Abstract

We introduce an effective method for measuring the spatial distribution of complex correlation matrix of a partially coherent vector light field obeying Gaussian statistics by extending our recently advanced generalized Hanbury Brown–Twiss experiment. The method involves a combination of the partially coherent vector light with a pair of fully coherent reference vector fields and a measurement of the intensity-intensity cross-correlation of the combined fields. We show the real and imaginary parts of the complex correlation matrix can be recovered through a judicious control of the phase delay between two reference fields. We test the feasibility of our method by measuring the complex correlation matrix of a specially correlated radially polarized vector beam and we find the consistence between the experimental results and our general theory. We further show that our complex correlation matrix measurement can be used in reconstructing the polarization states hidden behind a thin-layer diffuser.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
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    [Crossref]

2020 (5)

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Optical coherence and electromagnetic surface waves,” Prog. Opt. 65, 105–172 (2020).
[Crossref]

O. Korotkova and G. Gbur, “Applications of optical coherence theory,” Prog. Opt. 65, 43–104 (2020).
[Crossref]

K. Saastamoinen, H. Partanen, A. T. Friberg, and T. Setälä, “Probing the electromagnetic degree of coherence of light beams with nanoscatterers,” ACS Photonics 7(4), 1030–1035 (2020).
[Crossref]

Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020).
[Crossref]

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

2019 (4)

H. Mao, Y. Chen, C. Liang, L. Chen, Y. Cai, and S. A. Ponomarenko, “Self-steering partially coherent vector beams,” Opt. Express 27(10), 14353–14368 (2019).
[Crossref]

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

H. Partanen, A. T. Friberg, T. Setälä, and J. Turunen, “Spectral measurement of coherence Stokes parameters of random broadband light beams,” Photonics Res. 7(6), 669–677 (2019).
[Crossref]

G. Wu, D. Kuebel, and T. D. Visser, “Generalized Hanbury Brown-Twiss effect in partially coherent electromagnetic beams,” Phys. Rev. A 99(3), 033846 (2019).
[Crossref]

2018 (6)

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

K. A. Sharma, G. Costello, E. Vélez-Juárez, T. G. Brown, and M. A. Alonso, “Measuring vector field correlations using diffraction,” Opt. Express 26(7), 8301–8313 (2018).
[Crossref]

Y. K. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018).
[Crossref]

H. Partanen, B. J. Hoenders, A. T. Friberg, and T. Setälä, “Young’s interference experiment with electromagnetic narrowband light,” J. Opt. Soc. Am. A 35(8), 1379–1384 (2018).
[Crossref]

D. Singh and R. K. Singh, “Lensless Stokes holography with the Hanbury Brown–Twiss approach,” Opt. Express 26(8), 10801–10812 (2018).
[Crossref]

D. Singh, Z. Chen, J. Pu, and R. K. Singh, “Recovery of polarimetric parameters from non-imaged laser-speckle,” J. Opt. 20(8), 085605 (2018).
[Crossref]

2017 (1)

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

2016 (1)

2015 (4)

2014 (3)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

R. K. Singh, R. V. Vinu, and A. M. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

M. Takeda, W. Wang, D. N. Naik, and R. K. Singh, “Spatial statistical optics and spatial correlation holography: a review,” Opt. Rev. 21(6), 849–861 (2014).
[Crossref]

2012 (1)

2011 (1)

2009 (1)

E. Wolf, “Solution of the phase problem in the theory of structure determination of crystals from X-ray diffraction experiments,” Phys. Rev. Lett. 103(7), 075501 (2009).
[Crossref]

2008 (1)

2003 (2)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref]

1998 (1)

Alonso, M. A.

Benz, A.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Bhatti, D.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Bocklage, L.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Brenner, G.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Brown, D. P.

Brown, T. G.

Cai, Y.

Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020).
[Crossref]

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

H. Mao, Y. Chen, C. Liang, L. Chen, Y. Cai, and S. A. Ponomarenko, “Self-steering partially coherent vector beams,” Opt. Express 27(10), 14353–14368 (2019).
[Crossref]

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Chen, L.

Chen, Y.

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Optical coherence and electromagnetic surface waves,” Prog. Opt. 65, 105–172 (2020).
[Crossref]

H. Mao, Y. Chen, C. Liang, L. Chen, Y. Cai, and S. A. Ponomarenko, “Self-steering partially coherent vector beams,” Opt. Express 27(10), 14353–14368 (2019).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Chen, Z.

D. Singh, Z. Chen, J. Pu, and R. K. Singh, “Recovery of polarimetric parameters from non-imaged laser-speckle,” J. Opt. 20(8), 085605 (2018).
[Crossref]

Classen, A.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Costello, G.

Depeursinge, C.

Y. K. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018).
[Crossref]

Fischer, B.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Friberg, A. T.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Optical coherence and electromagnetic surface waves,” Prog. Opt. 65, 105–172 (2020).
[Crossref]

K. Saastamoinen, H. Partanen, A. T. Friberg, and T. Setälä, “Probing the electromagnetic degree of coherence of light beams with nanoscatterers,” ACS Photonics 7(4), 1030–1035 (2020).
[Crossref]

H. Partanen, A. T. Friberg, T. Setälä, and J. Turunen, “Spectral measurement of coherence Stokes parameters of random broadband light beams,” Photonics Res. 7(6), 669–677 (2019).
[Crossref]

H. Partanen, B. J. Hoenders, A. T. Friberg, and T. Setälä, “Young’s interference experiment with electromagnetic narrowband light,” J. Opt. Soc. Am. A 35(8), 1379–1384 (2018).
[Crossref]

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33(12), 2431–2442 (2016).
[Crossref]

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19(16), 15188–15195 (2011).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref]

Gbur, G.

O. Korotkova and G. Gbur, “Applications of optical coherence theory,” Prog. Opt. 65, 43–104 (2020).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

Gori, F.

Gorobtsov, O.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Hassinen, T.

Hoenders, B. J.

Huang, Z.

Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020).
[Crossref]

Konijnenberg, S.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Korotkova, O.

O. Korotkova and G. Gbur, “Applications of optical coherence theory,” Prog. Opt. 65, 43–104 (2020).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Kuebel, D.

G. Wu, D. Kuebel, and T. D. Visser, “Generalized Hanbury Brown-Twiss effect in partially coherent electromagnetic beams,” Phys. Rev. A 99(3), 033846 (2019).
[Crossref]

Lazarev, S.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Liang, C.

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

H. Mao, Y. Chen, C. Liang, L. Chen, Y. Cai, and S. A. Ponomarenko, “Self-steering partially coherent vector beams,” Opt. Express 27(10), 14353–14368 (2019).
[Crossref]

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Liu, X.

Lu, X.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mao, H.

Mehringer, T.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Mercurio, G.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Naik, D. N.

M. Takeda, W. Wang, D. N. Naik, and R. K. Singh, “Spatial statistical optics and spatial correlation holography: a review,” Opt. Rev. 21(6), 849–861 (2014).
[Crossref]

Ni, H.

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

Norrman, A.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Optical coherence and electromagnetic surface waves,” Prog. Opt. 65, 105–172 (2020).
[Crossref]

Obukhov, Y.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Park, Y. K.

Y. K. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018).
[Crossref]

Partanen, H.

K. Saastamoinen, H. Partanen, A. T. Friberg, and T. Setälä, “Probing the electromagnetic degree of coherence of light beams with nanoscatterers,” ACS Photonics 7(4), 1030–1035 (2020).
[Crossref]

H. Partanen, A. T. Friberg, T. Setälä, and J. Turunen, “Spectral measurement of coherence Stokes parameters of random broadband light beams,” Photonics Res. 7(6), 669–677 (2019).
[Crossref]

H. Partanen, B. J. Hoenders, A. T. Friberg, and T. Setälä, “Young’s interference experiment with electromagnetic narrowband light,” J. Opt. Soc. Am. A 35(8), 1379–1384 (2018).
[Crossref]

Ponomarenko, S. A.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Optical coherence and electromagnetic surface waves,” Prog. Opt. 65, 105–172 (2020).
[Crossref]

Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020).
[Crossref]

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

H. Mao, Y. Chen, C. Liang, L. Chen, Y. Cai, and S. A. Ponomarenko, “Self-steering partially coherent vector beams,” Opt. Express 27(10), 14353–14368 (2019).
[Crossref]

Popescu, G.

Y. K. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018).
[Crossref]

Pu, J.

D. Singh, Z. Chen, J. Pu, and R. K. Singh, “Recovery of polarimetric parameters from non-imaged laser-speckle,” J. Opt. 20(8), 085605 (2018).
[Crossref]

Raghunathan, S. B.

Röhlsberger, R.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Saastamoinen, K.

K. Saastamoinen, H. Partanen, A. T. Friberg, and T. Setälä, “Probing the electromagnetic degree of coherence of light beams with nanoscatterers,” ACS Photonics 7(4), 1030–1035 (2020).
[Crossref]

Schlage, K.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Schneider, R.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Schouten, H. F.

Senthilkumaran, P.

Setälä, T.

Shao, Y.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Sharma, A. M.

R. K. Singh, A. M. Sharma, and P. Senthilkumaran, “Vortex array embedded in a partially coherent beam,” Opt. Lett. 40(12), 2751–2754 (2015).
[Crossref]

R. K. Singh, R. V. Vinu, and A. M. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

Sharma, K. A.

Singh, D.

D. Singh and R. K. Singh, “Lensless Stokes holography with the Hanbury Brown–Twiss approach,” Opt. Express 26(8), 10801–10812 (2018).
[Crossref]

D. Singh, Z. Chen, J. Pu, and R. K. Singh, “Recovery of polarimetric parameters from non-imaged laser-speckle,” J. Opt. 20(8), 085605 (2018).
[Crossref]

Singh, R. K.

D. Singh and R. K. Singh, “Lensless Stokes holography with the Hanbury Brown–Twiss approach,” Opt. Express 26(8), 10801–10812 (2018).
[Crossref]

D. Singh, Z. Chen, J. Pu, and R. K. Singh, “Recovery of polarimetric parameters from non-imaged laser-speckle,” J. Opt. 20(8), 085605 (2018).
[Crossref]

R. V. Vinu and R. K. Singh, “Synthesis of statistical properties of a randomly fluctuating polarized field,” Appl. Opt. 54(21), 6491–6497 (2015).
[Crossref]

R. K. Singh, A. M. Sharma, and P. Senthilkumaran, “Vortex array embedded in a partially coherent beam,” Opt. Lett. 40(12), 2751–2754 (2015).
[Crossref]

R. V. Vinu and R. K. Singh, “Experimental determination of generalized Stokes parameters,” Opt. Lett. 40(7), 1227–1230 (2015).
[Crossref]

R. K. Singh, R. V. Vinu, and A. M. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

M. Takeda, W. Wang, D. N. Naik, and R. K. Singh, “Spatial statistical optics and spatial correlation holography: a review,” Opt. Rev. 21(6), 849–861 (2014).
[Crossref]

Skopintsev, P.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Takeda, M.

M. Takeda, W. Wang, D. N. Naik, and R. K. Singh, “Spatial statistical optics and spatial correlation holography: a review,” Opt. Rev. 21(6), 849–861 (2014).
[Crossref]

Tang, Y.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Tervo, J.

Turunen, J.

H. Partanen, A. T. Friberg, T. Setälä, and J. Turunen, “Spectral measurement of coherence Stokes parameters of random broadband light beams,” Photonics Res. 7(6), 669–677 (2019).
[Crossref]

Urbach, H.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Vartanyants, I. A.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Vélez-Juárez, E.

Vinu, R. V.

Visser, T. D.

G. Wu, D. Kuebel, and T. D. Visser, “Generalized Hanbury Brown-Twiss effect in partially coherent electromagnetic beams,” Phys. Rev. A 99(3), 033846 (2019).
[Crossref]

S. B. Raghunathan, H. F. Schouten, and T. D. Visser, “Correlation singularities in partially coherent electromagnetic beams,” Opt. Lett. 37(20), 4179–4181 (2012).
[Crossref]

von Zanthier, J.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Wagner, J.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Waldmann, F.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Wang, F.

Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020).
[Crossref]

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Wang, W.

M. Takeda, W. Wang, D. N. Naik, and R. K. Singh, “Spatial statistical optics and spatial correlation holography: a review,” Opt. Rev. 21(6), 849–861 (2014).
[Crossref]

Wenthaus, L.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Willing, S.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Wolf, E.

E. Wolf, “Solution of the phase problem in the theory of structure determination of crystals from X-ray diffraction experiments,” Phys. Rev. Lett. 103(7), 075501 (2009).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wu, G.

G. Wu, D. Kuebel, and T. D. Visser, “Generalized Hanbury Brown-Twiss effect in partially coherent electromagnetic beams,” Phys. Rev. A 99(3), 033846 (2019).
[Crossref]

Wurth, W.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Zaluzhnyy, I.

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Zhang, M.

Zhao, C.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Zhu, X.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

ACS Photonics (1)

K. Saastamoinen, H. Partanen, A. T. Friberg, and T. Setälä, “Probing the electromagnetic degree of coherence of light beams with nanoscatterers,” ACS Photonics 7(4), 1030–1035 (2020).
[Crossref]

Adv. Photonics (1)

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. K. Singh, R. V. Vinu, and A. M. Sharma, “Recovery of complex valued objects from two-point intensity correlation measurement,” Appl. Phys. Lett. 104(11), 111108 (2014).
[Crossref]

Chin. Phys. B (1)

H. Ni, C. Liang, F. Wang, Y. Chen, S. A. Ponomarenko, and Y. Cai, “Non-Gaussian statistics of partially coherent light in atmospheric turbulence,” Chin. Phys. B 29(6), 064203 (2020).
[Crossref]

J. Opt. (1)

D. Singh, Z. Chen, J. Pu, and R. K. Singh, “Recovery of polarimetric parameters from non-imaged laser-speckle,” J. Opt. 20(8), 085605 (2018).
[Crossref]

J. Opt. Soc. Am. A (3)

Nat. Photonics (1)

Y. K. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018).
[Crossref]

Nat. Phys. (1)

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14(2), 126–129 (2018).
[Crossref]

Opt. Express (6)

Opt. Lett. (4)

Opt. Rev. (1)

M. Takeda, W. Wang, D. N. Naik, and R. K. Singh, “Spatial statistical optics and spatial correlation holography: a review,” Opt. Rev. 21(6), 849–861 (2014).
[Crossref]

Photonics Res. (1)

H. Partanen, A. T. Friberg, T. Setälä, and J. Turunen, “Spectral measurement of coherence Stokes parameters of random broadband light beams,” Photonics Res. 7(6), 669–677 (2019).
[Crossref]

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

Phys. Rev. A (2)

G. Wu, D. Kuebel, and T. D. Visser, “Generalized Hanbury Brown-Twiss effect in partially coherent electromagnetic beams,” Phys. Rev. A 99(3), 033846 (2019).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Phys. Rev. Appl. (1)

Z. Huang, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Measuring complex degree of coherence of random light fields with generalized Hanbury Brown–Twiss experiment,” Phys. Rev. Appl. 13(4), 044042 (2020).
[Crossref]

Phys. Rev. Lett. (1)

E. Wolf, “Solution of the phase problem in the theory of structure determination of crystals from X-ray diffraction experiments,” Phys. Rev. Lett. 103(7), 075501 (2009).
[Crossref]

Prog. Opt. (3)

O. Korotkova and G. Gbur, “Applications of optical coherence theory,” Prog. Opt. 65, 43–104 (2020).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Optical coherence and electromagnetic surface waves,” Prog. Opt. 65, 105–172 (2020).
[Crossref]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

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Figures (5)

Fig. 1.
Fig. 1. The schematic of the experimental setup for generating a SCRP vector beam and measuring the complex correlation matrix. BS, beam splitter; NDF, neutral-density filter; BE, beam expander; RPC, radial polarization converter; L1, L2, L3, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; M, reflecting mirror; LP, linear polarizer; QWP, quarter-wave plant; PBS, polarization beam splitter; CCD1, CCD2, charge-coupled devices.
Fig. 2.
Fig. 2. Calculated (top row) and experimental (bottom row) results for the real, imaginary, and square modulus of the complex correlation matrix element $\mu _{xx}(\Delta \mathbf {r})$ of a SCRP vector beam.
Fig. 3.
Fig. 3. Calculated (top row) and experimental (bottom row) results for the real, imaginary, and square modulus of the complex correlation matrix element $\mu _{yy}(\Delta \mathbf {r})$ of a SCRP vector beam.
Fig. 4.
Fig. 4. Calculated (top row) and experimental (bottom row) results for the real, imaginary, and square modulus of the complex correlation matrix element $\mu _{xy}(\Delta \mathbf {r})$ of a SCRP vector beam.
Fig. 5.
Fig. 5. Recovery of the polarization state of the field hidden behind a RGGD through the complex correlations measurement. (a)–(d), the recovered Stokes parameters $\mathcal {S}_0$, $\mathcal {S}_1$, $\mathcal {S}_2$, and $\mathcal {S}_3$; (e), the spatial distribution of the reconstructed polarization state.

Equations (22)

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W ( r 1 , r 2 , ω ) = E ( r 1 , ω ) E T ( r 2 , ω ) ,
μ α β ( r 1 , r 2 ) = W α β ( r 1 , r 2 ) S α ( r 1 ) S β ( r 2 ) ,
I α ( r 1 ) I β ( r 2 ) S α ( r 1 ) S β ( r 2 ) = 1 + | μ α β ( r 1 , r 2 ) | 2 ,
E (C1) ( r ) = E (R1) ( r ) + E ( r ) ,
E (C2) ( r ) = E (R2) ( r ) + E ( r ) .
G α β (C) ( r 1 , r 2 ) = I α (C1) ( r 1 ) I β (C2) ( r 2 ) ,
G α β (C) ( r 1 , r 2 ) = I α (U1) ( r 1 ) I β (U2) ( r 2 ) + | W α β ( r 1 , r 2 ) | 2 + 2 I α (R1) ( r 1 ) I β (R2) ( r 2 ) R e [ e i Δ ϕ α β W α β ( r 1 , r 2 ) ] ,
G α β (U) ( r 1 , r 2 ) = I α (U1) ( r 1 ) I β (U2) ( r 2 ) ,
G α β (U) ( r 1 , r 2 ) = I α (U1) ( r 1 ) I β (U2) ( r 2 ) + | W α β ( r 1 , r 2 ) | 2 .
Δ G α β ( r 1 , r 2 , Δ ϕ α β ) = G α β (C) ( r 1 , r 2 ) G α β (U) ( r 1 , r 2 ) , = 2 I α (R1) ( r 1 ) I β (R2) ( r 2 ) R e [ e i Δ ϕ α β W α β ( r 1 , r 2 ) ] ,
μ α β ( r 1 , r 2 ) = Δ G α β ( r 1 , r 2 , Δ ϕ α β = 0 ) 2 I α (R1) ( r 1 ) I β (R2) ( r 2 ) S α ( r 1 ) S β ( r 2 ) ,
μ α β ( r 1 , r 2 ) = Δ G α β ( r 1 , r 2 , Δ ϕ α β = π 2 ) 2 I α (R1) ( r 1 ) I β (R2) ( r 2 ) S α ( r 1 ) S β ( r 2 ) ,
W α α ( r 1 , r 2 ) =   S 0 exp ( r 1 2 + r 2 2 4 σ 0 2 ) [ 1 ( α 2 α 1 ) 2 δ 0 2 ] × exp [ ( r 1 r 2 ) 2 2 δ 0 2 ] exp [ i ( r 1 r 2 ) v 0 ] ,
W x y ( r 1 , r 2 ) = S 0 exp ( r 1 2 + r 2 2 4 σ 0 2 ) ( x 2 x 1 ) ( y 2 y 1 ) δ 0 2 × exp [ ( r 1 r 2 ) 2 2 δ 0 2 ] exp [ i ( r 1 r 2 ) v 0 ] ,
W y x ( r 1 , r 2 ) =   W x y ( r 2 , r 1 ) ,
W ( ρ 1 , ρ 2 ) = δ ( ρ 1 ρ 2 ) ω 0 2 exp ( ρ 1 2 + ρ 2 2 ω 0 2 ) ( ρ 1 x ρ 2 x ρ 1 x ρ 2 y ρ 1 y ρ 2 x ρ 1 y ρ 2 y ) .
W ( r 1 , r 2 ) = W ( ρ 1 ρ 0 , ρ 2 ρ 0 ) H ( r 1 , ρ 1 ) H ( r 2 , ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
G α β (C) ( r 1 , r 2 , Δ ϕ α β = 0 ) = I α (C1) ( r 1 ) I β (C1) ( r 2 ) s ,
G α β (C) ( r 1 , r 2 , Δ ϕ α β = π 2 ) = I α (C1) ( r 1 ) I β (C2) ( r 2 ) s ,
G α β (U) ( r 1 , r 2 ) = I α (U1) ( r 1 ) I β (U2) ( r 2 ) s ,
W ( ρ 1 ρ 0 , ρ 2 ρ 0 ) = δ ( ρ 1 ρ 2 ) Φ ( ρ 1 ρ 0 , ρ 1 ρ 0 ) ,
W ( r 1 , r 2 ) = T ( r 1 ) T ( r 2 ) λ 2 f 2 2 Φ ( ρ ρ 0 , ρ ρ 0 ) exp [ i 2 π λ f 2 ( r 1 r 2 ) ρ ] d 2 ρ .

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