Abstract

We introduce a class of self-steering partially coherent vector optical beams with the aid of a generalized complex Gaussian representation. We show that such partially coherent vector beams have mobile guiding centers of their intensity and polarization state distributions on the beam free space propagation that could be employed to generate far-field polarization arrays. Further, we introduce theoretically and realize experimentally a class of vector beams with inhomogeneous statistical and nontrivial far-field angular distributions, which we term cylindrically correlated partially coherent (CCPC) vector beams. We find that such novel beams possess, in general, cylindrically polarized, far-field patterns of an adjustable degree of polarization. The steering control of the intensity and polarization of the self-steering CCPC vector beam is also demonstrated in experiment. Our findings can find important applications, such as trapping of neutral microparticles and excitation of novel surface waves.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  63. Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
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2019 (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. Photon.  1, 016005 (2019).
[Crossref]

2018 (11)

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent axiconic surface plasmon polariton fields,” Phys. Rev. A 97, 041801 (2018).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Coherence lattices in surface plasmon polariton fields,” Opt. Lett. 43, 3429–3432 (2018).
[Crossref] [PubMed]

H. Mao, Y. Chen, S. A. Ponomarenko, and A. T. Friberg, “Coherent pseudo-mode representation of partially coherent surface plasmon polaritons,” Opt. Lett. 43, 1395–1398 (2018).
[Crossref] [PubMed]

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] [PubMed]

X. Chen, J. Li, S. M. H. Rafsanjani, and O. Korotkova, “Synthesis of Im-Bessel correlated beams via coherent modes,” Opt. Lett. 43, 3590–3593 (2018).
[Crossref] [PubMed]

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97, 053838 (2018).
[Crossref]

R. Chriki, S. Mahler, C. Tradonsky, V. Pal, A. A. Friesem, and N. Davidson, “Spatiotemporal supermodes: Rapid reduction of spatial coherence in highly multimode lasers,” Phys. Rev. A 98, 023812 (2018).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
[Crossref] [PubMed]

R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43, 1627–1630 (2018).
[Crossref] [PubMed]

O. Korotkova and X. Chen, “Phase structuring of the complex degree of coherence,” Opt. Lett. 43, 4727–4730 (2018).
[Crossref] [PubMed]

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
[Crossref] [PubMed]

2017 (10)

T. Stangner, H. Zhang, T. Dahlberg, K. Wiklund, and M. Andersson, “Step-by-step guide to reduce spatial coherence of laser light using a rotating ground glass diffuser,” Appl. Opt. 56, 5427–5435 (2017).
[Crossref] [PubMed]

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

F. Wang, Y. Chen, L. Guo, L. Liu, and Y. Cai, “Complex Gaussian representations of partially coherent beams with nonconventional degrees of coherence,” J. Opt. Soc. Am. A 34, 1824–1829 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
[Crossref]

C. Liang, G. Wu, F. Wang, W. Li, Y. Cai, and S. A. Ponomarenko, “Overcoming the classical Rayleigh diffraction limit by controlling two-point correlations of partially coherent light sources,” Opt. Express 25, 28352–28362 (2017).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep.  7, 39957 (2017).
[Crossref] [PubMed]

C. Ding, M. Koivurova, J. Turunen, T. Setälä, and A. T. Friberg, “Coherence control of pulse trains by spectral phase modulation,” J. Opt.  19, 095501 (2017).
[Crossref]

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett.  111, 101106 (2017).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42, 3279–3282 (2017).
[Crossref] [PubMed]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Partially coherent sources with circular coherence,” Opt. Lett. 42, 1512–1515 (2017).
[Crossref] [PubMed]

2016 (8)

A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polaritons,” Europhys. Lett. 116, 64001 (2016).
[Crossref]

M. W. Hyde, S. R. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl.  6, 064030 (2016).
[Crossref]

S. Knitter, C. Liu, B. Redding, M. K. Khokha, M. A. Choma, and H. Cao, “Coherence switching of a degenerate VECSEL for multimodality imaging,” Optica 3, 403–406 (2016).
[Crossref]

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55, 3442–3446 (2016).
[Crossref] [PubMed]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41, 4182–4185 (2016).
[Crossref] [PubMed]

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

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett.  109, 061107 (2016).
[Crossref]

2015 (4)

2014 (5)

2013 (3)

2012 (1)

2011 (4)

2010 (1)

J. Tervo, J. Turunen, and F. Gori, “Impossibility of Stokes decomposition for a class of light beams,” Opt. Commun. 283, 4448–4451 (2010).
[Crossref]

2009 (4)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
[Crossref]

J. Tervo and J. Turunen, “Comment on “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett.  34, 1001 (2009).
[Crossref]

E. Wolf, “Reply to Comment on ”Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett.  34, 1002 (2009).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11, 085706 (2009).
[Crossref]

2008 (2)

P. Meemon, M. Salem, K.-S. Lee, M. Chopra, and J. P. Rolland, “Determination of the coherency matrix of a broadband stochastic electromagnetic light beam,” J. Mod. Opt. 55, 2765-2776 (2008).
[Crossref]

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008).
[Crossref] [PubMed]

2007 (1)

2003 (1)

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

2002 (1)

1998 (1)

1994 (1)

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Alonzo, M.

Andersson, M.

Arif, M.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
[Crossref] [PubMed]

Auñón, J. M.

Avramov-Zamurovic, S.

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55, 3442–3446 (2016).
[Crossref] [PubMed]

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

Borghi, R.

Bose-Pillai, S. R.

M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett.  111, 101106 (2017).
[Crossref]

M. W. Hyde, S. R. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl.  6, 064030 (2016).
[Crossref]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach(Wiley, 1998).

Cai, 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. Photon.  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]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep.  7, 39957 (2017).
[Crossref] [PubMed]

F. Wang, Y. Chen, L. Guo, L. Liu, and Y. Cai, “Complex Gaussian representations of partially coherent beams with nonconventional degrees of coherence,” J. Opt. Soc. Am. A 34, 1824–1829 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
[Crossref]

C. Liang, G. Wu, F. Wang, W. Li, Y. Cai, and S. A. Ponomarenko, “Overcoming the classical Rayleigh diffraction limit by controlling two-point correlations of partially coherent light sources,” Opt. Express 25, 28352–28362 (2017).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41, 4182–4185 (2016).
[Crossref] [PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett.  109, 061107 (2016).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013823 (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, 013801 (2014).
[Crossref]

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

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38, 1814–1816 (2013).
[Crossref] [PubMed]

Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, Vectorial Optical Fields: Fundamentals and Applications, Q. Zhen, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.

Cao, H.

Chen, S.

Chen, X.

Chen, Y.

H. Mao, Y. Chen, S. A. Ponomarenko, and A. T. Friberg, “Coherent pseudo-mode representation of partially coherent surface plasmon polaritons,” Opt. Lett. 43, 1395–1398 (2018).
[Crossref] [PubMed]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Coherence lattices in surface plasmon polariton fields,” Opt. Lett. 43, 3429–3432 (2018).
[Crossref] [PubMed]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent axiconic surface plasmon polariton fields,” Phys. Rev. A 97, 041801 (2018).
[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, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep.  7, 39957 (2017).
[Crossref] [PubMed]

F. Wang, Y. Chen, L. Guo, L. Liu, and Y. Cai, “Complex Gaussian representations of partially coherent beams with nonconventional degrees of coherence,” J. Opt. Soc. Am. A 34, 1824–1829 (2017).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42, 3279–3282 (2017).
[Crossref] [PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett.  109, 061107 (2016).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013823 (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, 013801 (2014).
[Crossref]

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

Choma, M. A.

Chopra, M.

P. Meemon, M. Salem, K.-S. Lee, M. Chopra, and J. P. Rolland, “Determination of the coherency matrix of a broadband stochastic electromagnetic light beam,” J. Mod. Opt. 55, 2765-2776 (2008).
[Crossref]

Chriki, R.

R. Chriki, S. Mahler, C. Tradonsky, V. Pal, A. A. Friesem, and N. Davidson, “Spatiotemporal supermodes: Rapid reduction of spatial coherence in highly multimode lasers,” Phys. Rev. A 98, 023812 (2018).
[Crossref]

Clark, Charles W.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
[Crossref] [PubMed]

Cory, D. G.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
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Dahlberg, T.

Davidson, N.

R. Chriki, S. Mahler, C. Tradonsky, V. Pal, A. A. Friesem, and N. Davidson, “Spatiotemporal supermodes: Rapid reduction of spatial coherence in highly multimode lasers,” Phys. Rev. A 98, 023812 (2018).
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de Sande, J. C. G.

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29, 256–260 (1979).
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Ding, C.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97, 053838 (2018).
[Crossref]

C. Ding, M. Koivurova, J. Turunen, T. Setälä, and A. T. Friberg, “Coherence control of pulse trains by spectral phase modulation,” J. Opt.  19, 095501 (2017).
[Crossref]

C. Ding, O. Korotkova, and L. Z. Pan, “The control of pulse profiles with tunable temporal coherence,” Phys. Lett. A 378, 1687–1690 (2014).
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Dong, Y.

Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, Vectorial Optical Fields: Fundamentals and Applications, Q. Zhen, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.

Fonseca, E. J. S.

I. Vidal, E. J. S. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84, 033836 (2011).
[Crossref]

Friberg, A. T.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent axiconic surface plasmon polariton fields,” Phys. Rev. A 97, 041801 (2018).
[Crossref]

H. Mao, Y. Chen, S. A. Ponomarenko, and A. T. Friberg, “Coherent pseudo-mode representation of partially coherent surface plasmon polaritons,” Opt. Lett. 43, 1395–1398 (2018).
[Crossref] [PubMed]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Coherence lattices in surface plasmon polariton fields,” Opt. Lett. 43, 3429–3432 (2018).
[Crossref] [PubMed]

C. Ding, M. Koivurova, J. Turunen, T. Setälä, and A. T. Friberg, “Coherence control of pulse trains by spectral phase modulation,” J. Opt.  19, 095501 (2017).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42, 3279–3282 (2017).
[Crossref] [PubMed]

A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polaritons,” Europhys. Lett. 116, 64001 (2016).
[Crossref]

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

A. Norrman, T. Setälä, and A. T. Friberg, “Partial spatial coherence and partial polarization in random evanescent fields on lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).
[Crossref]

Friesem, A. A.

R. Chriki, S. Mahler, C. Tradonsky, V. Pal, A. A. Friesem, and N. Davidson, “Spatiotemporal supermodes: Rapid reduction of spatial coherence in highly multimode lasers,” Phys. Rev. A 98, 023812 (2018).
[Crossref]

Gori, F.

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013823 (2015).
[Crossref]

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P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29, 256–260 (1979).
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Guth, S.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55, 3442–3446 (2016).
[Crossref] [PubMed]

Hickmann, J. M.

I. Vidal, E. J. S. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84, 033836 (2011).
[Crossref]

Hincks, I.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
[Crossref] [PubMed]

Huber, M. G.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
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M. W. Hyde, S. R. Bose-Pillai, and R. A. Wood, “Synthesis of non-uniformly correlated partially coherent sources using a deformable mirror,” Appl. Phys. Lett.  111, 101106 (2017).
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M. W. Hyde, S. R. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl.  6, 064030 (2016).
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Khokha, M. K.

Knitter, S.

Koivurova, M.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97, 053838 (2018).
[Crossref]

C. Ding, M. Koivurova, J. Turunen, T. Setälä, and A. T. Friberg, “Coherence control of pulse trains by spectral phase modulation,” J. Opt.  19, 095501 (2017).
[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. Photon.  1, 016005 (2019).
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X. Chen, J. Li, S. M. H. Rafsanjani, and O. Korotkova, “Synthesis of Im-Bessel correlated beams via coherent modes,” Opt. Lett. 43, 3590–3593 (2018).
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O. Korotkova and X. Chen, “Phase structuring of the complex degree of coherence,” Opt. Lett. 43, 4727–4730 (2018).
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C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55, 3442–3446 (2016).
[Crossref] [PubMed]

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, 013801 (2014).
[Crossref]

C. Ding, O. Korotkova, and L. Z. Pan, “The control of pulse profiles with tunable temporal coherence,” Phys. Lett. A 378, 1687–1690 (2014).
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S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37, 2970–2972 (2012).
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Lajunen, H.

Lee, K.-S.

P. Meemon, M. Salem, K.-S. Lee, M. Chopra, and J. P. Rolland, “Determination of the coherency matrix of a broadband stochastic electromagnetic light beam,” J. Mod. Opt. 55, 2765-2776 (2008).
[Crossref]

Leger, J. R.

Lehtolahti, J.

Li, J.

Li, W.

Liang, C.

Ling, X.

Liu, C.

Liu, L.

F. Wang, Y. Chen, L. Guo, L. Liu, and Y. Cai, “Complex Gaussian representations of partially coherent beams with nonconventional degrees of coherence,” J. Opt. Soc. Am. A 34, 1824–1829 (2017).
[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, 013801 (2014).
[Crossref]

Liu, X.

Liu, Y.

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. Photon.  1, 016005 (2019).
[Crossref]

Luo, H.

Ma, L.

Mahler, S.

R. Chriki, S. Mahler, C. Tradonsky, V. Pal, A. A. Friesem, and N. Davidson, “Spatiotemporal supermodes: Rapid reduction of spatial coherence in highly multimode lasers,” Phys. Rev. A 98, 023812 (2018).
[Crossref]

Malek-Madani, R.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).
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Mandel, L.

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

Martínez-Herrero, R.

Mazumder, J.

W. M. Steen and J. Mazumder, Laser Material Processing(Springer, 2007).

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P. Meemon, M. Salem, K.-S. Lee, M. Chopra, and J. P. Rolland, “Determination of the coherency matrix of a broadband stochastic electromagnetic light beam,” J. Mod. Opt. 55, 2765-2776 (2008).
[Crossref]

Miguel, P.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
[Crossref] [PubMed]

Nelson, C.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell Model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55, 3442–3446 (2016).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

Norrman, A.

Nsofini, J.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
[Crossref] [PubMed]

Pal, V.

R. Chriki, S. Mahler, C. Tradonsky, V. Pal, A. A. Friesem, and N. Davidson, “Spatiotemporal supermodes: Rapid reduction of spatial coherence in highly multimode lasers,” Phys. Rev. A 98, 023812 (2018).
[Crossref]

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Pan, L.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97, 053838 (2018).
[Crossref]

Pan, L. Z.

C. Ding, O. Korotkova, and L. Z. Pan, “The control of pulse profiles with tunable temporal coherence,” Phys. Lett. A 378, 1687–1690 (2014).
[Crossref]

Ping, C.

Piquero, G.

Ponomarenko, S. A.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent axiconic surface plasmon polariton fields,” Phys. Rev. A 97, 041801 (2018).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Coherence lattices in surface plasmon polariton fields,” Opt. Lett. 43, 3429–3432 (2018).
[Crossref] [PubMed]

H. Mao, Y. Chen, S. A. Ponomarenko, and A. T. Friberg, “Coherent pseudo-mode representation of partially coherent surface plasmon polaritons,” Opt. Lett. 43, 1395–1398 (2018).
[Crossref] [PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep.  7, 39957 (2017).
[Crossref] [PubMed]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42, 3279–3282 (2017).
[Crossref] [PubMed]

C. Liang, G. Wu, F. Wang, W. Li, Y. Cai, and S. A. Ponomarenko, “Overcoming the classical Rayleigh diffraction limit by controlling two-point correlations of partially coherent light sources,” Opt. Express 25, 28352–28362 (2017).
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X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41, 4182–4185 (2016).
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A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polaritons,” Europhys. Lett. 116, 64001 (2016).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett.  109, 061107 (2016).
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S. A. Ponomarenko, “Self-imaging of partially coherent light in graded-index media,” Opt. Lett. 40, 566–568 (2015).
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L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23, 1848–1856 (2015).
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L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39, 6656–6659 (2014).
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S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express 19, 17086–17091 (2011).
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Pushin, D. A.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
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Rafsanjani, S. M. H.

Redding, B.

Rolland, J. P.

P. Meemon, M. Salem, K.-S. Lee, M. Chopra, and J. P. Rolland, “Determination of the coherency matrix of a broadband stochastic electromagnetic light beam,” J. Mod. Opt. 55, 2765-2776 (2008).
[Crossref]

Saastamoinen, T.

Sahin, S.

Salem, M.

P. Meemon, M. Salem, K.-S. Lee, M. Chopra, and J. P. Rolland, “Determination of the coherency matrix of a broadband stochastic electromagnetic light beam,” J. Mod. Opt. 55, 2765-2776 (2008).
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F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11, 085706 (2009).
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Sarenac, D.

D. Sarenac, D. G. Cory, J. Nsofini, I. Hincks, P. Miguel, M. Arif, Charles W. Clark, M. G. Huber, and D. A. Pushin, “Generation of a lattice of spin-orbit beams via coherent averaging,” Phys. Rev. Lett.  121, 183602 (2018).
[Crossref] [PubMed]

Setälä, T.

C. Ding, M. Koivurova, J. Turunen, T. Setälä, and A. T. Friberg, “Coherence control of pulse trains by spectral phase modulation,” J. Opt.  19, 095501 (2017).
[Crossref]

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

A. Norrman, T. Setälä, and A. T. Friberg, “Partial spatial coherence and partial polarization in random evanescent fields on lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).
[Crossref]

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. Photon.  1, 016005 (2019).
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Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11, 085706 (2009).
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Steen, W. M.

W. M. Steen and J. Mazumder, Laser Material Processing(Springer, 2007).

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. Photon.  1, 016005 (2019).
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J. Lehtolahti, M. Kuittinen, J. Turunen, and J. Tervo, “Coherence modulation by deterministic rotating diffusers,” Opt. Express 23, 10453–10466 (2015).
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J. Tervo, J. Turunen, and F. Gori, “Impossibility of Stokes decomposition for a class of light beams,” Opt. Commun. 283, 4448–4451 (2010).
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R. Chriki, S. Mahler, C. Tradonsky, V. Pal, A. A. Friesem, and N. Davidson, “Spatiotemporal supermodes: Rapid reduction of spatial coherence in highly multimode lasers,” Phys. Rev. A 98, 023812 (2018).
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Turunen, J.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97, 053838 (2018).
[Crossref]

C. Ding, M. Koivurova, J. Turunen, T. Setälä, and A. T. Friberg, “Coherence control of pulse trains by spectral phase modulation,” J. Opt.  19, 095501 (2017).
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J. Lehtolahti, M. Kuittinen, J. Turunen, and J. Tervo, “Coherence modulation by deterministic rotating diffusers,” Opt. Express 23, 10453–10466 (2015).
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J. Tervo, J. Turunen, and F. Gori, “Impossibility of Stokes decomposition for a class of light beams,” Opt. Commun. 283, 4448–4451 (2010).
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J. Tervo and J. Turunen, “Comment on “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett.  34, 1001 (2009).
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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. Photon.  1, 016005 (2019).
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Vidal, I.

I. Vidal, E. J. S. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84, 033836 (2011).
[Crossref]

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M. W. Hyde, S. R. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl.  6, 064030 (2016).
[Crossref]

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F. Wang, Y. Chen, L. Guo, L. Liu, and Y. Cai, “Complex Gaussian representations of partially coherent beams with nonconventional degrees of coherence,” J. Opt. Soc. Am. A 34, 1824–1829 (2017).
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C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
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C. Liang, G. Wu, F. Wang, W. Li, Y. Cai, and S. A. Ponomarenko, “Overcoming the classical Rayleigh diffraction limit by controlling two-point correlations of partially coherent light sources,” Opt. Express 25, 28352–28362 (2017).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013823 (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, 013801 (2014).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,”. J. Opt. Soc. Am. A 31, 2083–2096 (2014).
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F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38, 1814–1816 (2013).
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Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
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Figures (8)

Fig. 1
Fig. 1 Calculated intensity distribution S0(R, Z) and degree of polarization P(R, Z) of a CCPC vector beam propagating in free space. (a) The intensity distribution in the XZ plane (Y = 0); (b), (c), and (d) The intensity distribution in the XY plane at propagation distances Z = 0, Z = 0.5, and Z = 1, respectively; (e) The distribution of the degree of polarization in the XZ plane (Y = 0); (f), (g), and (h) The distribution of the degree of polarization in the XY plane at propagation distances Z = 0, Z = 0.5, and Z = 1, respectively. The effective coherence length of the CCPC vector beam is ξc = 0.2 and the normalization factor is C0 = 1. The displayed intensities are normalized with respect to their maximum values.
Fig. 2
Fig. 2 Distribution of the CCPC vector beam polarization states at the distance Z = 1 with (a) θ = 0; (b) θ = π/4; (c) θ = −π/4; (d) θ = π/2. The effective spatial coherence length at the source is ξc = 0.2 and the normalization factor is C0 = 1. The blue lines in the figure denote linear polarization states.
Fig. 3
Fig. 3 Polarization purity ηp(Z) for a CCPC vector beam propagating in free space as a function of the effective coherence length ξc at the source and propagation distance Z.
Fig. 4
Fig. 4 Schematic of the experimental setup for generating and measuring the cylindrically correlated partially coherent vector beam. The laser source used in our experiment is a monochromatic He-Ne laser of wavelength being 633 nm. BE: beam expander; NDF: neutral density filter; LP: linear polarizer; RPC: radial-polarization converter; HH: double half-wave plates (HWPs) system as shown in the inset (b); L1, L2, L3: thin lenses; RGGD: rotating ground-glass disk; AF: Gaussian amplitude filter; BPA: beam profile analyzer. Inset (a): photograph of the LP (left) and RPC (right) that can convert a linearly polarized beam into a radially polarized beam. Inset (b): converting a radially polarized beam to a generalized cylindrically polarized beam by a double half-wave plates system. As shown in the inset φ is the angle between the fast axes of the two HWPs, while θ is the clockwise rotation angle of the linear polarization in the generalized cylindrical polarization distribution with respect to that in the radial polarization distribution. Here the relation between θ and φ is given by θ = 2φ. Inset (c): photograph of the RGGD that is driven by the controller of a optical chopper system.
Fig. 5
Fig. 5 Experimental intensity distribution of a CCPC vector beam transmitted through a thin lens with the focal distance f3 = 400 mm at several propagation distances. The source transverse coherence length and beam width are σc = 0.2 mm and σI = 1 mm, respectively. The beam wavelength is 633 nm.
Fig. 6
Fig. 6 Experimental results for the total intensities (top panel), the x-polarized (middle panel), and y-polarized (bottom panel) component intensities of CCPC vector beams with four different polarization distributions in the focal plane. The focal distance of the lens is 400 mm. The source coherence length and beam width are 0.2 mm and 1 mm, respectively. The beam wavelength is 633 nm. The white arrows in the figure denote the transmission direction of the linear polarizer.
Fig. 7
Fig. 7 Schematic of the experimental setup for generating the self-steering CCPC vector source from a focused off-axis cylindrically polarized beam. Here ρ0 is the off-axis displacement of the cylindrically polarized beam on the rotating ground-glass disk (RGGD). L and AF in the figure denote the thin lenses and Gaussian amplitude filter, respectively.
Fig. 8
Fig. 8 Experimental results for the total intensity (top panel), the x-polarized (middle panel), and y-polarized (bottom panel) component intensities of a self-steering CCPC vector beam transmitted through a thin lens with focal distance f = 400 mm at several propagation distances. The source coherence and beam width are 0.2 mm and 1 mm, respectively. The phase shift for the source is v0 = (10, 10). The beam wavelength is 633 nm. The white arrows in the figure denote the transmission direction of the linear polarizer.

Equations (52)

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W ( r 1 , r 2 , 0 ) = [ W x x ( r 1 , r 2 , 0 ) W x y ( r 1 , r 2 , 0 ) W y x ( r 1 , r 2 , 0 ) W y y ( r 1 , r 2 , 0 ) ] ,
W m n ( r 1 , r 2 , 0 ) = E m * ( r 1 ) E n ( r 2 ) ,
W ( R 1 , R 2 , 0 ) = d 4 α P ( α ) Ψ α * ( R 1 , 0 ) Ψ α ( R 2 , 0 ) .
P ( α ) = [ P x x ( α ) P x y ( α ) P y x ( α ) P y y ( α ) ] ,
Ψ α ( R , 0 ) = e v 2 / 2 π exp [ ( R 2 α ) 2 2 ] .
d 2 R Ψ α * ( R , 0 ) Ψ α ( R , 0 ) = 1 ,
d 4 α Ψ α * ( R 1 , 0 ) Ψ α ( R 2 , 0 ) = δ ( R 1 R 2 ) ,
W ( R 1 , R 2 , Z ) = d 4 α P ( α ) Ψ α * ( R 1 , Z ) Ψ α ( R 2 , Z ) ,
Ψ α ( R , Z ) = σ I 2 λ | B | d 2 R Ψ α ( R , 0 ) e i k σ I 2 2 B ( A R 2 2 R R + D R 2 )
Ψ α ( R , Z ) = e v 2 / 2 π i B Z A B | B | exp { 1 2 Z A B [ 2 A α 2 2 2 α R + Z A B D 1 i Z R 2 ] } ,
P ( s ) ( α ) = P ( α α 0 ) ,
W ( s ) ( R 1 , R 2 , 0 ) = W ( R 1 u 0 , R 2 u 0 , 0 ) e i ( R 1 R 2 ) v 0 ,
W ( s ) ( R 1 , R 2 , Z ) = W ( R 1 D 0 , R 2 D 0 , Z ) e i ( R 1 R 2 ) · P 0 ,
D 0 = A u 0 + Z v 0 ,
P 0 = A D 1 Z u 0 + D v 0 .
D 0 = u 0 + Z v 0 ,
P 0 = v 0 ,
S j ( R , Z ) = t r [ Φ ( R , Z ) σ j ] , j ( 0 , , 3 ) ,
S j ( s ) ( R , Z ) = S j ( R D 0 , Z ) , j ( 0 , , 3 ) ,
P ( α ) = δ ( u ) exp ( ξ c 2 v 2 2 ) [ a 2 ( v ) a ( v ) b ( v ) a ( v ) b ( v ) b 2 ( v ) ] ,
[ a ( v ) b ( v ) ] = M ( θ ) [ v x v y ] ,
S 0 ( R , Z ) = 2 C 0 exp ( R 2 S 0 2 ) ( 1 S 0 2 2 Z 2 ξ c 2 S 0 4 + 2 Z 2 R 2 ξ c 2 S 0 6 ) ,
S 1 ( R , Z ) = 4 C 0 Z 2 ξ c 2 S 0 6 exp ( R 2 S 0 2 ) ( X 2 cos 2 θ Y 2 cos 2 θ + 2 X Y sin 2 θ ) ,
S 2 ( R , Z ) = 4 C 0 Z 2 ξ c 2 S 0 6 exp ( R 2 S 0 2 ) ( X 2 sin 2 θ + Y 2 sin 2 θ + 2 X Y cos 2 θ ) ,
S 3 ( R , Z ) = 0 .
S 0 ( R , Z ) = S 0 p ( R , Z ) + S 0 u ( R , Z ) ,
P ( R , Z ) = S 0 p ( R , Z ) S 0 ( R , Z ) ,
S 0 p ( R , Z ) = S 1 2 ( R , Z ) + S 2 2 ( R , Z ) + S 3 2 ( R , Z ) .
S 0 p ( R , Z ) = 4 C 0 Z 2 R 2 ξ c 2 S 0 6 exp ( R 2 S 0 2 ) ,
S 0 u ( R , Z ) = 2 C 0 ( ξ c 2 S 0 2 2 Z 2 ) ξ c 2 S 0 4 exp ( R 2 S 0 2 ) ,
P ( R , Z ) = 2 Z 2 R 2 ξ c 2 S 0 4 2 Z 2 S 0 2 + 2 Z 2 R 2 .
S 1 ( R , Z ) S 2 ( R , Z ) = X 2 cos 2 θ Y 2 cos 2 θ + 2 X Y sin 2 θ X 2 sin 2 θ + Y 2 sin 2 θ + 2 X Y cos 2 θ ,
η p ( Z ) = d 2 R S 0 p ( R , Z ) d 2 R S 0 ( R , Z ) .
J ( φ ) = [ cos ( 2 φ ) sin ( 2 φ ) sin ( 2 φ ) cos ( 2 φ ) ] .
E ( v ) = E 0 ( v ) [ cos θ e ^ r + sin θ e ^ ϕ ] ,
W ( r 1 , r 2 , ω ) = Φ ( v ) H * ( r 1 , v ) H ( r 2 , v ) d 2 v ,
H ( r , v ) = i λ f 2 T ( r ) exp [ i π λ f 2 ( v 2 2 r v ) ]
v 0 = 2 π σ I ρ 0 / ( λ f 2 ) .
W x x ( R 1 , R 2 , 0 ) = C 0 [ 1 1 ξ c 2 a 2 ( | R 1 R 2 | ) ] exp [ ( R 1 R 2 ) 2 2 ξ c 2 ] exp ( R 1 2 + R 2 2 2 ) ,
W y y ( R 1 , R 2 , 0 ) = C 0 [ 1 1 ξ c 2 b 2 ( | R 1 R 2 | ) ] exp [ ( R 1 R 2 ) 2 2 ξ c 2 ] exp ( R 1 2 + R 2 2 2 ) ,
W x y ( R 1 , R 2 , 0 ) = C 0 1 ξ c 2 a ( | R 1 R 2 | ) b ( | R 1 R 2 | ) exp [ ( R 1 R 2 ) 2 2 ξ c 2 ] exp ( R 1 2 + R 2 2 2 ) ,
W y x ( R 1 , R 2 , 0 ) = W x y * ( R 2 , R 1 , 0 ) ,
a ( | R 1 R 2 | ) = | X 1 X 2 | cos θ + | Y 1 Y 2 | sin θ ,
b ( | R 1 R 2 | ) = | X 1 X 2 | sin θ + | Y 1 Y 2 | cos θ .
W x x ( R 1 , R 2 , Z ) = C 0 S 0 2 S ( R 1 , R 2 , Z ) { 1 + 2 B 2 k 2 σ c 2 S 0 2 σ I 2 [ σ I 2 2 S 0 2 ( T x x cos θ + T y y sin θ ) 2 1 ] } ,
W y y ( R 1 , R 2 , Z ) = C 0 S 0 2 S ( R 1 , R 2 , Z ) { 1 + 2 B 2 k 2 σ c 2 S 0 2 σ I 2 [ σ I 2 2 S 0 2 ( T x x sin θ + T y y cos θ ) 2 1 ] } ,
W x y ( R 1 , R 2 , Z ) = C 0 B 2 S 0 6 k 2 σ c 2 S ( R 1 , R 2 , Z ) ( T x x cos θ + T y y sin θ ) ( T x x sin θ + T y y cos θ ) ,
W y x ( R 1 , R 2 , Z ) = W x y * ( R 2 , R 1 , Z ) ,
S ( R 1 , R 2 , Z ) = exp [ i ( S 0 2 D A ) 2 Z S 0 2 ( R 2 2 R 1 2 ) 1 4 S 0 2 ( R 1 + R 2 ) 2 ξ c 2 + 2 4 ξ c 2 S 0 2 ( R 1 R 2 ) 2 ] ,
S 0 2 = A 2 + Z 2 + 2 Z 2 ξ c 2 ,
T x x = 1 σ I [ ( X 1 + X 2 ) + i A Z ( X 2 X 1 ) ] ,
T y y = 1 σ I [ ( Y 1 + Y 2 ) + i A Z ( Y 2 Y 1 ) ] .

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