Abstract

Experimental demonstrations of deterministic phase retrieval based on the Teague-Streibl irradiance transport equation are presented. A new technique is proposed, in which the transport equation is solved by the Fourier transform method for a periodic boundary condition with high spatial carrier frequency, which is created by making a light beam with unknown phase distribution pass through a grating. Quantitative phase measurements were performed by experiments without recourse to interferometry, and the results were found to be in good agreement with theory.

© 1988 Optical Society of America

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References

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  1. For a review of various phase retrieval schemes see, for example, H. A. Ferwerda, “The Phase Reconstruction Problems for Wave Amplitudes and Coherence Functions,” in Inverse Source Problems in Optics, H. P. Baltes, Ed. (Springer-Verlag, Heidelberg, 1978), p. 13; J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  2. M. R. Teague, “Irradiance Moments: Their Propagation and Use for Unique Retrieval of Phase,” J. Opt. Soc. Am. 72, 1199 (1982).
    [CrossRef]
  3. M. R. Teague, “Deterministic Phase Retrieval: a Green’s Function Solution,” J. Opt. Soc. Am. 73, 1434 (1983).
    [CrossRef]
  4. N. Streibl, “Phase Imaging by the Transport Equation of Intensity,” Opt. Commun. 49, 6 (1984).
    [CrossRef]
  5. N. Streibl, “Phase Imaging Based on the Transport Equation of Intensity,” in ICO-13 Conference Digest, 352 (1984).
  6. M. Takeda, H. Ina, S. Kobayashi, “Fourier-Transform Method of Fringe-Pattern Analysis for Computer-Based Topography and Interferometry,” J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  7. M. Takeda, S. Kobayashi, “Lateral Aberration Measurements with a Digital Talbot Interferometer,” Appl. Opt. 23, 1760 (1984).
    [CrossRef] [PubMed]
  8. M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl. Opt. 22, 3977 (1983).
    [CrossRef] [PubMed]

1984 (3)

N. Streibl, “Phase Imaging by the Transport Equation of Intensity,” Opt. Commun. 49, 6 (1984).
[CrossRef]

N. Streibl, “Phase Imaging Based on the Transport Equation of Intensity,” in ICO-13 Conference Digest, 352 (1984).

M. Takeda, S. Kobayashi, “Lateral Aberration Measurements with a Digital Talbot Interferometer,” Appl. Opt. 23, 1760 (1984).
[CrossRef] [PubMed]

1983 (2)

1982 (2)

Ferwerda, H. A.

For a review of various phase retrieval schemes see, for example, H. A. Ferwerda, “The Phase Reconstruction Problems for Wave Amplitudes and Coherence Functions,” in Inverse Source Problems in Optics, H. P. Baltes, Ed. (Springer-Verlag, Heidelberg, 1978), p. 13; J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758 (1982).
[CrossRef] [PubMed]

Ina, H.

Kobayashi, S.

Mutoh, K.

Streibl, N.

N. Streibl, “Phase Imaging by the Transport Equation of Intensity,” Opt. Commun. 49, 6 (1984).
[CrossRef]

N. Streibl, “Phase Imaging Based on the Transport Equation of Intensity,” in ICO-13 Conference Digest, 352 (1984).

Takeda, M.

Teague, M. R.

Appl. Opt. (2)

ICO-13 Conference Digest (1)

N. Streibl, “Phase Imaging Based on the Transport Equation of Intensity,” in ICO-13 Conference Digest, 352 (1984).

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

N. Streibl, “Phase Imaging by the Transport Equation of Intensity,” Opt. Commun. 49, 6 (1984).
[CrossRef]

Other (1)

For a review of various phase retrieval schemes see, for example, H. A. Ferwerda, “The Phase Reconstruction Problems for Wave Amplitudes and Coherence Functions,” in Inverse Source Problems in Optics, H. P. Baltes, Ed. (Springer-Verlag, Heidelberg, 1978), p. 13; J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758 (1982).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Spatial frequency spectra of the irradiance transport function −(∂I/∂z)z=0.

Fig. 2
Fig. 2

Schematic diagram of the experimental setup: L1, condenser lens; P, pinhole; L2, field lens as a phase object; G, grating.

Fig. 3
Fig. 3

Irradiance profile of I(x,0;0).

Fig. 4
Fig. 4

Irradiance transport function (∂I/∂z)z=0 for y = 0.

Fig. 5
Fig. 5

Spatial frequency spectra of the irradiance profile [I(x,0;0)].

Fig. 6
Fig. 6

Spatial frequency spectra of the irradiance transport function [−(∂I/∂z)z=0] for y = 0.

Fig. 7
Fig. 7

Measured first derivative of the spherical phase distribution (∂W/∂x)y=0,z=0.

Fig. 8
Fig. 8

Retrieved spherical phase distribution W(x,0;0) (solid line), and theoretically predicted phase distribution (broken line with solid circles).

Fig. 9
Fig. 9

Retrieved phase distribution for a bifocal wavefront W(x,0;0).

Fig. 10
Fig. 10

Measured second derivative of the spherical phase distribution and that predicted from theory.

Equations (11)

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T I · T W + I T 2 W + I / z = 0 ,
I ( x , y ; 0 ) = n = - C n exp ( 2 π i n f 0 x ) ,
n = - q n ( x , y ) exp ( 2 π i n f 0 x ) = - ( I / z ) z = 0 ,
q n ( x , y ) = [ 2 π i n f 0 W ( x , y ; 0 ) / x + T 2 W ( x , y ; 0 ) ] C n .
F [ - ( I / z ) z = 0 ] = n = - Q n ( f x - n f 0 , f y ) ,
Q n ( f x , f y ) = F [ q n ( x , y ) ] ,
F - 1 { T 1 ( f x , f y ) · F [ - ( I / z ) z = 0 ] } = q 1 ( x , y ) exp ( 2 π i f 0 x ) = [ 2 π i f 0 W ( x , y ; 0 ) / x + T 2 W ( x , y ; 0 ) ] C 1 exp ( 2 π i f 0 x ) .
F - 1 { T 1 ( f x , f y ) · F [ I ( x , y ; 0 ) ] } = C 1 exp ( 2 π i f 0 x ) .
W ( x , y ; 0 ) / x = ( 2 π f 0 ) - 1 Im [ p ( x , y ) ] ,
T 2 W ( x , y ; 0 ) = Re [ p ( x , y ) ] ,
p ( x , y ) F - 1 { T 1 ( f x , f y ) · F [ - ( I / z ) z = 0 ] } / F - 1 { T 1 ( f x , f y ) · F [ I ( x , y ; 0 ) ] } ,

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