Considerations on parametric instability in Fabry–Perot interferometer

We evaluate the parametric phenomenon for the main optical mode coupled to other resonant waves excited by acoustic interaction with the mirrors of a Fabry–Perot. We apply this to the arm cavity of a gravitational wave antenna. Under certain assumptions we find that the amplitude of these parasitic modes is expressed by analytic solutions that are always damped. We analyze both the zero detuning and the detuned case and solve the equations. The form of the solution shows that for equally spaced and excited cavity modes the instability is expressed by a threshold condition, which is well approximated for LIGO arm resonator parameters.