Abstract

We introduce a new kind of partially coherent source whose cross-spectral density (CSD) function is described as the incoherent superposition of elliptical twisted Gaussian Schell-model sources with different beam widths and transverse coherence widths, named twisted elliptical multi-Gaussian Schell-model (TEMGSM) beams. Analytical expression for the CSD function propagating through a paraxial ABCD optical system is derived with the help of the generalized Collins formula. Our results show that the TEMGSM beam is capable of generating a flat-topped elliptical beam profile in the far field, and the beam spot during propagation exhibits clockwise/anti-clockwise rotation with respect to its propagation axis. In addition, the analytical expressions for the orbital angular momentum (OAM) and the propagation factor are also derived by means of the Wigner distribution function. The influences of the twisted factor and the beam index on the OAM and the propagation factor are studied and discussed in detail.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  36. 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]
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    [Crossref]
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2019 (1)

2018 (9)

2017 (2)

2016 (1)

2015 (1)

2014 (1)

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16, 075704 (2014).
[Crossref]

2013 (3)

2012 (5)

2009 (2)

2007 (1)

2006 (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

2002 (2)

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738 (2002).
[Crossref]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19, 2036–2042 (2002).
[Crossref]

2001 (1)

2000 (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

1997 (1)

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

1994 (1)

1993 (4)

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, 8185–8189 (1992).
[Crossref]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

Ahmed, N.

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

Alarify, Y. S.

Alkelly, A. A.

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, 8185–8189 (1992).
[Crossref]

Alonzo, M.

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

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, 8185–8189 (1992).
[Crossref]

Borghi, R.

Cai, Y.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref]

X. Peng, L. Liu, F. Wang, S. Popov, and Y. Cai, “Twisted Laguerre-Gaussian Schell-model beam and its orbital angular moment,” Opt. Express 26, 33956–33969 (2018).
[Crossref]

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16, 075704 (2014).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38, 1814–1816 (2013).
[Crossref]

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37, 184–186 (2012).
[Crossref]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17, 2453–2464 (2009).
[Crossref]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17, 21472–21487 (2009).
[Crossref]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19, 2036–2042 (2002).
[Crossref]

Cao, P.

Chen, Y.

Dholakia, K.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

Dolinar, S.

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

Duan, M.

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]

Eyyuboglu, H. T.

Fazal, I.

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

Feng, Z.

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]

Friberg, A. T.

Fu, W.

Gbur, G.

Ge, D.

Gori, F.

Guattari, G.

Guo, M.

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]

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Hirano, T.

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

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

Huang, H.

J. Zhang, J. Wang, H. Huang, H. Wang, S. Zhu, Z. Li, and J. Lu, “Propagation characteristics of a twisted cosine-Gaussian correlated radially polarized beam,” Appl. Sci. 8, 1485 (2018).
[Crossref]

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref]

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

Korotkova, O.

Kuga, T.

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

Li, J.

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]

Li, Z.

Lin, Q.

Liu, L.

Liu, X.

Lu, J.

J. Zhang, J. Wang, H. Huang, H. Wang, S. Zhu, Z. Li, and J. Lu, “Propagation characteristics of a twisted cosine-Gaussian correlated radially polarized beam,” Appl. Sci. 8, 1485 (2018).
[Crossref]

Lü, B.

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738 (2002).
[Crossref]

Luo, S.

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738 (2002).
[Crossref]

Martinez-Herrero, R.

Mei, Z.

Mejias, P. M.

Movilla, J. M.

Mukunda, N.

Peng, X.

Popov, S.

Ren, Y.

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

Sahin, S.

Santarsiero, M.

Serna, J.

Shiokawa, N.

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

Shukri, M. A.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

Simon, R.

Spreeuw, R. J. C.

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, 8185–8189 (1992).
[Crossref]

Stahl, C. S. D.

Sundar, K.

Tervonen, B.

Tong, Z.

Torii, Y.

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

Tur, M.

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

Turunen, J.

Wang, F.

Wang, H.

Wang, J.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref]

J. Wang, H. Wang, S. Zhu, and Z. Li, “Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence,” J. Opt. Soc. Am. A 35, 1173–1179 (2018).
[Crossref]

J. Zhang, J. Wang, H. Huang, H. Wang, S. Zhu, Z. Li, and J. Lu, “Propagation characteristics of a twisted cosine-Gaussian correlated radially polarized beam,” Appl. Sci. 8, 1485 (2018).
[Crossref]

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

Willner, A.

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

Woerdman, J. P.

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, 8185–8189 (1992).
[Crossref]

Wu, G.

Yan, Y.

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

Yang, J.

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

Yang, W.

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]

Yuan, Y.

Yue, Y.

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

Zhang, H.

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]

Zhang, J.

J. Zhang, J. Wang, H. Huang, H. Wang, S. Zhu, Z. Li, and J. Lu, “Propagation characteristics of a twisted cosine-Gaussian correlated radially polarized beam,” Appl. Sci. 8, 1485 (2018).
[Crossref]

Zhang, Y.

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16, 075704 (2014).
[Crossref]

Zhao, C.

Zhao, D.

Zhou, Y.

Zhu, S.

Appl. Opt. (1)

Appl. Phys. B (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Appl. Sci. (1)

J. Zhang, J. Wang, H. Huang, H. Wang, S. Zhu, Z. Li, and J. Lu, “Propagation characteristics of a twisted cosine-Gaussian correlated radially polarized beam,” Appl. Sci. 8, 1485 (2018).
[Crossref]

J. Mod. Opt. (1)

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738 (2002).
[Crossref]

J. Opt. (1)

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16, 075704 (2014).
[Crossref]

J. Opt. Soc. Am. A (10)

G. Wu, “Propagation properties of a radially polarized partially coherent twisted beam in free space,” J. Opt. Soc. Am. A 33, 345–350 (2016).
[Crossref]

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

R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[Crossref]

K. Sundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
[Crossref]

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

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19, 2036–2042 (2002).
[Crossref]

M. A. Shukri, A. A. Alkelly, and Y. S. Alarify, “Spatial correlation properties of twisted partially coherent light focused by diffractive axicons,” J. Opt. Soc. Am. A 29, 2019–2027 (2012).
[Crossref]

C. S. D. Stahl and G. Gbur, “Twisted vortex Gaussian Schell-model beams,” J. Opt. Soc. Am. A 35, 1899–1906 (2018).
[Crossref]

W. Fu and P. Cao, “Second-order statistics of a radially polarized partially coherent twisted beam in a uniaxial crystal,” J. Opt. Soc. Am. A 34, 1703–1710 (2017).
[Crossref]

J. Wang, H. Wang, S. Zhu, and Z. Li, “Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence,” J. Opt. Soc. Am. A 35, 1173–1179 (2018).
[Crossref]

Nat. Photonics (1)

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

Opt. Express (5)

Opt. Lett. (13)

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

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37, 2970–2972 (2012).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38, 91–93 (2013).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. (a)–(d) Density plots of normalized spectral density of the TEMGSM source for different values of $u$ with $M={10}$. (a-1)–(d-1) The corresponding modulus of the DOC. The solid curves in (a)–(d) denotes the cross line ($y={0}$) of spectral density.
Fig. 2.
Fig. 2. Modulus of the DOC for a TEMGSM source with different values of twisted factor $u$ (set beam index $M={10}$): (a) $| \gamma ({x_1},{y_1}=0;{\bf r}_2=0) |$; (b) $| {\gamma ({{x_1}=0,{y_1};{{\bf r}_2}=0})} |$.
Fig. 3.
Fig. 3. (a)–(d) Density plots of normalized spectral density of the TEMGSM source for different values of beam index $M$ with $u={18}\,\,{\text{mm}^{-2}}$. (a-1)–(d-1) The corresponding modulus of the DOC. The solid curves in (a)–(d) denote the cross line ($y={0}$) of spectral density.
Fig. 4.
Fig. 4. Density plots of normalized spectral density of the TEMGSM beam with (a)–(e) $u=-{20}\,\,{\text{mm}^{-2}}$ and (a-1)–(e-1) $u={20}\,\,{\text{mm}^{-2}}$ at several propagation distances in free space.
Fig. 5.
Fig. 5. Density plots of normalized spectral density and the corresponding cross line of the TEMGSM beam for (a)–(d) different values of beam index $M$ with $u={20}\,\,{\text{mm}^{-2}}$ and (e)–(h) for different values of $u$ with $M={16}$ at $\text{z}={1000}\,\,\text{m}$ (in the far field).
Fig. 6.
Fig. 6. Variation of the kurtosis parameter with the twisted factor for different values of $M$ in the far field.
Fig. 7.
Fig. 7. Variation of the OAM of the TEMGSM beam for different values of twisted factor $u$ with different beam index $M$.
Fig. 8.
Fig. 8. Variation of the propagation factor of a TEMGSM beam in the $x$ and $y$ directions for different values of twisted factor $u$ with different beam index $M$.

Equations (25)

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W 0 ( r 1 , r 2 ) = p ( v ) H 0 ( r 1 , v ) H 0 ( r 2 , v ) d v ,
H 0 ( r , v ) = exp ( σ r 2 ) × exp { [ ( u α y + i x ) v x ( u β x i y ) v y ] } ,
p ( v ) = α β π 1 C 0 m = 1 M ( 1 ) m 1 ( M m ) exp ( m α v x 2 m β v y 2 ) ,
W 0 ( r 1 , r 2 ) = 1 C 0 m = 1 M ( 1 ) m 1 m ( M m ) × exp [ ( x 1 2 + x 2 2 ) 4 ω m x 2 ( y 1 2 + y 2 2 ) 4 ω m y 2 ] × exp [ ( x 1 x 2 ) 2 2 δ m x 2 ( y 1 y 2 ) 2 2 δ m y 2 ] × exp [ i μ m ( x 1 y 2 x 2 y 1 ) ] ,
1 4 ω m x 2 = ( σ β u 2 2 m ) , 1 4 ω m y 2 = ( σ α u 2 2 m ) , 1 2 δ m x 2 = ( 1 4 m α + β u 2 4 m ) , 1 2 δ m y 2 = ( 1 4 m β + α u 2 4 m ) , μ m = u m .
γ ( r 1 , r 2 ) = W 0 ( r 1 , r 2 ) W 0 ( r 1 , r 1 ) W 0 ( r 2 , r 2 ) .
W ( ρ 1 , ρ 2 ) = 1 λ 2 B 2 exp [ i k D 2 B ( ρ 1 2 ρ 2 2 ) ] W 0 ( r 1 , r 2 ) × exp { i k 2 B [ A ( r 1 2 r 2 2 ) 2 r 1 ρ 1 + 2 r 2 ρ 2 ] } d 2 r 1 d 2 r 2 ,
W ( ρ 1 , ρ 2 ) = 1 λ 2 B 2 1 C 0 m = 1 M ( 1 ) m 1 m ( M m ) π 2 α 1 α 2 α y γ 1 × exp [ i k D 2 B ( ρ 1 2 ρ 2 2 ) ] × exp [ k 2 ρ 2 y 2 4 α 2 B 2 k 2 ρ 2 x 2 4 α 1 B 2 + β y 2 4 α y + γ 2 2 4 γ 1 ] ,
α 1 = ( 1 4 ω m x 2 + 1 2 δ m x 2 + i k A 2 B ) , α 2 = ( 1 4 ω m y 2 + 1 2 δ m y 2 + i k A 2 B ) , α x = ( α 1 1 4 α 1 δ m x 4 + μ m 2 4 α 2 ) , α y = ( α 2 1 4 α 2 δ m y 4 + μ m 2 4 α 1 ) ,
β x = i k ρ 1 x B k ρ 2 y μ m 2 α 2 B i k ρ 2 x 2 α 1 B δ m x 2 , β y = i k ρ 1 y B + k ρ 2 x μ m 2 α 1 B i k ρ 2 y 2 α 2 B δ m y 2 , β x y = ( i μ m 2 α 1 δ m x 2 i μ m 2 α 2 δ m y 2 ) , γ 1 = ( α x β x y 2 4 α y ) , γ 2 = ( β x + β y β x y 2 α y ) .
K x = ρ x 4 ρ x 2 2 , K y = ρ y 4 ρ y 2 2 ,
ρ x n = 1 I ρ x n S ( ρ x , ρ y ) d ρ x d ρ y , n = 2 , 4 , ρ y n = 1 I ρ y n S ( ρ x , ρ y ) d ρ x d ρ y , n = 2 , 4 ,
I = S ( ρ x , ρ y ) d ρ x d ρ y .
W 0 ( r s , r d ) = 1 C 0 m = 1 M ( 1 ) m 1 m ( M m ) exp [ x s 2 2 ω m x 2 y s 2 2 ω m y 2 ] × exp [ ( 1 8 ω m x 2 + 1 2 δ m x 2 ) x d 2 ( 1 8 ω m y 2 + 1 2 δ m y 2 ) y d 2 ] exp [ i μ m ( x d y s x s y d ) ] ,
ξ a ξ b = 1 I ξ a ξ b h ( r s , θ ) d 2 r s d 2 θ ,
h ( r s , θ ) = k 2 4 π 2 W 0 ( r s , r d ) exp ( i k r d θ ) d r d .
x ^ : W ( r s , r d ) x W ( r s , r d ) , θ ^ x : W ( r s , r d ) 1 i k W ( r s , r d ) x d , y ^ : W ( r s , r d ) y W ( r s , r d ) , θ ^ y : W ( r s , r d ) 1 i k W ( r s , r d ) y d ,
ξ a ξ b = 1 I ξ ^ a ξ ^ b W 0 ( r s , r d ) | r d = 0 d 2 r s ,
x 2 = m = 1 M ( 1 ) m 1 m ( M m ) ω m x 3 ω m y m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y , y 2 = m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y 3 m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y ,
θ x 2 = 1 k 2 m = 1 M ( 1 ) m 1 m ( M m ) [ μ m 2 ω m y 3 ω m x + ( 1 4 ω m x 2 + 1 δ m x 2 ) ω m x ω m y ] m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y ,
θ y 2 = 1 k 2 m = 1 M ( 1 ) m 1 m ( M m ) [ μ m 2 ω m x 3 ω m y + ( 1 4 ω m y 2 + 1 δ m y 2 ) ω m x ω m y ] m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y ,
x θ y = 1 k m = 1 M ( 1 ) m 1 m ( M m ) μ m ω m x 3 ω m y m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y , y θ x = 1 k m = 1 M ( 1 ) m 1 m ( M m ) μ m ω m y 3 ω m x m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y ,
x θ x = y θ y = x y = θ x θ y = 0.
J z = ( I c ) ( x θ y y θ x ) = n m = 1 M ( 1 ) m 1 m ( M m ) μ m ω m x ω m y ( ω m x 2 + ω m y 2 ) m = 1 M ( 1 ) m 1 m ( M m ) ω m x ω m y ,
M x 2 = 2 k ( x 2 θ x 2 x θ x 2 ) 1 / 2 , M y 2 = 2 k ( y 2 θ y 2 y θ y 2 ) 1 / 2 .

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