Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 61 / 4 / d / Solution

The beam splitter relation and the cavity output equation giveFor $\alpha \ne -1$, solve this scalar algebraic relation and substitute back:The cavity equation becomesFinally, use the other beam-splitter output and ${\alpha }^2+{\beta }^2=1$:These derivations retain the printed input $b_{min}$ in both beam-splitter equations; the converted TeX changes one of these subscripts. Define ${\gamma }_{meff}=\gamma (1-\alpha )/(1+\alpha )$ and $c=\beta \sqrt{\gamma }/(1+\alpha )$, so $\dot{a}=-{\gamma }_{meff}a/2-c{b}_{min}$ and $b_2=b_{min}+ca$. In the nondegenerate range $|\alpha |<1$, ${\gamma }_{meff}>0$ and $c^2={\gamma }_{meff}$, the expected passive input-output relation. This is instantaneous coherent cavity feedback through a beam splitter; it uses the zero-delay interconnection assumed by the equations.