Abstract

The splitting of a linear or circular edge dislocation by a vortex nested in a nonparaxial Gaussian beam is studied. It is found that the edge dislocations are unstable and vanish, while new vortices may take place and the total topological charge may change with propagation. The location of the vortex, the intercept and slope of the edge dislocation, or the location and radius of the circular edge dislocation in the initial beam may affect the number and positions of vortices occurring in the field. The results are compared with previous works.

© 2019 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence

Jinhong Li, Penghui Gao, Ke Cheng, and Meiling Duan
Opt. Express 25(3) 2895-2908 (2017)

Dynamic evolution of an edge dislocation through aligned and misaligned paraxial optical ABCD systems

Hongwei Yan and Baida Lü
J. Opt. Soc. Am. A 26(4) 985-992 (2009)

Vortex revivals with trapped light

Gabriel Molina-Terriza, Lluis Torner, Ewan M. Wright, Juan J. García-Ripoll, and Víctor M. Pérez-García
Opt. Lett. 26(20) 1601-1603 (2001)

References

  • View by:
  • |
  • |
  • |

  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [Crossref]
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
    [Crossref]
  3. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
    [Crossref]
  4. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467, 301–304 (2010).
    [Crossref]
  5. M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
    [Crossref]
  6. M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
    [Crossref]
  7. R. Paez-Lopez, U. Ruiz, V. Arrizon, and R. Ramos-Garcia, “Optical manipulation using optimal annular vortices,” Opt. Lett. 41, 4138–4141 (2016).
    [Crossref]
  8. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
    [Crossref]
  9. J. Li, W. Wang, M. Duan, and J. Wei, “Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams,” Opt. Express 24, 20413–20423 (2016).
    [Crossref]
  10. J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4, B14–B28 (2016).
    [Crossref]
  11. Y. S. Kivshar, A. Nepomnyashchy, V. Tikhonenko, J. Christou, and B. Luther-Davies, “Vortex-stripe soliton interactions,” Opt. Lett. 25, 123–125 (2000).
    [Crossref]
  12. D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307–312 (2001).
    [Crossref]
  13. D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759–773 (2002).
    [Crossref]
  14. D. He, H. W. Yan, and B. D. Lü, “Interaction of the vortex and edge dislocation embedded in a cosh-Gaussian beam,” Opt. Commun. 282, 4035–4044 (2009).
    [Crossref]
  15. D. D. Li, X. Peng, Y. L. Peng, L. P. Zhang, and D. M. Deng, “Nonparaxial evolution of the Airy–Gaussian vortex beam in uniaxial crystal,” J. Opt. Soc. Am. B 34, 891–898 (2017).
    [Crossref]
  16. J. B. Zhang, K. Z. Zhou, J. H. Lian, Z. Y. Lai, X. L. Yang, and D. M. Deng, “Nonparaxial propagation of the chirped Airy vortex beams in uniaxial crystal orthogonal to the optical axis,” Opt. Express 26, 1290–1304 (2018).
    [Crossref]
  17. C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
    [Crossref]
  18. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).
  19. K. L. Duan and B. D. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. 36, 489–496 (2004).
    [Crossref]
  20. K. L. Duan and B. D. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21, 1924–1932 (2004).
    [Crossref]
  21. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [Crossref]
  22. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
    [Crossref]

2019 (1)

C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
[Crossref]

2018 (1)

2017 (1)

2016 (3)

2015 (1)

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

2013 (1)

2011 (1)

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

2010 (1)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467, 301–304 (2010).
[Crossref]

2009 (2)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

D. He, H. W. Yan, and B. D. Lü, “Interaction of the vortex and edge dislocation embedded in a cosh-Gaussian beam,” Opt. Commun. 282, 4035–4044 (2009).
[Crossref]

2004 (3)

2002 (1)

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759–773 (2002).
[Crossref]

2001 (2)

D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307–312 (2001).
[Crossref]

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

2000 (1)

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref]

1974 (1)

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

Agrawal, A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

Anderson, I. M.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

Arita, Y.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

Arrizon, V.

Barnett, S.

Berry, M. V.

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

Chen, M.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

Christou, J.

Courtial, J.

Deng, D. M.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Dholakia, K.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

Duan, K. L.

K. L. Duan and B. D. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21, 1924–1932 (2004).
[Crossref]

K. L. Duan and B. D. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. 36, 489–496 (2004).
[Crossref]

Duan, M.

Franke-Arnold, S.

Freund, I.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref]

Gibson, G.

He, D.

D. He, H. W. Yan, and B. D. Lü, “Interaction of the vortex and edge dislocation embedded in a cosh-Gaussian beam,” Opt. Commun. 282, 4035–4044 (2009).
[Crossref]

Herzing, A. A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

Hong, W. Y.

C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
[Crossref]

Kivshar, Y. S.

Lai, Z. Y.

Lezec, H. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

Li, D. D.

Li, J.

Lian, J. H.

Liu, H. Z.

C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
[Crossref]

Lü, B. D.

D. He, H. W. Yan, and B. D. Lü, “Interaction of the vortex and edge dislocation embedded in a cosh-Gaussian beam,” Opt. Commun. 282, 4035–4044 (2009).
[Crossref]

K. L. Duan and B. D. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. 36, 489–496 (2004).
[Crossref]

K. L. Duan and B. D. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21, 1924–1932 (2004).
[Crossref]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

Luther-Davies, B.

Lv, X.

C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
[Crossref]

Ma, B. B.

C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
[Crossref]

Mazilu, M.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

McClelland, J. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

McMorran, B. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

Nepomnyashchy, A.

Nye, J. F.

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

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Padgett, M.

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Paez-Lopez, R.

Pas’ko, V.

Peng, X.

Peng, Y. L.

Petrov, D. V.

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759–773 (2002).
[Crossref]

D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307–312 (2001).
[Crossref]

Ramos-Garcia, R.

Ruiz, U.

Schattschneider, P.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467, 301–304 (2010).
[Crossref]

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref]

Soskin, M. S.

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

Sun, C.

C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
[Crossref]

Tian, H.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467, 301–304 (2010).
[Crossref]

Tikhonenko, V.

Unguris, J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

Vasnetsov, M.

Vasnetsov, M. V.

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

Verbeeck, J.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467, 301–304 (2010).
[Crossref]

Wang, J.

Wang, W.

Wei, J.

Wright, E. M.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[Crossref]

Yan, H. W.

D. He, H. W. Yan, and B. D. Lü, “Interaction of the vortex and edge dislocation embedded in a cosh-Gaussian beam,” Opt. Commun. 282, 4035–4044 (2009).
[Crossref]

Yang, X. L.

Zhang, J. B.

Zhang, L. P.

Zhou, K. Z.

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

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

Nature (1)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467, 301–304 (2010).
[Crossref]

Opt. Commun. (3)

D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307–312 (2001).
[Crossref]

C. Sun, X. Lv, D. M. Deng, B. B. Ma, H. Z. Liu, and W. Y. Hong, “Nonparaxial propagation of the radially polarized Airy-Gaussian beams with different initial launch angles in uniaxial crystals,” Opt. Commun. 445, 147–154 (2019).
[Crossref]

D. He, H. W. Yan, and B. D. Lü, “Interaction of the vortex and edge dislocation embedded in a cosh-Gaussian beam,” Opt. Commun. 282, 4035–4044 (2009).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

K. L. Duan and B. D. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. 36, 489–496 (2004).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759–773 (2002).
[Crossref]

Opt. Rev. (1)

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22, 162–165 (2015).
[Crossref]

Photon. Res. (1)

Phys. Rev. A (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref]

Proc. R. Soc. London Ser. A (1)

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

Prog. Opt. (2)

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

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Science (1)

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
[Crossref]

Other (1)

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1. Propagation of the nonparaxial Gaussian beam carrying a vortex and a linear edge dislocation, where (a)  $z = {{0}}$ , (b)  $z = 0.0001{z_R}$ , (c)  $z = 0.08{z_R}$ , (d)  $z = 0.084{z_R}$ , (e)  $z = 1.58{z_R}$ , and (f)  $z = 1.84{z_R}$ .
Fig. 2.
Fig. 2. Contour lines of phase of the beam at the plane $z = {z_R}$ versus the off-axis distance of the vortex in the initial beam (a)  ${s_1} = - {w_0}$ , (b)  ${s_1} = - 0.5{w_0}$ , (c)  ${s_1} = - 0.221{w_0}$ , and (d)  ${s_1} = {w_0}$ .
Fig. 3.
Fig. 3. Contour lines of phase of the beam at the plane $z = {z_R}$ versus the intercept of the linear edge dislocation (a)  ${s_2} = - {w_0}$ , (b)  ${s_2} = - 0.5{w_0}$ , (c)  ${s_2} = 0.195{w_0}$ , and (d)  ${s_2} = 0.198{w_0}$ .
Fig. 4.
Fig. 4. Contour lines of phase of the beam at the plane $z = {z_R}$ versus the slope of the linear edge dislocation (a)  $b = - 0.3$ , (b)  $b = - 0.17$ , (c)  $b = 0.4$ , and (d)  $b = 0.75$ .
Fig. 5.
Fig. 5. Propagation of the nonparaxial Gaussian beam carrying a vortex and a circular edge dislocation, where (a)  $z = {{0}}$ , (b)  $z = 0.0001{z_R}$ , (c)  $z = 1.94{z_R}$ , and (d)  $z = {10}{z_R}$ .
Fig. 6.
Fig. 6. Contour lines of phase of the beam at the plane $z = {{5}}{z_R}$ versus the off-axis distance of the vortex in the initial beam (a)  ${s_1} = - {w_0}$ , (b)  ${s_1} = - 0.92{w_0}$ , (c)  ${s_1} = - 0.905{w_0}$ , and (d)  ${s_1} = {w_0}$ .
Fig. 7.
Fig. 7. Contour lines of phase of the beam at the plane $z = {5}{z_R}$ versus the coordinate of the circular edge dislocation’s center (a)  ${s_2} = - {w_0}$ , (b)  ${s_2} = - 0.23{w_0}$ , (c)  ${s_2} = - 0.221{w_0}$ , and (d)  ${s_2} = 0.105{w_0}$ .
Fig. 8.
Fig. 8. Contour lines of phase of the beam at the plane $z = {5}{z_R}$ versus the radius of the circular edge dislocation (a)  $R = {0}$ , (b)  $R = 0.8{w_0}$ , (c)  $R = 0.854{w_0}$ , and (d)  $R = 0.938{w_0}$ .

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E x ( x 0 , y 0 , 0 ) = ( x 0 s 1 + i y 0 ) ( b x 0 y 0 + s 2 ) w 0 2 × exp ( x 0 2 + y 0 2 w 0 2 ) ,
E y ( x 0 , y 0 , 0 ) = 0.
E x ( x , y , z ) = 1 2 π E x ( x 0 , y 0 , 0 ) G z d x 0 d y 0 ,
E y ( x , y , z ) = 1 2 π E y ( x 0 , y 0 , 0 ) G z d x 0 d y 0 ,
G ( r , r 0 ) = exp ( i k | r r 0 | ) / | r r 0 | ,
| r r 0 | = . r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ,
E x ( x , y , z ) = i z e i k r λ r 2 d y 0 E x ( x 0 , y 0 , 0 ) × exp ( x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) d x 0 ,
E y ( x , y , z ) = i z e i k r λ r 2 d y 0 E y ( x 0 , y 0 , 0 ) × exp ( x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) d x 0 .
x n exp ( u x 2 + 2 v x ) d x = n ! exp ( v 2 u ) π u ( v u ) n t = 0 [ n / 2 ] 1 ( n 2 t ) ! t ! ( u 4 v 2 ) t ,
E x ( x , y , z ) = N 1 [ b v x 2 + ( i b 1 ) v x v y i v y 2 u 2 + v x ( s 2 b s 1 ) u + v y ( s 1 i s 2 ) u + b i 2 u s 1 s 2 ] ,
E y ( x , y , z ) = 0 ,
N 1 = i π z λ u w 0 2 r 2 exp ( v x 2 + v y 2 u + i k r ) ,
u = 1 w 0 2 i k 2 r , v x = i k x 2 r , a n d v y = i k y 2 r .
{ Re [ E x ( x , y , z ) ] = 0 Im [ E x ( x , y , z ) ] = 0 ,
E x ( x 0 , y 0 , 0 ) = ( x 0 s 1 + i y 0 ) [ ( x 0 s 2 ) 2 + y 0 2 R 2 ] w 0 3 × exp ( x 0 2 + y 0 2 w 0 2 ) ,
E y ( x 0 , y 0 , 0 ) = 0 ,
E x ( x , y , z ) = N 2 [ ( v x 2 + v y 2 ) ( v x + i v y ) u 3 + 2 ( 1 s 2 v x ) ( v x + i v y ) s 1 ( v x 2 + v y 2 ) u 2 + R 2 ( v x i v y ) + s 2 2 ( v x + i v y ) + 2 s 1 s 2 v x ( s 1 + s 2 ) u + s 1 ( R 2 s 2 2 ) ] ,
E y ( x , y , z ) = 0 ,
N 2 = i π z λ u w 0 3 r 2 exp ( v x 2 + v y 2 u + i k r ) .

Metrics