Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 2 / d / Solution

Equal waist sizes imply equal positive Rayleigh ranges, so c gives $z_R=z_R^{\prime }=L/2$. Since $z_R=\pi w_0^2/\lambda$, the matched radii areThese are radii; the corresponding spot diameters are twice these values. One practical disadvantage is the transverse-mode degeneracy of the exact confocal geometry: mode frequencies coincide in families, making clean selection of a single transverse mode more difficult. Its $M=-I$ condition also lies at the strict-stability boundary. For example, perturbing the two radii in opposite directions, $R_1=L(1+\delta )$ and $R_2=L(1-\delta )$, gives $g_1{g}_2=-{\delta }^2/(1-{\delta }^2)<0$ for $0<|\delta |<1$, an unstable perturbation. This does not mean that all perturbations, or the ideal confocal resonator itself, are unstable.