## Paraxial localized waves in free space

Optics Express, Vol. 12, Issue 16, pp. 3848-3864 (2004)

http://dx.doi.org/10.1364/OPEX.12.003848

Acrobat PDF (412 KB)

### 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

## 1. Introduction

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

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

8. 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]

9. 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]

## 2. Paraxial localized waves based on a narrow angular spectrum approximation of the scalar wave equation

*û±*(

*r*⃗,

*ω*) exp[±

*i*(

*ω*/

*c*)

*z*]

*v̂*±(

*r*⃗,

*ω*) with

*v̂*±(

*r*⃗,

*ω*) governed by the complex parabolic equations

*z*– axis. In Eqs. (2.1) and (2.2),

*c*is the speed of light in vacuum and

*ω*denotes an angular frequency. The space-time paraxial solutions

*u*

_{±}(

*r*⃗,

*t*) can be expressed in terms of the Fourier spectral representations

*τ*

_{±}=

*t*∓

*z*/

*c*,κ⃗(

*k*

_{x}

*,k*

_{y}) and

*κ*=|

*K*⃗|. These representations allow one to determine the equations governing

*u*

_{±}(

*ρ⃗*,

_{τ}±,

*z*); specifically,

*pulsed beam equations*[10

10. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. **42**, 311–319 (1994). [CrossRef]

*u*

_{+}(

*ρ⃗*,τ,

*z*) has been used extensively recently (cf., e.g., [11

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

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

### 2.1 Paraxial luminal pulsed beams

*v*

_{+}(

*ρ⃗,z;α*) obeys the Schrödinger-like equation

*a*

_{1}is a free positive parameter and

*nth*order Laguerre polynomial. A general solution to the forward pulsed beam equation can be obtained by using Eq. (2.7) in conjunction with Eq. (2.5) and superimposing over the free parameter

*α*; specifically,

*F̃*(

*α*)=exp(-2

*α*

_{2}

*αc*),

*a*

_{2}a being a real positive parameter. Then, from Erdelyi (cf., Ref. [13], p. 174), one obtains

*z*– direction with speed

*c*, it sustains loss of amplitude as well as broadening. However, these distortions can be minimized by tweaking the free parameters

*a*

_{1}and

*a*

_{2}. An interesting property of the solution given in Eq. (2.9) is that with the formal replacementz

*z*→

*ς*

_{−}/2;

*ς*

_{−}≡

*z*+

*ct*, it becomes the

*nth*order

*splash mode*[1

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

*focus wave mode*(FWM)-type exact solutions to the homogeneous 3D scalar wave equation. Recently, the splash mode corresponding to

*n*=0 has been used as a Hertz potential in an extensive study of the spatio-temporal evolution of focused single-cycle terahertz electromagnetic pulses [14

14. 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]

*a*

_{1}<<

*a*

_{2}corresponding to the paraxial regime. This is a particular situation whereby an exact solution to the homogeneous 3D scalar wave equation behaves as a paraxial pulsed beam under certain parametrization. An analogous result, but in a different setting, has been discussed by Saari [15

15. P. Saari, “Evolution of subcycle pulses in nonparaxial Gaussian beams,” Opt. Express , **8**, 590–598 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-11-590. [CrossRef] [PubMed]

*u*

_{-}(

*ρ,z,τ*

_{-}); however, these wavepackets propagate in the negative

*z*– direction.

### 2.2 Paraxial superluminal localized waves

*u*(

*ρ⃗,z,t*) :

*ς*

*+z-ct*,η+=

*z-vt;v*>

*c*. One, then, obtains the transformed equation

*α*and

*β*are real positive free parameters with units of

*m*

^{-1}. Substitution into Eq. (2.11) leads to the dispersion relation

*δ*(·) denotes a Dirac delta function. For an azimuthally independent spectrum, viz.,

*u*

_{0}(

*α,β,κ⃗*)=

*u*(

*α,β,κ*), one obtains, in particular, the axisymmetric solution

*J*

_{0}(·) is the zero-order ordinary Bessel function. If

*ũ*

_{1}(

*α,β*)=exp(-α

_{1},

*β*)

*F̃*(

*α*),

*α*

_{1}a being a positive parameter, the integration over

*β*can be carried out explicitly (cf. [13], p. 192). As a result, one has

*v*

_{ph}=

*v*/[2(

*v*/

*c*)-1]. It should be noted that

*v*

_{ph}→

*c*/2 as

*v*→∞. The solution given in Eq. (2.17) is the paraxial version of the

*focus X wave*(FXW) (cf. Ref. [3

3. 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]

*F̃*(

*α*)=

*δ*(

*α*), one obtains the paraxial version of the infinite-energy zero-order X wave [16

16. 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]

17. 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**, 75–87 (1993). [CrossRef]

3. 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]

### 2.3 Paraxial subluminal localized waves

*v*

_{ph}=

*v*/[2(

*c*/

*v*)-1]. The following should be noted: [0,)

*v*

_{ph}→[0,

*c*/2) and

*v*

_{ph}→(∞,

*c*] as

*v*→(

*c*/2,

*c*].The solution in Eq. (2.23) consists of a superposition of paraxial MacKinnon-type wavepackets (cf. Ref. [3

3. 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]

*F̃*(

*α*).

### 2.4 Localized waves for u_{-}(ρ⃗,z,t)

*u*

_{-}(

*ρ⃗,z,t*) in Eq. (2.10):

*ς*

_{-}=

*z*+

*ct*,

*η*

_{+}=

*z-vt*. Then, one obtains

*a*and

*β*are real positive free parameters. Substitution into Eq. (2.24) leads to the dispersion relation

*ũ*

_{0}(

*α,β,κ⃗*)=

*ũ*

_{0}(

*α,β,κ*), one obtains, in particular, the axisymmetric solution

*paraxial monochromatic Bessel beam*. Its difference from an exact monochromatic Bessel beam solution to the homogeneous 3D scalar wave equation is discussed in Appendix A. If the spectrum

*ũ*

_{1}(

*α,β*) in Eq. (2.28) equals exp(-

*a*

_{1}

*β*)

*F̃*(

*a*),

*a*

_{1}being a positive parameter, the integration over

*β*can be carried out explicitly [13]. As a result, one has

*v*

_{ph}=

*c*(2+

*c/v*).

*v*<∞. If

*v*=

*c*, for example, with

*F̃*(

*α*)=d(

*α-α*

_{0}), one obtains the “luminal FXW”

*F̃*(

*α*)=

*δ*(

*α*), the “luminal zero-order X wave,” viz.,

*F̃*(

*α*) in Eq. (2.29).

## 3. Paraxial localized waves based on a narrowband temporal spectrum approximation of the forward and backward pulsed beam equations

*ũ*

_{0}(

*κ⃗,ω*)≡

*ũ*

_{1}(

*κ⃗*,

*ω*-

*ω*

_{0}) in Eq. (2.3) is assumed to be narrowband around the frequency

*ω*=

*ω*

_{0}. Furthermore, the phase term

*β*(

*κ,ω*)≡(

*ck*

^{2})/(2

*ω*)is expanded in a Taylor series around

*ω*=

*ω*

_{0}and only the first term in the expansion is retained; i.e.,

*ω*-

*ω*

_{0}, or

*ϕ*

_{±}(

*ρ⃗z,t*) governed by the equations

### 3.1 Subluminal and superluminal pulsed beams

*η*

_{+}=

*z-vt, v*≠

*c*, and

*f*(

*τ*

_{+}) is an arbitrary function (at least differentiable). It follows, then, that the wave function Φ(

*ρ*⃗,

*η*

_{+}) obeys the equation

18. A. Wunsche, “Embedding of focus wave modes into a wider class of approximate wave functions,” J. Opt. Soc. Am. A **6**, 1661–1668 (1989). [CrossRef]

*et al*. [19]. The special case with

*f*(

*τ*

_{+}) constant=was rediscovered by Longhi [20] recently. Longhi mistakenly attributed his solution to a “generalized paraxial approximation,” instead of to a narrowband approximation of the forward pulsed beam equation.

*σ*

_{±}=2

*η*

_{+}/(

*k*

_{0}|(

*v/c*)-1|). The plus sign is associated with the superluminal case

*v*>

*c*and the minus sign to the subluminal case

*v*<

*c*. In terms of the new variables, Eqs. (3.6) and (3.7) assume the simpler forms

*γ*

_{1,2}are free positive parameters and

*H*

_{m}(·) denotes the

*mth*order Hermite polynomial.

*γ*

_{0}is a free positive parameter and

*nth*order Laguerre polynomial.

*n*=0, the Laguerre-Gauss solution in Eq. (3.11) becomes the axisymmetric “modified” fundamental Gaussian pulsed beam

*γ*

_{0}=2

*a*/[

*k*

_{0}|(

*v/c*)-1|],

*a*being a real positive parameter, this solution can be rewritten as

*ρ,τ*

_{+},η

_{+}) multiplying exp(

*iω*

_{0}

*τ*

_{+})

*f*(

*τ*

_{+}) is an infinite energy invariant wavepacket propagating along the positive

*z*– direction with fixed speed

*v*, either superluminal or subluminal. The arbitrary time-limiting function

*f*(

*τ*

_{+}) in Eq. (3.13) can be chosen so that the entire wavepacket

*ρ,τ*

_{+},

*η*

_{+}) has finite energy and propagates to a large distance

*z*with almost no distortion, except for local deformations. For example, the function

*v*close to c and a large values of

*ω*

_{0}so that

*ω*

_{0}|(

*v*/

*c*)-1/

*c*=

*O*(1).

*J*

_{0}(·) denotes the zero-order ordinary Bessel function and

*θ*is an arbitrary real angle. It should be noted that for

*θ*=0, this solution reduces to the pulsed Gaussian beam in Eq. (3.12).

### 3.2 Luminal pulsed beams

*σ*-

_{z}=-(2

*z*)/

*k*

_{0}. It follows, then, that the wave function Φ(

*ρ⃗,σ-*

_{z}) obeys the Schrödinger equation

*σ*

_{±}must be replaced by

*σ*-

_{z}=-(2

*z*)/

*k*

_{0}for the luminal case under consideration.

*a*

_{1}is a real positive parameter and

*α*is an arbitrary real positive quantity. A superposition over the latter, e.g.,

*f̂*(

*t*) denotes the complex analytic signal associated with the spectrum

*F̃*(

*ω*).

*z*– direction at the speed of light

*in vacuo*and a “standing” fundamental Gaussian mode. In Eq. (3.22),

*ω*

_{0}=

*ck*

_{0}is fixed and

*γ*

_{0}is an arbitrary positive parameter. Thus, the pulsed beams given in Eqs. (3.21) and (3.22) differ substantially.

### 3.3 Localized waves for ψ-(ρ⃗,z,t)

*f*(

*τ*-) is an arbitrary function (at least differentiable). It follows, then, that the wave function Φ(

*ρ⃗,η*

_{+}) obeys the equation

*σ̄*

_{+}=2

*η*

_{+}/[

*k*

_{0}(1+

*v/c*)]. Then Eq. (3.24) changes to

*v*<∞. Since Φ(

*ρ⃗,*σ ¯
+) obeys the Schrödinger equation (3.25), one can have in a single setting Hermite-Gauss, Laguerre-Gauss and Bessel-Gauss subluminal, luminal and superluminal solutions. It must be pointed out, however, that whereas the “envelope” function (

*ρ⃗,*σ ¯

_{+}) moves in the +

*z*-direction with an arbitrary speed

*v*∈ [0,∞), the factor 0 exp(

*iω*

_{0}

*τ*-)

*f*(

*τ*-) in Eq. (3.26) travels in the opposite direction. For

*v*=0, one obtains

## 4. Derivation of paraxial subluminal and superluminal localized waves by means of Lorentz relativistic boosts

### 4.1 Subluminal boosts

*x*=

*x*′,

*y*=

*y*′,

*z*=

*z*′+

*vt*′),

*t*=

*t*′+(

*v*/

*c*2)

*z*′], where

*v*<

*c*and

*ψ*

_{±}(

*ρ⃗,z,t*)=exp (

*iω*

_{0}

*τ*

_{±})

*ϕ*

_{±}(

*ρ⃗,z,t*) [cf. Eq. (3.3)] transforms to

*ψ*

_{±}(

*ρ⃗,z′,t′*)=exp [

*iω*

_{0}

*v/c*)

*τ*′±]

*ϕ*

_{±}(

*ρ⃗,z,t*), where

*τ*′

_{±}=

*t*′∓

*z*′/

*c*and

*ρ⃗,*σ ¯
-z′) satisfies the parabolic equation (3.17) with the interchange

*z*→

*z*′, and (

*ρ⃗,*σ ¯
+) is governed by an analogous equation. Application of the inverse boosting

_{z}′*x*′=

*x,y*′=

*y,z*′

*z-vt*),

*ct*′=-

*v/c*)[

*z*-(

*c*2/

*v*)t] to Eq. (4.2) yields the general paraxial subluminal solutions [cf. Eqs. (3.9) and (3.26)]

### 4.2 Superluminal boosts

*ς*

_{±}=

*z*∓

*ct*. It follows, then, that the wave function Ψ(

*ρ⃗,t*) obeys the Schrödinger equation

*ct*)/

*k*

_{0}. Thus, a narrow angular spectrum and a narrowband frequency spectrum result in the following approximate luminal solution to the homogeneous 3D scalar wave equation:

*x*=

*x*′,

*y*=

*y*′,

*z*=γ(

*v/c*)[

*z*′+(

*c*

^{2}/v)t′],

*t*=γ(

*z*′+

*vt*′)/

*c*, where

*v*>

*c*, the solution

*ψ*

_{±}(

*ρ⃗,z,t*)=exp (

*iω*

_{0}

*τ*

_{±})

*ϕ*

_{±}(

*ρ⃗,z,t*) [cf. Eq. (3.3)] transforms to

*ψ*

_{±}(

*ρ⃗,z′,t′*) exp {-

*ik*

_{0}γ[

*v/c*]∓1]

*ς*′±}

*ϕ*

_{±}(

*ρ⃗,z′,t′*), where

*ς*′

_{±}=

*z*′∓

*ct*′ and

*ϕ*

_{±}(

*ρ⃗,z′,t′*) is given in Eq. (4.1) with

*k*

_{±}→

*k̄*

_{±}≡γ[(

*v/c*)]∓1]

*k*

_{0}. Consider, next, the general luminal solutions

*ρ⃗,σ∓*

_{t}′) satisfies Eq. (4.6) with

*t*→

*t*′. Application of the inverse boosting

*x*′=

*x,y*′=

*y,ct*′=-γ(

*z-vt*),

*z*′=

*γ*(

*v/c*)[

*z*-(

*c*

^{2}/

*v*)

*t*] to Eq. (4.9) yields the general paraxial superluminal solutions [cf. Eqs. (3.9) and (3.26)]

## 5. Embedding of exact localized wave solutions of the scalar wave equation into approximate paraxial ones

*ς*=

*z-ct,z*-=

*z*+

*ct*is introduced in Eq. (3.4). As a consequence one obtains the Schrödinger equations

*χ*-=-

*ς*-/

*k*

_{0}and

*χ*

_{+}=ς+/

*k*

_{0}. Thus, under the assumption of a narrow angular spectrum and a narrowband frequency spectrum one obtains the general solutions

*ψ*-(

*ρ⃗,z,t*) in Eq. (5.5) and

*ψ*

_{+}(

*ρ⃗,z,t*) in Eq. (5.4) are, respectively, the fundamental focus wave mode (FWM) and a variant of it. Both are

*exact*solutions to the homogeneous 3D scalar wave equation for an arbitrary wavenumber

*k*

_{0}! More generally, the solutions in Eqs. (5.2) and (5.3) embody Hermite-Gauss, Laguerre-Gauss and Bessel-Gauss FWM-type solutions that are also exact. This “peculiarity”, whereby exact solutions of the scalar wave equation are

*embedded*into approximations to this equation, has been mentioned by Wunsche [18

18. A. Wunsche, “Embedding of focus wave modes into a wider class of approximate wave functions,” J. Opt. Soc. Am. A **6**, 1661–1668 (1989). [CrossRef]

*f*(

*τ*-) constant -=and

*v*=

*c*, this expression simplifies to

*ρ⃗, z*) obeys the complex parabolic equation (3.17), a simple solution in the place of the general one in Eq. (5.7) is given as follows:

*Modulo*the constant multiplier

*k*

_{0}and with 0

*a*=

*k*, one recovers the exact FWM solution given in Eq. (5.5). Retracing the steps, it follows that the FWM arises from a subluminal Lorentz transformation of the monochromatic luminal beam [cf. restriction of Eq. (4.2)]

21. P. A. Be′langer, “Lorentz transformations of packet-like solutions of the homogeneous wave equation,”J. Opt. Soc. Am. A **3**, 541–542 (1986). [CrossRef]

## 6. Concluding remarks

## Appendix A

*v*

_{ph}=

*c*[

*α*+(

*v/c*)

*β*]/(

*β-α*).As mentioned earlier, this is an axisymmetric paraxial monochromatic Bessel beam solution to the homogeneous 3D scalar wave equation. It differs significantly from the exact monochromatic Bessel beam solution

*ω*|/|

*k*

_{z}|>

*c*. Assuming

*ω*and

*k*

_{z}to be positive, this means that the exact Bessel beam propagates along the +

*z*-direction with the superluminal speed

*v*=

*ω/k*

_{z}. The situation is much different in Eq. (A-1) Three distinct cases will be considered in detail:

*Case (i): a*=0,

*β*>0

*z*– direction at any speed

*v*∈(0,∞).

*Case (ii):a*>0,

*β*>0

*β*=

*µ*(

*c/v*)

*α;µ*>0. Then,

*v*

_{ph}=

*c*(1+

*µ*)/[(

*µ/δ*)-1];

*δ*≡

*v/c*>0. For

*v*

_{ph}>0, the inequality

*δ*<

*µ*must hold. With these restrictions, one finds that

*v*

_{ph}in Eq. (A-1) is subluminal, luminal or superluminal if

*δ*<,=,>

*µ*/(2+

*µ*), respectively.

*Case (iii):*0, a<,

*β*>(

*c/v*)

*a*

*β*=

*µ*(

*c/v*)

*a*;

*µ*>1. Then,

*v*

_{ph}=

*c*(-1+

*µ*)/[(

*µ*/

*δ*)+1. In this case,

*v*

_{ph}in Eq. (A-1) is subluminal, luminal or superluminal if

*δ*<,=,>

*µ*/(

*µ*-2);

*µ*>2, respectively.

## References and Links

1. | R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A |

2. | R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE |

3. | 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) |

4. | E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A |

5. | J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E |

6. | P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E |

7. | S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E |

8. | C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. |

9. | 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 |

10. | E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. |

11. | M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E |

12. | M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. soc. Am. B |

13. | A. Erdelyi, |

14. | S. M. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E |

15. | P. Saari, “Evolution of subcycle pulses in nonparaxial Gaussian beams,” Opt. Express , |

16. | 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 |

17. | 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 |

18. | A. Wunsche, “Embedding of focus wave modes into a wider class of approximate wave functions,” J. Opt. Soc. Am. A |

19. | 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). |

20. | S. Longhi, “Gaussian pulsed beams with arbitrary speeds,” Opt. Express , |

21. | P. A. Be′langer, “Lorentz transformations of packet-like solutions of the homogeneous wave equation,”J. Opt. Soc. Am. A |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(320.5540) Ultrafast optics : Pulse shaping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 7, 2004

Revised Manuscript: July 29, 2004

Published: August 9, 2004

**Citation**

Ioannis Besieris and Amr Shaarawi, "Paraxial localized waves in free space," Opt. Express **12**, 3848-3864 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3848

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### References

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