The study of tight focusing characteristics of azimuthally polarized vortex beams and the implementation of ultra-long super-resolved optical needle

Abstract

<sec>The tight focusing characteristics of azimuthally polarized vortex beams are systematically investigated in this work. The azimuthally polarized vortex beam can be decomposed into left-handed circularly polarized (LHCP) wave and right-handed circularly polarized ( RHCP) wave. It is found that the longitudinal component of LHCP and RHCP at the focal plane are equal in magnitude but opposite in phase. Thus, the total longitudinal field disappears because of the completely destructive interference. In contrast, there is almost no interference between the transverse component of LHCP and RHCP. Thus, the total transverse field is the incoherent superposition of them. Since the absolute value of the topological charge of LHCP component and RHCP component are not equal, the transverse component of LHCP and RHCP will be concentrated in the different areas on the focal plane. It is the reason for the orbit-induced SAM to be localized on the focal plane. Then, we compare the focal spot characteristics of the radially polarized beam and the azimuthally polarized beam with a first-order vortex. The advantages and disadvantages of them are discussed in detail, respectively.</sec><sec>For the radially polarized beam, the central focal spot is mainly longitudinal component, and the sidelobe is mainly transverse component. For the azimuthally polarized vortex beam with <inline-formula><tex-math id="M3">\begin{document}$l = 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230304_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230304_M3.png"/></alternatives></inline-formula>, the central focal spot is mainly LHCP component, and the sidelobe is mainly RHCP component. In both cases, the field distributions of the central spots are the same, and both show a distribution similar to the zero-order Bessel function. The situation of the sidelobe is different. The sidelobe of the radially polarized beam shows a distribution similar to the first-order Bessel function and the sidelobe of the azimuthally polarized vortex beam indicates a distribution similar to the second-order Bessel function. Therefore, the sidelobe of the radially polarized beam is closer to that of the optical axis, resulting in a larger central focal spot size. On the other hand, the sidelobe of the radially polarized beam accounts for a much smaller proportion of the total energy than that of the azimuthally polarized vortex beam. So the sidelobe peak intensity of the radially polarized beam is lower. Finally, an optimal binary phase element is designed to obtain an ultra-long super-resolution optical needle. The transverse full weight of half maximum (FWHM) can achieve <inline-formula><tex-math id="M4">\begin{document}$0.391\lambda $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230304_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230304_M4.png"/></alternatives></inline-formula> and the longitudinal FWHM can reach to <inline-formula><tex-math id="M5">\begin{document}$25.5\lambda $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230304_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230304_M5.png"/></alternatives></inline-formula> by using only 6 belts.</sec>