  We investigate $\eta$ photoproduction off the neutron target, i.e., $\gamma n
\to \eta n$, employing an effective Lagrangian method combining with a Regge
approach. As a background, we consider nucleon exchange in the $s$-channel
diagram and $\rho$- and $\omega$-meson Regge trajectories in the $t$ channel.
The role of nucleon resonances given in the Review of Particle Data Group in
the range of $W \approx 1500 - 2100$ MeV and the narrow nucleon resonance
$N(1685,1/2^+)$ is extensively studied. The numerical results of the total and
differential cross sections, double polarization observable $E$, and
helicity-dependent cross sections $\sigma_{1/2}$, $\sigma_{3/2}$ are found to
be in qualitative agreement with the recent A2 experimental data. The
predictions of the beam asymmetry are also given.
