Nonparaxial Mathieu and Weber Accelerating Beams

We demonstrate both theoretically and experimentally nonparaxial Mathieu and Weber accelerating beams, generalizing the concept of previously found accelerating beams. We show that such beams bend into large angles along circular, elliptical, or parabolic trajectories but still retain nondiffracting and self-healing capabilities. The circular nonparaxial accelerating beams can be considered as a special case of the Mathieu accelerating beams, while an Airy beam is only a special case of the Weber beams at the paraxial limit. Not only do generalized nonparaxial accelerating beams open up many possibilities of beam engineering for applications, but the fundamental concept developed here can be applied to other linear wave systems in nature, ranging from electromagnetic and elastic waves to matter waves.

Figure 1

(a) Illustrations of different trajectories associated to Mathieu accelerating beams (MABs); (b) amplitude of the MABs at $a<b$ (blue, dashed), $a=b$ (black, dotted), and $a>b$ (red, solid) at $z=0$; (c) intensity (top) and phase (bottom) of the Fourier spectra of the MABs in (b) as a function of the normalized spatial frequency ($f_x/k$); (d)–(f) Side-view propagations of the MABs in (b), where the $z=0$ position is marked by the white dashed line. Note that the horizontal and vertical scales are identical in the figures.

Figure 2

Amplitude at $z=0$ (a) and phase of the Fourier spectra (b) of a Weber accelerating beam (WAB) (blue, solid) and an Airy beam (red, dotted); (c) and (d) depict the side-view propagation of the WAB (c) and the Airy beam (d) corresponding to the profiles in (a), where the white dashed line marks the $z=0$ position; (e) illustrates the FWHM of the main lobe of the beam versus the propagation length $z$ for different beams with the same main lobe size at input.

Figure 3

Experimental side-view photography of the generated MABs at $a<b$ (a), $a=b$ (b), and $a>b$ (c), a WAB (d) and an Airy beam (e) taken from scattered light, where the dotted curves plot the predesigned trajectories. (f) and (g) are the transverse intensity distributions recorded at $z=0$ (f) and $z=50\mu \mathrm{m}$ (g) as marked by the dashed lines in (d) and (e), where the top and middle rows correspond to the WAB and Airy beam, respectively, and the bottom depicts the intensity profiles taken along the dashed lines (marked in the transverse intensity patterns). The dashed (red) and solid (blue) lines correspond to the Airy beam and the WAB, respectively.

Figure 4

Numerical simulations (top) and experimental results (bottom) of the self-healing properties of a MAB (left) and a WAB (right). (a) and (b) are for beams corresponding to Figs. 1d, 2c, and (c) and (d) correspond to Figs. 3a, 3d, respectively. The main lobe of the beam is blocked by an opaque obstacle (white dot) at a certain propagation distance. The diameter of the white dots in (a) and (b) is $1\mu \mathrm{m}$, and that in (c) and (d) is about $50\mu \mathrm{m}$.