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.

A bimaterial compensator would apply, to a WGM resonator, a force that would slightly change the shape of the resonator and thereby change its resonance frequencies. Through suitable choice of the design of the compensator, it should be possible to make the temperature dependence of the force-induced frequency shift equal in magnitude and opposite in sign to the temperature dependence of the frequency shift of the uncompensated resonator so that, to first order, a change in temperature would cause zero net change in frequency.

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}.

According to the present proposal, to make it possible to obtain exact temperature compensation, one would add a component having a nonlinear stiffness to the mechanical load path and would place the entire resonator-and-compensator assembly on a thermoelectric controller, in an oven, or both. Then the temperature dependence of frequency would be approximately quadratic and the net derivative of frequency with respect to temperature would be given by

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

where *A* is a parameter that characterizes the nonlinearity to lowest order in temperature and *ΔT* is the difference between the present temperature and some other temperature, which could be a target temperature. To find the target temperature that gives exact temperature compensation, one sets the derivative equal to zero and solves for *ΔT*:

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

The oven and/or the thermoelectric controller could be used to set the temperature to the exact compensation temperature. Even if the exact values of *A*, *∂f/∂T*, *∂f/∂F*, *S*_{2}, *E*_{2}, α_{1}, and α_{2} were not known in advance, one could still determine the exact compensation temperature by measuring frequency as a function of temperature and finding the lowest point on the approximately quadratic frequency-versus-temperature curve.

*This work was done by Anatoliy Savchenkov, Andrey Matsko, Dmitry Strekalov, Lute Maleki, Nan Yu, and Vladimir Iltchenko of Caltech for NASA’s Jet Propulsion Laboratory.*

*In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to:*

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*Refer to NPO-44567, volume and number of this *NASA Tech Briefs* issue, and the page number.*