Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns

We use a high-level isotropic laser with off-axis focused and on-axis circular pumping to generate the high order polarization-entangled transverse modes. The main finding is that the complex transverse modes can be categorized into four types: square pattern, hyperbolic pattern, elliptic pattern, and circular pattern. Importantly, all types of the polarization-entangled modes can be well analyzed with the generalized coherent states. With the connection between theoretical analysis and experimental results, the formation of complex singularities can be clearly represented.