= Solution
For a <paraxial approximation>[paraxial] <Gaussian beam>, choose the <complex beam parameter> with
$$
\frac1q=\frac1{R_b}-i\frac{\lambda}{\pi w^2},\qquad
q=z+iz_R,\quad z_R>0.
$$
Here $R_b$ is the wavefront radius and $w$ the intensity $1/e^2$ radius. The <ABCD law> is
$$
\boxed{q_{\rm out}=\frac{Aq_{\rm in}+B}{Cq_{\rm in}+D}.}
$$
For a real, lossless round-trip <ray transfer matrix> $M$ in free space, $AD-BC=1$. A reproducing <optical cavity>[cavity] mode requires $q=(Aq+B)/(Cq+D)$, hence
$$
Cq^2+(D-A)q-B=0,\qquad
(D-A)^2+4BC=(A+D)^2-4.
$$
Except for a <scalar matrix>[scalar identity matrix], a finite physical solution requires a <complex conjugate>[complex-conjugate] root pair, one with positive imaginary part. The generic strict stability condition is therefore
$$
\boxed{-2<A+D<2.}
$$
Equivalently, 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.
\b[The strict inequality has a degenerate exception relevant to this paper.] If $M=-I$, then $q'=q$ for every physical $q$ even though $A+D=-2$. A symmetric <confocal optical resonator>[confocal resonator] has exactly this <matrix> and supports Gaussian modes; it is not excluded as a physical <optical cavity>[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 <2/c>.
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