First-order correction to the Casimir force within an inhomogeneous medium

For the Casimir piston filled with an inhomogeneous medium, the Casimir energy is regularized and expressed with cylinder kernel coefficients by using the first-order perturbation theory. When the refraction index of the medium is smoothly inhomogeneous (i.e., derivatives of all orders exist), logarithmically cutoff-dependent term in Casimir energy is found. We show that in the piston model this term vanishes in the force and thus the Casimir force is always cutoff-independent, but this term will remain in the force in the half-space model and must be removed by additional regularization. We investigate the inhomogeneity of an exponentially decaying profile, and give the first-order corrections to both free Casimir energy and Casimir force. The present method can be extended to other inhomogeneous profiles. Our results should be useful for future relevant calculations and experimental studies.