Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section B / B6 / b / Solution

Write $C_R=4R/(1-R{)}^2$, keeping this coefficient distinct from the finesse $\mathcal{F}$. The transmission has maxima at $\delta =2\pi m$. At a displacement $ϵ$ from a maximum, its ratio to peak transmission is $[1+C_R{\sin }^2(ϵ/2){]}^{-1}$. Half maximum therefore requiresThis is the half-width in wave phase; the full width at half maximum is $2{\delta }_{1/2}$. Successive resonances are $2\pi$ apart, so the exact ideal-model result isFor narrow peaks, $(1-R)/(2\sqrt{R})\ll 1$, the small-angle approximation to the inverse sine givesThus the formula requested in the paper is the high-reflectivity approximation, not an exact identity at arbitrary $R$. Ordinary half-maximum peaks exist only when $(1-R)/(2\sqrt{R})\le 1$, or $R\ge 3-2\sqrt{2}$; below that threshold the transmission minimum never reaches half the peak. The prefactor $T^2/(1-R{)}^2$ cancels from this relative-width calculation.