The Casimir Energy of a Medium Containing a Permittivity Gradient

Abstract

The zero-point energy of a nodispersive but inhomogeneous dielectric sandwiched between perfectly conducting plates is considered. It is assumed that the permittivity increases lineally as separating from one of the plates, and the gradient is small. With the assumption it is shown that the eigenmodes of the electric and magnetic field between plates are determined by an equation including the Airy functions. Using asymptotic expansions of the Airy functions the obtained equation is solved. The Casimir energy is defined as the energy of the radiation at zero temperature, and it is calculated by the mode summation method.