We report on, in this letter, a phenomenon that the central zerointensity point of a doughnut beam, caused by phase singularity, disappears in the focus, when such a beam is focused by a high numerical-aperture objective in free space. In addition, the focal shape of the doughnut beam of a given topological charge exhibits the increased ring intensity in the direction orthogonal to the incident polarization state and an elongation in the polarization direction. These phenomena are caused by the effect of depolarization, associated with a high numerical-aperture objective, and become pronounced by the use of a central obstruction in the objective aperture.
It is shown that embedding of flipped $\mathrm{SU}(5)$ in a five-dimensional $\mathrm{SO}(10)$ enables exact unification of the gauge coupling constants. The demand for unification uniquely determines both the compactification scale and the cutoff scale. These are found to be ${M}_{C}\ensuremath{\approx}5.5\ifmmode\times\else\texttimes\fi{}{10}^{14}\mathrm{GeV},$ and ${M}_{*}\ensuremath{\approx}1.0\ifmmode\times\else\texttimes\fi{}{10}^{17}\mathrm{GeV},$ respectively. The theory explains the absence of $d=5$ proton-decay operators through the implementation of the missing partner mechanism. On the other hand, the presence of $d=6$ proton-decay operators points towards the bulk localization of the first and second families of matter fields.
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