  The exclusive reaction $\gamma p \to p K^+ K^-$ was studied in the photon
energy range $3.0 - 3.8 \mbox{ GeV}$ and momentum transfer range $0.6<-t<1.3
\mbox{ GeV}^2$. Data were collected with the CLAS detector at the Thomas
Jefferson National Accelerator Facility. In this kinematic range the integrated
luminosity was approximately 20 pb$^{-1}$. The reaction was isolated by
detecting the $K^+$ and the proton in CLAS, and reconstructing the $K^-$ via
the missing-mass technique. Moments of the di-kaon decay angular distributions
were extracted from the experimental data. Besides the dominant contribution of
the $\phi$ meson in the $P$-wave, evidence for $S-P$ interference was found.
The differential production cross sections $d\sigma/dt$ for individual waves in
the mass range of the $\phi$ resonance were extracted and compared to
predictions of a Regge-inspired model. This is the first time the $t$-dependent
cross section of the $S$-wave contribution to the elastic $K^+K^-$
photoproduction has been measured.
