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.
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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 equationEqs. (1) and (2), is the speed of light in vacuum and denotes the angular frequency. The space-time paraxial solutions can be expressed in terms of the Fourier representations15]. The equation for 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 . 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  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  and subsequently demonstrated experimentally by Siviloglou, Broky, Dogariu and Christodoulides . Their work was motivated by the infinite-energy (nonspreading) accelerating Airy solution to the Schrödinger equation introduced by Berry and Balazs  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 , Piksarv, Valmann, Valta-Lukner and Saari  and Kaganovsky and Heyman .
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 in Eq. (3) is assumed to have a narrow band around the frequency . Furthermore, the phase is expanded in a Taylor series around and only the first term in the expansion is retained, i.e., Within the framework of this additional approximation, the expressions in Eq. (3) assume the forms
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 where and as well as the superluminal Lorentz transformation , where and Based on this invariance, it has been shown by Besieris and Shaarawi  that the following narrowband paraxial subluminal and superluminal solutions to Eq. (5) can be obtained:30] has derived an expression for , except for the additional factor appearing in Eq. (7a).
4. Derivation of finite-energy paraxial subluminal, luminal and superluminal narrowband spatiotemporal wavepackets by a method due to Wűnsche
Wűnsche  introduced two sets of transformations. The first one,Eq. (6) invariant; specifically, The second one, viz,Eq. (6) as follows:Eq. (12) depends on the value of the parameter is smaller than for is equal to for and is larger than for is superluminal for and subluminal for but in a reverse direction. On the other hand, is subluminal for is equal to for and is superluminal for in the forward direction. is superluminal for and subluminal for but in a reverse direction. It should be noted that for The solution in Eq. (14) becomes bidirectional in this case; specifically,
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 equation22]Eqs. (7a) and (8a) with and respectively. One, then, obtains the subluminal and superluminal narrowband Airy wave packet solutionsEqs. (7b) and (8b) with and one obtains the subluminal and superluminal narrowband Airy wave packet solutionsEqs. (17)-(20) move along the direction linearly with a subluminal or superluminal speed At each instant of time, a plot of the intensity of the wave packets versus 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 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 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
It should be noted that and are, essentially, identical and are valid for 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 in Eqs. (17) - (21), the corresponding narrowband paraxial wave packets can contain finite energy. For all graphical results, the function is chosen.
Figure 1 shows a density plot of the modulus square of the wave packet given in Eq. (17) versus and for three values of the position at the pulse center 0, 20 and 40. The function that ensures finite energy is chosen as the Gaussian The remaining normalized parameters are given as follows: and The wave packet is relatively undistorted until due to a value of the subluminal speed close to the speed of light.
Figure 2 shows a density plot of the modulus square of the wave packet given in Eq. (18) versus and for three values of the position at the pulse center 0, 1 and 2. The function is chosen as the Gaussian The remaining normalized parameters are given as follows: and 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 and the speed of light.
6. Embedding of exact localized wave solutions of the scalar wave equation into approximate paraxial ones32] and Wűnsche  previously.
For example, one obtains from Eq. (20)Eq. (22). It turns out thatEq. (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 . 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 Specifically,Eq. (22), which can be expressed as
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 formEq. (26).1]. It will be referred to as the Airy splash mode.
The parameter in the Bateman conformal transformation is arbitrary. On the other hand, the free positive parameters and entering the solution given in Eq. (29) are critical. As discussed originally by Ziolkowski (; See, also ), their presence ensures finite energy. Their relative values measure the size of the forward and backward wave components. Only when the backward components are minimized, and the solution is almost undistorted. This is further explained in , 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 and for various values of the latter defined by the relationship The parameters and have the values and respectively. The wave packet is relatively undistorted because
If the transition from Eq. (25) to (29) is undertaken by means of the modified complexification the wave function is a finite-energy forward pulsed beam governed by the dimensionless version of Eq. (4), viz.,
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 scalar wave equation. Extensions to the 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 in Eq. (5):20], the narrow angular spectrum and narrowband temporal spectrum approximations are preserved provided that which imposes a limitation in group delays; specifically, This condition excludes subluminal and superluminal speeds close to
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 scalar wave equation given in Eq. (26) can be written as
Luminal FWM spatiotemporally localized Airy wave packet solutions to the scalar wave equation, such as those given in Eqs. (23) and (24) (see also , and ), 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 An alternative path is provided in Sec. 7, where a finite-energy Airy splash mode solution to the 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 and then it slows down to a luminal speed , with significant accompanying distortion. Although the peak of the pulse does move superluminally up to it is not causally related at two distinct ranges . Thus, no information can be transferred superluminally from to . The physical significance of the 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 form39-43] with chosen as an appropriate Airy-type solution.
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