Abstract

Partially coherent pseudo-Schell model sources are introduced and analyzed. They present radial symmetry and coherence characteristics depending on the difference between the radial distances of two points from the source center. As a consequence, all points belonging to circles centered on the symmetry center of the source are perfectly correlated. We show that such sources radiate fields with peculiar behaviors in paraxial propagation. In particular, when compared to beams produced by Gaussian Schell-model sources with comparable coherence parameters, their irradiance can present sharper profiles and higher peak valuesmono and a better beam quality parameter. Furthermore, when a pseudo-Schell model source presents a vortex, the propagated beam preserves a null of the intensity along its axis.

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

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References

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2018 (3)

2017 (6)

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. 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]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Synthesis of circularly coherent sources,” Opt. Lett. 42, 4115–4118 (2017).
[Crossref]

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

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Self-focusing of a partially coherent beam with circular coherence,” J. Opt. Soc. Am. A 34, 1441–1447 (2017).
[Crossref]

Y. Chen, S.A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Scientific Reports 739957 (2017).
[Crossref] [PubMed]

2016 (1)

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

2015 (2)

J. C. G. de Sande, M. Santarsiero, G. Piquero, and F. Gori, “The subtraction of mutually displaced Gaussian Schell-model beams,” J. Opt. 17, 125613 (2015).
[Crossref]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23, 1848–1856 (2015).
[Crossref]

2014 (5)

2013 (3)

2012 (3)

2011 (3)

2010 (2)

S. B. Raghunathan, T. van Dijk, E. J. G. Peterman, and T. D. Visser, “Experimental demonstration of an intensity minimum at the focus of a laser beam created by spatial coherence: application to the optical trapping of dielectric particles,” Opt. Lett. 35, 4166–4168 (2010).
[Crossref]

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref]

2009 (2)

2008 (5)

2007 (2)

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref]

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Physics Letters A 369, 157–166 (2007).
[Crossref]

2004 (3)

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]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

J. Pu, S. Nemoto, and X. Liu, “Beam shaping of focused partially coherent beams by use of the spatial coherence effect,” Appl. Opt. 43, 5281–5286 (2004).
[Crossref]

2003 (2)

2002 (1)

1999 (1)

1998 (1)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

1994 (1)

G. Piquero, P. M. Mejías, and R. Martínez-Herrero, “Quality changes of Gaussian beams propagating through axicons,” Opt. Commun. 105, 289–291 (1994).
[Crossref]

1992 (3)

1991 (1)

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

1984 (1)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

1982 (3)

1980 (1)

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

1979 (2)

R. Martínez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento B 54, 205–210 (1979).

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

1978 (1)

Alonzo, M.

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[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]

Boyd, R. W.

Brychkov, I.

A. Prudnikov, I. Brychkov, and O. Marichev, Integrals and Series: Special functions (Gordon and Breach Science Publishers, 1986).

Cai, Y.

D. Wu, F. Wang, and Y. Cai, “High-order nonuniformly correlated beams,” Opt. Laser Technol. 99, 230 – 237 (2018).
[Crossref]

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

Y. Chen, S.A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Scientific Reports 739957 (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. 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]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39, 769–772 (2014).
[Crossref]

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36, 1939–1941 (2011).
[Crossref] [PubMed]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre–Gaussian beam,” Opt. Lett. 36, 2251–2253 (2011).
[Crossref] [PubMed]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33, 1795–1797 (2008).
[Crossref] [PubMed]

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Physics Letters A 369, 157–166 (2007).
[Crossref]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
[Crossref]

Chen, Y.

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

Y. Chen, S.A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Scientific Reports 739957 (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. 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]

Cincotti, G.

Collett, E.

Davidson, F. M.

de Sande, J. C. G.

Ding, C.

Feng, G.

W. Li, G. Feng, and S. Zhou, “The M2 factor matrix of a paraxial Laguerre-Gaussian beam,” Journal of Modern Optics 60, 704–712 (2013).
[Crossref]

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref]

Friberg, A. T.

E. Tervonen, J. Turunen, and A. T. Friberg, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Gbur, G.

Gori, F.

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]

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]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Synthesis of circularly coherent sources,” Opt. Lett. 42, 4115–4118 (2017).
[Crossref]

J. C. G. de Sande, M. Santarsiero, G. Piquero, and F. Gori, “The subtraction of mutually displaced Gaussian Schell-model beams,” J. Opt. 17, 125613 (2015).
[Crossref]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39, 1713–1716 (2014).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and C.-F. Li, “Partially correlated thin annular sources: the scalar case,” J. Opt. Soc. Am. A 25, 2826–2832 (2008).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref]

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

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

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

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

Hyde, M. W.

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]

Khonina, S. N.

A. V. Ustinov and S. N. Khonina, “Calculating the complex transmission function of refractive axicons,” Opt. Mem. Neural Networks 21, 133–144 (2012).
[Crossref]

Koivurova, M.

Korotkova, O.

Lajunen, H.

Li, C.-F.

Li, W.

W. Li, G. Feng, and S. Zhou, “The M2 factor matrix of a paraxial Laguerre-Gaussian beam,” Journal of Modern Optics 60, 704–712 (2013).
[Crossref]

Liang, C.

Lin, Q.

Liu, L.

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

Liu, X.

Lu, X.

Lü, X.

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Physics Letters A 369, 157–166 (2007).
[Crossref]

Ma, L.

Magaña-Loaiza, O. S.

Maleev, I. D.

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]

Maluenda, D.

Mandel, L.

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

Mao, Y.

Marathay, A. S.

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]

Marichev, O.

A. Prudnikov, I. Brychkov, and O. Marichev, Integrals and Series: Special functions (Gordon and Breach Science Publishers, 1986).

Martínez-Herrero, R.

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]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Synthesis of circularly coherent sources,” Opt. Lett. 42, 4115–4118 (2017).
[Crossref]

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]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, “Elementary-field expansions of genuine cross-spectral density matrices,” Opt. Lett. 34, 2303–2305 (2009).
[Crossref]

G. Piquero, P. M. Mejías, and R. Martínez-Herrero, “Quality changes of Gaussian beams propagating through axicons,” Opt. Commun. 105, 289–291 (1994).
[Crossref]

J. Serna, P. Mejías, and R. Martínez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1651 (1992).
[Crossref] [PubMed]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. A 1, 556–558 (1984).
[Crossref]

R. Martínez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento B 54, 205–210 (1979).

R. Martínez-Herrero, D. Maluenda, G. Piquero, and J. C. G. de Sande, “Vortex pseudo Schell-model source: A proposal,” in “2016 15th Workshop on Information Optics (WIO),” (2016), pp. 1–3.

Mei, Z.

Mejías, P.

J. Serna, P. Mejías, and R. Martínez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
[Crossref]

Mejías, P. M.

Mirhosseini, M.

Nemoto, S.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

Palacios, D. M.

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]

Palma, C.

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

Pan, L.

Peterman, E. J. G.

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

Piquero, G.

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]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Synthesis of circularly coherent sources,” Opt. Lett. 42, 4115–4118 (2017).
[Crossref]

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]

J. C. G. de Sande, M. Santarsiero, G. Piquero, and F. Gori, “The subtraction of mutually displaced Gaussian Schell-model beams,” J. Opt. 17, 125613 (2015).
[Crossref]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39, 1713–1716 (2014).
[Crossref] [PubMed]

G. Piquero, P. M. Mejías, and R. Martínez-Herrero, “Quality changes of Gaussian beams propagating through axicons,” Opt. Commun. 105, 289–291 (1994).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1651 (1992).
[Crossref] [PubMed]

R. Martínez-Herrero, D. Maluenda, G. Piquero, and J. C. G. de Sande, “Vortex pseudo Schell-model source: A proposal,” in “2016 15th Workshop on Information Optics (WIO),” (2016), pp. 1–3.

Ponomarenko, S. A.

Ponomarenko, S.A.

Y. Chen, S.A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Scientific Reports 739957 (2017).
[Crossref] [PubMed]

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

Prudnikov, A.

A. Prudnikov, I. Brychkov, and O. Marichev, Integrals and Series: Special functions (Gordon and Breach Science Publishers, 1986).

Pu, J.

Raghunathan, S. B.

Ramírez-Sánchez, V.

Ricklin, J. C.

Rodenburg, B.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

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]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Synthesis of circularly coherent sources,” Opt. Lett. 42, 4115–4118 (2017).
[Crossref]

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]

J. C. G. de Sande, M. Santarsiero, G. Piquero, and F. Gori, “The subtraction of mutually displaced Gaussian Schell-model beams,” J. Opt. 17, 125613 (2015).
[Crossref]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39, 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and C.-F. Li, “Partially correlated thin annular sources: the scalar case,” J. Opt. Soc. Am. A 25, 2826–2832 (2008).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref]

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

Santis, P. D.

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

Serna, J.

J. Serna, P. Mejías, and R. Martínez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
[Crossref]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Shchepakina, E.

Siegman, A. E.

A. E. Siegman, Lasers(University Science Books, 1986).

Singh, M.

Starikov, A.

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Swartzlander, G. A.

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]

Tervo, J.

Tervonen, E.

Turunen, J.

Ustinov, A. V.

A. V. Ustinov and S. N. Khonina, “Calculating the complex transmission function of refractive axicons,” Opt. Mem. Neural Networks 21, 133–144 (2012).
[Crossref]

Vahimaa, P.

van Dijk, T.

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

Visser, T. D.

Wang, F.

Wang, L.-G.

L.-G. Wang and L.-Q. Wang, “Hollow Gaussian Schell-model beam and its propagation,” Opt. Commun. 281, 1337 – 1342 (2008).
[Crossref]

Wang, L.-Q.

L.-G. Wang and L.-Q. Wang, “Hollow Gaussian Schell-model beam and its propagation,” Opt. Commun. 281, 1337 – 1342 (2008).
[Crossref]

Wang, Y.

Wolf, E.

Wood, R. A.

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]

Wu, D.

D. Wu, F. Wang, and Y. Cai, “High-order nonuniformly correlated beams,” Opt. Laser Technol. 99, 230 – 237 (2018).
[Crossref]

Wu, G.

Yu, J.

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

Zhao, C.

Zhou, S.

W. Li, G. Feng, and S. Zhou, “The M2 factor matrix of a paraxial Laguerre-Gaussian beam,” Journal of Modern Optics 60, 704–712 (2013).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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

J. Mod. Opt. (2)

J. Serna, P. Mejías, and R. Martínez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

J. Opt. (1)

J. C. G. de Sande, M. Santarsiero, G. Piquero, and F. Gori, “The subtraction of mutually displaced Gaussian Schell-model beams,” J. Opt. 17, 125613 (2015).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (13)

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[Crossref]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and C.-F. Li, “Partially correlated thin annular sources: the scalar case,” J. Opt. Soc. Am. A 25, 2826–2832 (2008).
[Crossref]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
[Crossref]

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
[Crossref]

T. van Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008).
[Crossref]

M. Singh, J. Tervo, and J. Turunen, “Elementary-field analysis of partially coherent beam shaping,” J. Opt. Soc. Am. A 30, 2611–2617 (2013).
[Crossref]

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Self-focusing of a partially coherent beam with circular coherence,” J. Opt. Soc. Am. A 34, 1441–1447 (2017).
[Crossref]

E. Tervonen, J. Turunen, and A. T. Friberg, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. A 1, 556–558 (1984).
[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]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29, 2159–2164 (2012).
[Crossref]

J. Opt. Soc. Am. B (1)

Journal of Modern Optics (1)

W. Li, G. Feng, and S. Zhou, “The M2 factor matrix of a paraxial Laguerre-Gaussian beam,” Journal of Modern Optics 60, 704–712 (2013).
[Crossref]

Nuovo Cimento B (1)

R. Martínez-Herrero, “Expansion of complex degree of coherence,” Nuovo Cimento B 54, 205–210 (1979).

Opt. Commun. (7)

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

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

L.-G. Wang and L.-Q. Wang, “Hollow Gaussian Schell-model beam and its propagation,” Opt. Commun. 281, 1337 – 1342 (2008).
[Crossref]

G. Piquero, P. M. Mejías, and R. Martínez-Herrero, “Quality changes of Gaussian beams propagating through axicons,” Opt. Commun. 105, 289–291 (1994).
[Crossref]

Opt. Eng. (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

D. Wu, F. Wang, and Y. Cai, “High-order nonuniformly correlated beams,” Opt. Laser Technol. 99, 230 – 237 (2018).
[Crossref]

Opt. Lett. (19)

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]

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]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Synthesis of circularly coherent sources,” Opt. Lett. 42, 4115–4118 (2017).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1651 (1992).
[Crossref] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “Elementary-field expansions of genuine cross-spectral density matrices,” Opt. Lett. 34, 2303–2305 (2009).
[Crossref]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
[Crossref]

S. B. Raghunathan, T. van Dijk, E. J. G. Peterman, and T. D. Visser, “Experimental demonstration of an intensity minimum at the focus of a laser beam created by spatial coherence: application to the optical trapping of dielectric particles,” Opt. Lett. 35, 4166–4168 (2010).
[Crossref]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre–Gaussian beam,” Opt. Lett. 36, 2251–2253 (2011).
[Crossref] [PubMed]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33, 1795–1797 (2008).
[Crossref] [PubMed]

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36, 1939–1941 (2011).
[Crossref] [PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36, 4104–4106 (2011).
[Crossref]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37, 2970–2972 (2012).
[Crossref] [PubMed]

G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28, 1627–1629 (2003).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38, 2578–2580 (2013).
[Crossref]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39, 1713–1716 (2014).
[Crossref] [PubMed]

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39, 769–772 (2014).
[Crossref]

Opt. Mem. Neural Networks (1)

A. V. Ustinov and S. N. Khonina, “Calculating the complex transmission function of refractive axicons,” Opt. Mem. Neural Networks 21, 133–144 (2012).
[Crossref]

Phys. Rev. Lett. (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]

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref]

Physics Letters A (1)

X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Physics Letters A 369, 157–166 (2007).
[Crossref]

Prog. Opt. (1)

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

Scientific Reports (1)

Y. Chen, S.A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Scientific Reports 739957 (2017).
[Crossref] [PubMed]

Other (5)

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

R. Martínez-Herrero, D. Maluenda, G. Piquero, and J. C. G. de Sande, “Vortex pseudo Schell-model source: A proposal,” in “2016 15th Workshop on Information Optics (WIO),” (2016), pp. 1–3.

M. Abramowitz and I. Stegun, eds., Handbook of mathematical functions (Dover Publications Inc, 1972).

A. E. Siegman, Lasers(University Science Books, 1986).

A. Prudnikov, I. Brychkov, and O. Marichev, Integrals and Series: Special functions (Gordon and Breach Science Publishers, 1986).

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Figures (13)

Fig. 1
Fig. 1 Absolute value of the spectral degree of coherence along a radius and relative to a point at different distances ρ2 from the source center.
Fig. 2
Fig. 2 Absolute value of the spectral degree of coherence relative to a point at distance ρ 2 = 0 (left), ρ 2 = δ c (center) and ρ 2 = 2 δ c (right), from the source center.
Fig. 3
Fig. 3 (a) Evolution of rms beam width along propagation distance and (b) quality factor for different values of m and δ c = w 0 / 2. For comparison purposes the same quantities for the corresponding GSM or VGSM beams are also shown (dashed linesand squares).
Fig. 4
Fig. 4 Irradiance profile of a GPSM at z = 0, z = 0.1 z R, z = 0.4 z R, and z = 0.7 z R, from left to right for a coherence parameter δ c = 0.2 w 0.
Fig. 5
Fig. 5 (a) Normalized irradiance profile at different propagation distances for δ c = w 0 / 2 and (b) evolution of the FWHM of the beam with propagation distance for several values of the coherence parameter for a GPSM source.
Fig. 6
Fig. 6 Maximum irradiance of a GPSM beam during propagation for several values of the coherence parameter.
Fig. 7
Fig. 7 Evolution of the irradiance at several distances for the case of a VGPSM source with topological charge m = 1 and coherence parameter δ c = w 0 / 8.
Fig. 8
Fig. 8 Irradiance profile at several distances for the case of a VGPSM source with topological charge m = 1 and (a) δ c = w 0 / 4; (b) δ c = w 0 / 8.
Fig. 9
Fig. 9 Evolution of the FWHM of the irradiance profile of the beam radiated from a VGPSM source with topological charge m = 1, as a function of the propagation distance for several values of the coherence parameter δc.
Fig. 10
Fig. 10 Evolution with propagation distance of the maximum irradiance at each transverse plane, for the case of a VGPSM source with topological charge m = 1 and several values of the coherence parameter δc.
Fig. 11
Fig. 11 Coefficients of the pseudo-modal expansion for the GPSM source as a function of the coherence parameter δc.
Fig. 12
Fig. 12 Coefficients of the pseudo-modal expansion for the VGPSM source as a function of the coherence parameter δc.
Fig. 13
Fig. 13 Coefficients of the pseudo-modal expansion for (a) the GPSM and (b) VGPSM source for different values of the coherence parameter δc.

Equations (28)

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W ( ρ 1 , ρ 2 , 0 ) = τ * ( ρ 1 ) τ ( ρ 2 ) A * ( ρ 1 , υ ) A ( ρ 2 , υ ) d υ
A ( ρ , υ ) = H ( υ ) exp  ( i k ρ υ cos  ψ ) ,
W ( ρ 1 , ρ 2 , 0 ) = τ * ( ρ 1 ) τ ( ρ 2 ) g ( ρ 2 ρ 1 ) ,
g ( ρ ) = 2 π 0 | H ( υ ) | 2 J 0 ( k ρ υ ) υ d υ ,
μ ( ρ 1 , ρ 2 , 0 ) = W ( ρ 1 , ρ 2 , 0 ) W ( ρ 1 , ρ 1 , 0 ) W ( ρ 2 , ρ 2 , 0 ) } = τ * ( ρ 1 ) τ ( ρ 2 ) | τ ( ρ 1 ) τ ( ρ 2 ) | g ( ρ 2 ρ 1 ) ,
H ( υ ) = k δ c 2 π exp  ( 1 8 k 2 δ c 2 υ 2 ) ,
| μ ( ρ 1 , ρ 2 , 0 ) | = | g ( ρ 2 ρ 1 ) | = exp  [ ( ρ 2 ρ 1 ) 2 δ c 2 ] .
τ ( ρ ) = A 0 ( 2 ρ w 0 ) m exp  ( ρ 2 w 0 2 ) exp  ( i m φ ) ,
W m ( ρ 1 , ρ 2 , 0 ) = I 0 ( 2 ρ 1 ρ 2 w 0 2 ) m exp  ( ρ 1 2 + ρ 2 2 w 0 2 ) exp  [ ( ρ 2 ρ 1 ) 2 δ c 2 ] exp  [ i m ( φ 2 φ 1 ) ] ,
W m ( s ) ( ρ 1 , ρ 2 , 0 ) = I 0 ( 2 ρ 1 ρ 2 w 0 2 ) m exp  ( ρ 1 2 + ρ 2 2 w 0 2 ) exp  [ ( ρ 2 ρ 1 ) 2 δ c 2 ] exp  [ i m ( φ 2 φ 1 ) ] ,
W ( r 1 , r 2 , z ) = k 2 4 π 2 z 2 W ( ρ 1 , ρ 2 , 0 ) exp  [ i k 2 z ( | r 1 ρ 1 | 2 | r 2 ρ 2 | 2 ) ] d ρ 1 d ρ 2 ,
W ( r 1 , r 2 , z ) = 2 π I 0 λ 2 z 2 exp  [ i k 2 z ( r 2   2 r 1   2 ) ] exp  [ i m ( θ 2 θ 1 ) ] × 0 0 ( 2 ρ 1 ρ 2 w 0 2 ) m exp  ( ρ 1 2 + ρ 2 2 w 0 2 ) exp  [ ( ρ 2 ρ 1 ) 2 δ c 2 ] × exp  [ i k 2 z ( ρ 2   2 ρ 1   2 ) ] J m ( k z r 1 ρ 1 ) J m ( k z r 2 ρ 2 ) ρ 1 ρ 2 d ρ 1 d ρ 2 ,
η 2 = 1 P η 2 W ( ξ , η ; ξ , η ; 0 ) d ξ d η ,
P = W ( ξ , η ; ξ , η ) d ξ d η .
Q = ξ 2 + η 2 u 2 + v 2 ξ u + η v 2 ,
r 2   z = w 0 2 2 [ 1 + | m | + ( z 2 z R 2 ) ( 1 + w 0 2 δ c 2 + | m | ) ] ,
Q = 1 + | m | k 2 ( 1 + w 0 2 δ c 2 + | m | ) .
r 2   z ( s ) = w 0 2 2 [ 1 + | m | + ( z 2 z R 2 ) ( 1 + 2 w 0 2 δ c 2 + | m | ) ] ,
Q ( s ) = 1 + | m | k 2 ( 1 + 2 w 0 2 δ c 2 + | m | ) .
W ( ρ 1 , ρ 2 ) = n = 0 λ n Ψ n * ( ρ 1 ) Ψ n ( ρ 2 )
W m ( ρ 1 , ρ 2 , 0 ) = A 0 n = 0 γ n 2 n n ! ψ m , n * ( ρ 1 ) ψ m , n ( ρ 2 ) ,
γ = [ 1 + δ c 2 w 0 2 + 2 δ c w 0 ( 1 + δ c 2 2 w 0 2 ) 1 / 2 ] 1 ,
ψ m , n ( ρ ) = ( 2 ρ w 0 ) m exp  ( i m φ ) H n ( 2 α ρ ) exp  ( α ρ 2 ) ,
α = 1 w 0 2 ( 1 + 2 w 0 2 δ c 2 ) 1 / 2 .
Ψ m , n ( ρ ) = 1 β m , n ψ m , n ( ρ ) ,
β m , n 2 = 2 π 0 ( 2 ρ w 0 ) 2 m H n 2 ( 2 γ ρ ) exp  ( 2 γ ρ 2 ) ρ d ρ     = 2 2 n 2 m 2 π 3 / 2 ( 2 m + 1 ) ! w 0 2 m α m + 1 Γ ( m n + 3 / 2 ) 2 F 1 ( n , n ; m n + 3 / 2 ; 1 / 2 ) ,
W m ( ρ 1 , ρ 2 , 0 ) = A 0 n = 0 λ m , n Ψ m , n * ( ρ 1 ) Ψ m , n ( ρ 2 ) ,
λ m , n = γ n 2 n n ! β m , n 2 .

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