Non-Stationary Electromagnetics, 2nd Edition

Chapter 4

3D+T Problems with Electromagnetic Transients

4.1  The 3D Resolvent for a Problem with a Plane Boundary of a Dielectric Half-Space

The resolvent construction used in the previous sections for the 1D case is developed here for the 3D one. The resolvent must satisfy Eq. (3.7)

$\langle p\left|\widehat{R}\right|p^{\prime }\rangle =\langle p\left|\widehat{K}\right|p^{\prime }\rangle +{\displaystyle \int d{p}_1\langle p\left|\widehat{K}\right|p_1\rangle \langle p_1\left|\widehat{R}\right|p^{\prime }\rangle }$

(4.1)

The initial problem in an unbounded medium is described by the resolvent

${\langle x\left|{\widehat{R}}_e\right|x^{\prime }\rangle }_{ij}=\frac{{\epsilon }_1-\epsilon }{{\epsilon }_1}\left(\frac{{\partial }^2}{\partial x_i\partial x_j}-\frac{1}{v_1{2}^{}}{\delta }_{ij}\frac{{\partial }^2}{\partial t^2}\right)\frac{\delta \left(t-t^{\prime }-\left|r-r^{\prime }\right|/v_1\right)}{\left|r-r^{\prime }\right|},$

(4.2)

where the support domain of the δ-function is the light cone Γ. Let us consider now the situation when there is a boundary that divides space into two half-spaces; the left half-space is occupied by the background dielectric with ...