Abstract

Analytical expressions to describe the phase gradient of monochromatic light by means of the three-dimensional intensity distribution are derived. With these formulas it is shown that the two-dimensional phase gradient in a plane can be completely determined from noninterferometric intensity measurements if the light propagates strictly in one direction. The analytical expressions are verified by means of numerical investigations on simulated speckle fields, and the results are discussed with respect to common deterministic phase retrieval approaches.

© 2005 Optical Society of America

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References

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  1. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]
  2. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  3. N. Shvartsman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1995).
    [CrossRef]
  4. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  5. G. Vdovin, “Reconstruction of an object shape from the near-field intensity of a reflected paraxial beam,” Appl. Opt. 36, 5508–5513 (1997).
    [CrossRef] [PubMed]
  6. W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Recursive algorithm for phase retrieval in the fractional Fourier transform domain,” Appl. Opt. 37, 6906–6910 (1998).
    [CrossRef]
  7. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  8. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  9. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1983).
    [CrossRef]
  10. K. Ichikawa, A. W. Lohmann, M. Takeda, “Phase retrieval based on irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [CrossRef] [PubMed]
  11. K. G. Larkin, C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
    [CrossRef]
  12. M. J. Bastiaans, K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003).
    [CrossRef]
  13. G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
    [CrossRef]
  14. M. R. Teague, “Image formation in terms of the transport equation,” J. Opt. Soc. Am. A 2, 2019–2026 (1985).
    [CrossRef]
  15. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
    [CrossRef] [PubMed]
  16. 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. A 12, 1932–1941 (1995).
    [CrossRef]
  17. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  18. M. Fernández-Guasti, J. L. Jiménez, F. Granados-Augustín, A. Cornejo-Rodríguez, “Amplitude and phase representation of monochromatic fields in physical optics,” J. Opt. Soc. Am. A 20, 1629–1634 (2003).
    [CrossRef]
  19. G. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–407 (1977).
  20. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 402.
  21. B. Saleh, M. Teich, Fundamentals of Photonics, 1st ed. (Wiley-Interscience, New York, 1991), p. 52.

2003 (2)

1999 (1)

1998 (1)

1997 (1)

1995 (3)

1993 (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1990 (1)

1988 (1)

1985 (2)

M. R. Teague, “Image formation in terms of the transport equation,” J. Opt. Soc. Am. A 2, 2019–2026 (1985).
[CrossRef]

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

1983 (3)

1982 (2)

1977 (1)

G. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–407 (1977).

Ade, G.

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

Baranova, N. B.

Bastiaans, M. J.

Chen, N.-X.

Cong, W.-X.

Cornejo-Rodríguez, A.

Fernández-Guasti, M.

Fienup, J. R.

Freund, I.

N. Shvartsman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1995).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 402.

Granados-Augustín, F.

Gu, B.-Y.

Gureyev, T. E.

Ichikawa, K.

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Jiménez, J. L.

Larkin, K. G.

Lohmann, A. W.

Mamaev, A. V.

Nugent, K. A.

Pilipetsky, N. F.

Roberts, A.

Roddier, F.

Saleh, B.

B. Saleh, M. Teich, Fundamentals of Photonics, 1st ed. (Wiley-Interscience, New York, 1991), p. 52.

Sheppard, C. J. R.

Shkunov, V. V.

Shvartsman, N.

N. Shvartsman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1995).
[CrossRef]

Stoffregen, B.

G. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–407 (1977).

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1983).
[CrossRef]

Takeda, M.

Teague, M. R.

Teich, M.

B. Saleh, M. Teich, Fundamentals of Photonics, 1st ed. (Wiley-Interscience, New York, 1991), p. 52.

Vdovin, G.

Weigelt, G.

G. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–407 (1977).

Wolf, K. B.

Zel’dovich, B. Y.

Appl. Opt. (5)

J. Mod. Opt. (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Opt. Commun. (3)

N. Shvartsman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1995).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1983).
[CrossRef]

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

Optik (Stuttgart) (1)

G. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–407 (1977).

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 402.

B. Saleh, M. Teich, Fundamentals of Photonics, 1st ed. (Wiley-Interscience, New York, 1991), p. 52.

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Figures (4)

Fig. 1
Fig. 1

Phase (left) and intensity (right) of a simulated objective speckle field. The speckle field was simulated with a wavelength of 632.8 nm for an object with a circular illuminated area of 4 mm in diameter in a distance of 300 mm. The depicted area is 0.04 mm 2 .

Fig. 2
Fig. 2

Squared absolute value of the phase gradient T ϕ 2 of the speckle field in Fig. 1. The left image is evaluated directly from the phase map, while the right image is calculated according to Eq. (7) solely from the intensity distribution. Both images are scaled to the same maximum values. Bright values represent high phase gradients, and dark values low ones.

Fig. 3
Fig. 3

Direction β of the phase gradient of the speckle field in Fig. 1. The left image is evaluated directly from the phase image, and the right one shows the directional field of β obtained by numerical solution of Eq. (19) with initially known fields a and d. The values of β are displayed from zero (black values) to 2 π (brightest values).

Fig. 4
Fig. 4

Vector relations between isophase curves.

Equations (49)

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U ( r ) = A ( r ) exp [ i ϕ ( r ) ] ,
2 U ( r ) + k 2 U ( r ) = 0 .
2 A ( r ) + A ( r ) k 2 A ( r ) ϕ ( r ) 2
i [ 2 A ( r ) ϕ ( r ) + A ( r ) 2 ϕ ( r ) ] = 0 .
ϕ 2 = k 2 + 2 A A .
( ϕ x ) 2 + ( ϕ y ) 2 + ( ϕ z ) 2 = k 2 + 2 I I .
ϕ = ϕ 0 ( x , y ) + k z ,
T ϕ 2 ( ϕ x ) 2 + ( ϕ y ) 2 = 1 I ( 2 I x 2 + 2 I y 2 + 2 I z 2 )
2 I x 2 , 2 I y 2 2 I z 2 .
T ϕ 2 1 I ( 2 I x 2 + 2 I y 2 ) .
( I ϕ ) = 0 .
x ( I ϕ x ) + y ( I ϕ y ) = z ( I ϕ z ) .
ϕ x = T ϕ cos β , ϕ y = T ϕ sin β .
a = z ( I ϕ z ) , d = I T ϕ ,
x ( d cos β ) + y ( d sin β ) = a .
β x = ln d y ln I y cos 2 β + ln I x sin β cos β a d sin β ,
β y = ln d x + ln I x sin 2 β ln I y sin β cos β + a d cos β .
a = I z k , d = I ( 2 I I ) 1 2 .
y ( β x ) x ( β y ) = 0 .
c 0 = c 1 cos β + c 2 sin β + c 3 cos β sin β + c 4 cos 2 β + c 5 sin 2 β ,
c 0 = T 2 ln d + T ln I T ln d + ( a d ) 2 ,
c 1 = ( a d ) ln ( I d ) x ( a d ) x ,
c 2 = ( a d ) ln ( I d ) y ( a d ) y ,
c 3 = 2 ( 1 I 2 I x y ln I y ln d x ln I x ln d y ) ,
c 4 = 2 ln I y ln d y 1 I 2 I y 2 ,
c 5 = 2 ln I x ln d x 1 I 2 I x 2 .
cos β = ± ( 1 sin 2 β ) 1 2 or sin β = ± ( 1 cos 2 β ) 1 2 .
u μ = ( cos β sin β ) .
u ν = ( sin β cos β )
ln d x cos β + ln d y sin β + ( β y cos β β x sin β ) = a d .
β ν = a d ln d μ .
Δ β ν = ( β ν ) 1 Δ ν ,
Δ μ 1 = Δ ϕ T ϕ 1 .
Δ μ 2 = Δ ϕ T ϕ 2 .
Δ β μ = ( β μ ) 1 Δ μ 1 .
v p 3 p 4 = ( v μ v ν ) ;
Δ β μ = arctan ( v μ v ν ) .
p 3 = ( Δ μ 2 cos ( Δ β ν ) Δ ν + Δ μ 2 sin ( Δ β ν ) ) , p 4 = ( Δ μ 1 0 ) .
( β μ ) 1 Δ ϕ T ϕ 1 = arctan { Δ ϕ T ϕ 1 1 T ϕ 2 1 cos [ ( β ν ) 1 Δ ν ] Δ ν + Δ ϕ T ϕ 2 1 sin [ ( β ν ) 1 Δ ν ] } .
T ϕ 2 = T ϕ 1 + T ϕ 1 ν Δ ν .
β μ Δ ϕ T ϕ = arctan [ Δ ϕ 1 + ln T ϕ ν Δ ν cos ( β ν Δ ν ) T ϕ Δ ν + Δ T ϕ ν Δ ν 2 + Δ ϕ sin ( β ν Δ ν ) ] .
lim Δ ν 0 cos ( β ν Δ ν ) = 1 ,
lim Δ ν 0 sin ( β ν Δ ν ) = lim Δ ν 0 β ν Δ ν .
β μ T ϕ 1 = lim Δ ν 0 Δ ϕ 0 ln T ϕ ν T ϕ + T ϕ ν Δ ν + Δ ϕ β ν .
β μ = ln T ϕ ν .
β x cos β + β y sin β = ln T ϕ x sin β + ln T ϕ y cos β ,
β x sin β + β y cos β = a d ln d x cos β + ln d y sin β .
β x = ln d y ln I y cos 2 β + ln I x sin β cos β a d sin β ,
β y = ln d x + ln I x sin 2 β ln I y sin β cos β + a d cos β .

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