[show abstract][hide abstract] ABSTRACT: We present shape-preserving spatially accelerating electromagnetic wave packets in curved space: wave packets propagating along nongeodesic trajectories while periodically recovering their structure. These wave packets are solutions to the paraxial and nonparaxial wave equations in curved space. We analyze the dynamics of such beams propagating on surfaces of revolution, and find solutions that propagate along a variety of nongeodesic trajectories, with their intensity profile becoming narrower (or broader) in a scaled self-similar fashion. Such wave packets reflect the interplay between the curvature of space and interference effects. Finally, we extend this concept to nonlinear accelerating beams in curved space supported by the Kerr nonlinearity. Our study concentrates on optical settings, but the underlying concepts directly relate to general relativity. The complex dynamics of particles and of electromag-netic (EM) waves in curved space-time is still inaccessible to laboratory experiments. However, numerous physical systems have been suggested to demonstrate analogies of general-relativity phenomena, ranging from sound and gravity waves in flowing fluids [1–3] to Bose-Einstein [4–6] and optical systems, which have had a major success in demonstrating such phenomena [7–13]. For example, metamaterials enabled creating analogies to black holes, by engineering the (EM) properties of the material through which light is propagating [8–10]. Another example is using a moving dielectric medium that acts as an effective gravitational field on the light . This idea was dem-onstrated experimentally by employing ultrashort pulses in an optical fiber to create an artificial event horizon . Another route for such studies is to create curved space by engineering the geometry of the space itself. This idea, suggested in 1981 , started by exploring the dynamics of a free quantum particle constrained by an external potential to evolve within a thin sheet. More than 25 years later, these ideas were carried over to EM waves , where pioneering experiments studied light propagating in a thin-film waveguide attached to the curved surface area of a three-dimensional (3D) body . However, thus far, in all of these experiments and theoretical studies on general-relativity concepts with EM waves, the wave packets were propagating on geodesic trajectories, which are naturally the shortest path, analogous to straight lines in flat geometry. But, do wave packets propagating in curved space have to follow special geodesic paths, or can they exhibit other trajectories that are not predicted by the geodesic equation? Here, we show that wave packets can exhibit periodically shape-invariant spatially accelerating dynamics in curved space, propagating in nongeodesic trajectories that reflect the interplay between the curvature of space and interfer-ence effects arising from initial conditions. We study these beams in surfaces of revolution in the linear and nonlinear, paraxial and nonparaxial regimes and unravel a variety of new intriguing properties that are nonexistent in flat space. This study paves the way to accelerating-beams experi-ments in curved space to study basic concepts of general relativity, where the entire dynamics is nongeodesic. Before proceeding, we briefly recall the ideas underlying accelerating wave packets. They were first revealed in 1979 as a unique solution to the Schrödinger equation: a propagation-invariant wave packet shaped as an Airy function that accelerates in time . Almost 30 years later, the concept of accelerating wave packets was intro-duced into electromagnetism, demonstrating Airy beams that are spatially accelerating within the paraxial approxi-mation [18,19]. Following Refs. [18,19], accelerating wave packets have drawn extensive interest and initiated many new ideas, such as accelerating ultrashort pulses and light bullets [20–22], two-dimensional (2D) accelerating beams , accelerating beams following arbitrary convex accel-eration trajectories [24–26], accelerating beams in photon-ics potentials [27–30], and accelerating beams in nonlinear media [31–34]. These recent works were followed by many applications such as manipulating microparticles [35,36], self-bending plasma channels , and accelerating elec-tron beams . For some time, shape-preserving accel-erating wave packets were believed to exist strictly within the domain of the Schrödinger-type paraxial wave equation [17–19]. However, in 2012, we presented accelerating
Physical Review X 03/2014; 4(1):011038. · 6.71 Impact Factor
[show abstract][hide abstract] ABSTRACT: We present shape-preserving spatially accelerating electromagnetic
wavepackets in curved space: wavepackets propagating along non-geodesic
trajectories while recovering their structure periodically. These wavepackets
are solutions to the paraxial and non-paraxial wave equation in curved space.
We analyze the dynamics of such beams propagating on surfaces of revolution,
and find solutions that carry finite power. These solutions propagate along a
variety of non-geodesic trajectories, reflecting the interplay between the
curvature of space and interference effects, with their intensity profile
becoming narrower (or broader) in a scaled self-similar fashion Finally, we
extend this concept to nonlinear accelerating beams in curved space supported
by the Kerr nonlinearity. Our study concentrates on optical settings, but the
underlying concepts directly relate to General Relativity.
[show abstract][hide abstract] ABSTRACT: In 1958, a revolutionary paper by Aharonov and Bohm predicted a phase
difference between two parts of an electron wavefunction even when being
confined to a regime with no EM field. The Aharonov-Bohm effect was
groundbreaking: proving that the EM vector potential is a real physical
quantity, affecting the outcome of experiments not only through the EM
fields extracted from it. But is the EM potential a real necessity for
an Aharonov-Bohm-type effect? Can it exist in a potential-free system
such as free-space? Here, we find self-accelerating wavepackets that are
solutions of the free Dirac equation, for massive/massless
fermions/bosons. These accelerating Dirac particles mimic the dynamics
of a free-charge moving under a ``virtual'' EM field, even though no
field is acting and there is no charge: the entire dynamics is a direct
result of the initial conditions. We show that such particles display an
effective Aharonov-Bohm effect caused by exactly the same ``virtual''
potential that also ``causes'' the acceleration. Altogether, along the
trajectory, there is no way to distinguish between a real force and the
self-induced force - it is real by all measurable quantities. This
proves that one can create all effects induced by EM fields by only
controlling the initial conditions of a wave pattern, while the dynamics
is in free-space. These phenomena can be observed in various settings:
e.g., optical waves in honeycomb photonic lattices or in hyperbolic
metamaterials, and matter waves in honeycomb interference structures.
[show abstract][hide abstract] ABSTRACT: We present the nondiffracting spatially accelerating solutions of the Maxwell equations. Such beams accelerate in a circular trajectory, thus generalizing the concept of Airy beams to the full domain of the wave equation. For both TE and TM polarizations, the beams exhibit shape-preserving bending which can have subwavelength features, and the Poynting vector of the main lobe displays a turn of more than 90°. We show that these accelerating beams are self-healing, analyze their properties, and find the new class of accelerating breathers: self-bending beams of periodically oscillating shapes. Finally, we emphasize that in their scalar form, these beams are the exact solutions for nondispersive accelerating wave packets of the most common wave equation describing time-harmonic waves. As such, this work has profound implications to many linear wave systems in nature, ranging from acoustic and elastic waves to surface waves in fluids and membranes.
[show abstract][hide abstract] ABSTRACT: We present the spatially accelerating solutions of the Maxwell equations.
Such non-paraxial beams accelerate in a circular trajectory, thus generalizing
the concept of Airy beams. For both TE and TM polarizations, the beams exhibit
shape-preserving bending with sub-wavelength features, and the Poynting vector
of the main lobe displays a turn of more than 90 degrees. We show that these
accelerating beams are self-healing, analyze their properties, and compare to
the paraxial Airy beams. Finally, we present the new family of periodic
accelerating beams which can be constructed from our solutions.
[show abstract][hide abstract] ABSTRACT: We find self-accelerating beams in highly nonlocal nonlinear optical media, and show that their propagation dynamics is strongly affected by boundary conditions. Specifically for the thermal optical nonlinearity, the boundary conditions have a strong impact on the beam trajectory: they can increase the acceleration during propagation, or even cause beam bending in a direction opposite to the initial trajectory. Under strong self-focusing, the accelerating beam decomposes into a localized self-trapped beam propagating on an oscillatory trajectory and a second beam which accelerates in a different direction. We augment this study by investigating the effects caused by a finite aperture and by a nonlinear range of a finite extent.