On the generalized linear response functions

To discuss the response of a system to an electromagnetic field the minimal coupling Hamiltonian is assumed. The quadratic term in the potential is kept. The Kubo formalism for perturbation is applied and perturbations to all orders in the internal fields are considered. However only the first-order perturbation for the external probe is taken. Thus the linear response function is calculated. The inclusion of the quadratic term contributes terms to this function. Due to these terms the Dyson-like propagator equation is corrected. This equation is developed to obtain the different susceptibilities. It is found that the contributions from the density-current and current-current propagators must be taken into account, in addition to the density-density propagator, when the polarizability is considered to obtain the correct result. The polarizability of the system is then calculated in two approximations. When the quadrupole contribution is considered, the number of propagators increases. They are given through a coupled set of generalized Dyson-like equations. The different fields are defined and discussed and the interplay between them is manifested. The simplest coupled set is considered and developed to give two coupled integral equations in an extension to the classical integral equation for the electric polarization field. They are solved for a finite crystal and the refractive index is calculated for the propagating waves when the probing field is light. The phenomena of birefringence and optical rotation are discussed through simplified models. The extinction theorem is found to hold in a generalized form. Hence the dependence on the geometry of the system is discussed. The effects of the quadrupole terms on these phenomena are shown and discussed. Some other optical phenomena and higher multipole contributions are briefly discussed.