  Elastic and proton-dissociative photoproduction of $\rho^0$, $\phi$ and
$J/\psi$ vector mesons ($\gamma p\to Vp$, $\gamma p\to VN$, respectively) have
been measured in e^+p interactions at HERA up to -t=3 GeV$^2$, where t is the
four-momentum transfer squared at the photon-vector meson vertex. The analysis
is based on a data sample in which photoproduction reactions were tagged by
detection of the scattered positron in a special-purpose calorimeter. This
limits the photon virtuality, Q^2, to values less than 0.01 GeV$^2$, and
selects a $\gamma p$ average center-of-mass energy of <W> = 94 GeV. Results for
the differential cross sections, d$\sigma$/d$t$, for $\rho^0$, $\phi$ and
$J/\psi$ mesons are presented and compared to the results of recent QCD
calculations. Results are also presented for the t-dependence of the pion-pair
invariant-mass distribution in the $\rho^0$ mass region and of the spin-density
matrix elements determined from the decay-angle distributions. The Pomeron
trajectory has been derived from measurements of the W dependence of the
elastic differential cross sections d$\sigma$/d$t$ for both $\rho^0$ and $\phi$
mesons.
