Optical properties of carbon nanotubes

Using the extended Hubbard model and sum-over-state method, we have calculated the linear polarizability α and the third-order nonlinear polarizability γ for carbon nanotubes with finite lengths. We find that the chiral symmetry of nanotubes with finite length has a great effect on their optical properties. For example, the finite length $(n,m)$ tubes with n-m being not a multiple of 3, will have smaller α and much smaller γ values than other finite length tubes.