Relativistic aberration and null Doppler shift within the framework of superluminal and subluminal nondiffracting waves

The relativistic aberration of a wavevector and the corresponding Doppler shift are examined in connection with superluminal and subluminal spatiotemporally localized pulsed optical waves. The requirement of a null Doppler shift is shown to give rise to a speed associated with the relativistic velocity composition law of a double (two-step) Lorentz transformation. The effects of such a transformation are examined both in terms of four-coordinates and in the spectral domain. It is established that a subluminal pulse reverses its direction. In addition to a change in direction, the propagation term of a superluminal pulse becomes negative. The aberration due to a double Lorentz transformation is examined in detail for propagation invariant superluminal waves (X wave, Bessel X wave), as well as intensity-invariant superluminal and subluminal waves. Detailed symmetry considerations are provided for the superluminal focus X wave and the subluminal MacKinnon wavepacket.


Introduction
The aberration of the wave vector due to the relative motion of reference frames and the corresponding Doppler shift in frequency are subjects not easily comprehended by students and are commonly associated with astrophysical and high-energy-physics fields. In particular, the treatment of the transverse Doppler effect and conditions of a null frequency shift depending on the choice of reference are not discussed in detail in textbooks.
The goals of the present study are, first, to consider these effects in connection with nondiffracting spatiotemporally localized pulsed optical waves (LWs) which not only can be treated as Lorentz-transformed versions of certain simple fields, but also are easily feasible in a laboratory of ultrafast optics. The second goalwhich is of interest for the theory of LWs-is to reveal the symmetry properties of LWs with respect to certain double Lorentz transformations.
As a specific class of structured electromagnetic fields, LWs were mathematically discovered in the late 1980s, and since then a massive literature has been devoted to their theoretical and experimental study (see collective monographs [1, 2] and reviews [3][4][5][6][7][8]).
LWs are characterized by a specific type of space-time coupling. For all monochromatic plane-wave constituents of such pulsed waves in free space, there is a linear functional dependence between their temporal frequency (or wavenumber / k c   ) and the component z k of the wave vector in the direction of propagation of the pulses, rendering them propagation-invariant. This means that the spatial distribution of the pulse energy density does not change during propagation-it does not spread either in the lateral or in the longitudinal direction (or temporally).
In reality, such a non-diffracting (non-spreading) propagation occurs over a large but still finite distance, because the aforementioned frequency-wavenumber functional dependence is not strict for practically realizable (finiteenergy and finite-aperture) pulses.
The group velocity of LWs in empty space without the presence of any resonant medium not only can be smaller or equal to the vacuum speed of light , c but also can exceed , c i.e., it can be superluminal-in contradistinction with common superficial textbook treatments of the notion of group velocity of light pulses. This intriguing property has been widely discussed in the literature referred to above and has been experimentally verified by several groups [9][10][11][12] for cylindrically symmetric pulses. LWs that are localized in all three spatial dimensions and also temporally -which we shortly label as (3+1)D wavepackets-are in practice easily realizable only for certain types of superluminal and extremely subluminal pulses. At the same time, nondiffracting (2+1)D wave packets (pulsed light sheets), in which case the localization is sacrificed in one spatial dimension, can be flexibly generated by using spatial light modulators. Such technique has been recently introduced by Abouraddy's group at the University of Central Florida, resulting in the development of a wide program of study of such space-time wave packets (see, e.g., [8,13,14] and, particularly [15], where group velocities up to 30c were realized).
In our recent work [16], it was established that the velocity e v (normalized with respect to 1 c  ) at which the energy of LWs flows in the direction of propagation, is not equal to the propagation velocity of the pulse itself (i.e., to the normalized group velocity g v ). Instead, on the symmetry axis and/or on the locations of energy density maxima, these two quantities obey a simple relationship: specifically, . This is a physically contentrich result, because the right-hand-side expression arises from a two-step Lorentz transformation involving the normalized group velocity g v . In section 2, where a brief introduction to the notion of X waves-the simplest representatives of superluminal LWs (or space-time wave packets) is given, it will be shown that this relationship arises naturally from the requirement of a null Doppler effect. In section 3, the double (two-step) Lorentz transformation of a general expression of LWs will be derived, first within the framework of four-coordinate transformations and then using transformations in the spectral domain. The symmetry properties of propagation-invariant subluminal and superluminal LWs under the double Lorentz transformations will be also considered there.
Specific illustrative examples of the symmetry properties under the double Lorentz transformations will be given in section 4 for two typical propagation-invariant localized waves: the superluminal focus X wave and the subluminal MacKinnon wave packet, both of which have well-known closed-form analytical expressions. It should be noted that for the sake of simplicity the pulses are treated as scalar-valued fields, i.e., they may describe one component of an electromagnetic vector field. Nevertheless, the results can be extended to vector fields without loss of their validity, as was shown for the energy flow velocity in [16,17]. In section 4 we also briefly discuss the differences in symmetry properties of sub-and superluminal LWs. Concluding remarks are given in section 5.

Propagation-invariant X waves in terms of aberration
A primed reference frame is assumed to be moving along the z -axis with a speed u with respect to an unprimed laboratory frame. Then, the wavevectors and frequency of a (3+1)D (three-dimensional + time) plane wave in free space transform as follows: Let us first consider a monochromatic plane wave propagating at an angle  with respect to the z -axis, as shown in figure 1. The angle  , called axicon angle, is defined by the relationship cos .   In order to consider X waves, let us make two generalizations. First, we take a symmetrical pair of plane wavesthe propagation direction of the first one lies on the   In the primed reference frame the slope and the group velocity are infinitely large because the pulses are counterpropagating with respect to each other along the x -axis.
In the unprimed frame, the pulses form an X wave whose apex propagates without any spreading along the z -axis In figure 2a, the apex is twice (intensity 4-fold) higher than that of the constituent pulses, while in the transverse direction the field is constant. However, if we make a generalization to a 3D setting, viz., to axisymmetric cylindrical waves, x k has to be replaced by the transverse wavenumber  Bessel-X pulses have been studied in a great number of papers (see reviews [1, 2, 6]), including their experimental realization in optics [10][11][12]. Although 2D X-type waves are poorly localized, they can be experimentally generated in a wide range of group velocities without encountering substantial technological obstacles, and they have become a subject of intensive study recently [8,[13][14][15]. For the present study, the most important conclusion from figure 1 and this introductory survey of X-waves is that they can be obtained through a Lorentz transformation from a pair of counterpropagating pulsed plane waves (in the 2D case), or from a radially collapsing and thereupon expanding cylindrical pulse (in the axisymmetric 3D case) [4,6].

The double Lorentz transformation
The Doppler effect vanishes if the frequencies in the laboratory and moving frames are equal, i.e., ' As will be shown in detail in the next subsection, Eq. (2) defines the speed of a double (or two-step) Lorentz transformation from the laboratory reference to a second frame moving relative to the primed reference with speed cos    , or with the relativistically doubled superluminal group velocity of a X-type pulse. It is thoughtprovoking that Eq. (2), as shown in [16,17], also gives the on-axis energy flow speed e c  of (2+1)D and (3+1)D It should be noted that (7a) expresses in terms of the composition of velocities the same result that Eq. (3) does in terms of the Lorentz-transformed z -component of the wavevector.

Transformation of 4-coordinates
In the case of propagation-invariant localized waves (e.g., the X or Bessel-X pulses considered above), a if the positive signs are used in Eq. (8) corresponding to a double Lorentz boost in the direction of the pulse propagation in the laboratory frame. In Eq. (9) and from here on we omit the double primes on the right-hand side of transition expressions for brevity. It is seen, then, that a subluminal signal reverses its direction. In addition to a change in direction, the superluminal propagation variable becomes negative.

Transformation of the spectral parameters.
An expression for a localized solution to the scalar wave equation in free space is given by the spectral synthesis [4,9,16]    , . 1 1 Next, we let the moving frame be the doubly primed frame, i.e.,   Thus, for both subluminal and superluminal pulses a double Lorentz transformation reverses the propagation direction. The sign of the parameter b stays the same for subluminal waves but is reversed for superluminal ones.

General expressions of subluminal and superluminal localized waves
The spectral synthesis in Eq. (10) can be used to obtain the following forms of general subluminal and superluminal localized wave solutions to the scalar wave equation in free space: Here,  15) and (16) with identical parameters, differ substantially. Specifically, the wavelength of the carrier of the transformed FXW for the given parameters is 4 times shorter and, correspondingly, its frequency 4 times higher than those of the original FXW. Mathematically, the differences arise from the symmetry relations between Eqs. (15) and (16). To reveal the physical reasons and study the differences in more detail, let us first look at the transformation of the supports of the spectrum of FXW in figure 4.