  Nonlinear optical generation has been a well-established way to realize
frequency conversion in nonlinear optics, whereas previous studies were just
focusing on the scalar light fields. Here we report a concise yet efficient
experiment to realize frequency conversion from vector fields to vector fields
based on the vectorial nonlinear optical process, e.g., the second-harmonic
generation. Our scheme is based on two cascading type-I phase-matching BBO
crystals, whose fast axes are configured elaborately to be perpendicular to
each other. Without loss of generality, we take the full Poincar\'e beams as
the vectorial light fields in our experiment, and visualize the structured
features of vectorial second-harmonic fields by using Stokes polarimetry. The
interesting doubling effect of polarization topological index, i.e., a
low-order full Poincar\'e beam is converted to a high-order one are
demonstrated. However, polarization singularities of both C-points and L-lines
are found to keep invariant during the SHG process. Our scheme can be
straightforwardly generalized to other nonlinear optical effects. Our scheme
can offer a deeper understanding on the interaction of vectorial light with
media and may find important applications in optical imaging, optical
communication and quantum information science.
