Observation of Accelerating Wave Packets in Curved Space

We present the first experimental observation of accelerating beams in curved space. More specifically, we demonstrate, experimentally and theoretically, shape-preserving accelerating beams propagating on spherical surfaces: closed-form solutions of the wave equation manifesting nongeodesic self-similar evolution. Unlike accelerating beams in flat space, these wave packets change their acceleration trajectory due to the interplay between interference effects and the space curvature, and they focus and defocus periodically due to the spatial curvature of the medium in which they propagate.

Popular Summary

Light can be described as rays that travel in straight lines, which is convenient for explaining a large number of phenomena such as reflection or propagation through free space. Consequently, natural intuition about light relies on rays, and we expect light to go from one point to another in a straight line. However, light is actually a wave and therefore exhibits numerous features that are unique to waves. In recent years, researchers have shown that optical wave packets (beams) can propagate in a self-accelerating manner, where the structure of a beam is engineered to move along a curved trajectory. This field has attracted major interest, with many potential applications. Here, we take these accelerating beams one step further, demonstrating them in a medium that has a curved space geometry, where the trajectory of the accelerating beam is determined by the interplay between the curvature of space and interference effects arising from the beam’s structure.

The simplest example of a curved object is a sphere because it has the same constant curvature everywhere. Normally, optical beams that are confined to propagate on the surface of a sphere would move along geodesic paths, the largest circle on the sphere’s surface. But, as we show theoretically and experimentally, one can shape the structure of a beam such that it will accelerate and evolve in a shape-preserving manner on a nongeodesic line, such as a circle close to the North Pole. We use a thin hemispheric glass shell as the curved-space landscape for the light, and we couple a specifically shaped beam into this glass waveguide. The brightest lobe of this beam bends away from the shortest (geodesic) path, which is the trajectory that light would normally take on the sphere.

These experiments provide new avenues for controlling trajectories of light in nonplanar 3D settings and offer new opportunities for emulating general relativity.

Figure 1

(a)–(c) Calculated intensity cross sections of (a) a single transverse eigenmode (mode number $l=1000$, $m=950$) of the spherical shell, (b) the beam formed by abruptly cutting this mode, and (c) the shape-preserving accelerating beam constructed from a Gaussian superposition of modes of the same phase centered around $l_0=1000$, $m_0=950$; the lobes of this mode propagate in nongeodesic trajectories in a self-similar fashion. Experimentally, this beam can be generated by truncating the $l_0=1000$, $m_0=950$ eigenmode with an exponential density filter. The transverse coordinate is given in units of the radius $R$. (d)–(f) Corresponding spatial power spectra of the beams depicted in (a)–(c) as a function of mode number.

Figure 2

(a) Sketch of a spherical shell with the coordinate system marked. The beams propagate in the $\varphi$ direction and their transverse structure, shown in Fig. 1, extends in the $\theta$ direction. The thickness of the spherical shell is marked by the red arrows. In this region the refractive index $n_0$ is higher than in the surrounding medium $n$, and therefore the mode is confined within the shell. The structure of the beams in the radial direction is dictated by Eq. (4). (b) Three accelerating beams that follow three different nongeodesic trajectories. A beam with a larger $l_0-m_0$ index follows an acceleration trajectory with a higher curvature. The larger $l_0-m_0$, the higher the density of the lobes, which corresponds to higher transverse momentum. The red dashed lines are the acceleration trajectories of the beams, while the white dashed line shows the geodesic trajectory, which is the trajectory that a Gaussian beam propagating on a sphere would follow. Notice that each of these beams forms a caustic: the field on one side of the main lobe quickly decays to zero, while the other side of the beam consists of multiple lobes propagating in parallel on nongeodesic trajectories.

Figure 3

(a) Simulated propagation on the surface of a sphere of radius $R=1\mathrm{cm}$, for the accelerating beam of $l_0=1000$, $m_0=950$ (red lines) whose structure is shown in Fig. 2, and for a beam composed of the main lobe only (blue lines). The accelerating beam follows a nongeodesic path, whereas the isolated main lobe follows the expected geodesic line. The trajectories are marked by gray arrows. (b) Propagation evolution of the accelerating beam over a hemisphere (180° on the sphere). The spatial profile of the beam flips after propagating over half the sphere. (c) Angle difference $\mathrm{\Delta }\mathrm{\Theta }$ between the geodesic line and the trajectory of the main lobe of the accelerating beam of (a), as function of the main lobe width, in angular units (deg). When the lobes are very narrow, $\mathrm{\Delta }\mathrm{\Theta }$ has a large absolute value and therefore the acceleration is highly visible, whereas when the width of the lobes is increased, the lobes tend to approach geodesic trajectories. The same principle holds if the size of the input beam is kept fixed but the radius of the sphere is varied. Namely, increasing the radius of the shell has the same effect on the accelerating beam as decreasing the beam’s transverse size.

Figure 4

(a) Experimental setup. A collimated laser beam ($\lambda =532\mathrm{nm}$) is reflected off a spatial light modulator to generate the desired 1D spectrum (Gaussian modal superposition) of the beam, which is then Fourier transformed by a cylindrical lens to form the predesigned shape-preserving accelerating beam. This beam is then coupled to the hemispherical thin glass shell (3 cm radius, $550\mu \mathrm{m}$ width). (b) Experimentally observed propagation trajectory of the beam accelerating on the surface of a spherical shell. The propagation inside the shell is visible as a green beam thanks to the scattering from the red paint of the shell. (c) Zoom-in photograph of the lobes of the accelerating beam, as it propagates within the spherical shell.

Figure 5

(a) Experimentally measured intensity profiles of the self-similar accelerating beam (red lines) and of the single-lobe beam (blue lines), at various propagation stages on the spherical surface. The single-lobe beam is generated by taking just the main lobe of the accelerating beam at the input plane. This beam experiences diffraction broadening while propagating on a geodesic line. On the other hand, the accelerating beam is shape preserving up to 50°, while it propagates on a nongeodesic trajectory. This is clearly observed by the increasing shift between the main lobe of the accelerating beam and the single-lobed beam. (b) The experimentally measured angular difference between the single-lobed beam and the main lobe of the accelerating beam, as a function of angular propagation distance on the spherical surface. The parabolic relation of this distance, and the fact that our structured beam maintains its shape for a large distance, experimentally confirm the accelerating shape-preserving nature of our beam.