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Three-mode parametric instability is a threat to attaining design powerlevels in Advanced gravitational wave detectors. The first observation ofthree-mode parametric instability in a long optical cavity revealed thatinstabilities could be suppressed by time variation of the mirror radius ofcurvature. In this paper, we present three dimensional finite element analysisof this thermo-acousto-optics system to determine whether thermal modulationcould provide sufficient instability's suppression without degrading timeaveraged optical performance. It is shown that deformations due to the timeaveraged heating profile on the mirror surface can be compensated by rearsurface heating of the test mass. Results show that a $CO_2$ laser heating beamwith a modulation amplitude of $1$ Watt at 0.01 Hz is sufficient to stabilizeacoustic mode with parametric gain up to 3. The parametric gain suppressionfactor is linearly proportional to the peak modulation power.

Thermal modulation for suppression of parametric instability in advanced gravitational wave detectors