Creating Complex Optical Longitudinal Polarization Structures

In this Letter, we show that it is possible to structure the longitudinal polarization component of light. We illustrate our approach by demonstrating linked and knotted longitudinal vortex lines acquired upon nonparaxially propagating a tightly focused subwavelength beam. The remaining degrees of freedom in the transverse polarization components can be exploited to generate customized topological vector beams.

Figure 1

The profiles $f^{\text{Hopf}}$ and $f^{\text{trefoil}}$ of Eqs. (4) and (5) (a),(e) at narrow widths contain evanescent waves. This is demonstrated in (c) and (g), where the profiles are shown in the transverse Fourier domain together with a circle of radius $k_0$. A spectral attenuation (d),(h) according to Eq. (3) removes the evanescent amplitudes as well as amplitudes close to ${\mathit{k}}_{\perp }=0$ (see the text for details) and alters $E_z^f$ in the focal plane significantly (b),(f).

Figure 2

Propagation of the spectrally attenuated field shown in Figs. 1 and 1 gives rise to the vortex lines (black) in the forms of a Hopf link (a) and a trefoil (b). A phase slice is shown in the $xy$ plane at $z=0$. As a comparison, the idealized Hopf link and trefoil are shown as insets.

Figure 3

Amplitude and phase in the $z=0$ plane for the linearly polarized transverse components Eqs. (6) and (7) that give rise to a longitudinal component forming the Hopf link [Figs. 1 and 2] are shown in (a)–(d) and the trefoil [Figs. 1 and 2] is shown in (e)–(h). The color maps for each figure are on the right of the two rows of plots. Both transverse fields ${\mathbf{E}}_{\perp }^f=(e_x^f,0)$ and ${\mathbf{E}}_{\perp }^f=(0,e_y^f)$ shown in (a)–(d) and (e)–(h), respectively, yield the same longitudinal field, as well as superpositions ${\mathbf{E}}_{\perp }^f=(\alpha e_x^f,\beta e_y^f)$ as long as the coefficients fulfill $\alpha +\beta =1$.

Figure 4

Irrotational transverse amplitude and phase profiles in the $z=0$ plane producing a Hopf link (a)–(d) and a trefoil (e)–(h) in the longitudinal polarization component. In contrast to Fig. 3, here the transverse field producing the desired longitudinal component is fixed by Eqs. (8) and (9) to ${\mathbf{E}}_{\perp }^f=-{\nabla }_{\perp }V({\overrightarrow{r}}_{\perp })$, and we cannot change the weight of the components shown in (a)–(d) and (e)–(h), respectively.