Laguerre–Gaussian modes

Summary

The Laguerre–Gaussian modes are a family of beam solutions of the paraxial wave equation in cylindrical coordinates. Each one is a Gaussian beam multiplied by a generalised Laguerre polynomial in the radius and by a phase factor $e^{i\ell\varphi}$ in the angle. Like the Hermite–Gaussian modes they keep their shape while they propagate and form a complete orthogonal family. The modes with $\ell \ne 0$ have a dark centre and a helical wavefront, and they carry orbital angular momentum.