Confocal optical cavity ($L=R,z_R=L/2$)

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.

For two equal mirrors of radius $L$ separated by $L$, a Gaussian waist at the centre is reflected to another waist at the same plane. Putting $a=L/2$, the mirror maps $q=a+i{z}_R$ to $-a+i{a}^2/z_R$, so the reflected Rayleigh range is $a^2/z_R$. Both beams have equal spots on the mirrors although their waist radii differ. Equal waist sizes select $z_R=L/2$.

Confocal transverse-mode degeneracy occurs when distinct spatial modes of an ideal confocal optical cavity have the same resonance frequency. It follows from the rational Gouy phase advance of the equal-radius, equal-separation geometry, rather than from the modes having identical transverse field patterns.

For mirror separation $L=R$, the paraxial resonance frequencies areThe primary MIT notes derive the confocal phase shift and mode-frequency families.^[ref] Let total transverse order be $N=m+n$. Replacing $(q,N)$ by $(q-j,N+2j)$ leaves the frequency unchanged whenever the resulting indices describe permitted modes. Orders that are even integers therefore share the fundamental-order family, while orders that are odd integers form an interleaved family halfway between.

There is also degeneracy among patterns with the same total order. At $N=2$, the pairs $(m,n)=(2,0),(1,1),(0,2)$ have the same ideal frequency while their transverse distributions differ. A transverse optical cavity mode needs its spatial field as well as its resonance frequency to be identified.

An original frequency example uses $L=0.100\mathrm{m}$ and $c=299792458m/s$. Holding one transverse order fixed gives spacing $c/(2L)=1.49896229\mathrm{GHz}$. Relative to a chosen fundamental resonance, the mode with $N=2$ and longitudinal index one smaller has offset zero; the $N=1$ mode at the original longitudinal index has offset $749.481145\mathrm{MHz}$. Thus a spectrum containing both families can show half the fixed-order longitudinal spacing. It does not follow that every visible peak is a separate laser frequency.

The excitation strength depends on spatial coupling. In an ideal orthonormal cavity-mode basis, the incident field can be expanded with coefficients $a_j=\int u_j^{\ast }(x,y)E_{\mathrm{in}}(x,y)dxdy$, with consistent field normalization. Matching the fundamental field suppresses overlap with orthogonal modes, even if those modes share its frequency. Conversely, transmission at the common resonance can contain several patterns when they are excited; frequency selection alone cannot separate a degenerate subspace.

A practical test scans the cavity length while recording both transmitted power and spatial beam shape, then changes alignment or mode matching. Luhs’s primary apparatus discussion identifies unwanted extra peaks as transverse modes of the analyzer and distinguishes them from the probe’s own longitudinal spectrum.^[ref] A photodetector integrating the whole output can conceal spatial differences that a beam image reveals.

Mirror-spacing error, unequal curvatures, aberration and finite apertures perturb the ideal degeneracy. For a general one-way Gouy phase $\psi$, the corresponding phase model gives ${\nu }_{qN}=(c/(2L))[q+(N+1)\psi /\pi ]$. The nominally paired orders $N$ and $N+2$ then differ by $(c/(2L))(2\psi /\pi -1)$ when their longitudinal indices differ by one. This makes near-degenerate peaks a sensitive geometry probe, but attributing a measured splitting requires separating those perturbations from source structure and polarization effects.