Dispersive correction for the second- and the third-order nonlinear terms of the electrical energy density for a light-dielectric system

The classical energy of a light-dielectric system plays a key role in presenting a quantum theory for the system as the Hamiltonian resulted from the Lagrangian of the system should be equal to it. It is normally, assumed that the second- and the third-order nonlinear terms of the electrical energy are not frequency dependent. However, for some conditions (including the real case) their frequency dependence should be taken into account. Here, we consider the pulse propagation through a lossless, isotropic and inhomogeneous dielectric. Considering the properties of the second- and the third-order susceptibility tensors of the isotropic dielectric, we obtain the dispersive correction for the second and the third-order nonlinear terms of the electromagnetic energy density for this medium as in all practical cases the susceptibility tensors of the dielectrics are frequency dependent.