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Nonlinear Thermal Compensators for WGM Resonators
 Created: Tuesday, 01 September 2009
At one target temperature, thermal frequency fluctuations would vanish to first order.
In an alternative version of a proposed bimaterial thermal compensator for a whisperinggallerymode (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” (NPO44441), 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 temperaturerelated 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 forceinduced 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 thermalexpansion 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 crosssectional area of the frame greatly exceeded the crosssectional 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 crosssectional 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 crosssectional 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}.
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