Abstract

We study the paraxial propagation of the radially polarized Airy beams (RPAiBs) in uniaxial crystals orthogonal to the optical axis analytically and numerically. The propagation trajectory, the intensity and the radiation forces of the RPAiBs are investigated and the properties are elucidated by numerical examples in this paper. Results show that the RPAiBs evolve into the beams produced by the x-direction electric field (RPAiXBs) and the y-direction electric field (PRAiYBs) which are totally different in uniaxial crystals. During the propagation, the intensity of the RPAiXBs transfers from the side lobe in the x-direction to the main lobe and finally returns to the side lobe in the x-direction again, but that of the RPAiYBs transfers from the side lobe in the y-direction to the main lobe and flows to the side lobe in the x-direction at last. The effect of the intensity focusing for the RPAiXBs can be modulated by the ratio of the extraordinary index (ne) to the ordinary index (no) in anisotropic medium, which contributes to the intensity focusing of the RPAiBs in a short distance a lot. We can adjust the intensity distribution especially the focusing position, the propagation trajectory and the radiation forces distributions of the RPAiXBs through choosing an appropriate value of the ratio of ne to no to meet the actual usage accordingly.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  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]
  3. 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]
  4. 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]
  5. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2, 675–678 (2008).
    [Crossref]
  6. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
    [Crossref] [PubMed]
  7. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of Femtosecond Laser Airy Beams in Water,” Phys. Rev. Lett. 103(12), 123902 (2009).
    [Crossref] [PubMed]
  8. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
    [Crossref]
  9. P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
    [Crossref] [PubMed]
  10. T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
    [Crossref] [PubMed]
  11. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
    [Crossref]
  12. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  13. C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
    [Crossref] [PubMed]
  14. C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
    [Crossref]
  15. Y. I. Salamin, “Mono-energetic Gev electrons from ionization in a radially polarized laser beam,” Opt. Lett. 32(1), 90–92 (2007).
    [Crossref]
  16. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
    [Crossref]
  17. K. T. Gahagan and G. A. Swartzlander, “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16(4), 533–537 (1999).
    [Crossref]
  18. C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
    [Crossref] [PubMed]
  19. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
    [Crossref] [PubMed]
  20. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
    [Crossref]
  21. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
    [Crossref]
  22. D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24(3), 636–643 (2007).
    [Crossref]
  23. D. M. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32(18), 2711–2713 (2007).
    [Crossref] [PubMed]
  24. B. Gu, B. Wen, G. H. Rui, Y. X. Xue, Q. W. Zhan, and Y. P. Cui, “Varying polarization and spin angular momentum flux of radially polarized beams by anisotropic Kerr media,” Opt. Lett. 41(7), 1566–1569 (2016).
    [Crossref] [PubMed]
  25. S. N. Khonina, S. V. Karpeev, V. D. Paranin, and A. A. Morozov, “Polarization conversion under focusing of vortex laser beams along the axis of anisotropic crystals,” Phys. Lett. A 381, 2444–2455 (2017).
    [Crossref]
  26. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18(7), 1656–1661 (2001).
    [Crossref]
  27. A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19(7), 1422–1431 (2002).
    [Crossref]
  28. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003).
    [Crossref]
  29. G. Q. Zhou, R. P. Chen, and X. X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
    [Crossref] [PubMed]
  30. G. L. Zheng, X. Q. Deng, S. X. Xu, and Q. Y. Wu, “Propagation dynamics of a circular Airy beam in a uniaxial crystal,” Appl. Opt. 56(9), 2444–2448 (2017).
    [Crossref] [PubMed]
  31. G. L. Zheng, S. X. Xu, Q. Y. Wu, Q. Wang, and Z. B. Ouyang, “Electro-optical coupling of a circular Airy beam in a uniaxial crystal,” Opt. Express 25(13), 14654–14667 (2017).
    [Crossref] [PubMed]
  32. D. M. Deng, C. D. Chen, X. Zhao, and H. G. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
    [Crossref]
  33. M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
    [Crossref]
  34. F. Deng and D. M. Deng, “Nonparaxial propagation of an Airy-Gaussian beam in uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 380, 280–286 (2016).
    [Crossref]
  35. 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(4), 891–898 (2017).
    [Crossref]
  36. J. B. Zhang, K. Z. Zhou, J. H. Liang, 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(2), 1290–1304 (2018).
    [Crossref] [PubMed]
  37. S. N. Khonina, V. D. Paranin, A. V. Ustinov, and A. P. Krasnov, “Astigmatic transformation of Bessel beams in a uniaxial crystal,” Opt. Applicate XLVI(1), 5–18 (2016).
  38. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
    [Crossref]

2018 (1)

2017 (4)

2016 (3)

F. Deng and D. M. Deng, “Nonparaxial propagation of an Airy-Gaussian beam in uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 380, 280–286 (2016).
[Crossref]

S. N. Khonina, V. D. Paranin, A. V. Ustinov, and A. P. Krasnov, “Astigmatic transformation of Bessel beams in a uniaxial crystal,” Opt. Applicate XLVI(1), 5–18 (2016).

B. Gu, B. Wen, G. H. Rui, Y. X. Xue, Q. W. Zhan, and Y. P. Cui, “Varying polarization and spin angular momentum flux of radially polarized beams by anisotropic Kerr media,” Opt. Lett. 41(7), 1566–1569 (2016).
[Crossref] [PubMed]

2015 (1)

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

2014 (1)

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

2013 (3)

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[Crossref] [PubMed]

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

D. M. Deng, C. D. Chen, X. Zhao, and H. G. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[Crossref]

2012 (1)

2011 (1)

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[Crossref] [PubMed]

2010 (1)

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

2009 (2)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of Femtosecond Laser Airy Beams in Water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref] [PubMed]

2008 (3)

2007 (5)

2003 (2)

2002 (2)

2001 (3)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[Crossref]

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18(7), 1656–1661 (2001).
[Crossref]

2000 (1)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

1999 (2)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Baumgartl, J.

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

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Broky, J.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Chen, B.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Chen, C. D.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

D. M. Deng, C. D. Chen, X. Zhao, and H. G. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[Crossref]

Chen, R. P.

Christodoulides, D. N.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[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]

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]

Chu, X. X.

Ciattoni, A.

Cincotti, G.

Cižmár, T.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Coll-Lladó, C.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Couairon, A.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[Crossref] [PubMed]

Crosignani, B.

Cui, Y. P.

Dalgarno, H.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Deng, D. M.

Deng, F.

F. Deng and D. M. Deng, “Nonparaxial propagation of an Airy-Gaussian beam in uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 380, 280–286 (2016).
[Crossref]

Deng, X. Q.

Dholakia, K.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

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

Dogariu, A.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[Crossref]

Du, L. P.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[Crossref]

Fang, H.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Ferrier, D.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Gahagan, K. T.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[Crossref]

Gu, B.

Gunn-Moore, F.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Guo, Q.

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[Crossref] [PubMed]

Karpeev, S. V.

S. N. Khonina, S. V. Karpeev, V. D. Paranin, and A. A. Morozov, “Polarization conversion under focusing of vortex laser beams along the axis of anisotropic crystals,” Phys. Lett. A 381, 2444–2455 (2017).
[Crossref]

Khonina, S. N.

S. N. Khonina, S. V. Karpeev, V. D. Paranin, and A. A. Morozov, “Polarization conversion under focusing of vortex laser beams along the axis of anisotropic crystals,” Phys. Lett. A 381, 2444–2455 (2017).
[Crossref]

S. N. Khonina, V. D. Paranin, A. V. Ustinov, and A. P. Krasnov, “Astigmatic transformation of Bessel beams in a uniaxial crystal,” Opt. Applicate XLVI(1), 5–18 (2016).

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of Femtosecond Laser Airy Beams in Water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref] [PubMed]

Krasnov, A. P.

S. N. Khonina, V. D. Paranin, A. V. Ustinov, and A. P. Krasnov, “Astigmatic transformation of Bessel beams in a uniaxial crystal,” Opt. Applicate XLVI(1), 5–18 (2016).

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[Crossref] [PubMed]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Lai, Z. Y.

Lei, T.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[Crossref]

Li, D. D.

Li, H. G.

D. M. Deng, C. D. Chen, X. Zhao, and H. G. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[Crossref]

Liang, J. H.

Mazilu, M.

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

Min, C. J.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Moloney, J.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of Femtosecond Laser Airy Beams in Water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref] [PubMed]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Morozov, A. A.

S. N. Khonina, S. V. Karpeev, V. D. Paranin, and A. A. Morozov, “Polarization conversion under focusing of vortex laser beams along the axis of anisotropic crystals,” Phys. Lett. A 381, 2444–2455 (2017).
[Crossref]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Nylk, J.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Ouyang, Z. B.

Palma, C.

Panagiotopoulos, P.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[Crossref] [PubMed]

Papazoglou, D. G.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[Crossref] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Paranin, V. D.

S. N. Khonina, S. V. Karpeev, V. D. Paranin, and A. A. Morozov, “Polarization conversion under focusing of vortex laser beams along the axis of anisotropic crystals,” Phys. Lett. A 381, 2444–2455 (2017).
[Crossref]

S. N. Khonina, V. D. Paranin, A. V. Ustinov, and A. P. Krasnov, “Astigmatic transformation of Bessel beams in a uniaxial crystal,” Opt. Applicate XLVI(1), 5–18 (2016).

Peng, X.

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(4), 891–898 (2017).
[Crossref]

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Peng, Y. L.

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(4), 891–898 (2017).
[Crossref]

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Piché, M.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
[Crossref]

Polynkin, P.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of Femtosecond Laser Airy Beams in Water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Porto, P. D.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[Crossref]

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[Crossref] [PubMed]

Rui, G. H.

Salamin, Y. I.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Shen, J. F.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Shen, Z.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[Crossref] [PubMed]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[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]

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]

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Swartzlander, G. A.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Tzortzakis, S.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[Crossref] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Ustinov, A. V.

S. N. Khonina, V. D. Paranin, A. V. Ustinov, and A. P. Krasnov, “Astigmatic transformation of Bessel beams in a uniaxial crystal,” Opt. Applicate XLVI(1), 5–18 (2016).

Varin, C.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
[Crossref]

Vettenburg, T.

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Wang, Q.

Wen, B.

Wu, L. J.

Wu, Q. Y.

Xu, S. X.

Xue, Y. X.

Yang, X. B.

Yang, X. L.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Yuan, G. H.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Yuan, X. C.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Zhan, Q. W.

Zhang, J. B.

Zhang, L. P.

Zhang, Y. Q.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Zhao, X.

D. M. Deng, C. D. Chen, X. Zhao, and H. G. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[Crossref]

Zheng, G. L.

Zhou, G. Q.

Zhou, K. Z.

Zhou, M. L.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Zhu, S. W.

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (3)

D. M. Deng, C. D. Chen, X. Zhao, and H. G. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[Crossref]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[Crossref]

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, S83–S88 (2002).
[Crossref]

Chin. Phys. B (1)

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

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

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

J. Phys. D (2)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Nat. Commun. (2)

C. J. Min, Z. Shen, J. F. Shen, Y. Q. Zhang, H. Fang, G. H. Yuan, L. P. Du, S. W. Zhu, T. Lei, and X. C. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[Crossref] [PubMed]

Nat. Methods (1)

T. Vettenburg, H. Dalgarno, J. Nylk, C. Coll-Lladó, D. Ferrier, T. Čižmár, F. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

Nat. Photon. (1)

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

Opt. Applicate (1)

S. N. Khonina, V. D. Paranin, A. V. Ustinov, and A. P. Krasnov, “Astigmatic transformation of Bessel beams in a uniaxial crystal,” Opt. Applicate XLVI(1), 5–18 (2016).

Opt. Commun. (2)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

F. Deng and D. M. Deng, “Nonparaxial propagation of an Airy-Gaussian beam in uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 380, 280–286 (2016).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Lett. A (1)

S. N. Khonina, S. V. Karpeev, V. D. Paranin, and A. A. Morozov, “Polarization conversion under focusing of vortex laser beams along the axis of anisotropic crystals,” Phys. Lett. A 381, 2444–2455 (2017).
[Crossref]

Phys. Rev. Lett. (7)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[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]

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of Femtosecond Laser Airy Beams in Water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 Intensity distributions in different observation planes and the maximum intensity distribution of the RPAiXBs in uniaxial crystals orthogonal to the optical axis with ne = 1.5no.
Fig. 2
Fig. 2 Intensity distributions in x-z and y-z planes and the maximum intensity distribution of the RPAiXBs in uniaxial crystals orthogonal to the optical axis with different ratios of ne to no.
Fig. 3
Fig. 3 Intensity distributions in different observation planes and the maximum intensity distribution of the RPAiYBs in uniaxial crystals orthogonal to the optical axis with no = 2.616.
Fig. 4
Fig. 4 Propagation trajectories of the RPAiBs in the (a1)–(a3) x- and (b1)–(b3) y-directions through uniaxial crystals orthogonal to the optical axis with different ratios of ne to no.
Fig. 5
Fig. 5 Transverse intensity distributions of the (a1)–(d1) RPAiXBs, (a2)–(d2) RPAiYBs and (a3)–(d3) RPAiBs at different propagation distances with n0 = 1.5no.
Fig. 6
Fig. 6 (a1)–(a5) The transverse gradient force patterns and flows with ne = 1.5no and (b) the maximum gradient force distribution of the RPAiXBs during the propagation with different ratios of ne to no.
Fig. 7
Fig. 7 (a1)–(a5) The transverse gradient force patterns and flows and (b) the maximum gradient force distribution of the RPAiYBs during the propagation with no = 2.616.
Fig. 8
Fig. 8 (a) Maximum scattering force distributions of the RPAiXBs with different ratios of ne to no and (b) the RPAiYBs with no = 2.616 during the propagation.

Equations (14)

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

ε = ( n e 2 0 0 0 n o 2 0 0 0 n o 2 ) ,
[ E x ( x , y , 0 ) E y ( x , y , 0 ) ] = [ x w 1 Ai ( x w 1 ) Ai ( y w 2 ) exp ( a x w 1 + b y w 2 ) y w 2 Ai ( x w 1 ) Ai ( y w 2 ) exp ( a x w 1 + b y w 2 ) ] ,
E x ( x , y , z ) = i k n o 2 π z exp ( i k n e z ) + E x ( x 0 , y 0 , 0 ) × exp { i k [ n o 2 ( x x 0 ) 2 + n o 2 ( y y 0 ) 2 ] 2 z n e } d x 0 d y 0 ,
E y ( x , y , z ) = i k n o 2 π z exp ( i k n o z ) + E y ( x 0 , y 0 , 0 ) × exp { i k n o [ ( x x 0 ) 2 + ( y y 0 ) 2 ] 2 z } d x 0 d y 0 ,
E x ( x , y , z ) = k n o z w 1 w 2 M 1 N 1 exp [ Q ( x , y , z ) ] ( K 1 + K 2 ) ,
E y ( x , y , z ) = k n o z w 1 w 2 M 2 N 2 exp [ P ( x , y , z ) ] ( K 3 + K 4 ) ,
Q ( x , y , z ) = i k n e z + i 12 ( M 1 3 + N 1 3 ) 1 2 ( a M 1 2 + b N 1 2 ) i 2 ( a 2 M 1 + x w 1 M 1 + b 2 N 1 + y w 2 N 1 ) + a x w 1 + b y w 2 , P ( x , y , z ) = i k n o z + i 12 ( M 2 3 + N 2 3 ) 1 2 ( a M 2 2 + b N 2 2 ) i 2 ( a 2 M 2 + x w 1 M 2 + b 2 N 2 + y w 2 N 2 ) + a x w 1 + b y w 2 , K 1 = ( M 1 2 2 i a M 1 + x w 1 ) Ai ( M 1 2 4 i a M 1 + x w 1 ) Ai ( N 1 2 4 i b N 1 + y w 2 ) , K 2 = i M 1 A i ( M 1 2 4 i a M 1 + x w 1 ) Ai ( N 1 2 4 i b N 1 + y w 2 ) , K 3 = ( N 2 2 2 i b N 2 + y w 2 ) Ai ( N 2 2 4 i b N 2 + y w 2 ) Ai ( M 2 2 4 i a M 2 + x w 1 ) , K 4 = i N 2 A i ( N 2 2 4 i b N 2 + y w 2 ) Ai ( M 2 2 4 i a M 2 + x w 1 ) , M 1 = z n e k w 1 2 n o 2 , N 1 = z k w 2 2 n e , M 2 = z k w 1 2 n o , N 2 = z k w 2 2 n o ,
I x = | E x ( x , y , z ) | 2 , I y = | E y ( x , y , z ) | 2 .
I = I x + I y .
x 1 = z 2 n e 2 4 k 2 w 1 3 n o 4 , y 1 = z 2 4 k 2 w 2 3 n e 2 ,
x 2 = z 2 4 k 2 w 1 3 n o 2 , y 2 = z 2 4 k 2 w 2 3 n o 2 .
F g r a d ( x , y , z ) = 2 π n 2 r 0 3 c ( m 2 1 m 2 + 2 ) I ( x , y , z ) ,
F s c a t ( x , y , z ) = 8 π n 2 r 0 6 3 c ( m 2 1 m 2 + 2 ) 2 I ( x , y , z ) e z ,
I ( x , y , z ) = c n 2 ε 0 | E ( x , y , z ) | 2 2 ,

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