Abstract

We have previously demonstrated that Mie scattering of partially coherent plane waves can create coherence vortices, namely screw-type dislocations in the phase of the spectral degree of coherence. However, plane waves are an idealization and in practice, optical beams are often much closer to reality. Thus, in this paper, we consider coherence vortices created by Mie scattering of partially coherent focused beams. We demonstrate that Mie scattering of partially coherent complex focused beams can give rise to coherence vortices. As the scattered fields propagate coherence vortex-antivortex pairs are annihilated thus creating hair-pin structures in the coherence-vortex nodal lines. The evolution of correlation singularities in the scattered field with the variation of the complex focus point of the incident beam is also discussed. The variation of the degree of polarization of the scattered field is also studied.

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    [CrossRef]
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    [CrossRef] [PubMed]
  55. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: birth and evolution of phase singularities in the spatial coherence function,” in Fringe 2005, W. Osten, ed. (Springer BerlinHeidelberg, 2006), 46–53.
    [CrossRef]
  56. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
    [CrossRef] [PubMed]

2011 (2)

2010 (3)

M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010).
[CrossRef] [PubMed]

G. Gouesbet, “Hypotheses on the a priori rational necessity of quantum mechanics,” Principia, an international journal of epistemology 14, 393–404 (2010).

G. Gouesbet, “T-matrix formulation and generalized Lorenz-Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521 (2010).
[CrossRef]

2009 (1)

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[CrossRef]

2008 (2)

2007 (1)

2006 (3)

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[CrossRef]

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

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

2005 (2)

R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case I: symmetrically polarized beams,” J. Mod. Opt. 52, 2067–2092 (2005).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

2004 (6)

2003 (4)

2002 (1)

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

2000 (1)

M. Berry, “Making waves in physics,” Nature (London) 403, 21 (2000).
[CrossRef]

1999 (2)

1998 (3)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature (London) 394, 348–350 (1998).
[CrossRef]

M. V. Berry, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices...,” Proc SPIE 3487, 1–5 (1998).
[CrossRef]

1995 (2)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–571 (1995).
[CrossRef] [PubMed]

1994 (1)

M.V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A: Math. Gen. 27, L391–L398 (1994).
[CrossRef]

1988 (2)

1985 (1)

G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

1982 (2)

1980 (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

1979 (1)

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. Lond. A 366, 155–171 (1979).
[CrossRef]

1976 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974).
[CrossRef]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

1833 (1)

W. Whewell, “Essay towards a first approximation to a map of cotidal lines,” Phil. Trans. R. Soc. Lond. 123, 147–236 (1833).
[CrossRef]

Alonso, M. A.

Andersen, M. F.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Barnett, S. M.

S. M. Barnett, “On the quantum core of an optical vortex,” J. Mod. Opt. 55, 2279–2292 (2008).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Berry, M.

M. Berry, “Making waves in physics,” Nature (London) 403, 21 (2000).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A: Pure Appl. Opt. 6, S178–S180 (2004).
[CrossRef]

M. V. Berry, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices...,” Proc SPIE 3487, 1–5 (1998).
[CrossRef]

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974).
[CrossRef]

M. V. Berry, “Singularities in waves and rays,” in R. Balian, Kléman, and J-P Poirier eds., Les Houches Lecture Series, session XXXV, Physics of defects (North Holland, Amsterdam) 453–543 (1981).

Berry, M.V.

M.V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A: Math. Gen. 27, L391–L398 (1994).
[CrossRef]

Bogatyryova, G. V.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge University Press, Cambridge, 1999).

Cladé, P.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Cui, Z.

Cullen, F. A. L.

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. Lond. A 366, 155–171 (1979).
[CrossRef]

Dennis, M. R.

M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A: Pure Appl. Opt. 6, S178–S180 (2004).
[CrossRef]

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Duan, Z.

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

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: birth and evolution of phase singularities in the spatial coherence function,” in Fringe 2005, W. Osten, ed. (Springer BerlinHeidelberg, 2006), 46–53.
[CrossRef]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Felde, C. V.

Fischer, D. G.

Friberg, A.

Friberg, A. T.

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature (London) 394, 348–350 (1998).
[CrossRef]

Gbur, G.

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[CrossRef]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[CrossRef]

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

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

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

G. Gbur, “Optical and coherence vortices and their relationships,” in Eighth International Conference on Correlation Optics, M. Kujawinska and O. V. Angelsky, eds. Proc. SPIE7008, 70080N-1–70080N-7 (2008).

Gouesbet, G.

G. Gouesbet, “T-matrix formulation and generalized Lorenz-Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521 (2010).
[CrossRef]

G. Gouesbet, “Hypotheses on the a priori rational necessity of quantum mechanics,” Principia, an international journal of epistemology 14, 393–404 (2010).

G. Gouesbet, G. Gréhan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories, (Springer, Berlin, 2011) 48–50.
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories, (Springer, Berlin, 2011).
[CrossRef]

Gréhan, G.

G. Gouesbet, G. Gréhan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories, (Springer, Berlin, 2011) 48–50.
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories, (Springer, Berlin, 2011).
[CrossRef]

Gu, Y.

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[CrossRef]

Han, Y.

Hanson, S. G.

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

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: birth and evolution of phase singularities in the spatial coherence function,” in Fringe 2005, W. Osten, ed. (Springer BerlinHeidelberg, 2006), 46–53.
[CrossRef]

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature (London) 394, 348–350 (1998).
[CrossRef]

Helmerson, K.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Kant, R.

R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case I: symmetrically polarized beams,” J. Mod. Opt. 52, 2067–2092 (2005).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York, 1969).

Lock, J. A.

Maheu, B.

Maleev, I.

I. Maleev, “Partial coherence and optical vortices,” Ph.D. dissertation, Worcester Polytechnic Institute, 2004.

Maleev, I. D.

Mandel, L.

L. Mandel and E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

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

Marasinghe, M. L.

Marathay, A. S.

Miyamoto, Y.

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

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: birth and evolution of phase singularities in the spatial coherence function,” in Fringe 2005, W. Osten, ed. (Springer BerlinHeidelberg, 2006), 46–53.
[CrossRef]

Moore, N. J.

Natarajan, V.

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of the scattering sphere centered at the origin of the coordinate system. The incident CF beam propagates in the positive z-direction.

Fig. 2
Fig. 2

Observation plane (XY) of the spectral degree of coherence.

Fig. 3
Fig. 3

Variation of the spectral degree of coherence (μ) of the scattered field in XY plane (z = 5), when the incident beam is radially polarized. As described in Section 2.1, the incident beam is focused at a complex point ρ0 = r + iq, corresponding to the waist of the resulting focused beam being centered at r. (a) Observation plane XY (z = R). (b) Arg(μ) when ρ0 = (2, 0, 4i). (c) Abs(μ) when ρ0 = (2, 0, 4i). (d) Arg(μ) when ρ0 = (5, 0, 4i). (e) Abs(μ) when ρ0 = (5, 0, 4i). (f) Arg(μ) when ρ0 = (8, 0, 4i). (g) Abs(μ) when ρ 0 = (8, 0, 4i).

Fig. 4
Fig. 4

Variation of the spectral degree of coherence (μ) of the scattered field in XY plane (z = 5), when the azimuthally polarized beam is focused at different points ρ0. (a) Observation plane XY (z = R). (b) Arg(μ) when ρ0 = (5, 0, 4i). (c) Abs(μ) when ρ0 = (5, 0, 4i). (d) Arg(μ) when ρ0 = (0, 5, 4i). (e) Abs(μ) when ρ0 = (0, 5, 4i). (f) Arg(μ) when ρ0 = (6, 0, 4i). (g) Abs(μ) when ρ0 = (6, 0, 4i). (h) Arg(μ) when ρ0 = (8, 0, 4i). (i) Abs(μ) when ρ0 = (8, 0, 4i).

Fig. 5
Fig. 5

Variation of spectral degree of coherence of the scattered field in XY plane (z = 5), when the azimuthally polarized beam is focused beam at (3, 0, 4i). Panels (a) and (b) respectively show the phase and the magnitude of the complex measure μ in Eq. (3) [48], with panel (c) giving the real measure μξ in Eq. (4) [4951].

Fig. 6
Fig. 6

Variation of μ of propagating scattered field in different XY planes, when the azimuthally polarized incident beam is focused at (5, 0, 4i). (a) Observation plane XY (z > R). (b) Arg(μ) in z = 1.02kR plane. (c) Abs(μ) in z = 1.02kR plane. (d) Arg(μ) in z = 1.1kR plane. (e) Abs(μ) in z = 1.1kR plane. (f) Arg(μ) in z = 1.5kR plane. (g) Abs(μ) in z = 1.5kR plane.

Fig. 7
Fig. 7

Nodal lines of the coherence vortices in μ for the scattered field. Phase in upper and lower planes is indicated by contour lines colored according to the bar on right.

Fig. 8
Fig. 8

Degree of polarization of the scattered field in the XY plane (z = 5) when the azimuthally polarized incident field is focused at (5, 0, 4i). (a) P calculated using the Eq. (8) [53]. (b) Pξ calculated using Eq. (9) [49, 54].

Equations (19)

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U C F ( r ; ρ 0 ) = 4 π U 0 sin { k [ ( r ρ 0 ) ( r ρ 0 ) ] 1 / 2 } k [ ( r ρ 0 ) ( r ρ 0 ) ] 1 / 2 ,
E ( C F ) ( E ) ( r ; ρ 0 , p ) = U 0 l = 1 m = l l [ γ l m ( I ) ( ρ 0 , p ) Λ l m ( I ) ( r ) + γ l m ( II ) ( ρ 0 , p ) Λ l m ( II ) ( r ) ] ,
E ( C F ) ( B ) ( r ; ρ 0 , p ) = U 0 l = 1 m = l l [ γ l m ( II ) ( ρ 0 , p ) Λ l m ( I ) ( r ) γ l m ( I ) ( ρ 0 , p ) Λ l m ( II ) ( r ) ] ,
γ l m ( I , II ) ( ρ 0 , p ) = p [ Λ l m ( I , II ) ( r ) ] * .
E ( s c ) ( E ) ( r ; ρ 0 , p ) = U 0 l = 1 m = l l [ a l γ l m ( I ) ( ρ 0 , p ) Π l m ( I ) ( r ) + b l γ l m ( II ) ( ρ 0 , p ) Π l m ( II ) ( r ) ] ,
E ( s c ) ( B ) ( r ; ρ 0 , p ) = U 0 l = 1 m = l l [ a l γ l m ( II ) ( ρ 0 , p ) Π l m ( I ) ( r ) + b l γ l m ( I ) ( ρ 0 , p ) Π l m ( II ) ( r ) ] ,
W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) ,
μ ( r 1 , r 2 , ω ) = Tr [ W ( r 1 , r 2 , ω ) ] S ( r 1 , ω ) S ( r 2 , ω ) ,
μ ξ ( r 1 , r 2 , ω ) = Tr [ W ( r 1 , r 2 , ω ) W ( r 2 , r 1 , ω ) ] S ( r 1 , ω ) S ( r 2 , ω ) .
S ( r 1 , ω ) S ( r 2 , ω ) | μ ( r 1 , r 2 , ω ) | 2 = | i W i i ( r 1 , r 2 , ω ) | 2 ,
S ( r 1 , ω ) S ( r 2 , ω ) | μ ξ ( r 1 , r 2 , ω ) | 2 = i , j | W i j ( r 1 , r 2 , ω ) | 2 .
μ ( r , r , ω ) = 1 , μ ξ ( r , r , ω ) = Tr [ J 2 ( r , ω ) ] S ( r , ω ) 1 ,
W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω )
μ ( r 1 , r 2 , ω ) = E * ( r 1 , ω ) E ( r 2 , ω ) | E ( r 1 , ω ) | | E ( r 2 , ω ) | 1 , μ ξ ( r 1 , r 2 , ω ) = 1.
P = 1 4 Det J [ Tr ( J ) ] 2 = 2 [ Tr ( J 2 ) [ Tr ( J ) ] 2 1 2 ] = λ 1 λ 2 λ 1 + λ 2 ,
P = λ 1 λ 2 λ 1 + λ 2 + λ 3 ,
P ξ = 3 2 [ Tr ( J 2 ) [ Tr ( J ) ] 2 1 3 ] = λ 1 2 + λ 2 2 + λ 3 2 λ 1 λ 2 λ 1 λ 3 λ 2 λ 3 λ 1 + λ 2 + λ 3 .
μ ξ ( r , r , ω ) = 2 P ξ 2 + 1 3 .
2 π q = Γ r 1 arg [ μ ( r 1 , r 2 ; ω ) ] d t 1 ,

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