= Solution
A longitudinal mode makes an integer round-trip phase accumulation. For the same transverse mode and an empty nondispersive cavity, increasing that integer by one changes the optical frequency by the <free spectral range>
$$
\boxed{\Delta\nu=\frac{c}{2l},\qquad\Delta\omega=\frac{\pi c}{l}.}
$$
Frequency-dependent mirror phases or a dispersive intracavity medium would change the round-trip group delay used here.
For a paraxial ray, use the height-slope column $(x,\theta)^T$. Free propagation and reflection from a concave mirror are represented, in unfolded coordinates, by
$$
P=\begin{pmatrix}1&l\\0&1\end{pmatrix},\qquad C=\begin{pmatrix}1&0\\-2/R&1\end{pmatrix}.
$$
The reflection changes the slope by $-2x/R$. Because the mirrors are identical, the same half-round-trip matrix is applied repeatedly:
$$
A=CP=\begin{pmatrix}1&l\\-2/R&1-2l/R\end{pmatrix},\qquad\det A=1,\quad\frac12\operatorname{tr}A=1-\frac lR.
$$
Its two <eigenvalues> are reciprocal. They lie on the <unit circle> without a growing component when $|1-l/R|<1$, whereas $l>2R$ gives a real eigenvalue of modulus greater than one. At $l=2R$, the repeated eigenvalue has a nontrivial <Jordan block> and generic rays drift out. Thus the nondegenerate stable interval is
$$
\boxed{0<l<2R.}
$$
The confocal value $l=R$ is included: there $A^2=-I$, so the repeated round trip is bounded. Applying a strict trace inequality to the full-round-trip matrix without checking this exact matrix would incorrectly discard that point. These are the paraxial conditions for a <symmetric concave-mirror resonator>.
Symmetry places the <Gaussian beam> waist halfway between the mirrors. Its <Rayleigh range> is $z_R=\pi w_0^2/\lambda$, and its wavefront radius at distance $z$ from the waist is $R_b(z)=z+z_R^2/z$. Matching that wavefront to the mirror at $z=l/2$ gives
$$
R=\frac l2+\frac{z_R^2}{l/2},\qquad z_R^2=\frac l2\left(R-\frac l2\right).
$$
Hence
$$
\boxed{\pi w_0^2=\lambda\sqrt{\frac l2\left(R-\frac l2\right)}.}
$$
A real positive Rayleigh range requires the same stable interval. At its limiting geometries the finite confined Gaussian mode degenerates; the formula is not a way to realize an arbitrarily small physical waist without leaving the paraxial model.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2016-symmetric-resonator-stability.png]
{title=Symmetric cavity geometry and the fundamental waist across its stable interval, including the confocal point}
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