Hydrodynamics of noncircular vortices in beams of light and other two-dimensional fluids

The motion of noncircular two-dimensional vortices is shown to depend on a form of coupling between vortex ellipticity and the gradient of fluid density. The approach is based on the perspective that an elliptic vortex can be described as the projection of a virtual construct, a circular vortex with a symmetry axis that is tilted with respect to the direction of propagation. The resulting kinetic equation offers insights into how tilt and vortex velocity coevolve in few-body nonequilibrium settings such as vortex pair nucleation and annihilation. The model is developed and applied in association with optical vortices, and optical experiments are used to verify its predictive power. It is valid for quantum fluids and classical hydrodynamics settings as well.

Figure 1

Tilted vortex perspective. A 2D elliptical vortex in the transverse plane can be viewed as the projection, along the tilt axis, of a 3D vortex. The ellipticity can then be described by an azimuthal orientation angle $\xi$ and a polar lean $\theta$.

Figure 2

Experimental generation and measurement of a tilted optical vortex. (a) Schematic of the experimental setup. Here DO denotes diffracted order. (b)–(e) Projected holograms for vortices with $x_0=0.8$ mm and four sets of encoded tilt angles with corresponding (f)–(i) measured amplitudes and (j)–(m) phases. A 2D amplitude fit of (f) at $z=0$ mm using Eq. (17) gives a measurement of $w_0=1.1$ mm. (j)–(m) show measured tilt angles (insets) at $z=0$ mm calculated by fitting the phase data to Eq. (17) with fitting parameters $x_0$, $y_0$, $\xi$, and $\theta$.

Figure 3

Motion of a single vortex driven by tilt and background field gradients. Shown on top are the mode snapshots of an off-center tilted vortex calculated from Eq. (23) with its location (dots) and predicted trajectory (white arrow) for beam parameters $\lambda =633$ nm, $w_0=1$ mm, $x_0=0.75w_0$, ${\xi }_i={80}^{\circ }$, and ${\theta }_i={60}^{\circ }$. Brightness is amplitude and color shading represents phase. The middle shows a cropped window of the background field evolution, along with background gradient vectors to visualize each contribution to vortex velocity: Cyan (farthest right) arrows show velocity from the first term of Eq. (22), $v_{\phi }$, while orange (farthest left) arrows show the second term $v_{\xi ,\theta ,\rho }$; green (middle) arrows are the sum of these terms ($v_{\mathrm{total}}$). Shown on the bottom is the evolution of individual components and resulting total velocity for both the $x$ and $y$ directions, labeled by the corresponding component.

Figure 4

Vortex on the shoulder of the Gaussian beam. Velocity predictions in mm/m for both the $x$ and $y$ directions are shown for a single vortex with various tilts. Small circles are theoretical predictions from Eq. (24) and crosshairs are the measured vortex velocities with elliptical regions bounding the experimental uncertainty. Dashed semicircles show the predicted velocities for a specified polar angle ${\theta }_i$ as the azimuthal orientation ${\xi }_i$ is varied. Dotted lines are predictions for a fixed ${\xi }_i$ and varying ${\theta }_i$. In the absence of tilt, the vortex moves vertically, but one should note that specific combinations of tilt can also lead to a purely upward velocity. For severe tilt (${\theta }_i$ near ${90}^{\circ }$), the vortex can be made to move essentially orthogonal to its untilted trajectory.

Figure 5

Experimentally measured and analytically predicted vortex trajectory with $\xi =22.5^{\circ }$ and $\theta ={50}^{\circ }$. The experimentally measured vortex position as a function of propagation is shown for both the $x$ [in red (light gray)] and $y$ [in blue (black)] directions. Error bars represent one standard deviation of five independent measurements. The dashed lines are the analytical predictions from Eq. (24). Fitted slopes (not shown) calculated using the propagated uncertainties for both the $x$ and $y$ motions are well aligned with the analytical prediction and are used to create a single data ellipse in Fig. 4.

Figure 6

Two vortex motion from background field components at $z=0$. For a same-charge vortex pair (top row) and opposite-charge pair (bottom row), the background field components for vortex 2 are shown. Columns 2 and 3 show velocity contributions from the background phase and amplitude of vortex 1. The last column shows the background Gaussian amplitude contribution to velocity.

Figure 7

Evolution of tilt for an oppositely charged vortex pair as described by Eq. (27). For the left vortex in the pair, located at $x=-x_0$ at $z=0$, the evolutions of ${\xi }_{\mathrm{left}}$ and ${\theta }_{\mathrm{left}}$ are shown. Similarly, for the right vortex located at $x=+x_0$ at $z=0$, ${\xi }_{\mathrm{right}}$ and ${\theta }_{\mathrm{right}}$ are also plotted. Each curve is labeled by the corresponding angle.

Figure 8

Tilt perspective on nucleation and annihilation. A pair of identical vortices nucleates at the top left, but subsequently evolves so that the vortices are of opposite charge at the midpoint before becoming once again parallel as they annihilate at the lower right. The envelope of orientations is a Möbius strip. The results shown were obtained by combining Eqs. (28) and (27) and the tilt vectors shown are of equal magnitude.

Figure 9

Experimental dynamics of opposite-charge vortex annihilation. Measured (symbols) and analytical (lines) vortex trajectories for (a) the $x\text{−}y$ plane measurement for which each green (circle) and red (triangle) symbol pair is a measurement at a specific $z$, (b) the vortex separation along $z$, and (c) the $y$ position as a function of propagation.

Figure 10

Two-vortex tilt evolution. The left plot shows a comparison of the measured (dots) $\xi$ for the left and right vortices with propagation compared to the predicted (solid line) $\xi$ from Eq. (27) for each vortex. Similarly, the right plot shows the comparison of $\theta$ for both experiment and theory. The shaded regions show the rms error between the data and the predicted values.

Figure 11

Comparison of fluid models for predicting vortex-pair motion. The quarter-circle trajectory of a vortex in an opposite-pair annihilation event (blue solid line) with arrows shows the predicted velocities calculated from three different background fields, as described in the text and in Table 1. The black incompressible fluid prediction (vertical arrows), addition of fluid compressibility in green (light gray arrows), and inclusion of vortex tilt in red (tangent arrows) are shown for the full kinetic theory of Eq. (21). Only the compressible fluid plus tilt prediction accurately describes the vortex motion.

Figure 12

Vortex-pair holograms and measured $z=0$ mm field. The left column shows a vortex-pair hologram with a close-up view of the vortex configuration. A white dashed outline traces over each fork to highlight them. The measured field amplitude and phase are shown in the right column. The inset on the amplitude plot shows the measured beam waist and vortex displacement, from a 2D amplitude fit using Eq. (29).

Figure 13

Snapshots of the measured amplitude and phase with propagation show the vortex trajectory and annihilation event. Amplitude data are plotted with limited contours to highlight tilt evolution in the amplitude structure, but the data have resolution equal to that of the phase measurement. Data far beyond the annihilation point confirm that the vortices do not reemerge.