Far-field diffraction of truncated higher-order Laguerre-Gaussian beams

A simple analytic expression for the far-field diffraction pattern of a general high-order Laguerre-Gaussian mode from a circular aperture is derived. Aperturing of the fundamental Gaussian beam is shown as a special case, where in the limit oi waist size much larger than the aperture the Airy pattern is recovered and in the opposite limit we get a Gaussian intensity pattern. The diffraction pattern is shown to be an infinite series of Bessel functions and is strongly dependent on the ratio of the aperture area to the beam waist area (also known as the truncation parameter). This formulation leads to a more intuitive understanding of the far field of these modes, and the effect of the aperture as a spatial filter is discussed.