  Recent scanning tunneling microscopy measurements of the proximity effect in
Au/La$_{2-x}$Sr$_{x}$CuO$_{4}$ and
La$_{1.55}$Sr$_{0.45}$CuO$_{4}$/La$_{2-x}$Sr$_{x}$CuO$_{4}$ bilayers showed a
proximity-induced pseudogap [Yuli et al., Phys. Rev. Lett. {\bf 103}, 197003
(2009)]. We describe the proximity effect in mesoscopic
superconductor/normal-metal bilayers by using the Bogoliubov-de Gennes
equations for a tight-binding Hamiltonian with competing antiferromagnetic and
d-wave superconductivity orders . The temperature dependent local density of
states is calculated as a function of the distance from the interface. Bound
state due to both d-wave and spin density wave gaps are formed in the normal
metal for energies less than the respective gaps. If there is a mismatch
between the Fermi velocities in the two layers we observe that these states
will shift in energy when spin density wave order is present, thus inducing a
minigap at finite energy. We conclude that the STM measurement in the proximity
structures is able to distinguish between the two scenarios proposed for the
pseudogap (competing or precursor to superconductivity).
