Abstract

A coherence vortex (CV) carrying topological-charge information in its correlation dimension is a new option for optical manipulation and communication. CV generation by directly modulating the correlation function enables a way to control the light field in this dimension. However, few experimental realizations on this issue have been reported because of the difficulty in phase modulation when the light arrays are of low coherence. In this paper, we propose a method for generating a CV by utilizing partially coherent light arrays. A proper design of random arrays at the input plane leads to a complex CV field at the output plane after free-space propagation. This generation mechanism works well for beamlets of low coherence.

© 2019 Chinese Laser Press

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2019 (6)

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
[Crossref]

L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C.-T. Liao, J. San Román, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernández-García, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364, eaaw9486 (2019).
[Crossref]

H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
[Crossref]

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

A. B. Ortega, S. Bucio-Pacheco, S. Lopez-Huidobro, L. Perez-Garcia, F. J. Poveda-Cuevas, J. A. Seman, A. V. Arzola, and K. Volke-Sepúlveda, “Creation of optical speckle by randomizing a vortex-lattice,” Opt. Express 27, 4105–4115 (2019).
[Crossref]

J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44, 4371–4374 (2019).
[Crossref]

2018 (11)

Y. Shao, X. Lu, S. Konijnenberg, C. Zhao, Y. Cai, and H. P. Urbach, “Spatial coherence measurement and partially coherent diffractive imaging using self-referencing holography,” Opt. Express 26, 4479–4490 (2018).
[Crossref]

G. Piquero, M. Santarsiero, R. Martínez-Herrero, J. C. G. de Sande, M. Alonzo, and F. Gori, “Partially coherent sources with radial coherence,” Opt. Lett. 43, 2376–2379 (2018).
[Crossref]

Z. Mei and O. Korotkova, “Sources for random arrays with structured complex degree of coherence,” Opt. Lett. 43, 2676–2679 (2018).
[Crossref]

Y. Yuan, D. Liu, Z. Zhou, H. Xu, J. Qu, and Y. Cai, “Optimization of the probability of orbital angular momentum for Laguerre-Gaussian beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 26, 21861–21871 (2018).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26, 26830–26844 (2018).
[Crossref]

N. Montaut, O. S. Magaña-Loaiza, T. J. Bartley, V. B. Verma, S. W. Nam, R. P. Mirin, C. Silberhorn, and T. Gerrits, “Compressive characterization of telecom photon pairs in the spatial and spectral degrees of freedom,” Optica 5, 1418–1423 (2018).
[Crossref]

J. Chen and Y. Li, “Discrimination of incoherent vortex states of light,” Opt. Lett. 43, 5595–5598 (2018).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

A. Lizana, H. Zhang, A. Turpin, A. V. Eeckhout, F. A. Torres-Ruiz, A. Vargas, C. Ramirez, F. Pi, and J. Campos, “Generation of reconfigurable optical traps for microparticles spatial manipulation through dynamic split lens inspired light structures,” Sci. Rep. 8, 11263 (2018).
[Crossref]

Y. Altmann, S. McLaughlin, M. J. Padgett, V. K. Goyal, A. O. Hero, and D. Faccio, “Quantum-inspired computational imaging,” Science 361, eaat2298 (2018).
[Crossref]

Y. T. Zhang, C. L. Ding, L. Z. Pan, and Y. J. Cai, “Laser arrays of partially coherent beams with multi-Gaussian correlation function,” J. Quant. Spectrosc. Radiat. Transfer 218, 1–11 (2018).
[Crossref]

2017 (5)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

G. Kulkarni, R. Sahu, O. S. Magaña-loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1 (2017).
[Crossref]

Z. Yang, O. S. Magaña-loaiza, M. Mirhosseini, Y. Zhou, B. Gao, L. Gao, S. Mohammad, H. Rafsanjani, G.-l. Long, and R. W. Boyd, “Digital spiral object identification using random light,” Light Sci. Appl. 6, e17013 (2017).
[Crossref]

M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref]

V. Kumar, B. Piccirillo, S. G. Reddy, and R. P. Singh, “Topological structures in vector speckle fields,” Opt. Lett. 42, 466–469 (2017).
[Crossref]

2016 (6)

2015 (3)

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9, 822–826 (2015).
[Crossref]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review,” Prog. Electromagn. Res. 150, 123–143 (2015).
[Crossref]

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

2014 (2)

2013 (1)

2011 (2)

2009 (2)

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboglu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753–1765 (2009).
[Crossref]

L.-G. Wang, L.-Q. Wang, and S.-Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[Crossref]

2008 (1)

Y. D. Liu, C. Q. Gao, M. W. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).
[Crossref]

2006 (2)

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
[Crossref]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref]

2004 (2)

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
[Crossref]

2003 (1)

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

2002 (3)

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[Crossref]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[Crossref]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
[Crossref]

2000 (1)

1997 (1)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1997).
[Crossref]

Alonzo, M.

Altmann, Y.

Y. Altmann, S. McLaughlin, M. J. Padgett, V. K. Goyal, A. O. Hero, and D. Faccio, “Quantum-inspired computational imaging,” Science 361, eaat2298 (2018).
[Crossref]

Alves, C. R.

Arlt, J.

Arzola, A. V.

Aunon, J. M.

Banerji, J.

Banzer, P.

M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref]

Bartley, T. J.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1997).
[Crossref]

Boyd, R. W.

G. Kulkarni, R. Sahu, O. S. Magaña-loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1 (2017).
[Crossref]

Z. Yang, O. S. Magaña-loaiza, M. Mirhosseini, Y. Zhou, B. Gao, L. Gao, S. Mohammad, H. Rafsanjani, G.-l. Long, and R. W. Boyd, “Digital spiral object identification using random light,” Light Sci. Appl. 6, e17013 (2017).
[Crossref]

M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref]

O. S. Magaña-Loaiza, M. Mirhosseini, R. M. Cross, S. M. H. Rafsanjani, and R. W. Boyd, “Hanbury Brown and Twiss interferometry with twisted light,” Sci. Adv. 2, e1501143 (2016).
[Crossref]

Brooks, N. J.

L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C.-T. Liao, J. San Román, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernández-García, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364, eaaw9486 (2019).
[Crossref]

Bucio-Pacheco, S.

Cai, Y.

J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44, 4371–4374 (2019).
[Crossref]

H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
[Crossref]

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Y. Shao, X. Lu, S. Konijnenberg, C. Zhao, Y. Cai, and H. P. Urbach, “Spatial coherence measurement and partially coherent diffractive imaging using self-referencing holography,” Opt. Express 26, 4479–4490 (2018).
[Crossref]

Y. Yuan, D. Liu, Z. Zhou, H. Xu, J. Qu, and Y. Cai, “Optimization of the probability of orbital angular momentum for Laguerre-Gaussian beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 26, 21861–21871 (2018).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26, 26830–26844 (2018).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review,” Prog. Electromagn. Res. 150, 123–143 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review invited,” J. Opt. Soc. Am. A 31, 2083–2096 (2014).
[Crossref]

F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36, 3281–3283 (2011).
[Crossref]

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboglu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753–1765 (2009).
[Crossref]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
[Crossref]

Cai, Y. J.

Y. T. Zhang, C. L. Ding, L. Z. Pan, and Y. J. Cai, “Laser arrays of partially coherent beams with multi-Gaussian correlation function,” J. Quant. Spectrosc. Radiat. Transfer 218, 1–11 (2018).
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L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C.-T. Liao, J. San Román, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernández-García, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364, eaaw9486 (2019).
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L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
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I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier/Academic, 2007).

Sahu, R.

G. Kulkarni, R. Sahu, O. S. Magaña-loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1 (2017).
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San Román, J.

L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C.-T. Liao, J. San Román, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernández-García, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364, eaaw9486 (2019).
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G. Piquero, M. Santarsiero, R. Martínez-Herrero, J. C. G. de Sande, M. Alonzo, and F. Gori, “Partially coherent sources with radial coherence,” Opt. Lett. 43, 2376–2379 (2018).
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F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1997).
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Seman, J. A.

Senthilkumaran, P.

Shao, Y.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
[Crossref]

Y. Shao, X. Lu, S. Konijnenberg, C. Zhao, Y. Cai, and H. P. Urbach, “Spatial coherence measurement and partially coherent diffractive imaging using self-referencing holography,” Opt. Express 26, 4479–4490 (2018).
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Sharma, A. M.

Silberhorn, C.

Singh, R. K.

Singh, R. P.

Swartzlander, G. A.

B. Perez-Garcia, A. Yepiz, R. I. Hernandez-Aranda, A. Forbes, and G. A. Swartzlander, “Digital generation of partially coherent vortex beams,” Opt. Lett. 41, 3471–3474 (2016).
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D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
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Takeda, M.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
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W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
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H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
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Torres-Ruiz, F. A.

A. Lizana, H. Zhang, A. Turpin, A. V. Eeckhout, F. A. Torres-Ruiz, A. Vargas, C. Ramirez, F. Pi, and J. Campos, “Generation of reconfigurable optical traps for microparticles spatial manipulation through dynamic split lens inspired light structures,” Sci. Rep. 8, 11263 (2018).
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Turpin, A.

A. Lizana, H. Zhang, A. Turpin, A. V. Eeckhout, F. A. Torres-Ruiz, A. Vargas, C. Ramirez, F. Pi, and J. Campos, “Generation of reconfigurable optical traps for microparticles spatial manipulation through dynamic split lens inspired light structures,” Sci. Rep. 8, 11263 (2018).
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Urbach, H. P.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
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Y. Shao, X. Lu, S. Konijnenberg, C. Zhao, Y. Cai, and H. P. Urbach, “Spatial coherence measurement and partially coherent diffractive imaging using self-referencing holography,” Opt. Express 26, 4479–4490 (2018).
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Vargas, A.

A. Lizana, H. Zhang, A. Turpin, A. V. Eeckhout, F. A. Torres-Ruiz, A. Vargas, C. Ramirez, F. Pi, and J. Campos, “Generation of reconfigurable optical traps for microparticles spatial manipulation through dynamic split lens inspired light structures,” Sci. Rep. 8, 11263 (2018).
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Verma, V. B.

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1997).
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G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
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G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
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T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
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G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics (Elsevier B.V., 2010), Vol. 55, pp. 285–341.

Volke-Sepúlveda, K.

Wan, X.

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

Wang, F.

J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44, 4371–4374 (2019).
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J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26, 26830–26844 (2018).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
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F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review,” Prog. Electromagn. Res. 150, 123–143 (2015).
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Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review invited,” J. Opt. Soc. Am. A 31, 2083–2096 (2014).
[Crossref]

F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36, 3281–3283 (2011).
[Crossref]

Wang, H.

H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
[Crossref]

Wang, J.

Wang, L.-G.

L.-G. Wang, L.-Q. Wang, and S.-Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[Crossref]

Wang, L.-Q.

L.-G. Wang, L.-Q. Wang, and S.-Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[Crossref]

Wang, W.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref]

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
[Crossref]

Wang, Y.

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

Wei, D.

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
[Crossref]

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
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L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wu, G.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

Xiao, Q.

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

Xu, H.

Xu, J.

H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
[Crossref]

Xu, W.

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

Yang, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Yang, Z.

Z. Yang, O. S. Magaña-loaiza, M. Mirhosseini, Y. Zhou, B. Gao, L. Gao, S. Mohammad, H. Rafsanjani, G.-l. Long, and R. W. Boyd, “Digital spiral object identification using random light,” Light Sci. Appl. 6, e17013 (2017).
[Crossref]

Yepiz, A.

Yi Chen, T.

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

Yu, J.

Yuan, Y.

Zeng, J.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26, 26830–26844 (2018).
[Crossref]

Zhang, H.

A. Lizana, H. Zhang, A. Turpin, A. V. Eeckhout, F. A. Torres-Ruiz, A. Vargas, C. Ramirez, F. Pi, and J. Campos, “Generation of reconfigurable optical traps for microparticles spatial manipulation through dynamic split lens inspired light structures,” Sci. Rep. 8, 11263 (2018).
[Crossref]

Zhang, L.

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

Zhang, M.

H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
[Crossref]

Zhang, P.

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

Zhang, Q.

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

Zhang, Y. T.

Y. T. Zhang, C. L. Ding, L. Z. Pan, and Y. J. Cai, “Laser arrays of partially coherent beams with multi-Gaussian correlation function,” J. Quant. Spectrosc. Radiat. Transfer 218, 1–11 (2018).
[Crossref]

Zhao, C.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26, 26830–26844 (2018).
[Crossref]

Y. Shao, X. Lu, S. Konijnenberg, C. Zhao, Y. Cai, and H. P. Urbach, “Spatial coherence measurement and partially coherent diffractive imaging using self-referencing holography,” Opt. Express 26, 4479–4490 (2018).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
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C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboglu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753–1765 (2009).
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Zhao, N.

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9, 822–826 (2015).
[Crossref]

Zhou, C.

H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
[Crossref]

Zhou, Y.

Z. Yang, O. S. Magaña-loaiza, M. Mirhosseini, Y. Zhou, B. Gao, L. Gao, S. Mohammad, H. Rafsanjani, G.-l. Long, and R. W. Boyd, “Digital spiral object identification using random light,” Light Sci. Appl. 6, e17013 (2017).
[Crossref]

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

Zhou, Z.

Zhu, S.

Zhu, S.-Y.

L.-G. Wang, L.-Q. Wang, and S.-Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[Crossref]

Zhu, X.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
[Crossref]

J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44, 4371–4374 (2019).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114, 201106 (2019).
[Crossref]

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

J. Mod. Opt. (1)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1997).
[Crossref]

J. Opt. A (1)

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
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J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

Y. T. Zhang, C. L. Ding, L. Z. Pan, and Y. J. Cai, “Laser arrays of partially coherent beams with multi-Gaussian correlation function,” J. Quant. Spectrosc. Radiat. Transfer 218, 1–11 (2018).
[Crossref]

Light Sci. Appl. (3)

X. Wan, Q. Zhang, T. Yi Chen, L. Zhang, W. Xu, H. Huang, C. Kun Xiao, Q. Xiao, and T. J. Cui, “Multichannel direct transmissions of near-field information,” Light Sci. Appl. 8, 60 (2019).
[Crossref]

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
[Crossref]

Z. Yang, O. S. Magaña-loaiza, M. Mirhosseini, Y. Zhou, B. Gao, L. Gao, S. Mohammad, H. Rafsanjani, G.-l. Long, and R. W. Boyd, “Digital spiral object identification using random light,” Light Sci. Appl. 6, e17013 (2017).
[Crossref]

Nanophotonics (2)

H. Wang, L. Liu, C. Zhou, J. Xu, M. Zhang, S. Teng, and Y. Cai, “Vortex beam generation with variable topological charge based on a spiral slit,” Nanophotonics 8, 317–324 (2019).
[Crossref]

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7, 677–682 (2018).
[Crossref]

Nat. Commun. (1)

G. Kulkarni, R. Sahu, O. S. Magaña-loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1 (2017).
[Crossref]

Nat. Photonics (1)

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9, 822–826 (2015).
[Crossref]

Opt. Commun. (4)

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[Crossref]

L.-G. Wang, L.-Q. Wang, and S.-Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
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Y. D. Liu, C. Q. Gao, M. W. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).
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Opt. Express (5)

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J. M. Aunon and M. Nieto-Vesperinas, “Partially coherent fluctuating sources that produce the same optical force as a laser beam,” Opt. Lett. 38, 2869–2872 (2013).
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J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44, 4371–4374 (2019).
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Optica (1)

Photon. Res. (2)

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W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
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W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
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F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review,” Prog. Electromagn. Res. 150, 123–143 (2015).
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O. S. Magaña-Loaiza, M. Mirhosseini, R. M. Cross, S. M. H. Rafsanjani, and R. W. Boyd, “Hanbury Brown and Twiss interferometry with twisted light,” Sci. Adv. 2, e1501143 (2016).
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A. Lizana, H. Zhang, A. Turpin, A. V. Eeckhout, F. A. Torres-Ruiz, A. Vargas, C. Ramirez, F. Pi, and J. Campos, “Generation of reconfigurable optical traps for microparticles spatial manipulation through dynamic split lens inspired light structures,” Sci. Rep. 8, 11263 (2018).
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G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics (Elsevier B.V., 2010), Vol. 55, pp. 285–341.

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of a PCL array at the source with N=3. Relative CV spectrum of the array at (b) z=10zr and (c) z=20zr. The beam parameters are λ=632.8  nm, w=0.157  mm, σ=0.1w, r0=1.5w, and L0=1. Here, zr=πw2/λ=0.122  m is the Rayleigh distance.
Fig. 2.
Fig. 2. Numerical simulation of (a) spatial distribution of the averaged intensity for N=3 and z=0; (b) spatial distribution of the averaged intensity for N=3 and z=20zr; (c) correlation phase of a radial array for N=3 and z=0; and (d) correlation phase of a radial array for N=3 and z=20zr. The other beam parameters are the same as those in Fig. 1.
Fig. 3.
Fig. 3. (a) Experimental setup to generate a CV with a radial array of N fundamental PCL beams. Examples of the CGH displayed on the SLM for the production of the complex field (N=3) under the following conditions: (b) total coherence with c0=0, δp=0; and (c) poor coherence with c0=3w, δp(0,2π). (d) Instantaneous intensity I(k)(x,y) of the PCL array produced with c0=3w, δp(0,2π). (e) Phase distribution of the complex light field E(k)(x,y) produced with the CGH of (c). The array is in the random state of independent perturbation. The beam parameters are w=0.157  mm, r0=1.5w, and L0=1. BE, beam expander (Newport T81-3X); BS, beam splitter; SLM, spatial light modulator; NF, neutral density filter (Absorptive, optical density 1.0); RM, reflecting mirror; P1, pinhole; MO, microscope objective; CMOS, complementary metal oxide semiconductor camera; PC1, PC2, personal computers.
Fig. 4.
Fig. 4. Experimental results for the averaged intensity of the independently perturbed PCL array at different propagation distances: (a) z=0.2  m, (b) z=0.57  m, and (c) z=1.14  m. (d) The averaged intensity of the PCL array focused by the MO with a propagation distance z=1.14  m. The beam parameters are the same as those in Fig. 3(c). In the 2D graphs, the x and y axes are in the unit of pixels, and 1 pixel is 5.2 μm. In the 1D graphs, the vertical axes representing the averaged intensity are normalized. The diameters of the dark centers are marked in the 1D graphs. The camera is a CinCam CMOS-1201 (1280×1024 at 19.8 fps, 5.2 μm).
Fig. 5.
Fig. 5. Experimental results of the PCL array under different perturbation conditions. (a) and (b) are for c0=0, δp=0 and L0=1 with (a) z=0.57  m, (b) z=1.14  m; (c) and (d) are for the case of independent perturbation with c0=3w and L0=1 at (c) z=0.57  m, (d) z=1.14  m; (e) and (f) are for the case of uniform perturbation with c0=3w and L0=1 at (e) z=0.57  m, (f) z=1.14  m; and (g) and (h) are for the independent perturbation with c0=3w and L0=0 at (g) z=0.57  m, (h) z=1.14  m. Other parameters are w=0.157  mm and r0=1.5w. The horizontal and vertical axes representing the x and y axes are scaled in pixels with 1  pixel=5.2  μm. The camera is a CinCam CMOS-1201 (1280×1024 at 19.8 fps).
Fig. 6.
Fig. 6. Computational realizations of (a) the second-order correlation G(2)(x,y,x,y) of the random array; (b) the magnitude of the CCF, |W(x,y,x,y)|; and (c) the Fourier transform of the averaged intensity of the array. (d) Experimental realization of the second-order correlation obtained with a Mach–Zehnder interferometer with two Dove prisms. The parameters of the array are the same as those in Fig. 5(d). In (a) and (b), the shooting camera is a TUCSEN ISH500 (600×800 at 49.6 fps), and 1 pixel is 2.2 μm; while in (d), the shooting camera is a CinCam CMOS-1201 (1280×1024 at 19.8 fps), and 1 pixel is 5.2 μm. (c) is in spatial frequency domain where 1 pixel is 1/(5.2×1280)  μm1.

Equations (17)

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E(x,y;0)=eiδj=1NEj(x,y;0),
Ej(x,y;0)=exp[(xaj)2+(ybj)2w2]exp(iφ0j),
W(ρ1,ρ2;0)=E*(ρ1;0)E(ρ2;0)=s=1Nq=1NWsq(ρ1,ρ2;0),
Wsq(ρ1,ρ2;0)=exp[(x1as)2+(y1bs)2w2]exp[(x2aq)2+(y2bq)2w2]×exp(iφsq)g(ρ1,ρ2;0),
Wsq(r1,r2;z)Wsq(ρ1,ρ2;0)h1*(r1,ρ1)h2(r2,ρ2)d2ρ1d2ρ2,
hp(rp,ρp)=exp[ik2B(Axp22xpup+Dup2)+ik2B(Ayp22ypvp+Dvp2)],
W(r1,r2;z)=s=1Nq=1NWsq(r1,r2;z),
Wsq(r1,r2;z)=π2M1M2(1λ|B|)2exp(iφsq)exp(2r02w2)exp(N22+N124M2)×exp[14M1(2bsw2+ikB*v1)2+14M1(2asw2+ikB*u1)2]×exp[ikD*2B*(u12+v12)+ikD2B(u22+v22)],
M1=1w2+12σ2+ikA*2B*,M2=1w2+12σ214σ4M1ikA2B,N1=1w2(2aq+asσ2M1)+ik(u12σ2M1B*u2B),N2=1w2(2bq+bsσ2M1)+ik(v12σ2M1B*v2B).
P(L)|0002π02πW(r1,r2;z)exp(iL·Δθ)r1r2dr1dr2dθ1dθ2|,
P(L)(12π)|s=1Nq=1NPsq(L)|,
Psq(L)=Q1×exp[i(φsqL·αsq(+))]exp(4r02cos2αsq()w4M2)×n=0[4k2r02cos2αsq()w4M2k(k4iM2z)]n+|L|/2Γ(n+|L|/2+1)Q2×n!Γ(n+|L|+1),
Q1=2π3M1M2λ2z2exp(2r02w02+r02M1w04),Q2=π2M2λ2z2iπλz,
02πexp[inϕ1+ξcos(ϕ1ϕ2)]dϕ1=2πexp(inϕ2)In(ξ),In(ξ)=k=01k!Γ(k+n+1)(ξ2)2k+n,with  Re(n)>1,0x2m+1exp[qx2]dx=12qm+1Γ(m+1),with  Re(q)>0  and  m>1.
E(k)(x,y)=p=1M1[E1(p)(x,y)+E2(p)(x,y)+E3(p)(x,y)]exp(iδp),
uj(p)(x,y)=exp{[xajxj(p)]2+[ybjyj(p)]2w2}.
W(x1,y1,x2,y2)=E(k)*(x1,y1)E(k)(x2,y2)=1M2k=1M2E(k)*(x1,y1)E(k)(x2,y2).