Shape-Preserving Accelerating Electromagnetic Wavepackets in Curved Space

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.

The complex dynamics of particles and of electromagnetic (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][2][3], to Bose-Einstein [4][5][6] and optical systems, which have had a major success in demonstrating such phenomena [7][8][9][10][11][12][13]. For example, metamaterials enabled creating analogies to black holes, by engineering the (EM) properties of the material through which light is propagating [8][9][10]. Another example is using a moving dielectric medium that acts as an effective gravitational field on the light [12]. This idea was demonstrated experimentally by employing ultrashort pulses in an optical fiber to create an artificial event horizon [13]. Another route for such studies is to create curved space by engineering the geometry of the space itself. This idea, suggested in 1981 [14], 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 [15], where pioneering experiments studied light propagating in a thin film waveguide attached to the curved surface area of a three-dimensional (3D) body [16]. However, thus far, in all of these experiments and theoretical studies on General Relativity concepts with EM waves -the wavepackets were propagating on geodesic trajectories, which are naturally the shortest path, analogous to straight lines in flat geometry. But, do wavepackets 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 wavepackets can exhibit periodically-shape-invariant spatiallyaccelerating dynamics in curved space, propagating in non-geodesic trajectories that reflect interplay between the curvature of space and interference effects arising from initial conditions.
We study these beams 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 experiments in curved space to study basic concepts of General Relativity, where the entire dynamics in non-geodesic.
Thus far, however, accelerating wavepackets remained strictly within the realm of flat space.
Since the dynamics of EM waves in curved space is significantly different from that in flat space, a natural question to ask is whether accelerating wavepackets can at all exist in curved space, and if they do how do their features differ from those in flat space. In other words, are there wavepackets that travel along non-geodesic trajectories in free-space without contradicting the basic concepts of General Relativity?
Consider EM waves restricted to exist in 2D curved surface. This can be achieved by covering the surface area of a 3D shape (a sphere, for example) with a thin homogenous layer of a material with a higher refractive index. Such a layer acts as a waveguide, keeping the light confined inside it due to total internal reflection (Fig. 1). The dynamics of EM fields in curved space can be described by the 3D Maxwell equations in general coordinates [40]: where the polarization P D E α α α ≡ − can generally be nonlinear in the electric field. Notice that the second term does not appear in homogenous flat space: it arises strictly due to the curved space geometry.
We are interested in the evolution of the electric field in a general surface of revolution.
First, we introduce the metric of such a surface. These surfaces are parameterized by is the angle of rotation and u −∞ < < ∞ is a general parameterization of the surface along its axis of the revolution. . Every point in 3D space ( ) r can be described by the two coordinates on the curved surface ( ) Here 0 n is the refractive index in the surface layer, 0 k is the vacuum wavenumber and q has First, we focus on the paraxial regime, and derive the equation for the slowly varying varies with the algebraic factor γ , for the power to be conserved. This yields the paraxial equation for a general surface of revolution: where the effective one-dimensional potential depends on the determinant of the surface , and find the accelerating beam in curved space to be: Where a is a constant with units [ ] Equation (7) defines acceleration trajectories that depend on the metric determinant.
Consequently, the acceleration trajectory is different for every surface of revolution, and can even become non-convex in x , as shown in Fig. 1. Notice that, generally, the accelerating solution of Eq. (5) is not shape-preserving because 2 ψ varies with z . However, it is self-similar and can become narrower or broader during propagation, according to the geometry of the specific surface [41].
To understand the origin of the non-geodesic trajectory, we introduce a particle model to describe the trajectory of the main lobe of the accelerating beam. We account for the interference effect through an inhomogeneous term in the geodesic equation: where λ is an affine parameter which, in this case, can be the line element. This equation describes the motion of a particle in a surface of revolution under the influence of a force, where F has the dimensions of force per unit of mass when λ is taken to be time. Obviously, F is a "fictitious" force, because no real force is acting here. Constraining the motion of the particle to Here, the fictitious force, Thus far, we generalized the paraxial accelerating beam to curved space, and showed the various trajectories possible which are not the natural geodesics of these surfaces, but we did not find the actual solutions as of yet. To do that, we construct a beam propagating on the trajectories defined by Eq. (7) and also fulfills periodic boundary conditions, as necessary for surfaces of revolution. Such solutions are naturally periodic [42] and they are obtained from the Airy solution defined on the universal covering space, using Eq. (4): This finite range within which the spatial frequencies of the accelerating beam can exist has immediate physical consequences: such a curved-space accelerating beam carries finite power, because it is constrained to a circular perimeter and constructed from a finite number of spatial frequencies, due to the cut-off. This finding has an important implication: having a finite power, one can now define a center of mass for the accelerating beam. It is important to emphasize that although the self-reconstructing structure of the wavepacket travels along a nongeodesic trajectory, the center of mass travels along a geodesic trajectory as in [17,18].
However, almost all the applications of accelerating beams rely on light-matter interactions, where the important parameter is the local intensity and not the center of mass, e.g., acceleration of particles [30], formation of curved plasma channels [31], laser machining [43], to name a few out of many. For all such applications, what matters is the accelerating main lobe where the intensity is the highest, while the fact that the center of mass is propagating on a straight line is unimportant.
In examining the structure of the curved-space accelerating beam, we notice that it can be different from the Airy beam whose envelope is monotonically decaying. Here, the shapepreserving wavepacket accelerating in curved space can have several parallel beams whose number is set by the initial choice of the spectral components m C .
The accelerating solution in curved space is propagating on the curve defined by Eq. (7)  We apply transformation of coordinates that simplifies the equation for any surface of revolution.
We set: Clearly, the non-paraxial case is more complicated than the paraxial one: Eq. (11) We choose 0 n D = for any n q that is not between 0 and π , meaning that we allow only forwardpropagating waves (i.e., we assume that the backward-propagating waves are not excited at 0 z = ). This wavepacket is constructed from a discrete set of spatial frequencies that fulfill the periodic boundary conditions: ( ) 0 arccos n q n qR = (see Fig. 2). The spectrum is now limited from above, because at a high enough spatial frequency the propagation constant becomes imaginary and the spectral function becomes evanescent. Here, we are not interested in the evanescent waves, hence we set their initial population to zero ( 0 n D = for those modes). This non-paraxial accelerating beam carries finite power. In fact, the solution can support several parallel beams accelerating (bending) in parallel, as in the paraxial case, for a suitable choice of n D . As for the nonparaxial flat-space accelerating beams [33] this nonparaxial curved-space wavepacket is approximately shape-invariant because it is a superposition of only forward propagating waves ( 0 n q π < < ). If the counter-propagating waves were to be taken in the superposition, the beam would have been fully shape-preserving. Nevertheless, this wavepacket (Fig. 2)  where every β gives a beam with a different structure. Thus, every superposition of such beams (of various values of β ) also forms a periodically shape preserving accelerating beam.
Having solved for the simplest non-paraxial surface of revolution (a cylinder, where the metric is not z-dependent), the natural question to ask is whether a non-paraxial accelerating shape-invariant solution can exist for surfaces with a z-dependent curvature. Finding these kinds of solutions is especially challenging, because they cannot rely on the symmetry between all space coordinates, since this symmetry is inherently broken. Going back to Eq. (3.1), we simplify the equation using: ( ) ( ) , which cancels the term with the first derivative in respect to z , yielding: This equation is a Helmholtz type equation with two differences: (1) there is an additional term that gives a z -dependent addition to the effective wavenumber, and (2) the z -dependent metric multiplies all the terms except for the derivative with respect to x . We want to transform Eq. ( ) where the ± sign stands for the positive and negative curvatures, respectfully. Following the same approximation regarding the slow change in curvature on the scale of a wavelength, we assume that . We find the accelerating wavepackets on these surfaces to be: These accelerating solutions are traveling along a non-circular trajectory, bending to very large angles. This can be easily seen in k -space, where the transverse spatial frequencies vary while the beam is propagating in the z direction. The change in the spatial frequencies can cause a propagating mode become evanescent while propagating in z . When this disappearance of modes occurs, the wavepacket is no longer shape-invariant. One of the most fascinating features is that the trajectory can even flip to the other direction -and accelerates towards the direction of the other lobes. The reason is that the metric changes in the z direction, however after some z value this is no longer the direction normal to the wavefront, due to non-paraxial trajectory. This interesting feature could not be seen in the paraxial case. Naturally, this wavepackets is also constructed only from a discrete set of spatial frequencies: The trajectory is the same as in the linear case (Eq. (7)) .
The only free parameter in our solution is the initial conditions. To find the wavefunction, we assume that the nonlinear accelerating beam decays for x → ∞ , thus the nonlinear term in Eq.
(16) is negligible for x → ∞ . Therefore we choose the initial condition to be ( ) u C Airy x = ⋅ for x → ∞ . Typical shape-preserving solutions are shown in Fig. 3. These wavepackets are propagating in a self-similar fashion, similar to their linear counterparts. However, for the focusing case we find beams with narrower lobes than for the linear beam, and for the defocusing case we find beams with broader lobes. The solution for the focusing case exists for  [26]). Next, we check the stability of our solutions and find the solution for the defocusing case to be stable under random "white" noise, whereas the selffocusing solutions become unstable after some propagation distance. The most interesting feature is that the stability of the beam depends on the curvature: by changing the parameters of the surface we can make the beam stable for considerably larger distances (possibly even indefinite), as shown in Fig 3. This suggest on option for stabilizing nonlinear accelerating beams using the curvature of space, this is directly related to the instability of solitons in negatively curved space [44]. To augment this nonlinear section, we note that other saturable nonlinearities can be handled in a similar fashion, as was done in [26] for flat space.
To summarize, we have found linear and nonlinear, paraxial and non-paraxial, spatiallyaccelerating wavepackets in curved space, thereby introducing the concept of accelerating beams to curved space geometry. This work raises many further interesting ideas. The relation of this work to General Relativity opens up many ideas for future exploration. For example, the current work shows that wavepackets in curved space can be controlled by specifically designing their input wavefront. In principle, this means that one can predesign a wavefront that would be able to overcome (compensate for) effects of Gravity. Indeed, we are currently working on the nonlinear version of this idea, where the accelerating wavepacket is what causes the effective curving in space, in an optically nonlinear medium.