Non-degenerate optical four-wave mixing in single-walled carbon nanotubes

A theory is presented that describes the generation of radiation at the anti-Stokes frequency 2ω1 − ω2 created by two coherent light beams, with a difference in frequency ω1 − ω2, injected into an array of uniformly sized and closely packed semiconducting single-walled carbon nanotubes (SWCNTs), which are oriented along the polarization vector of all involved optical waves. Both the real and imaginary parts of the third-order nonlinear susceptibility χ(3), responsible for such an optical-frequency mixing, are calculated within a simple model based on the two-band approximation of the electronic structure of SWCNTs. It is shown that one can maximize the nonlinear optical response by tuning independently the frequencies ω1 and ω2 in resonance with the lowest interband transitions of the semiconducting SWCNTs. The highest peak value of ∣χ(3)∣ under such a resonance excitation is found to be about 10−2 esu, which may be important for nonlinear optical applications of SWCNTs.