Abstract

A scheme for recovering phase using irradiance data alone, without interferometric techniques, is developed using the transport equations for phase and irradiance. For the case of one transverse dimension a general solution, for an arbitrary irradiance distribution, of the transport equation for the optical phase is already given by an application of the divergence theorem. Numerical simulation results are given that indicate that the phase-recovery scheme works well even in the presence of large pupil-plane aberrations if the signal-to-noise ratio is sufficiently high. In particular, pupil-plane phase aberrations may be determined from irradiance measurements in two planes that are near the image plane.

© 1985 Optical Society of America

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References

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  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  2. R. H. T. Bates, W. R. Fright, “Reconstructing images from their Fourier intensities,” in Signal and Image Reconstruction from Incomplete Data. Theory and Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.
  3. H. P. Baltes, ed., Inverse Source Problems in Optics (Springer-Verlag, New York, 1978);Inverse Scattering Problems in Optics (Springer-Verlag,New York, 1980).
    [CrossRef]
  4. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
    [CrossRef]
  5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  6. B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
    [CrossRef]
  7. A. M. Huiser, H. A. Ferweda, “The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images,” Opt. Acta 23, 445–456 (1976).
    [CrossRef]
  8. J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
    [CrossRef]
  9. J. M. Wood, M. A. Fiddy, R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. 6, 514–516 (1981).
    [CrossRef] [PubMed]
  10. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  11. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6 (1984).
    [CrossRef]
  12. M. R. Teague, “Deterministic phase retrievel: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  13. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  15. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  16. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.
  17. M. R. Teague, “Image formation in terms of the transport equation,” Rep. UCRL-91408 (Lawrence Livermore National Laboratory, Livermore, Calif., October1984).
  18. K.-J. Hanssen, G. Ade, Übertragungs- und transporttheoretische Aussagen über Phasenkontrastbilding bei partiell kohaerenter Beleuchtung,” PTB-Berich APh-22 (Physikalisch-Technische Bundesanstalt, Braunschweig, 1984).
  19. G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
    [CrossRef]

1985 (1)

G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
[CrossRef]

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6 (1984).
[CrossRef]

1983 (1)

1982 (2)

1981 (2)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

J. M. Wood, M. A. Fiddy, R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. 6, 514–516 (1981).
[CrossRef] [PubMed]

1978 (1)

1976 (2)

R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
[CrossRef]

A. M. Huiser, H. A. Ferweda, “The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images,” Opt. Acta 23, 445–456 (1976).
[CrossRef]

1975 (1)

B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

Ade, G.

G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
[CrossRef]

K.-J. Hanssen, G. Ade, Übertragungs- und transporttheoretische Aussagen über Phasenkontrastbilding bei partiell kohaerenter Beleuchtung,” PTB-Berich APh-22 (Physikalisch-Technische Bundesanstalt, Braunschweig, 1984).

Bates, R. H. T.

R. H. T. Bates, W. R. Fright, “Reconstructing images from their Fourier intensities,” in Signal and Image Reconstruction from Incomplete Data. Theory and Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

Burge, R. E.

Ferweda, H. A.

A. M. Huiser, H. A. Ferweda, “The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images,” Opt. Acta 23, 445–456 (1976).
[CrossRef]

Fiddy, M. A.

Fienup, J. R.

Fright, W. R.

R. H. T. Bates, W. R. Fright, “Reconstructing images from their Fourier intensities,” in Signal and Image Reconstruction from Incomplete Data. Theory and Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.

Gonsalves, R. A.

Goodman, J. W.

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

Hanssen, K.-J.

K.-J. Hanssen, G. Ade, Übertragungs- und transporttheoretische Aussagen über Phasenkontrastbilding bei partiell kohaerenter Beleuchtung,” PTB-Berich APh-22 (Physikalisch-Technische Bundesanstalt, Braunschweig, 1984).

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Hoenders, B. J.

B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

Huiser, A. M.

A. M. Huiser, H. A. Ferweda, “The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images,” Opt. Acta 23, 445–456 (1976).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6 (1984).
[CrossRef]

Teague, M. R.

Walker, J. G.

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

Wood, J. M.

Appl. Opt. (1)

J. Math. Phys. (1)

B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Acta (2)

A. M. Huiser, H. A. Ferweda, “The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images,” Opt. Acta 23, 445–456 (1976).
[CrossRef]

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Opt. Commun. (2)

G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6 (1984).
[CrossRef]

Opt. Lett. (2)

Other (8)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Sec. 8.8.4.

M. R. Teague, “Image formation in terms of the transport equation,” Rep. UCRL-91408 (Lawrence Livermore National Laboratory, Livermore, Calif., October1984).

K.-J. Hanssen, G. Ade, Übertragungs- und transporttheoretische Aussagen über Phasenkontrastbilding bei partiell kohaerenter Beleuchtung,” PTB-Berich APh-22 (Physikalisch-Technische Bundesanstalt, Braunschweig, 1984).

R. H. T. Bates, W. R. Fright, “Reconstructing images from their Fourier intensities,” in Signal and Image Reconstruction from Incomplete Data. Theory and Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.

H. P. Baltes, ed., Inverse Source Problems in Optics (Springer-Verlag, New York, 1978);Inverse Scattering Problems in Optics (Springer-Verlag,New York, 1980).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Image-plane point-spread function in the absence of aberrations.

Fig. 2
Fig. 2

(log10I/Io)−1 in the image plane for the aberration-free case. Io = 1.0 μW/m2.

Fig. 3
Fig. 3

Image-plane phase for no pupil-plane aberrations. The π shifts in phase are solely to account for the ± sign of the wave amplitude.

Fig. 4
Fig. 4

Point-spread function (psf) at a defocus plane; appearance of psf a distance 10 times the depth of focus behind the image plane. There are no pupil-plane aberrations.

Fig. 5
Fig. 5

(log10I/Io)−1 at the phase-recovery plane. Same plane as Fig. 4.

Fig. 6
Fig. 6

Optical phase at the phase-recovery plane. Discontinuities are artificial and produced by requiring phase to be in interval (−π, +π] for this plot.

Fig. 7
Fig. 7

Longitudinal gradient of irradiance at the phase-recovery plane, δz used in obtaining ∂I/∂z was 1/40 × (depth of focus).

Fig. 8
Fig. 8

Retrieved phase at the recovery plane. All cross-hatched curves in this paper are obtained from phase recovery based on transport equation; non-cross-hatched curves in same figure are predictions of Fresnel diffraction theory.

Fig. 9
Fig. 9

Recovered pupil-plane phase in the aberration-free case. The phase aberration is 0.0053 wave and reflects the accuracy of the numerical simulation.

Fig. 10
Fig. 10

Recovered pupil-plane irradiance. Riplets are due to inverse Fresnel transforming a finite, rather than infinite, region of the image plane.

Fig. 11
Fig. 11

Recovered pupil-plane phase in the aberration-free case. There is no noise, and the noise-optimized parameters discussed in Section 4 are used. The rms wave-front error is 0.0395 wave.

Fig. 12
Fig. 12

Noise effects in the aberration-free case. Photon-limited noise is present in the detector plane and the signal-to-noise ratio at the central pixel is 10:1. The rms plane error is 0.0838 wave.

Fig. 13
Fig. 13

Severe aberrations but no detector noise. (a) The recovered pupil-plane phase. rms phase error is 0.193 wave. (b) The psf implied by the difference of curves in (a).

Fig. 14
Fig. 14

Severe aberrations in the presence of noise. Severe aberrations in the presence of noise. (a) Recovered phase, rms phase error is 0.277 wave. (b) psf implied by difference of waves in (a). Signal-to-noise ratio at central pixel was 1000:1 in Figs. 1416, and Figs. 1416 differ only because different statistical noise realizations were used—all with the same rms photon-number fluctuations—in these last three figures.

Fig. 15
Fig. 15

Severe aberrations in the presence of noise. (a) Recovered phase, rms phase error is 0.547 wave. (b) psf implied by difference of curves in (a).

Fig. 16
Fig. 16

Severe aberrations in the presence of noise. (a) Recovered phase, rms phase error is 1.29 waves. (b) Implied by difference of curves in (a).

Equations (11)

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u z ( r ) = exp [ i ϕ z ( r ) ] [ I z ( r ) ] 1 / 2 ,
u z ( r ) = e ikz u o ( r ) * * [ exp ( i π r 2 / λ z ) ] i λ z ,
( i z + 2 2 k + k ) u z ( r ) = 0 ,
I ϕ = k z I ,
1 2 I 2 I 1 4 ( I ) 2 I 2 ( ϕ ) 2 + 2 k 2 I 2 = k 2 I 2 z ϕ .
2 ϕ + [ ln ( I I o ) ] ϕ = k z ln ( I I o ) ,
x I ϕ x = k z I ,
I z ( x ) ϕ z x ( x ) = I z ( x o ) ϕ z ( x o ) x o k x o x d x z I z ( x ) ,
ϕ ( x ) = ϕ ( x o ) + x o x d x I ( x ) [ I ( x o ) ϕ ( x o ) x o k x o x d x z I z ( x ) ] ,
c d s F = k R d x d y I / z ,
ϕ o ( x ) = 2 π n = 0 8 a n P n ( x ) ,

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