Reexamination of Bessel beams: A generalized scheme to derive optical vortices

The electromagnetic field of optical vortices is in most cases derived from vector and scalar potentials using either a procedure based on the Lorenz or the Coulomb gauge. The former procedure has been typically used to derive paraxial solutions with Laguerre-Gauss radial profiles, while the latter procedure has been used to derive full solutions of the wave equation with Bessel radial profiles. We investigate the differences in the derivation procedures applying each one to both Bessel and Laguerre-Gauss profiles. We show that the electromagnetic fields thus derived differ in the relative strength of electric and magnetic contributions. The new solution that arises from the Lorenz procedure in the case of Bessel beams restores a field symmetry that previous work failed to resolve. Our procedure is further generalized and we find a spectrum of fields beyond the Lorenz and Coulomb gauge types. Finally, we describe a possible experiment to test our findings.

Figure 1

Normalized transition probabilities as a function of the position of the ion for different beams all with waist $w=\lambda /2$. The probabilities are calculated for a dipole transition with change in magnetic quantum number $\mathrm{\Delta }m=1$, assuming a negligible position spread for the center of mass of the ion. Panel (a) refers to beams with chirality $\ell =2$; panels (b) and (c) refer to beams with $\ell =0$ and 1, respectively. Blue (light gray) and red (dark gray) predictions are for the two different circular polarizations $\sigma =\pm 1$, while solid and dashed lines correspond to the prediction expected for a 2CA or 3CA model, respectively.