Abstract

The dynamics of the degree of polarization and the degree of cross-polarization for electromagnetic pulsed vortex beams in dispersive media are explored both in the space-time and space-frequency domains. It is shown that the impacts, arising from the second-order dispersion coefficient and the temporal coherence length, on the variations of the temporal degree of polarization are distinctly different from those on the spectral degree of polarization. Besides, we also suggest a method to access the measurement of the orbital angular momentum of pulsed vortex beams through the mapping relationship between the distribution of the temporal degree of cross-polarization and the number of topological charge.

© 2015 Optical Society of America

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J. Opt. Soc. Am. A 32(5) 741-750 (2015)

References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  21. M. Luo and D. Zhao, “Determining the topological charge of stochastic electromagnetic vortex beams with the degree of cross-polarization,” Opt. Lett. 39(17), 5070–5073 (2014).
    [Crossref] [PubMed]
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    [Crossref]

2014 (2)

2013 (2)

C. Ding, Y. Cai, Y. Zhang, H. Wang, Z. Zhao, and L. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

T. Voipio, T. Setala, and A. T. Friberg, “Partial polarization theory of pulsed optic al beams,” J. Opt. Soc. Am. A 30(1), 71–81 (2013).
[Crossref]

2011 (1)

2010 (1)

2009 (2)

2007 (3)

2005 (4)

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
[Crossref] [PubMed]

A. Matos-Abiague and J. Berakdar, “Photoinduced charge currents in mesoscopic rings,” Phys. Rev. Lett. 94(16), 166801 (2005).
[Crossref] [PubMed]

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[Crossref] [PubMed]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

2003 (3)

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

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

2002 (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

1998 (1)

Agrawal, G. P.

Berakdar, J.

A. Matos-Abiague and J. Berakdar, “Photoinduced charge currents in mesoscopic rings,” Phys. Rev. Lett. 94(16), 166801 (2005).
[Crossref] [PubMed]

Bertolotti, M.

Brixner, T.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[Crossref] [PubMed]

Cada, M.

Cai, Y.

C. Ding, Y. Cai, Y. Zhang, H. Wang, Z. Zhao, and L. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Chen, H.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Ding, C.

Du, X.

Ferrari, A.

Friberg, A. T.

T. Voipio, T. Setala, and A. T. Friberg, “Partial polarization theory of pulsed optic al beams,” J. Opt. Soc. Am. A 30(1), 71–81 (2013).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

García de Abajo, F. J.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[Crossref] [PubMed]

Hnatovsky, C.

Huang, W.

Korotkova, O.

Krolikowski, W.

Kuebel, D.

D. Kuebel, “Properties of the degree of cross-polarization in the space–time domain,” Opt. Commun. 282(17), 3397–3401 (2009).
[Crossref]

Lajunen, H.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
[Crossref] [PubMed]

Lin, Q.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Lü, B.

Luo, M.

Matos-Abiague, A.

A. Matos-Abiague and J. Berakdar, “Photoinduced charge currents in mesoscopic rings,” Phys. Rev. Lett. 94(16), 166801 (2005).
[Crossref] [PubMed]

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Padgett, M. J.

Pan, L.

Pfeiffer, W.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[Crossref] [PubMed]

Ponomarenko, S. A.

Rode, A. V.

Schneider, J.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[Crossref] [PubMed]

Sereda, L.

Setala, T.

Shirai, T.

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]

Shvedov, V. G.

Tervo, J.

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
[Crossref] [PubMed]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

Turunen, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Vahimaa, P.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
[Crossref] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Voipio, T.

Wang, H.

C. Ding, Y. Cai, Y. Zhang, H. Wang, Z. Zhao, and L. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Wang, L.

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Wolf, E.

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]

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

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Yao, A. M.

Zhang, Y.

C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014).
[Crossref] [PubMed]

C. Ding, Y. Cai, Y. Zhang, H. Wang, Z. Zhao, and L. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Zhao, D.

Zhao, Z.

C. Ding, Y. Cai, Y. Zhang, H. Wang, Z. Zhao, and L. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Zhu, S.

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Zhu, S. Y.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Adv. Opt. Photon. (1)

J. Opt. Soc. Am. A (4)

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

Opt. Commun. (5)

D. Kuebel, “Properties of the degree of cross-polarization in the space–time domain,” Opt. Commun. 282(17), 3397–3401 (2009).
[Crossref]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[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]

Opt. Express (2)

Opt. Lett. (2)

Phys. Lett. A (2)

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

C. Ding, Y. Cai, Y. Zhang, H. Wang, Z. Zhao, and L. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

A. Matos-Abiague and J. Berakdar, “Photoinduced charge currents in mesoscopic rings,” Phys. Rev. Lett. 94(16), 166801 (2005).
[Crossref] [PubMed]

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[Crossref] [PubMed]

Other (3)

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

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

Q. Lin and S. Wang, Tensor Optics-Generalized Beam Transformation and Beam Quality (Hangzhou University, 1994).

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Figures (4)

Fig. 1
Fig. 1

Changes in the temporal degree of polarization on-axis as a function of z for the choice of (a) second-order dispersion coefficient β 2 while T cx =0.4ps , T cy =0.1ps and (b) temporal coherence length T cx with β 2 =20 ps 2 / km .

Fig. 2
Fig. 2

Behaviors in the temporal degree of polarization on-axis at the propagation distance z=20m versus t when β 2 =20 ps 2 / km for different parameters of (a) topological charge m while T cx =0.4ps , T cy =0.1ps and (b) temporal coherence length T cx with m=1 .

Fig. 3
Fig. 3

The temporal degree of cross-polarization of spatially and temporally partially coherent EM vortex pulses for different topological charges (a) m=1 , (b) m=2 , (c) m=3 at z=20m with t 1 =0.1ps , t 2 =0.2ps , and T cx =0.4ps , T cy =0.1ps .

Fig. 4
Fig. 4

Variations in the spectral degree of polarization on-axis as a function of z, the other parameters are the same with Fig. 1.

Equations (24)

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Γ ij (0) ( x 10 , y 10 , t 10 ; x 20 , y 20 , t 20 )= R ij ( x 10 , y 10 , x 20 , y 20 ) T ij ( t 10 , t 20 ),
R ij ( x 10 , y 10 , x 20 , y 20 )= A i A j B ij ( x 10 i y 10 ) m ( x 20 +i y 20 ) m σ 2m exp( x 10 2 + y 10 2 + x 20 2 + y 20 2 σ 2 ) ×exp[ ( x 10 x 20 ) 2 + ( y 10 y 20 ) 2 δ ij 2 ],(i=x,y;j=x,y),
B ii = B jj =1, | B ij |=| B ji |.
T ij ( t 10 , t 20 )=exp[ t 10 2 + t 20 2 2 T 0 2 ( t 10 t 20 ) 2 2 T cij 2 +i ω 0 ( t 10 t 20 ) ],
Γ ij ( ρ 1 , ρ 2 , t 1 , t 2 ,z )= [ k(ω) 2πz ] 2 ω 0 2πa Γ ij (0) ( x 10 , y 10 , t 10 ; x 20 , y 20 , t 20 ) ×exp{ ik(ω) 2z [ ( ρ 1 r 1 ) 2 ( ρ 2 r 2 ) 2 ] } d 2 r 1 d 2 r 2 exp{ i ω 0 2a [ ( t 1 t 10 ) 2 ( t 2 t 20 ) 2 ] }d t 10 d t 20 ,
A=D=ε,C=0,B=( z/ n(ω) 0 0 0 z/ n(ω) 0 0 0 λ 0 ω 0 2 z v g 2 v g ω )
Γ ij ( ρ 1 , ρ 2 , t 1 , t 2 ,z )= ( k(ω) 2πz ) 2 R ij ( r 1 , r 2 ) exp{ ik(ω) 2z [ ( ρ 1 r 1 ) 2 ( ρ 2 r 2 ) 2 ] } d 2 r 1 d 2 r 2 × ω 0 2πa T ij ( t 10 , t 20 ) exp{ i ω 0 2a [ ( t 1 t 10 ) 2 ( t 2 t 20 ) 2 ] }d t 10 d t 20 = R tij ( ρ 1 , ρ 2 ,z) Y t i ( t 1 , t 2 ,z).
R tij ( ρ 1 , ρ 2 ,z) = A i A j B ij z 2m Δ ij m+1 k (ω) 2m σ 2m exp[ β ij ρ 1 2 + α ij ρ 2 2 2( x 1 x 2 + y 1 y 2 ) / δ ij 2 Δ ij ik(ω) 2z ( ρ 1 2 ρ 2 2 ) ] × n=0 m ( C m n ) 2 n! 4 n δ ij 2n [ 4( β ij ρ 1 2 + α ij ρ 2 2 ) Δ ij δ ij 2 +i k (ω) 2 z 2 ( x 1 y 2 x 2 y 1 )+( k (ω) 2 z 2 + 8 Δ ij δ ij 4 )( x 1 x 2 + y 1 y 2 ) ] mn ,
Y t i ( t 1 , t 2 ,z)= 1 Λ i exp[ ( t 1 + t 2 ) 2 4 T 0 2 Λ i ]exp[ i ω 0 2a ( 1 1 Λ i )( t 1 2 t 2 2 ) ]exp[ α ti Λ i ( t 1 t 2 ) 2 ],
P t ( ρ,t,z )= 1 4Det[ Γ ( ρ,ρ,t,t,z ) ] [ Tr Γ ( ρ,ρ,t,t,z ) ] 2
P t ( ρ,t,z )=| Γ xx ( ρ,ρ,t,z ) Γ yy ( ρ,ρ,t,z ) Γ xx ( ρ,ρ,t,z )+ Γ yy ( ρ,ρ,t,z ) |,
Γ ii ( ρ,ρ,t,z )= A i 2 z 2m Δ ii m+1 k (ω) 2m σ 2m 1 Λ i exp( β ii + α ii 2/ δ ii 2 Δ ii ρ 2 ) × n=0 m ( C m n ) 2 n! 4 n δ ii 2n [ [ 4( β ii + α ii ) Δ ii δ ii 2 + k (ω) 2 z 2 + 8 Δ ii δ ii 4 ] ρ 2 ] mn exp[ t 2 T 0 2 Λ i ] .
P t ( ρ,ρ, t 1 , t 2 ,z )=| Γ xx ( ρ,ρ, t 1 , t 2 ,z ) Γ yy ( ρ,ρ, t 1 , t 2 ,z ) Γ xx ( ρ,ρ, t 1 , t 2 ,z )+ Γ yy ( ρ,ρ, t 1 , t 2 ,z ) |.
W ij (0) ( x 10 , y 10 , ω 1 ; x 20 , y 20 , ω 2 )= 1 (2π) 2 × Γ ij (0) ( x 10 , y 10 , t 10 ; x 20 , y 20 , t 20 )exp[ i( ω 1 t 10 ω 2 t 20 ) ]d t 10 d t 20 ,
W ij (0) ( x 10 , y 10 , ω 1 ; x 20 , y 20 , ω 2 )= A i A j B ij T 0 2π Ω 0ij σ 2m ( x 10 i y 10 ) m ( x 20 +i y 20 ) m ×exp( x 10 2 + y 10 2 + x 20 2 + y 20 2 σ 2 )exp[ ( x 10 x 20 ) 2 + ( y 10 y 20 ) 2 δ ij 2 ] ×exp[ ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 2 Ω 0ij 2 ( ω 1 ω 2 ) 2 2 Ω cij 2 ].
W ij ( ρ 1 , ρ 2 , ω 1 , ω 2 ,z )= k 1 ( ω 1 ) k 2 ( ω 2 ) 4 π 2 z 2 exp{ i[ k 2 ( ω 2 ) k 1 ( ω 1 ) ]z } × W ij (0) ( r 1 , r 2 , ω 1 , ω 2 ) exp{ i 2z [ k 1 ( ω 1 ) ( ρ 1 r 1 ) 2 k 2 ( ω 2 ) ( ρ 2 r 2 ) 2 ] } d 2 r 1 d 2 r 2 .
W ij ( ρ 1 , ρ 2 , ω 1 , ω 2 ,z )= R wij ( ρ 1 , ρ 2 ,z) Y wij ( ω 1 , ω 2 ,z).
R wij ( ρ 1 , ρ 2 ,z)= A i A j B ij z 2m Δ wij m+1 k 1 m ( ω 1 ) k 2 m ( ω 2 ) σ 2m n=0 m ( C m n ) 2 n! 4 n δ ij 2n ×exp[ β wij ρ 1 2 + α wij ρ 2 2 2( x 1 x 2 + y 1 y 2 ) / δ ij 2 Δ wij i 2z [ k 1 ( ω 1 ) ρ 1 2 k 2 ( ω 2 ) ρ 2 2 ] ] × [ 4( β wij ρ 1 2 + α wij ρ 2 2 ) Δ wij δ ij 2 +i k 1 ( ω 1 ) k 2 ( ω 2 ) z 2 ( x 1 y 2 x 2 y 1 )+[ k 1 ( ω 1 ) k 2 ( ω 2 ) z 2 + 8 Δ wij δ ij 4 ]( x 1 x 2 + y 1 y 2 ) ] mn .
Y wij ( ω 1 , ω 2 ,z)= T 0 2π Ω 0ij exp{ i[ k 2 ( ω 2 ) k 1 ( ω 1 ) ]z }exp[ ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 2 Ω 0ij 2 ( ω 1 ω 2 ) 2 2 Ω cij 2 ].
W ij ( ρ,ρ,ω,z )= A i A j B ij z 2m T 0 2π Ω 0ij Δ wij m+1 k (ω) 2m σ 2m exp[ β wij + α wij 2/ δ ij 2 Δ wij ρ 2 ] × n=0 m ( C m n ) 2 n! 4 n δ ij 2n { [ 4( β wij + α wij ) Δ wij δ ij 2 + k (ω) 2 z 2 + 8 Δ wij δ ij 4 ] ρ 2 } mn exp[ ( ω ω 0 ) 2 Ω 0ij 2 ]
P ω ( ρ,ω,z )=| W xx ( ρ,ρ,ω,z ) W yy ( ρ,ρ,ω,z ) W xx ( ρ,ρ,ω,z )+ W yy ( ρ,ρ,ω,z ) |,
P ω ( ρ,ρ, ω 1 , ω 2 ,z )=| W xx ( ρ,ρ, ω 1 , ω 2 ,z ) W yy ( ρ,ρ, ω 1 , ω 2 ,z ) W xx ( ρ,ρ, ω 1 , ω 2 ,z )+ W yy ( ρ,ρ, ω 1 , ω 2 ,z ) |.
A x 2 Ω 0xx Δ wxx m+1 δ xx 2m exp[ ( ω ω 0 ) 2 Ω 0xx 2 ]= A y 2 Ω 0yy Δ wyy m+1 δ yy 2m exp[ ( ω ω 0 ) 2 Ω 0yy 2 ]
Δ wxx Δ wyy = ( A x A y ) 2 Ω 0y Ω 0x exp[ ( ω ω 0 ) 2 Ω 0y 2 ( ω ω 0 ) 2 Ω 0x 2 ].

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