Nonlinear Thermal Compensators for WGM Resonators
- 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 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 α1 and (2) a spacer made of another material having a thermalexpansion coefficient α2. 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)S2E2(α2 – α1)
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, S2 is the effective cross-sectional area of the spacer, and E2 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
S2 ≈ (∂f/∂T)/[(∂f/∂F)E2(α1 – α2)].
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 S2.