Universal analytic expression of electric-dipole matrix elements for carbon nanotubes

The optical properties of carbon nanotubes strongly depend on the polarization direction of the incident light. The challenge of describing the optical properties analytically is mainly associated with the calculation of dipole matrix elements. Although an analytic expression has been obtained for dipole matrix elements for light polarized parallel to the nanotube axis no expression has been found so far for the perpendicular case. Based on the structural symmetry we obtain an analytic expression for the electric-dipole matrix elements of single wall carbon nanotubes with arbitrary chirality for linearly polarized light with polarization perpendicular to the nanotube axis. This expression is used to calculate the short axis linear susceptibility semianalytically. Excellent agreement with numerical calculations is demonstrated.