Abstract

We investigate the propagation properties and the radiation forces of Airy Gaussian vortex (AiGV) beams in a harmonic potential analytically and numerically in this paper. Obtaining the propagation expression of AiGV beams by solving the dimensionless linear (2+1) D Schrödinger equation in a harmonic potential, we perform the track, the intensity and phase distributions, the propagation shapes, the energy flow and the angular momentum of AiGV beams in a harmonic potential with the method of numerical simulations. The trajectory acting like a cosine curve is shown. Periodic inversion and phase oscillation are demonstrated during propagation. The influence of the distribution factor and the vortex factor on the propagation of AiGV beams in a harmonic potential are discussed. Likewise, the motion of the Poynting vector and the angular momentum is elucidated respectively. As for the radiation forces, we explore the gradient and scattering forces on Rayleigh dielectric particles induced by AiGV beams. In particular, it’s found that the value of the scattering force is approximately seven orders of magnitude larger than that of the gradient force during the propagation in a harmonic potential.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Propagation properties of Airy–Gaussian vortex beams through the gradient-index medium

Ruihuang Zhao, Fu Deng, Weihao Yu, Jiayao Huang, and Dongmei Deng
J. Opt. Soc. Am. A 33(6) 1025-1031 (2016)

Propagation of Airy Gaussian vortex beams through slabs of right-handed materials and left-handed materials

Bo Chen, Chidao Chen, Xi Peng, and Dongmei Deng
J. Opt. Soc. Am. B 32(1) 173-178 (2015)

Propagation properties of right-hand circularly polarized Airy–Gaussian beams through slabs of right-handed materials and left-handed materials

Jiayao Huang, Zijie Liang, Fu Deng, Weihao Yu, Ruihuang Zhao, Bo Chen, Xiangbo Yang, and Dongmei Deng
J. Opt. Soc. Am. A 32(11) 2104-2109 (2015)

References

  • View by:
  • |
  • |
  • |

  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [Crossref]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
    [Crossref] [PubMed]
  5. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
    [Crossref] [PubMed]
  6. Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, Self-accelerating Airy Beams: Generation, Control, and Applications (Springer, 2012) Vol. 170, pp. 1–46.
  7. L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23(3), 2566–2576 (2015).
    [Crossref] [PubMed]
  8. R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. 39(19), 5523–5526 (2014).
    [Crossref] [PubMed]
  9. N. K. Efremidis, “Airy trajectory engineering in dynamic linear index potentials,” Opt. Lett. 36(15), 3006–3008 (2011).
    [Crossref] [PubMed]
  10. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011).
    [Crossref] [PubMed]
  11. Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of airy beams with optically induced refractiveindex gradient,” Opt. Lett. 36(16), 3230–3232 (2011).
    [Crossref] [PubMed]
  12. N. K. Efremidis, “Accelerating beam propagation in refractive-index potentials,” Phys. Rev. A 89(2), 023841 (2014).
    [Crossref]
  13. Y. Zhang, M. R. Belic, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015).
    [Crossref] [PubMed]
  14. Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
    [Crossref] [PubMed]
  15. Y. Gu and G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
    [Crossref] [PubMed]
  16. D. N. Christodoulides, “Optical trapping riding along an airy beam,” Nat. Photonics 2(11), 652–653 (2008).
    [Crossref]
  17. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  18. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
    [Crossref] [PubMed]
  19. P. Coulet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
    [Crossref]
  20. 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(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  21. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” P. Roy. Soc. Lon. A Mat. 336(1605), 165–190 (1974).
    [Crossref]
  22. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22(5), 849–861 (2005).
    [Crossref]
  23. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6(11), 1748–1754 (1989).
    [Crossref] [PubMed]
  24. D. Ganic, X. S. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100,” Opt. Lett. 27(15), 1351–1353 (2002).
    [Crossref]
  25. E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141(12), 59–64 (1997).
    [Crossref]
  26. Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. 30(18), 2472–2474 (2005).
    [Crossref] [PubMed]
  27. Y. Zhang, Z. Nie, Y. Zhao, C. Li, R. Wang, J. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
    [Crossref] [PubMed]
  28. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
    [Crossref] [PubMed]
  29. Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
    [Crossref]
  30. Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
    [Crossref] [PubMed]
  31. Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
    [Crossref]
  32. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276(5318), 1538–1541 (1997).
    [Crossref]
  33. A. Bekshaev, M. S. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum, (Nova Science, 2008).
  34. J. D. Jackson, Classical Electrodynamics, (Wiley, 1962).
  35. M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).
    [Crossref]
  36. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
    [Crossref]
  37. H.I. Sztul and R.R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
    [Crossref] [PubMed]
  38. B. Chen, C. Chen, X. Peng, and D. Deng, “Propagation of airy gaussian vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. B 32(1), 173–178 (2015).
    [Crossref]
  39. J. Huang, Z. Liang, F. Deng, W. Yu, R. Zhao, B. Chen, X. Yang, and D. Deng, “Propagation properties of right-hand circularly polarized Airy-Gaussian beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. A 32(11), 2104–2109 (2015).
    [Crossref]
  40. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
    [Crossref]
  41. J. A. Stratton, Electromagnetic Theory, (McGraw-Hill, 1941).
  42. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, 1969).
  43. Z. Zheng, B. Zhang, H Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50(1), 43–49 (2011).
    [Crossref] [PubMed]

2016 (1)

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

2015 (7)

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

B. Chen, C. Chen, X. Peng, and D. Deng, “Propagation of airy gaussian vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. B 32(1), 173–178 (2015).
[Crossref]

J. Huang, Z. Liang, F. Deng, W. Yu, R. Zhao, B. Chen, X. Yang, and D. Deng, “Propagation properties of right-hand circularly polarized Airy-Gaussian beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. A 32(11), 2104–2109 (2015).
[Crossref]

L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23(3), 2566–2576 (2015).
[Crossref] [PubMed]

Y. Zhang, M. R. Belic, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
[Crossref] [PubMed]

2014 (2)

N. K. Efremidis, “Accelerating beam propagation in refractive-index potentials,” Phys. Rev. A 89(2), 023841 (2014).
[Crossref]

R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. 39(19), 5523–5526 (2014).
[Crossref] [PubMed]

2011 (5)

2010 (3)

2008 (5)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

2005 (2)

2002 (1)

1999 (1)

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

1997 (2)

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141(12), 59–64 (1997).
[Crossref]

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[Crossref]

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(11), 8185–8189 (1992).
[Crossref] [PubMed]

1989 (2)

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” P. Roy. Soc. Lon. A Mat. 336(1605), 165–190 (1974).
[Crossref]

Abramochkin, E.

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141(12), 59–64 (1997).
[Crossref]

Alfano, R.R.

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[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(11), 8185–8189 (1992).
[Crossref] [PubMed]

Almazov, A. A.

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[Crossref]

Babiker, M.

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

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

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(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bekshaev, A.

A. Bekshaev, M. S. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum, (Nova Science, 2008).

Belic, M. R.

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Y. Zhang, M. R. Belic, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” P. Roy. Soc. Lon. A Mat. 336(1605), 165–190 (1974).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).
[Crossref]

Broky, J.

Chen, B.

Chen, C.

Chen, H

Chen, Z.

Chen, Z. G.

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, Self-accelerating Airy Beams: Generation, Control, and Applications (Springer, 2012) Vol. 170, pp. 1–46.

Christodoulides, D. N.

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

D. N. Christodoulides, “Optical trapping riding along an airy beam,” Nat. Photonics 2(11), 652–653 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, Self-accelerating Airy Beams: Generation, Control, and Applications (Springer, 2012) Vol. 170, pp. 1–46.

Coulet, P.

P. Coulet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Deng, D.

Deng, F.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Ding, J.

Dogariu, A.

Driben, R.

Efremidis, N. K.

N. K. Efremidis, “Accelerating beam propagation in refractive-index potentials,” Phys. Rev. A 89(2), 023841 (2014).
[Crossref]

N. K. Efremidis, “Airy trajectory engineering in dynamic linear index potentials,” Opt. Lett. 36(15), 3006–3008 (2011).
[Crossref] [PubMed]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
[Crossref] [PubMed]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, Self-accelerating Airy Beams: Generation, Control, and Applications (Springer, 2012) Vol. 170, pp. 1–46.

Elfstrom, H.

Fan, D.

Friberg, A. T.

Gan, X. S.

Ganic, D.

Gbur, G.

Gil, L.

P. Coulet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Gu, M.

Gu, Y.

Hain, M.

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[Crossref]

Hu, Y.

Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of airy beams with optically induced refractiveindex gradient,” Opt. Lett. 36(16), 3230–3232 (2011).
[Crossref] [PubMed]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, Self-accelerating Airy Beams: Generation, Control, and Applications (Springer, 2012) Vol. 170, pp. 1–46.

Huang, J.

Izdebskaya, Y.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, (Wiley, 1962).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, 1969).

Khonina, S. N.

Kivshar, Y. S.

Konotop, V. V.

Kotlyar, V. V.

Li, C.

Li, Y.

Liang, Z.

Liu, K.

Liu, S.

Liu, W.

Liu, X

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Liu, X.

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
[Crossref] [PubMed]

Losevsky, N.

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141(12), 59–64 (1997).
[Crossref]

Lou, C.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Meier, T.

Mills, M. S.

Miroshnichenko, A. E.

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Neshev, D. N.

Nie, Z.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” P. Roy. Soc. Lon. A Mat. 336(1605), 165–190 (1974).
[Crossref]

Padgett, M. J.

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

Peng, X.

Petrovic, M. S.

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Prakash, J.

Rocca, F.

P. Coulet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Schattschneider, P.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Shadrivov, I. V.

Shvedov, V.

Si, J.

Siviloglou, G. A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, Self-accelerating Airy Beams: Generation, Control, and Applications (Springer, 2012) Vol. 170, pp. 1–46.

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Soifer, V. A.

Somalingam, S.

Song, D.

Soskin, M. S.

A. Bekshaev, M. S. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum, (Nova Science, 2008).

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(11), 8185–8189 (1992).
[Crossref] [PubMed]

Stankovic, S.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory, (McGraw-Hill, 1941).

Sztul, H.I.

Tian, H.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Tschudi, T.

Turunen, J.

Vasara, A.

Vasnetsov, M.

A. Bekshaev, M. S. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum, (Nova Science, 2008).

Verbeeck, J.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Volostnikov, V.

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141(12), 59–64 (1997).
[Crossref]

Volyar, A.

Wang, H.

Wang, R.

Wen, F.

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(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).
[Crossref]

Xiao, M.

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Y. Zhang, Z. Nie, Y. Zhao, C. Li, R. Wang, J. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
[Crossref] [PubMed]

Yang, X.

Ye, Z.

Yu, W.

Zhang, B.

Zhang, J.

Zhang, L.

Zhang, P.

Zhang, Y.

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Y. Zhang, M. R. Belic, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
[Crossref] [PubMed]

Y. Zhang, M. R. Belic, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
[Crossref] [PubMed]

Y. Zhang, Z. Nie, Y. Zhao, C. Li, R. Wang, J. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
[Crossref] [PubMed]

Zhang, Z.

Zhao, J.

Zhao, R.

Zhao, Y.

Zheng, Z.

Zhong, H.

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23(3), 2566–2576 (2015).
[Crossref] [PubMed]

Zhong, W.

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
[Crossref] [PubMed]

Y. Zhang, M. R. Belic, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015).
[Crossref] [PubMed]

Zhu, D.

Zhu, Y.

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Ann. Phys. (1)

Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. 363, 305–315 (2015).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

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

Laser Photon. Rev. (1)

Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Nat. Photonics (2)

D. N. Christodoulides, “Optical trapping riding along an airy beam,” Nat. Photonics 2(11), 652–653 (2008).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Nature (1)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Opt. Commun. (3)

E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141(12), 59–64 (1997).
[Crossref]

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[Crossref]

P. Coulet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Opt. Express (5)

Opt. Lett. (11)

Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. 30(18), 2472–2474 (2005).
[Crossref] [PubMed]

D. Ganic, X. S. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100,” Opt. Lett. 27(15), 1351–1353 (2002).
[Crossref]

R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. 39(19), 5523–5526 (2014).
[Crossref] [PubMed]

N. K. Efremidis, “Airy trajectory engineering in dynamic linear index potentials,” Opt. Lett. 36(15), 3006–3008 (2011).
[Crossref] [PubMed]

W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011).
[Crossref] [PubMed]

Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of airy beams with optically induced refractiveindex gradient,” Opt. Lett. 36(16), 3230–3232 (2011).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015).
[Crossref] [PubMed]

Y. Gu and G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
[Crossref] [PubMed]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
[Crossref] [PubMed]

P. Roy. Soc. Lon. A Mat. (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” P. Roy. Soc. Lon. A Mat. 336(1605), 165–190 (1974).
[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(11), 8185–8189 (1992).
[Crossref] [PubMed]

N. K. Efremidis, “Accelerating beam propagation in refractive-index potentials,” Phys. Rev. A 89(2), 023841 (2014).
[Crossref]

Phys. Rev. Lett. (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

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 (1)

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Other (6)

A. Bekshaev, M. S. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum, (Nova Science, 2008).

J. D. Jackson, Classical Electrodynamics, (Wiley, 1962).

M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).
[Crossref]

J. A. Stratton, Electromagnetic Theory, (McGraw-Hill, 1941).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, 1969).

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, Self-accelerating Airy Beams: Generation, Control, and Applications (Springer, 2012) Vol. 170, pp. 1–46.

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 (10)

Fig. 1
Fig. 1

(a) Intensity and (b) Phase distribution of the AiGV beams at the input plane z = 0.

Fig. 2
Fig. 2

Numerical demonstrations of the AiGV beams propagating in a harmonic potential. (a1)–(a8) Longitudinal normalized intensity distribution at the positions 0.25π, 0.5π, 0.75π, 1π, 1.25π, 1.5π, 1.75π, and 2π, respectively. (b) Numerically simulated propagation path of the AiGV beams in the observation plane x = y. (c) Numerically simulated phase evolution of the AiGV beams in the observation plane x = y.

Fig. 3
Fig. 3

The initial intensity distribution of the AiGV beams in a harmonic potential with (a1) χ0 = 0.01, (b1) χ0 = 0.18, (c) χ0 = 1. The corresponding 3D propagation of the AiGV beams along the z axis with (a1)–(c1).

Fig. 4
Fig. 4

The iso-surface plot of the beam during propagation corresponding to (a) χ0 = 0.01, (b1) χ0 = 0.18, (c) χ0 = 1.

Fig. 5
Fig. 5

(a) The variation of the center of the gravity of the beams as α varies from 0 to 20 at the planes z = 1.25Z0 and z = Z0; (b) the variation of the center of mass versus the axial propagation distance is depicted with the different distribution factor χ0.

Fig. 6
Fig. 6

Numerical demonstrations of the AiGV beams propagating in a harmonic potential. (a)–(h) the Poynting vector of the AiGV beams with χ0 = 0.18 at the same positions as those in Figs. 2(a1)–2(a8).

Fig. 7
Fig. 7

Numerical demonstrations of the AiGV beams propagating in a harmonic potential. (a)–(h) Longitudinal normalized angular momentum density of the AiGV beams with χ0 = 0.18 at the same positions as those in Figs. 2(a1)–2(a8).

Fig. 8
Fig. 8

The transverse pattern (background) and plots (white line) of the gradient force on a Rayleigh particle with n1 = 1.50, r0 = 60nm at (a)–(j) the positions 0, 2π, 4π, 6π, 8π, 12π, 14π, 16π, 18π and 20π, respectively.

Fig. 9
Fig. 9

The transverse pattern (background) and plots (white line) of the scattering force on a Rayleigh particle with n1 = 1.50, r0 = 60nm at the same positions as those in Fig. 8.

Fig. 10
Fig. 10

The distribution of (a) the gradient force and (b) the scattering force on a Rayleigh particle with n1 = 1.50, r0 = 60nm at three different z planes.

Equations (17)

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

E ( x 0 , y 0 , 0 ) = A 0 Ai ( x 0 ) exp ( a x 0 ) Ai ( y 0 ) exp ( a y 0 ) × exp ( χ 0 2 x 0 2 χ 0 2 y 0 2 ) × [ x 0 x 1 + i ( y 0 y 1 ) ] m × [ x 0 x 2 i ( y 0 y 2 ) ] l
i E ( x , y , z ) z + 1 2 2 E ( x , y , z ) 1 2 α 2 ( x 2 + y 2 ) E ( x , y , z ) = 0
E ( x , y , z ) = i A 0 2 P Q exp ( J ( x , y , z ) ) ( F 1 + F 2 + F 3 )
J ( x , y , z ) = i 2 P Q cos ( α z ) + 1 4 P 2 Q ( x 2 + y 2 ) i 4 Q a + 1 8 P Q 2 ( x + y ) + 1 48 Q 3 + a 4 Q 2 + a 2 2 Q
F 1 = Ai ( s ( x ) ) Ai ( s ( y ) ) × [ ( i x 2 P Q + 1 8 Q 2 x 1 ) + i ( i y 2 P Q + 1 8 Q 2 y 1 ) ]
F 2 = 1 2 Q Ai ( s ( y ) ) [ a Ai ( s ( x ) ) + Ai ( s ( x ) ) ]
F 3 = 1 2 Q Ai ( s ( x ) ) [ a Ai ( s ( y ) ) + Ai ( s ( y ) ) ]
P = sin ( α z ) / α
Q = χ 0 2 i α cot ( α z ) / 2
s ( v ) = 1 16 Q 2 + a 2 Q i v 2 P Q , ( v = x , y )
x c = + x | E ( x , y , z ) | 2 d x d y + | E ( x , y , z ) | 2 d x d y , y c = + y | E ( x , y , z ) | 2 d x d y + | E ( x , y , z ) | 2 d x d y
S = c 4 π E × B = c 8 π ( i ω ( E E * E * E ) + 2 ω k | E | 2 e z ) ,
J = r × E × B = ω 2 ( 2 y k | E | 2 z i S y ) e x ^ + ( z i S x 2 x k | E | 2 ) e y ^ + i ( x S y y S x ) e z ^
F grad ( x , y , z , t ) = [ p ( x , y , z , t ) ] E ( x , y , z , t )
F grad ( x , y , z ) = 2 π n 2 r 0 3 c ( m 2 1 m 2 + 2 ) I ( x , y , z ) ,
F scat ( x , y , z ) = n 2 c C pr 0 I ( x , y , z ) e z ,
C pr 0 = C scat = 8 3 π ( k a ) 4 r 0 2 ( m 2 1 m 2 + 2 ) 2 .

Metrics