Casimir-Lifshitz formula for rectangular dielectric waveguide

We analyze the Casimir-Lifshitz effect associated with the electromagnetic field in the presence of a rectangular waveguide consisting of two distinct dielectric materials in $(3+1)$-dimensional space-time. We employ the surface-mode technique to derive a generalized Lifshitz formula for this specific geometry. Our formulation accounts for the unique dielectric properties of the materials composing the waveguide, leading to a precise calculation of the Casimir-Lifshitz energy. In the asymptotic limit, our results recover the classical expressions for perfectly reflecting boundaries. This work extends the applicability of the Lifshitz formula to more complex systems and provides valuable insights into the influence of dielectric materials on the electromagnetic Casimir effect.