Abstract

We embed a pair of vortices with different topological charges in a Gaussian beam and study its evolution through an astigmatic optical system, a tilted lens. The propagation dynamics are explained by a closed-form analytical expression. Furthermore, we show that a careful examination of the intensity distribution at a predicted position past the lens can determine the charge present in the beam. To the best of our knowledge, our method is the first noninterferometric technique to measure the charge of an arbitrary vortex pair. Our theoretical results are well supported by experimental observations.

© 2014 Optical Society of America

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  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).
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  4. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
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  5. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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  9. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
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  25. J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
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  26. A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19, 5760–5771 (2011).
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  27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
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  28. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16, 4991–4999 (2008).
    [CrossRef]
  29. A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012).
    [CrossRef]
  30. A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. 35, 3841–3843 (2010).
    [CrossRef]
  31. A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express 19, 6182–6190 (2011).
    [CrossRef]
  32. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
    [CrossRef]
  33. A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
    [CrossRef]
  34. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  35. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  36. J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003).
    [CrossRef]
  37. S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).
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    [CrossRef]
  40. G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), pp. 147–164.
  41. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).
  42. G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003).
    [CrossRef]
  43. R. S. Sirohi, A Course of Experiments with He-Ne Lasers (New Age International, 1991).

2013 (4)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013).
[CrossRef]

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).

2012 (2)

A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012).
[CrossRef]

P. Vaity and R. P. Singh, “Topological charge dependent propagation of optical vortices under quadratic phase transformation,” Opt. Lett. 37, 1301–1303 (2012).
[CrossRef]

2011 (5)

2010 (2)

A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. 35, 3841–3843 (2010).
[CrossRef]

J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

2008 (6)

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008).
[CrossRef]

M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16, 4991–4999 (2008).
[CrossRef]

S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008).
[CrossRef]

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25, 1279–1286 (2008).
[CrossRef]

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008).
[CrossRef]

2007 (1)

G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

2004 (3)

F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236, 433–440 (2004).
[CrossRef]

F. S. Roux, “Canonical vortex dipole dynamics,” J. Opt. Soc. Am. B 21, 655–663 (2004).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

2003 (3)

G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003).
[CrossRef]

2001 (1)

R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001).
[CrossRef]

2000 (1)

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

1999 (2)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef]

1998 (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

1996 (1)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

1993 (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (2)

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

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]

1974 (1)

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

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).

Angom, D.

S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

Banerji, J.

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012).
[CrossRef]

S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36, 4398–4400 (2011).
[CrossRef]

A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. 35, 3841–3843 (2010).
[CrossRef]

R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001).
[CrossRef]

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).

Baumgartl, J.

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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]

Bekshaev, A. Y.

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

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]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Carpentier, A. V.

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008).
[CrossRef]

Chen, H.

H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013).
[CrossRef]

Chen, M.

Chen, Z.

Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011).
[CrossRef]

Chvez-Cerda, S.

J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Dattoli, G.

G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003).
[CrossRef]

G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), pp. 147–164.

Davies, A. R.

R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001).
[CrossRef]

Dholakia, K.

Dienerowitz, M.

Dunlop, H. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Fonseca, E.

J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Gahagan, K. T.

Gao, Z.

H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013).
[CrossRef]

Gautam, S.

S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).

Ghai, D. P.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008).
[CrossRef]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

Hassan, N. A. M.

S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Hickmann, J.

J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Huang, X.

H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Jenkins, R. M.

R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001).
[CrossRef]

Karamoch, A. I.

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

Khan, S.

S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008).
[CrossRef]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Krauss, T. F.

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Kumar, A.

Laakkonen, P.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).

Marchiano, R.

J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003).
[CrossRef]

Mazilu, M.

Michinel, H.

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

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W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).

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A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008).
[CrossRef]

Orenstein, M.

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
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L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

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[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).

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[CrossRef]

Prabhakar, S.

Pu, J.

Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011).
[CrossRef]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Reece, P. J.

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Roux, F. S.

Salgueiro, J. R.

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008).
[CrossRef]

Scheuer, J.

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef]

Senthilkumaran, P.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008).
[CrossRef]

Shanor, C.

Siegman, A. E.

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

Singh, R. P.

Sirohi, R. S.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008).
[CrossRef]

R. S. Sirohi, A Course of Experiments with He-Ne Lasers (New Age International, 1991).

Skidanov, R. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Soares, W.

J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
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W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).

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A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

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]

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Terriza, G. M.

G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

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J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003).
[CrossRef]

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G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Torres, J. P.

G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Turunen, J.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
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Vaity, P.

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M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
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Vasnetsov, M. V.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
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V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Vyas, S.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008).
[CrossRef]

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H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013).
[CrossRef]

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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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]

Wright, E. M.

Yang, H.

H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013).
[CrossRef]

Yasmin, G.

S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008).
[CrossRef]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Zhao, D.

Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011).
[CrossRef]

Am. J. Phys. (1)

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008).
[CrossRef]

J. Math. Anal. Appl. (2)

S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008).
[CrossRef]

G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003).
[CrossRef]

J. Mod. Opt. (3)

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

J. Opt. A (1)

R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001).
[CrossRef]

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

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

J. Phys. B (1)

S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).

Nat. Phys. (1)

G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

Opt. Commun. (5)

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236, 433–440 (2004).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Opt. Express (3)

Opt. Lasers Eng. (1)

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008).
[CrossRef]

Opt. Lett. (5)

Optik (1)

H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013).
[CrossRef]

Phys. Lett. A (2)

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011).
[CrossRef]

Phys. Rev. A (2)

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]

A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012).
[CrossRef]

Phys. Rev. Lett. (3)

J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003).
[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. (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

Science (2)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef]

Other (5)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).

R. S. Sirohi, A Course of Experiments with He-Ne Lasers (New Age International, 1991).

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

G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), pp. 147–164.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).

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Figures (13)

Fig. 1.
Fig. 1.

Experimental setup for the determination of the net charge of an arbitrary vortex pair embedded in a Gaussian beam.

Fig. 2.
Fig. 2.

Intensity distributions of a vortex pair embedded in a Gaussian beam and the corresponding interferograms at 96 cm (top) and 147 cm (bottom) from the SLM with the orders (left) m=n=1; (right) m=n=2.

Fig. 3.
Fig. 3.

Theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of a vortex pair with topological charges of the same sign, at z=zc for x0=0.1w0.

Fig. 4.
Fig. 4.

Theoretical results for the intensity patterns of a vortex pair with topological charges of the same sign with varying separation at z=zc, (top) m=4, n=1; (bottom) m=n=2.

Fig. 5.
Fig. 5.

Experimental images corresponding to Fig. 4.

Fig. 6.
Fig. 6.

Line profiles of intensity distributions along the center of the lobes corresponding to Fig. 5 at x0=0.4w0, (left) m=4, n=1; (right) m=n=2.

Fig. 7.
Fig. 7.

Theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of an off-axis vortex of charge 2, at z=zc for different values of x0 as labeled in the figures.

Fig. 8.
Fig. 8.

Theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of a vortex pair with topological charges of opposite signs, at z=zc for x0=0.1w0.

Fig. 9.
Fig. 9.

Theoretical (top row) and experimental (bottom row) results for the intensity patterns of a dipole vortex of charge (2,2), at z=zc for different values of x0 as labeled in the figures.

Fig. 10.
Fig. 10.

Theoretical results for the intensity patterns of a vortex pair with topological charges (top) m=n=2; (bottom) m=2, n=3, at z=zc corresponding to the tilt angle θ=6°. As the tilt angle moves away from 6°, the sharpness of the patterns decreases, as expected.

Fig. 11.
Fig. 11.

Experimental images corresponding to Fig. 10 at z=zc.

Fig. 12.
Fig. 12.

Theoretical intensity patterns of a vortex pair of different charges (as given on the top) at various values of the propagation distance z (as given on the left).

Fig. 13.
Fig. 13.

Experimental images corresponding to Fig. 12.

Equations (40)

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

E1(x1,y1)=(x1+x0+iϵ1y1)m(x1x0+iϵ2y1)n×exp[(x12+y12w02)].
Mtot=(ABC/fD),
c1=secθ,c2=cosθ,aj=1zcj/f,dj=1z0cj/f,bj=z0+zdj,j=1,2.
E2(x2,y2)=i/λ|B|1/2dx1dy1E1(x1,y1)e(iπ/λ)ϕ(r1,r2),
ϕ(r1,r2)=r1TB1Ar1+r2TDB1r22r1TB1r2=x12a1/b1+y12a2/b2+x22d1/b1+y22d2/b22(x1x2/b1+y1y2/b2).
E1(x1,y1)=limt0t0[mtmntnexp{f(t,t)}],
f(t,t)=t(x1+x0+iϵ1y1)+t(x1x0+iϵ2y1)x12+y12w02.
Hn(x)=ntnexp(2xtt2)|t=0,
djdxjHn(x)=2jn!(nj)!Hnj(x),
E2(x2,y2)=kw1w2(i/2)m+n+1γm+n(b1b2)1/2×exp[(β1x22+β2y22)]Fm,n(x2,y2),
Fm,n(x2,y2)=j=0min(m,n)(mj)(nj)Δjj!Hmj[f1(x2,y2)]Hnj[f2(x2,y2)],
1wj2=1w02+ikaj2bj,
γ=(w12w22)1/2,
Δ=2(w12w22ϵ1ϵ2)/γ2,
αj=kwj22bj,
βj=(kwj2bj)2+ikdj2bj,
[f1(x2,y2)f2(x2,y2)]=1γ[α1x2+i(ϵ1α2y2x0)α1x2+i(ϵ2α2y2+x0)]=1γ[ϕ1(x2,y2)ϕ2(x2,y2)].
E2(x2,y2)=kw1w2(i/2)m+1(b1b2)1/2exp[(β1x22+β2y22)]×γmHm[(α1x2+iϵ1α2y2ix0)/γ];
Hn(x,y)=n!r=0[n/2]xn2ryr(n2r)!r!,
Hn(x)=Hn(2x,1).
Hm,n(x,z;y,w|τ)=s=0min(m,n)τss!(ms)(ns)Hms(x,z)Hns(y,w).
Fm,n=Hm,n(2f1,1;2f2,1|Δ),
exp[(u2+v2)+2(f1u+f2v)+Δuv]=m,n=0umvnm!n!Hm,n(2f1,1;2f2,1|Δ).
ka12b1|z=zc=ka22b2|z=zc=1w02.
z0=zR(1+2fcosθzRsin2θ)1/2zc=zR(1+cos2θ)+z0sin2θ2(zR/f)cosθsin2θ.
Δ={2ifϵ1ϵ2=1,2iifϵ1ϵ2=1;(w12w22)=w022(exp(iπ/4)exp(iπ/4));γ=w0exp(iπ/4);f1=δ1x2ϵ1δ2y2+(x0/w0)exp(iπ/4),f2=δ1x2ϵ2δ2y2(x0/w0)exp(iπ/4),
δj=kw022bi.
θ=δ1x2δ2y2,θ0=(x0/w0)exp(iπ/4).
Hn(x+y)=Hn(x)+2nyHn1(x)+O(y2).
r=0min(m,n)(2)rr!(mr)(nr)Hmr(x)Hnr(x)=Hm+n(x),
Fm,n=Hm+n(θ)+2θ0(mn)Hm+n1(θ)+O(θ02).
f1=θ+θ0,f2=θ+θ0.
Fm,n=Hm,n(2θ,1;2θ+,1|2i)+2mθ0Hm1,n(2θ,1;2θ+,1|2i)2nθ0Hm,n1(2θ,1;2θ+,1|2i)+O(θ02).
hm,n(x,y|τ)=m!n!j=0min(m,n)τjxmjynjj!(mj)!(nj)!={m!τmxnmLm(nm)(xy/τ),n>m,n!τnymnLn(mn)(xy/τ),m>n.
E2(x2,y2)=kw1w2im+n+12(b1b2)1/2exp[(β1x22+β2y22)]×{m!τmϕ1nmLm(nm)(ϕ1ϕ2/τ),n>m,n!τnϕ2mnLn(mn)(ϕ1ϕ2/τ),m>n,
cj=1,aj=a=1z/f,dj=d=1z0/f,bj=b=z0+zd,1wj2=1w2=1w02+ika2b,αj=α=kw22b,βj=β=(kw2b)2+ikd2b,Δ=2w2(1ϵ1ϵ2)/γ2,andγ0.
E2(x2,y2)=ikw22bexp[β(x22+y22)]×j=0min(m,n)(mj)(nj)Γjj!Δ1mjΔ2nj,
Δ1=x0+ikw22b(x2+iϵ1y2),Δ2=x0+ikw22b(x2+iϵ2y2),Γ=w22(1ϵ1ϵ2).
E2(x2,y2)=ikw22bexp[β(x22+y22)]Δ1mΔ2n.
E2(x2,y2)=ikw22bexp[β(x22+y22)]×{m!w2mΔ1nmLm(nm)(Δ1Δ2/w2),n>m,n!w2nΔ2mnLn(mn)(Δ1Δ2/w2),m>n.

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