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

Derivative expansion for the electromagnetic and Neumann Casimir effects in $2+1$ dimensions with imperfect mirrors