The longitudinal components of the focal field can be greatly enhanced by the tightly autofocusing beams, and can be effectively eliminated by the hybrid vector light field with “8-type” polarization. Whether the combination of the two methods leads to sound results needs further research. In this paper, we study the focusing characteristics of the tightly autofocusing beams with “8-type” polarization to better manipulate the longitudinally polarization component of the focal field. Moreover, how the parameters of the “8-type” polarization affect the transversely polarization component is explored in this paper. We investigate the influence of the parameters, including polarization order, base vectors, and polarization of the cross point, on the focusing performance such as the size of the uniform polarization, the dependence of the transverse polarization with the input field, and the intensity control of the longitudinal field. This work has great application potential in optical imaging, optical manipulation, and optical machining.

Based on the Rayleigh-Sommerfeld vectorial diffraction theory described by Eq. (3), the propagation process of the tightly autofocusing beams with “8-type” polarization expressed by Eq. (2) is simulated by calculating the three polarization components of the optical field at different distances step by step. By changing the parameters of the “8-type” polarization given by Eq. (1), the focal field on the focal plane (z=35.18λ) is calculated, on which the effects of polarization order, base vectors, and cross-point polarization are analyzed.

Figure 1 shows the propagation processes of transverse and longitudinal components [Figs. 1(c1) and 1(c2)] and the total field [Fig. 1(c3)] of a tightly autofocusing beam of which the input intensity profile and polarization distribution are shown in Fig. 1(b). The projection trajectory of the “8-type” polarization on the Poincaré sphere is shown in Fig. 1(a). The intensity profiles of the focal field are shown in Fig. 1(d). It can be seen that the beam focuses along a spherical surface, and the longitudinal field is greatly reduced compared with the result in Ref. [31]. The transverse polarizations of the focal fields at the central areas (white circles in Fig. 2) are uniformly distributed and the same as the cross-point polarizations, which are set as the left-handed polarization. With the increase of the order l of the “8-type” polarization, the side lobe intensity of the transverse component and the longitudinal component are both reduced, while the uniform polarization region is extended (Fig. 2). Figure 3 further shows that the central polarization on the focal plane is only dependent on the cross-point polarization. When cross-point polarization changes [Figs. 3(a) and 3(b)], the transverse polarization also changes and consists with the cross-point polarization. When the cross-point polarization remains unchanged [Figs. 3(c) and 3(d)], the transverse polarization remains the same even though the input polarization is changed. The order l, the cross points, and the base vectors of the “8-type” polarization can all affect the energy proportion and peak intensity of the longitudinal field (Fig. 4). The energy proportion of the longitudinal field tends to be constant when l>1, while the peak intensity keeps unchanged when l>3. The longitudinal field intensity is closely related to the crossing-point polarization, as shown in Fig. 5. With the unchanged base vectors and the changed cross point (case I in Fig. 5), the peak intensity of the longitudinal field varies periodically and reaches the maximum (minimum) when the cross point has horizontal or vertical polarization (circular polarization). With the unchanged cross point and the changed base vectors (case Ⅱ in Fig. 5), the longitudinal field remains stable.

We study the focusing characteristics of the tightly autofocusing beams with “8-type” polarization. The effects of the parameters on the focus fields including the polarization order, base vectors, and cross points are analyzed. The results show that the transverse polarization of the central focal field is consistent with the cross-point polarization, which can be used to control the focusing polarization; the “8-type” polarization can eliminate the on-axis longitudinal field at the focal point, and greatly weaken the off-axis longitudinal field, of which the proportion and peak intensity can be controlled via the polarization order and the cross point. Our work has great application potential in optical imaging, optical manipulation, and optical machining.