Abstract

Wavefront-sensing performance is assessed for focus-diverse phase retrieval as the aberration spatial frequency and the diversity defocus are varied. The analysis includes analytical predictions for optimal diversity values corresponding to the recovery of a dominant spatial-frequency component in the pupil. The calculation is shown to be consistent with the Cramér–Rao lower bound by considering a sensitivity analysis of the point-spread function to the spatial frequency being estimated. A maximum value of diversity defocus is also calculated, beyond which wavefront-sensing performance decreases as diversity defocus is increased. The results are shown to be consistent with the Talbot imaging phenomena, explaining multiple periodic regions of maximum and minimum contrast as a function of aberration spatial frequency and defocus. Wavefront-sensing performance for an iterative-transform phase-retrieval algorithm is also considered as diversity defocus and aberration spatial frequency are varied.

© 2003 Optical Society of America

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  1. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  2. The purpose of a diversity function in wavefront sensing is to “spread out” the optical response to expose aberration information. Since defocus has only angular dependence, no one part of the point-spread function is emphasized more than any other.
  3. J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
    [CrossRef] [PubMed]
  4. J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
    [CrossRef]
  5. D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramér–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A 16, 1005–1015 (1999).
    [CrossRef]
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  8. S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  9. R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).
  10. J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
    [CrossRef]
  11. J. N. Cederquist, C. C. Wackerman, “Phase-retrieval error: a lower bound,” J. Opt. Soc. Am. A 4, 1788–1792 (1987).
    [CrossRef]
  12. J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
    [CrossRef]
  13. B. H. Dean, R. Lyon, “Cramér–Rao bounds for focus diverse wavefront sensing with deformable mirror ‘print-through’,” in Computational Optics and Imaging for Space Applications, R. Lyon, ed. (NASA/Goddard Space Flight Center, Greenbelt, Md., 2000), pp. 127–142.
  14. B. H. Dean, “Cramér–Rao analysis for phase-diverse-phase-retrieval: diversity functions and broadband phase-retrieval,” presented at the Wavefront Sensing & Controls Conference, sponsored by Kamuela Optical Associates, Kohala Coast, Hawaii, November 13–16, 2000.
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    [CrossRef] [PubMed]
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  19. C. Bowers, “Some considerations regarding the propagation of spectral power from pupil to defocused image planes,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., April 10, 2001.
  20. B. H. Dean, “Fresnel zone propagation of spectral power with applications to image-based wavefront sensing,” presented for the NASANext Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., July 8, 2001.
  21. The frequency content in this instance is associated with the finite spacing of actuators placed in a periodic array behind a thin mirror face sheet. Ironically, the “print-through” associated with such systems is a result of the high-frequency correction capability of these many-actuator control systems. For example, for a Xinetics 349 channel deformable mirror, there are 21 actuators spanning the clear aperture, yielding approximately 10 cycles/aperture along a single spatial-frequency direction.
  22. R. Paxman, B. Thelen, “Wavefront sensing for deployable-optic systems,” presented at the Wavefront Sensing and Controls Conference, sponsored by Kamuela Optical Associates, Kohala Coast, Hawaii, November 13–16, 2000.
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  25. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  26. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  27. C. Roddier, F. Roddier, “Wavefront reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993).
    [CrossRef]
  28. C. Roddier, F. Roddier, “Combined approach to the Hubble Space Telescope wavefront distortion analysis,” Appl. Opt. 32, 2992–3008 (1993).
    [CrossRef] [PubMed]
  29. J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).
  30. R. G. Lyon, “DCATT wavefront sensing and optical control study,” , February22, 1999, http://jansky.gsfc.nasa.gov/OSCAR/ .
  31. D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
    [CrossRef]
  32. D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
    [CrossRef]
  33. B. H. Dean, “White-light phase-retrieval analysis,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., October 11, 2000.
  34. T. E. Gureyev, A. Roberts, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with the use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995).
    [CrossRef]
  35. B. H. Dean, R. Boucarut, “Deformable mirror optical calibration and test results,” presented at the Next Generation Space Telescope Science and Technology Exposition, Woods Hole, Mass., September 13–16, 1999.
  36. C. Roddier, F. Roddier, “Reconstruction of Hubble Space Telescope wavefront distortion from stellar images taken at various focal positions,” in Final Report: Jet Propulsion Laboratory Contract 958893 on Hubble Space Telescope Optical Telescope Assembly Analysis, May 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991), pp. 2–3.
  37. D. Malacara, S. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, New York, 1997).
  38. B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, T. Bruno, “Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications,” Appl. Opt. 37, 4553–4560 (1998).
    [CrossRef]
  39. If inaccurate pupil data are applied in the iterative-transform sense (Fig. 1), the pupil amplitude and the phase information become mixed, resulting in artificial aberrations in the phase estimate. For example, a pupil amplitude function consisting of a shifted central obscuration in a two-mirror system can lead to phase-retrieval estimates with artificial coma. But this ambiguity is resolved when utilizing diversity images on both sides of focus since the defocused PSF is asymmetric about focus (for a shifted central obscuration), while true coma produces a symmetric PSF about focus.

1999 (1)

1998 (1)

1995 (1)

1993 (3)

1989 (1)

1987 (1)

1986 (1)

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

1982 (2)

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

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1977 (1)

W. H. Southwell, “Wave-front analyzer using a maximum likelihood algorithm,” J. Opt. Soc. Am. A 3, 396–399 (1977).
[CrossRef]

1973 (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

1972 (1)

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

Basinger, S.

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Bely, P.

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

Boucarut, R.

B. H. Dean, R. Boucarut, “Deformable mirror optical calibration and test results,” presented at the Next Generation Space Telescope Science and Technology Exposition, Woods Hole, Mass., September 13–16, 1999.

Bowers, C.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

C. Bowers, “Some considerations regarding the propagation of spectral power from pupil to defocused image planes,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., April 10, 2001.

Bruno, T.

Burg, R.

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

Burns, L.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Carrara, D. A.

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Cederquist, J. N.

Craig, A. T.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

Davila, P.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Dean, B.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Dean, B. H.

B. H. Dean, “Phase-retrieval performance as a function of defocus and aberration frequency,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., August 20, 2000.

B. H. Dean, R. Lyon, “Cramér–Rao bounds for focus diverse wavefront sensing with deformable mirror ‘print-through’,” in Computational Optics and Imaging for Space Applications, R. Lyon, ed. (NASA/Goddard Space Flight Center, Greenbelt, Md., 2000), pp. 127–142.

B. H. Dean, “Fresnel zone propagation of spectral power with applications to image-based wavefront sensing,” presented for the NASANext Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., July 8, 2001.

B. H. Dean, R. Boucarut, “Deformable mirror optical calibration and test results,” presented at the Next Generation Space Telescope Science and Technology Exposition, Woods Hole, Mass., September 13–16, 1999.

B. H. Dean, “Cramér–Rao analysis for phase-diverse-phase-retrieval: diversity functions and broadband phase-retrieval,” presented at the Wavefront Sensing & Controls Conference, sponsored by Kamuela Optical Associates, Kohala Coast, Hawaii, November 13–16, 2000.

B. H. Dean, “White-light phase-retrieval analysis,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., October 11, 2000.

DeVore, S.

D. Malacara, S. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, New York, 1997).

Fienup, J. R.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef] [PubMed]

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).

Fitzmaurice, M.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Gerchberg, R. W.

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

Gonsalves, R. A.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 87–89.

Gureyev, T. E.

Hogg, R. V.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

Jankevics, A.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Kryskowski, D.

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

Landers, F.

Lee, D. J.

Levine, B. M.

Lowman, A.

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Lyon, R.

B. H. Dean, R. Lyon, “Cramér–Rao bounds for focus diverse wavefront sensing with deformable mirror ‘print-through’,” in Computational Optics and Imaging for Space Applications, R. Lyon, ed. (NASA/Goddard Space Flight Center, Greenbelt, Md., 2000), pp. 127–142.

Lyon, R. G.

R. G. Lyon, “DCATT wavefront sensing and optical control study,” , February22, 1999, http://jansky.gsfc.nasa.gov/OSCAR/ .

Malacara, D.

D. Malacara, S. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, New York, 1997).

Marron, J. C.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef] [PubMed]

J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).

Martinsen, E. A.

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Mosier, G.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

Norton, T.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Nugent, K. A.

Paxman, R.

R. Paxman, B. Thelen, “Wavefront sensing for deployable-optic systems,” presented at the Wavefront Sensing and Controls Conference, sponsored by Kamuela Optical Associates, Kohala Coast, Hawaii, November 13–16, 2000.

Paxman, R. G.

J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Perkins, B.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Petrone, P.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Redding, D.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

Roberts, A.

Robinson, S. R.

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

Roddier, C.

C. Roddier, F. Roddier, “Wavefront reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993).
[CrossRef]

C. Roddier, F. Roddier, “Combined approach to the Hubble Space Telescope wavefront distortion analysis,” Appl. Opt. 32, 2992–3008 (1993).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Reconstruction of Hubble Space Telescope wavefront distortion from stellar images taken at various focal positions,” in Final Report: Jet Propulsion Laboratory Contract 958893 on Hubble Space Telescope Optical Telescope Assembly Analysis, May 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991), pp. 2–3.

Roddier, F.

C. Roddier, F. Roddier, “Combined approach to the Hubble Space Telescope wavefront distortion analysis,” Appl. Opt. 32, 2992–3008 (1993).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Wavefront reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993).
[CrossRef]

C. Roddier, F. Roddier, “Reconstruction of Hubble Space Telescope wavefront distortion from stellar images taken at various focal positions,” in Final Report: Jet Propulsion Laboratory Contract 958893 on Hubble Space Telescope Optical Telescope Assembly Analysis, May 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991), pp. 2–3.

Roggemann, M. C.

Saxton, W. O.

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

Schulz, T. J.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef] [PubMed]

J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).

Seldin, J. H.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef] [PubMed]

J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).

Shi, F.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

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[CrossRef]

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W. Sun, “Cramér–Rao lower bound analysis on estimation accuracy for phase retrieval,” M. S. thesis, directed by T. Schulz (Michigan Technological University, Houghton, Mich., 1998).

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J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).

R. Paxman, B. Thelen, “Wavefront sensing for deployable-optic systems,” presented at the Wavefront Sensing and Controls Conference, sponsored by Kamuela Optical Associates, Kohala Coast, Hawaii, November 13–16, 2000.

Thelen, B. J.

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

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Wetherell, W. B.

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Whalen, A. D.

A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971).

Wheeler, L.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

Wilson, M.

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

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J. Phys. D (1)

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[CrossRef]

Opt. Eng. (2)

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[CrossRef]

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[CrossRef]

Optik (Stuttgart) (1)

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

Other (25)

The purpose of a diversity function in wavefront sensing is to “spread out” the optical response to expose aberration information. Since defocus has only angular dependence, no one part of the point-spread function is emphasized more than any other.

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971).

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1993).

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H. Seldin, B. Thelen, “Image inversion analysis of the Hubble Space Telescope,” in Final Report: Jet Propulsion Laboratory Contract 958892 on the Hubble Space Telescope Optical Telescope Assembly Analysis, August 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991).

R. G. Lyon, “DCATT wavefront sensing and optical control study,” , February22, 1999, http://jansky.gsfc.nasa.gov/OSCAR/ .

D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R. Burg, G. Mosier, “Wavefront sensing and control for a Next Generation Space Telescope,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
[CrossRef]

D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L. Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B. Perkins, P. Petrone, T. Norton, M. Wilson, L. Wheeler, “Wavefront control for a segmented deployable space telescope,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).
[CrossRef]

B. H. Dean, “White-light phase-retrieval analysis,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., October 11, 2000.

B. H. Dean, R. Boucarut, “Deformable mirror optical calibration and test results,” presented at the Next Generation Space Telescope Science and Technology Exposition, Woods Hole, Mass., September 13–16, 1999.

C. Roddier, F. Roddier, “Reconstruction of Hubble Space Telescope wavefront distortion from stellar images taken at various focal positions,” in Final Report: Jet Propulsion Laboratory Contract 958893 on Hubble Space Telescope Optical Telescope Assembly Analysis, May 1991 (Jet Propulsion Laboratory, Pasadena, Calif., 1991), pp. 2–3.

D. Malacara, S. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, New York, 1997).

B. H. Dean, R. Lyon, “Cramér–Rao bounds for focus diverse wavefront sensing with deformable mirror ‘print-through’,” in Computational Optics and Imaging for Space Applications, R. Lyon, ed. (NASA/Goddard Space Flight Center, Greenbelt, Md., 2000), pp. 127–142.

B. H. Dean, “Cramér–Rao analysis for phase-diverse-phase-retrieval: diversity functions and broadband phase-retrieval,” presented at the Wavefront Sensing & Controls Conference, sponsored by Kamuela Optical Associates, Kohala Coast, Hawaii, November 13–16, 2000.

If inaccurate pupil data are applied in the iterative-transform sense (Fig. 1), the pupil amplitude and the phase information become mixed, resulting in artificial aberrations in the phase estimate. For example, a pupil amplitude function consisting of a shifted central obscuration in a two-mirror system can lead to phase-retrieval estimates with artificial coma. But this ambiguity is resolved when utilizing diversity images on both sides of focus since the defocused PSF is asymmetric about focus (for a shifted central obscuration), while true coma produces a symmetric PSF about focus.

W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, R. Shannon, J. Wyant, eds. (Academic, New York, 1980), Vol. VIII, p. 172.

B. H. Dean, “Phase-retrieval performance as a function of defocus and aberration frequency,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., August 20, 2000.

C. Bowers, “Some considerations regarding the propagation of spectral power from pupil to defocused image planes,” presented for the NASA Next Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., April 10, 2001.

B. H. Dean, “Fresnel zone propagation of spectral power with applications to image-based wavefront sensing,” presented for the NASANext Generation Space Telescope Technical Memoranda, NASA/Goddard Space Flight Center, Greenbelt, Md., July 8, 2001.

The frequency content in this instance is associated with the finite spacing of actuators placed in a periodic array behind a thin mirror face sheet. Ironically, the “print-through” associated with such systems is a result of the high-frequency correction capability of these many-actuator control systems. For example, for a Xinetics 349 channel deformable mirror, there are 21 actuators spanning the clear aperture, yielding approximately 10 cycles/aperture along a single spatial-frequency direction.

R. Paxman, B. Thelen, “Wavefront sensing for deployable-optic systems,” presented at the Wavefront Sensing and Controls Conference, sponsored by Kamuela Optical Associates, Kohala Coast, Hawaii, November 13–16, 2000.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 87–89.

W. Sun, “Cramér–Rao lower bound analysis on estimation accuracy for phase retrieval,” M. S. thesis, directed by T. Schulz (Michigan Technological University, Houghton, Mich., 1998).

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Figures (18)

Fig. 1
Fig. 1

Iterative-transform phase retrieval [F(⋅) denotes Fourier transform].

Fig. 2
Fig. 2

Defocus + sinusoidal “aberration.”

Fig. 3
Fig. 3

Plot of sin(πν^02/8aˆ) versus defocus for ν0=10 cycles/aperture.

Fig. 4
Fig. 4

Position of maximum diversity defocus for ν0=10 cycles/aperture.

Fig. 5
Fig. 5

Comparison of transfer functions and analytical prediction for contrast maxima for n=1 and ν0=10 cycles/aperture; defocus= 8.33 waves.

Fig. 6
Fig. 6

Comparison of transfer functions and analytical prediction for contrast minima for n=1 and ν0=10 cycles/aperture; defocus= 12.5 waves.

Fig. 7
Fig. 7

Fisher information versus defocus for a phase grating assuming ν0=10 cycles/aperture. Values predicted by Eqs. (9) and (11) are also identified.

Fig. 8
Fig. 8

Phase-retrieval result for minimum-contrast case; n=1, ν0=10 cycles/aperture; defocus=12.5 waves.

Fig. 9
Fig. 9

Phase-retrieval result for maximum-contrast case; n=1, ν0=10 cycles/aperture; defocus=8.33 waves.

Fig. 10
Fig. 10

Phase-retrieval convergence comparison for maximum- and minimum-contrast cases.

Fig. 11
Fig. 11

Trefoil power spectral density (PSD).

Fig. 12
Fig. 12

Diversity point-spread function (PSF) data and phase-retrieval (PR) results for the trefoil aberration.

Fig. 13
Fig. 13

Phase-retrieval convergence comparison for phase-retrieval results shown in Fig. 12.

Fig. 14
Fig. 14

Deformable mirror “print-through” PSD.

Fig. 15
Fig. 15

Diversity PSF data and phase-retrieval results for a deformable mirror “print-through” phase residual.

Fig. 16
Fig. 16

Dominant PSD versus Zernike order.

Fig. 17
Fig. 17

Rms (Zernike fit minus data) versus Zernike order for a deformable mirror “print-through” phase residual.

Fig. 18
Fig. 18

Diversity defocus values implied by Zernike basis order.

Tables (1)

Tables Icon

Table 1 Summary of Defocus Values Corresponding to Contrast Maxima and Minima for n=0, 1, 2, Assuming ν^0=10 Cycles/Aperture

Equations (26)

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PSFca=F(A)=F(A(x)exp{ik[ax2+m sin(2πν0x)]}).
A[1+imk sin(2πν0x)]A(x)exp(ikax2),
PSFca(ν)=F {[1+imk sin(2πν0x)]}  F [exp(ikax2)]  F[A(x)],
PSFca=F(A)=δ(νˆ)+mk2 [δ(νˆ+ν^0)-δ(νˆ-ν^0)]exp(-iπλν^2/2a),
PSFca=12exp-iπλ2aν2+mk2exp-iπλ2a (ν+ν0)2-exp-iπλ2a (ν-ν0)2.
(a/λ)x2aˆx^2=aˆx2(D/2)2a=4λD2 aˆ,
PSFca=12exp-iπ8aˆν^2+mk2exp-iπ8aˆ (νˆ+ν^0)2-exp-iπ8aˆ (νˆ-ν^0)2.
I(νˆ)=PSFca × PSFca¯=1-2mk sin(πν^02/8aˆ)sin[2π(ν^0/8aˆ)νˆ],
πν^028aˆ=π2 (2n±1)a^max=±ν^024(2n±1),
n=0, 1, 2,,
sinπ2 (2n±1)=±cos(nπ),n=0, 1, 2, .
πν^028aˆ=nπa^min=±ν^028n,n=0, 1, 2, .
a^rev,max=±ν^024,
limaˆ{sin(πν^02/8aˆ)}=0,
Imin=1,
OTF(ωˆ)=F [I(νˆ)]=δ(ωˆ)+imk sin(πν^02/8aˆ)×[δ(ωˆ-ω^0)-δ(ωˆ+ω^0)],
ωˆ=λf#ω=ω/ωc,
ω^0ν^0/8aˆ.
Fαβ-E(αβ2L)=Npm=1Mi,j(αPSFm)(βPSFm)/PSF,
Var(a^β)σaβ2Fββ-1(a),
F(νˆI)2|νˆν^0=[kmπν^0cos(πν^02/4aˆ)sin(πν^02/8aˆ)/2aˆ]2.
Fmax2km(1-4n)ν^02.
Fmin=0,
PSDmean=bZorder+c,
b=0.1787,c=1.64cycles/aperture.
a^max=±(bZorder+c)2/(16n4),n=0, 1, 2,,

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