Contoured thermal deformation of mirror surface for detuning parametric instability in an optical cavity

Independent mode frequency control will improve the ability to suppress parametric instabilities in gravitational wave detectors. In this article we investigate a formalism that allows multiple cavity optical modes to be thermally tuned independently based on the transformation between the change in mode spacing and the mirror surface deformation. An FFT based cavity simulation software is used to demonstrate the effectiveness of the formalism. A case study showed that it is possible to tune the HG₀₃ mode spacing by 100 Hz while maintaining other mode spacings with minimum change. Our investigation also indicates that the desired mirror surface deformation is experimentally realisable with a matrix heater.