Modal Majorana Sphere and Hidden Symmetries of Structured-Gaussian Beams

Structured-Gaussian beams are shown to be fully and uniquely represented by a collection of points (or a constellation) on the surface of the modal Majorana sphere, providing a complete generalization of the modal Poincaré sphere to higher-order modes. The symmetries of this Majorana constellation translate into invariances to astigmatic transformations, giving way to continuous or quantized geometric phases. The experimental amenability of this system is shown by verifying the existence of both these symmetries and geometric phases.

Figure 1

Modal Poincaré sphere for $N=4$ and (left) $\ell =4$, (right) $\ell =2$. Also shown are the intensity distributions with phase coded in hue, for several HLG beams with their corresponding modal spot. The same color coding is used for all subsequent complex field plots.

Figure 2

(First row) $Q$ function and corresponding MC with the size indicating the number of stars. (Second row) Field distribution with hue representing phase for HLG beams of order $N=6$ along $\theta =\pi /4$ and $\varphi =\pi /4$ with (from left to right) $l=6$, 4, 2, 0. The same color coding is used for all subsequent plots of $Q$ functions.

Figure 3

First row: $Q$ function and corresponding MC. Second row: field distribution with the phase encoded in hue for four SG beams. The MCs were chosen as (from left to right) arbitrary with no symmetry, a pentagonal prism, an icosahedron, and a disdyakis dodecahedron.

Figure 4

Left to right: icosahedral MC with four stars lying on the equator, corresponding theoretical and experimental intensity distributions before and after a rotation of the MC along the marked axes. The theoretical intensity distribution is shown only once since it is the same for all three cases. The same gray scale is used for subsequent intensity plots.

Figure 5

Simplified experimental setup. The input SG beams are generated by illuminating SLM1 with a collimated laser beam with $\lambda =795\mathrm{nm}$ polarized along the SLM’s preferred axis. The beam is then relayed through a $4f$ system (not shown) to the pair of SLMs that perform the mode transformation. After the mode transformation the beam is relayed to the CCD for detection. When performing interferometric measurements, the input beam is also sent to a reference arm by using a beam splitter and then recombined with the transformed beam.

Figure 6

Left: $Q$ function, MC and field distribution for a SG beam with a discrete phase along the $u_1$ axis varying in steps of $8\pi /7$. Right: theoretical and measured geometric phases gained by the beam for different values of the rotation angle along with the intensity distribution of the transformed beam for three rotation angles.