Power content M 2-values smaller than one

The M ² beam propagation factor or times-diffraction-factor is widely used to characterize the quality of laser radiation and its propagation. When M ² is defined by the normalized product of the second moments, it is easily to prove that, for each radiation field in the paraxial approach, M ²≥1, with the equality in the case of the fundamental mode. For many applications, it is more convenient to use the power content values, also proposed by ISO. They are defined as the radii of the circles which contain a certain amount of the total power, normally η=86.5%. For the corresponding power content M _pc ² , it is often assumed that its minimum is again obtained for the fundamental mode, but no proof exists. In this paper it is shown that fields can be generated with M _pc ² <1 and that it strongly depends on the power content η. One example is the superposition of two coherent Gauss–Laguerre modes with radial symmetry. The beam radius as a function of the propagation distance is calculated, and for the 86.5% power content, the value M _pc ² =0.95 is obtained. This does not mean that such a beam is of higher quality than the fundamental mode but rather that the M _pc ² is not a reliable parameter for beam characterization.