Abstract

Partially coherent radially polarized (PCRP) beam was introduced and generated in recent years. In this paper, we investigate the statistical properties of a PCRP beam embedded with a vortex phase (i.e., PCRP vortex beam). We derive the analytical formula for the cross-spectral density matrix of a PCRP vortex beam propagating through a paraxial ABCD optical system and analyze the statistical properties of a PCRP vortex beam focused by a thin lens. It is found that the statistical properties of a PCRP vortex beam on propagation are much different from those of a PCRP beam. The vortex phase induces not only the rotation of the beam spot, but also the changes of the beam shape, the degree of polarization and the state of polarization. We also find that the vortex phase plays a role of resisting the coherence-induced degradation of the intensity distribution and the coherence-induced depolarization. Furthermore, we report experimental generation of a PCRP vortex beam for the first time. Our results will be useful for trapping and rotating particles, free-space optical communications and detection of phase object.

© 2016 Optical Society of America

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References

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2016 (2)

Y. Yang and Y. Liu, “Measuring azimuthal and radial mode indices of a partially coherent vortex field,” J. Opt. 18(1), 015604 (2016).
[Crossref]

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108(5), 051107 (2016).
[Crossref]

2015 (1)

2014 (4)

C. S. D. Stahl and G. Gbur, “Complete representation of a correlation singularity in a partially coherent beam,” Opt. Lett. 39(20), 5985–5988 (2014).
[Crossref] [PubMed]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[Crossref] [PubMed]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[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(1), 013801 (2014).
[Crossref]

2013 (5)

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

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
[Crossref]

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013).
[Crossref] [PubMed]

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] [PubMed]

2012 (6)

V. P. Aksenov and C. E. Pogutsa, “Increase in laser beam resistance to random inhomogeneities of atmospheric permittivity with an optical vortex included in the beam structure,” Appl. Opt. 51(30), 7262–7267 (2012).
[Crossref] [PubMed]

Y. Yang, M. Mazilu, and K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012).
[Crossref] [PubMed]

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] [PubMed]

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

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

J. Wang, J. 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]

2011 (2)

2010 (1)

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

2009 (5)

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[Crossref] [PubMed]

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

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] [PubMed]

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

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
[Crossref]

2008 (2)

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

2007 (1)

2006 (1)

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

2005 (1)

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).
[Crossref]

2004 (2)

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref] [PubMed]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[Crossref] [PubMed]

2003 (5)

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
[Crossref]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597(2), 1266–1270 (2003).
[Crossref]

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

2002 (2)

2001 (2)

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

2000 (1)

1998 (1)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1996 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1970 (1)

Abeysinghe, D. C.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

Ahmed, N.

J. Wang, J. 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]

Aksenov, V. P.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
[Crossref]

V. P. Aksenov and C. E. Pogutsa, “Increase in laser beam resistance to random inhomogeneities of atmospheric permittivity with an optical vortex included in the beam structure,” Appl. Opt. 51(30), 7262–7267 (2012).
[Crossref] [PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

Baykal, Y.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Bogatyryova, G. V.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

Brown, T.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Cai, Y.

S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015).
[Crossref] [PubMed]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[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(1), 013801 (2014).
[Crossref]

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

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

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

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[Crossref] [PubMed]

F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[Crossref] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

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Figures (12)

Fig. 1
Fig. 1 Spectral intensity distribution I and its components Ix and Iy of a focused PCRP vortex beam with m = 0 in the x-y plane at several propagation distances with δ0 = 0.5mm.The solid curve denotes the cross line (y = 0).
Fig. 2
Fig. 2 Spectral intensity distribution I and its components Ix and Iy of a focused PCRP vortex beam with m = 2 in the x-y plane at several propagation distances with δ0 = 0.5mm. The solid curve denotes the cross line (y = 0).
Fig. 3
Fig. 3 Spectral intensity distribution I and its components Ix and Iy of a focused PCRP vortex beam with m = −2 in the x-y plane at several propagation distances with δ0 = 0.5mm. The solid curve denotes the cross line (y = 0).
Fig. 4
Fig. 4 Spectral intensity distribution I and the corresponding cross line (y = 0) of a focused PCRP vortex beam in the focal plane for different values of the topological charge m with δ0 = 0.5mm.
Fig. 5
Fig. 5 Degree of polarization of a focused PCRP vortex beam at point (0.05mm, 0.05mm) versus the propagation distance z for different values of (a) the topological charge m with δ0 = 0.5 mm and (b) the coherence width δ0 with m = 2.
Fig. 6
Fig. 6 Degree of polarization of a focused PCRP vortex beam (cross line y = 0) in the focal plane for different values of (a) the topological charge m with δ0 = 0.5 mm and (b) the coherence width δ0 with m = 2.
Fig. 7
Fig. 7 Variation of the normalized powers of the completely unpolarized part (ηu) and the completely polarized part (ηp) of a focused PCRP vortex beam versus the propagation distance z for different values of the topological charge m with δ0 = 0.5mm.
Fig. 8
Fig. 8 Variation of the state of polarization of a focused PCRP vortex beam at several propagation distances in free space for different values of the topological charge m with δ0 = 0.5mm. The yellow line denotes linear polarization and the red (or green) ellipsoid denotes right- (or left-) handed elliptical polarization..
Fig. 9
Fig. 9 Variation of the state of polarization of a focused PCRP vortex beam in the focal plane for different values of the topological charge m with δ0 = 0.5mm.
Fig. 10
Fig. 10 Experimental setup for generating a PCRP vortex beam and measuring its focused intensity. L1, L2, L3, thin lenses; M, reflecting mirror; BE, beam expander; RGGP, rotating ground-glass plate; GAF, Gaussian amplitude filter; RPC, radial polarization converter; SPP, spiral phase plate; BPA, beam profile analyzer; PC, personal computer.
Fig. 11
Fig. 11 Experimental results of the intensity distribution I and its components Ix and Iy of a focused PCRP vortex beam with m = 2, σ0 = 1mm and δ0 = 0.8mm at several propagation distances.
Fig. 12
Fig. 12 Experimental result (dotted curve) of the degree of polarization of a focused PCRP vortex beam with m = 2, σ0 = 1mm and δ0 = 0.8mm versus the coordinate x in the focal plane. The solid curve denotes the theoretical result.

Equations (31)

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W 0 x x ( ρ 01 , θ 01 , ρ 02 , θ 02 ) = ρ 01 ρ 02 cos θ 01 cos θ 02 4 σ 0 2 exp ( ρ 01 2 + ρ 02 2 4 σ 0 2 ) × exp [ ρ 01 2 + ρ 02 2 2 ρ 01 ρ 02 cos ( θ 01 θ 02 ) 2 δ 0 2 ] ,
W 0 x y ( ρ 01 , θ 01 , ρ 02 , θ 02 ) = ρ 01 ρ 02 cos θ 01 sin θ 02 4 σ 0 2 exp ( ρ 01 2 + ρ 02 2 4 σ 0 2 ) × exp [ ρ 01 2 + ρ 02 2 2 ρ 01 ρ 02 cos ( θ 01 θ 02 ) 2 δ 0 2 ] ,
W 0 y x ( ρ 01 , θ 01 , ρ 02 , θ 02 ) = W 0 x y * ( ρ 02 , θ 02 , ρ 01 , θ 01 )
W 0 y y ( ρ 01 , θ 01 , ρ 02 , θ 02 ) = ρ 01 ρ 02 sin θ 01 sin θ 02 4 σ 0 2 exp ( ρ 01 2 + ρ 02 2 4 σ 0 2 ) × exp [ ρ 01 2 + ρ 02 2 2 ρ 01 ρ 02 cos ( θ 01 θ 02 ) 2 δ 0 2 ] ,
W α β ( ρ 01 , θ 01 , ρ 02 , θ 02 ) = W 0 α β ( ρ 01 , θ 01 , ρ 02 , θ 02 ) exp ( i m θ 01 + i m θ 02 ) , ( α = x , y ; β = x , y ) .
W α β ( ρ 1 , θ 1 , ρ 2 , θ 2 ) = 1 λ 2 B B * exp ( i k D * ρ 2 2 2 B * i k D ρ 1 2 2 B ) 0 2 π 0 2 π 0 0 W α β ( ρ 01 , θ 01 , ρ 02 , θ 02 ) × exp { i k [ A * ρ 02 2 2 B * A ρ 01 2 2 B + ρ 1 ρ 01 B cos ( θ 1 θ 01 ) ] × exp [ i k ρ 2 ρ 02 B * cos ( θ 2 θ 02 ) ] ρ 01 ρ 02 d ρ 01 d ρ 02 d θ 01 d θ 02 ,
W x x ( ρ 1 , θ 1 , ρ 2 , θ 2 ) = ( ρ 1 , ρ 2 ) l = s = 0 exp [ i l ( θ 1 θ 2 ) ] [ exp ( 2 i θ 2 ) Ω 1 ( p 1 , l ) Ω 2 * ( p 1 , l 2 ) s ! Γ ( s + | m + l 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l 1 | Ω 1 ( p 1 , l ) Ω 2 * ( p 1 , l ) s ! Γ ( s + | m + l 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l 1 | Ω 1 ( p 2 , l ) Ω 2 * ( p 2 , l ) s ! Γ ( s + | m + l + 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l + 1 | + exp ( 2 i θ 2 ) Ω 1 ( p 2 , l ) Ω 2 * ( p 2 , l + 2 ) s ! Γ ( s + | m + l + 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l + 1 | ] ,
W x y ( ρ 1 , θ 1 , ρ 2 , θ 2 ) = i ( ρ 1 , ρ 2 ) l = s = 0 exp [ i l ( θ 1 θ 2 ) ] [ exp ( 2 i θ 2 ) Ω 1 ( p 1 , l ) Ω 2 * ( p 1 , l 2 ) s ! Γ ( s + | m + l 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l 1 | + Ω 1 ( p 1 , l ) Ω 2 * ( p 1 , l ) s ! Γ ( s + | m + l 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l 1 | Ω 1 ( p 2 , l ) Ω 2 * ( p 2 , l ) s ! Γ ( s + | m + l + 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l + 1 | exp ( 2 i θ 2 ) Ω 1 ( p 2 , l ) Ω 2 * ( p 2 , l + 2 ) s ! Γ ( s + | m + l + 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l + 1 | ] ,
W y x ( ρ 1 , θ 1 , ρ 2 , θ 2 ) = W x y * ( ρ 1 , θ 1 , ρ 2 , θ 2 ) ,
W y y ( ρ 1 , θ 1 , ρ 2 , θ 2 ) = ( ρ 1 , ρ 2 ) l = s = 0 exp [ i l ( θ 1 θ 2 ) ] [ exp ( 2 i θ 2 ) Ω 1 ( p 1 , l ) Ω 2 * ( p 1 , l 2 ) s ! Γ ( s + | m + l 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l 1 | + Ω 1 ( p 1 , l ) Ω 2 * ( p 1 , l ) s ! Γ ( s + | m + l 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l 1 | + Ω 1 ( p 2 , l ) Ω 2 * ( p 2 , l ) s ! Γ ( s + | m + l + 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l + 1 | + exp ( 2 i θ 2 ) Ω 1 ( p 2 , l ) Ω 2 * ( p 2 , l + 2 ) s ! Γ ( s + | m + l + 1 | +1 ) ( 2 δ 0 2 ) 2 s + | m + l + 1 | ] ,
α 1 = 1 4 σ 0 2 + 1 2 δ 2 + i k A 2 B , α 2 = 1 4 σ 0 2 + 1 2 δ 2 i k A * 2 B * , p 1 = 2 s + | m + l 1 | + 3 , p 2 = 2 s + | m + l + 1 | + 3 , ( ρ 1 , ρ 2 ) = 1 B B * ( k 4 σ 0 ) 2 exp [ i ( k D * ρ 2 2 2 B * k D ρ 1 2 2 B ) ] exp ( k 2 ρ 1 2 4 α 1 B 2 k 2 ρ 2 2 4 α 2 B * 2 ) , Ω 1 ( p , q ) = ( α 1 ) p / 2 2 q ! Γ ( p + q 2 ) ( k 2 ρ 1 2 4 α 1 B 2 ) q / 2 F 1 1 ( q p 2 + 1 ; q + 1 ; k 2 ρ 1 2 4 α 1 B 2 ) , Ω 2 * ( p , q ) = ( α 2 ) p / 2 2 q ! Γ ( p + q 2 ) ( k 2 ρ 2 2 4 α 2 B * 2 ) q / 2 F 1 1 ( q p 2 + 1 ; q + 1 ; k 2 ρ 2 2 4 α 2 B * 2 ) .
exp [ i k ρ 1 ρ 01 B cos ( θ 1 θ 01 ) ] = l = i l J l ( k ρ 1 ρ 01 B ) exp [ i l ( θ 1 θ 01 ) ] ,
J l ( χ ) = ( i ) l 2 π 0 2 π exp [ i l θ 0 + i χ cos ( θ 0 ) ] d θ 0 ,
I v ( α ρ 01 ρ 02 ) = s = 0 1 s ! Γ ( s + | v | +1 ) ( α ρ 01 ρ 02 2 ) 2 s + | v | ,
0 2 π exp [ i m θ 01 + α ρ 01 ρ 02 cos ( θ 01 θ 02 ) ] = 2 π exp ( i m θ 02 ) I m ( α ρ 01 ρ 02 ) ,
0 x μ exp ( α x 2 ) J l ( β x ) d t = β l Γ [ ( l + μ + 1 ) / 2 ] 2 l + 1 α ( μ + l + 1 ) / 2 Γ ( l + 1 ) F 1 1 ( l + μ + 1 2 ; l + 1 ; β 2 4 α ) , ( Re α > 0 , Re ( μ + l ) > 1 ) .
I ( ρ , θ ) = Tr W ( ρ , θ , ρ , θ ) = I x ( ρ , θ ) + I y ( ρ , θ ) ,
P ( ρ , θ ) = 1 4 Det W ( ρ , θ , ρ , θ ) [ Tr W ( ρ , θ , ρ , θ ) ] 2 .
W ( ρ , θ , ρ , θ ) = W ( u ) ( ρ , θ , ρ , θ ) + W ( p ) ( ρ , θ , ρ , θ ) ,
W ( u ) ( ρ , θ , ρ , θ ) = ( A ( ρ , θ , ρ , θ ) 0 0 A ( ρ , θ , ρ , θ ) ) ,
W ( p ) ( ρ , θ , ρ , θ ) = ( B ( ρ , θ , ρ , θ ) D ( ρ , θ , ρ , θ ) D * ( ρ , θ , ρ , θ ) C ( ρ , θ , ρ , θ ) ) ,
  A ( ρ , θ , ρ , θ ) = 1 2 [ W x x ( ρ , θ , ρ , θ ) + W y y ( ρ , θ , ρ , θ ) 1 2 [ W x x ( ρ , θ , ρ , θ ) W y y ( ρ , θ , ρ , θ ) ] 2 + 4 | W x y ( ρ , θ , ρ , θ ) | 2 ] ,
  B ( ρ , θ , ρ , θ ) = 1 2 [ W x x ( ρ , θ , ρ , θ ) W y y ( ρ , θ , ρ , θ ) + 1 2 [ W x x ( ρ , θ , ρ , θ ) W y y ( ρ , θ , ρ , θ ) ] 2 + 4 | W x y ( ρ , θ , ρ , θ ) | 2 ] ,
  C ( ρ , θ , ρ , θ ) = 1 2 [ W y y ( ρ , θ , ρ , θ ) W x x ( ρ , θ , ρ , θ ) + 1 2 [ W x x ( ρ , θ , ρ , θ ) W y y ( ρ , θ , ρ , θ ) ] 2 + 4 | W x y ( ρ , θ , ρ , θ ) | 2 ] ,
D ( ρ , θ , ρ , θ ) = W x y ( ρ , θ , ρ , θ )
I ( u ) ( ρ , θ ) = Tr W ( u ) ( ρ , θ , ρ , θ ) , I ( p ) ( ρ , θ ) = Tr W ( p ) ( ρ , θ , ρ , θ ) ,
  A 1 , 2 ( ρ , θ ) = 1 2 [ [ W x x ( ρ , θ , ρ , θ ) W y y ( ρ , θ , ρ , θ ) ] 2 + 4 | W x y ( ρ , θ , ρ , θ ) | 2 ± 1 2 [ W x x ( ρ , θ , ρ , θ ) W y y ( ρ , θ , ρ , θ ) ] 2 + 4 | Re W x y ( ρ , θ , ρ , θ ) | 2 ] 1 / 2 ,
ε ( ρ , θ ) = A 2 ( ρ , θ , ρ , θ ) A 1 ( ρ , θ , ρ , θ ) ,
ϑ ( ρ , θ ) = 1 2 arc tan [ 2 Re W x y ( ρ , θ , ρ , θ ) W x x ( ρ , θ , ρ , θ ) W y y ( ρ , θ , ρ , θ ) ] .
( A B C D ) = ( 1 z 0 1 ) ( 1 0 1 / f 1 ) = ( 1 z / f z 1 / f 1 ) .
η ( l ) ( z ) = I ( l ) ( ρ , θ ) ρ d ρ d θ I ( ρ , θ ) ρ d ρ d θ , ( l = u , p ) ,

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