User:Mrivelli/sandbox

The spectral-domain approach ( SDA) is a method for the resolution of electric field's spatial propagation in particular structures. The SDA is only applicable on a planar stratified dielectric structure with presence of metallization of negligible thickness (printed-circuit techniques) and with possible presence of (metallic) ground plane and top cover. The Spectral-Domain method exploits the fact that the basic dielectric structure is uniform on the planes perpendicular to the stratication direction z: that suggests to express the field as a spectrum of plane waves. The modes of the structure are determined solving integral equations formulated imposing the suitable boundary conditions on the conductors. Therefore, the first step consists in obtaining the link between the electromagnetic field in the stratified structure and the currents present. Let us suppose that in a certain volume V'there are electric and magnetic (impressed) currents J and M. The linearity of Maxwell's equations allows to state that the electromagnetic field can be expressed in the following way:


 * $$\underline{E}(\underline{r})= \int_{V'}\underline{\underline{G_{ee}}}(\underline{r},\underline{r'})\cdot \underline{J}(\underline{r'})dV'+ \int_{V'}\underline{\underline{G_{eh}}}(\underline{r},\underline{r'})\cdot \underline{M}(\underline{r'})dV'\,$$


 * $$\underline{H}(\underline{r})= \int_{V'}\underline{\underline{G_{he}}}(\underline{r},\underline{r'})\cdot \underline{J}(\underline{r'})dV'+ \int_{V'}\underline{\underline{G_{hh}}}(\underline{r},\underline{r'})\cdot \underline{M}(\underline{r'})dV'\,$$

If we impose that the previous expressions be solutions of Maxwell's equations, we apply the linearity to consider separately the contribution of impressed electric and magnetic currents, and we exploit the arbitrariness of the currents themselves, we obtain that the dyadic Green's functions Gee, Geh , Ghe , and Ghh are defined as solutions of the following equations (let us note the analogy with Maxwell's equations):


 * $$\nabla \times \underline{\underline{G_{ee}}}= -j\omega\mu\underline{\underline{G_{eh}}}\,$$