 Solution

For a paraxial Gaussian beam, choose the complex beam parameter withHere $R_b$ is the wavefront radius and $w$ the intensity $1/e^2$ radius. The ABCD law isFor a real, lossless round-trip ray transfer matrix $M$ in free space, $AD-BC=1$. A reproducing cavity mode requires $q=(Aq+B)/(Cq+D)$, henceExcept for a scalar identity matrix, a finite physical solution requires a complex-conjugate root pair, one with positive imaginary part. The generic strict stability condition is thereforeEquivalently, the ray eigenvalues are distinct unit-modulus numbers $e^{\pm i\mu }$, so repeated round trips do not cause unbounded ray excursions. Adequate mirror apertures and longitudinal resonance are also needed for a low-loss resonant mode.

The strict inequality has a degenerate exception relevant to this paper. If $M=-I$, then $q^{\prime }=q$ for every physical $q$ even though $A+D=-2$. A symmetric confocal resonator has exactly this matrix and supports Gaussian modes; it is not excluded as a physical cavity. Thus the printed strict condition should be understood as the generic, nondegenerate stability criterion, not a universal necessary condition covering the confocal case in c.