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
Since many years there has been a continuous interest in the existence of exact spatial-temporal localized packetlike solutions to the scalar or vectorial wave equations and in their finite-energy realizations (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and reference therein). Such an interest has increased in recent years especially after the experimental observations in acoustics  and optics  of non-diffracting and non-dispersive waves. These studies were initiated by Brittingham  and Ziolkowski  who introduced a particular three-dimensional packetlike solution of the homogeneous Maxwell’s equations in the form of a moving modified Gaussian beam which was called focus wave mode (FWM). It consists of an envelope, with a transverse Gaussian profile and algebraic longitudinal localization, traveling undistorted along the positive 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, 10], Bessel-Gauss modified pulses  and focused single-cycle electromagnetic pulses . Ultrashort pulsed Gaussian light beams have been also introduced as nonseparable solutions of the paraxial wave equation in order to represent few-cycle ultrashort light beams with a nearly Gaussian cross section and propagating in free-space at a luminal velocity [16, 17]. Finally, using a generalized bidirectional representation, FWM-like solutions traveling undistorted (or relatively undistorted) at a superluminal velocity, with either infinite or finite energy, were recently introduced [10, 13]. The close connection between FWM solutions to the scalar wave equation and the monochromatic Gaussian beam solutions of the diffractive paraxial wave equation was pointed out by Belanger [3, 3], who showed that, in terms of the bidirectional 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 . By considering wave propagation in a dispersive medium instead of vacuum, 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 .
In this work it is shown that Gaussian-like envelope beams moving at an 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, 13], such modified moving Gaussian beams are composed solely by forward traveling-wave components. Gaussian beams moving at a subluminal velocity 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
The starting point of our analysis is provided by the scalar wave equation for the field ψ(x,y,z,t) in free space:
where =∂ 2/∂x 2+∂ 2/∂y 2 is the transverse Laplacian. By extending the Ansatz originally proposed by Belanger [3, 4], we search for a solution of the scalar wave equation in the form of an envelope Φ, propagating undistorted along the z direction with a velocity v, modulated by a plane wave of frequency ω, i.e. we set:
where k=ω/c is the wavenumber of the carrier plane wave. Usual FWM solutions are retrieved in the case v=-c (see [4, 6]). In this case the form of Eq. (3) reduces 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 |∂ 2Φ/∂ξ2|≪2k|∂Φ/∂ξ| (see, for instance, ). 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:
where: ρ=(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 2k. Within this approximation we may hence assume:
In terms of the original envelope equation (3), the extended paraxial approximation corresponds to neglecting the second-order derivative term of Φ with respect to ξ, i.e., Eq. (3) reduces to the form:
Localized solutions to Eq. (7) with axial symmetry are given by Gauss-Laguerre modes:
where n=0,1,2, …, is the generalized Laguerre polynomial of order n, ξ0 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 
for v < c, and:
for 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 Φ=0, i.e., Φ is an harmonic function with respect to the transverse 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).
As a final remark it should be noted that, as for ordinary FWMs and stationary (non-moving) Gaussian beams, the Gauss-Laguerre moving beams, expressed by Eqs. (2) and (8), have an infinite energy content. A finite energy pulsed solution, i.e. for which ∫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:
Note that the integral entering in Eq. (11) has the form of a Laplace transform. Analogously to superpositions of FWMs, one can show that the resulting pulsed solution ψ(ρ,z,t) has finite energy if at least G(ω)/√ω is square integrable (see Appendix B of ). Typically one can consider a spectral amplitude G(ω) which is nonvanishing in a small interval Δω around a carrier frequency ω 0 (Δω≪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 :
(α>0) which yields the finite-energy pulsed solution:
Note that this solution differs from the simple “monochromatic” solution, given by Eqs. (2) and (8) with 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:
where 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 kz =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
The existence of both subluminal and superluminal moving Gaussian beams can be physically understood by exploiting the invariance of the non-paraxial scalar wave equation (1) under a Lorentz transformation (see, e.g., [4, 10]). Indeed, let us introduce an inertial moving reference frame (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
where we have set 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:
i.e., one observes a standard monochromatic non-moving Gaussian beam, with a frequency-shifted carrier wave and a modified Rayleigh range due to relativistic space contraction. It is worth observing that, in the moving reference frame, one can write 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:
We may call such a solution a “temporal” diffracting Gaussian beam, in the sense that the role of longitudinal spatial variable in the ordinary monochromatic Gaussian beam, which accounts for spatial diffraction, is replaced here by the temporal variable, yielding a sort of “temporal” beam diffraction instead of spatial (longitudinal) diffraction. Indeed, one can write in this case 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 ∑-(1/c 2)∂ 2∑/∂t′2=k′2∑, which in the paraxial approximation reduces again to a two-dimensional Scrödinger equation for the envelope Φ.
In conclusion, we have shown that, under a paraxial approximation which extends to the non-monochromatic regime the usual one used in the paraxial diffraction theory, the scalar wave equation admits of moving Gaussian beam solutions with an arbitrary speed, either subluminal or superluminal. These solutions appear, in a moving reference frame, either as an ordinary stationary monochromatic Gaussian beam or as a “temporal diffracting” Gaussian beam.
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