Instantaneous coherent cavity feedback through a beam splitter (${\gamma }_{\mathrm{eff}}=\gamma (1-\alpha )/(1+\alpha )$)

With $b_0=\beta b_{\mathrm{in}}-\alpha b_1$ and $b_2=\alpha b_{\mathrm{in}}+\beta b_1$, real ${\alpha }^2+{\beta }^2=1$ and $|\alpha |<1$, eliminate the internal fields to obtain $\dot{a}=-{\gamma }_{\mathrm{eff}}a/2-c{b}_{\mathrm{in}}$, where $c=\beta \sqrt{\gamma }/(1+\alpha )$ and $c^2={\gamma }_{\mathrm{eff}}$. The internal output transfer is $b_1/b_{\mathrm{in}}=\beta G/(1+\alpha G)$; the external output transfer is $b_2/b_{\mathrm{in}}=(\alpha +G)/(1+\alpha G)=(s-{\gamma }_{\mathrm{eff}}/2)/(s+{\gamma }_{\mathrm{eff}}/2)$. 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.