  We calculate the Casimir interaction energy in $d=2$ spatial dimensions
between two (zero-width) mirrors, one flat, and the other slightly curved, upon
which {\em imperfect\/} conductor boundary conditions are imposed for an
Electromagnetic (EM) field. Our main result is a second-order Derivative
Expansion (DE) approximation for the Casimir energy, which is studied in
different interesting limits. In particular, we focus on the emergence of a
non-analyticity beyond the leading-order term in the DE, when approaching the
limit of perfectly-conducting mirrors. We also show that the system considered
is equivalent to a dual one, consisting of a massless real scalar field
satisfying imperfect Neumann conditions (on the very same boundaries).
Therefore, the results obtained for the EM field hold also true for the scalar
field model
