  We study the $\gamma p \to p f_0$ [$f_0(980)$ and $f_0(1500)$] reaction close
to threshold within an effective Lagrangian approach. The production process is
described by $s$-channel nucleon pole or $t$-channel $\rho$ and $\omega$
exchange. The $K^0 \bar{K}^0$ invariant mass distributions of the $\gamma p\to
f_0(980)p \to K^0 \bar{K}^0$ and $\gamma p\to f_0(1500)p \to K^0 \bar{K}^0 p$
reactions are investigated, where the two kaons have been separated in $S$ wave
decaying from $f_0(980)$ and $f_0(1500)$. It is shown that the $s$-channel
process is favored for the production of $f_0(980)$, while for the $f_0(1500)$
production, the experimental measurements can be described quite well by the
$t$-channel process. It is expected that the theoretical results can be tested
by further experiments at CLAS.
