    [Crossref]
  16. L. Meynadier, V. Michan, M.-T. Velluet, J.-M. Conan, L. M. Mugnier, G. Rousset, “Noise propagation in wave-front sensing with phase diversity,” Appl. Opt. 38, 4967–4979 (1999).
    [Crossref]
  17. F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19.
    [Crossref]
  18. R. Avila, J. Vermin, E. Masciadri, “Whole atmospheric-turbulence profiling with generalized SCIDAR,” Appl. Opt. 36, 7898–7905 (1997).
    [Crossref]
  19. R. A. Johnston, R. G. Lane, “Modeling scintillation from a periodic Kolmogorov phase screen,” Appl. Opt. 39, 4761–4769 (2000).
    [Crossref]
  20. N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).
  21. D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A 16, 1005–1015 (1999).
    [Crossref]
  22. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [Crossref]

2000 (1)

1999 (3)

1998 (1)

1997 (1)

1996 (1)

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[Crossref]

1994 (3)

1993 (1)

1992 (1)

1990 (1)

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

1988 (1)

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Acton, D. S.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Avila, R.

Baba, N.

N. Baba, H. Tomita, N. Miura, “Iterative reconstruction method in phase-diversity imaging,” Appl. Opt. 33, 4428–4433 (1994).
[Crossref] [PubMed]

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

Citterio, O.

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

Conan, J.-M.

Conti, G.

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

Duncan, A. L.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Fienup, J. R.

Flanmery, B. D.

W. H. Press, B. D. Flanmery, S. A. Tenkolsky, W. T. Vallerling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Franza, F.

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

Fried, D. L.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Ghiglia, D. C.

Giordano, P.

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Harwit, A.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Johnston, R. A.

Keller, C. U.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[Crossref]

Kendrick, R. L.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Kenmochi, E.

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

Lane, R. G.

Lee, D. J.

Loefdahl, M. G.

M. G. Loefdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).

Löfdahl, M. G.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[Crossref]

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Masciadri, E.

Mattaini, E.

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
[Crossref]

Meynadier, L.

Michan, V.

Mitchell, K. E.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Miura, N.

Mugnier, L. M.

Noethe, L.

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

Paxman, R. G.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[Crossref]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[Crossref]

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Pratt, M. D.

D. C. Ghiglia, M. D. Pratt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

Press, W. H.

W. H. Press, B. D. Flanmery, S. A. Tenkolsky, W. T. Vallerling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Roddier, C.

Roddier, F.

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]

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19.
[Crossref]

Roggemann, M. C.

Romero, L. A.

Rousset, G.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Scharmer, G. B.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[Crossref]

M. G. Loefdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).

Schulz, T. J.

Seldin, J. H.

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[Crossref]

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

Strand, J.

Taxt, T.

Tenkolsky, S. A.

W. H. Press, B. D. Flanmery, S. A. Tenkolsky, W. T. Vallerling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Tomita, H.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, New York, 1991).

Vallerling, W. T.

W. H. Press, B. D. Flanmery, S. A. Tenkolsky, W. T. Vallerling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Velluet, M.-T.

Vermin, J.

Welsh, B. M.

Wilson, R. N.

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

Appl. Opt. (5)

Astron. Astrophys. Suppl. Ser. (1)

M. G. Loefdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).

Astrophys. J. (1)

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase-diverse techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[Crossref]

J. Mod. Opt. (1)

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, O. Citterio, G. Conti, E. Mattaini, “Active optics II. Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Optik (2)

N. Baba, E. Kenmochi, “Wavefront retrieval with use of defocused PSF data,” Optik 84, 70–72 (1990).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other (6)

R. K. Tyson, Principles of Adaptive Optics (Academic, New York, 1991).

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19.
[Crossref]

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, D. S. Acton, “A phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” in Space Telescopes and Instruments V, P. V. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 1190–1201 (1998).
[Crossref]

D. C. Ghiglia, M. D. Pratt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

W. H. Press, B. D. Flanmery, S. A. Tenkolsky, W. T. Vallerling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

G-S-based algorithm to retrieve unity-magnitude phasor from focused and defocused specklegrams.

Fig. 2
Fig. 2

(a) Phase distribution with 1λ coma and 1λ astigmatism aberrations on a pupil plane. The pupil is an annular shape with outer and inner diameters of 80 and 12 pixels, respectively. (b) The PSF resulting from the pupil-plane phase in (a).

Fig. 3
Fig. 3

(a) Atmospherically distorted phase on the pupil plane with the existence of 1λ coma and 1λ astigmatism aberrations, (b), (c) specklegrams at in-focus and out-of-focus planes. Specklegrams are displayed with an 8-bit gray scale on 128 × 128 pixels.

Fig. 4
Fig. 4

(a) Specklegram after the G-S-based procedure. The RMSE between this specklegram and that of Fig. 3(b) is 0.071. (b) Specklegram after the phase-unwrapping procedure. The RMSE is 0.169. (c) Specklegram after the phase-refinement procedure. The RMSE recovers to 0.071. (d) Final reconstructed phase distribution, which is similar to Fig. 3(a).

Fig. 5
Fig. 5

(a) Specklegram with a Strehl ratio of 0.013 at the in-focus plane; (b) specklegram at the out-of-focus plane; (c) phase distribution on a pupil plane, which produces the specklegrams shown in (a) and (b); (d) wrapped phase representation of (c); (e) wrapped phase distribution after the G-S- based procedure; (f) phase distribution after the phase-unwrapping procedure; (g) wrapped phase representation of (f); (h) phase distribution after the phase-refinement procedure; (i) wrapped phase representation of (h).

Fig. 6
Fig. 6

(a) Estimated phase distribution obtained by the averaging of 853 wave fronts. The rmse between this phase distribution and that of Fig. 2(a) is 0.154 rad. (b) The PSF calculated from the phase in (a). The RMSE between this intensity distribution and that of Fig. 2(b) is 0.141.

Fig. 7
Fig. 7

(a) Phase distribution on a pupil plane; (b) nonuniform amplitude distribution with a mean of 1.0 and σ = 0.077; (c), (d) specklegrams at in-focus and out-of-focus planes derived from (a) and (b); (e) reconstructed phase by use of nonuniform amplitude information; (f) reconstructed phase when the amplitude on the pupil plane is forced to unity in the G-S procedure.

Equations (10)

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

Pu, v=Au, vexpiΘ0fu, v.
pfx, y=-1Pu, v=|pfx, y|expiΦfx, y,
pfx, y=Ifx, y expiΦfx, y.
RMSE: i,jIi,j-I˜i,j2i,jI˜i,j21/2,
θi+1,j-2θi,j+θi-1,j+θi,j+1-2θi,j-θi,j-1=ρi,j,
ρi,j=δi,ju-δi-1,ju+δi,jv-δi,j-1v,
δi,ju=WΘi+1,j-Θi,j,  δi,jv=WΘi,j+1-Θi,j,
θˆi,j=ρˆi,j2cosπiN+cosπjN-2, i=0,  , N-1,  j=0,  , N-1,
E=i,jexpiΘi,j-expiθi,j2,
rmse: i,jθi,j-θ˜i,j2NT1/2,

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