Winding light beams along elliptical helical trajectories

Conventional caustic methods in real or Fourier space produced accelerating optical beams only with convex trajectories. We developed a superposition caustic method capable of winding light beams along nonconvex trajectories. We ascertain this method by constructing a one-dimensional (1D) accelerating beam moving along a sinusoidal trajectory, and subsequently extending to two-dimensional (2D) accelerating beams along arbitrarily elliptical helical trajectories. We experimentally implemented the method with a compact and robust integrated optics approach by fabricating micro-optical structures on quartz glass plates to perform the spatial phase and amplitude modulation to the incident light, generating beam trajectories highly consistent with prediction. The theoretical and implementation methods can in principle be extended to the construction of accelerating beams with a wide variety of nonconvex trajectories, thereby opening up a route of manipulating light beams for fundamental research and practical applications.

Figure 1

Schematic view and construction principle of an elliptical helical beam. (a) An illustration of the realization of a helical beam. (b) The transverse acceleration dynamics of 1D sinusoidal beam described as a bundle of rays compared with (c) the well-known Airy beam accelerating along a parabolic trajectory. The rays represent key spatial frequencies of light emanating from the initial plane (the thick black line at the bottom), and their interference forms a local intensity maximum along a curve.

Figure 2

Experimental implementation. (a) 1D and 2D binary patterns employed to realize grayscale amplitude modulation. (b) Enlarged view of the fabricated sample under the optical microscope and scanning electron microscope (SEM). The scale bar from left to right and from top to bottom is 5 mm, 50 $\mu \mathrm{m}$, 5 $\mu \mathrm{m}$, and 1 $\mu \mathrm{m}$, respectively. (c) Three fabricated samples for generating the 1D, 2D sinusoidal beams and circular helical beam, respectively, with a scale bar of 5 mm. (d) Experimental setup used to generate the accelerating beams as well as measuring their propagation dynamics in free space.

Figure 3

Generation of the basic 1D sinusoidal beam. (a) The designed phase and amplitude modulation. (b) Simulated and experimental intensity distribution for the beam propagating in free space. (c) Quantitative analysis of the main lobe trajectory. Here red dots mark the positions of the intensity maxima at the cross sections and the blue curves are the fitted trajectories. The fitted curve in simulation is $x=50.6+0.49z+54.2sin\left(2\pi z/69.3\right)$, while in experiment is $x=51.5+0.61z+53.4sin\left(2\pi z/70\right)$. The spatial unit is micron for $x$ and millimeter for $z$.

Figure 4

Realization of a 2D sine beam and a helical beam based on the 1D sine beam. (a) The 3D plot of the trajectory for the main lobe of the 2D sine beam, and its projection to $x-z$ and $y-z$ planes (with an error bar showing possible deviation in determining the position of the intensity maximum limited by our CMOS camera), as well as three selected cross sectional intensity distributions, with a scale bar of $100\mu \mathrm{m}$. Here the fitted curves in $x-z$ and $y-z$ planes are $x=51.5+0.65z+51.5sin\left(2\pi z/71+0.09\pi \right)$ and $y=51.5+0.69z+54.3sin\left(2\pi z/71\right)$ ($x$ and $y$ in microns, and $z$ in millimeters), respectively. (b) The 3D illustration of the trajectory for the main lobe of the helical beam with four cross sectional intensity distributions presented and its projection to $x-z$ and $y-z$ planes. In this case, the fitted curves in $x-z$ and $y-z$ planes become $x=51.5+1.08z+57.6sin\left(2\pi z/70+\pi /2\right)$ and $y=51.5+0.35z+49.6sin\left(2\pi z/70\right)$ ($x$ and $y$ in microns, and $z$ in millimeters), respectively, with an introduced $\pi$/2 phase shift as designed.