Abstract

The use of the term differential intensity measurements implies that two or more intensity measurements are made, with a slight change in a parameter of the measurement-taking system between measurements. For example, the position of the focal plane or the transmission of the aperture might be changed. In either of these cases it is possible to find a closed-form solution to the phase-retrieval problem, as opposed to the usual iterative solution. The first case corresponds to a quadratic phase shift in the unknown phase and has been studied previously by several authors. This examination of the problem leads to the discovery of some details that may be useful. In the second case a new result leads to a simpler solution for the phase. The results in this paper apply to the one-dimensional phase-retrieval problem. An extension to the more important two-dimensional problem can be made by means described in this paper.

© 1987 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. C. L. Mehta, “New approach to the phase problem of optical coherence theory,”J. Opt. Soc. Am. 58, 1233–1234 (1968).
    [CrossRef]
  3. J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
    [CrossRef]
  4. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,”J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  5. M. R. Teague, “Image formation in terms of the transport equation,” presented at the Workshop on Unconventional Imagery, Rigi-Kaltbad-First, Switzerland, September 1984.
  6. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 29, 6–10 (1984).
    [CrossRef]
  7. P. Kiedron, “Conditions sufficient for a one-dimensional unique recovery of the phase under assumption that the image intensity distributions |f(x)|2and |df(x)/dx|2are known,” Opt. Appl. 10, 149–154 (1980).
  8. P. Kiedron, “Phase recovery from intensity distributions generated by differential operators in one-dimensional coherent imaging,” Opt. Appl. 10, 253–265 (1980).
  9. P. Kiedron, “On a possibility of the phase recovery from intensity distributions generated by differential operators in two-dimensional coherent imaging,” Opt. Appl. 10, 483–486 (1080).
  10. P. Kiedron, “Phase retrieval method using differential filters,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 189–196 (1983).
    [CrossRef]
  11. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  12. R. A. Gonsalves, “Phase retrieval,” in Digital Image Processing, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrun. Eng.528, 202–215 (1985).
    [CrossRef]

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 29, 6–10 (1984).
[CrossRef]

1982 (3)

1981 (1)

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

1980 (2)

P. Kiedron, “Conditions sufficient for a one-dimensional unique recovery of the phase under assumption that the image intensity distributions |f(x)|2and |df(x)/dx|2are known,” Opt. Appl. 10, 149–154 (1980).

P. Kiedron, “Phase recovery from intensity distributions generated by differential operators in one-dimensional coherent imaging,” Opt. Appl. 10, 253–265 (1980).

1968 (1)

Fienup, J. R.

Gonsalves, R. A.

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

R. A. Gonsalves, “Phase retrieval,” in Digital Image Processing, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrun. Eng.528, 202–215 (1985).
[CrossRef]

Kiedron, P.

P. Kiedron, “Conditions sufficient for a one-dimensional unique recovery of the phase under assumption that the image intensity distributions |f(x)|2and |df(x)/dx|2are known,” Opt. Appl. 10, 149–154 (1980).

P. Kiedron, “Phase recovery from intensity distributions generated by differential operators in one-dimensional coherent imaging,” Opt. Appl. 10, 253–265 (1980).

P. Kiedron, “On a possibility of the phase recovery from intensity distributions generated by differential operators in two-dimensional coherent imaging,” Opt. Appl. 10, 483–486 (1080).

P. Kiedron, “Phase retrieval method using differential filters,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 189–196 (1983).
[CrossRef]

Mehta, C. L.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 29, 6–10 (1984).
[CrossRef]

Teague, M. R.

M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,”J. Opt. Soc. Am. 72, 1199–1209 (1982).
[CrossRef]

M. R. Teague, “Image formation in terms of the transport equation,” presented at the Workshop on Unconventional Imagery, Rigi-Kaltbad-First, Switzerland, September 1984.

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]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Acta (1)

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

Opt. Appl. (3)

P. Kiedron, “Conditions sufficient for a one-dimensional unique recovery of the phase under assumption that the image intensity distributions |f(x)|2and |df(x)/dx|2are known,” Opt. Appl. 10, 149–154 (1980).

P. Kiedron, “Phase recovery from intensity distributions generated by differential operators in one-dimensional coherent imaging,” Opt. Appl. 10, 253–265 (1980).

P. Kiedron, “On a possibility of the phase recovery from intensity distributions generated by differential operators in two-dimensional coherent imaging,” Opt. Appl. 10, 483–486 (1080).

Opt. Commun. (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 29, 6–10 (1984).
[CrossRef]

Opt. Eng. (1)

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

Other (3)

R. A. Gonsalves, “Phase retrieval,” in Digital Image Processing, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrun. Eng.528, 202–215 (1985).
[CrossRef]

P. Kiedron, “Phase retrieval method using differential filters,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 189–196 (1983).
[CrossRef]

M. R. Teague, “Image formation in terms of the transport equation,” presented at the Workshop on Unconventional Imagery, Rigi-Kaltbad-First, Switzerland, September 1984.

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Figures (5)

Fig. 1
Fig. 1

Generation of the wave front w(f).

Fig. 2
Fig. 2

Basis functions for the wave front w(f).

Fig. 3
Fig. 3

Unweighted aperture A(f) and a weighted aperture A(f, B).

Fig. 4
Fig. 4

Example of the differential aperture weighting algorithm.

Fig. 5
Fig. 5

Example of the differential quadratic phase algorithm.

Equations (26)

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S ( u ) = K u - ( 11 / 3 ) ,
w ( f ) = a ( k ) g ( f , k ) ,
g ( f , k ) = 2 π cos ( k π f / D ) ,
H ( f ) = A ( f ) e i w ( f ) ,
h ( x ) = H ( f ) exp ( i 2 π f x ) d f ,
p ( x ) = h ( x ) 2 .
A ( f , B ) = A ( f ) exp ( 2 π B f ) ,
h ( x , B ) = - A ( f ) exp [ 2 π B f + i w ( f ) + i 2 π f x ] d f ,
p ( x , B ) = h ( x , B ) 2 = h ( x , B ) h * ( x , B ) .
d p ( x , B ) d B | B = 0 = p ( x , B ) - p ( x ) B             for B small .
d p ( x , B ) d B | B = 0 = 2 Re [ h * ( x ) d h ( x , B ) d B | B = 0 ] .
d h ( x , B ) d B | B = 0 = - A ( f ) 2 π f exp ( i w ( f ) + i 2 π f x ) d f = - i d h ( x ) d x ,
d p ( x , B ) d B | B = 0 = 2 Re [ - i h * ( x ) d h ( x ) d x ] .
h ( x ) = r ( x ) e i t ( x ) .
d h ( x ) d x = h ( x ) = r ( x ) e i t ( x ) + i t ( x ) r ( x ) e i t ( x ) ,
d p ( x , B ) d B | B = 0 = 2 Re { - i r ( x ) e - i t ( x ) [ r ( x ) + i r ( x ) t ( x ) ] e i t ( x ) } = 2 r 2 ( x ) t ( x ) .
r 2 ( x ) = p ( x ) ,
d p ( x , B ) d B | B = 0 = 2 p ( x ) t ( x ) .
t ( x ) = 1 2 p ( x ) p ( x , B ) - p ( x ) B ,             B small .
t ( x ) = - x 1 2 p ( y ) p ( y , B ) - p ( y ) B d y .
h ( x ) = r ( x ) e i t ( x ) ,
h ( x , B ) = - A ( f ) exp [ - i 2 π B f 2 + i w ( f ) + i 2 π f x ] d f .
t ( x ) = - x 1 π p ( y ) - y p ( z ) - p ( z , B ) B d z d y .
x p t x + y p t y = 1 2 Π p ( x , y ) - p ( x , y , B ) B ,
d 1 ( x , y ) = t ( x , y ) x ,             d 2 ( x , y ) = t ( x , y ) y .
T ( f , l ) = ( i 2 Π f ) D 1 ( f , l ) + ( i 2 Π l ) D 2 ( f , l ) ( 2 Π f ) 2 + ( 2 Π l ) 2 .

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