Abstract

Phase-retrieval algorithms have been used for wave-front sensing to determine the aberrations of an optical system from system point-spread functions (blurred images of point sources). Previously, computationally efficient algorithms were developed and applied to data from the Hubble Space Telescope [Appl. Opt. 32, 1737 (1993); Appl. Opt. 32, 1747 (1993)], but those algorithms, which employ analytic expressions for the gradient of an error metric, required narrow-band light and adequately sampled images. Generalizations of those phase-retrieval algorithms, which accommodate broadband light, allow for undersampled images, permit fitting of multiple images simultaneously, and have a flexible description of the aberrations, are described in this study.

© 1999 Optical Society of America

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References

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  1. Space Optics for Astrophysics and Earth and Planetary Remote Sensing, 1991, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).
  2. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  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, 1747–1768 (1993).
    [CrossRef] [PubMed]
  4. C. Roddier, F. Roddier, “Combined approach to Hubble Space Telescope wave-front distortion analysis,” Appl. Opt. 32, 2992–3008 (1993).
    [CrossRef] [PubMed]
  5. J. E. Krist, C. J. Burrows, “Phase-retrieval analysis of pre- and post-repair Hubble Space Telescope images,” Appl. Opt. 34, 4951–4964 (1995).
    [CrossRef] [PubMed]
  6. R. G. Lyon, J. E. Dorband, J. M. Hollis, “Hubble Space Telescope faint object camera calculated point-spread functions,” Appl. Opt. 36, 1752–1765 (1997).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  8. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  9. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
    [CrossRef]
  10. J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
    [CrossRef]
  11. J. R. Fienup, “Phase retrieval for multiple undersampled polychromatic images,” in Signal Recovery and Synthesis, Vol. 11 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 41–43.
  12. J. R. Fienup, “White-light phase retrieval,” presented at the Annual Meeting of the Optical Society of America, Baltimore, Md., October 4–9, 1998.

1997 (1)

1995 (1)

1993 (3)

1988 (1)

1982 (1)

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

Burrows, C. J.

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 Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Dorband, J. E.

Fienup, J. R.

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1768 (1993).
[CrossRef] [PubMed]

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

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

J. R. Fienup, “Phase retrieval for multiple undersampled polychromatic images,” in Signal Recovery and Synthesis, Vol. 11 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 41–43.

J. R. Fienup, “White-light phase retrieval,” presented at the Annual Meeting of the Optical Society of America, Baltimore, Md., October 4–9, 1998.

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 (McGraw-Hill, San Francisco, Calif., 1968).

Hollis, J. M.

Krist, J. E.

Lyon, R. G.

Marron, J. C.

Paxman, R. G.

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

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

Roddier, C.

Roddier, F.

Schulz, T. J.

Seldin, J. H.

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 Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Appl. Opt. (5)

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

Opt. Eng. (1)

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

Other (5)

Space Optics for Astrophysics and Earth and Planetary Remote Sensing, 1991, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

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

J. R. Fienup, “Phase retrieval for multiple undersampled polychromatic images,” in Signal Recovery and Synthesis, Vol. 11 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 41–43.

J. R. Fienup, “White-light phase retrieval,” presented at the Annual Meeting of the Optical Society of America, Baltimore, Md., October 4–9, 1998.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

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Figures (4)

Fig. 1
Fig. 1

Simulated PSF’s. (a) Polychromatic PSF with +1.5 waves (rms) defocus, (b) polychromatic PSF with -0.75 waves defocus, (c) monochromatic PSF with +1.5 waves defocus, (d) monochromatic PSF with -0.75 waves defocus.

Fig. 2
Fig. 2

Normalized error as a function of iteration number. Upper curve, calculated with a monochromatic PSF calculation; lower curve, calculated with a polychromatic PSF calculation.

Fig. 3
Fig. 3

PSF’s with photon noise. (a), (b) 1,000,000 total photons; (c), (d) 1000 total photons.

Fig. 4
Fig. 4

Residual rms phase error versus light level.

Tables (2)

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Table 1 Simulated Wavelengths and Spectral Response for WF/PC F555W Filter

Tables Icon

Table 2 Values of Zernike Coefficients a

Equations (37)

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Glk(p, q)=λlλomnAl(m, n)×expi λoλl ϕokλoλl m, λoλl n×exp-i2πmpM+nqN,
Glk(p, q)=λoλlmnglk(m, n)×exp-i2πmpMl+nqNl,
glk(m, n)=Ao(m, n)expi λoλl ϕok(m, n)
Ml=ΔuΔxλlzf=Mo λoλl,
ϕok(m, n)=jd=2J1ajd,kZjd(m, n)+js=J1+1JajsZjs(m, n)+ϕopp(m, n),
Ik(p, q)=grid(p, q)l=1LSl|Glk( p, q)|2*D(p, q),
E2=K-1k=1KΦk-1pqWk(p, q)grid(p, q)×αkl=1LSl|Glk(p, q)|2 * D(p, q)1/2-|F|k(p, q)2,
E2β=K-1 k=1Kαk2Φk-1pqWk( p, q)grid(p, q)×1-|F|k(p, q)αkl=1LSl|Glk(p, q)|2*D(p, q)1/2×l=1LSl|Glk(p, q)|2β*D(p, q).
|Glk(p, q)|2β=λoλlGlk*(p, q)mn glk(m, n)β×exp-i2πmpMl+nqNl+c.c.,
|Glk(p, q)|2β*D(p, q)
=λoλlmn glk(m, n)β pqGlk*(p, q)
×exp-i2πmpMl+nqNl×D(p-p, q-q)+c.c.,
E2β=-K-1l=1LSlλoλlk=1Kαk2Φk-1×mn glk(m, n)β (glkD)*(m, n)+c.c.,
(glkD)*(m, n)
=pq exp-i2πmpMl+nqNlGlk*(p, q)
×pqD( p-p, q-q)Wk(p, q)grid(p, q)
×|F|k(p, q)αkl=1LSl|Glk(p, q)|2*D(p, q)1/2-1.
glk(m, n)ajd,k1=iλoλlglk(m, n)Zjd(m, n)δ(k, k1),
glk(m, n)ajs=iλoλlglk(m, n)Zjs(m, n),
glk(m, n)ϕpp(m1, n1)=iλoλlglk(m1, n1)δ(m, m1)δ(n, n1),
glk(m, n) Re[gl1,k1(m1, n1)]
=δ(m, m1)δ(n, n1)δ(l, l1)δ(k, k1),
glk(m, n) Im[gl1,k1(m1, n1)]
=iδ(m, m1)δ(n, n1)δ(l, l1)δ(k, k1),
glk(m, n)Ao(m1, n1)
=expi λoλl ϕok(m1, n1)δ(m, m1)δ(n, n1),
δ(m, m1)=1,m=m10,mm1.
E2ajd,k1=2K-1αk12Φk1-1mnZjd(m, n)l=1LSlλoλl2×Im[glk1(m, n)(glk1D)*(m, n)],
E2ajs=2K-1mnZjs(m, n)k=1Kαk2Φk-1l=1LSlλoλl2×Im[glk(m, n)(glkD)*(m, n)],
E2ϕopp(m1, n1)=2K-1k=1Kαk2Φk-1l=1LSlλoλl2×Im[glk(m1, n1)(glkD)*(m1, n1)],
E2gl1,k1(m1, n1)
E2 Re[gl1,k1(m1, n1)]+i E2 Im[gl1,k1(m1, n1)]=-2K-1k=1Kαk2Φk-1l=1LSlλoλlglkD(m1, n1),
E2Ao(m1, n1)
=-2K-1k=1Kαk2Φk-1l=1LSlλoλl
×Reexpi λoλl ϕok(m1, n1)(glkD)*(m1, n1).
E2αk1=2K-1Φk1-1pqWk1( p, q)grid( p, q)×αk1l=1LSl|Glk( p, q)|2*D( p, q)1/2-|F|k1( p, q)×l=1LSl|Glk(p, q)|2*D( p, q)1/2.
αk1opt=pqWk1( p, q)grid( p, q)|F|k1( p, q)l=1LSl|Glk( p, q)|2*D( p, q)1/2pqWk1( p, q)grid( p, q)l=1LSl[|Glk1( p, q)|2*D( p, q)].

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