Based on the conclusion that azimuthally polarized vector beam with topological nuclei has radial component after tight focusing, the tight focusing formula of the azimuthally polarized vortex beam is amended. We afresh study the tight focusing properties of the azimuthally polarized higher-order Laguerre-Gauss vortex beams through a diffractive optical element and a high numerical aperture lens. The results show that a new three-dimensional (3D) triple optical chain, composed of a 3D main optical chain and two symmetrical 3D paraxial secondary optical chains, is obtained along optical axis near the focal plane. The effects of the radial mode number of associated Laguerre polynomial and the interception ratio of incident beam, the structure of the diffractive optical element, and the numerical aperture of the focusing lens on the triple optical chain are analyzed in detail. Results show that the change of radial mode number will destroy the structure of triple optical chain. By adjusting the interception ratio and the structure of the diffractive optical element, the triple optical chain with higher symmetry can be obtained again, and the high degree of freedom control of the triple optical chain can be achieved.