= Instantaneous coherent cavity feedback through a beam splitter
{title2=$\gamma_{\rm eff}=\gamma(1-\alpha)/(1+\alpha)$}
With $b_0=\beta b_{\rm in}-\alpha b_1$ and $b_2=\alpha b_{\rm in}+\beta b_1$, real $\alpha^2+\beta^2=1$ and $|\alpha|<1$, eliminate the internal fields to obtain $\dot a=-\gamma_{\rm eff}a/2-cb_{\rm in}$, where $c=\beta\sqrt\gamma/(1+\alpha)$ and $c^2=\gamma_{\rm eff}$. The internal output transfer is $b_1/b_{\rm in}=\beta G/(1+\alpha G)$; the external output transfer is $b_2/b_{\rm in}=(\alpha+G)/(1+\alpha G)=(s-\gamma_{\rm eff}/2)/(s+\gamma_{\rm eff}/2)$. The loop is asymptotically stable in this nondegenerate range. At $\alpha=1$ the cavity mode is undamped and decoupled; at $\alpha=-1$ the instantaneous algebraic feedback connection is singular.
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