  We introduce and study generalized holographic superconductors with higher
derivative couplings between the field strength tensor and a complex scalar
field, in four dimensional AdS black hole backgrounds. We study this theory in
the probe limit, as well as with backreaction. There are multiple tuning
parameters in the theory, and with two non-zero parameters, we show that the
theory has a rich phase structure, and in particular, the transition from the
normal to the superconducting phase can be tuned to be of first order or of
second order within a window of one of these. This is established numerically
as well as by computing the free energy of the boundary theory. We further
present analytical results for the critical temperature of the model, and
compare these with numerical analysis. Optical properties of this system are
also studied numerically in the probe limit, and our results show evidence for
negative refraction at low frequencies.
