Abstract

A new kind of partially coherent vector beam, named a partially coherent radially polarized fractional vortex (PCRPFV) beam, is introduced as a natural extension of the recently introduced scalar partially coherent fractional vortex beams [Zeng et al., Opt. Express 26, 26830 (2018) [CrossRef]  ]. Realizability conditions and propagation formulas for a PCRPFV beam are derived. Statistical properties of a focused PCRPFV beam, such as average intensity, degree of polarization, state of polarization and cross-spectral density matrix, are illustrated in detail and compared with that of a partially coherent radially polarized integer vortex beam and a scalar partially coherent fractional vortex beam. It is found that the statistical properties of a PCRPFV beam are qualitatively different from these simpler beam classes and are strongly determined by the vortex phase (i.e., fractional topological charge) and initial coherence width. We demonstrate experimental generation of PCRPFV beams and confirm their behavior. Our results will be useful for the rotating and trapping of particles, the detection of phase objects, and polarization lidar systems.

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

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References

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2020 (2)

X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14(2), 102–108 (2020).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. V. Volyar, “Topological charge of a linear combination of optical vortices: topological competition,” Opt. Express 28(6), 8266–8281 (2020).
[Crossref]

2019 (8)

F. Gu, L. Li, C. Chang, C. Yuan, S. Feng, S. Nie, and J. Ding, “Generation of fractional ellipse perfect vector beams,” Opt. Commun. 443, 44–47 (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(20), 201106 (2019).
[Crossref]

J. Zeng, X. Lu, L. Liu, X. Zhu, C. Zhao, and Y. Cai, “Simultaneous measurement of the radial and azimuthal mode indices of a higher-order partially coherent vortex beam based on phase detection,” Opt. Lett. 44(15), 3881–3884 (2019).
[Crossref]

D. Deng, M. Lin, Y. Li, and H. Zhao, “Precision Measurement of Fractional Orbital Angular Momentum,” Phys. Rev. Appl. 12(1), 014048 (2019).
[Crossref]

S. Li, B. Shen, W. Wang, Z. Bu, H. Zhang, H. Zhang, and S. Zhai, “Diffraction of relativistic vortex harmonics with fractional average orbital angular momentum,” Chin. Opt. Lett. 17(5), 050501 (2019).
[Crossref]

J. Wen, L. G. Wang, X. Yang, J. Zhang, and S. Y. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
[Crossref]

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

J. Zeng, X. Liu, C. Zhao, F. Wang, G. Gbur, and Y. Cai, “Spiral spectrum of a Laguerre-Gaussian beam propagating in anisotropic non-Kolmogorov turbulent atmosphere along horizontal path,” Opt. Express 27(18), 25342–25356 (2019).
[Crossref]

2018 (6)

2017 (2)

X. Liu, T. Wu, L. Liu, C. Zhao, and Y. Cai, “Experimental determination of the azimuthal and radial mode orders of a partially coherent LGpl beam,” Chin. Opt. Lett. 15(3), 030002 (2017).
[Crossref]

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

2016 (3)

2015 (2)

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

A. T. Friberg and T. D. Visser, “Scintillation of electromagnetic beams generated by quasi-homogeneous sources,” Opt. Commun. 335, 82–85 (2015).
[Crossref]

2014 (1)

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

2013 (3)

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. J. A. P. L. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

2012 (5)

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. J. A. P. L. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express 20(14), 15908–15927 (2012).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

2011 (2)

2010 (2)

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

C. S. Guo, Y. N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
[Crossref]

2009 (5)

G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 26(8), 1788–1797 (2009).
[Crossref]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing Radially Polarized Light by a Concentrically Corrugated Silver Film without a Hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
[Crossref]

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

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of the Beam Coherence-Polarization Matrix of a Random Electromagnetic Beam,” IEEE J. Quantum Electron. 45(9), 1163–1167 (2009).
[Crossref]

2008 (5)

2007 (2)

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[Crossref]

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[Crossref]

2006 (2)

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh Criterion Limit with Optical Vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

S. S. R. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73(3), 032339 (2006).
[Crossref]

2005 (3)

2004 (3)

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref]

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004).
[Crossref]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

2003 (3)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

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

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

2000 (1)

1992 (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

1979 (1)

P. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

’t Hooft, G. W.

S. S. R. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73(3), 032339 (2006).
[Crossref]

Ahmed, N.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Aiello, A.

S. S. R. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73(3), 032339 (2006).
[Crossref]

Antosiewicz, T. J.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing Radially Polarized Light by a Concentrically Corrugated Silver Film without a Hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
[Crossref]

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh Criterion Limit with Optical Vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

Barbieri, C.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh Criterion Limit with Optical Vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

Baykal, Y.

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

Bianchini, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh Criterion Limit with Optical Vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

Borghi, R.

Brown, T. G.

Bu, Z.

Cai, Y.

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P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing Radially Polarized Light by a Concentrically Corrugated Silver Film without a Hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
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H. Zhang, J. Li, M. Guo, M. Duan, Z. Feng, and W. Yang, “Optical trapping two types of particles using a focused vortex beam,” Optik 166, 138–146 (2018).
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Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
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Youngworth, K. S.

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Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
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J. Zeng, X. Liu, C. Zhao, F. Wang, G. Gbur, and Y. Cai, “Spiral spectrum of a Laguerre-Gaussian beam propagating in anisotropic non-Kolmogorov turbulent atmosphere along horizontal path,” Opt. Express 27(18), 25342–25356 (2019).
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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(20), 201106 (2019).
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J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
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Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
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Zhu, X.

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Appl. Phys. Lett. (3)

F. Wang, Y. Cai, Y. Dong, and O. J. A. P. L. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
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Nanophotonics (1)

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(3), 677–682 (2018).
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A. T. Friberg and T. D. Visser, “Scintillation of electromagnetic beams generated by quasi-homogeneous sources,” Opt. Commun. 335, 82–85 (2015).
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T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
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O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
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F. Gu, L. Li, C. Chang, C. Yuan, S. Feng, S. Nie, and J. Ding, “Generation of fractional ellipse perfect vector beams,” Opt. Commun. 443, 44–47 (2019).
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Opt. Express (14)

J. Wen, L. G. Wang, X. Yang, J. Zhang, and S. Y. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
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Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
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X. Wang, G. Rui, L. Gong, B. Gu, and Y. Cui, “Manipulation of resonant metallic nanoparticle using 4Pi focusing system,” Opt. Express 24(21), 24143–24152 (2016).
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J. Zeng, X. Liu, C. Zhao, F. Wang, G. Gbur, and Y. Cai, “Spiral spectrum of a Laguerre-Gaussian beam propagating in anisotropic non-Kolmogorov turbulent atmosphere along horizontal path,” Opt. Express 27(18), 25342–25356 (2019).
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S. Li, B. Shen, X. Zhang, Z. Bu, and W. Gong, “Conservation of orbital angular momentum for high harmonic generation of fractional vortex beams,” Opt. Express 26(18), 23460–23470 (2018).
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J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
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Optik (1)

H. Zhang, J. Li, M. Guo, M. Duan, Z. Feng, and W. Yang, “Optical trapping two types of particles using a focused vortex beam,” Optik 166, 138–146 (2018).
[Crossref]

Phys. Lett. A (1)

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

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
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Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
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Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
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Phys. Rev. Appl. (2)

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
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D. Deng, M. Lin, Y. Li, and H. Zhao, “Precision Measurement of Fractional Orbital Angular Momentum,” Phys. Rev. Appl. 12(1), 014048 (2019).
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Phys. Rev. Lett. (2)

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing Radially Polarized Light by a Concentrically Corrugated Silver Film without a Hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
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Figures (13)

Fig. 1.
Fig. 1. Average intensity distribution $I$ and its component $I_x$ and $I_y$ of a focused PCRPFV beam with $l\textrm { = }1.5$ in the $x-y$ plane at several propagation distances $z$ with ${\sigma _g} = 3\rm {mm}$ .
Fig. 2.
Fig. 2. Average intensity distribution $I$ and its component $I_x$ and $I_y$ of a focused PCRPFV beam with $l\textrm { = }1.5$ in the $x-y$ plane at several propagation distances $z$ with ${\sigma _g} = 0.8\rm {mm}$ .
Fig. 3.
Fig. 3. Degree of polarization of a focused partially coherent radially polarized vortex beam (cross line $y=0$ ) in the focal plane for different values of the topological charge $l$ with ${\sigma _g} = 0.8\rm {mm}$ .
Fig. 4.
Fig. 4. Cross lines of the normalized intensity distributions $I^p\left (x,0\right )/\textrm {max}\left [I\left (x,0\right )\right ]$ (solid black line), $I^u\left (x,0\right )/\textrm {max}\left [I\left (x,0\right )\right ]$ (short dashed red line), $I\left (x,0\right )/\textrm {max}\left [I\left (x,0\right )\right ]$ (long dashed blue line) of a focused partially coherent radially polarized vortex beam at several propagation distances for different values of topological charge $l$ .
Fig. 5.
Fig. 5. Variation of the integrated normalized intensities of the polarized part $\left ({\eta ^{ p }}\right )$ and the unpolarized part $\left ({\eta ^{ u }}\right )$ of a focused partially coherent radially polarized vortex beam for (a-b) different values of the topological charge $l$ with ${\sigma _g} = 0.6\rm {mm}$ versus the propagation distance $z$ , (c-d) different values of the initial coherence width $\sigma _g$ versus the propagation distance $z$ and (e-f) different propagation distance $z$ versus the topological charge $l$ .
Fig. 6.
Fig. 6. Variation of the state of polarization of a focused partially coherent radially polarized vortex beam with different values of the topological charge $l$ at several propagation distances for ${\sigma _g} = 0.8\rm {mm}$ .
Fig. 7.
Fig. 7. Variation of the state of polarization of a focused partially coherent radially polarized vortex beam with different values of the topological charge $l$ at focal plane for different values of $\sigma _g$ .
Fig. 8.
Fig. 8. Density plot of the modulus of the trace of the CSD matrix $\left | {\textrm {Tr}\left [ {W\left ( {0,\boldsymbol {\rho } } \right )} \right ]} \right |$ of a partially coherent radially polarized vortex beam with different $l$ (both integral and fractional) focused by a lens at focal plane for different values of $\sigma _g$ .
Fig. 9.
Fig. 9. Density plot of the modulus of the trace of the CSD matrix $\left | {\textrm {Tr}\left [ {W\left ( {0,\boldsymbol {\rho } } \right )} \right ]} \right |$ , toghther with the modulus of the elements of the CSD matrix $\left | {{W_{xx}}\left ( {0,\boldsymbol {\rho } } \right )} \right |$ , $\left | {{W_{yy}}\left ( {0,\boldsymbol {\rho } } \right )} \right |$ and $\left | {{W_{xy}}\left ( {0,\boldsymbol {\rho } } \right )} \right |$ of a partially coherent radially polarized vortex beam with different $l$ ( $l=2$ and $l=1.5$ ) focused by a lens at focal plane for different values of $\sigma _g$ .
Fig. 10.
Fig. 10. Experimental setup for generating a PCRPFV beam and measuring its focused intensity. LASER, He-Ne laser; LP, linear polarizer; BE, beam expander; RM, reflecting mirror; $\rm {L}_1$ , $\rm {L}_2$ , $\rm {L}_3$ , thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; RPC, radial polarization converter; SPP, spiral phase plate; CCD, charge-coupled device.
Fig. 11.
Fig. 11. Experimental results of the average intensity distribution $I$ and its component $I_x$ and $I_y$ of a focused PCRPFV beam with $l=1.5$ , $w_0=1\rm {mm}$ and ${\sigma _g} = 3\rm {mm}$ in the $x-y$ plane at several propagation distances $z$ .
Fig. 12.
Fig. 12. Experimental results of the average intensity distribution $I$ and its component $I_x$ and $I_y$ of a focused PCRPFV beam with $l=1.5$ , $w_0=1\rm {mm}$ and ${\sigma _g} =0.8\rm {mm}$ in the $x-y$ plane at several propagation distances $z$ .
Fig. 13.
Fig. 13. Experimental results (red dotted curve) of the degree of polarization of a focused PCRPFV beam with $w_0=1\rm {mm}$ and ${\sigma _g} =0.8\rm {mm}$ versus the coordinate $x$ in the focal plane.

Equations (47)

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W α β ( r 1 , r 2 ) = E α ( r 1 ) E β ( r 2 ) , ( α , β = x , y ) ,
W α β ( r 1 , r 2 ) = α β w 0 2 exp ( r 1 2 + r 2 2 w 0 2 ) exp [ i l ( φ 1 φ 2 ) ] μ α β ( r 1 r 2 ) , ( α , β = x , y ) ,
μ α β ( r 1 r 2 ) = B α β exp [ ( r 1 r 2 ) 2 2 σ α β 2 ] , ( α , β = x , y ) ,
l = 1 2 π C ψ ( r ) d r ,
| B α β | = 1 , ϕ α β = 0 , ( α = β ) , | B α β | 1 , | B x y | = | B y x | , ϕ x y = ϕ y x , ( α β ) , σ x y = σ y x .
W α β ( r 1 , r 2 ) = p α β ( v ) H α ( r 1 , v ) H β ( r 2 , v ) d 2 v ,
p α β ( v ) = ( p x x ( v ) p x y ( v ) p y x ( v ) p y y ( v ) ) .
p x x ( v ) 0 , p y y ( v ) 0 , p x x ( v ) p y y ( v ) p x y ( v ) p y x ( v ) 0.
H α ( r , v ) = α w 0 exp ( r 2 w 0 2 ) exp ( i l φ ) exp ( i k r v ) , ( α = x , y ) ,
p α β ( v ) = k 2 B α β σ α β 2 2 π exp ( k 2 σ α β 2 v 2 / 2 ) ,
B x x B y y σ x x 2 σ y y 2 exp [ k 2 ( σ x x 2 + σ y y 2 ) v 2 2 ] | B x y | 2 σ x y 4 exp ( k 2 σ x y 2 v 2 ) .
σ x x 2 + σ y y 2 2 σ x y σ x x σ y y | B x y | ,
| B x y | 2 σ x x σ y y σ x x 2 + σ y y 2 .
A 1 , 2 ( r ) = 1 2 { [ W x x ( r , r ) W y y ( r , r ) ] 2 + 4 | W x y ( r , r ) | 2 ± [ W x x ( r , r ) W y y ( r , r ) ] 2 + 4 | Re [ W x y ( r , r ) ] | 2 } 1 / 2 ,
ε ( r ) = A 2 ( r ) / A 1 ( r ) ,
θ ( r ) = 1 2 arctan { 2 Re [ W x y ( r , r ) ] W x x ( r , r ) W y y ( r , r ) } .
tan [ 2 θ ( r ) ] = Re [ B x y ] 2 y / x 1 ( y / x ) 2 .
Im [ B x y ] = 0.
Re [ B x y ] = 1.
σ x x = σ y y = σ x y = σ y x .
B x x = B y y = B x y = B y x = 1 , σ x x = σ y y = σ x y = σ y x = σ g .
W α β ( r 1 , r 2 ) = α β w 0 2 exp ( r 1 2 + r 2 2 w 0 2 ) exp [ i l ( φ 1 φ 2 ) ] exp [ ( r 1 r 2 ) 2 2 σ g 2 ] , ( α , β = x , y ) .
W α β ( ρ 1 , ρ 2 ) = k 2 4 π 2 B 2 W α β ( r 1 , r 2 ) exp ( i k A 2 B r 1 2 + i k B r 1 ρ 1 i k D 2 B ρ 1 2 ) × exp ( i k A 2 B r 2 2 i k B r 2 ρ 2 + i k D 2 B ρ 2 2 ) d 2 r 1 d 2 r 2 ,
W x x ( ρ , ρ ) = I x ( ρ ) = λ 2 k 2 4 π 2 B 2 | A ~ x ( v λ + ρ λ B ) | 2 p ( v λ ) d 2 v λ ,
W y y ( ρ , ρ ) = I y ( ρ ) = λ 2 k 2 4 π 2 B 2 | A ~ y ( v λ + ρ λ B ) | 2 p ( v λ ) d 2 v λ ,
W x y ( ρ , ρ ) = λ 2 k 2 4 π 2 B 2 A ~ x ( v λ + ρ λ B ) A ~ y ( v λ + ρ λ B ) p ( v λ ) d 2 v λ ,
W y x ( ρ , ρ ) = λ 2 k 2 4 π 2 B 2 A ~ y ( v λ + ρ λ B ) A ~ x ( v λ + ρ λ B ) p ( v λ ) d 2 v λ
W x x ( 0 , ρ ) = 1 B 2 exp [ i π λ B D ( ρ λ B ) 2 ] A ~ x ( v λ ) p ( v λ ) A ~ x ( v λ  +  ρ λ B ) d 2 v λ ,
W y y ( 0 , ρ )  =  1 B 2 exp [ i π λ B D ( ρ λ B ) 2 ] A ~ y ( v λ ) p ( v λ ) A ~ y ( v λ  +  ρ λ B ) d 2 v λ ,
W x y ( 0 , ρ )  =  1 B 2 exp [ i π λ B D ( ρ λ B ) 2 ] A ~ x ( v λ ) p ( v λ ) A ~ y ( v λ  +  ρ λ B ) d 2 v λ ,
W y x ( 0 , ρ )  =  1 B 2 exp [ i π λ B D ( ρ λ B ) 2 ] A ~ y ( v λ ) p ( v λ ) A ~ x ( v λ  +  ρ λ B ) d 2 v λ ,
A x ( r )  =  x w 0 exp ( x 2 + y 2 w 0 2 ) exp ( i l φ ) exp ( i k A r 2 / 2 B ) ,
A y ( r )  =  y w 0 exp ( x 2 + y 2 w 0 2 ) exp ( i l φ ) exp ( i k A r 2 / 2 B ) ,
p ( v )  =  k 2 σ g 2 2 π exp ( k 2 σ g 2 v 2 / 2 ) ,
I ( ρ )  =  W x x ( ρ , ρ ) + W y y ( ρ , ρ ) ,
P ( ρ )  =  1 4 Det [ W ( ρ , ρ ) ] { Tr [ W ( ρ , ρ ) ] } 2 ,
W ( ρ , ρ ) = W p ( ρ , ρ ) + W u ( ρ , ρ ) ,
W p ( ρ , ρ ) = ( B ( ρ , ρ ) D ( ρ , ρ ) D ( ρ , ρ ) C ( ρ , ρ ) ) ,
W u ( ρ , ρ ) = ( A ( ρ , ρ ) 0 0 A ( ρ , ρ ) ) ,
A ( ρ , ρ )  =  1 2 [ W x x ( ρ , ρ ) + W y y ( ρ , ρ ) [ W x x ( ρ , ρ ) W y y ( ρ , ρ ) ] 2 + 4 | W x y ( ρ , ρ ) | 2 ] ,
B ( ρ , ρ )  =  1 2 [ W x x ( ρ , ρ ) W y y ( ρ , ρ ) + [ W x x ( ρ , ρ ) W y y ( ρ , ρ ) ] 2 + 4 | W x y ( ρ , ρ ) | 2 ] ,
C ( ρ , ρ )  =  1 2 [ W y y ( ρ , ρ ) W x x ( ρ , ρ ) + [ W x x ( ρ , ρ ) W y y ( ρ , ρ ) ] 2 + 4 | W x y ( ρ , ρ ) | 2 ] ,
D ( ρ , ρ ) = W x y ( ρ , ρ ) .
I p ( ρ )  =  W x x p ( ρ , ρ ) + W y y p ( ρ , ρ ) ,
I u ( ρ )  =  W x x u ( ρ , ρ ) + W y y u ( ρ , ρ ) .
( A B C D ) = ( 1 z 0 1 ) ( 1 0 1 / f 1 ) = ( 1 z / f z 1 / f 1 ) .
η m  =  I m ( ρ ) d 2 ρ I ( ρ ) d 2 ρ , ( m = p , u ) ,

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