Accelerating Self-Imaging: the Airy-Talbot Effect

Abstract

We present self-imaging of optical waves along curved trajectories, theoretically and experimentally. Unlike the Talbot effect, the field wave need not be periodic. Paraxially, selfimaging persists indefinitely, while non-paraxially is limited by overall bending angle. OCIS codes: (050.1960) Diffraction theory; (070.3185) Invariant optical fields;In the Talbot effect, first observed in 1836, a periodic paraxial optical field pattern self-reproduces itself at constant intervals. The effect was explained by Lord Rayleigh in 1881 , who gave the mathematical conditions of the effect, which is also called self-imaging. The Talbot effect has many manifestation in optics, from interferometry and optical testing [1], quantum optics [2], high numerical aperture, nonparaxial illumination [2] and can also appear with incoherent light, under certain conditions [1]. The Talbot effect has also attracted research interest in other physical fields such as matter waves [3] and x-ray phase imaging [4].So far, research was focused on self-imaging along a straight propagation line [5]. That is, the field is reproduced strictly in the plane normal to the propagation direction. Unrelated to self-imaging, the recent years have seen a surge in the study of accelerating beams -optical beams whose trajectory is bent, as a consequence of interference effects. These accelerating beams, were introduced into the optical domain in 2007 [6] and subsequently attracted much attention. Applications of such accelerating beams include particle manipulation [7], imaging and microscopy [8] and accelerating pulses [9]. The acceleration is not limited to small angles: in 2012, self-accelerating wavepackets of Maxwell's equations were discovered [10]. Such beams that bend almost all the way to 180°, as demonstrated in experiments [11]. Soon thereafter, other kinds of non-paraxial accelerating beams were discovered [12]. It is therefore natural to ask: is it possible to have an accelerating (self-bending) self-imaging effect?Here, we present the Airy-Talbot effect: self-imaging of wavepackets along accelerated (bent) trajectories. While the concept seems similar to the Talbot effect, the underlying physics is different. Unlike the Talbot effect, the selfimaging accelerating wavepacket need not be periodic, but instead can have an almost arbitrary shape and is described as a sum of shifted fundamental accelerating beams, at a constant interval. We present examples in two regimes. The first is the paraxial regime, where accelerating beams are Airy beams, and the acceleration is along a parabolic trajectory. In this case, the ideal (infinite energy) beams can self-image indefinitely, while finite energy beams can self-reproduce a finite number of times, in clear correspondence to the Talbot effect. The second regime is the non-paraxial regime, where the fundamental accelerating beams display a half-Bessel shape, and the acceleration is along a half-circle. In the non-paraxial regime, the self-imaging is azimuthal -the beam selfreproduces along planes of cons...