Fast oscillations of orbital angular momentum and Shannon entropy caused by radial numbers of structured vortex beams

We address theoretical and experimental considerations of two-parameter excitation of each Hermite-Gaussian (HG) mode in composition of a structured Laguerre-Gaussian (sLG) beam. The complex amplitude of the sLG beam is shaped in such a way that the radial and azimuthal numbers of eigenmodes are entangled with each other. As a result, variations in the amplitude and phase parameters of mode excitation, although dramatically changing the intensity and phase patterns, do not change the structural stability of the beam. We reveal that the radial number of the sLG beam can cause fast oscillations of the orbital angular momentum and Shannon entropy, dramatically increasing the uncertainty of detecting the beam in some particular state. We found that despite the fast oscillations, the sLG beam has an invariant in the form of a module of the total topological charge (TC), with the exception of narrow intervals of the phase parameter, where the measurement error does not allow us to accurately measure the sign of the TC. The difference between the interpretation of informational entropy as a measure of uncertainty and a measure of information capacity is considered on the example of the measurement of Shannon entropy in the bases of LG and HG modes.