Vectorial complex-source vortex beams

The scalar complex source vortex model is an accurate description of highly focused scalar vortices. We use it to construct a variety of vectorial solutions of Maxwell's equations describing highly focused and variously polarized vector vortex beams accurately. Three different families of optical vector vortex beams are presented and studied in detail. In this model, optical vortices derived within Cartesian symmetry correspond to circularly and linearly polarized highly focused vortex beams in the focus of a high numerical aperture focusing system. In addition, we report on vortical complex-source beams derived within cylindrical and spherical symmetries which exhibit very special and intriguing properties.

Figure 1

Distribution of the amplitude (a)–(c) and the phase (d),(e) of a regular (a),(d) and irregular (b),(c),(e),(f) scalar complex-source vortex for $k{z}_0=1.5$ and for $m=1$. The gray lines depict the branch cuts for a “source” (b),(e) and “beam” (c),(f) choices and the points ${\mathrm{O}}_1$, ${\mathrm{O}}_2$ depict the position of the virtual sources. The white lines in (a)–(c) depict the wave fronts.

Figure 2

Distribution of the modulus of the electric field $\left|\mathbf{E}\right|$ of transverse electric (a),(b) and transverse magnetic (c),(d) regular complex-source vortices within spherical symmetry for $k{z}_0=1.5$ and for $m=1$. The white streamlines depict the in-plane components of the electric field.

Figure 3

Projection of circularly polarized Cartesian vectors ${\mathbf{e}}_{\beta ,l}^{\left(M\right)}$ (a),(c),(e), ${\mathbf{e}}_{\beta ,l}^{\left(N\right)}$ (b),(d),(f) onto the entrance pupil plane of an aplanatic lens. The handedness $\beta =1$, and the topological charge is $l=-1$ (a),(b), $l=0$ (c),(d), $l=1$ (e),(f). The blue darts (red drops) show the streamlines of the real (imaginary) parts of the electric field.

Figure 4

Distributions of $\left|\mathbf{E}\right|$ of transverse electric (a),(b) and transverse magnetic (c),(d) regular complex-source vortices within cylindrical symmetry for $k{z}_0=1.5$ and for $m=1$ (a),(b). The white streamlines depict the in-plane components of the electric field.

Figure 5

Distributions of $\left|\mathbf{E}\right|$ of circularly polarized regular complex source $U_M$ (a),(c),(e),(g), $U_N$ (b),(d),(f),(h) vortices for $k{z}_0=1.5$, $\beta =1$ (a),(b),(e),(f), $\beta =-1$ (c),(d),(g),(h) and $m=1$. The white streamlines depict the in-plane components of the electric field.

Figure 6

Projection of the linearly polarized Cartesian vectors ${\mathbf{e}}_x^{\left(M\right)}$ (a), ${\mathbf{e}}_x^{\left(N\right)}$ (b) onto the entrance pupil plane of an aplanatic lens. The topological charge is $l=1$.