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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  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.

2003 (1)

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

2002 (1)

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).

2000 (1)

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]

1999 (2)

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]

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

1998 (3)

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

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]

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

1997 (1)

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

1992 (2)

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]

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]

1991 (2)

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]

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

1989 (1)

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

1986 (1)

1985 (1)

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

1984 (1)

1983 (1)

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

Abdel-Rahman, M.

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]

Belanger, P.A.

Besieris, I.M.

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]

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]

Brittingham, J.N.

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

Chatzipetros, A.

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]

Donnelly, R.

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]

Fagerholm, J.

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]

Feng, S.

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]

Friberg, A.T.

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]

Gradshteyn, I.S.

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

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, 19–31 (1992).
[CrossRef] [PubMed]

Hellwarth, R.W.

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]

Hernandez-Figueroa, H.E.

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).

Longhi, S.

S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 1–6 (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, 19–31 (1992).
[CrossRef] [PubMed]

Overfelt, P.L.

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

Porras, M.A.

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

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

Ratsep, M.

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

Recami, E.

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).

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

Ryzhik, I.M.

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

Saari, P.

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

Salo, J.

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]

Salomaa, M.M.

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]

Shaarawi, A.

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]

Shaarawi, A.M.

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]

Siegman, A.E.

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

Winful, H.G.

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]

Zamboni-Rached, M.

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).

Ziolkowski, R.

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]

Ziolkowski, R.W.

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]

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

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

Eur. Phys. J. (1)

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. 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, 19–31 (1992).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

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. (1)

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 (2)

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

Phys. Rev. A (2)

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 (4)

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

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]

Phys. Rev. Lett. (1)

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

Physica A (1)

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

Proc. IEEE (1)

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 (1)

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) (1)

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 (2)

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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