Accelerating Self-Imaging: The Airy-Talbot Effect

We present a new type of self-imaging phenomenon: self-imaging along curved trajectories. Unlike the Talbot effect, where self-imaging occurs for periodic wave patterns propagating along a straight line, here the field is generally not periodic and is self-imaged along curved trajectories. In the paraxial regime, self-imaging along a parabolic trajectory can ideally go on indefinitely. In the nonparaxial regime the self-imaging is along a circular trajectory and lasts as long as the beam bends. We demonstrate this accelerating self-imaging effect experimentally, and discuss generalizations to higher dimensions.

Figure 1

(a) Intensity plot of the one-dimensional Airy beam. (b) Intensity plot of the beam of the form of Eq. (4), exhibiting the Airy-Talbot effect. The beam is comprised of an addition of 20 Airy beams, each multiplied by a random coefficient, and shifted along $u$ with $\mathrm{\Delta }=3$, corresponding to a longitudinal periodicity of $\mathrm{\Gamma }=4\pi /3$. (c) Intensity cross sections at three planes of self-imaging, $z=\mathrm{\Gamma }$, $2\mathrm{\Gamma }$, and $3\mathrm{\Gamma }$, marked by dashed lines in (b). The cross sections are shifted along the parabolic trajectory in order to make the comparison clearer.

Figure 2

Experimental realization of the Airy-Talbot effect. (a) Sketch of the experimental setup. (b) Intensity plot in the xz plane of the beam exhibiting the Airy-Talbot effect. The beam was generated using a SLM, by imposing a periodic phase profile in addition to the cubic phase profile. (c) Intensity cross sections along three planes of self-imaging, marked by dashed lines in (b). In order to make the comparison clearer, the cross sections are shifted along the parabolic trajectory. In addition, the intensity was normalized to account for the intensity attenuation caused by residual diffraction broadening due to the finite aperture of the beam.

Figure 3

(a) Intensity pattern arising from the sum of 20 nonparaxial half-Bessel-function beams with random coefficients. The nonparaxial, seemingly random field self-reproduces periodically with period π/10. Dashed lines mark two angles of self-imaging. (b) Intensity profile at cross sections defined by the dashed lines in (a).