The role of angular slit number and angular slit width on the OAM spectrum

The uncertainty principle sets the limit for simultaneous measurements of position and momentum, and its angular analogue is realized through angular diffraction. When a beam is spatially confined by angular slits, the uncertainty in its orbital angular momentum (OAM) increases, leading to the generation of OAM sidebands. Both the angular slit number and the angular slit width shapes the spatial confinement of the beam. In this study, we investigate this dependence of the OAM sidebands by obstructing a Gaussian beam with $N$ number of evenly spaced angular slits with angular separation $\Delta\theta_{sep}$. The power distribution among the OAM sidebands exhibits oscillatory behavior as a function of $\Delta\theta_{sep}$. We find that the number of oscillations over the full range $0 \leq \Delta\theta_{sep} \leq 2\pi/N$ is given by the ratio $\frac{|l|}{N}$. Furthermore, each OAM sideband acquires power only when $\frac{|l|}{N}$ takes an integer value, thereby demonstrating the role of the angular slit geometry to the structure of the OAM sidebands.