Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects

Abstract

This paper is concerned with linear and nonlinear magneto-optical effects in multilayered magnetic systems when treated by the simplest phenomenological model that allows their response to be represented in terms of electric polarization. The problem is addressed by formulating a set of boundary conditions at infinitely thin interfaces, taking into account the existence of surface polarizations. Essential details are given that describe how the formalism of distributions ͑generalized functions͒ allows these conditions to be derived directly from the differential form of Maxwell's equations. Using the same formalism we show the origin of alternative boundary conditions that exist in the literature. The boundary value problem for the wave equation is formulated, with an emphasis on the analysis of second harmonic magneto-optical effects in ferromagnetically ordered multilayers. An associated problem of conventions in setting up relationships between the nonlinear surface polarization and the fundamental electric field at the interfaces separating anisotropic layers through surface susceptibility tensors is discussed. A problem of self-consistency of the model is highlighted, relating to the existence of rescaling procedures connecting the different conventions. The linear approximation with respect to magnetization is pursued, allowing rotational anisotropy of magneto-optical effects to be easily analyzed owing to the invariance of the corresponding polar and axial tensors under ordinary point groups. Required representations of the tensors are given for the groups ϱm, 4mm, mm2, and 3m. With regard to centrosymmetric multilayers, nonlinear volume polarization is also considered. A concise expression is given for its magnetic part, governed by an axial fifth-rank susceptibility tensor being invariant under the Curie group ϱϱm.

Magnetization-induced-second-harmonic generation from surfaces and interfaces

“…40 Most often, however, bulk anisotropy of layers can be neglected, together with the linear MO effects that are argued to be small enough.…”

Response theory for time-resolved second-harmonic generation and two-photon photoemission

“…A simplified description of the surface using Fresnel factors has been employed successfully to describe SFG from metals [25][26][27][28]. A detailed discussion of boundary conditions at the surface, focusing on the nonlinear optical response of magnetic systems, can be found in [29].…”

The Field of an Electric Dipole and the Polarizability of a Conducting Object Embedded in the Interface Between Dielectric Materials

“…Previous authors [14,[20][21][22][23][24] have considered the problem of obtaining jump conditions for the fields across such a polarization sheet, but the distinction between excess and bound dipole moments in these treatments has not been made clear. We follow the method of [24], adapted for our situation.…”

“…We should note that these conclusions would be not be the same if a different microscopic model were chosen for the dipole. For instance, in [20], a vacuum layer of small thickness on either side of the plane z = 0 is postulated, the dipole is placed at z = 0 and the thickness of the vacuum layer is allowed to approach zero. In this case, no additional bound charge will be associated with that of the dipole.…”