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.

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Using a universal density of states (Mintmire and White 1998 Phys. Rev. Lett. 81 2506), we have found an analytic expression for the long-axis linear susceptibility of single-walled carbon nanotubes valid for arbitrary diameter and chirality. The applicability of our general expression has been assessed by comparison with numerical calculations. Excellent agreement is demonstrated in the low-energy range for semiconducting carbon nanotubes having a moderate or large diameter. The agreement is less convincing for metallic nanotubes having the same diameter as semiconducting ones and the reason for this difference has been clarified. Based on the simple closed-form expression for the linear susceptibility and using the perturbation treatment developed by Aspnes and Rowe (1972 Phys. Rev. B 5 4022), an analytic expression for the third-order nonlinear optical susceptibility χ((3))(ω;0,0,ω) has been derived for arbitrary semiconducting single-walled carbon nanotubes.

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Analytical expressions for dipole matrix elements describing interband optical transitions in carbon nanotubes are obtained for arbitrary light polarization and nanotube chiralities. The effect of the symmetry with respect to the time reversal on the dependences of the optical matrix elements on the quantum numbers of electronic states in carbon nanotubes is studied.