Nondiffracting Accelerating Wave Packets of Maxwell’s Equations

We present the nondiffracting spatially accelerating solutions of the Maxwell equations. Such beams accelerate in a circular trajectory, thus generalizing the concept of Airy beams to the full domain of the wave equation. For both TE and TM polarizations, the beams exhibit shape-preserving bending which can have subwavelength features, and the Poynting vector of the main lobe displays a turn of more than 90°. We show that these accelerating beams are self-healing, analyze their properties, and find the new class of accelerating breathers: self-bending beams of periodically oscillating shapes. Finally, we emphasize that in their scalar form, these beams are the exact solutions for nondispersive accelerating wave packets of the most common wave equation describing time-harmonic waves. As such, this work has profound implications to many linear wave systems in nature, ranging from acoustic and elastic waves to surface waves in fluids and membranes.

Figure 1

Accelerating beams of the Maxwell equations. (a) The forward-accelerating beam of TE polarization ($\alpha =150$) reaches a trajectory which is almost vertical, while exhibiting nondiffracting (shape-preserving) acceleration. (b) The paraxial approximation yields an Airy beam which accelerates only for a short distance before breaking up. (c), (d) The forward propagating beam of TM polarization ($\alpha =150$). The power transfers from the $x$ polarization (c) to the $z$ polarization (d). All figures are simulated with $\lambda =1\mu \mathrm{m}$ and in a square of $35\mu \mathrm{m}\times 35\mu \mathrm{m}$. The diagram in the center shows the Fourier plane where the beam is confined in a circle describing the propagating plane waves. The top (bottom) half stands for the forward (backward) beam.

Figure 2

Properties of the Bessel-like accelerating beams. (a) The Poynting vector of the TE/TM polarization, when launched with an initial angle and bends by 130°. (b) The self-healing property “reviving” the first lobe that was initially cut out. The dashed black curve describes the original trajectory of the first lobe of a perfect accelerating beam. (c) A periodic accelerating beam composed of two jointly accelerating components $\alpha =150$ and $\alpha =165$ with equal coefficients. All figures are simulated with $\alpha =150$, $\lambda =1\mu \mathrm{m}$ and in a square of $30\mu \mathrm{m}\times 45\mu \mathrm{m}$ (a) or $35\mu \mathrm{m}\times 35\mu \mathrm{m}$ (b), (c).