Confocal transverse-mode degeneracy

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.