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Finite-energy spatiotemporally localized Airy wavepackets

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Abstract

The time-diffraction technique introduced by Porras recently is motivated in this article in terms of the Lorentz invariance of the equation governing the narrow angular spectrum and narrowband temporal spectrum paraxial approximation and it is used to derive finite-energy spatiotemporally confined subluminal, luminal and superluminal Airy wave packets. In addition, a novel exact finite-energy luminal Airy splash mode-type solution to the scalar wave equation is derived using Bateman’s conformal transformation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Luminal, or focus wave modes (FWM), superluminal, or X waves (XWs), and subluminal spatiotemporally localized solutions to various hyperbolic equations governing acoustic, electromagnetic and quantum wave phenomena have been studied intensively during the past few years (see [1–12] for pertinent literature). Further details can be found in two recent edited monographs on the subject [13,14]. In general, both linear and nonlinear LW pulses exhibit distinct advantages in comparison to conventional quasi-monochromatic signals. Their spatiotemporal confinement and extended field depths render them very useful in diverse physical applications.

There exist physical situations where a paraxial approximation is appropriate. The conventional paraxial approximation to a solution of the free-space 2D Helmholtz equation

(2x2+2z2+ω2c2)u^(x,z,ω)=0,
viz., u^(x,z,ω)=exp[i(ω/c)z]v^±(x,z,ω),with v^±(x,z,ω)governed by the complex parabolic equations
izv^±(x,,z,ω)=±c2ω2x2v^±(x,z,ω),
is based on a narrow angular spectrum with respect to the zaxis. In Eqs. (1) and (2), cis the speed of light in vacuum and ω denotes the angular frequency. The space-time paraxial solutions u±(x,z,t) can be expressed in terms of the Fourier representations
u±(x,z,t)=R1dωR1dkxexp[i(ωτ±kxx)]exp[±i(ckx2z)/(2ω)ψ˜0(kx,ω)],
where τ±=tz/c. These representations lead to the equations governing u±(x,z,t);specifically,
(2x22c2τ±z)u±(x,z,t)=0.
These relations are known as the forward and backward pulsed beam equations [15]. The equation for u+(x,z,t) has been used extensively recently (cf., e.g., [16,17]), especially in connection with ultra-wideband (few-cycle) signals.

A systematic approach for deriving spatiotemporally localized waves to the forward and backward pulsed beam Eqs. (4) has been undertaken in [18]. Finite-energy narrow angular spectrum subluminal, luminal and superluminal wave packets have been derived.

A primary motivation for this article has been the recent work on time diffraction by Porras [19–21]. This framework, based on a narrow angular spectrum and narrowband temporal spectrum approximation, allows one to obtain finite-energy subluminal, luminal and superluminal spatiotemporally confined paraxial wave packets. The method is similar to one expounded by Besieris and Shaarawi [18] and it is intimately related to the Lorentz invariance of the relevant governing equation.

The Airy beam, a remarkable finite-energy solution to the paraxial Eq. (2), was first formulated analytically by Siviloglou and Christodoulides [21] and subsequently demonstrated experimentally by Siviloglou, Broky, Dogariu and Christodoulides [22]. Their work was motivated by the infinite-energy (nonspreading) accelerating Airy solution to the Schrödinger equation introduced by Berry and Balazs [23] in the context of quantum mechanics. The Airy beam is slowly diffracting while bending laterally along a parabolic path even though its centroid is constant, it can perform ballistic dynamics akin to those of projectiles moving under the action of gravity, and it is self-healing, that is, it regenerates when part of the aperture is obstructed; this is due to the reinforcement of the main lobe by the side lobes.

An important question is whether spatiotemporally localized versions of Airy beams are feasible. For luminal solutions, this question has been answered affirmatively by Saari [24], Piksarv, Valmann, Valta-Lukner and Saari [25] and Kaganovsky and Heyman [26].

More recently, however, work on nonluminal spatiotemporally confined Airy wave packets has appeared in the literature [27–29]. Finite-energy versions of such wave packets can arise only within the framework of the narrow angular spectrum and narrowband temporal spectrum paraxial approximation where the time-diffraction method is valid. The Airy solution is intimately related to a parabolic equation. Only for luminal spatiotemporally localized wave solutions to the scalar wave equation can such an association be made. For this reason, subluminal and superluminal spatiotemporally confined Airy solutions to the wave equation do not exist. As a special case, broadband subluminal and superluminal spatiotemporally localized Airy wave packets based only on the narrow angular spectrum paraxial approximation do not exist.

The main purpose in this article is to provide a different motivation for the time-diffraction method and use it to derive finite-energy spatiotemporally localized subluminal, luminal and superluminal paraxial Airy wave packets. Also, a situation will be addressed whereby exact localized wave solutions to the scalar wave equation are embedded into approximate paraxial ones. Finally, a novel exact finite-energy luminal Airy splash mode-type solution to the scalar wave equation will be derived using a conformal transformation technique due to Bateman.

2. Narrow angular spectrum and narrowband temporal spectrum approximation

The spectral function ψ˜0(kx,ω)ψ˜1(kx,ωω0)in Eq. (3) is assumed to have a narrow band around the frequency ω0. Furthermore, the phase β(kx,ω)=(xkx2)/(2ω) is expanded in a Taylor series around ω=ω0 and only the first term in the expansion is retained, i.e., β(kx,ω)(xkx2)/(2ω0).Within the framework of this additional approximation, the expressions in Eq. (3) assume the forms

ψ±(x,z,t)=exp(iω0τ±)R1dΩR1dkxexp(iΩτ±)exp(ikxx)×exp[±i(ckx2z)/(2ω0)]ψ˜1(kx,Ω);Ωωω0,
or ψ±(x,z,t)=exp(iω0τ±)ϕ±(x,z,t),with ϕ±(x,z,t)governed by the equations

i(z±1ct)ϕ±(x,z,t)=±12k02x2ϕ±(x,z,t);k0ω0/c.

3. Derivation of finite-energy paraxial subluminal and superluminal narrowband spatiotemporal wavepackets by means of Lorentz relativistic boots

It is well known that the scalar wave equation is invariant under a Lorentz transformation; also, that the parabolic equation [cf. Equation (2)] is invariant under a Galilei transformation. Perhaps, less known is that Eq. (6) is, also, invariant under a Lorentz transformation.

Equation (6) is invariant under the subluminal Lorentz transformation x=x,z=γ¯(zvt),ct=γ¯(v/c)[z(c2/v)t], where v<c and γ¯=1/1(v/c)2, as well as the superluminal Lorentz transformation x=x,ct=γ(zvt),z=γ(v/c)[z(c2/v)t], where v>cand γ=1/(v/c)21. Based on this invariance, it has been shown by Besieris and Shaarawi [18] that the following narrowband paraxial subluminal and superluminal solutions to Eq. (5) can be obtained:

ψ+sub(x,τ+,σ+)=exp(iω0τ+)f(τ+)Φ(x,σ+);τ+=tz/c,σ+=2(zvt)k0(1v/c),ψsub(x,τ,σ)=exp(iω0τ)f(τ)Φ(x,σ);τ=t+z/c,σ=2(zvt)k0(1+v/c);
ψ+sup(x,ς+,σ¯)=exp(ik0ς+)f(ς+)Φ(x,σ¯);ς+=zct,σ¯=2(zvt)k0(1v/c),ψsup(x,ς,σ¯+)=exp(ik0ς)f(ς)Φ(x,σ¯+);ς=z+ct,σ¯+=2(zvt)k0(1+v/c).
In these expressions, f() is, essentially, an arbitrary function and Φ(x,σ) satisfies the parabolic equation
i4σΦ(x,σ)+2x2Φ(x,σ)=0.
It should be mentioned that Longhi [30] has derived an expression for ψ+sub(x,τ+,σ+), except for the additional factor f(τ+) appearing in Eq. (7a).

Spatiotemporally confined luminal narrowband localized are bidirectional. Using v=cin Eq. (8a), and v=c in Eq. (8b), one obtains

ψ+(x,ς+,ς)=exp(ik0ς+)f(ς+)Φ(x,ς);ς+=zct,ς=(z+ct)/k0,ψ(x,ς,ς+)=exp(ik0ς)f(ς)Φ(x,ς+);ς=z+ct,ς+=(zct)/k0.

4. Derivation of finite-energy paraxial subluminal, luminal and superluminal narrowband spatiotemporal wavepackets by a method due to Wűnsche

Wűnsche [31] introduced two sets of transformations. The first one,

z±=(1+μ/2)z(μ/2)ct,t±=±(μ/2c)z(1+μ/2)t,
leaves the operator on the left-hand side of Eq. (6) invariant; specifically, (z±(1/c)t)(z±(1/c)t). The second one, viz,
ξ±=±(z±ct)/2=(μ+1)(zveff±t)/2;veff±±c(μ1)/(μ+1),τ±=tz/c=tz/c,
changes Eq. (6) as follows:
iξ±ϕW±(x,ξ±,τ±)=±12k02x2ϕW±(x,ξ±,τ±);k0ω0/c.
The solution of the latter assumes the form
ϕW±(x,ξ±,τ±)=ϕ±(x,ξ±)f(τ±).
The effective velocity defined in Eq. (12) depends on the value of the parameter μ. veff+ is smaller than c(subluminal) for μ>1, is equal to c for μ=0, and is larger than c(superluminal)for μ<1. veff+is superluminal for 1<μ<0 and subluminal for 0<μ<1,but in a reverse direction. On the other hand, veff is subluminal for 0<μ<1, is equal to c for μ=0, and is superluminal for 1<μ<0 in the forward direction. veffis superluminal for μ>1, and subluminal for μ<1,but in a reverse direction. It should be noted that ξ±=(z±ct)/2 for μ=0. The solution in Eq. (14) becomes bidirectional in this case; specifically, ϕW±(x,ξ±,τ±)=ϕ±(x,z±ct)f(tz/c).

5. Finite-energy paraxial subluminal and superluminal narrowband spatiotemporally localized Airy wavepackets

The subluminal and superluminal paraxial wave packets given in Eqs. (7) and (8) can be associated with any solutions of the regular parabolic equation

iZΨ(X,Z)+122X2Ψ(X,Z)=0,
written in terms of the dimensionless transverse variable X=x/x0 and the normalized range Z=z/(k0x02). Consider, for example, the Siviloglou-Christodoulides finite-energy “accelerating” Airy beam solution [22]
Ψ(X,Z)=e112(2a+iZ)(2a26X4iaZ+Z2)Ai(X+iaZZ24),
where a is a dimensionless positive parameter ensuring finite energy. The expression for Ψ(X,Z) is used in Eqs. (7a) and (8a) with Z=σ+/(2x02) and Z=σ¯+/(2x02),respectively. One, then, obtains the subluminal and superluminal narrowband Airy wave packet solutions
ψ+(sub)(x,τ+,σ+)=exp(iω0τ+)f(τ+)×e112(2a+iσ+2x02)[2a26xx04iaσ+2x02+14(σ+x02)2]Ai[(xx0+iaσ+2x02116(σ+x02)2)],
ψ+(sup)(x,ς+,σ¯)=exp(ik0ς+)f(ς+)×e112(2a+iσ¯2x02)[2a26xx04iaσ¯2x02+14(σ¯x02)2]Ai[(xx0+iaσ¯2x02116(σ¯x02)2)],
If, on the other hand, the Siviloglou-Christodoulides Airy beam solution is used in Eqs. (7b) and (8b) with Z=σ/(2x02) and Z=σ¯+/(2x02),one obtains the subluminal and superluminal narrowband Airy wave packet solutions
ψ(sub)(x,τ,σ)=exp(iω0τ)f(τ)×e112(2a+iσ2x02)[2a26xx04iaσ2x02+14(σx02)2]Ai[(xx0+iaσ2x02116(σx02)2)],
ψ(sup)(x,ς,σ¯+)=exp(ik0ς)f(ς)×e112(2a+iσ¯+2x02)[2a26xx04iaσ¯+2x02+14(σ¯+x02)2]Ai[(xx0+iaσ¯+2x02116(σ¯+x02)2)].
The solutions given in Eqs. (17)-(20) move along the z direction linearly with a subluminal or superluminal speed v. At each instant of time, a plot of the intensity of the wave packets versus xandz will show nonlinear lateral bending as in the case of the monochromatic Siviloglou-Christodoulides Airy beam [cf. Eq.(16)]. As time increases, the lateral bending will become distorted due to the presence in the solutions of the functions f(τ±)andf(ς±)which ensure finite energy content.

The subluminal and superluminal narrowband paraxial wave packets given in Eq. (14) can be associated with any solutions of the regular parabolic Eq. (15) with Z±Z. Consider, specifically, the Siviloglou-Christodoulides Airy beam solution given in Eq. (16). Wűnsche-type spatiotemporally confined Airy wave packet can be written in the form

ψW±(x,ξ±,τ±)=exp(iω0τ±)f(τ±)Φ(x/x0,±ξ±/(2k0x02)).

It should be noted that ψ(sub)(x,τ,σ)and ψ(sup)(x,ς,σ¯+)are, essentially, identical and are valid for 0v<.Iff()=1, the subluminal and superluminal paraxial Airy wavepackets derived in this section are characterized by a finite instantaneous power independent of time. Consequently, they contain infinite energy. Choosing, however, appropriate functions f() in Eqs. (17) - (21), the corresponding narrowband paraxial wave packets can contain finite energy. For all graphical results, the function f(s)=exp[s2/(2σ2)] is chosen.

Figure 1 shows a density plot of the modulus square of the wave packet ψ+(sub)(x,τ+,σ+) given in Eq. (17) versus ςsub=[v(zvt)/(cv)] and x for three values of the position at the pulse centervt: 0, 20 and 40. The function f(τ+) that ensures finite energy is chosen as the Gaussian exp(τ+2/10). The remaining normalized parameters are given as follows: ω0=1, c=1 and v=0.9. The wave packet is relatively undistorted until vt30 due to a value of the subluminal speed vclose to the speed of light.

 figure: Fig. 1

Fig. 1 A density plot of |ψ+(sub)(x,τ+,σ+)| versus ςsub=[v(zvt)/(cv)] and x for vtequal to 0, 20 and 30. f(τ+)=exp(τ+2/10) and the remaining normalized parameters are given as follows: ω0=1, c=1 and v=0.9.

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Figure 2 shows a density plot of the modulus square of the wave packet ψ+(sup)(x,ς+,σ¯) given in Eq. (18) versus ςsup=[v(zvt)/(vc)] and x for three values of the position at the pulse centervt: 0, 1 and 2. The function f(ς+) is chosen as the Gaussian exp(ς+2/10). The remaining normalized parameters are given as follows: ω0=1, c=1 and v=2. The wave packet is very quickly distorted by comparison to the subluminal one due to the large difference of the value of the superluminal speed vand the speed of light.

 figure: Fig. 2

Fig. 2 A density plot of |ψ+(sup)(x,ς+,σ¯+)| versus ςsup=[v(zvt)/(vc)] and x for vt equal to 0, 1 and 3. f(ς+)=exp(ς+2/10) and the remaining normalized parameters are given as follows: k0=ω0/c=1, c=1 and v=2.

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

It should be noted that the expressions ψsub(x,τ,σ) in Eq. (7) and ψsup(x,ς,σ¯+) in Eq. (8), with f()=1 and v=c, satisfy exactly the (2+1)D scalar wave equation

(2x2+2z21c22t2)ψ(x,z,t)=0.
In this situation, k0 is a free parameter. Appropriate superpositions over this parameter can lead to finite-energy spatiotemporally localized luminal wave packets. This “peculiarity”, whereby exact solutions of the scalar wave equation are embedded into approximations of this equation, has been mentioned by Belanger [32] and Wűnsche [31] previously.

For example, one obtains from Eq. (20)

ψ+(x,z,t)=exp[ik0ς]×e112(2a+iς+2k0x02)[2a26xx04iaς+2k0x02+14(ς+k0x02)2]Ai[(xx0+iaς+2k0x02116(ς+k0x02)2)];ς+=zct,ς=z+ct,
which is an exact solution to the scalar wave Eq. (22). It turns out that
ψ(x,z,t)=exp[ik0ς+]×e112(2a+iς2k0x02)[2a26xx04iaς2k0x02+14(ςk0x02)2]Ai[(xx0+iaς2k0x02116(ςk0x02)2)]
is also an exact solution to the scalar wave Eq. (22). Although the Siviloglou-Christodoulides Airy beam given in Eq. (16), as well as the subluminal and superluminal narrowband paraxial wave packets derived in the previous section contain finite energy, the exact solutions of the scalar wave equation given in Eqs. (23) and (24), do not have this property. This is the case for all focus wave mode luminal localized waves [1]. Only appropriate superpositions of such modes lead to spatiotemporally localized finite-energy solutions.

7. Exact finite-energy Airy splash mode solution to the (2 + 1)D scalar wave equation

Consider the solution in Eq. (23) in dimensionless form and with a=0.Specifically,

Ψ(X,Λ+,Λ)=exp[iΛ]ei124Λ+(6X+14Λ+2)Ai(X116Λ+2),
where Λ±=ZT are dimensionless variables corresponding to ς±. Ψ(X,Λ+,Λ) satisfies the dimensionless version of the scalar wave Eq. (22), which can be expressed as

(2X2+42Λ+Λ)Ψ(X,Λ+,Λ)=0.

In 1910, Bateman [33,34] discovered a transformation, more general than a conformal change of the metric, which could be used to transform solutions of Maxwell equations into similar ones. In the case of the scalar wave equation, the Bateman transformation assumes the form

Ψ1(X,Λ+,Λ)=1ΛΨ[XΛ,AΛ,X2+Λ+ΛAΛ],
with A a free parameter. The function Ψ1(X,Λ+,Λ)also obeys the scalar wave Eq. (26).

The Bateman transformation is applied twice to the solution given in Eq. (25). These two sequential operations result in the following new solution to Eq. (26):

Ψ2(X,Λ+,Λ)=1X2+Λ+ΛAi[Λ+2X(X2+Λ+Λ)16(X2+Λ+Λ)2]×exp[iΛ+3+24AXΛ+(X2+Λ+Λ)+96A2Λ(X2+Λ+Λ)296(X2+Λ+Λ)3].
Next, this expression is complexified by means of the changes Λ+Λ+ia1,ΛΛ+ia2, where a1,2 are two positive parameters. As a consequence, one obtains
Ψ2(X,Λ+,Λ;a1,a2)=exp[iQ96[X2+(a1+iΛ+)(a2iΛ)]3]X2+(a1+iΛ+)(a2iΛ)Ai[(a1+iΛ+)2+16AX3+16AX(a1+iΛ+)(a2iΛ)16[X2+(a1+iΛ+)(a2iΛ)]2],
with the exponent Q given as
Q=i(a1+iΛ+)3i24AX(a1+iΛ+)[X2+(a2iΛ)(a2iΛ)]+i96A2(a2iΛ)[X2+(a2iΛ)(a2iΛ)]2.
This is a novel finite-energy spatiotemporally localized luminal wave packet belonging to the class of splash modes studied by Ziolkowski [1]. It will be referred to as the Airy splash mode.

The parameter A in the Bateman conformal transformation is arbitrary. On the other hand, the free positive parameters a1 and a2 entering the solution given in Eq. (29) are critical. As discussed originally by Ziolkowski ([3]; See, also [4]), their presence ensures finite energy. Their relative values measure the size of the forward and backward wave components. Only when a1a2 the backward components are minimized, and the solution is almost undistorted. This is further explained in [6], where it is shown that very close replicas of localized waves, such as the one in Eq. (29), can be launched causally from apertures constructed on the basis of the Huygens principle.

Figure 3 shows surface plots of the intensity of Airy splash mode versus Λ+ and X for various values of T, the latter defined by the relationship Λ=Λ++2T. The parameters a1 and a2 have the values 8×102 and 100, respectively. The wave packet is relatively undistorted because a1>>a2.

 figure: Fig. 3

Fig. 3 Surface plots of the modulus of Ψ2(X,Λ+,Λ;a1,a2)versus Λ+ and X for various values of T, the latter defined by the relationship Λ=Λ++2T. The parameters a1 and a2 have the values 8×102 and 100, respectively.

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If the transition from Eq. (25) to (29) is undertaken by means of the modified complexification Λ+Λ+ia1,Λ2Z+ia2,the wave functionU2+(X,Λ+,Z)Ψ2(X,Λ+,2Z;a1,a2) is a finite-energy forward pulsed beam governed by the dimensionless version of Eq. (4), viz.,

(2X2+2Λ+Z)U2+(X,Λ+,Z)=0.

8. Discussion and concluding remarks

The time-diffraction technique introduced by Porras recently [19–21] has been motivated in this article in terms of the Lorentz invariance of the equation governing the narrow angular spectrum and narrowband temporal spectrum paraxial approximation and it is used to derive finite-energy spatiotemporally confined subluminal, luminal and superluminal Airy wave packets. For simplicity, all the work has been limited to appropriate narrowband paraxial solutions of the (2+1)D scalar wave equation. Extensions to the (3+1)D case are straightforward.

An important question is whether the parabolic and narrowband nature of Eqs. (6) is preserved under the subluminal and superluminal Lorentz transformations carried out in Sec. 3. To answer this question, consider specifically the respective boosted versions of ϕ+(x,z,t) in Eq. (5):

ϕ+sub(x,z,t)=R1dΩR1dkxexp[iΩγ¯(1+v/c)(tz/c)]exp(ikxx)×exp[ikx22k0asub(tzv)]ψ˜1(kx,Ω);asub(1c1v),v<c;
ϕ+sup(x,z,t)=R1dΩR1dkxexp[iΩγ(1+v/c)(tz/c)]exp(ikxx)×exp[ikx22k0asup(tvc2z)]ψ˜1(kx,Ω);asup(1cvc2),v>c;
As discussed by Porras [20], the narrow angular spectrum and narrowband temporal spectrum approximations are preserved provided that ΔΩ=Δkx2/(2k0|asub,sup|)ω0, which imposes a limitation in group delays; specifically, |asub,sup|Δkx2/(2ko2c). This condition excludes subluminal and superluminal speeds close to c.

The Airy solution is intimately related to a parabolic equation. Only for luminal spatiotemporally localized wave solutions to the scalar wave equation such an association can be made. Specifically, FWM solutions to the dimensionless (2+1)D scalar wave equation given in Eq. (26) can be written as

Ψ(X,Λ+,Λ)=e±iΛΦ(Λ±),
with Φ(Λ±) governed by the parabolic equations
4iΛ±Φ(Λ±)+2X2Φ(Λ±)=0,
with Λ±=Z±T.For this reason, subluminal and superluminal spatiotemporally confined Airy solutions to the wave equation do not exist. As a special case, broadband subluminal and superluminal spatiotemporally localized Airy wave packets based only on the narrow angular spectrum paraxial approximation do not exist.

Luminal FWM spatiotemporally localized Airy wave packet solutions to the scalar wave equation, such as those given in Eqs. (23) and (24) (see also [35], and [36]), as well as similar ones for the Klein-Gordon equation [37,38], are well known. However, as pointed out in Sec. 6, they are characterized by infinite energy content. In the case of the solutions given in Eqs. (23) and (24), finite-energy solutions can be obtained by means of superpositions over the free parameter k0. An alternative path is provided in Sec. 7, where a finite-energy Airy splash mode solution to the (2+1)D scalar wave equation has been derived using Bateman’s conformal transformation.

A note on superluminality is appropriate. The presence of a superluminal speed in the finite-energy solution given, for example, in Eq. (18) does not contradict relativity. If the parameters are chosen appropriately, the pulse moves superluminally with almost no distortion up to a certain distance zd=vt,and then it slows down to a luminal speed c, with significant accompanying distortion. Although the peak of the pulse does move superluminally up to zd, it is not causally related at two distinct ranges z1,z2[0,zd). Thus, no information can be transferred superluminally from z1 to z2. The physical significance of the (2+1)D narrowband Airy wave packet is due to its spatiotemporal localization.

In closing, the work in this article has been confined to establishing finite-energy spatiotemporally confined subluminal, luminal and superluminal Airy wave packets in free space. Such solutions are invariably nonseparable. On the other hand, separable finite-energy spatiotemporal Airy solutions have been studied in the presence of temporal dispersion and, possibly, with the inclusion of quadratic inhomogeneity. The relevant paraxial equation assumes the form

iβ0zu(x,y,z,τ)+122u(x,y,z,τ)+12β0β22τ2u(x,y,z,τ)+12α(x2+y2)u(x,y,z,τ)=0,
where n(ω) is the index of refraction, β0=k0n0=(ω0/c)n0,n0=n(ω0),τ=tβ1z, β1=ddωn(ω)|ωω0is the inverse of the group speed, and β2=d2dω2n(ω)|ωω0 is the dispersion index in the case of anomalous dispersion. A separable solution of the formu(x,y,z,τ)=U(z,τ)ψ(x,y,z) is governed by the equations
iβ0zU(z,τ)+12β0β22τ2U(z,τ)=0,iβ0zψ(x,y,z)+122ψ(x,y,z)+(x2+y2)ψ(x,y,z)=0.
Several finite-energy spatiotemporal results have appeared in the literature [39-43] with U(z,τ) chosen as an appropriate Airy-type solution.

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10. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004). [CrossRef]  

11. S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066612 (2003). [CrossRef]  

12. C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003). [CrossRef]  

13. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Localized Waves (Wiley-Interscience, Hoboken, NJ, 2008).

14. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Non-Diffracting Waves (Wiley-VCH, 2014).

15. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994). [CrossRef]  

16. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998). [CrossRef]  

17. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16(9), 1468–1474 (1999). [CrossRef]  

18. I. Besieris and A. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12(16), 3848–3864 (2004). [CrossRef]   [PubMed]  

19. M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017). [CrossRef]   [PubMed]  

20. M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A (Coll. Park) 97(6), 063803 (2018). [CrossRef]  

21. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]   [PubMed]  

22. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]   [PubMed]  

23. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]  

24. P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009). [CrossRef]  

25. P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014). [CrossRef]  

26. Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28(6), 1243–1255 (2011). [CrossRef]   [PubMed]  

27. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017). [CrossRef]  

28. H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018). [CrossRef]   [PubMed]  

29. H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018). [CrossRef]   [PubMed]  

30. S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004). [CrossRef]   [PubMed]  

31. A. Wűnsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A 6(11), 1661–1668 (1989). [CrossRef]  

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

33. H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910). [CrossRef]  

34. H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910). [CrossRef]  

35. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994). [CrossRef]  

36. N. K. Efremidis, “Spatiotemporal diffraction-free pulsed beams in free-space of the Airy and Bessel type,” Opt. Lett. 42(23), 5038–5041 (2017). [CrossRef]   [PubMed]  

37. A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990). [CrossRef]  

38. D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015). [CrossRef]  

39. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]  

40. X. Peng, Y. Peng, D. Li, L. Zhang, J. Zhuang, F. Zhao, X. Chen, X. Yang, and D. Deng, “Propagation properties of spatiotemporal chirped Airy Gaussian vortex wave packets in a quadratic index medium,” Opt. Express 25(12), 13527–13538 (2017). [CrossRef]   [PubMed]  

41. J. Zhuang, D. Deng, X. Chen, F. Zhao, X. Peng, D. Li, and L. Zhang, “Spatiotemporal sharply autofocused dual-Airy-ring Airy Gaussian vortex wave packets,” Opt. Lett. 43(2), 222–225 (2018). [CrossRef]   [PubMed]  

42. S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019). [CrossRef]  

References

  • View by:

  1. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
    [Crossref]
  2. A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).
  3. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
    [Crossref] [PubMed]
  4. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
    [Crossref]
  5. 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(1), 19–31 (1992).
    [Crossref] [PubMed]
  6. 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(1), 75–87 (1993).
    [Crossref]
  7. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
    [Crossref]
  8. I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagnetics Res. 19, 1–48 (1998).
    [Crossref]
  9. R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
    [Crossref]
  10. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
    [Crossref]
  11. S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066612 (2003).
    [Crossref]
  12. C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
    [Crossref]
  13. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Localized Waves (Wiley-Interscience, Hoboken, NJ, 2008).
  14. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Non-Diffracting Waves (Wiley-VCH, 2014).
  15. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994).
    [Crossref]
  16. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
    [Crossref]
  17. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16(9), 1468–1474 (1999).
    [Crossref]
  18. I. Besieris and A. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12(16), 3848–3864 (2004).
    [Crossref] [PubMed]
  19. M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017).
    [Crossref] [PubMed]
  20. M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A (Coll. Park) 97(6), 063803 (2018).
    [Crossref]
  21. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  22. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref] [PubMed]
  23. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  24. P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
    [Crossref]
  25. P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
    [Crossref]
  26. Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28(6), 1243–1255 (2011).
    [Crossref] [PubMed]
  27. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
    [Crossref]
  28. H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
    [Crossref] [PubMed]
  29. H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
    [Crossref] [PubMed]
  30. S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004).
    [Crossref] [PubMed]
  31. A. Wűnsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A 6(11), 1661–1668 (1989).
    [Crossref]
  32. P. A. Belanger, “Lorentz transformations of packet-like solutions of the homogeneous wave equation,” J. Opt. Soc. Am. A 3(4), 541–542 (1986).
    [Crossref]
  33. H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
    [Crossref]
  34. H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
    [Crossref]
  35. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
    [Crossref]
  36. N. K. Efremidis, “Spatiotemporal diffraction-free pulsed beams in free-space of the Airy and Bessel type,” Opt. Lett. 42(23), 5038–5041 (2017).
    [Crossref] [PubMed]
  37. A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
    [Crossref]
  38. D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
    [Crossref]
  39. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
    [Crossref]
  40. X. Peng, Y. Peng, D. Li, L. Zhang, J. Zhuang, F. Zhao, X. Chen, X. Yang, and D. Deng, “Propagation properties of spatiotemporal chirped Airy Gaussian vortex wave packets in a quadratic index medium,” Opt. Express 25(12), 13527–13538 (2017).
    [Crossref] [PubMed]
  41. J. Zhuang, D. Deng, X. Chen, F. Zhao, X. Peng, D. Li, and L. Zhang, “Spatiotemporal sharply autofocused dual-Airy-ring Airy Gaussian vortex wave packets,” Opt. Lett. 43(2), 222–225 (2018).
    [Crossref] [PubMed]
  42. S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
    [Crossref]

2019 (1)

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

2018 (4)

2017 (4)

2015 (1)

D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
[Crossref]

2014 (1)

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

2011 (1)

2010 (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

2009 (1)

P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
[Crossref]

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

2004 (3)

S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004).
[Crossref] [PubMed]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

I. Besieris and A. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12(16), 3848–3864 (2004).
[Crossref] [PubMed]

2003 (3)

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066612 (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(17), 170406 (2003).
[Crossref]

1999 (1)

1998 (2)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 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,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

1994 (2)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[Crossref]

1993 (1)

1992 (1)

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(1), 19–31 (1992).
[Crossref] [PubMed]

1990 (1)

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

1989 (3)

A. Wűnsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A 6(11), 1661–1668 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

1986 (1)

1983 (2)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

1910 (2)

H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
[Crossref]

H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
[Crossref]

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,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

Abouraddy, A. F.

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
[Crossref] [PubMed]

H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref] [PubMed]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bateman, H.

H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
[Crossref]

H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
[Crossref]

Belanger, P. A.

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Besieris, I.

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,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[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(1), 75–87 (1993).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

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,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

Chen, S.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Chen, X.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

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(17), 170406 (2003).
[Crossref]

Deng, D.

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(17), 170406 (2003).
[Crossref]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Efremidis, N. K.

Fairchild, S. R.

Fortin, M.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Greenleaf, J. F.

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(1), 19–31 (1992).
[Crossref] [PubMed]

Grunwald, R.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Heyman, E.

Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28(6), 1243–1255 (2011).
[Crossref] [PubMed]

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 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(17), 170406 (2003).
[Crossref]

Kaganovsky, Y.

Karlovets, D. V.

D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
[Crossref]

Kebbel, V.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Kiselev, A. P.

A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).

Kondakci, H. E.

H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref] [PubMed]

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
[Crossref] [PubMed]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

Kummrow, A.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Li, D.

Lin, G.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Longhi, S.

S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004).
[Crossref] [PubMed]

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

Lu, J. Y.

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(1), 19–31 (1992).
[Crossref] [PubMed]

Ma, S.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Meem, M.

Menon, R.

Neumann, U.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Nibbering, E. T.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Peng, X.

Peng, Y.

Piche, M.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Piksarv, P.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[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(17), 170406 (2003).
[Crossref]

Porras, M. A.

M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A (Coll. Park) 97(6), 063803 (2018).
[Crossref]

M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017).
[Crossref] [PubMed]

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16(9), 1468–1474 (1999).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
[Crossref]

Reivelt, K.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Reyes, D.

Richardson, M.

Rini, M.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Rousseau, G.

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Saari, P.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Shaarawi, A.

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,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[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(1), 75–87 (1993).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

Siviloglou, G. A.

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Thul, D.

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(17), 170406 (2003).
[Crossref]

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(17), 170406 (2003).
[Crossref]

Valdman, A.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

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(17), 170406 (2003).
[Crossref]

Valtma-Lukner, H.

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Wunsche, A.

Xie, J.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Yang, X.

Yessenov, M.

Zhan, Y.

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Zhang, L.

Zhao, F.

Zhuang, J.

Ziolkowski, R. W.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[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(1), 75–87 (1993).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

Am. J. Phys. (2)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62(6), 519–521 (1994).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antenn. Propag. 42(3), 311–319 (1994).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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(1), 19–31 (1992).
[Crossref] [PubMed]

J. Appl. Phys. (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

J. Math. Phys. (2)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30(6), 1254–1269 (1989).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations,” J. Math. Phys. 31(10), 2511–2519 (1990).
[Crossref]

J. Opt. Soc. Am. A (4)

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

J. Phys. Conf. Ser. (1)

P. Piksarv, A. Valdman, H. Valtma-Lukner, and P. Saari, “Ultrabroadband Airy light bullets,” J. Phys. Conf. Ser. 497, 012003 (2014).
[Crossref]

Laser Phys. (1)

P. Saari, “Airy pulse-A new member of family of localized waves,” Laser Phys. 19(4), 725–729 (2009).
[Crossref]

Nat. Photonics (2)

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Opt. Commun. (1)

S. Chen, G. Lin, J. Xie, Y. Zhan, S. Ma, and D. Deng, “Propagation properties of chirped Airy hollow Gaussian wave packets in a harmonic potential,” Opt. Commun. 430, 364–373 (2019).
[Crossref]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. A (2)

D. V. Karlovets, “Gaussian and Airy wave packets of massive particles with optical angular momentum,” Phys. Rev. A 91(1), 013847 (2015).
[Crossref]

R. Grunwald, V. Kebbel, U. Neumann, A. Kummrow, M. Rini, E. T. 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(6), 063820 (2003).
[Crossref]

Phys. Rev. A (Coll. Park) (1)

M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A (Coll. Park) 97(6), 063803 (2018).
[Crossref]

Phys. Rev. A Gen. Phys. (1)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A Gen. Phys. 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036612 (2004).
[Crossref]

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

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(1), 1086–1093 (1998).
[Crossref]

Phys. Rev. Lett. (4)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

H. E. Kondakci and A. F. Abouraddy, “Airy Wave Packets Accelerating in Space-Time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Proc. Lond. Math. Soc. (2)

H. Bateman, “The transformation of the electrodynamical equations,” Proc. Lond. Math. Soc. 8(1), 223–264 (1910).
[Crossref]

H. Bateman, “The transformations of coordinates which can be used to transform one physical problem into another,” Proc. Lond. Math. Soc. 8(1), 469–488 (1910).
[Crossref]

Prog. Electromagnetics Res. (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,” Prog. Electromagnetics Res. 19, 1–48 (1998).
[Crossref]

Radio Phys. Quant. Electron. (1)

A. P. Kiselev, “Modulated Gaussian beams,” Radio Phys. Quant. Electron. 26, 1014 (1983).

Other (2)

H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Localized Waves (Wiley-Interscience, Hoboken, NJ, 2008).

H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Non-Diffracting Waves (Wiley-VCH, 2014).

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Figures (3)

Fig. 1
Fig. 1 A density plot of | ψ + (sub) ( x, τ + , σ + ) | versus ς sub =[ v( zvt )/( cv ) ] and x for vtequal to 0, 20 and 30. f( τ + )=exp( τ + 2 /10 ) and the remaining normalized parameters are given as follows: ω 0 =1, c=1 and v=0.9.
Fig. 2
Fig. 2 A density plot of | ψ + (sup) ( x, ς + , σ ¯ + ) | versus ς sup =[ v( zvt )/( vc ) ] and x for vt equal to 0, 1 and 3. f( ς + )=exp( ς + 2 /10 ) and the remaining normalized parameters are given as follows: k 0 = ω 0 /c=1, c=1 and v=2.
Fig. 3
Fig. 3 Surface plots of the modulus of Ψ 2 ( X, Λ + , Λ ; a 1 , a 2 )versus Λ + and X for various values of T, the latter defined by the relationship Λ = Λ + +2T. The parameters a 1 and a 2 have the values 8× 10 2 and 100, respectively.

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

( 2 x 2 + 2 z 2 + ω 2 c 2 ) u ^ ( x,z,ω )=0,
i z v ^ ± ( x,,z,ω )=± c 2ω 2 x 2 v ^ ± ( x,z,ω ),
u ± ( x,z,t )= R 1 dω R 1 d k x exp[ i( ω τ ± k x x ) ]exp[ ±i( c k x 2 z )/( 2ω ) ψ ˜ 0 ( k x ,ω ) ],
( 2 x 2 2 c 2 τ ± z ) u ± ( x,z,t )=0.
ψ ± ( x,z,t )=exp( i ω 0 τ ± ) R 1 dΩ R 1 d k x exp( iΩ τ ± )exp( i k x x ) ×exp[ ±i( c k x 2 z )/( 2 ω 0 ) ] ψ ˜ 1 ( k x ,Ω );Ωω ω 0 ,
i( z ± 1 c t ) ϕ ± ( x,z,t )=± 1 2 k 0 2 x 2 ϕ ± ( x,z,t ); k 0 ω 0 /c.
ψ + sub ( x, τ + , σ + )=exp( i ω 0 τ + )f( τ + )Φ( x, σ + ); τ + =tz/c, σ + = 2( zvt ) k 0 ( 1v/c ) , ψ sub ( x, τ , σ )=exp( i ω 0 τ )f( τ )Φ( x, σ ); τ =t+z/c, σ = 2( zvt ) k 0 ( 1+v/c ) ;
ψ + sup ( x, ς + , σ ¯ )=exp( i k 0 ς + )f( ς + )Φ( x, σ ¯ ); ς + =zct, σ ¯ = 2( zvt ) k 0 ( 1v/c ) , ψ sup ( x, ς , σ ¯ + )=exp( i k 0 ς )f( ς )Φ( x, σ ¯ + ); ς =z+ct, σ ¯ + = 2( zvt ) k 0 ( 1+v/c ) .
i4 σ Φ( x,σ )+ 2 x 2 Φ( x,σ )=0.
ψ + ( x, ς + , ς )=exp( i k 0 ς + )f( ς + )Φ( x, ς ); ς + =zct, ς =( z+ct )/ k 0 , ψ ( x, ς , ς + )=exp( i k 0 ς )f( ς )Φ( x, ς + ); ς =z+ct, ς + =( zct )/ k 0 .
z ± =(1+μ/2)z(μ/2)ct, t ± =±(μ/2c)z(1+μ/2)t,
ξ ± =±( z ±c t )/2=( μ+1 )( z v eff ± t )/2; v eff ± ±c( μ1 )/( μ+1 ), τ ± = t z /c=tz/c,
i ξ ± ϕ W ± ( x, ξ ± , τ ± )=± 1 2 k 0 2 x 2 ϕ W ± ( x, ξ ± , τ ± ); k 0 ω 0 /c.
ϕ W ± ( x, ξ ± , τ ± )= ϕ ± ( x, ξ ± )f( τ ± ).
i Z Ψ( X,Z )+ 1 2 2 X 2 Ψ( X,Z )=0,
Ψ( X,Z )= e 1 12 ( 2a+iZ )( 2 a 2 6X4iaZ+ Z 2 ) Ai( X+iaZ Z 2 4 ),
ψ + (sub) ( x, τ + , σ + )=exp( i ω 0 τ + )f( τ + ) × e 1 12 ( 2a+i σ + 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ + 2 x 0 2 + 1 4 ( σ + x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ + 2 x 0 2 1 16 ( σ + x 0 2 ) 2 ) ],
ψ + (sup) ( x, ς + , σ ¯ )=exp( i k 0 ς + )f( ς + ) × e 1 12 ( 2a+i σ ¯ 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ ¯ 2 x 0 2 + 1 4 ( σ ¯ x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ ¯ 2 x 0 2 1 16 ( σ ¯ x 0 2 ) 2 ) ],
ψ (sub) ( x, τ , σ )=exp( i ω 0 τ )f( τ ) × e 1 12 ( 2a+i σ 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ 2 x 0 2 + 1 4 ( σ x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ 2 x 0 2 1 16 ( σ x 0 2 ) 2 ) ],
ψ (sup) ( x, ς , σ ¯ + )=exp( i k 0 ς )f( ς ) × e 1 12 ( 2a+i σ ¯ + 2 x 0 2 )[ 2 a 2 6 x x 0 4ia σ ¯ + 2 x 0 2 + 1 4 ( σ ¯ + x 0 2 ) 2 ] Ai[ ( x x 0 +ia σ ¯ + 2 x 0 2 1 16 ( σ ¯ + x 0 2 ) 2 ) ].
ψ W ± ( x, ξ ± , τ ± )=exp( i ω 0 τ ± )f( τ ± )Φ( x/ x 0 ,± ξ ± /( 2 k 0 x 0 2 ) ).
( 2 x 2 + 2 z 2 1 c 2 2 t 2 )ψ( x,z,t )=0.
ψ + ( x,z,t )=exp[ i k 0 ς ] × e 1 12 ( 2a+i ς + 2 k 0 x 0 2 )[ 2 a 2 6 x x 0 4ia ς + 2 k 0 x 0 2 + 1 4 ( ς + k 0 x 0 2 ) 2 ] Ai[ ( x x 0 +ia ς + 2 k 0 x 0 2 1 16 ( ς + k 0 x 0 2 ) 2 ) ]; ς + =zct, ς =z+ct,
ψ ( x,z,t )=exp[ i k 0 ς + ] × e 1 12 ( 2a+i ς 2 k 0 x 0 2 )[ 2 a 2 6 x x 0 4ia ς 2 k 0 x 0 2 + 1 4 ( ς k 0 x 0 2 ) 2 ] Ai[ ( x x 0 +ia ς 2 k 0 x 0 2 1 16 ( ς k 0 x 0 2 ) 2 ) ]
Ψ( X, Λ + , Λ )=exp[ i Λ ] e i 1 24 Λ + ( 6X+ 1 4 Λ + 2 ) Ai( X 1 16 Λ + 2 ),
( 2 X 2 +4 2 Λ + Λ )Ψ( X, Λ + , Λ )=0.
Ψ 1 ( X, Λ + , Λ )= 1 Λ Ψ[ X Λ , A Λ , X 2 + Λ + Λ A Λ ],
Ψ 2 ( X, Λ + , Λ )= 1 X 2 + Λ + Λ Ai[ Λ + 2 X( X 2 + Λ + Λ ) 16 ( X 2 + Λ + Λ ) 2 ] ×exp[ i Λ + 3 +24AX Λ + ( X 2 + Λ + Λ )+96 A 2 Λ ( X 2 + Λ + Λ ) 2 96 ( X 2 + Λ + Λ ) 3 ].
Ψ 2 ( X, Λ + , Λ ; a 1 , a 2 )= exp[ i Q 96 [ X 2 +( a 1 +i Λ + )( a 2 i Λ ) ] 3 ] X 2 +( a 1 +i Λ + )( a 2 i Λ ) Ai[ ( a 1 +i Λ + ) 2 +16A X 3 +16AX( a 1 +i Λ + )( a 2 i Λ ) 16 [ X 2 +( a 1 +i Λ + )( a 2 i Λ ) ] 2 ],
Q=i ( a 1 +i Λ + ) 3 i24AX( a 1 +i Λ + )[ X 2 +( a 2 i Λ )( a 2 i Λ ) ] +i96 A 2 ( a 2 i Λ ) [ X 2 +( a 2 i Λ )( a 2 i Λ ) ] 2 .
( 2 X 2 + 2 Λ + Z ) U 2 + ( X, Λ + ,Z )=0.
ϕ + sub ( x,z,t )= R 1 dΩ R 1 d k x exp[ iΩ γ ¯ ( 1+v/c )( tz/c ) ]exp( i k x x ) ×exp[ i k x 2 2 k 0 a sub ( t z v ) ] ψ ˜ 1 ( k x ,Ω ); a sub ( 1 c 1 v ),v<c;
ϕ + sup ( x,z,t )= R 1 dΩ R 1 d k x exp[ iΩγ( 1+v/c )( tz/c ) ]exp( i k x x ) ×exp[ i k x 2 2 k 0 a sup ( t v c 2 z ) ] ψ ˜ 1 ( k x ,Ω ); a sup ( 1 c v c 2 ),v>c;
Ψ( X, Λ + , Λ )= e ±i Λ Φ( Λ ± ),
4i Λ ± Φ( Λ ± )+ 2 X 2 Φ( Λ ± )=0,
i β 0 z u( x,y,z,τ )+ 1 2 2 u( x,y,z,τ )+ 1 2 β 0 β 2 2 τ 2 u( x,y,z,τ ) + 1 2 α( x 2 + y 2 )u( x,y,z,τ )=0,
i β 0 z U( z,τ )+ 1 2 β 0 β 2 2 τ 2 U( z,τ )=0, i β 0 z ψ( x,y,z )+ 1 2 2 ψ( x,y,z )+( x 2 + y 2 )ψ( x,y,z )=0.

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