  We investigate the $\Lambda(1405,1/2^-)\equiv\Lambda^*$ photoproduction off
the proton target, i.e. $\gamma p\to K^+\Lambda^*$, considering explicitly its
two-pole structure, the higher ($\Lambda^*_H:\,1430\,\mathrm{MeV}$) and lower
($\Lambda^*_L:\,1390\,\mathrm{MeV}$) mass-pole contributions, suggested by the
chiral-unitary model (ChUM) approaches. For this purpose, we construct a
two-body process model, which mimics the Dalitz process, $\gamma p\to
K^+\pi\Sigma$, assuming that the mass of $\Lambda^*$ as the invariant mass of
$\pi$ and $\Sigma$, i.e. $M_{\Lambda^*}\sim M_{\pi\Sigma}$. We employ the
effective Lagrangian method with the tree-level Born approximation, using the
gauge-invariant prescription for the phenomenological form factors. We provide
the numerical results for the energy and angular dependences, $\pi$-$\Sigma$
invariant-mass distribution, and so on. It turns out that the model parameters
determined from ChUM reproduce the experimental data qualitatively well,
supporting the two-pole structure. Moreover, the nucleon resonance contribution
near the threshold plays an important role to describe the data.
