 Confocal reciprocal-Rayleigh-range map

For two equal mirrors of radius $L$ separated by $L$, a Gaussian waist at the centre is reflected to another waist at the same plane. Putting $a=L/2$, the mirror maps $q=a+i{z}_R$ to $-a+i{a}^2/z_R$, so the reflected Rayleigh range is $a^2/z_R$. Both beams have equal spots on the mirrors although their waist radii differ. Equal waist sizes select $z_R=L/2$.