  We formulate the Babinet Principle (BP) as a relation between the scattering
amplitudes for electromagnetic waves, and combine it with multiple scattering
techniques to derive new properties of Casimir forces. We show that the Casimir
force exerted by a planar conductor or dielectric on a self- complementary
perforated planar mirror is approximately half that on a uniform mirror
independent of the distance between them. The BP suggests that Casimir edge
effects are anomalously small, supporting results obtained earlier in special
cases. Finally, we illustrate how the BP can be used to estimate Casimir forces
between perforated planar mirrors.
