 Transverse optical cavity mode

A transverse optical cavity mode is a resonant field pattern across the direction of propagation in an optical cavity. After propagation and reflection through a round trip, the mode reproduces its transverse shape up to a complex amplitude factor. Longitudinal order counts axial phase cycles; transverse order describes the pattern across the beam. Different transverse patterns can resonate at the same frequency, as in confocal transverse-mode degeneracy.

In an ideal paraxial cavity with suitable spherical mirrors, a Cartesian Hermite-Gaussian optical mode at its waist has field proportional towhere $H_m,H_n$ are Hermite polynomials, $m,n$ are nonnegative integers, and $w_0$ is the beam waist radius. The intensity is proportional to $|u_{mn}{|}^2$, not to the signed field. The primary photonics notes give these transverse solutions, their phase advance and lower-order patterns.^[ref]

For $(m,n)=(0,0)$, the intensity is a single Gaussian beam spot. For $(1,0)$, $H_1(s)=2s$, so intensity is proportional to $x^2{e}^{-2(x^2+y^2)/w_0^2}$. It vanishes on $x=0$ and its two maxima occur at $x=\pm w_0/\sqrt{2}$, $y=0$. The two lobes have opposite field signs even though both have positive intensity. The $(1,1)$ pattern has two crossing nodal lines and four intensity lobes.

For an ideal empty two-mirror cavity of length $L$, set $g_i=1-L/R_i$. On the near-planar stable branch with $g_1,g_2>0$, the resonance frequencies areThe transverse contribution is a Gouy phase correction to the round-trip phase. Other branches require a consistent phase choice; the square root alone should not erase that choice. For an original example, $L=0.10\mathrm{m}$ and $R_1=R_2=0.20\mathrm{m}$ give $g_1=g_2=1/2$ and a one-order transverse spacing $c/(6L)\approx 499.65\mathrm{MHz}$ at fixed $q$. Modes with the same $m+n$ are degenerate in this ideal symmetric model.

A proposed observation scans cavity length while recording both transmission peaks and a camera image of the transmitted beam. Translating or tilting the incident beam changes its overlap with different transverse modes; matching the incident wavefront and width favors the fundamental mode. A transmission peak alone cannot identify the spatial pattern, and an intensity image alone does not recover its phase. Finite apertures, astigmatism and mirror imperfections alter losses and split ideal degeneracies; state the cavity geometry and imaging plane when comparing observations.