Accurate characterization of the temperature coefficient of permittivity of sapphire utilizing the dual-mode frequency locked technique

In this work we present a new method to obtain accurate characterization of the temperature coefficient of permittivity (TCP) of an anisotropic crystal, which we name the dual-mode frequency locked (DMFL) technique. The technique requires precise simultaneous frequency measurements of two orthogonally polarized modes as a function of temperature. This was achieved by configuring a sapphire resonator as a pound locked dual-mode oscillator and measuring the frequencies with a frequency counter, while simultaneously recording the temperature with a platinum thermometer. From the two data files of frequency versus temperature, and the knowledge of electric and dimensional filling factors (obtained from finite element analysis), it was possible to calculate the integrals of the TCP without resorting to polynomial fits. This resulted in a precise model of the components of the TCP between 50 to 77 K for sapphire. The model is accurate enough to predict within a few degrees K, the frequency turnover temperatures of the difference frequency of a dual-mode oscillator.