On the derivation of the after-effect solution for the Kerr effect relaxation from the Langevin equation for rotation in three dimensions

Abstract

The Kerr effect response (KER) of an assembly of non-interacting dipolar molecules following the removal of a dc electric field is calculated by direct averaging of the Langevin equation. The Hermite polynomials in the angular velocity components (ω 1 and ω 2) and the associated Legendre functions of order 2 in the polar angle, ϑ, are chosen as variables. This leads directly, using the statistical properties of the white noise driving torques, to the differential-difference equations which govern in Kerr effect relaxation. The general term in the average of the product of Hermite polynomials and the associated Legendre function of order 2 is given. This allows us to calculate the Laplace transform of the average value of the second order Legendre polynomial; < P 2 (cos ϑ( t))>, following the removal of the a dc field, i.e. the affect-effect solution for the dynamic Kerr effect, up to any desired degree of accuracy using the Langevin equation method by converting the set of differential-difference equations into a set of ordinary differential equations and successively increasing the size of the transition matrix. The convergence of the procedure for various values of the inertial parameters is investigated by computing the Fourier transform of the response for successively increasing sizes of the transition matrix.