Abstract

It is shown that the homogeneous scalar wave equation under a generalized paraxial approximation admits of Gaussian beam solutions that can propagate with an arbitrary speed, either subluminal or superluminal, in free-space. In suitable moving inertial reference frames, such solutions correspond either to standard stationary Gaussian beams or to “temporal” diffracting Gaussian fields.

© 2004 Optical Society of America

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References

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  1. J.N. Brittingham, ???Focus waves modes in homogeneous Maxwell???s equations: transverse electric mode,??? J. Appl. Phys. 54, 1179???1189 (1983).
    [CrossRef]
  2. R.W. Ziolkowski, ???Exact solutions of the wave equation with complex source locations,??? J. Math. Phys. 26, 861???863 (1985).
    [CrossRef]
  3. P.A. Belanger, ???Packetlike solutions of the homogeneous-wave equation,??? J. Opt. Soc. Am. A 1, 723???724 (1984).
    [CrossRef]
  4. P.A. Belanger, ???Lorentz transformation of packetlike solutions of the homogeneous-wave equation,??? J . Opt. Soc. Am. A 3, 541???542 (1986).
    [CrossRef]
  5. R.W. Ziolkowski, ???Localized transmission of electromagnetic energy,??? Phys. Rev. A 39, 2005???2033 (1989).
    [CrossRef] [PubMed]
  6. R.W. Ziolkowski, I.M. Besieris, and A.M. Shaarawi, ???Localized wave representations of acoustic and electromagnetic radiation,??? Proc. IEEE 79 (10), 1371???1378 (1991).
    [CrossRef]
  7. P.L. Overfelt, ???Bessel-Gauss pulses,??? Phys. Rev. A 44, 3941-3947 (1991).
    [CrossRef] [PubMed]
  8. R. Donnelly and R. Ziolkowski, ???A method for constructing solutions of homogeneous partial differential equations: localized waves,??? Proc. R. Soc. Lond. A 437, 673???692 (1992).
    [CrossRef]
  9. E. Recami, ???On localized ???X-shaped??? superluminal solutions to Maxwell equations,??? Physica A 252, 586???610 (1998).
    [CrossRef]
  10. I.M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, ???Two fundamental representations of localized pulse solutions to the scalar wave equation,??? Progr. Electromagn. Res. (PIER) 19, 1???48 (1998).
    [CrossRef]
  11. S. Feng, H.G. Winful, and R.W. Hellwarth, ???Spatiotemporal evolution of focused single-cycle electromagnetic pulses,??? Phys. Rev. E 59, 4630???4649 (1999).
    [CrossRef]
  12. J. Salo, J. Fagerholm, A.T. Friberg, and M.M. Salomaa, ???Unified description of nondiffracting X and Y waves,??? Phys. Rev. E 62, 4261???4275 (2000).
    [CrossRef]
  13. M. Zamboni-Rached, E. Recami, and H.E. Hernandez-Figueroa, ???New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,??? Eur. Phys. J. 21, 217???228 (2002).
  14. 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, 19???31 (1992).
    [CrossRef] [PubMed]
  15. P. Saari and M. Ratsep, ???Evidence of X-Shaped Propagation-Invariant Localized Light Waves,??? Phys. Rev. Lett. 79, 4135???4138 (1997).
    [CrossRef]
  16. M.A. Porras, ???Ultrashort pulsed Gaussian light beams,??? Phys. Rev. E 58, 1086???1093 (1998).
    [CrossRef]
  17. M.A. Porras, ???Nonsinusoidal few-cycle pulsed light beams in free space,??? J. Opt. Soc. Am. B 16, 1468???1474 (1999).
    [CrossRef]
  18. S. Longhi, ???Spatial-temporal Gauss-Laguerre waves in dispersive media,??? Phys. Rev. E 68, 066612 1???6 (2003).
    [CrossRef]
  19. A.E. Siegman, Lasers (University Science Books, Sausalito, 1986).
  20. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965), Eq. 6.643.

Eur. Phys. J.

M. Zamboni-Rached, E. Recami, and H.E. Hernandez-Figueroa, ???New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,??? Eur. Phys. J. 21, 217???228 (2002).

IEEE Trans. Ultrason. Ferroelectr. Freq.

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, 19???31 (1992).
[CrossRef] [PubMed]

J . Opt. Soc. Am. A

P.A. Belanger, ???Lorentz transformation of packetlike solutions of the homogeneous-wave equation,??? J . Opt. Soc. Am. A 3, 541???542 (1986).
[CrossRef]

J. Appl. Phys.

J.N. Brittingham, ???Focus waves modes in homogeneous Maxwell???s equations: transverse electric mode,??? J. Appl. Phys. 54, 1179???1189 (1983).
[CrossRef]

J. Math. Phys.

R.W. Ziolkowski, ???Exact solutions of the wave equation with complex source locations,??? J. Math. Phys. 26, 861???863 (1985).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Phys. Rev. A

R.W. Ziolkowski, ???Localized transmission of electromagnetic energy,??? Phys. Rev. A 39, 2005???2033 (1989).
[CrossRef] [PubMed]

P.L. Overfelt, ???Bessel-Gauss pulses,??? Phys. Rev. A 44, 3941-3947 (1991).
[CrossRef] [PubMed]

Phys. Rev. E

S. Feng, H.G. Winful, and R.W. Hellwarth, ???Spatiotemporal evolution of focused single-cycle electromagnetic pulses,??? Phys. Rev. E 59, 4630???4649 (1999).
[CrossRef]

J. Salo, J. Fagerholm, A.T. Friberg, and M.M. Salomaa, ???Unified description of nondiffracting X and Y waves,??? Phys. Rev. E 62, 4261???4275 (2000).
[CrossRef]

M.A. Porras, ???Ultrashort pulsed Gaussian light beams,??? Phys. Rev. E 58, 1086???1093 (1998).
[CrossRef]

S. Longhi, ???Spatial-temporal Gauss-Laguerre waves in dispersive media,??? Phys. Rev. E 68, 066612 1???6 (2003).
[CrossRef]

Phys. Rev. Lett.

P. Saari and M. Ratsep, ???Evidence of X-Shaped Propagation-Invariant Localized Light Waves,??? Phys. Rev. Lett. 79, 4135???4138 (1997).
[CrossRef]

Physica A

E. Recami, ???On localized ???X-shaped??? superluminal solutions to Maxwell equations,??? Physica A 252, 586???610 (1998).
[CrossRef]

Proc. IEEE

R.W. Ziolkowski, I.M. Besieris, and A.M. Shaarawi, ???Localized wave representations of acoustic and electromagnetic radiation,??? Proc. IEEE 79 (10), 1371???1378 (1991).
[CrossRef]

Proc. R. Soc. Lond. A

R. Donnelly and R. Ziolkowski, ???A method for constructing solutions of homogeneous partial differential equations: localized waves,??? Proc. R. Soc. Lond. A 437, 673???692 (1992).
[CrossRef]

Progr. Electromagn. Res. (PIER)

I.M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, ???Two fundamental representations of localized pulse solutions to the scalar wave equation,??? Progr. Electromagn. Res. (PIER) 19, 1???48 (1998).
[CrossRef]

Other

A.E. Siegman, Lasers (University Science Books, Sausalito, 1986).

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965), Eq. 6.643.

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Equations (19)

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t 2 ψ + 2 ψ z 2 1 c 2 2 ψ t 2 = 0 ,
ψ ( x , y , z , t ) = Φ ( x , y , z , v t ) ext [ i ω ( t z c ) ] .
t 2 Φ + [ 1 ( v c ) 2 ] 2 Φ ξ 2 2 ik ( 1 v c ) Φ ξ = 0 ,
Φ ( ρ , ξ ) = d Q F ( Q ) J 0 ( k ( Q ) ρ ) exp ( i Q ξ ) ,
k ( Q ) = Q ( 1 v c ) [ 2 k Q ( 1 + v c ) ] ,
Φ ( ρ , ξ ) = d Q F ( Q ) J 0 ( 2 k Q ( 1 v c ) ρ ) exp ( i Q ξ ) .
t 2 Φ 2 ik ( 1 v c ) Φ ξ = 0 .
Φ ( ρ , ξ ) = 1 ( ξ 0 i ξ ) n + 1 L n 0 ( k 1 v c ρ 2 2 ( ξ 0 i ξ ) ) exp [ k 1 v c ρ 2 2 ( ξ 0 i ξ ) ] ,
F ( Q ) = { 1 n ! Q n exp ( ξ 0 Q ) Q > 0 0 Q < 0
F ( Q ) = { 1 n ! ( Q ) n exp ( ξ 0 Q ) Q < 0 0 Q > 0
ψ ( ρ , z , t ) = 1 ξ 0 i ( z vt ) 0 d ω G ( ω ) exp [ ω s ( ρ , z , t ) ] ,
s ( ρ , z , t ) = 1 v c 2 c ρ 2 ξ 0 i ( z vt ) i ( t z c ) .
G ( ω ) = { i Γ ( α ) ( ω ω 0 ) α 1 exp ( ω ω 0 Δ ω ) ω > ω 0 0 ω < ω 0 ,
ψ ( ρ , z , t ) = exp [ ω 0 s ( ρ , z , t ) ] ξ 0 i ( z vt ) × 1 [ s ( ρ , z , t ) + 1 Δ ω ] α .
ψ ( ρ , z , t ) = 0 d ω d Q F ( Q , ω ) J 0 ( k ( Q , ω ) ρ ) exp [ it ( ω Qv ) iz ( ω c Q ) ] ,
{ x = x y = y z = γ ( z + Vt ) t = γ ( t + V c 2 z )
ψ ( x , y , z , t ) = Φ ( x , y , γ ( 1 vV c 2 ) z γ ( v V ) t ) exp [ i ω ( t z c ) ] ,
ψ ( x , y , z , t ) = Φ ( x , y , z γ ) exp [ i ω ( t z c ) ] ,
ψ ( x , y , z , t ) = Φ ( x , y , vt γ ) exp [ i ω ( t z c ) ] ,

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