General expressions for the coupling coefficient, quality and filling factors for a cavity with an insert using energy coupled mode theory.

A cavity (CV) with a dielectric resonator (DR) insert forms an excellent probe for the use in electron paramagnetic resonance (EPR) spectrometers. The probe's coupling coefficient, κ, the quality factor, Q, and the filling factor, η are vital in assessing the EPR spectrometer's performance. Coupled mode theory (CMT) is used to derive general expressions for these parameters. For large permittivity the dominating factor in κ is the ratio of the DR and CV cross sectional areas rather than the dielectric constant. Thus in some cases, resonators with low dielectric constant can couple much stronger with the cavity than do resonators with a high dielectric constant. When the DR and CV frequencies are degenerate, the coupled η is the average of the two uncoupled ones. In practical EPR probes the coupled η is approximately half of that of the DR. The Q of the coupled system generally depends on the eigenvectors, uncoupled frequencies (ω1,ω2) and the individual quality factors (Q1,Q2). It is calculated for different probe configurations and found to agree with the corresponding HFSS® simulations. Provided there is a large difference between the Q1, Q2 pair and the frequencies of DR and CV are degenerate, Q is approximately equal to double the minimum of Q1 and Q2. In general, the signal enhancement ratio, Iwithinsert/Iempty, is obtained from Q and η. For low loss DRs it only depends on η1/η2. However, when the DR has a low Q, the uncoupled Qs are also needed. In EPR spectroscopy it is desirable to excite only a single mode. The separation between the modes, Φ, is calculated as a function of κ and Q. It is found to be significantly greater than five times the average bandwidth. Thus for practical probes, it is possible to excite one of the coupled modes without exciting the other. The CMT expressions derived in this article are quite general and are in excellent agreement with the lumped circuit approach and finite numerical simulations. Hence they can also be applied to a loop-gap resonator in a cavity. For the design effective EPR probes, one needs to consider the κ, Q and η parameters.