Abstract

Provided the intensity is not too high (for example, with I << 1018 W/cm2, for a wavelength of 1 μm), response of an under-dense plasma to the fields of a laser pulse can still be considered linear, and inhomogeneous wave equations for the vector and scalar potentials A and Φ, respectively, may be derived from Maxwell’s equations. A rigorous, but approximate, solution to the wave equation satisfied by a one-component, azimuthally symmetric, vector potential is developed using a Fourier transform method. It is found that an ultra-short and tightly-focused, radially-polarized laser pulse, described by this vector potential, propagates in the plasma like a laser bullet. The pulse is termed a Bessel-Bessel bullet because, to leading order in a power-series expansion, the vector potential, from which the pulse fields E and B are derived, is expressed in terms of a Bessel function of the first kind J0 and a spherical Bessel function of the first kind j0.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

Y. I. Salamin, “Fields of an ultrashort tightly focused radially polarized laser pulse in a linear response plasma,” Phys. Plasmas 24, 103107 (2017).
[Crossref]

Y. I. Salamin and J.-X. Li, “Electromagnetic fields of an ultra-short tightly-focused radially-polarized laser pulse,” Opt. Commun. 405, 265 (2017).
[Crossref]

2016 (4)

J.-X. Li, Y. I. Salamin, K. Z. Hatsagortsyan, and C. H. Keitel, “Fields of an ultrashort tightly focused laser pulse,” J. Opt. Soc. Am. B 33, 405 (2016).
[Crossref]

J. M. Urrutia and R. L. Stenzel, “Helicon waves in uniform plasmas. IV. Bessel beams, Gendrin beams, and helicons,” Phys. Plasmas 23, 052112 (2016).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “An optical tweezer in asymmetrical vortex Bessel-Gaussian beams,” J. Appl. Phys. 120, 023101 (2016).
[Crossref]

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6, 30765 (2016).
[Crossref] [PubMed]

2015 (3)

Y. I. Salamin, “Simple analytical derivation of the fields of an ultrashort tightly focused linearly polarized laser pulse,” Phys. Rev. A 92, 063818 (2015).
[Crossref]

Y. I. Salamin, “Fields and propagation characteristics in vacuum of an ultrashort tightly focused radially polarized laser pulse,” Phys. Rev. A 92, 053836 (2015).
[Crossref]

J. Mendoza-Hernández, M. Arroyo-Carrasco, M. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre-Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40, 3739 (2015).
[Crossref] [PubMed]

2013 (1)

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24, 22 (2013).
[Crossref]

2012 (1)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607 (2012).
[Crossref]

2011 (1)

2010 (3)

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828 (2010).
[Crossref] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nature Photonics 4, 103 (2010).
[Crossref]

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. A 82033834 (2010).
[Crossref]

2009 (1)

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

2007 (1)

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

2006 (1)

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

2003 (1)

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

2002 (1)

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82 (2002).
[Crossref]

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348 (1998).
[Crossref]

1995 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177 (1979).
[Crossref]

Arlt, J.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82 (2002).
[Crossref]

Arnold, C. B.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607 (2012).
[Crossref]

Arroyo-Carrasco, M.

Belic, M.

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. A 82033834 (2010).
[Crossref]

Bellei, C.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Broky, J.

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

Chávez-Cerda, S.

J. Mendoza-Hernández, M. Arroyo-Carrasco, M. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre-Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40, 3739 (2015).
[Crossref] [PubMed]

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82 (2002).
[Crossref]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nature Photonics 4, 103 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nature Photonics 4, 103 (2010).
[Crossref]

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

Clarke, R. J.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Conti, C.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Dangor, A. E.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177 (1979).
[Crossref]

Dholakia, K.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82 (2002).
[Crossref]

Di Trapani, P.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Dogariu, A.

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

Dudley, A.

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24, 22 (2013).
[Crossref]

Duocastella, M.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607 (2012).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

Esarey, E.

Forbes, A.

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24, 22 (2013).
[Crossref]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348 (1998).
[Crossref]

Fu, S.

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6, 30765 (2016).
[Crossref] [PubMed]

Gao, C.

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6, 30765 (2016).
[Crossref] [PubMed]

Garcés-Chávez, V.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82 (2002).
[Crossref]

Hatsagortsyan, K. Z.

Heathcote, R.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348 (1998).
[Crossref]

Huang, T.

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. A 82033834 (2010).
[Crossref]

Iturbe-Castillo, M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Jedrkiewicz, O.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Kaluza, M. C.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Kamperidis, C.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Karsch, S.

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Keitel, C. H.

Kneip, S.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “An optical tweezer in asymmetrical vortex Bessel-Gaussian beams,” J. Appl. Phys. 120, 023101 (2016).
[Crossref]

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “An optical tweezer in asymmetrical vortex Bessel-Gaussian beams,” J. Appl. Phys. 120, 023101 (2016).
[Crossref]

Kozawa, Y.

Krall, J.

Krushelnick, K.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Lancaster, K. L.

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Lavery, M.

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24, 22 (2013).
[Crossref]

Li, J.-X.

Y. I. Salamin and J.-X. Li, “Electromagnetic fields of an ultra-short tightly-focused radially-polarized laser pulse,” Opt. Commun. 405, 265 (2017).
[Crossref]

J.-X. Li, Y. I. Salamin, K. Z. Hatsagortsyan, and C. H. Keitel, “Fields of an ultrashort tightly focused laser pulse,” J. Opt. Soc. Am. B 33, 405 (2016).
[Crossref]

Lopes, N.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Mangles, S. P. D.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Mendoza-Hernández, J.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

Mori, W. B.

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Nagel, S. R.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Najmudin, Z.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Nazarov, W.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348 (1998).
[Crossref]

Nilson, P. M.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Padgett, M.

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24, 22 (2013).
[Crossref]

Pilloff, M.

Piskarskas, A.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Porfirev, A. P.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “An optical tweezer in asymmetrical vortex Bessel-Gaussian beams,” J. Appl. Phys. 120, 023101 (2016).
[Crossref]

Pu, J.

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nature Photonics 4, 103 (2010).
[Crossref]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348 (1998).
[Crossref]

Salamin, Y. I.

Y. I. Salamin and J.-X. Li, “Electromagnetic fields of an ultra-short tightly-focused radially-polarized laser pulse,” Opt. Commun. 405, 265 (2017).
[Crossref]

Y. I. Salamin, “Fields of an ultrashort tightly focused radially polarized laser pulse in a linear response plasma,” Phys. Plasmas 24, 103107 (2017).
[Crossref]

J.-X. Li, Y. I. Salamin, K. Z. Hatsagortsyan, and C. H. Keitel, “Fields of an ultrashort tightly focused laser pulse,” J. Opt. Soc. Am. B 33, 405 (2016).
[Crossref]

Y. I. Salamin, “Simple analytical derivation of the fields of an ultrashort tightly focused linearly polarized laser pulse,” Phys. Rev. A 92, 063818 (2015).
[Crossref]

Y. I. Salamin, “Fields and propagation characteristics in vacuum of an ultrashort tightly focused radially polarized laser pulse,” Phys. Rev. A 92, 053836 (2015).
[Crossref]

Sato, S.

Schreiber, J.

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Siviloglou, G. A.

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

Sprangle, P.

Stenzel, R. L.

J. M. Urrutia and R. L. Stenzel, “Helicon waves in uniform plasmas. IV. Bessel beams, Gendrin beams, and helicons,” Phys. Plasmas 23, 052112 (2016).
[Crossref]

Thomas, A. G. R.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Tian, B.

Trillo, S.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Urrutia, J. M.

J. M. Urrutia and R. L. Stenzel, “Helicon waves in uniform plasmas. IV. Bessel beams, Gendrin beams, and helicons,” Phys. Plasmas 23, 052112 (2016).
[Crossref]

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82 (2002).
[Crossref]

Wang, R.

R. Wang, Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis (Cambridge University, 2012).
[Crossref]

Wei, M. S.

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Willingale, L.

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nature Photonics 4, 103 (2010).
[Crossref]

Zhang, S.

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6, 30765 (2016).
[Crossref] [PubMed]

Zhong, W.-P.

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. A 82033834 (2010).
[Crossref]

J. Appl. Phys. (1)

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “An optical tweezer in asymmetrical vortex Bessel-Gaussian beams,” J. Appl. Phys. 120, 023101 (2016).
[Crossref]

J. Opt. B: Quantum Semiclass. Opt. (1)

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82 (2002).
[Crossref]

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

J. Opt. Soc. Am. B (2)

Laser Photon. Rev. (1)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607 (2012).
[Crossref]

Nature (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348 (1998).
[Crossref]

Nature Photonics (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nature Photonics 4, 103 (2010).
[Crossref]

Opt. Commun. (1)

Y. I. Salamin and J.-X. Li, “Electromagnetic fields of an ultra-short tightly-focused radially-polarized laser pulse,” Opt. Commun. 405, 265 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Opt. Photon. News (1)

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24, 22 (2013).
[Crossref]

Phys. Plasmas (2)

Y. I. Salamin, “Fields of an ultrashort tightly focused radially polarized laser pulse in a linear response plasma,” Phys. Plasmas 24, 103107 (2017).
[Crossref]

J. M. Urrutia and R. L. Stenzel, “Helicon waves in uniform plasmas. IV. Bessel beams, Gendrin beams, and helicons,” Phys. Plasmas 23, 052112 (2016).
[Crossref]

Phys. Rev. A (4)

Y. I. Salamin, “Fields and propagation characteristics in vacuum of an ultrashort tightly focused radially polarized laser pulse,” Phys. Rev. A 92, 053836 (2015).
[Crossref]

Y. I. Salamin, “Simple analytical derivation of the fields of an ultrashort tightly focused linearly polarized laser pulse,” Phys. Rev. A 92, 063818 (2015).
[Crossref]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177 (1979).
[Crossref]

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. A 82033834 (2010).
[Crossref]

Phys. Rev. Lett. (5)

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated x-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[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]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma,” Phys. Rev. Lett. 96, 245002 (2006).
[Crossref] [PubMed]

L. Willingale, S. R. Nagel, A. G. R. Thomas, C. Bellei, R. J. Clarke, A. E. Dangor, R. Heathcote, M. C. Kaluza, C. Kamperidis, S. Kneip, K. Krushelnick, N. Lopes, S. P. D. Mangles, W. Nazarov, P. M. Nilson, and Z. Najmudin, “Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams,” Phys. Rev. Lett. 102, 125002 (2009).
[Crossref] [PubMed]

Sci. Rep. (1)

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6, 30765 (2016).
[Crossref] [PubMed]

Other (6)

Central Laser Facility: http://www.clf.stfc.ac.uk/CLF/12248.aspx

Extreme Light Infrastructure: http://www.eli-beams.eu/

R. Wang, Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis (Cambridge University, 2012).
[Crossref]

K. T. McDonald, “Bessel beams,” http://puhep1.princeton.edu/~kirkmcd/examples/bessel.pdf

K. T. McDonald, “Gaussian laser beams with radial polarization,” http://puhep1.princeton.edu/~kirkmcd/examples/axicon.pdf

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

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Figures (6)

Fig. 1
Fig. 1 Surface plots of the pulse’s scaled vector potential |A(0)/a0|2 as a function of the axial coordinate z and the transverse coordinate r, in units of λ0 = 1μm. The figure displays snapshots at t = 0, 1 fs, 1 ps, and 1 ns. Note that r has been allowed to take on negative values in order to bring out the cylindrical symmetry of the fields, with the understanding that, in reality, r ≥ 0. This is also done in Figs. 35 below.
Fig. 2
Fig. 2 Density plots (upper panel) and surface plots (lower panel) of the initial (t = 0) axial, radial and azimuthal intensity profiles in the focal plane (z = 0) of an ultrashort (L = 0.8λ0) and tightly focused (w0 = 0.9λ0) laser pulse of central wavelength λ0 = 1μm, propagating in vacuum (n0 = 0). Other parameters used are φ0 = 0 and kr = x1/w0, where x1 = 2.40483 is the first zero of J0(x).
Fig. 3
Fig. 3 Same as Fig. 1, but for | E z ( 0 ) / E 0 | 2, and φ0 = π/2. The pulse is generated at t = 0, with its centroid at the origin of coordinates (r = z = 0). The centroid subsequently advances to positions consistent with zct, at (b) t = 1 fs, (c) t = 1 ps, and (d) t = 1 ns.
Fig. 4
Fig. 4 Same as Fig. 3, but for | E r ( 0 ) / E 0 | 2.
Fig. 5
Fig. 5 Same as Fig. 4, but for propagation in an under-dense plasma of ambient electron density n0 = 1020 cm−3, and for different times.
Fig. 6
Fig. 6 Snapshots showing the on-axis (r = 0) scaled axial intensity |Ez/E0|2 of a pulse of the same parameters as in Fig. 2, as a function of the propagation distance. Black: in vacuum, and red: in a plasma of ambient electron density n0 = 1020 cm−3.

Equations (25)

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( 2 1 c 2 2 t 2 k p 2 ) A = 0 ,
( 1 r r r r + 2 2 η ζ k p 2 ) A = 0 .
A ( r , η , ζ ) = z ^ a 0 a ( r , η , ζ ) e i k 0 ζ ,
a ( r , η , ζ ) = 1 2 π a k ( r , η , k ) e i k ζ d k ,
( 1 r r r r + 2 i ( k + k 0 ) η k p 2 ) a k = 0 ,
r d 2 F d r 2 + d F d r + k r 2 r F = 0 , and d G d η + i 2 ( k r 2 + k p 2 k + k 0 ) G = 0 ,
a k ( r , η , k ) ~ f k J 0 ( k r r ) exp [ i 2 ( k r 2 + k p 2 k + k 0 ) η ] ,
f k = { 2 π Δ k | k | Δ k 2 ; 0 , elsewhere .
a ( r , η , ζ ) = J 0 ( k r r ) Δ k Δ k 2 Δ k 2 ϕ k e i k ζ d k ; ϕ k ( η ) = exp [ ( i k 0 α k + k 0 ) η ] ; α = k r 2 + k p 2 2 k 0 .
ϕ k = m = 0 k m m ! ϕ 0 ( m ) ; ϕ 0 ( m ) = m ϕ k k m | k = 0 ,
A ( n ) ( r , η , ζ ) = a 0 J 0 ( k r r ) e i k 0 ζ m = 0 n ϕ 0 ( m ) m ! i m d m f d ζ m ,
A ( n ) = a 0 J 0 ( k r r ) e i k 0 ζ 2 π / L m = 0 n i m + 1 ϕ 0 ( m ) m ! ζ m + 1 [ Γ ( m + 1 , i π ζ L ) Γ ( m + 1 , i π ζ L ) ] .
A ( 0 ) ( r , η , ζ ) = a 0 J 0 ( k r r ) j 0 ( π ζ L ) e i φ ( 0 ) ,
φ ( 0 ) = φ 0 + k 0 ζ α η , φ 0 = constant .
ω 0 = φ ( 0 ) t = c k 0 ( 1 + ) ; = α 2 k 0 ,
k z ( 0 ) = φ ( 0 ) k = k 0 ( 1 ) .
[ ω ( 0 ) c ] 2 [ k z ( 0 ) ] 2 = 4 k 0 2 .
v p ( 0 ) = ω ( 0 ) k z ( 0 ) = c [ 1 + 1 ] > c .
v g ( 0 ) = d ω ( 0 ) d k z ( 0 ) = c 2 v p ( 0 ) v p ( 0 ) v g ( 0 ) = c 2 .
E z ( 0 ) = ( E 0 k 0 ) e i φ ( 0 ) J 0 ( k r r ) { [ Q 2 + Q 3 R Q 1 Q 4 R 2 ] j 0 ( π ζ L ) + 1 ζ [ 1 2 Q 1 R Q 4 R 2 ] cos ( π ζ L ) } ,
E r ( 0 ) = E 0 ( k r k 0 ) e i φ ( 0 ) J 1 ( k r r ) R [ Q 1 j 0 ( π ζ L ) + cos ( π ζ / L ) ζ ] ,
c B θ ( 0 ) = E 0 ( k r k 0 ) e i φ ( 0 ) J 1 ( k r r ) j 0 ( π ζ L ) .
Q 1 = i k 0 1 ζ i α 2 ; Q 2 = i k 0 1 ζ + i α 2 ,
R = Q 2 + π L cot ( π ζ L ) ,
Q 3 = 2 Q 1 ζ + k 0 2 + π 2 L 2 + α ( α 4 k 0 ) 4 ; Q 4 = 1 ζ 2 + π 2 L 2 csc 2 ( π ζ L ) .

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