Abstract

The time-diffraction technique introduced by Porras recently is motivated in this article in terms of the Lorentz invariance of the equation governing the narrow angular spectrum and narrowband temporal spectrum paraxial approximation and it is used to derive finite-energy spatiotemporally confined subluminal, luminal and superluminal Airy wave packets. In addition, a novel exact finite-energy luminal Airy splash mode-type solution to the scalar wave equation is derived using Bateman’s conformal transformation.

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

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References

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  1. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
    [Crossref]
  2. A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).
  3. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
    [Crossref] [PubMed]
  4. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
    [Crossref]
  5. 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. Control 39(1), 19–31 (1992).
    [Crossref] [PubMed]
  6. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10(1), 75–87 (1993).
    [Crossref]
  7. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
    [Crossref]
  8. I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
    [Crossref]
  9. R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
    [Crossref]
  10. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
    [Crossref]
  11. S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066612 (2003).
    [Crossref]
  12. C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
    [Crossref]
  13. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Localized Waves (Wiley-Interscience, Hoboken, NJ, 2008).
  14. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Non-Diffracting Waves (Wiley-VCH, 2014).
  15. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994).
    [Crossref]
  16. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
    [Crossref]
  17. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16(9), 1468–1474 (1999).
    [Crossref]
  18. I. Besieris and A. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12(16), 3848–3864 (2004).
    [Crossref] [PubMed]
  19. M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017).
    [Crossref] [PubMed]
  20. M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A (Coll. Park) 97(6), 063803 (2018).
    [Crossref]
  21. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  22. 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]
  23. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  24. P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
    [Crossref]
  25. P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
    [Crossref]
  26. Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28(6), 1243–1255 (2011).
    [Crossref] [PubMed]
  27. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
    [Crossref]
  28. H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
    [Crossref] [PubMed]
  29. H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
    [Crossref] [PubMed]
  30. S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004).
    [Crossref] [PubMed]
  31. A. Wűnsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A 6(11), 1661–1668 (1989).
    [Crossref]
  32. P. A. Belanger, “Lorentz transformations of packet-like solutions of the homogeneous wave equation,” J. Opt. Soc. Am. A 3(4), 541–542 (1986).
    [Crossref]
  33. H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
    [Crossref]
  34. H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
    [Crossref]
  35. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
    [Crossref]
  36. N. K. Efremidis, “Spatiotemporal diffraction-free pulsed beams in free-space of the Airy and Bessel type,” Opt. Lett. 42(23), 5038–5041 (2017).
    [Crossref] [PubMed]
  37. A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
    [Crossref]
  38. D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
    [Crossref]
  39. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
    [Crossref]
  40. X. Peng, Y. Peng, D. Li, L. Zhang, J. Zhuang, F. Zhao, X. Chen, X. Yang, and D. Deng, “Propagation properties of spatiotemporal chirped Airy Gaussian vortex wave packets in a quadratic index medium,” Opt. Express 25(12), 13527–13538 (2017).
    [Crossref] [PubMed]
  41. J. Zhuang, D. Deng, X. Chen, F. Zhao, X. Peng, D. Li, and L. Zhang, “Spatiotemporal sharply autofocused dual-Airy-ring Airy Gaussian vortex wave packets,” Opt. Lett. 43(2), 222–225 (2018).
    [Crossref] [PubMed]
  42. S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
    [Crossref]

2019 (1)

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

2018 (4)

2017 (4)

2015 (1)

D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
[Crossref]

2014 (1)

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

2011 (1)

2010 (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

2009 (1)

P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
[Crossref]

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

2004 (3)

S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004).
[Crossref] [PubMed]

I. Besieris and A. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12(16), 3848–3864 (2004).
[Crossref] [PubMed]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

2003 (3)

S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066612 (2003).
[Crossref]

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

1999 (1)

1998 (2)

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
[Crossref]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

1994 (2)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[Crossref]

1993 (1)

1992 (1)

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. Control 39(1), 19–31 (1992).
[Crossref] [PubMed]

1990 (1)

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

1989 (3)

A. Wűnsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A 6(11), 1661–1668 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

1986 (1)

1983 (2)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).

1979 (1)

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

1910 (2)

H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
[Crossref]

H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
[Crossref]

Abdel-Rahman, M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

Abouraddy, A. F.

H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref] [PubMed]

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
[Crossref] [PubMed]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

Balazs, N. L.

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

Bateman, H.

H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
[Crossref]

H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
[Crossref]

Belanger, P. A.

Berry, M. V.

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

Besieris, I.

Besieris, I. M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[Crossref]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10(1), 75–87 (1993).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

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]

Chatzipetros, A.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

Chen, S.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Chen, X.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (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,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

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]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Conti, C.

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

Deng, D.

di Trapani, P.

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

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]

Efremidis, N. K.

Fairchild, S. R.

Fortin, M.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

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. Control 39(1), 19–31 (1992).
[Crossref] [PubMed]

Grunwald, R.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Heyman, E.

Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28(6), 1243–1255 (2011).
[Crossref] [PubMed]

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994).
[Crossref]

Jedrkiewicz, O.

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

Kaganovsky, Y.

Karlovets, D. V.

D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
[Crossref]

Kebbel, V.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Kiselev, A. P.

A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).

Kondakci, H. E.

H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref] [PubMed]

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
[Crossref] [PubMed]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

Kummrow, A.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Li, D.

Lin, G.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Longhi, S.

S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004).
[Crossref] [PubMed]

S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066612 (2003).
[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. Control 39(1), 19–31 (1992).
[Crossref] [PubMed]

Ma, S.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Meem, M.

Menon, R.

Neumann, U.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Nibbering, E. T.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Peng, X.

Peng, Y.

Piche, M.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Piksarv, P.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

Piskarskas, A.

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

Porras, M. A.

M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A (Coll. Park) 97(6), 063803 (2018).
[Crossref]

M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017).
[Crossref] [PubMed]

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16(9), 1468–1474 (1999).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
[Crossref]

Reivelt, K.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Reyes, D.

Richardson, M.

Rini, M.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Rousseau, G.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Saari, P.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Shaarawi, A.

Shaarawi, A. M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[Crossref]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10(1), 75–87 (1993).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

Siviloglou, G. 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]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Thul, D.

Trillo, S.

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

Trull, J.

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

Valdman, A.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

Valiulis, G.

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

Valtma-Lukner, H.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Wunsche, A.

Xie, J.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Yang, X.

Yessenov, M.

Zhan, Y.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Zhang, L.

Zhao, F.

Zhuang, J.

Ziolkowski, R. W.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[Crossref]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10(1), 75–87 (1993).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

Am. J. Phys. (2)

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

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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. Control 39(1), 19–31 (1992).
[Crossref] [PubMed]

J. Appl. Phys. (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

J. Math. Phys. (2)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

J. Opt. Soc. Am. A (4)

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

J. Phys. Conf. Ser. (1)

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

Laser Phys. (1)

P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
[Crossref]

Nat. Photonics (2)

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Opt. Commun. (1)

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. A (2)

D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
[Crossref]

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820 (2003).
[Crossref]

Phys. Rev. A (Coll. Park) (1)

M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A (Coll. Park) 97(6), 063803 (2018).
[Crossref]

Phys. Rev. A Gen. Phys. (1)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066612 (2003).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
[Crossref]

Phys. Rev. Lett. (4)

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]

H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Proc. Lond. Math. Soc. (2)

H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
[Crossref]

H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
[Crossref]

Prog. Electromagnetics Res. (1)

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

Radio Phys. Quant. Electron. (1)

A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).

Other (2)

H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Localized Waves (Wiley-Interscience, Hoboken, NJ, 2008).

H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Non-Diffracting Waves (Wiley-VCH, 2014).

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Figures (3)

Fig. 1
Fig. 1 A density plot of | ψ + (sub) ( x, τ + , σ + ) | versus ς sub =[ v( zvt )/( cv ) ] and x for vtequal to 0, 20 and 30. f( τ + )=exp( τ + 2 /10 ) and the remaining normalized parameters are given as follows: ω 0 =1, c=1 and v=0.9.
Fig. 2
Fig. 2 A density plot of | ψ + (sup) ( x, ς + , σ ¯ + ) | versus ς sup =[ v( zvt )/( vc ) ] and x for vt equal to 0, 1 and 3. f( ς + )=exp( ς + 2 /10 ) and the remaining normalized parameters are given as follows: k 0 = ω 0 /c=1, c=1 and v=2.
Fig. 3
Fig. 3 Surface plots of the modulus of Ψ 2 ( X, Λ + , Λ ; a 1 , a 2 )versus Λ + and X for various values of T, the latter defined by the relationship Λ = Λ + +2T. The parameters a 1 and a 2 have the values 8× 10 2 and 100, respectively.

Equations (37)

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( 2 x 2 + 2 z 2 + ω 2 c 2 ) u ^ ( x,z,ω )=0,
i z v ^ ± ( x,,z,ω )=± c 2ω 2 x 2 v ^ ± ( x,z,ω ),
u ± ( x,z,t )= R 1 dω R 1 d k x exp[ i( ω τ ± k x x ) ]exp[ ±i( c k x 2 z )/( 2ω ) ψ ˜ 0 ( k x ,ω ) ],
( 2 x 2 2 c 2 τ ± z ) u ± ( x,z,t )=0.
ψ ± ( x,z,t )=exp( i ω 0 τ ± ) R 1 dΩ R 1 d k x exp( iΩ τ ± )exp( i k x x ) ×exp[ ±i( c k x 2 z )/( 2 ω 0 ) ] ψ ˜ 1 ( k x ,Ω );Ωω ω 0 ,
i( z ± 1 c t ) ϕ ± ( x,z,t )=± 1 2 k 0 2 x 2 ϕ ± ( x,z,t ); k 0 ω 0 /c.
ψ + sub ( x, τ + , σ + )=exp( i ω 0 τ + )f( τ + )Φ( x, σ + ); τ + =tz/c, σ + = 2( zvt ) k 0 ( 1v/c ) , ψ sub ( x, τ , σ )=exp( i ω 0 τ )f( τ )Φ( x, σ ); τ =t+z/c, σ = 2( zvt ) k 0 ( 1+v/c ) ;
ψ + sup ( x, ς + , σ ¯ )=exp( i k 0 ς + )f( ς + )Φ( x, σ ¯ ); ς + =zct, σ ¯ = 2( zvt ) k 0 ( 1v/c ) , ψ sup ( x, ς , σ ¯ + )=exp( i k 0 ς )f( ς )Φ( x, σ ¯ + ); ς =z+ct, σ ¯ + = 2( zvt ) k 0 ( 1+v/c ) .
i4 σ Φ( x,σ )+ 2 x 2 Φ( x,σ )=0.
ψ + ( x, ς + , ς )=exp( i k 0 ς + )f( ς + )Φ( x, ς ); ς + =zct, ς =( z+ct )/ k 0 , ψ ( x, ς , ς + )=exp( i k 0 ς )f( ς )Φ( x, ς + ); ς =z+ct, ς + =( zct )/ k 0 .
z ± =(1+μ/2)z(μ/2)ct, t ± =±(μ/2c)z(1+μ/2)t,
ξ ± =±( z ±c t )/2=( μ+1 )( z v eff ± t )/2; v eff ± ±c( μ1 )/( μ+1 ), τ ± = t z /c=tz/c,
i ξ ± ϕ W ± ( x, ξ ± , τ ± )=± 1 2 k 0 2 x 2 ϕ W ± ( x, ξ ± , τ ± ); k 0 ω 0 /c.
ϕ W ± ( x, ξ ± , τ ± )= ϕ ± ( x, ξ ± )f( τ ± ).
i Z Ψ( X,Z )+ 1 2 2 X 2 Ψ( X,Z )=0,
Ψ( X,Z )= e 1 12 ( 2a+iZ )( 2 a 2 6X4iaZ+ Z 2 ) Ai( X+iaZ Z 2 4 ),
ψ + (sub) ( x, τ + , σ + )=exp( i ω 0 τ + )f( τ + ) × e 1 12 ( 2a+i σ + 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ + 2 x 0 2 + 1 4 ( σ + x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ + 2 x 0 2 1 16 ( σ + x 0 2 ) 2 ) ],
ψ + (sup) ( x, ς + , σ ¯ )=exp( i k 0 ς + )f( ς + ) × e 1 12 ( 2a+i σ ¯ 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ ¯ 2 x 0 2 + 1 4 ( σ ¯ x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ ¯ 2 x 0 2 1 16 ( σ ¯ x 0 2 ) 2 ) ],
ψ (sub) ( x, τ , σ )=exp( i ω 0 τ )f( τ ) × e 1 12 ( 2a+i σ 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ 2 x 0 2 + 1 4 ( σ x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ 2 x 0 2 1 16 ( σ x 0 2 ) 2 ) ],
ψ (sup) ( x, ς , σ ¯ + )=exp( i k 0 ς )f( ς ) × e 1 12 ( 2a+i σ ¯ + 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ ¯ + 2 x 0 2 + 1 4 ( σ ¯ + x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ ¯ + 2 x 0 2 1 16 ( σ ¯ + x 0 2 ) 2 ) ].
ψ W ± ( x, ξ ± , τ ± )=exp( i ω 0 τ ± )f( τ ± )Φ( x/ x 0 ,± ξ ± /( 2 k 0 x 0 2 ) ).
( 2 x 2 + 2 z 2 1 c 2 2 t 2 )ψ( x,z,t )=0.
ψ + ( x,z,t )=exp[ i k 0 ς ] × e 1 12 ( 2a+i ς + 2 k 0 x 0 2 )[ 2 a 2 6 x x 0 4ia ς + 2 k 0 x 0 2 + 1 4 ( ς + k 0 x 0 2 ) 2 ] Ai[ ( x x 0 +ia ς + 2 k 0 x 0 2 1 16 ( ς + k 0 x 0 2 ) 2 ) ]; ς + =zct, ς =z+ct,
ψ ( x,z,t )=exp[ i k 0 ς + ] × e 1 12 ( 2a+i ς 2 k 0 x 0 2 )[ 2 a 2 6 x x 0 4ia ς 2 k 0 x 0 2 + 1 4 ( ς k 0 x 0 2 ) 2 ] Ai[ ( x x 0 +ia ς 2 k 0 x 0 2 1 16 ( ς k 0 x 0 2 ) 2 ) ]
Ψ( X, Λ + , Λ )=exp[ i Λ ] e i 1 24 Λ + ( 6X+ 1 4 Λ + 2 ) Ai( X 1 16 Λ + 2 ),
( 2 X 2 +4 2 Λ + Λ )Ψ( X, Λ + , Λ )=0.
Ψ 1 ( X, Λ + , Λ )= 1 Λ Ψ[ X Λ , A Λ , X 2 + Λ + Λ A Λ ],
Ψ 2 ( X, Λ + , Λ )= 1 X 2 + Λ + Λ Ai[ Λ + 2 X( X 2 + Λ + Λ ) 16 ( X 2 + Λ + Λ ) 2 ] ×exp[ i Λ + 3 +24AX Λ + ( X 2 + Λ + Λ )+96 A 2 Λ ( X 2 + Λ + Λ ) 2 96 ( X 2 + Λ + Λ ) 3 ].
Ψ 2 ( X, Λ + , Λ ; a 1 , a 2 )= exp[ i Q 96 [ X 2 +( a 1 +i Λ + )( a 2 i Λ ) ] 3 ] X 2 +( a 1 +i Λ + )( a 2 i Λ ) Ai[ ( a 1 +i Λ + ) 2 +16A X 3 +16AX( a 1 +i Λ + )( a 2 i Λ ) 16 [ X 2 +( a 1 +i Λ + )( a 2 i Λ ) ] 2 ],
Q=i ( a 1 +i Λ + ) 3 i24AX( a 1 +i Λ + )[ X 2 +( a 2 i Λ )( a 2 i Λ ) ] +i96 A 2 ( a 2 i Λ ) [ X 2 +( a 2 i Λ )( a 2 i Λ ) ] 2 .
( 2 X 2 + 2 Λ + Z ) U 2 + ( X, Λ + ,Z )=0.
ϕ + sub ( x,z,t )= R 1 dΩ R 1 d k x exp[ iΩ γ ¯ ( 1+v/c )( tz/c ) ]exp( i k x x ) ×exp[ i k x 2 2 k 0 a sub ( t z v ) ] ψ ˜ 1 ( k x ,Ω ); a sub ( 1 c 1 v ),v<c;
ϕ + sup ( x,z,t )= R 1 dΩ R 1 d k x exp[ iΩγ( 1+v/c )( tz/c ) ]exp( i k x x ) ×exp[ i k x 2 2 k 0 a sup ( t v c 2 z ) ] ψ ˜ 1 ( k x ,Ω ); a sup ( 1 c v c 2 ),v>c;
Ψ( X, Λ + , Λ )= e ±i Λ Φ( Λ ± ),
4i Λ ± Φ( Λ ± )+ 2 X 2 Φ( Λ ± )=0,
i β 0 z u( x,y,z,τ )+ 1 2 2 u( x,y,z,τ )+ 1 2 β 0 β 2 2 τ 2 u( x,y,z,τ ) + 1 2 α( x 2 + y 2 )u( x,y,z,τ )=0,
i β 0 z U( z,τ )+ 1 2 β 0 β 2 2 τ 2 U( z,τ )=0, i β 0 z ψ( x,y,z )+ 1 2 2 ψ( x,y,z )+( x 2 + y 2 )ψ( x,y,z )=0.

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