Self-accelerating beams, i.e. beams which maintain their shape while propagating along curved trajectories, are raising great interest in recent years.  The Airy wave-function was the first to exhibit the above property and was suggested in the framework of quantum mechanics by Berry and Balazs more than 30 years ago, as a solution to the Schrödinger equation for a free particle. However, only in 2007, optical Airy beams were suggested and observed in free-space light beams as a solution of the paraxial Helmholtz equation [1], which is analogues to the Schrödinger equation. Self-accelerating free-space beams which accelerate along arbitrary convex trajectories have also been realized. Recently, a new group of accelerating free-space beams were introduced, i.e. the Mathieu and Weber beams [2]. These beams are exact solutions of the non-paraxial Helmholtz equation, and therefore enable one to achieve rapid acceleration, having much sharper angles with respect to the optical axis. Self-accelerating beams usually have two additional interesting features: they are "diffraction-free" - i.e. preserve their shape while propagating,  and exhibit "self-healing" – even if part of the beam is blocked by an obstacle, it returns to its original shape. While these features describe infinite energy self-accelerating beams, a finite energy beam can be obtained by truncating the long transverse tails of these beams, e.g. using an exponential or Gaussian envelope. In this case, the generated beam will exhibit these features only over finite distance.

A unique feature of the Airy beam is that it maintains its shape also in a two-dimensional system, consisting of a single transverse coordinate in addition to the propagation coordinate. Therefore, it can also exist as a surface waves, such as surface-plasmon-polariton (SPP) wave. SPPs are electromagnetic-induced polarization waves that are coupled to collective electron oscillations in the metal and propagate along the interface between a dielectric and a metal layer. The Mathieu and Weber beams are solutions of the two-dimensional Helmholtz equation and are therefore defined with only a single transverse coordinate and a propagation coordinate. However, up until now all the demonstration of non-paraxial accelerating beams were done in free-space, with a broad beam (approximating a plane-wave) in the third dimension. The realization of these non-paraxial accelerating beams in a two dimensional system remained an open challenge, since they need to be excited from a free-space beam, having a different wave-vector and in addition their transverse amplitude and phase distribution should be defined with sub-wavelength resolution.

In this work, we demonstrate and examine, both numerically and experimentally, the first realization of both self-accelerating surface plasmon beams along arbitrary caustic curvatures [3],  and rapidly accelerating plasmonic Mathieu and Weber beams [4]. These demonstrations are realized in a low-dimensional system, as they are defined, and in particular, as plasmonic surface waves.

In order to realize these unique waves as surface plasmon waves, several fundamental challenges, owing to the plasmonic nature of the waves, should be addressed. First, coupling a free-space wave into a surface-plasmon wave requires a compensation momentum to match the two wave-vectors of the beams. Second, owing to the limited propagation length of surface-plasmons and the limited measurement range of their characterization tools, the resulting beams should be formed directly in the near-field, before they decay. Third, unlike planar phase plates, surface-plasmons are excited over a finite propagation distance and therefore their phase cannot be simply defined at a specific one-dimensional plane. Fourth, dynamic tools for controlling the wavefront of free-space beams, like spatial-light-modulators, do not exist for surface-plasmons. To overcome these fundamental challenges, we introduce a new class of binary plasmonic holograms, which are designed specifically for the near-filed [5]. This robust holographic scheme provides complete control over the amplitude and phase of surface-plasmons, thereby enabling the generation of any desired plasmonic light beam.

[1] G. Siviloglou, J. Broky, a. Dogariu, and D. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007)

[2] P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).

[3] I. Epstein and , “Arbitrary Bending Plasmonic Light Waves“, Phys. Rev. Lett. 112, 023903 (2014).

[4] Ana Libster-Hershko*, I. Epstein* and A. Arie, “Rapidly Accelerating Mathieu and Weber Surface Plasmon Beams“, Phys. Rev. Lett. 113, 123902 (2014).

[5] I. Epstein, Yigal Lilach and A. Arie, “Shaping Plasmonic Light Beams with Near-field Plasmonic Holograms“, JOSA B 31, 1642 (2014).