Admittance Space Geometry of N-Layer Dielectric Structures and Synthesis Problems

Abstract

It is known that the level of analytical modeling direct problems of electrodynamics of layered dielectric structures (LDSs) is seriously lagging behind the needs of optimization filter synthesis problems and inverse problem solving in this field. In this paper, we consider the problem formulation of synthesizing a $\left[ {{{\mathcal{K}}_{1}},{{\mathcal{K}}_{2}}} \right]$ band-pass optical filter based on the ideal of its energy reflection coefficient $\tilde {R}\left( \kappa \right)$ specified in the wavenumber band in the $\mathbb{C}\left[ {{{\mathcal{K}}_{1}},{{\mathcal{K}}_{2}}} \right]$ space metric. This problem statement is simplified and supplemented with conditions that are important for practice for the system admittances of $\vec {p}\mathop = \limits^{{\text{def}}} \left( {{{p}_{1}}, \ldots ,{{p}_{N}}} \right)$, where ${{{\text{K}}}_{{\vec {p}}}}\mathop = \limits^{{\text{def}}} \left\{ {\hat {p} \leqslant {{p}_{j}} \leqslant \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{p} ,\left( {j = 1, \ldots ,N} \right)} \right\}$, to belong to the cube of allowable admittances and electrical thicknesses of the system layers of $\vec {\nu }\mathop = \limits^{{\text{def}}} \left( {{{\nu }_{1}}, \ldots ,{{\nu }_{N}}} \right)$ to the parallelepiped ${{{{\Pi }}}_{{\vec {\nu }}}}\mathop = \limits^{{\text{def}}} \left\{ {{{{\hat {\nu }}}_{j}} \leqslant {{\nu }_{j}} \leqslant {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\nu } }}_{j}},\left( {j = 1, \ldots ,N} \right)} \right\}$ of LDS thickness restrictions. The interaction between the structures of the electrodynamic parameter spaces of the ${{\mathcal{P}}_{N}}\mathop = \limits^{{\text{def}}} \left\{ {\vec {p}} \right\}$ admittances and the ${{\mathcal{N}}_{N}}\mathop = \limits^{{\text{def}}} \left\{ {\vec {\nu }} \right\}$ electrical thicknesses of the LDS layers is studied, which was not performed until recently.