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 to
$$
u_{mn}(x,y)=H_m(\sqrt2x/w_0)H_n(\sqrt2y/w_0)
\exp[-(x^2+y^2)/w_0^2],
$$
where $H_m,H_n$ 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.<Fundamentals of Photonics and Quantum Electronics lecture notes by Franz X. Kärtner>[§2.6.2, printed pp. 114–116, equations 2.297–2.303, Figures 2.80–2.82]{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 $x^2e^{-2(x^2+y^2)/w_0^2}$. It vanishes on $x=0$ and its two maxima occur at $x=\pm w_0/\sqrt2$, $y=0$. 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.

\Image[/transverse-cavity-mode-patterns.png]
{title=Intensity patterns of four ideal transverse cavity modes}
{description=Computed Hermite-Gaussian waist-plane intensities for orders (0,0), (1,0), (2,0) and (1,1), each normalized to its own maximum. Equal color scales compare shape, not equal circulating <power>; intensity alone does not display field sign.}

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> are
$$
\nu_{qmn}=\frac{c}{2L}\left[q+\frac{m+n+1}{\pi}\arccos\sqrt{g_1g_2}\right].
$$
The 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 $c/(6L)\approx499.65\,\mathrm{MHz}$ 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.


 Transverse optical cavity mode (source code)