A unified operator approach is described for deriving Hermite-Gaussian and Laguerre-Gaussian laser beams by using as a starting point a plane-wave-spectrum representation of the electromagnetic field. We show that by using the plane-wave representation of the fundamental Gaussian mode as a seed function, all higher-order beam modes can be derived by acting with differential operators on this fundamental solution. The approach presented can be easily generalized to nonparaxial situations and to include vector effects of the electromagnetic field.