Alternative calculation of the Casimir forces between birefringent plates

Barash has calculated the Casimir forces between parallel birefringent plates with optical axes parallel to the plate boundaries [Izv. Vyssh. Uchebn. Zaved., Radiofiz. 12, 1637 (1978)]. The interesting feature of the solution compared to the case of isotropic plates is the existence of a Casimir torque which acts to line up the optical axes if they are not parallel or perpendicular. The forces were found from a calculation of the Helmholtz free energy of the electromagnetic field. Given the length of the calculations in this problem and hopes of an experimental measurement of the torque, it is important to check the results for the Casimir forces by a different method. We provide this check by calculating the electromagnetic stress tensor between the plates and showing that the resulting forces are in agreement with those found by Barash.

Figure 1

Two birefringent plates lie in the $yz$ plane with constant separation $a$. The lines on the plates represent the directions of the optical axes, which also lie in the $yz$ plane. The axis of plate 2 lies in the $z$ direction while that of plate 1 is rotated relative to this by $\theta$. Between the plates is an isotropic medium of permittivity $ϵ$ and permeability $\mu$.