Abstract

Recent papers have shown that there are different coherent and partially coherent fields that may have identical intensity distributions throughout space. On the other hand, the well-known transport-of-intensity equation allows the phase of a coherent field to be recovered from intensity measurements, and the solution is widely held to be unique. A discussion is given on the recovery of the structure of both coherent and partially coherent fields from intensity measurements, and we reconcile the uniqueness question by showing that the transport-of-intensity equation has a unique solution for the phase only if the intensity distribution has no zeros.

© 1995 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. M. G. Rayner, M. Beck, D. F. MacAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef]
  9. V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
    [CrossRef]
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    [CrossRef]
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  12. K. Dutta, J. W. Goodman, “Reconstructions of images of partially coherent objects from samples of mutual intensity,” J. Opt. Soc. Am. 67, 796–803 (1977).
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  13. K. A. Nugent, “Coherence induced spectral changes and generalised radiance,” Opt. Commun. 91, 13–17 (1992).
    [CrossRef]
  14. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  15. D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
    [CrossRef]
  16. P. Coullett, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
    [CrossRef]
  17. G. A. Swartzlander, C. T. Law, “Optical vortex solitons in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
    [CrossRef] [PubMed]
  18. J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London A336, 165–190 (1974).
  19. H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
    [CrossRef]
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    [CrossRef]

1995 (1)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

1994 (1)

M. G. Rayner, M. Beck, D. F. MacAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef]

1993 (2)

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

G. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

1992 (5)

S. R. Restaino, “Wavefront sensing and image deconvolution of solar data,” Appl. Opt. 31, 7442–7449 (1992).
[CrossRef] [PubMed]

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

K. A. Nugent, “Coherence induced spectral changes and generalised radiance,” Opt. Commun. 91, 13–17 (1992).
[CrossRef]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992); see also the comment by G. Hazak, Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef] [PubMed]

G. A. Swartzlander, C. T. Law, “Optical vortex solitons in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

P. Coullett, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

1988 (1)

1986 (1)

1984 (1)

N. Striebl, “Phase imaging by the transport of intensity equation,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (2)

1982 (1)

1977 (1)

1974 (1)

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London A336, 165–190 (1974).

Bagini, V.

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Baranova, N. B.

Bastiaans, M. J.

Beck, M.

M. G. Rayner, M. Beck, D. F. MacAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef]

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London A336, 165–190 (1974).

Coullett, P.

P. Coullett, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Dutta, K.

Fried, D. L.

Gil, L.

P. Coullett, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Gilbarg, D.

D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
[CrossRef]

Goodman, J. W.

Gori, F.

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Gori, G.

Guattari, G.

G. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

He, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Ichikawa, K.

Law, C. T.

G. A. Swartzlander, C. T. Law, “Optical vortex solitons in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

Lohmann, A. W.

MacAlister, D. F.

M. G. Rayner, M. Beck, D. F. MacAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef]

Mamaev, A. V.

Nugent, K. A.

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992); see also the comment by G. Hazak, Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef] [PubMed]

K. A. Nugent, “Coherence induced spectral changes and generalised radiance,” Opt. Commun. 91, 13–17 (1992).
[CrossRef]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London A336, 165–190 (1974).

Pilipetsky, N. F.

Rayner, M. G.

M. G. Rayner, M. Beck, D. F. MacAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef]

Restaino, S. R.

Rocca, F.

P. Coullett, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Roddier, F.

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Santarsiero, M.

G. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Schirripa Spagnolo, G.

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

Shkunov, V. V.

Striebl, N.

N. Striebl, “Phase imaging by the transport of intensity equation,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Swartzlander, G. A.

G. A. Swartzlander, C. T. Law, “Optical vortex solitons in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

Takeda, M.

Teague, M. R.

Trudinger, N. S.

D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
[CrossRef]

Vaughn, J. L.

Zel’dovich, B. Ya.

Appl. Opt. (4)

J. Mod. Opt. (1)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Commun. (4)

P. Coullett, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

N. Striebl, “Phase imaging by the transport of intensity equation,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

V. Bagini, F. Gori, M. Santarsiero, G. Guattari, G. Schirripa Spagnolo, “Space intensity distribution and projections of the cross spectral density,” Opt. Commun. 102, 495–504 (1993).
[CrossRef]

K. A. Nugent, “Coherence induced spectral changes and generalised radiance,” Opt. Commun. 91, 13–17 (1992).
[CrossRef]

Phys. Rev. Lett. (3)

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992); see also the comment by G. Hazak, Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef] [PubMed]

M. G. Rayner, M. Beck, D. F. MacAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef]

G. A. Swartzlander, C. T. Law, “Optical vortex solitons in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

Proc. R. Soc. London (1)

J. F. Nye, M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London A336, 165–190 (1974).

Other (1)

D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Propagation geometry that suggests a simple ray interpretation of the generalized radiance. As noted in the text, this interpretation is subject to some important limitations.

Fig. 2
Fig. 2

Phase function 2πϕ = plotted as a function of x and y for m = 2.

Fig. 3
Fig. 3

Identical intensity distributions that can be produced by two fields if the difference in their direction vectors forms closed loops in a plane perpendicular to the optic axis.

Equations (38)

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r 1 / 2 ( r 1 + r 2 ) ,             x ( r 1 - r 2 ) ,
B ( r , u , z ) = 1 λ 2 J ( r , x , z ) exp ( - 2 π i x · u / λ ) d x .
B ( r , u , z ) = B ( r - z u , u , 0 ) .
I ( r , z ) = B ( r , u , z ) d u .
I ( r , z ) = B ( r - z u , u , 0 ) d u .
I z = z B ( r - z u , u , 0 ) d u .
I z = - r · u B ( r , u , 0 ) d u ,
I z = 0.
J ( r , x , z ) = ψ ( r + x / 2 , z ) ψ * ( r - x / 2 , z ) ,
ψ ( r , z ) = A ( r , z ) exp [ 2 π i ϕ ( r , z ) ] ,
B coh ( r , u , z ) = 1 λ 2 A ( r + x / 2 , z ) A ( r - x / 2 , z ) × exp { 2 π i [ ϕ ( r + x / 2 , z ) - ϕ ( r - x / 2 , z ) ] } exp ( - 2 π i x · u / λ ) d x .
ϕ ( r + x / 2 , z ) - ϕ ( r - x / 2 , z ) x · r ϕ ( r , z ) .
B a ( r , u , 0 ) 1 λ 2 A ( r + x / 2 , 0 ) A ( r - x / 2 , 0 ) × exp ( - 2 π i x · u / λ ) d x ,
B ( r , u , 0 ) = B a [ r , u - λ r ϕ ( r , 0 ) ] ,
I z = - r · u B a [ r , u - λ r ϕ ( r , 0 ) ] d u ,
I z = - r · [ u + λ r ϕ ( r , 0 ) ] B a ( r , u , 0 ) d u .
I z = - r · [ λ r ϕ ( r , 0 ) I ( r , 0 ) + u B a ( r , u , 0 ) d u ] .
I z = - r · [ λ r ϕ ( r , 0 ) I ( r , 0 ) ] ,
ϕ ( r , z ) = 1 2 π arg [ ψ ( x , y ) ] = 1 2 π i n [ ψ ( r , z ) A ( r , z ) ] .
L - r · ( I r )
I ( x , y ) > 0             ( x , y ) Ω .
L φ = f ,
L ϕ = f ,             ϕ Γ = g .
L ϕ = f ,             I n ϕ Γ = g
Ω f ( x , y ) d x d y + Γ g ( s ) d s = 0.
1 λ z Ω I ( x , y ) d x d y = - Γ I n φ d s ,
ψ m ( r , θ , 0 ) = I 0 1 / 2 ( r ) exp ( i 2 π ϕ n ) ,
2 π φ m = m θ
- r · ( I r ϕ ) = - I Δ ϕ - r ϕ · r I = 0.
r ϕ · r I = 0.
ϕ ( θ ) = 0 ,
2 π ϕ ( θ ) = α θ + β ,
2 π ϕ ( θ ) = 2 π ϕ ( θ + 2 π ) + 2 π n ,
L ϕ m = 0 ,             n ϕ m Γ = 0 ,
r · { I ( r , 0 ) [ ϕ 1 ( r , 0 ) - ϕ 2 ( r , 0 ) ] } = 0.
I ( r , 0 ) ϕ 1 ( r , 0 ) = I ( r , 0 ) ϕ 2 ( r , 0 ) + × A ( r ) .
A ( r , z ) = A ( r ) k .
A ( r ) = m r I ( t ) t d t .

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