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Here we define a theoretical basis for the generalization of the beam quality factor M 2  to three-dimensional (3D) space, which we call M 6  formalism. The formalism is established through the use of examples of multifocal and Axicon optical systems to illustrate discrete and continuous axial beam shaping, respectively. For the continuous case, we expand the definition of the Rayleigh range to incorporate a quality factor having both axial and transverse componentsM   add   2$M_{{\rm{add}}}^2$    and M 2 . Using geometrical ray tracing simulations, a proportion factor C is found to empirically describe the axial quality factorM  z  2$M_z^2$    of an optical setup including an Axicon and a paraxial focusing lens with a Gaussian single mode input beam. Using our M 6  formalism depth of focus (DOF) ranges are calculated for higher M 2  beams, and are shown to be in good agreement with the simulated DOF range, demonstrating the usefulness of the M 6  formalism for the design of real optical systems.

M 6 formalism – generalization of the laser beam quality factor M 2 to the 3D domain