Kramers-Kronig relations in nonlinear optics

We review dispersion relations, which relate the real part of the optical susceptibility (refraction) to the imaginary part (absorption). We derive and discuss these relations as applied to nonlinear optical systems. It is shown that in the nonlinear case, for self-action effects the correct form for such dispersion relations is nondegenerate, i.e. it is necessary to use multiple frequency arguments. Nonlinear dispersion relations have been shown to be very useful as they usually only require integration over a limited frequency range (corresponding to frequencies at which the absorption changes), unlike the conventional linear Kramers-Kronig relation which requires integration over all absorbing frequencies. Furthermore, calculation of refractive index changes using dispersion relations is easier than a direct calculation of the susceptibility, as transition rates (which give absorption coefficients) are, in general, far easier to calculate than the expectation value of the optical polarization. Both resonant (generation of some excitation that is long lived compared with an optical period) and nonresonant 'instantaneous' optical nonlinearities are discussed, and it is shown that the nonlinear dispersion relation has a common form and can be understood in terms of the linear Kramers-Kronig relation applied to a new system consisting of the material plus some 'perturbation'. We present several examples of the form of this external perturbation, which can be viewed as the pump in a pump-probe experiment. We discuss the two-level saturated atom model and bandfilling in semiconductors among others for the resonant case. For the nonresonant case some recent work is included where the electronic nonlinear refractive coefficient, n,, is determined from the nonlinear absorption processes of two-photon absorption, Raman transitions and the a.c. Stark effect. We also review how the dispersion relations can be extended to give alternative forms for frequency summation which, for example, allows the real and imaginary parts of x!'l to be related.