Abstract

Airy-Bessel configuration wave packets are believed to be exotic localized linear light bullets (LLBs) without spatiotemporal spread during propagation in free space. By carefully studying the propagation of ideal Airy-Bessel wave packets (ABWs) in free space, several new results were obtained. Cubic spatially induced dispersion (SID) slightly broadens Airy pulses while quadratic SID cannot temporally change ABWs transmission modes. Hence, ABWs, although remaining as superior localized linear wave packets, cannot be regarded as absolute LLBs. Moreover, cubic SID also decreases the longitudinal acceleration of the Airy pulse peak during propagation.

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2012

D. Mihalachem, “Linear and nonlinear light bullets: recent theoretical and experimental studies,” Rom. J. Physiol.57, 352–371 (2012).

Z. J. Ren, H. Z. Jin, Y. L. Shi, J. C. Xu, W. D. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading Airy-Bessel wave packets,” J. Opt. Soc. Am. A29(6), 848–853 (2012).
[CrossRef] [PubMed]

2011

2010

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Opt. Express18, 566–574 (2010).
[CrossRef] [PubMed]

F. O. Fahrbach and A. Rohrbach, “A line scanned light-sheet microscope with phase shaped self-reconstructing beams,” Opt. Express18(23), 24229–24244 (2010).
[CrossRef] [PubMed]

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt.12(9), 093001 (2010).
[CrossRef]

F. W. Wise, “Generation of light bullets,” Physics3, 107 (2010).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (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(25), 253901 (2010).
[CrossRef] [PubMed]

2009

V. Pasiskevicius, “Engineering Airy beams,” Nat. Photonics3(7), 374–375 (2009).
[CrossRef]

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]

H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express17(17), 14948–14955 (2009).
[CrossRef] [PubMed]

2008

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A77(4), 043814 (2008).
[CrossRef]

I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(4), 046605 (2008).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express16(20), 15733–15740 (2008).
[CrossRef] [PubMed]

2007

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]

2005

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt.7(5), R53–R72 (2005).
[CrossRef]

2004

S. Longhi, D. Janner, and P. Laporta, “Propagating pulsed Bessel beams in periodic media,” J. Opt. B: Quantum Semiclass.6(11), 477–481 (2004).
[CrossRef]

M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A21(12), 2455–2463 (2004).
[CrossRef] [PubMed]

2003

B. D. Lü and Z. J. Liu, “Propagation properties of ultrashort pulsed Bessel beams in dispersive media,” J. Opt. Soc. Am. A20(3), 582–587 (2003).
[CrossRef] [PubMed]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
[CrossRef]

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(9), 093904 (2003).
[CrossRef] [PubMed]

2002

2001

2000

1999

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun.167(1-6), 15–22 (1999).
[CrossRef]

1998

A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B15(3), 951–956 (1998).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics58(1), 1086–1093 (1998).
[CrossRef]

1997

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys.7, 32–39 (1997).

1992

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control39(1), 19–31 (1992).
[CrossRef] [PubMed]

1987

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

1979

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 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(25), 253901 (2010).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun.167(1-6), 15–22 (1999).
[CrossRef]

Akturk, S.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt.12(9), 093001 (2010).
[CrossRef]

Ament, C.

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum Generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett.107(24), 243901 (2011).
[CrossRef] [PubMed]

Balazs, N. L.

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

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

Berry, M. V.

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

Besieris, I. M.

I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(4), 046605 (2008).
[CrossRef] [PubMed]

Bhuyan, M. K.

Bowlan, P.

Broky, J.

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]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
[CrossRef]

Christodoulides, D. N.

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19(23), 23132–23139 (2011).
[CrossRef] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
[CrossRef]

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]

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]

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(9), 093904 (2003).
[CrossRef] [PubMed]

Couairon, A.

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A77(4), 043814 (2008).
[CrossRef]

Courvoisier, F.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

Di Trapani, P.

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express16(20), 15733–15740 (2008).
[CrossRef] [PubMed]

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(9), 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(21), 213901 (2007).
[CrossRef] [PubMed]

Dudley, J. M.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Faccio, D.

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A77(4), 043814 (2008).
[CrossRef]

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express16(20), 15733–15740 (2008).
[CrossRef] [PubMed]

Fahrbach, F. O.

Fortin, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
[CrossRef]

Franco, M.

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A77(4), 043814 (2008).
[CrossRef]

Furfaro, L.

Greenleaf, J. F.

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control39(1), 19–31 (1992).
[CrossRef] [PubMed]

Griebner, U.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
[CrossRef]

Grunwald, R.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
[CrossRef]

Gu, X.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt.12(9), 093001 (2010).
[CrossRef]

Guo, H.

Hernández-Figueroa, H. E.

Hu, W.

Hwang, C. Y.

Jacquot, M.

Janner, D.

S. Longhi, D. Janner, and P. Laporta, “Propagating pulsed Bessel beams in periodic media,” J. Opt. B: Quantum Semiclass.6(11), 477–481 (2004).
[CrossRef]

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(9), 093904 (2003).
[CrossRef] [PubMed]

Jin, H. Z.

Kaminer, I.

Kaplan, A. E.

Kebbel, V.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
[CrossRef]

Kim, K. Y.

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]

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express16(20), 15733–15740 (2008).
[CrossRef] [PubMed]

Kummrow, A.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
[CrossRef]

Lacourt, P. A.

Laporta, P.

S. Longhi, D. Janner, and P. Laporta, “Propagating pulsed Bessel beams in periodic media,” J. Opt. B: Quantum Semiclass.6(11), 477–481 (2004).
[CrossRef]

Lee, B.

Liu, Z. J.

Lõhmus, M.

Longhi, S.

S. Longhi, D. Janner, and P. Laporta, “Propagating pulsed Bessel beams in periodic media,” J. Opt. B: Quantum Semiclass.6(11), 477–481 (2004).
[CrossRef]

Lu, J. Y.

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control39(1), 19–31 (1992).
[CrossRef] [PubMed]

Lü, B. D.

Lumer, Y.

Malomed, B. A.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt.7(5), R53–R72 (2005).
[CrossRef]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Mihalache, D.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt.7(5), R53–R72 (2005).
[CrossRef]

Mihalachem, D.

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R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
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R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
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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(9), 093904 (2003).
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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(9), 093904 (2003).
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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).
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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(9), 093904 (2003).
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A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
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R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A67(6), 063820 (2003).
[CrossRef]

Phys. Rev. E

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E65(2), 026606 (2002).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(4), 046605 (2008).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics58(1), 1086–1093 (1998).
[CrossRef]

Phys. Rev. Lett.

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[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]

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(9), 093904 (2003).
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Physics

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Rom. J. Physiol.

D. Mihalachem, “Linear and nonlinear light bullets: recent theoretical and experimental studies,” Rom. J. Physiol.57, 352–371 (2012).

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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 (3)

Fig. 1
Fig. 1

The trajectory of the Airy pulse peak for different cases.

Fig. 2
Fig. 2

Propagating in free space, temporal broadening factor of Gaussian-Bessel wave packet and ideal Airy-Bessel wave packet under the jointly influence both quadratic and cubic SID.

Fig. 3
Fig. 3

The temporal evolution of intensity distribution of an ideal ABW at different propagation distances.

Equations (34)

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[ 2 +k (ω) 2 ]U(x,y,z;ω)=0,
k(ω)=ω/c,
U(r,z;ω)= J 0 (r/ r 0 )exp[iβ(ω)z]f(0,ω ω 0 ),
β 2 (ω)= k 2 (ω)1/ r 0 2 = ω 2 / c 2 1/ r 0 2 ,
β(ω)= β 0 + m=1 β m m! (ω ω 0 ) m ,
β 0 = k 0 1 1 r 0 2 k 0 2 ,
β 1 = r 0 k 0 c r 0 2 k 0 2 1 ,
β 2 = r 0 c 2 ( r 0 2 k 0 2 1) 3 2 ,
β 3 = 3 r 0 3 k 0 c 3 ( r 0 2 k 0 2 1) 5 2 ,
U(r,z;ω)= J 0 (r/ r 0 )f(0,ω ω 0 )exp{iz[ β 0 + n=1 β n n! (ω ω 0 ) n ]},
Ψ(r,z;t)= 1 2π J 0 (r/ r 0 )× f(0,ω ω 0 )exp{iz[ β 0 + n=1 β n n! (ω ω 0 ) n ]}exp(iωt)dω .
Ψ(t;r,z)=E(r,z;t)exp[i( β 0 z ω 0 t)].
E(r,z;t)= J 0 (r/ r 0 )Φ(z,t).
Φ(z,t)= 1 2π f(0,ω ω 0 )exp[iz n=1 β n n! (ω ω 0 ) n i(ω ω 0 )t]dω .
Φ(z,T)= 1 2π × f(0,ω ω 0 )exp[ i 2 β 2 (ω ω 0 ) 2 z+ i 6 β 3 (ω ω 0 ) 3 zi(ω ω 0 )T]dω .
Φ(0,T)=exp( T 2 2 T 0 2 ),
η= T T 0 = [1+ ( z L 2 ) 2 + 1 4 ( z L 3 ) 2 ] 1/2 .
Φ(0,T)=Airy(T/ T 0 ),
f(0,ω ω 0 )= Ai(T/ T 0 ) e i(ω ω 0 )T dT= 1 2π T 0 e i T 0 3 (ω ω 0 ) 3 /3 .
Φ(z,T)= 1 2π exp{i[ (ω ω 0 ) 3 3 ( T 0 3 + β 3 z 2 )+ β 2 z 2 (ω ω 0 ) 2 T(ω ω 0 )]} dω,
L 2 = T 0 2 | β 2 | = c 2 T 0 2 r 0 ( r 0 2 k 0 2 1) 3 2 .
L 3 = T 0 3 | β 3 | = c 3 T 0 3 3 r 0 3 k 0 ( r 0 2 k 0 2 1) 5 2 .
ω ω 0 = ( T 0 3 + β 3 z 2 ) 1/3 φ= 1 T 0 (1+ z 2 L 3 ) 1/3 φ
Φ(z,T)= (1+z/2 L 3 ) 1/3 2π exp{i[ φ 3 3 + z 2 L 2 (1+ z 2 L 3 ) 2 3 φ 2 + (1+ z 2 L 3 ) 1 3 T T 0 φ]} dφ = (1+ z 2 L 3 ) 1/3 exp{i z 2 L 2 (1+ z 2 L 3 ) 2 3 [ z 2 6 L 2 2 (1+ z 2 L 3 ) 4 3 (1+ z 2 L 3 ) 1 3 T T 0 ]} ×Ai[ (1+ z 2 L 3 ) 1 3 T T 0 z 2 4 L 2 2 (1+ z 2 L 3 ) 4 3 ].
Φ(z,T)=exp[i z 2 L 2 ( z 2 6 L 2 2 T T 0 )]Ai( T T 0 z 2 4 L 2 2 ),
Φ(z,T)= (1+ z 2 L 3 ) 1/3 Ai[ (1+ z 2 L 3 ) 1 3 T T 0 ].
η= T T 0 = (1+z/2 L 3 ) 1/3 = [1+ 3z r 0 3 k 0 2 c 3 T 0 3 ( r 0 2 k 0 2 1) 5 2 ] 1/3 .
L 3 c 3 r 0 2 T 0 3 k 0 4 /3.
η [1+ 3z 2 c 3 r 0 2 T 0 3 k 0 4 ] 1 3 .
z r 0 2 T 0 3 2 3 ( η 3 1) c 3 k 0 4
z s r 0 2 T 0 3 0.1051 c 3 k 0 4 .
a= z 2 4 L 2 2 (1+ z 2 L 3 ) 4 3 .
L 3 = c T 0 ( r 0 2 k 0 2 1) 3 r 0 2 k 0 L 2 .
L 3 = c T 0 k 0 3 L 2 .

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