Accelerated rotation with orbital angular momentum modes

We introduce a class of light field that angularly accelerates during propagation. We show that the acceleration (deceleration) may be controlled by adjustment of a single parameter, and tuned continuously, down to no acceleration at all. As the angular acceleration takes place in a bounded space, the azimuthal degree of freedom, such fields accelerate periodically as they propagate. Notably, the amount of angular acceleration is not limited by paraxial considerations, may be tailored for large accelerations over arbitrarily long distances, and can be engineered independently of the beam's spatial extent. We discuss how such angularly accelerating light fields can maintain the conservation of angular momentum through an energy exchange mechanism across the field.

Figure 1

(a)–(c) Isotropic vortex phases for $\ell =1,2$, and 3, respectively. (d)–(f) Anisotropic vortex phases for $\ell =1,2$, and 3, respectively. (g) A plot of the phase as a function of azimuthal angle for an isotropic (blue) and an anisotropic (black) vortex of order $\ell =2$. (h) Schematic showing the superposition of two Bessel beams as intensities (with anisotropic vortex phase as insets) and the resulting intensity petal structure, for $\ell =3$.

Figure 2

(a) A Bessel beam can be formed by illuminating a ring slit placed in the Fourier plane of a lens. (b) To create a field comprising a single radial wave vector $k_r$, a single ring slit is illuminated. Superpositions of isotropic Bessel beams with differing radial wave vectors simply require multiple ring slits as shown in (c), while anisotropic superpositions require amplitude modulation inside each of these ring slits, as shown in (d). The corresponding intensities produced for each case are shown in the insets.

Figure 3

The intensity images in the top rows of (a) and (b) show the petal rotation at a fixed rate $\theta =0$, while the bottom rows show the nonlinear rotation $\theta =\pi /3$, for (a) $\ell =1$ and (b) $\ell =3$. The solid white lines indicate the movement of the petals in the linear case and are overlaid on the nonlinear images for reference. The angular difference between the two cases (linear and nonlinear) is given as text in each frame. The columns represent increasing propagation distances sampled from the zoomed-in data set of Fig. 4. The maximum beam intensities in each frame were normalized to unity to aid comparison.

Figure 4

The rotation angle ($\varphi$) of the petal structure as the field propagated. The measured data (symbols) are shown together with the theoretical predictions (lines) for various values of the control parameter $\theta$, from 0 (red), $\pi /5$ (black), $\pi /4$ (green), to $\pi /3$ (blue). The rotation rate was observed to transition from constant ($\theta =0$) to highly nonlinear ($\theta =\pi /3$). Results are shown for two different topological charge values of (a) $\ell =1$ and (b) $\ell =3$.

Figure 5

The nonlinear azimuthal phase of the anisotropic beam results in a rotation angle that varies nonlinearly with propagation distance. This varying angular velocity imparts angular acceleration to the field. The nonconstant angular velocity is shown in (a) and (b) for $\ell =1$ and $\ell =3$, respectively, while the resultant angular acceleration for these examples is shown in (c) and (d). The measured data (circles) are in good agreement with the theoretical predictions (solid curve).

Figure 6

As the beam accelerates the energy in the inner region (petals) is transported to the outer region (rings). This process is reversed during deceleration. Because the field oscillates between the two cases during propagation, there is a continuous flow of energy in and out of the central petal structure of the field. (a) The normalized energy in the two regions (petals and rings) of the field during propagation for $\theta =0$ (red), $\theta =\pi /5$ (black), $\theta =\pi /4$ (green), and $\theta =\pi /3$ (blue). (b) The measured maximum fractional energy exchange as a function of $\theta$. All data are for the $\ell =3$ case.