Abstract

Subluminal, luminal and superluminal localized wave solutions to the paraxial pulsed beam equation in free space are determined. A clarification is also made to recent work on pulsed beams of arbitrary speed which are solutions of a narrowband temporal spectrum version of the forward pulsed beam equation.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. P. Saari and K. Reivelt, ???Generation and classification of localized waves by Lorentz transformations in Fourier space,??? Phys. Rev. E 65, 036612 1-12 (2004).
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  20. S. Longhi, ???Gaussian pulsed beams with arbitrary speeds,??? Opt. Express, 12, 935-940 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-935.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-935.</a>
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IEEE Trans. Antennas Propag.

E. Heyman, ???Pulsed beam propagation in an inhomogeneous medium,??? IEEE Trans. Antennas Propag. 42, 311-319 (1994).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq.

J. Y. Lu and J. F. Greenleaf, ???Nondiffracting X waves-exact solutions to the free-space wave equation and their finite aperture realizations,??? IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Opt. soc. Am. B

Opt. Express

Phys. Rev. A

R. W. Ziolkowski, ???Localized transmission of electromagnetic energy,??? Phys. Rev. A 39, 2005-2033 (1989).
[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. A 67, 063820 1-5 (2003).
[CrossRef]

Phys. Rev. E

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]

P. Saari and K. Reivelt, ???Generation and classification of localized waves by Lorentz transformations in Fourier space,??? Phys. Rev. E 65, 036612 1-12 (2004).

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

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

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

Phys. Rev. Lett.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz and J. Trull, ???Nonlinear electromagnetic X waves,??? Phys. Rev. Lett. 90, 170406 1-4 (2003).
[CrossRef]

Physica A

E. Recami, ???On localized ???X-shaped??? superluminal solutions to Maxwell???s 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, 1371-1378 (1991).
[CrossRef]

Progr. Electromagn. Res.

I. M. Besieris, M. Abdel-Rahman, A. M. 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. Erdelyi, Tables of Integral Transforms (Academic Press, New York, 1980), Vol. I.

I. M. Besieris, M. Abdel-Rahman and A. M. Shaarawi, ???Symplectic (nonseparable) spectra and novel, slowly decaying beam solutions to the complex parabolic equation,??? URSI Digest, p. 281 (abstract), IEEE AP-S Intern. Symp. and URSI Natl. Meeting, Baltimore, MD, July 21-26 (1996).

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Equations (96)

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( Δ ρ 2 + 2 z 2 + ω 2 c 2 ) u ̂ ( r , ω ) = 0 ; ρ = ( x , y ) ,
i z ν ̂ ± ( r , ω ) = ± c 2 ω 2 ν ̂ ± ( r , ω ) ,
u ± ( ρ , τ ± , z ) = R 1 d ω R 2 d κ exp [ i ( ω τ ± κ · ρ ) exp [ ± i ( c κ 2 z ) / ( 2 ω ) ] u ˜ 0 ( κ , ω ) ,
( ρ 2 2 c 2 τ ± z ) u ± ( ρ , τ ± , z ) = 0 .
u + ( ρ , τ + , z ; α ) = exp ( i 2 α c τ + ) ν + ( ρ , z ; α ) .
i 4 α z v + ( ρ , z ; α ) = ρ 2 v + ( ρ , z ; α ) ,
ν n ( ρ , z ; α ) = A 0 a 1 ( a 1 + iz ) n + 1 exp ( α ρ 2 a 1 + iz ) L n ( 0 ) ( α ρ 2 a 1 + iz ) ; n = 0 , 1 , 2 ,
u + ( ρ , z , τ + ) = 0 d α exp ( 2 i α c τ + ) v n ( ρ , z ; α ) F ˜ ( α ) .
u + ( ρ , z , τ + ) = A 0 a 1 ( a 1 + iz ) n + 1 Γ ( n + 1 ) n ! [ 2 c ( a 2 + i τ + ) ] n [ 2 c ( a 2 + i τ + ) + ρ 2 / ( a 1 + iz ) ] n + 1 .
t ( z ± 1 c t ) u ± ( ρ , z , t ) = ± c 2 ρ 2 u ± ( ρ , z , t ) .
2 ( v c 1 ) 2 ζ + η + u + ( ρ , ζ + , η + ) 2 v c ( v c 1 ) 2 η + u + ( ρ , ζ + , η + ) = ρ 2 u + ( ρ , ζ + , η + ) .
u + ( e ) ( ρ , ζ + , η + ; α , β , κ ) = exp ( i κ · ρ ) exp ( i α ζ + ) exp ( i β η + ) ,
κ 2 = 2 v c ( v c 1 ) β 2 + 2 ( v c 1 ) α β .
u + ( ρ , z , t ) = 0 d α 0 d β R 2 d κ u + ( e ) ( ρ , ζ + , η + ; α , β , κ )
× δ [ κ 2 2 v c ( v c 1 ) β 2 2 ( v c 1 ) α β ] u ˜ 0 ( α , β , κ ) ,
u + ( ρ , z , t ) = 0 d α 0 d β exp [ i ( α ζ + + β η + ) ] J 0 [ ρ 2 v c ( v c 1 ) β 2 + c v α β ] u ˜ 1 ( α , β ) ,
u + ( ρ , z , t ) = 0 d α v + ( ρ , z , t ; α ) F ˜ ( α ) ,
v + ( ρ , z , t , α ) = exp [ i α ( 1 c 2 v ) ( z v ph t ) ] [ 2 ( v c ) ( v c 1 ) ρ 2 + ( a 1 + i ( z vt ) ) 2 ] 1 / 2
× exp [ c α 2 v 2 ( v c ) ( v c 1 ) ρ 2 + ( α 1 + i ( z vt ) ) 2 ] ,
u + ( ρ , z , t ) = { 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 } 1 / 2 .
F ˜ ( α ) = { 0 , b > α > 0 , 1 Γ ( q ) ( α b ) q 1 exp [ a 2 ( α b ) ] , α b ; b , q 0 ,
u + ( ρ , z , t ) = [ 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 ] 1 / 2 exp ( ib λ ) [ c / ( 2 v ) 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 + ( a 2 i λ ) ] q
× exp { ( bc ) / ( 2 v ) 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 } ,
κ 2 = 2 v c ( 1 v c ) [ ( α c 2 v ) 2 β ¯ 2 ] ; β ¯ β + α c 2 v .
u + ( ρ , ζ + , η + ) = 0 d α ( α c ) / ( 2 v ) d β ¯ J 0 { ρ [ 2 v c ( 1 v c ) ] 1 / 2 [ ( α c 2 v ) 2 β ¯ 2 ] 1 / 2 }
× exp ( i α ζ + ) exp ( i β ¯ η + ) exp [ i η + ( α c ) / ( 2 v ) ] u ˜ 2 ( α , β ¯ ) .
u + ( ρ , z , t ) = 0 d α sin [ ( α c ) / ( 2 v ) 2 ( v / c ) [ 1 ( v / c ) ] ρ 2 + ( z vt ) 2 ] 2 ( v / c ) [ 1 ( v / c ) ] ρ 2 + ( z vt ) 2
× exp { i α [ 1 c / ( 2 v ) ] ( z v ph t ) } F ˜ ( α ) ,
2 ( 1 + v c ) 2 ζ η + u ( ρ , ζ , η + ) 2 v c ( 1 + v c ) 2 η + 2 u ( ρ , ζ , η + ) = ρ 2 u ( ρ , ζ , η + ) .
u ( e ) ( ρ , ζ , η + ; α , β , κ ) = exp ( i κ · ρ ) exp ( i α ζ ) exp ( i β η + ) ,
κ 2 = 2 v c ( 1 + v c ) β 2 + 2 ( 1 + v c ) α β ,
u ( ρ , z , t ) = 0 d α 0 d β R 2 d κ u ( e ) ( ρ , ζ , η + ; , α , β , κ )
× δ [ κ 2 2 v c ( v c 1 ) β 2 2 ( v c 1 ) α β ] u ˜ 0 ( α , β , κ ) ,
u ( ρ , z , t ) = 0 d α 0 d β exp [ i ( α ζ β η + ) ] J 0 [ ρ 2 v c ( 1 + v c ) β 2 + c v α β ] u ˜ 1 ( α , β ) .
u ( ρ , z , t ) = 0 d α v ( ρ , z , t ; α ) F ˜ ( α ) ,
v ( ρ , z , t ; α ) = exp [ i α ( 1 + c 2 v ) ( z + v ph t ) ] [ 2 v c ( 1 + v c ) ρ 2 + ( a 1 + i ( z vt ) ) 2 ] 1 / 2
× exp { ( c α ) / ( 2 v ) 2 ( v / c ) [ ( 1 + v / c ) ] ρ 2 + ( a 1 + i ( z vt ) ) 2 } ,
u ( ρ , z , t ) = exp { i ( 3 / 2 ) α 0 [ z + ( c / 3 ) t ] } [ 4 ρ 2 + ( a 1 + i ( z ct ) ) 2 ] 1 / 2
× exp { α 0 / ( 2 c ) 4 ρ 2 + ( a 1 + i ( z ct ) ) 2 } ,
u ( ρ , z , t ) = [ 4 ρ 2 + ( a 1 + i ( z ct ) ) 2 ] 1 / 2
β ( κ , ω ) β ( κ , ω 0 ) = ( c κ 2 ) / ( 2 ω 0 ) .
ψ ± ( ρ , z , t ) exp ( i ω 0 τ ± ) R 1 d Ω R 2 d κ exp ( i Ω τ ± ) exp ( i κ · ρ )
× exp [ ± i ( c κ 2 z ) / ( 2 ω 0 ) ] u ˜ 1 ( κ , Ω ) ,
ψ ± ( ρ , z , t ) = exp ( i ω 0 τ ± ) ϕ ± ( ρ , z , t ) ,
i ( z ± 1 c t ) ϕ ± ( ρ , z , t ) = ± 1 2 k 0 ρ 2 ϕ ± ( ρ , z , t ) ; k 0 ω 0 / c .
ϕ + ( ρ , z , t ) = f ( τ + ) Φ ( ρ , η + ) ,
i ( 1 v c ) η + Φ ( ρ , η + ) = 1 2 k 0 ρ 2 Φ ( ρ , η + ) .
ψ + ( ρ , τ + , η + ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , η + ) ,
i 4 σ ± Φ ( ρ , σ ± ) = ρ 2 Φ ( ρ , σ ± ) ,
ψ + ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , σ ± ) ,
ψ + ( mn ) ( x , y , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) exp [ x 2 / ( γ 1 + i σ ± ) ] exp [ y 2 / ( γ 2 + i σ ± ) ] ( γ 1 + i σ ± ) ( m + 1 ) / 2 ( γ 2 + i σ ± ) ( n + 1 ) / 2
× H m ( x / γ 1 + i σ ± ) H n ( y / γ 2 + i σ ± ) .
ψ + ( n ) ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) γ 0 ( γ 0 + i σ ± ) n + 1 exp ( ρ 2 γ 0 + i σ ± ) L n ( 0 ) [ ρ 2 ( γ 0 + i σ ± ) ] .
ψ + ( 0 ) ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) γ 0 γ 0 + i σ ± exp [ ρ 2 ( γ 0 + i σ ± ) ] .
ψ + ( 0 ) ( ρ , τ + , η + ) = exp ( i ω 0 τ + ) f ( τ + ) a a ± i η + exp ( ω 0 2 c v c 1 ρ 2 a ± i η + ) .
f ( τ + ) = exp ( τ + 2 4 T 2 ) = exp { 1 4 T 2 [ ( t z v ) ( v c vc ) z ] 2 }
ψ + ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) γ 0 γ 0 + i σ ± J 0 ( γ 0 k 0 ρ sin θ γ 0 + i σ ± ) exp ( ρ 2 γ 0 + i σ ± )
× exp [ i σ ± 4 ( k 0 2 γ 0 sin 2 θ ) / ( γ 0 + i σ ± ) ] .
ϕ + ( ρ , τ + , σ z ) = f ( τ + ) Φ ( ρ , σ z ) ,
i 4 σ z Φ ( ρ , σ z ) = ρ 2 Φ ( ρ , σ z ) .
ψ + ( ρ , τ + , σ z ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , σ z ) .
u + ( ρ , z ; α ) = exp ( 2 i α c τ + ) a 1 ( a 1 + i z ) exp ( α ρ 2 a 1 + i z ) ,
u + ( ρ , z , t ) = 1 π 0 d α F ˜ ( α ) exp ( 2 i α c τ + ) a 1 ( a 1 + i z ) exp ( α ρ 2 a 1 + i z ) ,
u + ( ρ , z , t ) = a 1 ( a 1 + i z ) f ̂ ( t z / c i 1 2 c ρ 2 a 1 + i z ) ,
ψ + ( ρ , z , t ) = exp [ i ω 0 ( t z / c ) ] f ( t z / c ) γ 0 γ 0 i 2 z / k 0 exp ( ρ 2 γ 0 i 2 z / k 0 ) .
ϕ ( ρ , z , t ) = f ( τ ) Φ ( ρ , η + ) ,
i ( 1 + v c ) η + Φ ( ρ , η + ) = 1 2 k 0 ρ 2 Φ ( ρ , η + ) .
i 4 σ ¯ + Φ ( ρ , σ ¯ + ) = ρ 2 Φ ( ρ , σ ¯ + ) .
ψ ( ρ , τ , σ ¯ + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ ¯ + ) .
ψ ( ρ , τ , σ z + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ z + ) ; σ z + σ z = 2 z / k 0 ,
i ( z ± 1 c t ) ϕ ± ( ρ , z , t ) = ± 1 2 k ± ρ 2 ϕ ± ( ρ , z , t ) ; k ± γ ¯ ( 1 v / c ) k 0 .
ψ ± ( ρ , τ ± , σ ¯ z ) = exp [ i ω 0 γ ¯ ( 1 v c ) τ ± ] f [ γ ¯ ( 1 v c ) τ ± ] Φ ( ρ , σ ¯ z ) ;
τ ± t z c ; σ ¯ z 2 z k ± ,
ψ + ( ρ , τ + , σ ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , σ )
ψ ( ρ , τ , σ ¯ + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ ¯ + ) .
ϕ ± ( ρ , ς ± , t ) = g ( ς ± ) Ψ ( ρ , t ) ,
i 4 σ t Ψ ( ρ , σ t ) = ρ 2 Ψ ( ρ , σ t ) ,
ψ ± ( ρ , ς , σ t ) = exp ( i k 0 ς ± ) g ( ς ± ) Ψ ( ρ , σ t )
ψ ± ( ρ , z , t ) = exp [ i k 0 ( z c t ) ] g ( z c t ) γ 0 γ 0 i ( 2 c t ) / k 0 exp ( ρ 2 γ 0 i ( 2 c t ) / k 0 ) .
ψ ± ( ρ , ς ± , σ t ) = exp [ i k 0 γ ( v c 1 ) ς ± ] g [ γ ( v c 1 ) ς ± ] Ψ ( ρ , σ t ) ;
ς ± z c t ; σ t ( 2 c t ) / k ¯ ± ,
ψ + ( ρ , ς + , σ + ) = exp ( i k 0 ς + ) g ( ς + ) Ψ ( ρ , σ + )
ψ ( ρ , ς , σ ¯ + ) = exp ( i k 0 ς ) g ( ς ) Ψ ( ρ , σ ¯ + )
i 4 χ ϕ ± ( ρ , χ ) = ρ 2 ϕ ± ( ρ , χ ) ,
ψ + ( ρ , z , t ) = exp ( i k 0 ς + ) ϕ + ( ρ , χ ) = exp [ i k 0 ( z c t ) ] ϕ + [ ρ , ( z + c t ) / k 0 ] .
ψ ( ρ , z , t ) = exp ( i k 0 ς ) ϕ ( ρ , χ + ) = exp [ i k 0 ( z + c t ) ] ϕ [ ρ , ( z + c t ) / k 0 ] .
ψ + ( ρ , z , t ) = k 0 k 0 i ( z + c t ) exp [ i k 0 ( z c t ) ] exp [ k 0 ρ 2 k 0 i ( z + c t ) ] ,
ψ ( ρ , z , t ) = k 0 k 0 + i ( z c t ) exp [ i k 0 ( z + c t ) ] exp [ k 0 ρ 2 k 0 + i ( z c t ) ] .
ψ ( ρ , τ , σ ¯ + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ ¯ + ) ; τ = t + z c z , σ ¯ + = 2 ( z v t ) k 0 ( 1 + v / c ) .
ψ ( ρ , z , t ) = exp [ i k 0 ( z + c t ) ] Φ ( ρ , z c t k 0 ) .
ψ ( ρ , z , t ) = k 0 a + i ( z c t ) exp [ i k 0 ( z + c t ) ] exp [ k 0 ρ 2 a + i ( z c t ) ] , a > 0 .
ψ ( ρ , τ , σ ¯ z + ) = 1 a / k 0 + i σ ¯ z + exp [ i ω 0 γ ¯ ( 1 v c ) τ ] exp [ ρ 2 a / k 0 + i σ ¯ z + ]
τ = t + z c ; σ ¯ z + = 2 z k , k = γ ¯ ( 1 + v / c ) k 0 .
B p ( ρ , z , t ) = exp [ i ( β α ) ( z v ph t ) ] J 0 [ ρ 2 v c ( 1 + v c ) β 2 + c v α β ] ,
B e ( ρ , z , t ) = exp [ i k z ( z ω k z t ) ] J 0 [ ρ ( ω / c ) 2 k z 2 ] .
B p ( ρ , z , t ) = exp [ i β ( z v t ) ] J 0 [ ρ β 2 v c ( 1 + v c ) ]

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