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

In recent years, there has been increasing interest in novel classes of spatio-temporally localized solutions to various hyperbolic equations governing acoustic, electromagnetic and quantum mechanical wave phenomena. The bulk of the research along these lines has been performed in connection to the basic formulation, generation, propagation, guidance, scattering and diffraction properties of electromagnetic and acoustic localized waves (LWs) in free space (see [16] for pertinent review literature). However, some work has been done in the area of propagation of localized waves in dispersive (see [7] and references therein) and nonlinear (see [8] and references therein) media. This interest has been sustained by advancements in ultrafast acoustical, optical and electrical devices capable of generating and shaping very short pulsed wave fields (see, e.g., [9]). Localized wave pulses exhibit distinct advantages in their performance by comparison to conventional quasi-monochromatic signals. It has been shown, in particular, that such pulses have extended ranges of localization in the near-to-far field regions. These properties render LW fields very useful in diverse physical applications, such as remote sensing, secure signaling, nondestructive testing, ultrafast microscopy, high resolution imaging, tissue characterization and photodynamic therapy.

There exist physical situations where a paraxial approximation to the scalar wave equation is pertinent. In this paper, a systematic approach to deriving paraxial spatio-temporally localized waves in free space is provided. Two distinct classes of such packet-like solutions are identified. The first class, which is based on a narrow angular spectrum assumption, is discussed in Section 2. The second one, based on both a narrow angular spectrum and a narrowband temporal spectrum approximation, is described in Section 3. Both classes incorporate subluminal, luminal and superluminal paraxial localized waves. For the second class, the subluminal and superluminal paraxial localized waves are shown in Section 4 to arise from subluminal and superluminal Lorentz boosts of two distinct types of general luminal solutions. Finally, a situation is addressed in Section 5, whereby exact localized wave solutions to the scalar wave equation are embedded into approximate paraxial solutions. Concluding remarks are made in Section 6.

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

The conventional paraxial approximation to a solution of the free-space homogeneous 3D Helmholtz equation

(Δρ2+2z2+ω2c2)û(r,ω)=0;ρ=(x,y),

viz., û±(r⃗,ω) exp[±i(ω/c)z]±(r⃗,ω) with ±(r⃗,ω) governed by the complex parabolic equations

izν̂±(r,ω)=±c2ω2ν̂±(r,ω),

is based on the assumption of a narrow angular spectrum with respect to the 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 ±(r⃗,t) can be expressed in terms of the Fourier spectral representations

u±(ρ,τ±,z)=R1dωR2dκexp[i(ωτ±κ·ρ)exp[±i(cκ2z)/(2ω)]u˜0(κ,ω),

where τ ±=tz/c,κ⃗(kx,ky ) and κ=|K⃗|. These representations allow one to determine the equations governing u ±(ρ⃗,τ±,z); specifically,

(ρ22c2τ±z)u±(ρ,τ±,z)=0.

These relations are known as the forward and backward pulsed beam equations [10]. The equation for u +(ρ⃗,τ,z) has been used extensively recently (cf., e.g., [11, 12]), especially in connection with ultra-wideband (few-cycle) signals.

2.1 Paraxial luminal pulsed beams

A solution to the forward pulsed beam equation is assumed as follows:

u+(ρ,τ+,z;α)=exp(i2αcτ+)ν+(ρ,z;α).

Then, v +(ρ⃗,z;α) obeys the Schrödinger-like equation

i4αzv+(ρ,z;α)=ρ2v+(ρ,z;α),

which has a large number of known solutions. Among them are the Hermite-Gauss, the Laguerre-Gauss and Bessel-Gauss beams. A specific class of axisymmetric Laguerre-Gauss beams is given as follows:

νn(ρ,z;α)=A0a1(a1+iz)n+1exp(αρ2a1+iz)Ln(0)(αρ2a1+iz);n=0,1,2,

Here, ρ=ρ=x2+y2, a 1 is a free positive parameter and Ln(0) (·) denotes the 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,

u+(ρ,z,τ+)=0dαexp(2iαcτ+)vn(ρ,z;α)F˜(α).

As a particular example, let (α)=exp(-2α 2 αc), a 2 a being a real positive parameter. Then, from Erdelyi (cf., Ref. [13], p. 174), one obtains

u+(ρ,z,τ+)=A0a1(a1+iz)n+1Γ(n+1)n![2c(a2+iτ+)]n[2c(a2+iτ+)+ρ2/(a1+iz)]n+1.

As this finite-energy pulsed beam propagates in the positive 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 zς/2; ςz+ct, it becomes the nth order splash mode [1], which, in turn, belongs to the class of 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]. Special attention has been paid to the limiting case 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] recently.

Luminal pulsed beams analogous to those in Eq. (2.8) can be found for u-(ρ,z,τ-); however, these wavepackets propagate in the negative z – direction.

2.2 Paraxial superluminal localized waves

It will be more convenient in the subsequent discussion of paraxial localized waves to recast equations (2.4) into new, but equivalent, forms, viz.,

t(z±1ct)u±(ρ,z,t)=±c2ρ2u±(ρ,z,t).

The following change of variables is undertaken in the equation for u (ρ⃗,z,t) : ς +z-ct,η+=z-vt;v>c. One, then, obtains the transformed equation

2(vc1)2ζ+η+u+(ρ,ζ+,η+)2vc(vc1)2η+u+(ρ,ζ+,η+)=ρ2u+(ρ,ζ+,η+).

An elementary solution is chosen, next, in the form

u+(e)(ρ,ζ+,η+;α,β,κ)=exp(iκ·ρ)exp(iαζ+)exp(iβη+),

where α and β are real positive free parameters with units of m -1. Substitution into Eq. (2.11) leads to the dispersion relation

κ2=2vc(vc1)β2+2(vc1)αβ.

A general solution to the forward pulsed beam equation can be written as

u+(ρ,z,t)=0dα0dβR2dκu+(e)(ρ,ζ+,η+;α,β,κ)
×δ[κ22vc(vc1)β22(vc1)αβ]u˜0(α,β,κ),

where 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

u+(ρ,z,t)=0dα0dβexp[i(αζ++βη+)]J0[ρ2vc(vc1)β2+cvαβ]u˜1(α,β),

where J 0(·) is the zero-order ordinary Bessel function. If ũ1(α,β)=exp(-α1,β)(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

u+(ρ,z,t)=0dαv+(ρ,z,t;α)F˜(α),

where

v+(ρ,z,t,α)=exp[iα(1c2v)(zvpht)][2(vc)(vc1)ρ2+(a1+i(zvt))2]1/2
×exp[cα2v2(vc)(vc1)ρ2+(α1+i(zvt))2],

with vph =v/[2(v/c)-1]. It should be noted that vphc/2 as v→∞. The solution given in Eq. (2.17) is the paraxial version of the focus X wave (FXW) (cf. Ref. [3]). If, in Eq. (2.16), one chooses the singular spectrum (α)=δ(α), one obtains the paraxial version of the infinite-energy zero-order X wave [16,17]; specifically,

u+(ρ,z,t)={2(v/c)[(v/c)1]ρ2+(a1+i(zvt))2}1/2.

On the other hand, a smooth spectrum, e.g.,

F˜(α)={0,b>α>0,1Γ(q)(αb)q1exp[a2(αb)],αb;b,q0,

results in the paraxial version of the finite-energy modified focus X wave (MFXW) pulse [3]

u+(ρ,z,t)=[2(v/c)[(v/c)1]ρ2+(a1+i(zvt))2]1/2exp(ibλ)[c/(2v)2(v/c)[(v/c)1]ρ2+(a1+i(zvt))2+(a2iλ)]q
×exp{(bc)/(2v)2(v/c)[(v/c)1]ρ2+(a1+i(zvt))2},

where λ≡[1-c/(2v)](z-vpht). It should be noted that although v +(ρ,z,t;α) in Eq. (2.17) is unidirectional, this is not necessarily true of u +(ρ,z,t) in Eq. (2.20); the latter may contain both forward and backward propagating components. As in the case of the exact MFXW solution to the scalar wave equation, superluminality in the paraxial version given in Eq. (2.20) does not contradict relativity theory. If the parameters are chosen so that u + (ρ,z,t) contains mostly forward propagating components, the pulse moves superluminally with almost no distortion up to a certain distance zd , and then slows down to a luminal speed c, with significant accompanying distortion. Although the peak of the pulse does move superluminally up to d z , it is not causally related at two distinct ranges z 1,z 2 ∈ [0,zd ). Thus, no information can be transferred superluminally from z 1 to z 2. The physical significance of u + (ρ,z,t) is due to its spatio-temporal localization.

2.3 Paraxial subluminal localized waves

For v<c, the dispersion relation in Eq. (2.13) can be recast into the form

κ2=2vc(1vc)[(αc2v)2β¯2];β¯β+αc2v.

Then, a general axisymmetric solution to Eq. (2.10) can be written as follows:

u+(ρ,ζ+,η+)=0dα(αc)/(2v)dβ¯J0{ρ[2vc(1vc)]1/2[(αc2v)2β¯2]1/2}
×exp(iαζ+)exp(iβ¯η+)exp[iη+(αc)/(2v)]u˜2(α,β¯).

A particular solution is given by

u+(ρ,z,t)=0dαsin[(αc)/(2v)2(v/c)[1(v/c)]ρ2+(zvt)2]2(v/c)[1(v/c)]ρ2+(zvt)2
×exp{iα[1c/(2v)](zvpht)}F˜(α),

where vph =v/[2(c/v)-1]. The following should be noted: [0,) vph →[0,c/2) and vph →(∞,c] as v→(c/2,c].The solution in Eq. (2.23) consists of a superposition of paraxial MacKinnon-type wavepackets (cf. Ref. [3]). Finite-energy solutions can be obtained by choosing smooth spectra a (α).

2.4 Localized waves for u-(ρ⃗,z,t)

The following change of variables is undertaken in the equation for u-(ρ⃗,z,t) in Eq. (2.10): ς-=z+ct,η +=z-vt. Then, one obtains

2(1+vc)2ζη+u(ρ,ζ,η+)2vc(1+vc)2η+2u(ρ,ζ,η+)=ρ2u(ρ,ζ,η+).

An elementary solution is chosen, next, of the form

u(e)(ρ,ζ,η+;α,β,κ)=exp(iκ·ρ)exp(iαζ)exp(iβη+),

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

κ2=2vc(1+vc)β2+2(1+vc)αβ,

A general solution can be written as

u(ρ,z,t)=0dα0dβR2dκu(e)(ρ,ζ,η+;,α,β,κ)
×δ[κ22vc(vc1)β22(vc1)αβ]u˜0(α,β,κ),

For an azimuthally independent spectrum, viz., ũ0(α,β,κ⃗)=ũ0(α,β,κ), one obtains, in particular, the axisymmetric solution

u(ρ,z,t)=0dα0dβexp[i(αζβη+)]J0[ρ2vc(1+vc)β2+cvαβ]u˜1(α,β).

The integrand is a 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 ũ1(α,β) in Eq. (2.28) equals exp(-a 1 β)(a), a 1 being a positive parameter, the integration over β can be carried out explicitly [13]. As a result, one has

u(ρ,z,t)=0dαv(ρ,z,t;α)F˜(α),

where

v(ρ,z,t;α)=exp[iα(1+c2v)(z+vpht)][2vc(1+vc)ρ2+(a1+i(zvt))2]1/2
×exp{(cα)/(2v)2(v/c)[(1+v/c)]ρ2+(a1+i(zvt))2},

with vph =c(2+c/v).

An interesting property of the localized solution in Eq. (2.29) is that it is valid for 0<v<∞. If v=c, for example, with (α)=d(α-α 0), one obtains the “luminal FXW”

u(ρ,z,t)=exp{i(3/2)α0[z+(c/3)t]}[4ρ2+(a1+i(zct))2]1/2
×exp{α0/(2c)4ρ2+(a1+i(zct))2},

and for (α)=δ(α), the “luminal zero-order X wave,” viz.,

u(ρ,z,t)=[4ρ2+(a1+i(zct))2]1/2

Finite-energy paraxial localized waves can be determined by using smooth spectra (α) in Eq. (2.29).

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

The function ũ0(κ⃗,ω)≡ũ1(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)/(2ω)is expanded in a Taylor series around ω=ω 0 and only the first term in the expansion is retained; i.e.,

β(κ,ω)β(κ,ω0)=(cκ2)/(2ω0).

Within the framework of this additional approximation the expressions in Eq. (2.3) assume the forms

ψ±(ρ,z,t)exp(iω0τ±)R1dΩR2dκexp(iΩτ±)exp(iκ·ρ)
×exp[±i(cκ2z)/(2ω0)]u˜1(κ,Ω),

where Ω=ω-ω 0, or

ψ±(ρ,z,t)=exp(iω0τ±)ϕ±(ρ,z,t),

where ϕ ±(ρ⃗z,t) governed by the equations

i(z±1ct)ϕ±(ρ,z,t)=±12k0ρ2ϕ±(ρ,z,t);k0ω0/c.

3.1 Subluminal and superluminal pulsed beams

A solution to Eq. (3.4) for ϕ+ (ρ⃗,z,t) is assumed of the form

ϕ+(ρ,z,t)=f(τ+)Φ(ρ,η+),

where η +=z-vt, vc, and f(τ +) is an arbitrary function (at least differentiable). It follows, then, that the wave function Φ(ρ⃗,η +) obeys the equation

i(1vc)η+Φ(ρ,η+)=12k0ρ2Φ(ρ,η+).

Thus, a narrow angular spectrum and a narrowband temporal spectrum result in the following approximate nonluminal solution to the homogeneous 3D scalar wave equation:

ψ+(ρ,τ+,η+)=exp(iω0τ+)f(τ+)Φ(ρ,η+),

This general solution was originally reported by Wunsche [18] and, independently, by Besieris 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.

It will be convenient for the discussion in the sequel to introduce new variables as follows: σ ±=2η +/(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

i4σ±Φ(ρ,σ±)=ρ2Φ(ρ,σ±),
ψ+(ρ,τ+,σ±)=exp(iω0τ+)f(τ+)Φ(ρ,σ±),

respectively. Cited below are specific examples of superluminal/subluminal pulsed beams based on three distinct classes of solutions of Eq. (3.8).

Hermite-Gauss pulsed beams:

ψ+(mn)(x,y,τ+,σ±)=exp(iω0τ+)f(τ+)exp[x2/(γ1+iσ±)]exp[y2/(γ2+iσ±)](γ1+iσ±)(m+1)/2(γ2+iσ±)(n+1)/2
×Hm(x/γ1+iσ±)Hn(y/γ2+iσ±).

Here, γ 1,2 are free positive parameters and Hm (·) denotes the mth order Hermite polynomial.

Axisymmetric Laguerre-Gauss pulsed beams:

ψ+(n)(ρ,τ+,σ±)=exp(iω0τ+)f(τ+)γ0(γ0+iσ±)n+1exp(ρ2γ0+iσ±)Ln(0)[ρ2(γ0+iσ±)].

Here, γ 0 is a free positive parameter and Ln(0) (·) denotes the nth order Laguerre polynomial.

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

ψ+(0)(ρ,τ+,σ±)=exp(iω0τ+)f(τ+)γ0γ0+iσ±exp[ρ2(γ0+iσ±)].

With γ 0=2a/[k 0|(v/c)-1|], a being a real positive parameter, this solution can be rewritten as

ψ+(0)(ρ,τ+,η+)=exp(iω0τ+)f(τ+)aa±iη+exp(ω02cvc1ρ2a±iη+).

It should be noted that the factor in ψ+(0) (ρ,τ ++) multiplying exp( 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 ψ+(0) (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

f(τ+)=exp(τ+24T2)=exp{14T2[(tzv)(vcvc)z]2}

can be used to achieve this goal for values of the speed v close to c and a large values of ω 0 so that ω 0|(v/c)-1/c=O(1).

Axisymmetric Bessel-Gauss pulsed beam:

ψ+(ρ,τ+,σ±)=exp(iω0τ+)f(τ+)γ0γ0+iσ±J0(γ0k0ρsinθγ0+iσ±)exp(ρ2γ0+iσ±)
×exp[iσ±4(k02γ0sin2θ)/(γ0+iσ±)].

Here, 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

A solution to Eq. (3.4) for ϕ_ (ρ ⃗,z,t) is assumed of the form

ϕ+(ρ,τ+,σz)=f(τ+)Φ(ρ,σz),

where σ- z =-(2z)/k 0. It follows, then, that the wave function Φ(ρ⃗,σ-z ) obeys the Schrödinger equation

i4σzΦ(ρ,σz)=ρ2Φ(ρ,σz).

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

ψ+(ρ,τ+,σz)=exp(iω0τ+)f(τ+)Φ(ρ,σz).

The Hermite-Gauss solutions in Eq. (3.10), the Laguerre-Gauss solutions in Eq. (3.11) and the Bessel-Gauss solution in Eq. (3.15) are still applicable; however, σ ± must be replaced by σ- z =-(2z)/k 0 for the luminal case under consideration.

It is important to discuss the basic differences between the luminal solutions to the pulsed beam equation [cf. Sec. 1], which are based on the narrow angular spectrum approximation, and those given in Eq. (3.18). A particularly simple example of the former is the monochromatic Gaussian beam

u+(ρ,z;α)=exp(2iαcτ+)a1(a1+iz)exp(αρ2a1+iz),

where a 1 is a real positive parameter and α is an arbitrary real positive quantity. A superposition over the latter, e.g.,

u+(ρ,z,t)=1π0dαF˜(α)exp(2iαcτ+)a1(a1+iz)exp(αρ2a1+iz),

yields the forward pulsed beam solution

u+(ρ,z,t)=a1(a1+iz)f̂(tz/ci12cρ2a1+iz),

where (t) denotes the complex analytic signal associated with the spectrum (ω).

A particularly simple example of a luminal pulsed beam based on a narrow angular spectrum and a narrowband temporal spectrum is the following:

ψ+(ρ,z,t)=exp[iω0(tz/c)]f(tz/c)γ0γ0i2z/k0exp(ρ2γ0i2z/k0).

It consists of a product of two factors; a plane wave modulated by a longitudinal envelope function traveling along the 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)

A solution to Eq. (3.4) for ϕ- (ρ⃗,z,t) is assumed of the form

ϕ(ρ,z,t)=f(τ)Φ(ρ,η+),

where f (τ-) is an arbitrary function (at least differentiable). It follows, then, that the wave function Φ(ρ⃗,η +) obeys the equation

i(1+vc)η+Φ(ρ,η+)=12k0ρ2Φ(ρ,η+).

It is convenient to introduce a new variable as follows: σ̄+=2η +/[k 0 (1+v/c)]. Then Eq. (3.24) changes to

i4σ¯+Φ(ρ,σ¯+)=ρ2Φ(ρ,σ¯+).

Thus, a narrow angular spectrum and a narrowband temporal spectrum approximation result in the solution

ψ(ρ,τ,σ¯+)=exp(iω0τ)f(τ)Φ(ρ,σ¯+).

By construction, this solution is valid for 0≤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( 0 τ-)f(τ-) in Eq. (3.26) travels in the opposite direction. For v=0, one obtains

ψ(ρ,τ,σz+)=exp(iω0τ)f(τ)Φ(ρ,σz+);σz+σz=2z/k0,

an expression dual to that for ψ +(ρ⃗,τ,σ-z) in Eq. (3.18).

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

4.1 Subluminal boosts

It should be noted that Eq. (3.4) is Lorentz invariant. Specifically, under the subluminal Lorentz transformations x=x′,y=y′,z=γ¯ (z′+vt′),t=γ¯ [t′+(v/c2)z′], where v<c and γ¯=1/1(v/c)2 the solution ψ ±(ρ⃗,z,t)=exp ( 0 τ ±) ϕ ±(ρ⃗,z,t) [cf. Eq. (3.3)] transforms to ψ ± (ρ⃗,z′,t′)=exp [ 0 γ¯ (1+v/c)τ′±] ϕ ± (ρ⃗,z,t), where τ±=t′∓z′/c and

i(z±1ct)ϕ±(ρ,z,t)=±12k±ρ2ϕ±(ρ,z,t);k±γ¯(1v/c)k0.

Consider, next, the following general luminal solutions:

ψ±(ρ,τ±,σ¯z)=exp[iω0γ¯(1vc)τ±]f[γ¯(1vc)τ±]Φ(ρ,σ¯z);
τ±tzc;σ¯z2zk±,

where Φ(ρ⃗,σ¯ -z′) satisfies the parabolic equation (3.17) with the interchange zz′, and (ρ⃗,σ¯ +z) is governed by an analogous equation. Application of the inverse boosting x′=x,y′=y,zγ¯ =(z-vt),ct′=-γ¯ (v/c)[z-(c2/v)t] to Eq. (4.2) yields the general paraxial subluminal solutions [cf. Eqs. (3.9) and (3.26)]

ψ+(ρ,τ+,σ)=exp(iω0τ+)f(τ+)Φ(ρ,σ)

and

ψ(ρ,τ,σ¯+)=exp(iω0τ)f(τ)Φ(ρ,σ¯+).

This observation has also been made by Longhi [19] for ψ +(ρ⃗,τ′,σ-z′), except for the additional function f[(γ̄(1-v/c)τ+] appearing in Eq. (4.2).

4.2 Superluminal boosts

An interesting question is the following: Are there general luminal solutions to Eq. (3.4) which become the general paraxial superluminal solutions given in Eqs. (3.9) and (3.26) after a Lorentz transformation? In order to answer this question, we seek solutions to Eq. (3.4) of the form

ϕ±(ρ,ς±,t)=g(ς±)Ψ(ρ,t),

where, as defined earlier, ς ±=zct. It follows, then, that the wave function Ψ(ρ⃗,t) obeys the Schrödinger equation

i4σtΨ(ρ,σt)=ρ2Ψ(ρ,σt),

where σt =∓(2ct)/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:

ψ±(ρ,ς,σt)=exp(ik0ς±)g(ς±)Ψ(ρ,σt)

Simple examples of such solutions are the following:

ψ±(ρ,z,t)=exp[ik0(zct)]g(zct)γ0γ0i(2ct)/k0exp(ρ2γ0i(2ct)/k0).

Equation (3.4) is Lorentz invariant. More specifically, under the generalized (superluminal) Lorentz transformation x=x′,y=y′,z=γ(v/c)[z′+(c 2/v)t′],t=γ(z′+vt′)/c, where γ=1/(v/c)21 and v>c, the solution ψ ± (ρ⃗,z,t)=exp ( 0 τ ±) ϕ ± (ρ⃗,z,t) [cf. Eq. (3.3)] transforms to ψ ±(ρ⃗,z′,t′) exp {-ik 0γ[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 ±±≡γ[(v/c)]∓1]k 0. Consider, next, the general luminal solutions

ψ±(ρ,ς±,σt)=exp[ik0γ(vc1)ς±]g[γ(vc1)ς±]Ψ(ρ,σt);
ς±zct;σt(2ct)/k¯±,

where Ψ(ρ⃗,σ∓t ′) satisfies Eq. (4.6) with tt′. 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)]

ψ+(ρ,ς+,σ+)=exp(ik0ς+)g(ς+)Ψ(ρ,σ+)

and

ψ(ρ,ς,σ¯+)=exp(ik0ς)g(ς)Ψ(ρ,σ¯+)

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

The change of variables ς=z-ct,z-=z+ct is introduced in Eq. (3.4). As a consequence one obtains the Schrödinger equations

i4χϕ±(ρ,χ)=ρ2ϕ±(ρ,χ),

with the definition 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+(ρ,z,t)=exp(ik0ς+)ϕ+(ρ,χ)=exp[ik0(zct)]ϕ+[ρ,(z+ct)/k0].

and

ψ(ρ,z,t)=exp(ik0ς)ϕ(ρ,χ+)=exp[ik0(z+ct)]ϕ[ρ,(z+ct)/k0].

The simplest such solutions are the following:

ψ+(ρ,z,t)=k0k0i(z+ct)exp[ik0(zct)]exp[k0ρ2k0i(z+ct)],
ψ(ρ,z,t)=k0k0+i(zct)exp[ik0(z+ct)]exp[k0ρ2k0+i(zct)].

But 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] previously.

It is possible to provide a more physical explanation for the “peculiarity” described above. Consider, for example, the solution given in Eq. (4.4), viz.,

ψ(ρ,τ,σ¯+)=exp(iω0τ)f(τ)Φ(ρ,σ¯+);τ=t+zcz,σ¯+=2(zvt)k0(1+v/c).

With f (τ-) constant -=and v=c, this expression simplifies to

ψ(ρ,z,t)=exp[ik0(z+ct)]Φ(ρ,zctk0).

Since Φ(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:

ψ(ρ,z,t)=k0a+i(zct)exp[ik0(z+ct)]exp[k0ρ2a+i(zct)],a>0.

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)]

ψ(ρ,τ,σ¯z+)=1a/k0+iσ¯z+exp[iω0γ¯(1vc)τ]exp[ρ2a/k0+iσ¯z+]
τ=t+zc;σ¯z+=2zk,k=γ¯(1+v/c)k0.

The procedure is analogous to the one followed by Be′langer [21] who showed that certain Gaussian packet-like solutions to the homogeneous scalar wave equation could be explained as monochromatic Gaussian beams observed in a another inertial frame.

6. Concluding remarks

A systematic approach to deriving paraxial spatio-temporally localized waves has been introduced. Two distinct classes of such pulsed waves have been studied in detail. The first category deals with paraxial localized waves based on a narrow angular spectrum assumption. The second class is more restricted because it is based on both a narrow angular spectrum and a narrowband temporal spectrum approximation. Both classes allow subluminal, luminal and superluminal paraxial localized waves. For the second class, however, the subluminal and superluminal paraxial localized waves have been shown to arise from subluminal and superluminal Lorentz boosts of two types of general luminal solutions. Finally, the situation has been addressed, whereby exact localized wave solutions to the scalar wave equation are embedded into approximate paraxial solutions.

Appendix A

The integrand in Eq. (2.28) may be rewritten as

Bp(ρ,z,t)=exp[i(βα)(zvpht)]J0[ρ2vc(1+vc)β2+cvαβ],

where vph =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

Be(ρ,z,t)=exp[ikz(zωkzt)]J0[ρ(ω/c)2kz2].

In order for the argument of the Bessel function in the latter to be real, one must have the inequality |ω|/|kz |>c. Assuming ω and kz to be positive, this means that the exact Bessel beam propagates along the +z-direction with the superluminal speed v=ω/kz . The situation is much different in Eq. (A-1) Three distinct cases will be considered in detail:

Case (i): a=0,β>0

In this case, the paraxial Bessel beam simplifies as follows:

Bp(ρ,z,t)=exp[iβ(zvt)]J0[ρβ2vc(1+vc)]

It propagates in the positive z – direction at any speed v∈(0,∞).

Case (ii):a>0,β>0

Let β=µ(c/v)α;µ>0. Then, vph =c(1+µ)/[(µ/δ)-1]; δv/c>0. For vph >0, the inequality δ<µ must hold. With these restrictions, one finds that vph in Eq. (A-1) is subluminal, luminal or superluminal if δ<,=,>µ/(2+µ), respectively.

Case (iii): 0, a<, β>(c/v)a

Let β=µ(c/v)a ;µ>1. Then, vph =c(-1+µ)/[(µ/δ)+1. In this case, vph 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 39, 2005–2033 (1989). [CrossRef]   [PubMed]  

2. 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]  

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]  

4. E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998). [CrossRef]  

5. 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]  

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).

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]  

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

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]  

13. A. Erdelyi, Tables of Integral Transforms (Academic Press, New York, 1980), Vol. I.

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]  

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]  

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]  

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]  

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 , 12, 935–940 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-935. [CrossRef]   [PubMed]  

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]  

References

  • View by:

  1. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [Crossref] [PubMed]
  2. 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]
  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]
  4. E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998).
    [Crossref]
  5. 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]
  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).
  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]
  10. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [Crossref]
  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]
  13. A. Erdelyi, Tables of Integral Transforms (Academic Press, New York, 1980), Vol. I.
  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]
  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]
  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]
  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]
  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,  12, 935–940 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-935.
    [Crossref] [PubMed]
  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]

2004 (2)

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, “Gaussian pulsed beams with arbitrary speeds,” Opt. Express,  12, 935–940 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-935.
[Crossref] [PubMed]

2003 (3)

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

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]

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]

2001 (1)

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)

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

1998 (3)

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

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]

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998).
[Crossref]

1994 (1)

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

1993 (1)

1992 (1)

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]

1991 (1)

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]

1989 (2)

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

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]

1986 (1)

Abdel-Rahman, M.

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]

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).

Be'langer, P. A.

Besieris, I. M.

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]

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]

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]

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).

Chatzipetros, A.

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]

Conti, C.

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]

Di Trapani, P.

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]

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (Academic Press, New York, 1980), Vol. I.

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. M.

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]

Fortin, M.

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]

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]

Greenleaf, J. F.

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]

Griebner, U.

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]

Grunwald, R.

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]

Hellwarth, R. W.

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]

Heyman, E.

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

Jedrkiewicz, O.

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]

Kebbel, V.

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]

Kummrow, A.

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]

Longhi, S.

Lu, J. Y.

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]

Neumann, U.

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]

Nibbering, E. T. J.

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]

Piche, M.

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]

Piskarskas, A.

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]

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]

Recami, E.

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998).
[Crossref]

Reivelt, K.

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).

Rini, M.

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]

Rousseau, G.

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]

Saari, P.

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).

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.
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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).
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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).
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Shaarawi, A. M.

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).
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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).
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R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
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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).

Trillo, S.

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).
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Trull, J.

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).
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Valiulis, G.

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).
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Winful, H. G.

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).
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Wunsche, A.

Ziolkowski, R. W.

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).
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R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
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R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
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IEEE Trans. Antennas Propag. (1)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
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IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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).
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J. Opt. Soc. Am. A (3)

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

Opt. Express (2)

Phys. Rev. A (2)

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 (5)

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. (1)

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 (1)

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (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, 1371–1378 (1991).
[Crossref]

Progr. Electromagn. Res. (PIER) (1)

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 (2)

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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