Abstract

Because of mechanical aspects of fabrication, launch, and operational environment, space telescope optics can suffer from unforeseen aberrations, detracting from their intended diffraction-limited performance goals. We give the results of simulation studies designed to explore how wave-front aberration information for such near-diffraction-limited telescopes can be estimated through a regularized, low-pass filtered version of the Gonsalves (least-squares) phase-diversity technique. We numerically simulate models of both monolithic and segmented space telescope mirrors; the segmented case is a simplified model of the proposed next generation space telescope. The simulation results quantify the accuracy of phase diversity as a wave-front sensing (WFS) technique in estimating the pupil phase map. The pupil phase is estimated from pairs of conventional and out-of-focus photon-limited point-source images. Image photon statistics are simulated for three different average light levels. Simulation results give an indication of the minimum light level required for reliable estimation of a large number of aberration parameters under the least-squares paradigm. For weak aberrations that average a 0.10λ pupil rms, the average WFS estimation errors obtained here range from a worst case of 0.057λ pupil rms to a best case of only 0.002λ pupil rms, depending on the light level as well as on the types and degrees of freedom of the aberrations present.

© 1997 Optical Society of America

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References

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  1. D. J. Lee, M. C. Roggemann, B. M. Welsh, “Using wavefront sensor information in image post-processing to improve the resolution of telescope with small aberrations,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 42–53 (1996).
  2. R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, 1991).
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    [CrossRef]
  4. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
    [CrossRef]
  5. B. M. Welsh, M. C. Roggemann, “Signal-to-noise comparison of deconvolution from wave-front sensing with traditional linear and speckle image reconstruction,” Appl. Opt. 34, 2111–2119 (1995).
    [CrossRef] [PubMed]
  6. R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. J. Tescher, ed., Proc. SPIE207, 32–39 (1979).
  7. R. A. Golsalves, “Fundamentals of wavefront sensing by phase retrieval,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 56–65 (1982).
    [CrossRef]
  8. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  9. M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  10. R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescope using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. Haiè, eds., Proc. SPIE1351, 787–797 (1990).
    [CrossRef]
  11. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1994).
  13. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
    [CrossRef]
  14. “What will be the next Big Thing,” Nature 381, 465 (1996).
    [CrossRef]
  15. D. Dooling, “Beyond Hubble,” The Institute (IEEE monthly newsletter)1 (June1996).
  16. A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1998).
  18. 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]
  19. R. L. Langendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
    [CrossRef]
  20. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewoods Cliffs, N. J., 1993).
  21. I. J. D. Craig, J. C. Brown, Inverse Problems in Astronomy: a Guide to Inversion Strategies for Remotely Sensed Data (Hilger, Bristol, UK, 1986).
  22. M. C. Roggemann, D. W. Tyler, M. F. Bilmont, “Linear reconstruction of compensated images: theory and experimental results,” Appl. Opt. 31, 7429–7441 (1992).
    [CrossRef] [PubMed]
  23. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
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    [CrossRef]
  25. C. Chi, P. Mehta, A. Ostroff, “Vibrational modes of the primary mirror structure in the large space telescope system,” in Space Optics: Proceedings of the Ninth International Conference of the International Commission for OpticsB. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences, Washington, D.C., 1974), Vol. 9, pp. 209–238.
  26. L. Needels, B. M. Levine, M. Milman, “Limits on adaptive optics systems for lightweight space telescopes,” in Space Astronomical Telescopes and Instruments II, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE1945, 176–184 (1993).
  27. A. Grace, Optimization Toolbox for Use with MATLAB (Math Works, Inc., Natick, Mass., 1992).
  28. C. Kittel, H. Kroemer, Thermal Physics (Freeman, New York, 1980).
  29. R. G. Paxman, J. H. Seldin, M. G. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
    [CrossRef]
  30. M. G. Lofdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. Suppl. Ser. 107, 243–264 (1994).
  31. 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]
  32. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, UK, 1992).

1996

“What will be the next Big Thing,” Nature 381, 465 (1996).
[CrossRef]

A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
[CrossRef]

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

1995

1994

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

1993

1992

1990

R. L. Langendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

1988

1982

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

1976

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Biemond, J.

R. L. Langendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Bilmont, M. F.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1994).

Brown, J. C.

I. J. D. Craig, J. C. Brown, Inverse Problems in Astronomy: a Guide to Inversion Strategies for Remotely Sensed Data (Hilger, Bristol, UK, 1986).

Chi, C.

C. Chi, P. Mehta, A. Ostroff, “Vibrational modes of the primary mirror structure in the large space telescope system,” in Space Optics: Proceedings of the Ninth International Conference of the International Commission for OpticsB. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences, Washington, D.C., 1974), Vol. 9, pp. 209–238.

Chidlaw, R.

R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. J. Tescher, ed., Proc. SPIE207, 32–39 (1979).

Craig, I. J. D.

I. J. D. Craig, J. C. Brown, Inverse Problems in Astronomy: a Guide to Inversion Strategies for Remotely Sensed Data (Hilger, Bristol, UK, 1986).

Crippen, S. L.

R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescope using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. Haiè, eds., Proc. SPIE1351, 787–797 (1990).
[CrossRef]

Fienup, J. R.

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, UK, 1992).

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Fried, D. L.

D. L. Fried, “Post-detection wavefront distortion compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. SPIE828, 127–133 (1987).
[CrossRef]

Golsalves, R. A.

R. A. Golsalves, “Fundamentals of wavefront sensing by phase retrieval,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 56–65 (1982).
[CrossRef]

Gonsalves, R. A.

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

R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. J. Tescher, ed., Proc. SPIE207, 32–39 (1979).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1998).

Grace, A.

A. Grace, Optimization Toolbox for Use with MATLAB (Math Works, Inc., Natick, Mass., 1992).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewoods Cliffs, N. J., 1993).

Keller, C. U.

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

Kittel, C.

C. Kittel, H. Kroemer, Thermal Physics (Freeman, New York, 1980).

Kroemer, H.

C. Kittel, H. Kroemer, Thermal Physics (Freeman, New York, 1980).

Langendijk, R. L.

R. L. Langendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Lee, D. J.

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Using wavefront sensor information in image post-processing to improve the resolution of telescope with small aberrations,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 42–53 (1996).

Levine, B. M.

L. Needels, B. M. Levine, M. Milman, “Limits on adaptive optics systems for lightweight space telescopes,” in Space Astronomical Telescopes and Instruments II, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE1945, 176–184 (1993).

Lofdahl, M. G.

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

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

Marron, J. C.

Mehta, P.

C. Chi, P. Mehta, A. Ostroff, “Vibrational modes of the primary mirror structure in the large space telescope system,” in Space Optics: Proceedings of the Ninth International Conference of the International Commission for OpticsB. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences, Washington, D.C., 1974), Vol. 9, pp. 209–238.

Miller, M. I.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer, New York, 1991).
[CrossRef]

Milman, M.

L. Needels, B. M. Levine, M. Milman, “Limits on adaptive optics systems for lightweight space telescopes,” in Space Astronomical Telescopes and Instruments II, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE1945, 176–184 (1993).

Needels, L.

L. Needels, B. M. Levine, M. Milman, “Limits on adaptive optics systems for lightweight space telescopes,” in Space Astronomical Telescopes and Instruments II, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE1945, 176–184 (1993).

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Ostroff, A.

C. Chi, P. Mehta, A. Ostroff, “Vibrational modes of the primary mirror structure in the large space telescope system,” in Space Optics: Proceedings of the Ninth International Conference of the International Commission for OpticsB. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences, Washington, D.C., 1974), Vol. 9, pp. 209–238.

Paxman, R. G.

R. G. Paxman, J. H. Seldin, M. G. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase diversity 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]

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[CrossRef]

R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescope using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. Haiè, eds., Proc. SPIE1351, 787–797 (1990).
[CrossRef]

Press, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, UK, 1992).

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Roggemann, M. C.

B. M. Welsh, M. C. Roggemann, “Signal-to-noise comparison of deconvolution from wave-front sensing with traditional linear and speckle image reconstruction,” Appl. Opt. 34, 2111–2119 (1995).
[CrossRef] [PubMed]

M. C. Roggemann, D. W. Tyler, M. F. Bilmont, “Linear reconstruction of compensated images: theory and experimental results,” Appl. Opt. 31, 7429–7441 (1992).
[CrossRef] [PubMed]

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Using wavefront sensor information in image post-processing to improve the resolution of telescope with small aberrations,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 42–53 (1996).

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Scharmer, G. B.

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

M. G. Lofdahl, 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. Lofdahl, G. B. Scharmer, C. U. Keller, “Evaluation of phase diversity techniques for solar-image restoration,” Astrophys. J. 466, 1087–1099 (1996).
[CrossRef]

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]

Snyder, D. L.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer, New York, 1991).
[CrossRef]

Tekalp, A. M.

R. L. Langendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, UK, 1992).

Tyler, D. W.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, 1991).

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, UK, 1992).

Watson, A.

A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
[CrossRef]

Welsh, B. M.

B. M. Welsh, M. C. Roggemann, “Signal-to-noise comparison of deconvolution from wave-front sensing with traditional linear and speckle image reconstruction,” Appl. Opt. 34, 2111–2119 (1995).
[CrossRef] [PubMed]

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Using wavefront sensor information in image post-processing to improve the resolution of telescope with small aberrations,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 42–53 (1996).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1994).

Appl. Opt.

Astron. Astrophys. Suppl. Ser.

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

Astrophys. J.

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

J. Opt. Soc. Am. A

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[CrossRef]

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[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]

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Nature

“What will be the next Big Thing,” Nature 381, 465 (1996).
[CrossRef]

Opt. Eng.

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

R. L. Langendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Science

A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1998).

D. Dooling, “Beyond Hubble,” The Institute (IEEE monthly newsletter)1 (June1996).

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, UK, 1992).

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewoods Cliffs, N. J., 1993).

I. J. D. Craig, J. C. Brown, Inverse Problems in Astronomy: a Guide to Inversion Strategies for Remotely Sensed Data (Hilger, Bristol, UK, 1986).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer, New York, 1991).
[CrossRef]

C. Chi, P. Mehta, A. Ostroff, “Vibrational modes of the primary mirror structure in the large space telescope system,” in Space Optics: Proceedings of the Ninth International Conference of the International Commission for OpticsB. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences, Washington, D.C., 1974), Vol. 9, pp. 209–238.

L. Needels, B. M. Levine, M. Milman, “Limits on adaptive optics systems for lightweight space telescopes,” in Space Astronomical Telescopes and Instruments II, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE1945, 176–184 (1993).

A. Grace, Optimization Toolbox for Use with MATLAB (Math Works, Inc., Natick, Mass., 1992).

C. Kittel, H. Kroemer, Thermal Physics (Freeman, New York, 1980).

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescope using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. Haiè, eds., Proc. SPIE1351, 787–797 (1990).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1994).

R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. J. Tescher, ed., Proc. SPIE207, 32–39 (1979).

R. A. Golsalves, “Fundamentals of wavefront sensing by phase retrieval,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 56–65 (1982).
[CrossRef]

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Using wavefront sensor information in image post-processing to improve the resolution of telescope with small aberrations,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 42–53 (1996).

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, 1991).

D. L. Fried, “Post-detection wavefront distortion compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. SPIE828, 127–133 (1987).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Possible configuration of the proposed NGST.

Fig. 2
Fig. 2

Simplified approximation of the pupil of the proposed eight-petalled NGST: (a) gray-scale layout of the pupil components with the piston error of the petals indexed sequentially so they can be seen and the central obscuration artificially highlighted to show its location, (b) randomly misaligned NGST pupil phase map is shown in mesh surface format, where the z-axis values are given in radians of phase delay and pupils are sampled on a 64 × 64 discrete grid.

Fig. 3
Fig. 3

Simplified block diagram of one possible optical implementation of the phase-diversity WFS technique.

Fig. 4
Fig. 4

Demonstration of the effectiveness of the noise-rejection filter for the Gonsalves objective function. The plot shows a coarsely sampled one-dimensional slice of the objective function both with and without a noise-rejection filter. The original image suffered from approximately a quarter of a wave rms Zernike aberration spread through modes 4–36. The actual value of Zernike mode 7, which varies along the x axis of this plot, was 0.10 rad. The average number of photons was reduced to 200 per image for demonstration purposes. Without noise rejection, the objective function reaches a minimum at an incorrect point.

Fig. 5
Fig. 5

Computer simulation simplified block diagram.

Fig. 6
Fig. 6

Plots of the diagonals of the aberration coefficient covariance matrices Γα for the various experimental cases discussed in the text; these plots show the mean-squared values of the various aberration coefficients as generated for the Monte Carlo simulation realizations: (a) Zernike subcase, (b) NGST subcase.

Fig. 7
Fig. 7

Pupil-averaged rms phase-diversity WFS estimation errors for the 50 simulated realizations of Zernike case 1 and the 100 realizations of Zernike case 2 (sorted by error value): (a) case 1 simulations exhibited and estimated Zernike modes 4 through 11; (b) case 2 simulations exhibited Zernike modes 4 through 36 and estimated modes 4 through 22. The original average pupil aberrations were λ/10 (ensemble average of rms).

Fig. 8
Fig. 8

Pupil-averaged rms phase-diversity WFS estimation errors for the 50 simulated realizations of NGST case 1 and the 100 realizations of NGST case 2 (sorted by error value): (a) case 1 simulations exhibited and estimated eight NGST petal piston errors; (b) case 2 simulations expanded upon case 1 by exhibiting and estimating segment-tilt errors, for a total of 26 misalignment parameters. The original average pupil aberrations were λ/10 (ensemble average of rms).

Fig. 9
Fig. 9

Averages of the curves in Fig. 7, giving ensemble averages of the pupil-averaged rms estimation errors for cases 1 and 2 of the Zernike aberration simulations: (a) case 1 simulations exhibited and estimated Zernike modes 4 through 11; (b) case 2 simulations exhibited Zernike modes 4 through 36 and estimated modes 4 through 22. The original average pupil aberrations were λ/10 (ensemble average of rms).

Fig. 10
Fig. 10

Averages of the curves in Fig. 8, giving ensemble averages of the pupil-averaged rms estimation errors for cases 1 and 2 of the NGST aberration simulations; these are the NGST mirror segment zonal aberration cases [as noted in the text, the 20 divergent realizations for case 2(b) were excluded from averaging]: (a) case 1 simulations exhibited and estimated eight NGST petal piston errors; (b) case 2 simulations expanded on case 1 by exhibiting and estimating segment-tilt errors for a total of 26 misalignment parameters. The original average pupil aberrations were λ/10 (ensemble average of rms).

Fig. 11
Fig. 11

Case 1(a), monolithic/Zernike, low DOF, = 105; average Zernike aberration-mode strength and WFS rms error bars. Data are derived from averages of the pupil-averaged rms estimation errors for N different phase-diversity estimation-simulated realizations.

Fig. 12
Fig. 12

Case 2(a), monolithic/Zernike, high DOF, = 105; average Zernike aberration-mode strength and WFS rms error bars. Data are derived from averages of the pupil-averaged rms estimation errors for N different phase-diversity estimation-simulated realizations.

Equations (21)

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ix=ox*hx,
If=OfHf,
Wx=1for x  pupil0otherwise,
GPFx=Wxexpjϕx,
Hf=ACFGPFfλdiACFGPFfλdi|f=0,
ϕRr, θ=i=4NαiZir, θ,
α=α4, α5, , αNT,
α18=8 petal piston errors,α916=8 petalx-tilt errors,α1724=8 petal y-tilt errors,α25,26=2 central mirror tilts, x and y.
I1f=OfH1f; α,I2f=OfH2f; α+Δ.
D1f=OfH1f; α+N1f,D2f=OfH2f; α+Δ+N2f,
Jα=fD1-OH1α2+D2-OH2α+Δ2,
Jα=fD1H˜2α+Δ-D2H˜1α2H˜1α2+H˜2α+Δ2.
αWFS=argminαJα.
Jregα=fχ1FD1H˜2α+Δ-D2H˜1α2H˜1α2+H˜2α+Δ2.
SNRDf=DfDf2-Df21/2,
Γαi,j=αiαj
α=Rαb,
WFS,iin waves=12π dxWxϕix-ϕix2 dxWx1/2,
WFS=1Ni=1NWFS,i.
αnrmsin waves=12π1Ni=1Nαn,i21/2.
WFS,nrmsin waves=1Ni=1Nα˜n,i-αn,i21/2.

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