Abstract

We formulate coherence retrieval, the process of recovering via intensity measurements the two-point correlation function of a partially coherent field, as a convex weighted least-squares problem and show that it can be solved with a novel iterated descent algorithm using a coherent-modes factorization of the mutual intensity. This algorithm is more memory-efficient than the standard interior point methods used to solve convex problems, and we verify its feasibility by reconstructing the mutual intensity of a Schell-model source from both simulated data and experimental measurements.

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References

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2012 (2)

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express20, 8296–8308 (2012).
[CrossRef] [PubMed]

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

2007 (1)

2003 (1)

2002 (1)

1995 (1)

1994 (1)

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (1)

1988 (1)

Y. Li, G. Eichmann, and M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun.67, 177–179 (1988).
[CrossRef]

1985 (1)

1983 (1)

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

1982 (1)

1980 (1)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

1974 (1)

1972 (1)

T. Asakura, H. Fujii, and K. Murata, “Measurement of spatial coherence using speckle patterns,” Optica Acta19, 273–290 (1972).
[CrossRef]

1969 (1)

E. Polak and G. Ribiere, “Note sur la convergence de methodes de directions conjugées,” Rev. Fr. Inform. Rech. O.3, 35–43 (1969).

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.15, 187–188 (1967).
[CrossRef]

1957 (2)

Asakura, T.

T. Asakura, H. Fujii, and K. Murata, “Measurement of spatial coherence using speckle patterns,” Optica Acta19, 273–290 (1972).
[CrossRef]

Barbastathis, G.

Barreiro, J. C.

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Beck, M.

Bengtsson, J.

Born, M.

M. Born and E. Wolf, Principles of optics, 7th. ed. (Cambridge University, 2005).

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Clarke, L.

Conner, M.

Y. Li, G. Eichmann, and M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun.67, 177–179 (1988).
[CrossRef]

Dragoman, D.

Eichmann, G.

Y. Li, G. Eichmann, and M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun.67, 177–179 (1988).
[CrossRef]

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

Fujii, H.

T. Asakura, H. Fujii, and K. Murata, “Measurement of spatial coherence using speckle patterns,” Optica Acta19, 273–290 (1972).
[CrossRef]

Gamo, H.

Kutay, M. A.

Lee, J.

Li, Y.

Y. Li, G. Eichmann, and M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun.67, 177–179 (1988).
[CrossRef]

Lohmann, A. W.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Mayer, A.

McAlister, D.

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

McAlister, D. F.

Michalski, M.

Murata, K.

T. Asakura, H. Fujii, and K. Murata, “Measurement of spatial coherence using speckle patterns,” Optica Acta19, 273–290 (1972).
[CrossRef]

neda, J. O.-C.

Oh, S. B.

Ojeda-Castañeda, J.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

Ozaktas, H. M.

Papoulis, A.

Polak, E.

E. Polak and G. Ribiere, “Note sur la convergence de methodes de directions conjugées,” Rev. Fr. Inform. Rech. O.3, 35–43 (1969).

Rabal, H. J.

Raymer, M. G.

Ribiere, G.

E. Polak and G. Ribiere, “Note sur la convergence de methodes de directions conjugées,” Rev. Fr. Inform. Rech. O.3, 35–43 (1969).

Rydberg, C.

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.15, 187–188 (1967).
[CrossRef]

Sicre, E. E.

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

Thompson, B. J.

Tian, L.

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

Wolf, E.

Yüksel, S.

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.15, 187–188 (1967).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Nature Photon. (1)

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

Opt. Commun. (3)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Y. Li, G. Eichmann, and M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun.67, 177–179 (1988).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Optica Acta (1)

T. Asakura, H. Fujii, and K. Murata, “Measurement of spatial coherence using speckle patterns,” Optica Acta19, 273–290 (1972).
[CrossRef]

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

Rev. Fr. Inform. Rech. O. (1)

E. Polak and G. Ribiere, “Note sur la convergence de methodes de directions conjugées,” Rev. Fr. Inform. Rech. O.3, 35–43 (1969).

Other (2)

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

M. Born and E. Wolf, Principles of optics, 7th. ed. (Cambridge University, 2005).

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Figures (15)

Fig. 1
Fig. 1

Optical arrangement that generates a Schell-model beam. Uniform spatially incoherent quasi-monochromatic light is used to illuminate an amplitude mask at the front focal plane (I) of a convex lens (II) with focal length f. The partially coherent field immediately after an amplitude mask at the back focal plane (III) is that of a Schell-model source.

Fig. 2
Fig. 2

Using a convex lens, as shown in (a), we can capture both the image (I) and the Fourier transform intensity (II) of a partially coherent source located at (O) using a finitely deep focal stack. That is, the intensity at planes I and II correspond to the horizontal and vertical axes respectively of the ambiguity function (b), allowing more of the ambiguity function to be directly measured compared to a lensless approach.

Fig. 3
Fig. 3

Simulated noiseless data used as input to the factored form descent algorithm, visualized as focal stacks. The right image has been gamma-boosted with γ = 0.3 to enhance the visibility of lower energy parts of the stack.

Fig. 4
Fig. 4

Simulated noisy data used as input to the factored form descent algorithm, visualized as focal stacks (top row) and gamma-boosted focal stacks (bottom row).

Fig. 5
Fig. 5

Convergence of RMS error between the input intensity data set and the intensity computed from the current iterate of the mutual intensity in the factored form descent algorithm. Both the error axis and the iteration (time) axis are shown in log scale.

Fig. 6
Fig. 6

Convergence of RMS error between the theoretical mutual intensity and the current iterate of the mutual intensity. Both the error axis and the iteration (time) axis are shown in log scale.

Fig. 7
Fig. 7

At top is an image of the magnitude of the theoretical mutual intensity. Below the top image are three sets of images corresponding to three different uniform noise runs, with the left image being the magnitude of the resulting mutual intensity and the right image being the magnitude of the difference between the resulting mutual intensity and the theoretical mutual intensity.

Fig. 8
Fig. 8

Images corresponding to the runs using the Poisson shot noise data set yp. The left column contains images of the magnitude of the mutual intensity and the right column contains images of the magnitude of the difference between the attained mutual intensity and the theoretical mutual intensity. The error images in this case have been scaled the same to allow easier comparison between uniform weighting and matched weighting.

Fig. 9
Fig. 9

A plot of the energy contained in each mode after performing a coherence mode decomposition on the theoretical mutual intensity as well as the computed mutual intensity from each run. The horizontal axis gives the mode number on a log scale, with modes sorted by decreasing energy, and the vertical axis gives the energy on a log scale.

Fig. 10
Fig. 10

A plot of the magnitude of the field for the first five coherence modes of the theoretical mutual intensity as well as the computed mutual intensity from each run.

Fig. 11
Fig. 11

The data collected during the experiment, visualized as focal stacks (top row) and gamma-boosted focal stacks (bottom row).

Fig. 12
Fig. 12

Convergence of RMS error between the input experimental data and the intensity computed from the current iterate of the mutual intensity in the factored form descent algorithm. Both the error axis and the iteration (time) axis are shown in log scale.

Fig. 13
Fig. 13

Images corresponding to the resulting mutual intensity computed by runs on the experimental data sets. The left column contains images of the magnitude of the mutual intensity and the right column contains images of the magnitude of the difference between the attained mutual intensity and the theoretical mutual intensity.

Fig. 14
Fig. 14

A plot of the energy contained in each mode after performing a coherence mode decomposition on the theoretical mutual intensity as well as the computed mutual intensity from each experimental run. The horizontal axis gives the mode number, with modes sorted by decreasing energy, and the vertical axis gives the energy on a log scale.

Fig. 15
Fig. 15

A plot of the magnitude of the field for the first five coherence modes of the theoretical mutual intensity as well as the computed mutual intensity from each run.

Tables (2)

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Table 1 Algorithm Runs on Simulated Data

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Table 2 Algorithm Runs on Experimental Data

Equations (19)

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minimize f ( J ) = m = 1 M σ m 2 ( y m k m H J k m ) 2 subject to J 0
J = X X H
minimize f ^ ( X ) = m = 1 M σ m 2 ( y m k m H X X H k m ) 2
β ( i ) = max [ 0 , Re { G ( i ) , G ( i ) G ( i 1 ) G ( i 1 ) F 2 } ]
G ( i ) = n = 1 N D ( i ) σ n ( i ) u n ( i ) V H ( i )
0 = n = 1 N D ( i ) σ n ( i ) u n ( i )
u n H ( i ) D ( i ) u p ( i ) = u n ( i ) u p H ( i ) , D ( i )
J ^ = X ( i ) X H ( i ) + ε S
S ^ = B ^ H S B ^
v H J ^ v = v ^ H B ^ H X ( i ) X H ( i ) B ^ v ^ + ε v ^ H S ^ v ^
v H J ^ v = ε v ^ H S ^ v ^
S , D ( i ) > 0
D ( i ) = n e n e n H
S , D ( i ) = S , n e n e n H = n e n H S e n = n ( B ^ H e n ) H S ^ ( B ^ H e n ) < 0
X ( i + 1 ) = X ( i ) + ε ( max ( λ 1 , 0 ) v 1 , , max ( λ N , 0 ) v N )
J ( x 1 , x 2 ) = a ( x 1 ) a * ( x 2 ) μ ( x 1 x 2 )
J ( x 1 , x 2 ) = I 0 rect ( x 1 / W 2 ) rect ( x 2 / W 2 ) sinc ( W 1 ( x 1 x 2 ) / ( λ F ) )
λ = 532 nm , F = 100 mm , W 1 = 100 μ m , W 2 = 500 μ m
0 = S ( i ) , G ( i ) = G ( i ) + β ( i ) S ( i 1 ) , G ( i ) = G ( i ) , G ( i )

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