Angular Momentum of Topologically Structured Darkness

We theoretically analyze and experimentally measure the extrinsic angular momentum contribution of topologically structured darkness found within fractional vortex beams, and show that this structured darkness can be explained by evanescent waves at phase discontinuities in the generating optic. We also demonstrate the first direct measurement of the intrinsic orbital angular momentum of light with both intrinsic and extrinsic angular momentum, and explain why the total orbital angular momenta of fractional vortices do not match the winding number of their generating phases.

Figure 1

(a) Plot of average OAM as a function of generating spiral phase step height $\mu$, as drawn in (b). Three colored points along the curve in (a) correspond to the modes labeled in matching colors in (c). (c) Shows various theoretical fractional vortices for three different $\mu$, modeled at a propagation distance of $\frac{1}{25}{Z}_R$ from a fractional spiral phase plate. Intensity is represented by brightness, and phase is represented by color. The arrows show the flow of transverse Poynting vectors, which are within a scaling factor of local OAM per photon.

Figure 2

Blue dots: Direct experimental measurement of the intrinsic orbital angular momentum of fractional vortex beams generated by noninteger spiral phases of topological charge $\mu$. The blue-rimmed inset shows the experimental alignment: a fractional vortex beam is input into a cylindrical lens such that the vortex chain (green dots) is perpendicular to the axis of the lens; in this case, the vortices in the chain are all focused onto the optical axis in the Fourier plane (green rectangle) and extrinsic OAM contributions are eliminated. The measured total OAM of the same fractional vortex modes is shown by red dots achieved by a 45° rotation of the input mode, as shown in the red-rimmed inset. The extrinsic OAM (black dots) is then calculated as the difference between the total OAM and intrinsic OAM.

Figure 3

Intensity and phase of theoretical (top) and experimental (bottom) realizations of fractional vortex mode ${\psi }_{4.4}$ with 2 mm waists of a 633 nm He-Ne laser at a distance of 0.7 m, $\sim \frac{1}{25}{Z}_R$ from the generating optic. Notably, the vortex chain (a string of unit vortices of alternating sign) is resolved in both theoretical and experimental phase images.

Figure 4

Left to right we show the amplitude and phase of ${\psi }_{\mu }$, $\widetilde{{\psi }_{i\mu }}$, and $\widetilde{{\psi }_{d\mu }}$. Local transverse Poynting vectors are also shown as white arrows in plots of $\widetilde{{\psi }_{d\mu }}$. The first two rows demonstrate the match between experiment and theory for $\mu =1.5$, while the bottom row demonstrates the theoretical form of $\mu =1.75$ for comparison. The plotted transverse momentum flow for $\widetilde{{\psi }_{d1.5}}$ demonstrates that the dominant transverse momentum is made of off-axis linear momenta pointing in opposite directions, suggesting no net extrinsic contribution, which is consistent with observations in Fig. 2. The transverse momentum flow of $\widetilde{{\psi }_{d1.75}}$ has a net positive off-axis linear momentum, also consistent with Fig. 2.

Figure 5

(Top) Far field amplitude and phase of fractional vortex beams, calculated as the propagating part of beams with Gaussian amplitude and the fractional $2\pi \mu$ spiral phase. This shows that the dark stripe seen in any fractional vortex beam represents the region where the field was nonpropagating. (Bottom) Near field amplitude of beam centers of integer and fractional modes generated as the Gaussian excitations of spiral phases, showing that the lateral discontinuity of fractional beams comes from depletion into evanescent waves at the surface of the generating phase optic. The orientation of the dark stripe rotates by $(\pi /2)$ over propagation to the far field.