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.

Peeter Saari Ioannis M. Besieris

Electromagnetic energy backflow is a phenomenon occurring in regions where the direction of the Poynting vector is opposite to that of the propagation of the wave field. It is particularly remarkable in the nonparaxial regime and has been exhibited in the focal region of sharply focused beams, for vector Bessel beams, and vector-valued spatiotemporally localized waves. A detailed study is undertaken of this phenomenon and the conditions for its appearance are examined in detail in the case of a superposition of four plane waves in free space, the simplest electromagnetic arrangement for the observation of negative energy flow, as well as its comprehensive and transparent physical interpretation. It is shown that the state of polarization of the constituent components of the electromagnetic plane wave quartet determines whether energy backflow takes place or not and what values the energy flow velocity assumes. Depending on the polarization angles, the latter can assume any value from c (the speed of light in vacuum) to -c in certain spatiotemporal regions of the field.

Peeter Saari Ioannis Besieris

Backflow, or retro-propagation, is a counterintuitive phenomenon where for a forward-propagating wave the energy or probability density locally propagates backward. In this study the energy backflow has been examined in connection with relatively simple causal unidirectional finite-energy solutions of the wave equation which are derived from a factorization of the so-called basic splash mode. Specific results are given for the energy backflow arising in known azimuthally symmetric unidirectional wavepackets, as well as in novel azimuthally asymmetric extensions. Using the Bateman-Whittaker technique, a novel finite-energy unidirectional null localized wave has been constructed that is devoid of energy backflow and has some of the topological properties of the basic Hopfion.

Ioannis Besieris Peeter Saari

Hall and Abouraddy [1] have reported first experimental observation of optical de Broglie-Mackinnon wave packets, which is a seminal achievement in the study of so-called non-diffracting optical pulses. These wave packets propagate in free space without spreading with subluminal relativistic velocities, i.e., with speeds slower but close to the velocity of light in vacuum. The experiments in [1] became possible thanks to the application of quite a witty method. Unfortunately, the explanation of the physical nature of the wave packets and their graphical and mathematical descriptions in the theoretical part of [1] suffer from some ambiguities that need to be clarified.

Peeter Saari Ioannis M. Besieris

Backflow, or retropropagation, is a counterintuitive phenomenon whereby for a forward-propagating wave the energy locally propagates backward. In the context of backflow, physically most interesting are the so-called unidirectional waves, which contain only forward propagating plane wave constituents. Yet, very few such waves possessing closed-form analytic expressions for evaluation of the Poynting vector are known. In this study, we examine energy backflow in a novel (2+1)-dimensional unidirectional monochromatic wave and in a (2+1)D spatio-temporal wave packet, analytic expressions which we succeeded to find. We also present a detailed study of the backflow in the "needle" pulse. This is an interesting model object because well-known superluminal non-diffracting space-time wave packets can be derived from its factored wave function. Finally we study the backflow in an unidirectional version of the so-called focus wave mode--a pulse propagating luminally and without spread, which is the first and most studied representative of the (3+1)D non-diffracting space-time wave packets (also referred to as spatiotemporally localized waves).

Peeter Saari Ioannis Besieris

The behavior of wave signals in the far zone is not only of theoretical interest but also of paramount practical importance in communications and other fields of applications of optical, electromagnetic or acoustic waves. Long time ago T. T. Wu introduced models of 'electromagnetic missiles' whose decay could be made arbitrarily slower than the usual inverse distance by an appropriate choice of the high frequency portion of the source spectrum. Very recent work by Plachenov and Kiselev introduced a finite-energy scalar wave solution, different from Wu's, decaying slower than inversely proportional with the distance. A physical explanation for the unusual asymptotic behavior of the latter will be given in this article. Furthermore, two additional examples of scalar wave pulses characterized by abnormal slow decay in the far zone will be given and their asymptotic behavior will be discussed. A proof of feasibility of acoustic and electromagnetic fields with the abnormal asymptotics will be described.

Peeter Saari Ioannis Besieris

In recent years the topic of localized wave solutions of the homogeneous scalar wave equation, i.e., the wave fields that propagate without any appreciable spread or drop in intensity, has been discussed in many aspects in numerous publications. In this review the main results of this rather disperse theoretical material are presented in a single mathematical representation - the Fourier decomposition by means of angular spectrum of plane waves. This unified description is shown to lead to a transparent physical understanding of the phenomenon as such and yield the means of optical generation of such wave fields.

Kaido Reivelt Peeter Saari

In this paper we show that in the $n$-body problem with harmonic potential one can find a continuum of central configurations for $n=3$. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari's conjecture. This will help to refine our understanding and formulation of the Generalized Saari's conjecture, and in turn it might provide insight in how to solve the classical Saari's conjecture for $n\geq 4$.

Manuele Santoprete

We prove Saari's homographic conjecture for a large class of $n$-body problems with power law potentials, including the classical $n$-body problem.

Pieter Tibboel

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $n\ge 3$.

Florin Diacu Toshiaki Fujiwara Ernesto Perez-Chavela Manuele Santoprete