Focusing properties of the vector beam with complicated polarization and helical phase

We report a vector beam with complicated polarization and helical phase and derive the corresponding focusing model of the aplanatic system, where the 10 variables optimized provide a more powerful and flexible tool to construct various interesting focal spots. By optimizing these 10 variables, we directly construct some specific focal spots, including the single focal spot with lateral resolution beyond the Abbe diffraction limit and without axial electric field component and the focal spots with unique structures in the intensity, such as the square and triangular bright spots, the various double lines, the cross lines, the ring-shaped square, the ring-shaped hexagon, and so on. Some of those focal spots with unique structures in the intensity have high resolution, even beyond the Abbe diffraction limit. We conclude the functions of some variables affecting the structures of the focal spots, draw some regulars constructing the focal spot with high quality, and indicate how to find the new values of various variables optimized in order to rotate the focal spot distribution. Although only the focal spot structures of the intensity distribution are discussed in this paper, the ideas and the developed focusing model can be used for the investigations of the various focal spot structures of polarizations, gradient forces, and spin and orbital angular momentums.