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A review of theoretical results on electromagnetic properties of carbon nanotubes is presented, including the linear electrodynamics of nanotubes, nonlinear transport, optical effects in nanotubes and foundations of their quantum electrodynamics. The method of effective boundary conditions is shown to be a universal tool for the study of electrodynamic problems of nanotubes. The concept of nanotube as a surface wave nanowaveguide in the infrared is developed, and scattering of electromagnetic waves at the nanotube edge is discussed. The high-efficiency generation of high-order harmonics in the nanotubes under the action of strong pumping fields is predicted. Experimental results on third-harmonic generation in a nanotube ensemble is compared with theoretical predictions. It is shown that, in a strong static field, the nanotubes exhibit negative differential conductivity. The formalism of electrodynamics of lossy dispersive mediums is applied to the problem of spontaneous radiation of an excited atom in the carbon nanotube.
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