The Azimuthal Spectrum of Common Transmittance Functions and Random Media

Structured laser beam propagation through inhomogeneous media is of interest for remote sensing and spatial division multiplexed free-space optical communications. The structured light of interest contains orbital angular momentum (OAM) that inherently forms an orthogonal basis set. The decomposition of an optical wavefront into OAM modes is analogous to an azimuthal Fourier transform that provides angular information. When a wavefront propagates through an imperfect optical system or an inhomogeneous medium, the inhomogeneities manifest themselves as phase distortions along the optical wavefront. This study aims to understand how phase-only distortions shape a beam's OAM spectrum. The azimuthal information of the distortions will be explored using the Fourier duality between the angular position and azimuthal frequency. To present this perspective, the example of an azimuthal aperture will be used to display the angular uncertainty principle. This concept will be further discussed using an example of a phase-only distortion represented by a common lens aberration using Zernike polynomials and then to the more complicated scenario of a random medium. It is found that the Fourier duality can be used to calculate the OAM spectrum of a random phase distortion. For the case of a finite beam incident on the distortion, it is found that the beam size and spatial structure play a role in spreading the beam's OAM spectrum. It is seen that a beam's OAM spectrum spread is independent of the mode if the beam size is taken into account.