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

Let $a=L/2$, so each mirror in the symmetric confocal optical resonator has radius $R=L=2a$. At mirror 1, an incident beam whose waist is at the centre has $q_{\mathrm{in}}=a+i{z}_R$, with distances measured along its incoming direction. Reflection givesAfter propagating the distance $a$ back to the centre, its parameter is purely imaginary. Thus the reflected waist is also at the cavity centre andThe incident and reflected waist radii need not agree. With axial position $z$ measured from the centre, their radii obeyAt either mirror, $z^2=a^2$ and the two spot sizes agree, as reflection requires. Between the mirrors they generally have different waists while still reproducing themselves after a full round trip.