 Symmetric concave-mirror resonator

Two identical concave mirrors of curvature radius R separated by L have half-round-trip ray matrix $A=\left(\begin{array}{cc}1& L\\ -2/R& 1-2L/R\end{array}\right)$. Its determinant is one and its trace is $2(1-L/R)$. The eigenvalues are a conjugate unit-modulus pair for the displayed interval. At the confocal point $L=R$, $A^2=-I$, so it is bounded despite the full-round-trip trace reaching minus two. Matching a Gaussian beam's mirror wavefront gives $z_R^2=(L/2)(R-L/2)$ and $\pi w_0^2=\lambda z_R$. The endpoints are degenerate limiting geometries, not finite confined Gaussian modes.