Shaping Gaussian modes through truncation and apodization: theory and interpretation

The transformation of Gaussian beams into structured intensity profiles, such as flat-top and ring-shaped distributions, is a longstanding goal in beam shaping. Conventional methods using spatial light modulators, digital micromirror devices, or interferometry are effective but often bulky and expensive. This paper explores a simpler, low-cost alternative by shaping Gaussian beams through soft and hard truncation. We examine this approach in both Cartesian and cylindrical coordinate systems, covering beam types such as the cosine beam, cosine-Gaussian beam, elegant Hermite-Gaussian beam, truncated cosine beam, and truncated Hermite-Gaussian Beam, along with their cylindrical counterparts: the Bessel beam, Bessel-Gaussian beam, elegant Laguerre-Gaussian beam, truncated Bessel beam, and truncated Laguerre-Gaussian beam. Using mathematical asymptotics and Fourier optics, we provide theoretical insight into how truncation and spatial modulation shape the far-field beam profiles. This framework not only explains the formation of flat-top and ring-shaped beams but also supports the development of compact, passive beam-shaping systems.