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 $dsigma/dt$ 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.
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