The exclusive reactions $\gamma p \to \bar K^0 K^+ n$ and $\gamma p \to \bar
K^0 K^0 p$ have been studied in the photon energy range 1.6--3.8 GeV, searching
for evidence of the exotic baryon $\Theta^+(1540)$ in the decays $\Theta^+\to
nK^+$ and $\Theta^+\to p K^0$. Data were collected with the CLAS detector at
the Thomas Jefferson National Accelerator Facility. The integrated luminosity
was about 70 pb$^{-1}$. The reactions have been isolated by detecting the $K^+$
and proton directly, the neutral kaon via its decay to $K_S \to \pi^+ \pi^-$
and the neutron or neutral kaon via the missing mass technique. The mass and
width of known hyperons such as $\Sigma^+$, $\Sigma^-$ and $\Lambda(1116)$ were
used as a check of the mass determination accuracy and experimental resolution.
Approximately 100,000 $\Lambda^*(1520)$'s and 150,000 $\phi$'s were observed in
the $\bar K^0 K^+ n$ and $\bar K^0 K^0 p$ final state respectively. No evidence
for the $\Theta^+$ pentaquark was found in the $nK^+$ or $pK_S$ invariant mass
spectra. Upper limits were set on the production cross section of the reaction
$\gamma p \to \Theta^+ \bar K^0$ as functions of center-of-mass angle, $nK^+$
and $pK_S$ masses. Combining the results of the two reactions, the 95% C.L.
upper limit on the total cross section for a resonance peaked at 1540 MeV was
found to be 0.7 nb. Within most of the available theoretical models, this
corresponds to an upper limit on the $\Theta^+$ width, $\Gamma_{\Theta^{+}}$,
ranging between 0.01 and 7 MeV.
