Taylor series, nonlinear optics

We have not failed to recognize that appropriately designed (6,0) carbon and C/B/N nanotubes may display considerably enhanced nonlinear optical activity. This term refers to the response of the dipole moment of a molecule (or the polarization of bulk material) to the oscillating electric field of electromagnetic radiation.82 85 The component of the dipole moment along an axis i in the presence of an electric field e can be represented by a Taylor series ... 

TDHF [52, 53] is one of the most widely-employed ab initio techniques to evaluate nonlinear-optical response tensors. The fDHF approach is size consistent but cannot account for the finite lifetime of the excited states. The matrices of the TDHF equation are expanded in a Taylor series of the perturbation due to the static and/or dynamic electric fields and are solved for each order [52, 53], The so-obtained successive field-derivatives of the density matrix are then inserted into the expressions for the hyperpolarizability,... 

Molecular optical nonlinearities can be introduced in a similar way to bulk nonlinearities. The ground state electric dipole moment of a molecule upon application of an external electric field can be expanded as a Taylor series of the applied field, taking the unperturbed molecule as the zero point of the series [2] ... 

In centrosymmetric media the dominant nonlinearity is of the third order. This effect is represented by the third term in the Taylor s series (Equation 1) it is the result of the interaction of a number of optical fields (one to three) producing a new frequency + Uj + cdj. The third order polarization is given by... 

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