  In this paper, we study the polarization of the $\Lambda(1405)$ in the
$\gamma p \rightarrow K^+ \pi \Sigma$ reaction within an effective Lagrangian
approach and isobar model. In our model, the $\Lambda(1405)$ is excited through
the t-channel $K/K^*$ exchanges and u-channel hyperon exchange. Compared to
previous studies, we also include the contribution from a contact term, which
is necessary for our model to interpret the polarization of the
$\Lambda(1405)$. In addition, we also discuss the possibility to verify the
proposed two-pole structure of the $\Lambda(1405)$ using the polarization data.
We find that the polarization of the $\Lambda(1405)$ or the polarization of the
final $\Sigma$ in this reaction is sensitive to the invariant mass
$M_{\pi\Sigma}$. Thus the measurement of the dependence of the $\Lambda(1405)$
polarization on the $M_{\pi\Sigma}$ can offer valuable information about the
pole structure of the $\Lambda(1405)$.
