Calculation of the rise transient and relaxation time of the induced dipole Kerr effect

Abstract

The exact calculation of the rise transient of the birefringence and the corresponding relaxation times by different theoretical methods is described, in particular the Kerr-effect response of an assembly of nonpolar but anisotropicaily polarizable molecules following the imposition of a constant electric field is studied by solving the Smoluchowski equation. This equation is transformed into a set of differential recurrence relations containing Legendre polynomials of even order only. By taking the Laplace transform of the birefringence function, it is shown that the singularity at s=0 (zero-frequency limit) may be removed so that the relaxation time for the rise process may be exactly expressed as a sum of products of Kummer functions and its first derivatives. The second approach is based on a matrix method where the spectrum of eigenvalues λ2j and their associated amplitudes A2j (extracted from the first components of eigenvectors) are calculated allowing one to express the relaxation time as ∑A2j(λ2j-1). Numerical values of this time are tabulated for a large range of g values (0<g<40), g being the parameter measuring the ratio of the orientational energy arising from the electrical polarizabilities to the thermal energy. It is thus demonstrated that the lowest eigenvalue (λ2) dominates almost completely the rise process. The effective relaxation time is also calculated exactly and expressed very simply as the ratio of two Kummer functions. Its evolution as a function of g leads to behavior similar to that of the relaxation time obtained either from the Kummer functions or from the eigenvalue method. It is characterized by a maximum situated around g = 2, which is interesting in view of experimental applications.