Abstract

Phase retrieval from one or more intensity measurements is a potentially powerful and appealing technique for real-time adaptive-optics wave-front sensors. Under the assumption of small wave-front phase excursions, one is able to derive an exact solution to the inverse problem given three or more intensity measurements with known phase offsets. Applications include a high-order wave-front sensor to correct for residual aberrations in an adaptive-optics system in tandem with a low-resolution Hartmann–Shack wave-front sensor. The formula can also furnish mathematical insights into the full nonlinear phase-retrieval task.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. Gonsalves, Opt. Eng. 21, 829 (1982).
    [CrossRef]
  2. T. I. Kuznetsova, Sov. Phys. Usp. 31, 364 (1988).
    [CrossRef]
  3. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
    [CrossRef]
  4. A. Graham, Kronecker Products and Matrix Calculus:?With Applications (Wiley, New York, 1981).
  5. P. J. Davis, Circulant Matrices (Wiley, New York, 1979).
  6. E. Kibblewhite, Department of Astronomy and Astrophysics, University of Chicago, Chicago, Illinois 60637 (personal communication, December, 1997).
  7. K. Creath, Prog. Opt. 26, 349 (1988).
    [CrossRef]
  8. W. Wild, E. Kibblewhite, and R. Vuilleumier, Opt. Lett. 20, 955 (1995).
    [CrossRef]
  9. J. R. P. Angel, Nature (London) 368, 203 (1994).
    [CrossRef]

1995 (1)

1994 (1)

J. R. P. Angel, Nature (London) 368, 203 (1994).
[CrossRef]

1988 (2)

T. I. Kuznetsova, Sov. Phys. Usp. 31, 364 (1988).
[CrossRef]

K. Creath, Prog. Opt. 26, 349 (1988).
[CrossRef]

1982 (1)

R. A. Gonsalves, Opt. Eng. 21, 829 (1982).
[CrossRef]

Angel, J. R. P.

J. R. P. Angel, Nature (London) 368, 203 (1994).
[CrossRef]

Creath, K.

K. Creath, Prog. Opt. 26, 349 (1988).
[CrossRef]

Davis, P. J.

P. J. Davis, Circulant Matrices (Wiley, New York, 1979).

Gonsalves, R. A.

R. A. Gonsalves, Opt. Eng. 21, 829 (1982).
[CrossRef]

Graham, A.

A. Graham, Kronecker Products and Matrix Calculus:?With Applications (Wiley, New York, 1981).

Horn, R. A.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

Kibblewhite, E.

W. Wild, E. Kibblewhite, and R. Vuilleumier, Opt. Lett. 20, 955 (1995).
[CrossRef]

E. Kibblewhite, Department of Astronomy and Astrophysics, University of Chicago, Chicago, Illinois 60637 (personal communication, December, 1997).

Kuznetsova, T. I.

T. I. Kuznetsova, Sov. Phys. Usp. 31, 364 (1988).
[CrossRef]

Vuilleumier, R.

Wild, W.

Nature (London) (1)

J. R. P. Angel, Nature (London) 368, 203 (1994).
[CrossRef]

Opt. Eng. (1)

R. A. Gonsalves, Opt. Eng. 21, 829 (1982).
[CrossRef]

Opt. Lett. (1)

Prog. Opt. (1)

K. Creath, Prog. Opt. 26, 349 (1988).
[CrossRef]

Sov. Phys. Usp. (1)

T. I. Kuznetsova, Sov. Phys. Usp. 31, 364 (1988).
[CrossRef]

Other (4)

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

A. Graham, Kronecker Products and Matrix Calculus:?With Applications (Wiley, New York, 1981).

P. J. Davis, Circulant Matrices (Wiley, New York, 1979).

E. Kibblewhite, Department of Astronomy and Astrophysics, University of Chicago, Chicago, Illinois 60637 (personal communication, December, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (15)

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

i=FpF*p¯,
F=F1F1, =FR1FR1-FI1FI1+iFI1FR1+FR1FI1, FR+iFI,
F1=1n[11111tt2tn-11t2t4t2n-11tn-1t2n-1tn-1n-1]FR1+iFI1,
tm=exp2πmi/n=cos2πm/n+i sin2πm/n,
FR=FRT,  FI=FIT, FRFI=FIFR,  FR2+FI2=I
i=FRw-FIwφFRw-FIwφ+FIw+FRwφFIw+FRwφ.
1/2ik-i-ck=FRwφkFRwφ+FIwφkFIwφ,
ckFRwφkFRwφk+FIwφkFIwφk+2FRwφkFIw-2FIwφkFRw.
wφ=FI+FR-11Dkk1/2ik-i-ckFI-FRwφk-1/2ik-i-ckFI-FRwφk.
DkkFIwφkFRwφk-FIwφkFRwφk.
wφ=FI+FR-112kkDkkk=kik-i-ckFI-FRwφk-ik-i-ckFI-FRwφk,
aRb=a1TRb
2wφφk=α1TFR+β1TFI-1d4,
d41/24i2k-i-2k-c2k+c-2k-2ik-i-k-ck+c-k,
αFRwφk2,  βFIwφk2,

Metrics