## Gaussian pulsed beams with arbitrary speed

Optics Express, Vol. 12, Issue 5, pp. 935-940 (2004)

http://dx.doi.org/10.1364/OPEX.12.000935

Acrobat PDF (63 KB)

### 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

## 1. Introduction

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]

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]

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]

*z*-direction with speed

*c*(the speed of light in vacuum), modulated by a plane wave moving in the negative z-direction with speed c. Later on, modified, extended and superposition of FWM solutions to the scalar wave equation were introduced, including among others relatively undistorted finite-energy FWM [5

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

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]

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

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]

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]

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]

3. P.A. Belanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A **1**, 723–724 (1984). [CrossRef]

3. P.A. Belanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A **1**, 723–724 (1984). [CrossRef]

*c*-cone variables ξ=

*z-ct*, η=

*z*+

*ct*, the envelope wavepacket of the scalar wave equation satisfies the well-known paraxial wave equation of diffraction, which admits of Gauss-Laguerre (or Gauss-Hermite) invariant solutions. Using the invariance property of the scalar wave equation under the Lorentz transformations, the moving FWMs were also explained as monochromatic Gaussian beams observed in another reference inertial frame [4

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

*exact*Gauss-Laguerre beams, composed solely by forward traveling-waves moving with an almost luminal velocity and with a longitudinal localization determined by the material dispersion properties, have been recently studied as well [18

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

*arbitrary*velocity

*v*, either subluminal or superluminal, can be simply obtained as solutions of the scalar wave equation in free-space under a generalized paraxial condition. Contrary to FWMs or their generalizations studied in previous works [10

**19**, 1–48 (1998). [CrossRef]

*v*(

*v*<

*c*) can be viewed as steady monochromatic Gaussian beams observed in a moving inertial reference frame at velocity

*v*. Conversely, Gaussian beams moving at a superluminal velocity

*v*(

*v*>

*c*) appear as Gaussian fields with “temporal diffraction” in a reference inertial frame moving at a speed

*V*=

*c/v*

^{2}.

## 2. Moving Gauss-Laguerre beams

*ψ*(

*x,y,z,t*) in free space:

*∂*

^{2}/

*∂x*

^{2}+

*∂*

^{2}/

*∂y*

^{2}is the transverse Laplacian. By extending the Ansatz originally proposed by Belanger [3

**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]

*v*, modulated by a plane wave of frequency

*ω*, i.e. we set:

*x,y*,ξ), with ξ=

*z-vt*:

*k*=

*ω/c*is the wavenumber of the carrier plane wave. Usual FWM solutions are retrieved in the case

*v*=-

*c*(see [4

**3**, 541–542 (1986). [CrossRef]

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]

*exactly*to the paraxial wave equation of diffraction theory (or to the Schrödinger equation), which admits of Gauss-Laguerre (or Gauss-Hermite) solutions. The fundamental FWM solution thus corresponds to a wavepacket with transverse Gaussian profile and longitudinal algebraic localization, traveling undistorted in the backward

*z*direction at the speed

*c*, modulated by a monochromatic plane-wave traveling in the forward

*z*direction. For

*v*=0, Eq. (2) corresponds to the usual monochromatic beam solution of the wave equation, with a motionless envelope Φ(

*x,y,z*) which can be expressed again in terms of Gauss-Laguerre (or Gauss-Hermite) beams provided that the paraxial approximation is assumed, i.e. under the condition |

*∂*

^{2}Φ/

*∂*ξ

^{2}|≪2

*k*|

*∂*Φ/

*∂*ξ| (see, for instance, [19]). Here we consider the case

*v*≠±

*c*; in addition, in order to to be able to generate a completely causal forward traveling wave, we will assume

*v*>0 to avoid backward traveling wave components. The most general solution with axial symmetry to Eq. (3) is given by a superposition of Bessel beams according to:

*x*

^{2}+

*y*

^{2})

^{1/2},

*k*

_{⊥}(

*Q*) is the dispersion relation for the transverse wave number, given by:

*Q*represents a longitudinal wavenumber offset from the plane wave value

*k*=

*ω/c*,

*F*(

*Q*) is an arbitrary spectral amplitude function, and the integral is extended over values of

*Q*such that

*k*

_{⊥}is real-valued. We now introduce the extended paraxial approximation by assuming, in the spectral representation (4), that the amplitude

*F*(

*Q*) is nonvanishing in a narrow region around

*Q*=0, such that in Eq. (5) we may neglect the term

*Q*(1+

*v*/

*c*) as compared to 2

*k*. Within this approximation we may hence assume:

*n*=0,1,2, …,

*n*, ξ

_{0}is the Rayleigh range of the Gaussian beam that determines the longitudinal (diffractive) length and transverse beam size, and the upper (lower) sign applies if

*v*<

*c*(

*v*>

*c*). The spectral amplitude

*F*(

*Q*), entering in Eq. (6), that produces the Gauss-Laguerre solution given by Eq. (8) is a one-sided exponential-like spectrum given by [20]

*v*<

*c*, and:

*v*>

*c*. We thus have constructed a family of

*moving*Gaussian beams, with a velocity

*v*either subluminal or superluminal, under the analogous paraxial approximation used in paraxial diffraction theory. It is worth observing that Gaussian beams propagating

*in free space*at a luminal velocity, i.e. with

*v*=

*c*, do not exist. Indeed, if we set

*v*=

*c*in Eq. (3), one has

*x*and

*y*variables. For the properties of harmonic functions, one can not simultaneously satisfy the conditions of spatial localization and absence of singularities, so that for

*v*=

*c*there are not physically acceptable solutions of the scalar wave equation (1) satisfying the Ansatz (2).

*dxdydz*|

*ψ*(

*x,y,z,t*)|

^{2}<∞, can be obtained by suitable superpositions of these basic solutions corresponding to different frequencies

*ω*. For instance, if we consider the lowest-order Gaussian mode (

*n*=0) in Eq. (8) and assume a frequency-independent Rayleigh range ξ0, a spectral superposition of moving Gaussian beams with a spectral amplitude

*G*(

*ω*) yields:

*ψ*(

*ρ,z,t*) has finite energy if at least

*G*(

*ω*)/√

*ω*is square integrable (see Appendix B of [5

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

*G*(

*ω*) which is nonvanishing in a small interval Δ

*ω*around a carrier frequency

*ω*

_{0}(Δ

*ω*≪

*ω*

_{0}), so that the paraxial approximation can be safely satisfied for any Gaussian spectral component entering in Eq. (11). As for FWMs, a simple analytical solution can be obtained by considering the modified power spectrum [6

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]

*α*>0) which yields the finite-energy pulsed solution:

*ω*=

*ω*

_{0}, by the last factor on the right hand side in Eq. (14). The dependence of

*ψ*on time

*t*and propagation coordinate

*z*occurs through the two variables

*z*-

*vt*and

*z-ct*. Correspondingly, the angular spectrum of the solution does not contain backward (acasual) components. This circumstance can be seen, as a general rule, by adopting the following Bessel beam spectral decomposition, which follows directly from Eqs. (2) and (4) after an integration over the frequency:

*k*

_{⊥}(

*Q,ω*) is given by Eq. (5). Note that the overall field results from a Bessel beam superposition modulated by plane waves with frequency Ω=

*ω-vQ*and wave number

*k*

_{z}=

*ω/c-Q*. In the spirit of the paraxial approximation and assuming a spectral amplitude which is nonvanishing in a small interval around

*ω*=

*ω*

_{0}, one has

*ω*/

*c*≫

*Q*and

*ω*≫

*vQ*, so that only forward and near-paraxial plane waves enter in the Bessel integral representation of

*ψ*.

## 3. Moving Gaussian beams and Lorentz transformations

**3**, 541–542 (1986). [CrossRef]

**19**, 1–48 (1998). [CrossRef]

*x*′,

*y*′,

*z*′), which travels with a velocity

*V*(

*V*<

*c*) in the forward z direction, with

*z*′=

*z*. The Lorentz transformation that connects the space-time variables in the two reference frames reads

*γ*=[1-(

*V*/

*c*)

^{2}]

^{-1/2}. Using Eqs. (2) and (16), in the moving reference frame the scalar field ψ then becomes:

*ω*′=

*γω*(1-

*V*/

*c*). In the moving reference frame the beam envelope thus remains Gaussian, but with a Doppler-shifted carrier frequency

*ω*′ and a modified envelope velocity and longitudinal Rayleigh range. Let us first consider the case of a subluminal moving Gaussian beam (

*v*<

*c*) in the steady reference frame, and let us choose

*V*=

*v*; in this case, in the moving reference frame one obtains:

*ψ*(

*x*′,

*y*′,

*z*′,

*t*′)=∑(

*x*′,

*y*′,

*z*′)exp(

*iω′t*′), i.e., one has a monochromatic beam, and ∑(

*x*′,

*y*′,

*z*′) is a solution of the three-dimensional Helmholtz equation (

*∂*

^{2}/

*∂z*′

^{2})∑+(

*ω*′/

*c*)

^{2}∑=0, which in the paraxial approximation reduces to a two-dimensional Scrödinger equation for the envelope Φ. If we instead consider a pulsed Gaussian beam traveling, in the steady reference frame, at a superluminal velocity

*v*>

*c*, in any other moving reference frame we can not observe a steady monochromatic Gaussian beam, i.e., the superluminal pulsed Gaussian beam can not be explained in terms of a relativistic transformation of a steady monochromatic Gaussian beam. However, in this case if we choose

*V*=

*c*

^{2}/

*v*, from Eq. (17) it follows that in the moving reference frame the observed field reads:

*ψ*(

*x*′,

*y*′,

*z*′,

*t*′)=∑(

*x*′,

*y*′,

*t*′)exp(-

*ik*′

*z*′), where

*k*′=

*ω*′/

*c*and ∑(

*x*′,

*y*′,

*t*′) is now a solution of the two-dimensional Klein-Gordon equation

*c*

^{2})

*∂*

^{2}∑/

*∂t*′

^{2}=

*k*′

^{2}∑, which in the paraxial approximation reduces again to a two-dimensional Scrödinger equation for the envelope Φ.

## 4. Conclusions

## References and links

1. | J.N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. |

2. | R.W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. |

3. | P.A. Belanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A |

4. | P.A. Belanger, “Lorentz transformation of packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A |

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

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

7. | P.L. Overfelt, “Bessel-Gauss pulses,” Phys. Rev. A |

8. | R. Donnelly and R. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. Lond. A |

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

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) |

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

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

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. |

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 |

15. | P. Saari and M. Ratsep, “Evidence of X-Shaped Propagation-Invariant Localized Light Waves,” Phys. Rev. Lett. |

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

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

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

19. | A.E. Siegman, |

20. | I.S. Gradshteyn and I.M. Ryzhik, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(320.5540) Ultrafast optics : Pulse shaping

(320.5550) Ultrafast optics : Pulses

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 3, 2004

Revised Manuscript: March 1, 2004

Published: March 8, 2004

**Citation**

Stefano Longhi, "Gaussian pulsed beams with arbitrary speed," Opt. Express **12**, 935-940 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-935

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### References

- J.N. Brittingham, ???Focus waves modes in homogeneous Maxwell???s equations: transverse electric mode,??? J. Appl. Phys. 54, 1179???1189 (1983). [CrossRef]
- R.W. Ziolkowski, ???Exact solutions of the wave equation with complex source locations,??? J. Math. Phys. 26, 861???863 (1985). [CrossRef]
- P.A. Belanger, ???Packetlike solutions of the homogeneous-wave equation,??? J. Opt. Soc. Am. A 1, 723???724 (1984). [CrossRef]
- P.A. Belanger, ???Lorentz transformation of packetlike solutions of the homogeneous-wave equation,??? J . Opt. Soc. Am. A 3, 541???542 (1986). [CrossRef]
- R.W. Ziolkowski, ???Localized transmission of electromagnetic energy,??? Phys. Rev. A 39, 2005???2033 (1989). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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. 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).
- 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]
- P. Saari and M. Ratsep, ???Evidence of X-Shaped Propagation-Invariant Localized Light Waves,??? Phys. Rev. Lett. 79, 4135???4138 (1997). [CrossRef]
- M.A. Porras, ???Ultrashort pulsed Gaussian light beams,??? Phys. Rev. E 58, 1086???1093 (1998). [CrossRef]
- M.A. Porras, ???Nonsinusoidal few-cycle pulsed light beams in free space,??? J. Opt. Soc. Am. B 16, 1468???1474 (1999). [CrossRef]
- S. Longhi, ???Spatial-temporal Gauss-Laguerre waves in dispersive media,??? Phys. Rev. E 68, 066612 1???6 (2003). [CrossRef]
- 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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