### At one target temperature, thermal frequency fluctuations would vanish to first order.

In an alternative version of a proposed bimaterial thermal compensator for a whispering-gallery-mode (WGM) optical resonator, a mechanical element having nonlinear stiffness would be added to enable stabilization of a desired resonance frequency at a suitable fixed working temperature. The previous version was described in “Bimaterial Thermal Compensators for WGM Resonators” (NPO-44441), *NASA Tech Briefs*, Vol. 32, No. 10 (October 2008), page 96. Both versions are intended to serve as inexpensive means of preventing (to first order) or reducing temperature-related changes in resonance frequencies.

Because the version now proposed is similar to the previous version in most respects, it is necessary to recapitulate most of the description from the cited prior article, with appropriate modifications. In both the previous and present versions (see figure), a compensator as proposed would include (1) a frame made of one material having a thermal-expansion coefficient α_{1} and (2) a spacer made of another material having a thermalexpansion coefficient α_{2}. The WGM resonator would be sandwiched between disks, and the resulting sandwich would be squeezed between the frame and the spacer. Assuming that the cross-sectional area of the frame greatly exceeded the cross-sectional area of the spacer and that the thickness of the sandwich was small relative to the length of the spacer, the net rate of change of a resonance frequency with changing temperature would be given by

*df/dT* ≈ *∂f/∂T* + (*∂f/∂F*)*S*_{2}*E*_{2}(α_{2} – α_{1})

where *f* is the resonance frequency, *T* is temperature, *∂f/∂T* is the rate of change of resonance frequency as a function of temperature of the uncompensated resonator, *∂f/∂F* is the rate of change of frequency as a function of applied force *F* at constant temperature, *S*_{2} is the effective cross-sectional area of the spacer, and *E*_{2} is the modulus of elasticity of the spacer.

In principle, through appropriate choice of materials and geometry, one could obtain temperature compensation — that is, one could make *df/dT* ≈ 0. For example, the effective spacer cross-sectional area for temperature compensation is given by

*S*_{2} ≈ (*∂f/∂T*)/[(*∂f/∂F*)*E*_{2}(α_{1} – α_{2})].

In practice, because of inevitable manufacturing errors and imprecise knowledge of thermomechanical responses of structural components, it is difficult or impossible to obtain exact temperature compensation of frequency through selection of *S*_{2}.