Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space

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Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space

Rivka Bekenstein, Jonathan Nemirovsky, Ido Kaminer, and Mordechai Segev

Phys. Rev. X 4, 011038 – Published 13 March 2014

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.

Received 7 August 2013

DOI:https://doi.org/10.1103/PhysRevX.4.011038

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Rivka Bekenstein, Jonathan Nemirovsky, Ido Kaminer, and Mordechai Segev

Physics Department and Solid State Institute, Technion, Haifa 32000, Israel

Popular Summary

General relativity, proposed by Albert Einstein almost 100 years ago, describes gravity as the property of curved space-time. Its natural settings are those of stars, galaxies, and the Universe. Bringing general relativity into earth-bound small laboratories would seem like science fiction. This fiction has, however, been made into a reality during the past decade in optics labs, where a number of general-relativity phenomena have found their optical analogue, for example, light-absorbing “black holes” generated by metamaterials. This development has expanded the frontier of optics in new and fascinating ways. However, in all these analogue experiments, the electromagnetic (EM) wave packets always propagated along geodesic trajectories—paths of the shortest distance that connect two points in curved space-time. Can an optical beam travel along a path in curved space that is not geodesic?

In this theoretical paper, we show that the equations describing EM wave propagation in curved space actually contain solutions that correspond to accelerating optical beams traveling along nongeodesic paths.

A curved space for optical beams can be created by covering the surface area of a three-dimensional body (a sphere, for example) with a waveguiding layer that confines the light in it through total internal reflection. The dynamics of the EM field in such a curved space is a generalization of the flat-space Maxwell equations. The new solutions we have found are fascinating: They are wave packets accelerating along nongeodesic paths while changing and recovering their waveforms periodically. Our analysis reveals that these wave packets are the result of an interplay between the forces exerted on the optical beam: the true force originating from the curved space geometry and the interference effects arising from the shape of the wave packet which act as a fictitious force.

The work we have presented here offers a way to manipulate EM wave packets on curved surfaces through the design (or shaping) of the launch beam. It should find applications in near-field microscopy and EM-energy focused at predesigned locations, where control of EM fields and plasmonic effects on the surface of three-dimensional metallic bodies is essential.

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Issue

Vol. 4, Iss. 1 — January - March 2014

Subject Areas

Optics

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Images

Figure 1
(b) Sketch of a surface of revolution (cone). The EM field is restricted to propagate within the surface area by a thin waveguide layer. (a)–(c) The evolution of the envelope of an accelerating beam ( $ψ$ ) on the surface area of a cone (a) and of an hyperboloid (both with 10-cm height, 3-mm base radius, and a propagation constant of $1.2 e 7 m^{- 1}$ ). (c) The dashed white line displays the propagation of the same beam in flat space, projected on the surface of revolution. The beam apertures are (a) 9 mm and (c) 6 mm, with main lobes of widths of (a) $30 μ m$ and (c) $33 μ m$ .
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Figure 2
Periodic accelerating beams constructed from discrete spatial frequencies as they propagate on various cylinders. This wave packet travels along circular trajectories while bending to very large nonparaxial angles. (a) The wave packet consists of several parallel beams that depend on the initial choice of $D_{n}$ ; only every fifth spectral function is populated, each with $D_{5 n} = 1$ . (b) Schematic illustration of the periodic accelerating beam in $k$ space. The beam is constructed from discrete spatial frequencies that reside in a half-circle in $k$ space. This beam is a superposition of only forward-propagating waves. (d) Periodic accelerating beam on a cylinder, displaying a single intense main lobe ( $D_{n} = 1$ ). (c),(e) The beams from (a) and (d) propagating on a surface of a cylinder.
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Figure 3
Nonparaxial accelerating wave packets as they propagate on two kinds of surface, with (b) negative and (c) positive curvature. These wave packets propagate on various trajectories determined by the initial beam launched and the curvature of space. The propagation of the same beam on the surface of a cylinder is displayed in (a) for comparison. The spectral functions vary during propagation; in (b), at some propagation distance, the beam stops accelerating because all spectral frequencies become evanescent, and in (c), the trajectory of the beam becomes nonconvex. (The wavelength of the light in all figures is $0.5 μ m$ .)
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Figure 4
(a)–(c) Profile of a nonlinear accelerating wave packet in curved space under (a) defocusing and (b),(c) focusing Kerr nonlinearity. The profile differs from the nonlinear accelerating beam: For the defocusing case, the lobes are wider than the linear Airy beam, whereas for the focusing case, the lobes are much thinner. (d)–(f) Evolution of the nonlinear wave packets [of (a)–(c)] on the surface area of a hyperboloid with random noise (of 2%). (d) Defocusing nonlinearity supports stable propagation, whereas (e) for strong focusing, the beam become unstable to noise. (f) Evolution of the beam of (e) under different surface parameters can stabilize the beam.
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