Abstract

Adding a twist phase term to the cross-spectral density (CSD) function of a partially coherent source can be done if and only if the resulting function remains nonnegative definite. Constraints on the twist term that guarantee the validity of the resulting CSD have been derived only for Twisted Gaussian Schell-model (TGSM) sources. Here, an infinite family of higher-order TGSM sources is introduced, whose CSDs are expressed as products of the CSD of a standard TGSM source times Hermite polynomials of arbitrary orders and suitable arguments. All the members present the same twist term and for all of them the twist-coherence constraint keeps obeying the form valid for a standard TGSM source. They can be used as building blocks for constructing an endless number of valid twisted CSDs, with an assigned value of the twist parameter and intensity and/or coherence features that can be very different from those of a standard TGSM source. Through partial transposition, higher-order TGSM CSDs are converted into higher-order Astigmatic Gaussian Schell-model (AGSM) CSDs. The problem of the separability of higher-order TGSM and AGSM CSDs is addressed, and conditions ensuring their separability are derived.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  7. F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
    [Crossref] [PubMed]
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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]
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    [Crossref]
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    [Crossref]
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2018 (5)

2017 (3)

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42, 255–258 (2017).
[Crossref] [PubMed]

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

2015 (3)

2014 (1)

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

2013 (1)

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

2011 (1)

2010 (2)

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

2009 (1)

2007 (1)

2004 (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

2001 (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

1998 (2)

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

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1996 (1)

1994 (3)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

1993 (1)

Agarwal, G. S.

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

Aiello, A.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Ambrosini, D.

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Arvind,

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Bagini, V.

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Borghi, R.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

Cai, Y.

Chaturvedi, S.

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Chen, Y.

Chowdhury, P.

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Eberly, J. H.

Friberg, A. T.

Fu, W.

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

Gbur, G.

Ghose, P.

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

Giacobino, E.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 4th edition (W. H. Freeman, 2017).

Gori, F.

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
[Crossref] [PubMed]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

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

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

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

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Guattari, G.

Huang, H.

Korotkova, O.

Leuchs, G.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Li, Z.

Majumdar, A. S.

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Mandel L., L.

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

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

Marquardt, C.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Martinez-Herrero, R.

Mei, Z.

Mejías, P. M.

Mukherjee, A.

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

Mukunda, N.

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

Pacileo, A. M.

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

Qian, X.

Qian, X. F.

Santarsiero, M.

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
[Crossref] [PubMed]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

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

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

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Schirripa Spagnolo, G.

Siegman, A. E.

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

Simon, B. N.

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Simon, R.

Simon, S.

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Spreeuw, R. J. C.

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

Stahl, C. S. D.

Szegö, G.

G. Szegö, Orthogonal Polynomials, 4th edition (American Mathematical Society, 1975).

Tervonen, E.

Töppel, F.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Turunen, J.

Vamivakas, A. N.

Vicalvi, S.

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

Wang, H.

Wang, J.

Wolf, E.

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

Zhang, H.

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

Zhu, S.

Eur. Phys. J. D (1)

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

Fortschr. Phys. (1)

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

J. Mod. Optics (1)

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

J. Opt. Soc. Am. A (5)

New J. Phys. (1)

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Opt. Commun. (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

Opt. Express (1)

Opt. Lett. (8)

Optica (1)

Phys. Rev. A (1)

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Phys. Rev. E (2)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Phys. Rev. Lett. (1)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Rev. Theor. Sci. (1)

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

Other (5)

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

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

G. Szegö, Orthogonal Polynomials, 4th edition (American Mathematical Society, 1975).

J. W. Goodman, Introduction to Fourier Optics, 4th edition (W. H. Freeman, 2017).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

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Figures (2)

Fig. 1
Fig. 1 Plot of the intensity in Eq. (25) as a function of x/δ and y/δ, with a = b = α.
Fig. 2
Fig. 2 Contour plot of |μ10|, obtained from Eq. (26), as a function of (x1x2)/δ and (y1y2)/δ for a = b = α and different values of /δ: 0 (a); 0.5 (b); 1.0 (c); 1.5 (d).

Equations (46)

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W G ( r 1 , r 2 ) = I 0 exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 ] ,
W T ( r 1 , r 2 ) = I 0 exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 ] exp [ i k u ( x 1 y 2 x 2 y 1 ) ] ,
k | u | 1 / δ 2 ,
W ( r 1 , r 2 ) = p ( ρ ) H ( r 1 , ρ ) H * ( r 2 , ρ ) d ρ ,
H ( r , ρ ) = exp [ a ( r ρ ) 2 2 π i α ( s y t x ) ] ,
W ( r 1 , r 2 ) = p ( s , t ) exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] + 2 π i α t ( x 1 x 2 ) } × exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α s ( y 1 y 2 ) } d s d t .
p ( s , t ) exp [ b ( s 2 + t 2 ) ] ,
1 4 σ 2 = a b 2 a + b , 1 2 δ 2 = a 2 + π 2 α 2 2 a + b , k u = 4 π α a 2 a + b .
k | u | δ 2 1 = 2 π | α | a a 2 + π 2 α 2 1 = ( a π | α | 2 ) a 2 + π 2 α 2 0 ,
m = 0 M n = 0 N d m n s 2 m t 2 n ,
n = 0 N c n ( s 2 + t 2 ) n = n = 0 N c n k = 0 n ( n k ) s 2 k t 2 ( n k ) ,
W ( r 1 , r 2 ) = m = 0 M n = 0 N d m n W m n ( r 1 , r 2 ) ,
W mn ( r 1 , r 2 ) = s 2 m t 2 n exp [ b ( s 2 + t 2 ) ] × exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] 2 π i α s ( y 1 y 2 ) } × exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α t ( x 1 x 2 ) } d s d t .
W m n ( r 1 , r 2 ) = S m ( r 1 , r 2 ) T n ( r 1 , r 2 ) ,
S m ( r 1 , r 2 ) = s 2 m exp ( b s 2 ) exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] 2 π i α s ( y 1 y 2 ) } d s ,
T n ( r 1 , r 2 ) = t 2 n exp ( b t 2 ) exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α t ( x 1 x 2 ) } d t .
S 0 ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ a 2 2 a + b ( x 1 x 2 ) 2 π 2 α 2 2 a + b ( y 1 y 2 ) 2 ] × exp [ i 2 π a α 2 a + b ( x 1 y 1 x 2 y 2 x 1 y 2 + x 2 y 1 ) ] ,
W m n ( r 1 , r 2 ) = 4 π ( 1 ) m + n [ 4 ( 2 a + b ) ] n + m + 1 exp [ a b 2 a + b ( r 1 2 + r 2 2 ) ] × exp [ a 2 + π 2 α 2 2 a + b ( r 1 r 2 ) 2 + i 4 π a α 2 a + b ( x 1 y 2 x 2 y 1 ) ] × H 2 m [ π α ( y 1 y 2 ) + i a ( x 1 + x 2 ) 2 a + b ] H 2 n [ π α ( x 1 x 2 ) i a ( y 1 + y 2 ) 2 a + b ] ,
W m n ( r 1 , r 2 ) = A m n ( 1 ) m + n exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k u ( x 1 y 2 x 2 y 1 ) ] × H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] H 2 n [ c ( x 1 x 2 ) i c + ( y 1 + y 2 ) 2 δ ] ,
c + = cos ϕ ; c = sin ϕ ,
ϕ = arctan ( π α a ) .
I m n ( r ) = A m n ( 1 ) m + n exp ( r 2 2 σ 2 ) H 2 m ( i 2 c + x δ ) H 2 n ( i 2 c + y δ ) .
| μ m n ( r 1 , r 2 ) | = exp [ ( r 1 , r 2 ) 2 2 δ 2 ] | H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] | | H 2 m ( i 2 c + x 1 δ ) H 2 m ( i 2 c + x 2 δ ) | × | H 2 n [ c ( x 1 x 2 ) + i c + ( y 1 + y 2 ) 2 δ ] | | H 2 n ( i 2 c + y 1 δ ) H 2 n ( i 2 c + y 2 δ ) | .
W 10 ( r 1 , r 2 ) = 2 A 10 exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k u ( x 1 y 2 x 2 y 1 ) ] × { 1 2 [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] 2 } ,
I 10 ( r ) = 2 A 10 exp ( r 2 2 σ 2 ) ( 1 + 4 c + 2 x 2 δ 2 ) ,
| μ 10 ( r 1 , r 2 ) | = exp [ ( r 1 , r 2 ) 2 2 δ 2 ] | 1 2 [ c ( y 1 y 2 ) δ + i c + ( x 1 + x 2 ) δ ] 2 | ( 1 + 4 c + 2 x 1 2 δ 2 ) ( 1 + 4 c + 2 x 2 2 δ 2 )
W ¯ m n ( r 1 , r 2 ) = A m n ( 1 ) m + n exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k u ( x 2 y 2 x 1 y 1 ) ] × H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] H 2 n [ c ( x 1 x 2 ) + i c + ( y 1 + y 2 ) 2 δ ] ,
H ( r , ρ ) = exp [ a ( r , ρ ) 2 + 2 π i α ( x s ) ( y t ) ] .
W m n ( r 1 , r 2 ) = s 2 m t 2 n e b ( s 2 + t 2 ) exp [ a ( r 1 ρ ) 2 + 2 π i α ( x 1 s ) ( y 1 t ) ] × exp [ a ( r 2 ρ ) 2 2 π i α ( x 2 s ) ( y 2 t ) ] d s d t ,
W ¯ m n ( r 1 , r 2 ) = s 2 m t 2 n e b ( s 2 + t 2 ) exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] 2 π i α t ( x 1 x 2 ) } × exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α s ( y 1 y 2 ) } d s d t .
W m n ( r 1 , r 2 ) = W ¯ m n ( r 1 , r 2 ) exp [ 2 π i α ( x 1 y 1 x 2 y 2 ) ] ,
W m n ( r 1 , r 2 ) = A m n ( 1 ) m + n exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k v ( x 2 y 2 x 1 y 1 ) ] × H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] H 2 n [ c ( x 1 x 2 ) + i c + ( y 1 + y 2 ) 2 δ ] ,
k v = 4 π α a 2 a + b + 2 π α = 2 π b α 2 a + b .
k v δ 2 = π b α a 2 + π 2 α 2 ,
S m ( r 1 , r 2 ) = exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 ] × ( ζ + x 12 ) 2 m e ( 2 a + b ) ζ 2 e 2 π i γ 12 ζ d ζ ,
g = a 2 2 a + b , x 12 = a ( x 1 + x 2 ) 2 a + b , γ 12 = α ( y 1 y 2 ) .
S m ( r 1 , r 2 ) = exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 ] × k = 0 2 m ( 2 m k ) x 12 k ζ 2 m k e ( 2 a + b ) ζ 2 e 2 π i γ 12 ζ d ζ ,
S m ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 ] × k = 0 2 m ( 2 m k ) x 12 k ( 2 π i ) 2 m k d 2 m k d γ 12 2 m k [ e π 2 γ 12 2 / ( 2 a + b ) ] .
d n d x n e x 2 = ( 1 ) n e x 2 H n ( x ) ,
S m ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 π 2 γ 12 2 2 a + b ] × k = 0 2 m ( 2 m k ) x 12 k ( 2 i 2 a + b ) 2 m k H 2 m k ( π γ 12 2 a + b ) .
S m ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ a 2 ( x 1 x 2 ) 2 + π 2 α 2 ( y 1 y 2 ) 2 2 a + b ] × exp [ i 2 π a α 2 a + b ( x 1 y 1 x 2 y 2 x 1 y 2 + x 2 y 1 ) ] F m ( x 1 + x 2 , y 1 y 2 ) ,
F m ( ξ , η ) = k = 0 2 m ( 2 m k ) ( a ξ 2 a + b ) k ( i 2 2 a + b ) 2 m k H 2 m k ( π α η 2 a + b )
k = 0 n ( n k ) ( 2 y ) n k H k ( x ) = H n ( x + y ) ,
F m ( ξ , η ) = [ 1 4 ( 2 a + b ) ] m H 2 m ( π α η + i a ξ 2 a + b ) .
T n ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ a 2 ( y 1 y 2 ) 2 + π 2 α 2 ( x 1 x 2 ) 2 2 a + b ] × exp [ i 2 π a α 2 a + b ( y 1 x 1 y 2 x 2 y 1 x 2 + y 2 x 1 ) ] F n * ( y 1 + y 2 , x 1 x 2 ) .
W m n ( r 1 , r 2 ) = π ( 1 ) m + n 4 m + n ( 2 a + b ) m + n + 1 exp [ a b 2 a + b ( r 1 2 + r 2 2 ) ] × exp [ a 2 + π 2 α 2 2 a + b ( r 1 r 2 ) 2 + i 4 π a α 2 a + b ( x 1 y 2 x 2 y 1 ) ] × H 2 m [ π α ( y 1 y 2 ) + i a ( x 1 + x 2 ) 2 a + b ] H 2 n [ π α ( x 1 x 2 ) i a ( y 1 + y 2 ) 2 a + b ] .

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