“…Interestingly, when a path is formed on a sphere on which all modes carry the same amount of orbital angular momentum, like the second sphere of second order modes, no geometric phase is generated (Galvez and O'Connell, 2005). This would indicate that geometric phases are mediated by a variation of orbital angular momentum, in the same way as polarization transformations that generate a PB phase involve variation of the spin angular momentum (Van Enk, 1993;McWilliam et al, 2022;Tiwari, 1992). While the sphere-based representation is useful as it directly relates to transformations that are easily realisable in the laboratory, it is not suitable to describe generic transformations in the state space of higher order modes.…”