Propagation of high-order azimuthal Fourier terms of the amplitude distribution of a light beam: a useful feature

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Abstract
A high-order azimuthal Fourier component (with index l) of a light beam exhibits a feature that is sometimes useful in calculating the laser-beam-quality degradation that is due to index-of-refraction variation of the gain medium. If the Fourier component has wave fronts that are flat or spherical in the radial dimension and the radial intensity distribution is fairly smooth, then, after propagation, the intensity distribution will have a hole with nearly zero intensity in its center, and the (equivalent collimated) Fresnel number of the hole will be equal to l/π.