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A symmetric confocal optical resonator has two concave mirrors of equal curvature $R$, separated by $R$. Its symmetric fundamental Gaussian beam has a central beam waist and Rayleigh range $R/2$. Its ideal transverse spectrum has degeneracies, so transmission at one resonance need not certify exclusive excitation of the fundamental normal mode.

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.

Let $a=L/2$, so each mirror in the symmetric confocal optical resonator has radius $R=L=2a$. At mirror 1, an incident beam whose waist is at the centre has $q_{\mathrm{in}}=a+i{z}_R$, with distances measured along its incoming direction. Reflection givesAfter propagating the distance $a$ back to the centre, its parameter is purely imaginary. Thus the reflected waist is also at the cavity centre andThe incident and reflected waist radii need not agree. With axial position $z$ measured from the centre, their radii obeyAt either mirror, $z^2=a^2$ and the two spot sizes agree, as reflection requires. Between the mirrors they generally have different waists while still reproducing themselves after a full round trip.

Equal waist sizes imply equal positive Rayleigh ranges, so c gives $z_R=z_R^{\prime }=L/2$. Since $z_R=\pi w_0^2/\lambda$, the matched radii areThese are radii; the corresponding spot diameters are twice these values. One practical disadvantage is the transverse-mode degeneracy of the exact confocal geometry: mode frequencies coincide in families, making clean selection of a single transverse mode more difficult. Its $M=-I$ condition also lies at the strict-stability boundary. For example, perturbing the two radii in opposite directions, $R_1=L(1+\delta )$ and $R_2=L(1-\delta )$, gives $g_1{g}_2=-{\delta }^2/(1-{\delta }^2)<0$ for $0<|\delta |<1$, an unstable perturbation. This does not mean that all perturbations, or the ideal confocal resonator itself, are unstable.

For a symmetric confocal optical resonator, the mirror separation equals either mirror's curvature, $R_2=L_2$, so its symmetric fundamental Gaussian beam hasThe same optical mode matching calculation applies, now with $f_{\min }=\sqrt{z_A{L}_2/2}$. Keeping the example's $L_2=10\mathrm{m}$ gives $z_B=5\mathrm{m}$ and $f_{\min }=0.707107\mathrm{m}$, compared with $0.930605\mathrm{m}$ for the nonconfocal target. With $f=1.5\mathrm{m}$, the two branches becomeAgain only the positive-$v$ branch fits the stated downstream geometry. The confocal target's smaller Rayleigh range means a smaller fundamental beam waist at the same wavelength and a less stringent minimum $f$.

A confocal optical resonator also has transverse normal mode degeneracies: in the ideal paraxial spectrum, increasing transverse order by two shifts the resonance by one longitudinal spacing. Such a cavity can transmit combinations of higher transverse normal modes at a fundamental resonance, so transmitted power alone is a weaker diagnostic of pure fundamental optical mode matching. The degeneracy does not remove the complex beam parameter matching requirement if the objective is specifically the symmetric fundamental Gaussian beam.