Abstract

A novel algorithm for the retrieval of the spatial mutual coherence function of the optical field of a light beam in the quasimonochromatic approximation is presented. The algorithm only requires that the intensity distribution is known in a finite number of transverse planes along the beam. The retrieval algorithm is based on the observation that a partially coherent field can be represented as an ensemble of coherent fields. Each field in the ensemble is propagated with coherent methods between neighboring planes, and the ensemble is then subjected to amplitude restrictions, much in the same way as in conventional phase recovery algorithms for coherent fields. The proposed algorithm is evaluated both for one- and two-dimensional fields using numerical simulations.

©2007 Optical Society of America

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References

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  1. F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. 61, 158–164 (1948).
    [Crossref]
  2. B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. 47, 895–902 (1957).
    [Crossref]
  3. J. Schwider, “Continuous Lateral Shearing Interferometer,” Appl. Opt. 23, 4403–4409 (1983).
    [Crossref]
  4. C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
    [Crossref]
  5. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Letters 72, 1137–1140 (1993).
    [Crossref]
  6. M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
    [Crossref]
  7. H. Laabs, B. Eppich, and H. Weber, “Modal decomposition of partially coherent beams using the ambiguity function,” J. Opt. Soc. Am. A 19, 497–504 (2002).
    [Crossref]
  8. R. Borghi, G. Guattari, L. de la Torre, F. Gori, and M. Santarsiero, “Evaluation of the spatial coherence of a light beam through transverse intensity measurements,” J. Opt. Soc. Am. A 20, 1763–1770 (2003).
    [Crossref]
  9. M. Ježek and Z. Hradil, “Reconstruction of spatial, phase, and coherence properties of light,” J. Opt. Soc. Am. A 21, 1407–1416 (2004).
    [Crossref]
  10. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  11. B. Y. Gu, G. Z. Yang, and B. Z. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
    [Crossref] [PubMed]
  12. L. Mandel and E. Wolf, Optical coherence and quantum optics, Cambridge University Press, Cambridge (1995).
  13. F. Gori and M. Santarsiero, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [Crossref]
  14. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1825 (1994).
    [Crossref]
  15. D. Dragoman, “Unambiguous coherence retrieval from intensity measurements,” J. Opt. Soc. Am. A 20, 290–295 (2003).
    [Crossref]
  16. C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A 23, 1616–1625 (2006).
    [Crossref]
  17. H. Stark and Y.Y. Yang, Vector space projections: A numerical approach to signal and image processing, neural nets, and optics, Wiley, New York (1998).
    [PubMed]
  18. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [Crossref] [PubMed]
  19. J. W. Goodman, Introduction to Fourier optics, McGraw-Hill, New York, 1996.

2006 (1)

2004 (1)

2003 (2)

2002 (1)

2000 (1)

C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[Crossref]

1999 (1)

1994 (1)

1993 (3)

1986 (1)

1983 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1957 (1)

1948 (1)

F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. 61, 158–164 (1948).
[Crossref]

Anderson, E.

C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[Crossref]

Attwood, D.

C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[Crossref]

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Letters 72, 1137–1140 (1993).
[Crossref]

Bengtsson, J.

Borghi, R.

Chang, C.

C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[Crossref]

de la Torre, L.

Dong, B. Z.

Dragoman, D.

Eppich, B.

Fienup, J. R.

Friberg, A. T.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics, McGraw-Hill, New York, 1996.

Gori, F.

Gu, B. Y.

Guattari, G.

Hradil, Z.

Ježek, M.

Laabs, H.

Mandel, L.

L. Mandel and E. Wolf, Optical coherence and quantum optics, Cambridge University Press, Cambridge (1995).

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Letters 72, 1137–1140 (1993).
[Crossref]

Naulleau, p.

C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[Crossref]

Raymer, M. G.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Letters 72, 1137–1140 (1993).
[Crossref]

Rydberg, C.

Santarsiero, M.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schwider, J.

Stark, H.

H. Stark and Y.Y. Yang, Vector space projections: A numerical approach to signal and image processing, neural nets, and optics, Wiley, New York (1998).
[PubMed]

Tervonen, E.

Thompson, B. J.

Turunen, J.

Weber, H.

Wolf, E.

B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. 47, 895–902 (1957).
[Crossref]

L. Mandel and E. Wolf, Optical coherence and quantum optics, Cambridge University Press, Cambridge (1995).

Yang, G. Z.

Yang, Y.Y.

H. Stark and Y.Y. Yang, Vector space projections: A numerical approach to signal and image processing, neural nets, and optics, Wiley, New York (1998).
[PubMed]

Zernike, F.

F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. 61, 158–164 (1948).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

C. Chang, p. Naulleau, E. Anderson, and D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[Crossref]

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. Letters (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Letters 72, 1137–1140 (1993).
[Crossref]

Proc. Phys. Soc. (1)

F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. 61, 158–164 (1948).
[Crossref]

Other (3)

H. Stark and Y.Y. Yang, Vector space projections: A numerical approach to signal and image processing, neural nets, and optics, Wiley, New York (1998).
[PubMed]

J. W. Goodman, Introduction to Fourier optics, McGraw-Hill, New York, 1996.

L. Mandel and E. Wolf, Optical coherence and quantum optics, Cambridge University Press, Cambridge (1995).

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Figures (6)

Fig. 1.
Fig. 1.

Illustration of the operations in the coherence retrieval algorithm. In this example the intensity of the beam is measured in four transverse planes at longitudinal positions z = {z 1,…,z 4}. The ensemble of fields, representing the partially coherent field, is propagated one coherent field, Ui , at a time from one plane to the next, which is represented by the operator P. Each coherent field is then adjusted so that the intensity distribution of the ensemble at the new plane corresponds to the measured intensity distribution. This operation is denoted by+.

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Fig. 2.
Fig. 2.

The residual error in the retrieval of the mutual coherence function as a function of the number of fields, N, in the ensemble representing the partially coherent field.

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Fig. 3.
Fig. 3.

The residual error in the retrieval of the mutual coherence function as a function of the number of iterations in the retrieval algorithm. The three different cases are simulated with the same beam, but the locations of the four transverse planes from which the intensity information is collected are chosen differently, as indicated in the inserts.

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Fig. 4.
Fig. 4.

The residual error in the retrieval of the mutual coherence function as a function of the number of iterations for the case of a two-dimensional beam. The same twisted Gaussian-Schell beam is considered in two different set ups, shown in the two inserts, with the beam propagating from left to right. The top insert shows the intensity distribution generated by the twisted beam upon propagation through a rotationally symmetric lens. The retrieval algorithm converges but not to the correct mutual coherence function because of the multitude of possible solutions. The bottom insert illustrates how the beam propagates after the insertion of a cylinder lens with different focal lengths along the x- and y-axes. Above each beam-path diagram is indicated the intensity distribution in the different planes. The non-rotationally symmetric cylinder lens produces an elliptical intensity distribution so that the rotation also becomes visible, which restores the uniqueness of the problem. As the diagram shows, the retrieved mutual coherence function now tends toward the correct solution.

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Fig. 5.
Fig. 5.

Performance of the retrieval algorithm with noisy measured intensity data. The residual error in the retrieval of the mutual coherence function is shown as function of the intensity noise root-mean-square level expressed in percent of the noise-free intensity. The plot shows the average behavior of several simulations, each generated with different distortions of the intensity data.

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Fig. 6.
Fig. 6.

Performance of the retrieval algorithm when the planes where the transverse intensity data is measured are displaced compared to their locations assumed in the algorithm. The residual error in the retrieved mutual coherence function is shown as a function of the root-mean-square error, expressed in percent of the nominal distance between plane z 1 and z 4, of the longitudinal position of the transverse intensity planes. The plot shows the average behavior of several simulations, each generated with a different error for the location of each plane.

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Equations (12)

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Γ 12 ( ρ 1 , ρ 2 , t 1 , t 2 ) ≡ 〈 u ( ρ 1 , t 1 ) u * ( ρ 2 , t 2 ) 〉 ,
S ( ρ 1 , ρ 2 , v ) = ∫ − ∞ ∞ Γ 12 ( ρ 1 , ρ 2 , τ ) exp [ 2 πivτ ] dτ .
{ U 1 ( ρ , z ) , U 2 ( ρ , z ) ,… , U N ( ρ , z ) } .
I ( ρ , z ) ≡ ∑ i = 1 N U i ( ρ , z ) U i * ( ρ , z ) ,
S ( ρ 1 , ρ 2 , z ) ≡ ∑ i = 1 N U i ( ρ 1 , z ) U i * ( ρ 2 , z ) .
Corr ( ρ , z n ) = I ref ( ρ , z n ) I ( ρ , z n ) ,
U i + ( ρ , z n ) = Corr ( ρ , z n ) U i ( ρ , z n ) ,
U i ( ρ , z n + 1 ) = P ( U i + ( ρ , z n ) ) ,
S ref ( x 1 , x 2 ) = exp [ − x 1 2 w 2 ] exp [ − x 2 2 w 2 ] exp [ − ( x 2 − x 1 ) 2 2 σ g 2 ] exp [ ik ( x 2 2 − x 1 2 ) 2 f ]
E ( S calc , S ref ) ≡ ∫ ∫ ∣ S calc ( x 1 , x 2 ) − S ref ( x 1 , x 2 ) ∣ 2 d x 1 d x 2 ∫ ∫ ∣ S calc ( x 1 , x 2 ) ∣ 2 + ∣ S ref ( x 1 , x 2 ) ∣ 2 d x 1 d x 2 .
U i ( ρ , z n + 1 ) = P zn → z element → P element → P z element → z n + 1 ,
S ref ( ρ 1 , ρ 2 ) = exp [ − ρ 1 2 w 2 ] exp [ − ρ 2 2 w 2 ] exp [ − ( ρ 2 − ρ 1 ) 2 2 σ g 2 ] × exp [ ik ( ρ 2 2 − ρ 1 2 ) 2 f ] exp [ ik ( x 1 y 2 − x 2 y 1 ) u ] ,

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