The Hermite-Gaussian beams are a complete set of solutions to the paraxial Helmholtz equation. Any other solution of beams can be written as a superposition of these beams. This complete set of solutions known as Laguerre -Gaussian beams. Laguerre gaussian beams are described by writing the paraxial Helmholtz equation in cylindrical coordinate¬ (?, ?, z).
Then using the separation-of-variables in ? and ?, rather than in x and y. The complex amplitude of the Laguerre-Gaussian beam is. In equation of LG beam the phase has the same dependence, as the Gaussian beam on ? and z, but the phase has an extra term which is proportional to the azimuthal angle ?, and on a Gouy phase that is greater by the factor (l + 2m + 1). Because of this linear dependence of the phase on ? (for l ? 0) when the wave travels in the z direction, then wave front tilts helically as show in