Abstract

We introduce new kinds of beams, cos-Airy-Gaussian and the cos-Airy-Gaussian-vortex beams, which in theory can be achieved by adding a cosine complex variable function onto normal beams. The analytical expressions for these beams propagating in a chiral medium are deduced, and we focus on exploring the effects of the cosine factor on them. It is shown that the cosine factor can eliminate the central lobe and the x-direction side lobe of the origin intensity distribution when the Airy-Gaussian beams tend to be Airy beams. During propagation, the intensity of the cos-Airy-Gaussian beams transfers from the side lobe in the y-direction to a certain lobe and finally flows to the side lobe in the x-direction. Moreover, the cos-Airy-Gaussian beams have a special transverse displacement along the z-axis when the distribution factor χ0 is small, which is unpredictable in analytical expressions unlike the normal Airy-Gaussian beams. In addition, we have developed several new formulae about the ultimate transverse displacements and the overlap position of the beams and the optical vortex, which have not been used before, and we find that there are always ultimate transverse displacements of the Airy-Gaussian beams and vortex because of the existence of the distribution factor χ0.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [Crossref]
  5. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
    [Crossref] [PubMed]
  6. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
    [Crossref]
  7. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy beams,” Science 324, 229–232 (2009).
    [Crossref] [PubMed]
  8. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of Femtosecond Laser Airy Beams in Water,” Phys. Rev. Lett. 103, 123902 (2009).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (4)

J. Xie, J. Zhang, J. Ye, H. Liu, Z. liang, S. Long, K. Zhou, and D. Deng, “Paraxial propagation of the first-order chirped Airy vortex beams in a chiral medium,” Opt. Express 26, 5845–5856 (2018).
[Crossref]

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

H. Li, J. Wang, M. Tang, and X. Li, “Propagation properties of cosh-Airy beams,” J. Mod. Opt. 65, 314–320 (2018).
[Crossref]

H. Li, J. Wang, M. Tang, J. Cao, and X. Li, “Phase transition of cosh-Airy beams in inhomogeneous media,” Opt. Commun. 427, 147–151 (2018).
[Crossref]

2017 (1)

S. Hua, Y. Liu, H. Zhang, L. Tang, and Y. Feng, “Propagation of an Airy-Gaussian-Vortex beam in a chiral medium,” Opt. Commun. 388, 29–37 (2017).
[Crossref]

2016 (1)

F. Deng, W. Yu, J. Huang, R. Zhao, J. Lin, and D. Deng, “Propagation of Airy-Gaussian beams in a chiral medium,” Eur. Phys. J. D 70, 87 (2016).
[Crossref]

2014 (1)

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

2013 (3)

2012 (1)

2011 (2)

2010 (2)

H. Dai, Y. Liu, D. Luo, and X. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35, 4075–4077 (2010).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105, 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, 229–232 (2009).
[Crossref] [PubMed]

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

2008 (3)

2007 (3)

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[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, 253901 (2010).
[Crossref]

Balazs, N. L.

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

Bandres, M. A.

Baumgartl, J.

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

Berry, M. V.

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

Broky, J.

Cao, J.

H. Li, J. Wang, M. Tang, J. Cao, and X. Li, “Phase transition of cosh-Airy beams in inhomogeneous media,” Opt. Commun. 427, 147–151 (2018).
[Crossref]

Chen, J.

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

Chen, X.

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

Chen, Z.

Chremmos, I. D.

Christodoulides, D. N.

Cizmar, T.

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

Coll-Llado, C.

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

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]

Dai, H.

Dalgarno, H.

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

Deng, D.

Deng, F.

F. Deng, W. Yu, J. Huang, R. Zhao, J. Lin, and D. Deng, “Propagation of Airy-Gaussian beams in a chiral medium,” Eur. Phys. J. D 70, 87 (2016).
[Crossref]

Dholakia, K.

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

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

Dogariu, A.

Du, X.

Efremidis, N. K.

Feng, Y.

S. Hua, Y. Liu, H. Zhang, L. Tang, and Y. Feng, “Propagation of an Airy-Gaussian-Vortex beam in a chiral medium,” Opt. Commun. 388, 29–37 (2017).
[Crossref]

Ferrier, D.

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

Gao, Y.

Gunn-Moore, F.

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

Gutierrez-Vega, J. C.

Hua, S.

S. Hua, Y. Liu, H. Zhang, L. Tang, and Y. Feng, “Propagation of an Airy-Gaussian-Vortex beam in a chiral medium,” Opt. Commun. 388, 29–37 (2017).
[Crossref]

Huang, J.

F. Deng, W. Yu, J. Huang, R. Zhao, J. Lin, and D. Deng, “Propagation of Airy-Gaussian beams in a chiral medium,” Eur. Phys. J. D 70, 87 (2016).
[Crossref]

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, 229–232 (2009).
[Crossref] [PubMed]

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

Lai, Z.

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

Li, H.

H. Li, J. Wang, M. Tang, and X. Li, “Propagation properties of cosh-Airy beams,” J. Mod. Opt. 65, 314–320 (2018).
[Crossref]

H. Li, J. Wang, M. Tang, J. Cao, and X. Li, “Phase transition of cosh-Airy beams in inhomogeneous media,” Opt. Commun. 427, 147–151 (2018).
[Crossref]

Li, X.

H. Li, J. Wang, M. Tang, J. Cao, and X. Li, “Phase transition of cosh-Airy beams in inhomogeneous media,” Opt. Commun. 427, 147–151 (2018).
[Crossref]

H. Li, J. Wang, M. Tang, and X. Li, “Propagation properties of cosh-Airy beams,” J. Mod. Opt. 65, 314–320 (2018).
[Crossref]

liang, Z.

Lin, J.

F. Deng, W. Yu, J. Huang, R. Zhao, J. Lin, and D. Deng, “Propagation of Airy-Gaussian beams in a chiral medium,” Eur. Phys. J. D 70, 87 (2016).
[Crossref]

Liu, H.

Liu, J.

Liu, Y.

Long, S.

Luo, D.

Mazilu, M.

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

Mo, H.

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

Moloney, J.

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

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, 229–232 (2009).
[Crossref] [PubMed]

Nylk, J.

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

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]

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]

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

Polynkin, P.

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

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

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

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

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 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, 213901 (2007).
[Crossref]

Sun, X.

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, 253901 (2010).
[Crossref]

Tang, L.

S. Hua, Y. Liu, H. Zhang, L. Tang, and Y. Feng, “Propagation of an Airy-Gaussian-Vortex beam in a chiral medium,” Opt. Commun. 388, 29–37 (2017).
[Crossref]

Tang, M.

H. Li, J. Wang, M. Tang, and X. Li, “Propagation properties of cosh-Airy beams,” J. Mod. Opt. 65, 314–320 (2018).
[Crossref]

H. Li, J. Wang, M. Tang, J. Cao, and X. Li, “Phase transition of cosh-Airy beams in inhomogeneous media,” Opt. Commun. 427, 147–151 (2018).
[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]

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

Vettenburg, T.

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

Wang, J.

H. Li, J. Wang, M. Tang, J. Cao, and X. Li, “Phase transition of cosh-Airy beams in inhomogeneous media,” Opt. Commun. 427, 147–151 (2018).
[Crossref]

H. Li, J. Wang, M. Tang, and X. Li, “Propagation properties of cosh-Airy beams,” J. Mod. Opt. 65, 314–320 (2018).
[Crossref]

Xie, J.

Yang, X.

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

Ye, J.

Ye, Y.

Yu, W.

F. Deng, W. Yu, J. Huang, R. Zhao, J. Lin, and D. Deng, “Propagation of Airy-Gaussian beams in a chiral medium,” Eur. Phys. J. D 70, 87 (2016).
[Crossref]

Zhang, H.

S. Hua, Y. Liu, H. Zhang, L. Tang, and Y. Feng, “Propagation of an Airy-Gaussian-Vortex beam in a chiral medium,” Opt. Commun. 388, 29–37 (2017).
[Crossref]

Zhang, J.

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

J. Xie, J. Zhang, J. Ye, H. Liu, Z. liang, S. Long, K. Zhou, and D. Deng, “Paraxial propagation of the first-order chirped Airy vortex beams in a chiral medium,” Opt. Express 26, 5845–5856 (2018).
[Crossref]

Zhang, P.

Zhao, D.

Zhao, J.

Zhao, R.

F. Deng, W. Yu, J. Huang, R. Zhao, J. Lin, and D. Deng, “Propagation of Airy-Gaussian beams in a chiral medium,” Eur. Phys. J. D 70, 87 (2016).
[Crossref]

Zhou, K.

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

J. Xie, J. Zhang, J. Ye, H. Liu, Z. liang, S. Long, K. Zhou, and D. Deng, “Paraxial propagation of the first-order chirped Airy vortex beams in a chiral medium,” Opt. Express 26, 5845–5856 (2018).
[Crossref]

Zhuang, F.

Am. J. Phys. (1)

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

Eur. Phys. J. D (1)

F. Deng, W. Yu, J. Huang, R. Zhao, J. Lin, and D. Deng, “Propagation of Airy-Gaussian beams in a chiral medium,” Eur. Phys. J. D 70, 87 (2016).
[Crossref]

J. Mod. Opt. (1)

H. Li, J. Wang, M. Tang, and X. Li, “Propagation properties of cosh-Airy beams,” J. Mod. Opt. 65, 314–320 (2018).
[Crossref]

J. Opt. (1)

K. Zhou, J. Zhang, H. Mo, J. Chen, X. Yang, Z. Lai, X. Chen, X. Yang, and D. Deng, “Propagation of the chirped-Airy-Gaussian-vortex beams in the chiral medium,” J. Opt. 20, 075601 (2018).
[Crossref]

Nat. Commun. (1)

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]

Nat. Methods (1)

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

Nat. Photonics (1)

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

Opt. Commun. (2)

S. Hua, Y. Liu, H. Zhang, L. Tang, and Y. Feng, “Propagation of an Airy-Gaussian-Vortex beam in a chiral medium,” Opt. Commun. 388, 29–37 (2017).
[Crossref]

H. Li, J. Wang, M. Tang, J. Cao, and X. Li, “Phase transition of cosh-Airy beams in inhomogeneous media,” Opt. Commun. 427, 147–151 (2018).
[Crossref]

Opt. Express (3)

Opt. Lett. (8)

Phys. Rev. Lett. (3)

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

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

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy Light Bullets in the Linear and Nonlinear Regimes,” Phys. Rev. Lett. 105, 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, 229–232 (2009).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1 Intensity distributions of the Airy-Gaussian beams (a1)–(a3) and cos-Airy-Gaussian beams (b1)–(b3) at the z=0 plane with different distribution factors χ0.
Fig. 2
Fig. 2 Intensity distribution of the LCP beams of the Airy-Gaussian beams (a1)–(a3), (c1)–(c3) and cos-Airy-Gaussian beams (b1)–(b3), (d1)–(d3) at different propagation distances with different distribution factors χ0. (Because the results from the RCP beams are similar, we only present the snapshots of the LCP beams)
Fig. 3
Fig. 3 Intensity evolution along the z-axis of the Airy-Gaussian beams (a1)–(a4) and cos-Airy-Gaussian beams (b1)–(b4), and normalized maximum intensity evolution along the z-axis (c1)–(c2) of the cos-Airy-Gaussian beams with different distribution factors χ0.
Fig. 4
Fig. 4 Interference intensity (a1), (b1), total intensity (a2), (b2) and normalized interference intensity evolutions (c) along the z-axis of cos-Airy-Gaussian beams with different distribution factors χ0.
Fig. 5
Fig. 5 Central position (a1), (b1) and optical vortex position evolutions (a2), (b2) of the cos- or Airy-Gaussian-vortex beams along the z-axis with different distribution factors χ0.
Fig. 6
Fig. 6 Ultimate transverse displacement of the cos- or Airy-Gaussian beams and the optical vortex as a function of χ0 (a1)–(a2), and overlap position (beams center and vortex are superimposed) as a function of the transverse origin vortex position with different chiral parameters γ and distribution factors χ0 (b1)–(b2).
Fig. 7
Fig. 7 Intensity and phase distributions of the LCP beams of the cos-Airy-Gaussian-vortex beams with different origin vortex positions. The black arrows in the third row indicate the locations of vortex. (0, −0.4) (a1)–(a4), (0, −0.8) (b1)–(b4), (0, −1.2) (c1)–(c4), (0, −1.6) (d1)–(d4).

Equations (53)

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E ( x 0 , y 0 , 0 ) = A 0 Ai ( x 0 w 1 ) Ai ( y 0 w 2 ) exp ( a x 0 w 1 + a y 0 w 2 ) exp ( x 0 2 + y 0 2 w 0 2 ) × ( x 0 x 1 w 1 + i y 0 y 1 w 2 ) l cos m ( x 0 w 1 + i y 0 w 2 ) ,
E ( x , y , z ) = i k 2 π B + E ( x 0 , y 0 , 0 ) × exp { i k 2 B [ A ( x 0 2 + y 0 2 ) 2 ( x 0 x + y 0 y ) + D ( x 2 + y 2 ) ] } d x 0 d y 0 ,
E 00 ( x , y , z ) = i k A 0 2 BM exp [ Q 1 ( x , y , z ) ] L 1 ,
E 01 ( x , y , z ) = i k A 0 4 BM { exp [ Q 2 ( x , y , z ) ] L 2 + exp [ Q 3 ( x , y , z ) ] L 3 } ,
E 10 ( x , y , z ) = i k A 0 2 BM exp [ Q 1 ( x , y , z ) ] ( K 1 + K 2 + K 3 ) ,
E 11 ( x , y , z ) = k A 0 4 BM { exp [ Q 2 ( x , y , z ) ] ( P 1 + P 2 + P 3 ) + exp [ Q 3 ( x , y , z ) ] ( F 1 + F 2 + F 3 ) } ,
Q 1 ( x , y , z ) = i k D 2 B ( x 2 + y 2 ) + N 01 2 + N 02 2 4 M + 1 8 M 2 ( N 01 w 1 3 + N 02 w 2 3 ) + 1 96 M 3 ( 1 w 1 6 + 1 w 2 6 ) ,
Q 2 ( x , y , z ) = i k D 2 B ( x 2 + y 2 ) + N 11 2 + N 12 2 4 M + 1 8 M 2 ( N 11 w 1 3 + N 12 w 2 3 ) + 1 96 M 3 ( 1 w 1 6 + 1 w 2 6 ) ,
Q 3 ( x , y , z ) = i k D 2 B ( x 2 + y 2 ) + N 13 2 + N 14 2 4 M + 1 8 M 2 ( N 13 w 1 3 + N 14 w 2 3 ) + 1 96 M 3 ( 1 w 1 6 + 1 w 2 6 ) ,
L 1 = Ai [ f 1 ( x ) ] Ai [ g 1 ( y ) ] , L 2 = Ai [ f 2 ( x ) ] Ai [ g 2 ( y ) ] , L 3 = Ai [ f 3 ( x ) ] Ai [ g 3 ( y ) ] ,
K 1 = [ 1 w 1 ( N 01 2 M + 1 8 w 1 3 M 2 x 1 ) + i w 2 ( N 02 2 M + 1 8 w 2 3 M 2 y 1 ) ] L 1 ,
K 2 = 1 2 w 1 2 M Ai [ f 1 ( x ) ] Ai [ g 1 ( y ) ] , K 3 = i 2 w 2 2 M Ai [ f 1 ( x ) ] Ai [ g 1 ( y ) ] ,
P 1 = [ i w 1 ( x 1 N 11 2 M 1 8 w 1 3 M 2 ) 1 w 2 ( y 1 N 12 2 M 1 8 w 2 3 M 2 ) ] L 2 ,
P 2 = i 2 w 1 2 M Ai [ f 2 ( x ) ] Ai [ g 2 ( y ) ] , P 3 = 1 2 w 2 2 M Ai [ f 2 ( x ) ] Ai [ g 2 ( y ) ] ,
F 1 = [ i w 1 ( x 1 N 13 2 M 1 8 w 1 3 M 2 ) 1 w 2 ( y 1 N 14 2 M 1 8 w 2 3 M 2 ) ] L 3 ,
F 2 = i 2 w 1 2 M Ai [ f 3 ( x ) ] Ai [ g 3 ( y ) ] , F 3 = 1 2 w 2 2 M Ai [ f 3 ( x ) ] Ai [ g 3 ( y ) ] ,
f 1 ( x ) = 1 16 w 1 4 M 2 + N 01 2 w 1 M , g 1 ( y ) = 1 16 w 2 4 M 2 + N 02 2 w 2 M ,
f 2 ( x ) = 1 16 w 1 4 M 2 + N 11 2 w 1 M , g 2 ( y ) = 1 16 w 2 4 M 2 + N 12 2 w 2 M ,
f 3 ( x ) = 1 16 w 1 4 M 2 + N 13 2 w 1 M , g 3 ( y ) = 1 16 w 2 4 M 2 + N 14 2 w 2 M ,
M = 1 w 0 2 + i k A 2 B , N 01 = a w 1 + i k x B , N 02 = a w 2 + i k y B , N 11 = a + i w 1 + i k x B , N 12 = a 1 w 2 + i k y B , N 13 = a i w 1 + i k x B , N 14 = a + 1 w 2 + i k y B .
( A ( L ) B ( L ) C ( L ) D ( L ) ) = ( 1 z / n ( L ) 0 1 ) and ( A ( R ) B ( R ) C ( R ) D ( R ) ) = ( 1 z / n ( R ) 0 1 ) ,
I = | E ( L ) ( x , y , z ) | 2 + | E ( R ) ( x , y , z ) | 2 + I int ,
I int = E ( L ) ( x , y , z ) E ( R ) * ( x , y , z ) + E ( R ) ( x , y , z ) E ( L ) * ( x , y , z ) ,
x c = A 16 w 1 3 M M * ,
x o v = A x 1 + A 8 w 1 3 M M * ,
x cl = Aw 1 16 χ 0 4 , ( z )
x ovl = Aw 1 8 χ 0 4 , ( z )
z s ( L , R ) = An ( L , R ) ( w 1 16 x 1 + χ 0 4 ) Zr , ( w 1 16 χ 0 4 < x 1 0 )
E 01 ( x 0 , y 0 , 0 ) = A 0 Ai ( x 0 w 1 ) Ai ( y 0 w 2 ) exp ( a x 0 w 1 + a y 0 w 2 ) exp ( x 0 2 + y 0 2 w 0 2 ) cos ( x 0 w 1 + i y 0 w 2 ) .
E ( 1 ) ( x 0 , y 0 , 0 ) = 1 2 A 0 Ai ( x 0 w 1 ) Ai ( y 0 w 2 ) exp ( a x 0 w 1 + a y 0 w 2 ) exp ( x 0 2 + y 0 2 w 0 2 ) exp ( i x 0 w 1 y 0 w 2 ) ,
E ( 2 ) ( x 0 , y 0 , 0 ) = 1 2 A 0 Ai ( x 0 w 1 ) Ai ( y 0 w 2 ) exp ( a x 0 w 1 + a y 0 w 2 ) exp ( x 0 2 + y 0 2 w 0 2 ) exp ( i x 0 w 1 + y 0 w 2 ) .
E ( 1 ) ( x , y , z ) = i k A 0 4 π B exp [ i k D 2 B ( x 2 + y 2 ) ] × + Ai ( x 0 w 1 ) exp [ ( 1 w 0 2 + i k A 2 B ) x 0 2 + ( a + i w 1 + i k x B ) x 0 ] d x 0 × + Ai ( y 0 w 2 ) exp [ ( 1 w 0 2 + i k A 2 B ) y 0 2 + ( a 1 w w + i k y B ) y 0 ] d y 0 .
M = 1 w 0 2 + i k A 2 B , N 11 = a + i w 1 + i k x B , N 12 = a 1 w 2 + i k y B ,
E ( 1 ) ( x , y , z ) = i k A 0 4 π B exp [ i k D 2 B ( x 2 + y 2 ) ] × + Ai ( x 0 w 1 ) exp ( M x 0 2 + N 11 x 0 ) d x 0 + Ai ( y 0 w 2 ) exp ( M y 0 2 + N 12 y 0 ) d y 0 .
+ Ai ( x a ) exp ( b x 2 + c x ) d x = π b exp ( c 2 4 b + c 8 a 3 b 2 + 1 96 a 6 b 3 ) Ai ( 1 16 a 4 b 2 + c 2 a b ) ,
E ( 1 ) ( x , y , z ) = i k A 0 4 BM exp [ i k D 2 B ( x 2 + y 2 ) + N 11 2 + N 12 2 4 M + 1 8 M 2 ( N 11 w 1 3 + N 12 w 2 3 ) + 1 96 M 3 ( 1 w 1 6 + 1 w 2 6 ) ] Ai ( 1 16 w 1 4 M 2 + N 11 2 w 1 M ) Ai ( 1 16 w 2 4 M 2 + N 12 2 w 2 M ) .
E ( 2 ) ( x , y , z ) = i k A 0 4 BM exp [ i k D 2 B ( x 2 + y 2 ) + N 13 2 + N 14 2 4 M + 1 8 M 2 ( N 13 w 1 3 + N 14 w 2 3 ) + 1 96 M 3 ( 1 w 1 6 + 1 w 2 6 ) ] Ai ( 1 16 w 1 4 M 2 + N 13 2 w 1 M ) Ai ( 1 16 w 2 4 M 2 + N 14 2 w 2 M ) ,
N 13 = a i w 1 + i k x B , N 14 = a + 1 w 2 + i k y B .
E 01 ( x , y , z ) = E ( 1 ) ( x , y , z ) + E ( 2 ) ( x , y , z ) .
f 1 ( x ) = i k 2 Bw 1 M ( x + B 8 i k w 1 3 M + Ba i k w 1 ) .
x c = A 16 w 1 3 M M * .
N 01 2 M + 1 8 w 1 3 M 2 x 1 = i k 2 BM ( x + Ba i k w 1 + B 4 i k w 1 3 M 2 BM i k x 1 ) .
x ov = A x 1 + A 8 w 1 3 M M * .
f 2 ( x ) = 1 16 w 1 4 M 2 + N 11 2 w 1 M = i k 2 B w 1 M ( x + B 8 i k w 1 3 M + B ( a + i ) i k w 1 ) .
x c 1 = A 16 w 1 3 M M * B k w 1 .
f 3 ( x ) = 1 16 w 1 4 M 2 + N 13 2 w 1 M = i k 2 B w 1 M ( x + B 8 i k w 1 3 M + B ( a i ) i k w 1 ) ,
x c 2 = A 16 w 1 3 M M * + B k w 1 .
x c = x c 1 + x c 2 2 = A 16 w 1 3 M M * .
P 1 = [ i w 1 ( x 1 N 11 2 M 1 8 w 1 3 M 2 ) 1 w 2 ( y 1 N 12 2 M 1 8 w 2 3 M 2 ) ] L 2 , x 1 N 11 2 M 1 8 w 1 3 M 2 = i k 2 BM ( x + B ( a + i ) i k w 1 + B 4 i k w 1 3 M 2 BM i k x 1 ) .
x ov 1 = A x 1 + A 8 w 1 3 M M * B k w 1 .
F 1 = [ i w 1 ( x 1 N 13 2 M 1 8 w 1 3 M 2 ) 1 w 2 ( y 1 N 14 2 M 1 8 w 2 3 M 2 ) ] L 3 , x 1 N 13 2 M 1 8 w 1 3 M 2 = i k 2 BM ( x + B ( a i ) i k w 1 + B 4 i k w 1 3 M 2 BM i k x 1 ) ,
x ov 2 = A x 1 + A 8 w 1 3 M M * B k w 1 .
x ov = x ov 1 + x ov 2 2 = A x 1 + A 8 w 1 3 M M * .

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