مرکز منطقه ای اطلاع رساني علوم و فناوري - Asymptotic developments and scattering theory in terms of a vector combining the electric and magnetic fields

Abstract :

The vector combination $\\vec{M}=({\\mu \\over \\epsilon})^{1/4} \\vec{H} + j ({\\epsilon \\over \\mu})^{1/4} \\vec{E}$ which was in principle introduced by Bateman and Silberstein in order to shorten Maxwell\´s equations for homogeneous media, also proves to be useful for the treatment of inhomogeneous media $\\epsilon$ and $\\mu$ not depending on the time). The vector $M\\rightarrow$ is to be considered together with its conjugated quantity $M^{x}\\rightarrow$ obtained by replacing the imaginary unit $j$ by $-j$ . In a source-free medium the Maxwell equations reduce to $\\hbox{curl } \\vec{M} + {j \\over c} (\\epsilon \\mu)^{1/2}\\partial \\vec{M}/ \\partial^{f} = {1 \\over 4} \\hbox{ grad log } {\\mu \\over \\epsilon} \\wedge \\vec{M}^{x}$ and to the equation obtained by taking the conjugated complex value. This relation shows how an interaction between $M\\rightarrow$ and $M^{x}\\rightarrow$ is produced only by the inhomogeneity of the medium. The theory of scattering by special volume elements, as well as that of partial reflections against layers with rapidly changing $\\epsilon$ and $\\mu$ , can be based on the single relation (1) while fully accounting for the vectorial character of the field, The introduction of $M\\rightarrow$ and $M^{x}\\rightarrow$ also enables one to put many results of Luneberg-Kline\´s theory concerning asymptotic developments in a very simple form. As an example we mention the equation: $\\hbox{grad } S \\wedge \\vec{m}_{r} - i(_r \\mu )^{1/2}m_{r} = c \\hbox{ curl } \\vec{m}_{r-1} - {c \\over \\epsilon} \\hbox{ grad log } {\\mu \\over \\epsilon} \\wedge \\vec{m}_{r-1^{x}}$ , which fixes all recurrence relations between the consecutive terms of geometric-optical expansions; these expansions are defined by the asymptotic development $\\over\\rightarrow{M} =e^{ik^{s}} \\Sigma \\min{r=0} \\max {\\infty }(frac{i}{kc))^{r} \\overri- ghtarrow {m_{r}}$ , for monochromatic solutions corresponding to some eiconal function $S$ .