### 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}.