Tailoring focal plane component intensities of polarization singular fields in a tight focusing system

Abstract

The scientific community studies tight focusing of radially and azimuthally-polarized vector beams as it is a versatile solution for many applications. We offer a new method to produce tight focusing that ensures a more uniform intensity profile in multiple dimensions, providing a more versatile and stable solution. We manipulate the polarization of the radially and azimuthally polarized vector beams to find an optimal operating point. We examine in detail optical fields whose polarization states lie on the equator of the relevant Poincaré spheres namely, the fundamental Poincaré sphere, the hybrid order Poincaré sphere (HyOPS), and the higher order Poincaré sphere. We find via simulation that the fields falling on these equators have focal plane intensity distributions characterized by a single rotation parameter $\alpha$ determining the individual state of polarization. The strengths of the component field distributions vary with $\alpha$ and can be tuned to achieve equal strengths of longitudinal (z) and transverse (x and y) components at the focal plane. Without control of this parameter (e.g., using $\alpha =0$ in radially and $\alpha =\pi$ in azimuthally-polarized vector beams) intensity in x and y components are at 20% of the z component. In our solution with $\alpha =\pi /2$ , all components are at 80% of the maximum possible intensity of z. In examining the impact of $\alpha$ on a tightly focused beam, we also found that a helicity inversion of HyOPS beams causes a rotation of 180 degree in the axial intensity distribution.

Figure 1

Schematic diagram of circular basis FPS, HOPS, and HyOPS. In each sphere, the superscripts in the coordinate axes $S_1$, $S_2$, and $S_3$ denote the content of OAM in the right (${\widehat{e}}_R$) and left (${\widehat{e}}_L$) circular polarization eigenstates, respectively.

Figure 2

Schematic diagram of the tightly focused optical system.

Figure 3

For a FPS beam, focal plane distributions of tightly-focused optical fields for various $\alpha$; rows show incident beam spatial distributions of (I) polarization, (II) $S_{12}$-Stokes field phase, and the constituent focal plane intensity (III) $|E_x{|}^2$, (IV) $|E_y{|}^2$, and (V) $|E_z{|}^2$, and (VI) total intensity ${|E|}^2$.

Figure 4

For a HOPS beam of Poincaré–Hopf index $\eta =+1$, focal plane distributions of tightly-focused optical fields for various $\alpha$; rows show incident beam spatial distributions of (I) polarization, (II) $S_{12}$-Stokes field phase, and the constituent focal plane intensity (III) $|E_x{|}^2$, (IV) $|E_y{|}^2$, and (V) $|E_z{|}^2$, and (VI) total intensity ${|E|}^2$.

Figure 5

For a right-handed HyOPS beam of polarization singularity index $I_C=+1$, focal plane distributions of tightly-focused optical fields for various $\alpha$; rows show incident beam spatial distributions of (I) polarization, (II) $S_{12}$-Stokes field phase, and the constituent focal plane intensity (III) $|E_x{|}^2$, (IV) $|E_y{|}^2$, and (V) $|E_z{|}^2$, and (VI) total intensity ${|E|}^2$.

Figure 6

Variations of maximum intensity values of $|E_x{|}^2$, $|E_y{|}^2$, and $|E_z{|}^2$ with $\alpha$ for (a) Linearly polarized beams (FPS Beams); (b) HOPS beams with $\eta =+1$; (c) left- or right-handed HyOPS beams with $I_C=+1$.

Figure 7

Total intensity distributions in the xz-plane of a lens with $NA={75}^{\circ }$ for optical fields with various $\alpha$; rows show axial plane intensity distributions of (I) FPS beams, (II) HOPS beams with $\eta =+1$, (III) right-handed HyOPS beams with $I_C=+1$, and (IV) left-handed HyOPS beams with $I_C=+1$.