Due to translation invariance, the M and E polarizations decouple to Dirichlet and Neumann scalar fields:
"Casimir force at a knife’s edge,"
Graham, Shpunt, Emig, Jaffe, & Kardar, Phys. Rev. D 81, 061701(R) (2010)
The interaction energy between a parabolic cylinder and a mirror at a separation H:
The limit H >> R corresponds to a knife edge, where the interaction as a function of tilt angle is