Taylor series, electromagnetic field

Raman scattering has been discussed by many authors. As in the case of IR vibrational spectroscopy, the interaction is between the electromagnetic field and a dipole moment, however in this case the dipole moment is induced by the field itself The induced dipole is pj j = a E, where a is the polarizability. It can be expressed in a Taylor series expansion in coordinate isplacement... 

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 ... 

The paraxial approximation is essentially a Taylor series expansion of an exact solution of the wave equation in powers of p/w, terminated at ( p/tv), that allows us to exploit the rapid decay of a Gaussian beam away from the optical axis. We will develop a more precise criterion in the sequel. We will also show that the phase and amplitude modulation of the underlying plane wave structure of the electromagnetic field is a slowly varying function of distance from the point where the beam is launched. 

Following the same principle that was used in deriving the localized quasi-linear approximation for an electromagnetic field, we can assume that the dominant contribution to the integral G [As Ap ] in equation (14.50) is from the neighborhood of the point r = r. We can expand p (r,o ) into a Taylor series about r = Fj, similar to (14.44) ... 

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