Using tight-binding model with nearest neighbour interactions, the optical properties of carbon nanotubes under the influence of an external magnetic field are analyzed. First, dipole matrix elements for two cases of light polarized parallel as well as perpendicular to the nanotube axis are analyzed. A close form analytic expression for dipole matrix is obtained for carbon nanotubes with arbitrary chirality in the case of light polarized parallel to the nanotube axis. Then the diagonal and off-diagonal elements of the frequency-dependent susceptibility in the presence of an axial magnetic field are investigated. The off-diagonal elements are applied to calculate the interband Faraday rotation and the Verdet constant. These effects should be clearly detectable under realistic conditions using weak magnetic fields. 1. Introduction The rotation of the polarization of a plane-polarized electromagnetic wave passing through a substance under the influence of a static magnetic field along the direction of propagation is known as the Faraday rotation [1, 2]. It arises because right circular polarization and left circular polarization waves propagate with different phase velocities, and a phase difference results between the two waves. For a weak absorption and then ignoring the extinction coefficient, the wave vector components of a linearly polarized light have different refractive indices , and then the rotation of the polarization plane known as optical Faraday rotation is proportional to the difference in the refractive indices, . The Faraday rotation for solids, liquids, and gases has been investigated both theoretically and experimentally [2–8]. Optical Faraday rotation near the resonance regime has been proposed to measure a single-electron spin in a single quantum dot inside a microcavity [9]. Numerical calculations have been made [10] to calculate the Faraday rotation in graphite by extrapolation from high-field tight binding calculations. It has been demonstrated that the graphene turns the polarization by several degrees in modest magnetic fields [11]. Experimentally, Faraday rotation is measured mainly under nonresonant conditions, that is, for frequencies far from resonance, where the absorption is weak. In this case, the angle of rotation is proportional to the imaginary part of the off-diagonal elements of susceptibility. By studying the off-diagonal elements of susceptibility, the goal of this paper is to investigate the Faraday rotation in single-walled carbon nanotubes (SWCNs) threaded by a parallel magnetic field. Hence, we calculate the
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