Abstract

Phase-diverse wave-front sensing (PDWFS) is a methodology for estimating aberration coefficients from multiple incoherent images whose pupil phases differ from one another in a known manner. With the use of previous work by other authors, the Cramér–Rao lower-bound (CRLB) expression for the phase diversity aberration estimation problem is developed and is generalized slightly to allow for multiple phase-diverse images, various beam-splitting configurations, and imaging of known extended objects. The CRLB for a given problem depends implicitly on the true underlying value of the aberration being estimated. Therefore we use numerical evaluation and Monte Carlo analysis of the PDWFS CRLB expressions. The numerical evaluation is performed on an ensemble of aberration phase screens while simulating a number of different imaging configurations. We demonstrate the use of average CRLB values as figures of merit in comparing these various PDWFS configurations. For simulated point-source imaging we quantify the effects of varying the amounts and the types of diversity phase and briefly address the issue of the number of diversity images. Our results show that there is a diversity defocus configuration that is optimal in a Cramér–Rao sense for estimating certain aberrations. We also show that PDWFS Cramér–Rao squared-error values can be orders of magnitude higher for imaging of an extended target object than those from a point-source target.

© 1999 Optical Society of America

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References

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  1. R. A. Gonsalves, R. Chidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. SPIE207, 32–39 (1979).
    [CrossRef]
  2. 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]
  3. 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]
  4. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  5. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  6. M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  9. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  10. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
    [CrossRef]
  11. 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).
  12. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer, New York, 1991).
  13. D. L. Fried, “Least-square fitting a wave front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  14. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  15. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,” Appl. Opt. 34, 4186–4195 (1995).
    [CrossRef] [PubMed]
  16. R. A. Gonsalves, “Fundamentals of wavefront sensing by phase retrieval,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 56–65 (1982).
    [CrossRef]
  17. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]

1995 (1)

1994 (1)

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 (1)

1992 (1)

1988 (1)

1983 (1)

1982 (1)

1977 (1)

1976 (1)

Chidlaw, R.

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

Ellerbroek, B. L.

Fienup, J. R.

Fried, D. L.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gonsalves, R. A.

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

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

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Kay, S. M.

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

Lofdahl, M. G.

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.

Miller, M. I.

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

Noll, R. J.

Paxman, R. G.

Pennington, T. L.

Roggemann, M. C.

Scharmer, G. B.

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.

Snyder, D. L.

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

Wallner, E. P.

Welsh, B. M.

Appl. Opt. (3)

Astron. Astrophys., Suppl. Ser. (1)

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).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Other (8)

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

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Simplified schematic diagram of one possible phase diversity setup.

Fig. 2
Fig. 2

Graphical explanation of the procedure used to obtain the Cramér–Rao figure-of-merit metric values of Subsection 4.C. This general procedure is justified and explained in the text.

Fig. 3
Fig. 3

Ensemble average of pupil-averaged theoretical minimum mean-square phase diversity estimation error for 50 quarter-wave rms aberrated pupils. Data points represent the ensemble averages of the traces of the inverses of the 50 Fisher information matrices. The Fisher matrices are calculated with a 50/50 beam splitter, and the total photocount is normalized to K¯=1. Shown are results for the traditional phase diversity setup, with one image in focus and one image out of focus by the amount indicated on the x axis. Point-source imaging is modeled.

Fig. 4
Fig. 4

Schematic comparison of the standard and symmetrical diversity image collection schemes discussed in the text.

Fig. 5
Fig. 5

Ensemble average of pupil-averaged theoretical minimum mean-square estimation error for 50 quarter-wave rms aberrated pupils with the use of symmetrical defocus phase diversity. Data points represent the ensemble averages of the traces of the inverses of the 50 Fisher information matrices. The Fisher matrices are calculated with a 50/50 beam splitter, and the total photocount is normalized to K¯=1. Shown are results for a symmetrical phase diversity setup, with both images out of focus, each by the amount indicated on the x axis. Point source imaging is modeled.

Fig. 6
Fig. 6

Computer-aided design rendering of an orbiting satellite. The data shown are used to model the pristine object irradiance distribution for imaging of a space object.

Tables (1)

Tables Icon

Table 1 Summary of Results of the Various Phase Diversity CRLB Experiments, with Photocounts Normalized to Unity

Equations (47)

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

A(u)=1forupupil0otherwise.
xh(x)dx=1.
xW2(u)du=1.
g(u; α)=W(u)exp[jϕ(u; α)].
ϕ(u; α)=iαiZi(u).
G(x; α)=FT[g(u; α)],
h(x; α)=|G(x; α)|2=G(x; α)G*(x; α),
λ(x; α)=o(x) * h(x; α),
xλ(x; α)dx=K¯.
xonorm(x)dx=1.
λ(x; α)=K¯onorm(x) * h(x; α)
=K¯onorm(x) * [G(x; α)G*(x; α)].
λ(x; α)αk=K¯onorm(x) * h(x; α)αk,
h(x; α)αk=G*(x; α)αkG(x; α)+G(x; α)αkG*(x; α)=2 ReG(x; α)αkG*(x; α).
G(x; α)αk=αk{FT[g(u; α)]}=αkFTW(u)expji=1JαiZi(u)=αkuW(u)expji=1JαiZi(u)×exp(j2πu·x)du.
G(x; α)αk=ug(u; α)jZk(u)exp(j2πu·x)du=j FT[g(u; α)Zk(u)].
h(x; α)αk=2 ReG(x; α)αkG*(x; α)=2 Re{j FT[g(u; α)Zk(u)]G*(x; α)}=-2 Im{FT[g(u; α)Zk(u)]G*(x; α)},
λ(x; α)αk=-2K¯onorm(x) * Im{FT[g(u; α)Zk(u)]G*(x; α)}.
Var(α˜k)[F-1(α)]kk.
F(α)=-E[HL(α)],
[HL(α)]jk=2L(d; α)αjαk,
{λn(x; α+Δn)}={λ1(x; α+Δ1),λ2(x; α+Δ2),, λN(x; α+ΔN)},
n=1NK¯n=K¯.
αpd=argminαJ(α|{dn(x)).
PDF[dn(x)]=[λn(x; α+Δn)]dn(x) exp[-λn(x; α+Δn)]dn(x)!.
E[dn(x)]=λn(x; α+Δn).
PDF[{dn(x)}]=n=1NxPDF[dn(x)].
L(d; α)ln{PDF[dn(x)]}=lnn=1NxPDF[dn(x)]=n=1Nx ln{PDF[dn(x)]}
=n=1Nx{-λn(x; α+Δn)+dn(x)ln[λn(x; α+Δn)]-ln[dn(x)!]}.
L(d; α)=n=1Nx{-λn(x; α+Δn)+dn(x)ln[λn(x; α+Δn)]}.
L(d; α)αk=n=1Nx λn(x; α+Δn)αk×dn(x)λn(x; α+Δn)-1.
EL(d; α)α=0,
Edn(x)λn(x; α+Δn)=1λn(x; α+Δn)E[dn(x)]=λn(x; α+Δn)λn(x; α+Δn)
[HL(α)]jk=2L(d; α)αjαk
=n=1Nx2λn(x; α+Δn)αjαk×dn(x)λn(x; α+Δn)-1-λn(x; α+Δn)αjλn(x; α+Δn)αk×dn(x)[λn(x; α+Δn)]2.
[F(α)]jk=-E{[HL(α)]jk}=n=1Nx λn(x; α+Δn)αjλn(x; α+Δn)αk×1λn(x; α+Δn).
[F(α)]jk=n=1Nx 4(Kn¯)2λn(x; α+Δn)×onorm(x) * (Im{Gn*(x; α+Δn)×FT[gn(u; α+Δn)Zj(u)]})×onorm(x) * (Im{Gn*(x; α+Δn)×FT[gn(u; α+Δn)Zk(u)]})
=n,xKn¯ 4onorm(x) * hn(x; α+Δn)×a=j,konorm(x) * (Im{Gn*(x; α+Δn)×FT[gn(u; α+Δn)Za(u)]}).
Datapoint=i=1Ni2N=2.
xh(x; α)dx=1.
xh(x; α)dx=1,
x|G(x; α)|2 dx=1,
u|g(u; α)|2 du=1,
ug(u; α)g*(u; α)du=1,
uW(u)exp[jϕ(u; α)]W(u)exp[-jϕ(u; α)]du=1,
uW2(u)=1,
W(u)=1/aforupupil0otherwise.

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