Generation and classification of localized waves by Lorentz transformations in Fourier space

Abstract

The Lorentz transformations of propagation-invariant localized waves (also known as nondispersive or nondiffracting or undistorted progressive waves) are studied in the frequency-momentum space. For supports of wave functions in this space rules of transformation are derived which allow one to group all localized waves into distinct classes: subluminal, luminal, and superluminal localized waves. It is shown that for each class there is an inertial frame in which any given localized wave takes a particularly simple form. In other words, any localized wave is nothing but a relativistically aberrant and Doppler shifted version of a simple "seed" wave. Also discussed are the relations of the physical (subluminal) Lorentz tranformation to other mathematical tranformations used in the literature on localized waves, as well as physical interpretation of the substantial changes that localized waves undergo if observed and generated in different inertial frames.

Photon localization barrier can be overcome

“…(6), (9), and (10). However, at any spatial location the wave function is square integrable with respect to time, thus the condition of the Paley-Wiener theorem has been satisfied.…”

“…The result is a new independent solution but it can also be considered as the wave given by Eqs. (5) and (6), which is observed in another inertial reference frame [10]:…”

“…Since the FXW in the limit β → 1 becomes a FWM [10], Fig.2 gives also an idea how a FWM looks like. Multiplying Eq.…”

“…By making use of the historically first representative of localized waves -the so-called focus wave mode (FWM) [22]- [24] (see also Ref. [10] and reviews [11], [15] and references therein) one readily obtains an example of the field that exhibits even much stronger than exponential localization. FWM is given by the scalar function…”

SuperluminalX-wave propagation: Energy localization and velocity

“…Specifically, we analyze the field due to a ring-shaped aperture over a metallic screen on which a linearly polarized plane wave impinges. On this basis, and in the far field approximation, we can obtain information about the propagation of energy flux and the velocity of the energy.The motion of a Bessel beam is of great interest in physics both for its characteristic as a non-diffracting beam [1,2,3,4,5,6,7], and for its implications with regard to the topic of superluminality [8,9,10,11,12,13]. Extended studies have been devoted to these subjects from both an experimental and a theoretical point of view.…”