Modes and Fields of Two Stacked Dielectric Resonators in a Cavity of an Electron Paramagnetic Resonance Probe

An electron paramagnetic resonance (EPR) probe consisting of two dielectric resonators (DRs) and a cavity (CV) is ideal for EPR experiments where both signal enhancement and tuning capabilities are required. The coupling of two DRs, resonating in their TE01δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TE}}_{01\delta }$$\end{document} mode and a CV resonating in its TE011\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TE}}_{011}$$\end{document} mode, is studied using energy-coupled mode theory (ECMT). The frequencies and eigenvectors of the three coupled modes are analytically derived. As predicted numerically, ECMT confirms that the TE++-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TE}}^{ + + - }$$\end{document} and TE+--\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TE}}^{ + - - }$$\end{document} modes are indeed found to be degenerate at a specific distance between the two DRs d12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{12}$$\end{document}. Additionally, the condition at which degeneracy occurs is specified. For a considerable range, the calculated frequency of the TE+++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TE}}^{ + + + }$$\end{document} mode changes linearly with respect to d12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{12}$$\end{document}. The TE+++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TE}}^{ + + + }$$\end{document} mode showed a 500 MHz frequency change over a distance of 2 cm, when the resonance frequency is around 9.7 GHz. This enables the experimentalist to linearly tune the probe over this large frequency range. Finally the asymmetric configuration, where one of the resonators (DR2) is kept at the cavity center and the other one is allowed to move along the cavity axis, is studied. It is estimated that the frequency changes by 600 MHz over a distance of 1.5 cm. A formula for the magnitude of the magnetic field along the cavity axis, where the EPR samples are usually placed, is developed. This is crucial in determining the magnetic field in the vicinity of the sample and the probe’s filling factor.