Optical properties of carbon nanotubes

Using the extended Hubbard model and sum-over-state method, we have calculated the linear polarizability $\alpha$ and the third-order nonlinear polarizability $\gamma$ 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 $\mathrm{}(n,m)\mathrm{}$ tubes with $n-m$ being not a multiple of 3, will have smaller $\alpha$ and much smaller $\gamma$ values than other finite length tubes.