Metric for the measurement of the quality of complex beams: a theoretical study.

We present a theoretical study of various definitions of laser beam width in a given cross section. Quality of the beam is characterized by dimensionless beam propagation products (BPPs) Δx·Δθ(x)/λ, which are different for the 21 definitions presented, but are close to 1. Six particular beams are studied in detail. In the process, we had to review the properties for the Fourier transform of various modifications and the relationships between them: physical Fourier transform (PFT), mathematical Fourier transform (MFT), and discrete Fourier transform (DFT). We found an axially symmetric self-MFT function, which may be useful for descriptions of diffraction-quality beams. In the appendices, we illustrate the thesis "the Fourier transform lives on the singularities of the original."