Abstract

A method for reconstructing an unknown wave front from measurements of its intensity distribution on two planes along the direction of propagation is described. The method solves the intensity transport equation by use of Neumann boundary conditions, leading to a solution that requires only matrix multiplication. The method provides real-time wave-front reconstruction with high accuracy and is easily reposed to permit reconstruction of the wave front in any orthonormal basis set.

© 2003 Optical Society of America

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References

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  1. N. G. von Kampen, “S-matrix and causality condition. I. Maxwell field,” Phys. Rev. 89, 1072–1079 (1953).
    [CrossRef]
  2. D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2216 (1973).
    [CrossRef]
  3. 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).
  4. A. H. Greenaway, “Proposal for phase recovery from a single intensity distribution,” Opt. Lett. 1, 10–12 (1977).
    [CrossRef] [PubMed]
  5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  6. U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (Method CLEAN),” Astron. Astrophys. 65, 345–356 (1978).
  7. A. Lannes, “Backprojection mechanisms in phase-closure imaging,” Exp. Astron. 1, 47–76 (1989).
    [CrossRef]
  8. H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).
  9. G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 115–138.
  10. F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).
  11. D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
    [CrossRef]
  12. R. A. Muller, A. Buffington, “Real-time correction of atmospherically-degraded telescope images through image sharpness,” J. Opt. Soc. Am. 64, 1200–1210 (1974).
    [CrossRef]
  13. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  14. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  15. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  16. T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
    [CrossRef]
  17. 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. 12, 1932–1941 (1995).
    [CrossRef]
  18. E. Acosta, S. Rios, M. Soto, V. V. Voitsekhovich, “Role of boundary measurements in curvature sensing,” Opt. Commun. 169, 59–62 (1999).
    [CrossRef]
  19. P. M. Blanchard, S. Woods, D. Fisher, A. H. Greenaway, “Phase-diversity wave-front sensing with a distorted diffraction grating,” Appl. Opt. 39, 6649–6655 (2000).
    [CrossRef]
  20. P. M. Blanchard, A. H. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. 38, 6692–6699 (1999).
    [CrossRef]
  21. S. Djidel, A. H. Greenaway, “Nanometric wave-front metrology,” in Adaptive Optics for Industry and Medicine, S. R. Restaino, S. W. Teare, eds. (Starline, Albuquerque, N. Mex., 2002), pp. 213–219.

2000 (1)

1999 (2)

E. Acosta, S. Rios, M. Soto, V. V. Voitsekhovich, “Role of boundary measurements in curvature sensing,” Opt. Commun. 169, 59–62 (1999).
[CrossRef]

P. M. Blanchard, A. H. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. 38, 6692–6699 (1999).
[CrossRef]

1997 (1)

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

1995 (1)

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. 12, 1932–1941 (1995).
[CrossRef]

1994 (1)

1992 (1)

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

1991 (1)

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

1989 (1)

A. Lannes, “Backprojection mechanisms in phase-closure imaging,” Exp. Astron. 1, 47–76 (1989).
[CrossRef]

1988 (1)

1983 (1)

1982 (1)

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

1978 (2)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (Method CLEAN),” Astron. Astrophys. 65, 345–356 (1978).

1977 (1)

1974 (1)

1973 (1)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2216 (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).

1953 (1)

N. G. von Kampen, “S-matrix and causality condition. I. Maxwell field,” Phys. Rev. 89, 1072–1079 (1953).
[CrossRef]

Acosta, E.

E. Acosta, S. Rios, M. Soto, V. V. Voitsekhovich, “Role of boundary measurements in curvature sensing,” Opt. Commun. 169, 59–62 (1999).
[CrossRef]

Angel, J. R. P.

Barclay, H. T.

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

Blanchard, P. M.

Buffington, A.

Djidel, S.

S. Djidel, A. H. Greenaway, “Nanometric wave-front metrology,” in Adaptive Optics for Industry and Medicine, S. R. Restaino, S. W. Teare, eds. (Starline, Albuquerque, N. Mex., 2002), pp. 213–219.

Fienup, J. R.

Fisher, D.

Fontanella, J. C.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Gaffard, J. P.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

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]

Greenaway, A. H.

Gureyev, T. E.

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

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. 12, 1932–1941 (1995).
[CrossRef]

Kern, P.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Lannes, A.

A. Lannes, “Backprojection mechanisms in phase-closure imaging,” Exp. Astron. 1, 47–76 (1989).
[CrossRef]

Léna, P.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Lloyd-Hart, M.

Malyak, P.

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

McCarthy, D.

McGonagle, W.

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

Merkle, F.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

Muller, R. A.

Nugent, K. A.

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

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. 12, 1932–1941 (1995).
[CrossRef]

Reich, R.

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

Rigaut, F.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Rios, S.

E. Acosta, S. Rios, M. Soto, V. V. Voitsekhovich, “Role of boundary measurements in curvature sensing,” Opt. Commun. 169, 59–62 (1999).
[CrossRef]

Roberts, A.

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. 12, 1932–1941 (1995).
[CrossRef]

Roddier, F.

Rousset, G.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 115–138.

Rowe, G.

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

Sandler, D. G.

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).

Schwarz, U. J.

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (Method CLEAN),” Astron. Astrophys. 65, 345–356 (1978).

Soto, M.

E. Acosta, S. Rios, M. Soto, V. V. Voitsekhovich, “Role of boundary measurements in curvature sensing,” Opt. Commun. 169, 59–62 (1999).
[CrossRef]

Stahl, S.

Teague, M. R.

Twichell, J.

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

Voitsekhovich, V. V.

E. Acosta, S. Rios, M. Soto, V. V. Voitsekhovich, “Role of boundary measurements in curvature sensing,” Opt. Commun. 169, 59–62 (1999).
[CrossRef]

von Kampen, N. G.

N. G. von Kampen, “S-matrix and causality condition. I. Maxwell field,” Phys. Rev. 89, 1072–1079 (1953).
[CrossRef]

Woods, S.

Appl. Opt. (3)

Astron. Astrophys. (2)

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (Method CLEAN),” Astron. Astrophys. 65, 345–356 (1978).

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Léna, “Adaptive optics on a 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Exp. Astron. (1)

A. Lannes, “Backprojection mechanisms in phase-closure imaging,” Exp. Astron. 1, 47–76 (1989).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Phys. D (1)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

Lincoln Lab. J. (1)

H. T. Barclay, P. Malyak, W. McGonagle, R. Reich, G. Rowe, J. Twichell, “The SWAT wavefront sensor,” Lincoln Lab. J. 5, 115–130 (1992).

Opt. Commun. (2)

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

E. Acosta, S. Rios, M. Soto, V. V. Voitsekhovich, “Role of boundary measurements in curvature sensing,” Opt. Commun. 169, 59–62 (1999).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (2)

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).

Phys. Rev. (1)

N. G. von Kampen, “S-matrix and causality condition. I. Maxwell field,” Phys. Rev. 89, 1072–1079 (1953).
[CrossRef]

Other (2)

G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 115–138.

S. Djidel, A. H. Greenaway, “Nanometric wave-front metrology,” in Adaptive Optics for Industry and Medicine, S. R. Restaino, S. W. Teare, eds. (Starline, Albuquerque, N. Mex., 2002), pp. 213–219.

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Figures (2)

Fig. 1
Fig. 1

Principle of phase diversity–wave-front-curvature sensing.

Fig. 2
Fig. 2

Phase maps of a Kolmogorov turbulence wave front measured with (a) an interferometer and (b) our phase-diversity algorithm.

Equations (30)

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iz+22k+kuz(r)=0,
uz(r)=[Iz(r)]1/2 exp[iϕz(r)].
-kzIz(r)=·[Iz(r)ϕz(r)].
-kzIz(r)=Iz(r)2ϕz(r)+Iz(r)·ϕz(r),
-kIIz=2ϕ,
I=I0WA,I=-I0δCnˆ,
-kzIz(r)=I0WA2ϕz(r)-I0δCnˆ·ϕz(r).
Rf(r)2g(r)d2r=Rg(r)2f(r)d2r+Pf(r)g(r)·dnˆ-Pg(r)f(r)·dnˆ,
2G(r, r)=δ(r-r),
f(r)=RG(r, r)2f(r)d2r+Pf(r)G(r, r)·dnˆ-PG(r, r)f(r)·dnˆ.
nˆ·f(r)=h(r)forrP,
G(r, r)·nˆ=0forrP.
R2G(r, r)d2r=PG(r, r)·dnˆ.
2G(r, r)=δ(r-r)-A-1,
ϕ(r)=RG(r, r)2ϕ(r)d2r-PG(r, r)ϕ(r)·dnˆ.
S(r)=-kI0zIz(r).
S(r)G(r, r)d2r=(WA2ϕ-δCnˆ·ϕ)G(r, r)d2r.
S(r)G(r, r)d2r=AG(r, r)2ϕ(r)d2r-CG(r, r)ϕ(r)·dnˆ.
ϕ(r)=S(r)G(r, r)d2r
ai=ϕ(r)ui(r)d2r.
ai=S(r)G(r, r)ui(r)d2rd2r.
Gi(r)=G(r, r)ui(r)d2r,
ai=S(r)Gi(r)d2r.
ui(r)=δ(r-Ri),
ui(r)=δ(r-Ri)p(r),
I(r, z)zI(r, z+δz/2)-I(r, z-δz/2)δz.
S(r)jSj[δ(r-ρi)q(r)],
ai=jSj[δ(r-ρj)q(r)]Gi(r)d2r.
Gi,j=[δ(r-ρj)q(r)]Gi(r)d2r,
ai=jSjGi,j,

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