 Confocal optical cavity

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.^[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 $R_b(L/2)=L$ givesHere 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 $\lambda =633\mathrm{nm}$, $z_R=50\mathrm{mm}$, $w_0\approx 100.4\mu \mathrm{m}$ and the mirror spot radius is about $142.0\mu \mathrm{m}$. 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 $m,n\ge 0$ and longitudinal index $q$, the ideal paraxial resonance frequencies, with a common mirror-phase offset absorbed into the index convention, areThe primary MIT notes give this confocal resonance condition and the two interleaved transverse-order families.^[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.^[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.