Casimir-Polder repulsion near edges: Wedge apex and a screen with an aperture

Although repulsive effects have been predicted for quantum vacuum forces between bodies with nontrivial electromagnetic properties, such as between a perfect electric conductor and a perfect magnetic conductor, realistic repulsion seems difficult to achieve. Repulsion is possible if the medium between the bodies has a permittivity in value intermediate to those of the two bodies, but this may not be a useful configuration. Here, inspired by recent numerical work, we initiate analytic calculations of the Casimir-Polder interaction between an atom with anisotropic polarizability and a plate with an aperture. In particular, for a semi-infinite plate, and, more generally, for a wedge, the problem is exactly solvable, and for sufficiently large anisotropy, Casimir-Polder repulsion is indeed possible, in agreement with the previous numerical studies. In order to achieve repulsion, what is needed is a sufficiently sharp edge (not so very sharp, in fact) so that the directions of polarizability of the conductor and the atom are roughly normal to each other. The machinery for carrying out the calculation with a finite aperture is presented. As a motivation for the quantum calculation, we carry out the corresponding classical analysis for the force between a dipole and a metallic sheet with a circular aperture, when the dipole is on the symmetry axis and oriented in the same direction.