= Confocal optical cavity
{title2=$L=R,\quad z_R=L/2$}
= Confocal optical resonator
{synonym}
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>.
Each mirror’s paraxial focal length is $R/2$, so their focal points coincide halfway between them. It is used for optical feedback and scanned spectral analysis; the manufacturer’s apparatus manual identifies the equal-radius, equal-separation geometry explicitly.<LM-0300 Fabry Perot Spectrum Analyser by Walter Luhs>[§1.3, printed p. 8, Figure 11]{ref}
The usual symmetric fundamental <Gaussian beam> has a central <beam waist>. Its wavefront curvature a distance $z$ from the waist is $R_b(z)=z+z_R^2/z$, where $z_R$ is the <Rayleigh range>. Requiring $R_b(L/2)=L$ gives
$$
\boxed{z_R=L/2,\qquad w_0^2=\frac{\lambda L}{2\pi},\qquad w_{\rm mirror}=\sqrt2\,w_0.}
$$
Here the cavity is empty, $\lambda$ is its vacuum wavelength, and $w$ is the standard Gaussian intensity radius. For an original example with $L=0.100\,\mathrm m$ and $\lambda=633\,\mathrm{nm}$, $z_R=50\,\mathrm{mm}$, $w_0\approx100.4\,\mathrm{\mu m}$ and the mirror spot radius is about $142.0\,\mathrm{\mu m}$. Finite mirror apertures and imperfect wavefronts change the loss and usable mode structure.
A key feature is <confocal transverse-mode degeneracy>. The one-way <Gouy phase> change of this symmetric beam is $\pi/2$. For transverse orders $m,n\ge0$ and longitudinal index $q$, the ideal paraxial resonance <frequencies>, with a common mirror-phase offset absorbed into the index convention, are
$$
\nu_{qmn}=\frac{c}{2L}\left[q+\frac{m+n+1}{2}\right].
$$
The primary MIT notes give this confocal resonance condition and the two interleaved transverse-order families.<Fundamentals of Photonics and Quantum Electronics lecture notes by Franz X. Kärtner>[§2.6.2, printed p. 116, equation 2.303 and Figure 2.82]{ref} One way to obtain it is to set the round-trip propagation phase $4\pi\nu L/c$ minus the round-trip transverse phase $(m+n+1)\pi$ equal to $2\pi q$.
Holding $m,n$ fixed gives the <longitudinal cavity mode> spacing $c/(2L)$. Increasing $m+n$ by two while reducing $q$ by one leaves the ideal <frequency> unchanged; orders of the other parity give a family halfway between. Thus an analyzer exciting both families can show interleaved resonances spaced by $c/(4L)$. This confocal spacing is not the longitudinal spacing of one fixed transverse mode. For $L=0.100\,\mathrm m$, the two values are approximately $1.499\,\mathrm{GHz}$ and $749.5\,\mathrm{MHz}$.
This distinction has a direct apparatus test. Send a narrow-band beam through mode-matching optics into two confocal mirrors, scan one mirror with a piezoelectric drive, and record transmitted light with a <photodetector> while monitoring the scan. Compare well-matched on-axis excitation with deliberately changed alignment or beam shape. Extra peaks can be cavity transverse modes rather than extra source <frequencies>. Luhs’s worked scan uses $100\,\mathrm{mm}$ mirror curvature, a confocal calibration near $750\,\mathrm{MHz}$, and an estimated finesse near fifty; its observed $15\,\mathrm{MHz}$ width bounds rather than independently resolves the narrower probe linewidth.<LM-0300 Fabry Perot Spectrum Analyser by Walter Luhs>[§3.1.2, printed p. 20, Figures 20–21]{ref}
A measured peak spacing therefore needs a stated mode-excitation pattern, and one transmission peak does not prove that only the fundamental spatial mode was excited. Departures from confocal geometry can split degeneracies; mirror loss, aberration and scan nonlinearity require separate characterization. The confocal geometry is a particular <symmetric concave-mirror resonator>, rather than a rule that applies to every pair of curved mirrors.
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