  We use a functional approach to the Casimir effect in order to evaluate the
exact vacuum energy for a real scalar field in $d+1$ dimensions, in the
presence of backgrounds that, in a particular limit, impose Dirichlet boundary
conditions on one or two parallel surfaces. Outside of that limit, the
background may be thought of as describing finite-width mirrors with
frequency-dependent transmission and reflection coefficients. We provide new
explicit results for the Casimir energy in some particular backgrounds
