Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields

We present the propagation-dependent polarization vector fields by use of an isotropic microchip laser with the longitudinal-transverse coupling and the entanglement of the polarization states. With the coherent superposition of orthogonal circularly polarized vortex modes which are made up of two Laguerre-Gaussian modes with different order, the experimental three-dimensional vector fields can be reconstructed analytically. From the theoretical analyses, the generalized structures of singularities such as V points, C lines, and L surfaces can be clearly demonstrated. Importantly, the projections of C lines on the transverse plane are found to form the intriguing petal structures.