  Heterodyne cavity experiments for gravitational wave (GW) detection
experience a rising interest since recent studies showed that they allow to
probe the ultra high frequency regime above $10\,\text{kHz}$. In this paper, we
present a concise theoretical study of the experiment based on ideas from the
former MAGO collaboration which already started experiments in turn of the
millenium. It extends the former results via deriving an additional term
originating from a back-action of the electromagnetic field on the cavity
walls, also known as Lorentz Force Detuning. We argue that this term leads to a
complex dependence of the signal power $P_{\text{sig}}$ on the coupling
coefficient between the mechanical shell modes and the electromagnetic
eigenmodes of the cavity. It turns out that one has to adapt the coupling over
the whole parameter space since the optimal value depends on the mechanical
mode $\omega_l$ and the GW frequency $\omega_g$. This result is particularly
relevant for the design of future experiments.
