= Hermite-Gaussian optical mode
{c}
{title2=$u_{mn}$}
A <Hermite-Gaussian optical mode> is a separable paraxial field pattern whose transverse dependence is a product of <Hermite polynomials> and a <Gaussian beam> envelope. In an ideal aligned optical resonator with suitable spherical mirrors, these patterns form a family of <transverse optical cavity modes>. The two nonnegative integer orders $m,n$ count the straight nodal lines along the chosen transverse axes. The primary photonics notes give the mode fields and their order-dependent phase advance.<Fundamentals of Photonics and Quantum Electronics lecture notes by Franz X. Kärtner>[§2.6.2, equations 2.297–2.303, printed pp. 114–116]{ref}
At a common waist plane, choose a unit-normalized scalar field
$$
\boxed{u_{mn}(x,y)=\frac{\sqrt{2/\pi}}{w_0\sqrt{2^{m+n}m!n!}}
H_m(\sqrt2x/w_0)H_n(\sqrt2y/w_0)e^{-(x^2+y^2)/w_0^2}.}
$$
Here $H_j$ uses the physicists' <Hermite polynomial> convention, with $H_0(s)=1$, $H_1(s)=2s$ and $H_2(s)=4s^2-2$. The normalization follows from
$$
\int_{-\infty}^{\infty}H_m(s)H_p(s)e^{-s^2}\,ds
=\sqrt\pi\,2^m m!\,\delta_{mp}.
$$
After the two coordinate substitutions, $\iint u_{mn}^*u_{pq}\,dx\,dy=\delta_{mp}\delta_{nq}$. Thus coefficients in a common-plane expansion can be recovered by <inner products>; the modes are orthogonal field patterns, not independent bright spots.
For $(m,n)=(1,0)$, intensity is proportional to $x^2e^{-2(x^2+y^2)/w_0^2}$, so its maxima lie at $x=\pm w_0/\sqrt2$, $y=0$. At $w_0=100\,\mathrm{\mu m}$, the bright lobes are about $141.4\,\mathrm{\mu m}$ apart. The field changes sign across $x=0$, although an intensity camera records positive intensity on both sides. For $(2,0)$, the nodal lines instead lie at $x=\pm w_0/2$. These patterns are illustrated in <transverse optical cavity mode> using individually peak-normalized images; equal peak intensity is different from equal integrated power.
A <second-moment width of a Hermite-Gaussian optical mode> distinguishes these multi-lobed beams quantitatively. Applying the <Hermite polynomial> recurrence inside the normalized integrals gives
$$
\langle x^2\rangle=\frac{w_0^2}{4}(2m+1),\qquad
\langle y^2\rangle=\frac{w_0^2}{4}(2n+1).
$$
For order $(1,0)$, the RMS widths are $\sqrt3w_0/2$ horizontally and $w_0/2$ vertically, despite the same envelope parameter $w_0$. Reporting a fitted single-lobe radius without its definition can obscure this difference.
During propagation the beam radius and wavefront curvature evolve, and the <Gouy phase> is multiplied by $m+n+1$. Consequently, transverse order contributes to the resonator's frequency condition and to <confocal transverse-mode degeneracy>. A proposed experiment scans the resonator while imaging a specified plane, then interferes its output with a reference field to test the phase reversal across a node. An intensity fit alone does not establish a pure coherent mode: mixtures, astigmatism, finite apertures and degeneracy can alter what is observed.
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