  We study the photoproduction of the exotic pentaquark Theta^+ baryon with the
vector kaon, assuming that the quantum numbers of the Theta^+ to be
J^P=3/2^(+-) and J^P=1/2^+. Scalar meson kappa(800)-exchange is also taken into
account. In contrast with the gamma N -> Kbar Theta^+(3/2^(+-)) process, the
large suppression from the proton target is not observed in the total cross
sections. We also suggest a method to determine which meson exchange is the
most dominant by analyzing the polarizations of incident photon and outgoing
K^*. We find that kappa-exchange turns out to be prominent when the
polarizations of the photon and K^* are aligned to be parallel, whereas
K-exchange does when they are perpendicular to each other.
