A method to determine the M2 beam quality from the electric field in a single plane

Laser beam quality is a key parameter for both industry and science. However, the most common measure, the M2 parameter, requires numerous intensity spatial-profiles for its determination. This is particularly inconvenient for modelling the impact of photonic devices on M2, such as metalenses and thin-film stacks, since models typically output a single electric field spatial-profile. Such a profile is also commonly determined in experiments from e.g., Shack-Hartmann sensors, shear plates, or off-axis holography. We introduce and test the validity and limitations of an explicit method to calculate M2 from a single electric field spatial-profile of the beam in any chosen transverse plane along the propagation direction.