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Casimir torque between birefringent plates with metamaterials


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We investigate the Casimir torque between two coaxial plates separated by a small gap, each with positive or negative refractive index. Starting from the existing formula of two plates both with positive refractive index, we use suitable approximations and limits to extend it to the negative refractive index case. The torque found between two plates both with negative refractive index shows similar behavior in rotational motion to the case where both plates have positive refractive index, i.e., their principal axes in both x and y directions rotate to align them in their stable equilibrium-angle position of 0 or π, due to the “attractive torque“. The torque between two plates, one with positive and the other negative refractive index has a subtle behavior which depends on r value, the multiplication of the anisotropy for the plates. If r lies in the interval (−∞, −2), this system is identical to that where both plates carry positive or negative refractive index. However, if r lies in the interval math formula, generally three new equilibrium points arise additional to the usual four of math formula, and the former three are all unstable while the latter four are all stable. If r lies in the interval math formula, a different rotational behavior occurs due to the “repulsive torque”, i.e., the principal axes of each plate increase their misalignment to avoid the unstable equilibrium positions 0 or π. These effects probably cannot be measured experimentally at present due to their requirement of a wide range of frequencies for negative refractive index and a very small gap between the two plates, but may be in future.