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Uncertainty principle for periodic orbital angular momentum and angular position with infinity

The angular uncertainty principle (angular-UP) states the orbital angular momentum (OAM) is precisely defined in an optical vortex with angular position (AP) ranging over 2π azimuthal coordinate ( ϕ ). However, the pair of observable states is discretely selected and does not correspondent to the pair of unselected linear momentum and position states for the lower bound. This discrete selection is such that the pair of angular uncertainties is independent of n -fold symmetry. Herein, we demonstrate the smaller difference between mean OAM and the product of azimuthal phase-gradient (PG) and , the larger ϕ range of one periodic helical wavefront in a set of numerous singular light beams, each of which utilizes the superposition comprising two fractional OAM light beams that have a difference of δ in the azimuthal PG. This is a periodically angular UP (periodic-UP) for any pair of unselected states of periodic OAM and AP by a constant product 0 . 187 on their underlying uncertainties, or the pair of unlimited scale of periodic OAM and AP uncertainties. This constant lower bound corresponds to the Robeson bound held by linear UP for the pair of unselected observables; however, it is stronger by 2.67 times. We demonstrate a macroscopic example of the periodic-UP by illustrating a physical interpretation of the image constructed by phase shift. Moreover, we demonstrate that the pair of periodically angular uncertainties in this singular light beam is compatible with the pair of angular uncertainties by dividing and multiplying the periodic number ( ) c o s n n   , respectively. We demonstrate that both the OAM and AP uncertainties are two monotonic functions of δ and two various distribution types of the OAM spectrum and image intensity in this singular light with identically equivalent PGs. These monotonic functions and distribution types may be useful for existing OAM applications. We experimentally generate these singular light beams and show the azimuthally sinusoidal square variation in their intensity images.