Laguerre–Gaussian modes

Remarks

Each $u_{p\ell}$ solves the paraxial wave equation for the envelope $u_{p\ell} e^{-ikz}$. The mode $\mathrm{LG}_{00}$ is the Gaussian beam. The number $p$ is the number of zeros of the radial factor for $r > 0$, so the intensity has $p + 1$ bright rings. The index $\ell$ is the azimuthal index: the phase winds $\ell$ times around the axis. For each fixed $z$ the transverse profiles are orthogonal in $L^2(\R^2)$, and after normalisation they form an orthonormal basis. The modes of the same order $2p + \abs{\ell} = N$ are linear combinations of the Hermite–Gaussian modes with $m + n = N$, and conversely.

Example: $\mathrm{LG}_{01}$ at the waist is $(\sqrt2 \, r / w_0) e^{-r^2/w_0^2} e^{i\varphi}$, since $L_0^{(1)} = 1$. Its intensity $2 r^2 w_0^{-2} e^{-2r^2/w_0^2}$ is a ring of radius $w_0/\sqrt2$ with a zero at the centre.

Non-example: the profile $e^{-r^2/w_0^2} e^{i\varphi}$ is not a Laguerre–Gaussian mode. It is not continuous at $r = 0$, because its modulus tends to $1$ while its phase has no limit there.