Abstract

We examine selective reflection (SR) spectrum of an Airy beam at an interface between a dielectric and a homogeneous atomic medium. It is shown that both the general reflection (GR) and the SR of Airy beams exhibit accelerating dynamics with a parabolic trajectory, however, the accelerating rate for the SR is slightly greater than that for the GR. Due to interaction of atoms and the Airy beams at the interface between dielectric and resonant atoms, the SR beams can create far-field interference patterns. We also show that the amplitude of the SR can be dramatically modified by the detuning of the incident fields, the reduced x-coordinate and the distance from the interface. The SR at the resonant medium interface of Airy beams can probably be a powerful tool in the use of optical power delivering, resonant particle manipulation and spatial spectrum detection of a resonant medium at an interface.

© 2013 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  22. M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett.74(5), 666–669 (1995).
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    [CrossRef]

2012 (3)

2011 (3)

2010 (2)

2009 (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

2008 (3)

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[CrossRef] [PubMed]

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

Y. Li, X. Hou, J. Bai, J. Yan, C. Gan, and Y. Zhang, “Two-photon Dicke-narrowing selective reflection spectroscopy in thin atomic vapor,” Acta Opt. Sin.28(8), 1623–1627 (2008) (in Chinese).
[CrossRef]

2007 (3)

P. Chaves de Souza Segundo, I. Hamdi, M. Fichet, D. Bloch, and M. Ducloy, “Selective reflection spectroscopy on the UV third-resonance line of Cs: Simultaneous probing of a van der Waals atom-surface interaction sensitive to far IR couplings and interatomic collisions,” Laser Phys.17(7), 983–992 (2007).
[CrossRef]

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] [PubMed]

2006 (1)

Y. Li, G. Zhang, and Y. Zhou, “Selective reflection combined with Fabry-Perot effects from two-level atoms confined between two dielectric walls,” Chin. Phys.15(5), 985–991 (2006).
[CrossRef]

2005 (1)

M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

2003 (1)

H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D23(2), 237–255 (2003).
[CrossRef]

1997 (1)

S. Harris, “Electromagnetically induced transparency,” Phys. Today50(7), 36–42 (1997).
[CrossRef]

1995 (1)

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett.74(5), 666–669 (1995).
[CrossRef] [PubMed]

1994 (1)

N. Papageorgiou, M. Fichet, V. A. Sautenkov, D. Bloch, and M. Ducloy, “Doppler-free reflection spectroscopy of self-induced and krypton-induced collisional shift and broadening of cesium D2 line components in optically dense vapor,” Laser Phys.4, 392–395 (1994).

1992 (1)

M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France2(4), 631–657 (1992).
[CrossRef]

1991 (1)

K. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

1979 (1)

M. Berry and N. Balazs, “Non-spreading wave packets,” Am. J. Phys.47(3), 264–267 (1979).
[CrossRef]

Bai, J.

Y. Li, X. Hou, J. Bai, J. Yan, C. Gan, and Y. Zhang, “Two-photon Dicke-narrowing selective reflection spectroscopy in thin atomic vapor,” Acta Opt. Sin.28(8), 1623–1627 (2008) (in Chinese).
[CrossRef]

Balasanyan, N.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

Balazs, N.

M. Berry and N. Balazs, “Non-spreading wave packets,” Am. J. Phys.47(3), 264–267 (1979).
[CrossRef]

Berry, M.

M. Berry and N. Balazs, “Non-spreading wave packets,” Am. J. Phys.47(3), 264–267 (1979).
[CrossRef]

Bloch, D.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

P. Chaves de Souza Segundo, I. Hamdi, M. Fichet, D. Bloch, and M. Ducloy, “Selective reflection spectroscopy on the UV third-resonance line of Cs: Simultaneous probing of a van der Waals atom-surface interaction sensitive to far IR couplings and interatomic collisions,” Laser Phys.17(7), 983–992 (2007).
[CrossRef]

H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D23(2), 237–255 (2003).
[CrossRef]

N. Papageorgiou, M. Fichet, V. A. Sautenkov, D. Bloch, and M. Ducloy, “Doppler-free reflection spectroscopy of self-induced and krypton-induced collisional shift and broadening of cesium D2 line components in optically dense vapor,” Laser Phys.4, 392–395 (1994).

M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France2(4), 631–657 (1992).
[CrossRef]

Boller, K.

K. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[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] [PubMed]

Chaves de Souza Segundo, P.

P. Chaves de Souza Segundo, I. Hamdi, M. Fichet, D. Bloch, and M. Ducloy, “Selective reflection spectroscopy on the UV third-resonance line of Cs: Simultaneous probing of a van der Waals atom-surface interaction sensitive to far IR couplings and interatomic collisions,” Laser Phys.17(7), 983–992 (2007).
[CrossRef]

Chen, Z.

Chevrollier, M.

M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France2(4), 631–657 (1992).
[CrossRef]

Chremmos, I. D.

Christodoulides, D. N.

Chu, X.

Dan, D.

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[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] [PubMed]

Du, X.

Ducloy, M.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

P. Chaves de Souza Segundo, I. Hamdi, M. Fichet, D. Bloch, and M. Ducloy, “Selective reflection spectroscopy on the UV third-resonance line of Cs: Simultaneous probing of a van der Waals atom-surface interaction sensitive to far IR couplings and interatomic collisions,” Laser Phys.17(7), 983–992 (2007).
[CrossRef]

H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D23(2), 237–255 (2003).
[CrossRef]

N. Papageorgiou, M. Fichet, V. A. Sautenkov, D. Bloch, and M. Ducloy, “Doppler-free reflection spectroscopy of self-induced and krypton-induced collisional shift and broadening of cesium D2 line components in optically dense vapor,” Laser Phys.4, 392–395 (1994).

M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France2(4), 631–657 (1992).
[CrossRef]

Efremidis, N. K.

Failache, H.

H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D23(2), 237–255 (2003).
[CrossRef]

Fichet, M.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

P. Chaves de Souza Segundo, I. Hamdi, M. Fichet, D. Bloch, and M. Ducloy, “Selective reflection spectroscopy on the UV third-resonance line of Cs: Simultaneous probing of a van der Waals atom-surface interaction sensitive to far IR couplings and interatomic collisions,” Laser Phys.17(7), 983–992 (2007).
[CrossRef]

H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D23(2), 237–255 (2003).
[CrossRef]

N. Papageorgiou, M. Fichet, V. A. Sautenkov, D. Bloch, and M. Ducloy, “Doppler-free reflection spectroscopy of self-induced and krypton-induced collisional shift and broadening of cesium D2 line components in optically dense vapor,” Laser Phys.4, 392–395 (1994).

M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France2(4), 631–657 (1992).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Gan, C.

Y. Li, X. Hou, J. Bai, J. Yan, C. Gan, and Y. Zhang, “Two-photon Dicke-narrowing selective reflection spectroscopy in thin atomic vapor,” Acta Opt. Sin.28(8), 1623–1627 (2008) (in Chinese).
[CrossRef]

Gao, P.

Gea-Banacloche, J.

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett.74(5), 666–669 (1995).
[CrossRef] [PubMed]

Hamdi, I.

P. Chaves de Souza Segundo, I. Hamdi, M. Fichet, D. Bloch, and M. Ducloy, “Selective reflection spectroscopy on the UV third-resonance line of Cs: Simultaneous probing of a van der Waals atom-surface interaction sensitive to far IR couplings and interatomic collisions,” Laser Phys.17(7), 983–992 (2007).
[CrossRef]

Harris, S.

S. Harris, “Electromagnetically induced transparency,” Phys. Today50(7), 36–42 (1997).
[CrossRef]

Harris, S. E.

K. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Hou, X.

Y. Li, X. Hou, J. Bai, J. Yan, C. Gan, and Y. Zhang, “Two-photon Dicke-narrowing selective reflection spectroscopy in thin atomic vapor,” Acta Opt. Sin.28(8), 1623–1627 (2008) (in Chinese).
[CrossRef]

Hu, Y.

Huang, S.

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Imamolu, A.

K. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Janunts, N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107(11), 116802 (2011).
[CrossRef] [PubMed]

Jin, S.

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett.74(5), 666–669 (1995).
[CrossRef] [PubMed]

Kivshar, Y. S.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107(11), 116802 (2011).
[CrossRef] [PubMed]

Klein, A. E.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107(11), 116802 (2011).
[CrossRef] [PubMed]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Laliotis, A.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

Lei, M.

Li, Y.

Y. Li, X. Hou, J. Bai, J. Yan, C. Gan, and Y. Zhang, “Two-photon Dicke-narrowing selective reflection spectroscopy in thin atomic vapor,” Acta Opt. Sin.28(8), 1623–1627 (2008) (in Chinese).
[CrossRef]

Y. Li, G. Zhang, and Y. Zhou, “Selective reflection combined with Fabry-Perot effects from two-level atoms confined between two dielectric walls,” Chin. Phys.15(5), 985–991 (2006).
[CrossRef]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett.74(5), 666–669 (1995).
[CrossRef] [PubMed]

Lou, C.

Marangos, J.

M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Maurin, I.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

Mills, M. S.

Minovich, A.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107(11), 116802 (2011).
[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,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Neshev, D. N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107(11), 116802 (2011).
[CrossRef] [PubMed]

Oria, M.

M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France2(4), 631–657 (1992).
[CrossRef]

Papageorgiou, N.

N. Papageorgiou, M. Fichet, V. A. Sautenkov, D. Bloch, and M. Ducloy, “Doppler-free reflection spectroscopy of self-induced and krypton-induced collisional shift and broadening of cesium D2 line components in optically dense vapor,” Laser Phys.4, 392–395 (1994).

Pertsch, T.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107(11), 116802 (2011).
[CrossRef] [PubMed]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Prakash, J.

Rahmat, G.

M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France2(4), 631–657 (1992).
[CrossRef]

Saltiel, S.

H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D23(2), 237–255 (2003).
[CrossRef]

Sarkisyan, A.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

Sarkisyan, D.

A. Laliotis, I. Maurin, M. Fichet, D. Bloch, M. Ducloy, N. Balasanyan, A. Sarkisyan, and D. Sarkisyan, “Selective reflection spectroscopy at the interface between a calcium fluoride window and Cs vapor,” Appl. Phys. B90(3-4), 415–420 (2008).
[CrossRef]

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N. Papageorgiou, M. Fichet, V. A. Sautenkov, D. Bloch, and M. Ducloy, “Doppler-free reflection spectroscopy of self-induced and krypton-induced collisional shift and broadening of cesium D2 line components in optically dense vapor,” Laser Phys.4, 392–395 (1994).

Shen, J.

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P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324(5924), 229–232 (2009).
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G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
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Acta Opt. Sin. (1)

Y. Li, X. Hou, J. Bai, J. Yan, C. Gan, and Y. Zhang, “Two-photon Dicke-narrowing selective reflection spectroscopy in thin atomic vapor,” Acta Opt. Sin.28(8), 1623–1627 (2008) (in Chinese).
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[CrossRef]

Chin. Phys. (1)

Y. Li, G. Zhang, and Y. Zhou, “Selective reflection combined with Fabry-Perot effects from two-level atoms confined between two dielectric walls,” Chin. Phys.15(5), 985–991 (2006).
[CrossRef]

Eur. Phys. J. D (1)

H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D23(2), 237–255 (2003).
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[CrossRef] [PubMed]

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P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324(5924), 229–232 (2009).
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Figures (4)

Fig. 1
Fig. 1

(a) The geometrical configuration of probe and coupling fields impinging on the interface between a transparent dielectric window and a homogeneous sample of cold atoms. The diverse coordinate systems are also shown. (b) Lambda-type three-level system of atoms.

Fig. 2
Fig. 2

The real part contribution of the SR versus the probe detuning Δ p / γ 31 for G c =0.3 γ 31 and G c = γ 31 .

Fig. 3
Fig. 3

GR amplitude (a1) and SR amplitude (b1) for the Airy beam with an incident angle θ 1 =10° . (a2), (b2): Beam amplitudes versus the position χ r from the Snell’s reflection axis in (a1) and (b1) respectively at different depths inside the medium 1. (i), (ii), (iii), (iv) and (v) are for z r = 0, 1cm, 2cm, 3cm and 4cm respectively. Other parameters are G c = γ 31 , Δ p = γ 31 .

Fig. 4
Fig. 4

SR amplitudes versus the position χ r of the Airy beam for various probe detuning and propagating depth in medium 1. (a), (b), (c), (d), (e) and (f) are for z r = 0, 1cm, 2cm, 3cm, 4cm and 5cm respectively. Red, blue and green lines are for Δ p =0 , γ 31 and 3 γ 31 respectively. Other parameters are G c = γ 31 , Δ c =0 .

Equations (15)

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E p ( x i , z i =0)= u 0 ( x i )exp(i k 1 sin θ 1 x i ),
u 0 ( x i )Ai(± x i cos θ 1 x 0 )exp(±a x i cos θ 1 x 0 ),
R gr (k)=( q 1 q 2 )/( q 1 + q 2 ),
R sr (k)=i/[ E pin (k) ( q 1 + q 2 ] [ E p (k) (z)/z ] z=0 ,
T(k)2 q 1 /( q 1 + q 2 ),
E p (k) (z)= E p (x,z) exp(ikx)dx,
E p (x,z)= 1 2π + [ + E p (x,z=h)exp(ikx)dx ]T(k)exp(ikx)exp(i q 2 z+i q 1 h) dk.
2 E p (k) (z)/ z 2 +2i q 2 E p (k) (z)/z=4π q 2 2 P (k) (z).
[ E p (k) (z)/z ] z=0 =4π q 2 2 0 L P (k) ( z )exp(2i q 2 z )d z .
σ 31 /t= Λ 31 σ 31 +i G p (k) ( σ 11 σ 33 )+i G c σ 21 ,
σ 21 /t= Λ 21 σ 21 i G p (k) σ 23 +i G c σ 31 ,
σ 23 /t= Λ 32 * σ 23 i G p (k) σ 21 i G c ( σ 22 σ 33 ),
P (k) (z)=iN μ 31 G p (k) Λ 21 /( Λ 21 Λ 31 + G c 2 ).
R sr (k)= 4π q 2 2 E pin (k) ( q 1 + q 2 ) 0 L N μ 31 G p (k) Λ 21 /( Λ 21 Λ 31 + G c 2 )exp(2i q 2 z )d z .
u( χ r , z r )= 1 2πcos θ 1 dκ F 0 ( κ cos θ 1 ) R sr ( k 1 sin θ 1 +κ)exp[ iκ χ r i κ 2 ( z r +h) 2 k 1 cos 3 θ 1 ].

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