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We consider the Casimir pressure between two graphene sheets andcontributions to it determined by evanescent and propagating waves withdifferent polarizations. For this purpose, the derivation of the 2-dimensional(2D) Fresnel reflection coefficients on a graphene sheet is presented in termsof the transverse and longitudinal dielectric permittivities of graphene withdue account of the spatial dispersion. The explicit expressions for bothdielectric permittivities as the functions of the 2D wave vector, frequency,and temperature are written along the real frequency axis in the regions ofpropagating and evanescent waves and at the pure imaginary Matsubarafrequencies using the polarization tensor of graphene. It is shown that in theapplication region of the Dirac model nearly the total value of the Casimirpressure between two graphene sheets is determined by the electromagnetic fieldwith transverse magnetic (TM) polarization. By using the Lifshitz formulawritten along the real frequency axis, the contributions of the TM-polarizedpropagating and evanescent waves into the total pressure are determined. Byconfronting these results with the analogous results found for plates made ofreal metals, the way for bringing the Lifshitz theory using the realisticresponse functions in agreement with measurements of the Casimir force betweenmetallic test bodies is pointed out.

physics

The Casimir Force between Two Graphene Sheets: 2D Fresnel Reflection  Coefficients, Contributions of Different Polarizations, and the Role of  Evanescent Waves