This work presents an exact, analytical derivation of the ensemble-averaged orbital angular momentum (OAM) power spectrum for a circularly polarized Gaussian beam traversing a statistical Q-plate with Gaussian spatial disorder. Utilizing the Gaussian moment theorem, a new closed-form expression for the averaged mutual coherence is obtained. This coherence function is then rigorously projected onto OAM modes, yielding an exactly normalized series representation whose absolute convergence is formally proven. The analysis meticulously resolves limiting disorder regimes: for coarse disorder, an ideal OAM spectrum, sharply peaked at its nominal OAM mode, is demonstrated. Conversely, for fine disorder, a specific fraction of total power exponentially decays with disorder variance into the nominal OAM mode, with remaining power distributed among nearby OAM modes, exhibiting an effective width inversely proportional to the dimensionless correlation length. Crucially, a new universal scaling framework, defined by a master control parameter and a universal coordinate, is introduced. This framework, rigorously derived from the exact solution, unifies the spectrum's description across all relevant disorder regimes and enables robust data collapse. Monte Carlo simulations, implemented with the same normalization and OAM projection, substantiate these claims by corroborating the predicted scaling and universal behavior, offering a foundational, device-level perspective on OAM universality in complex media.
The recent work of Dafydd and Porter [2024] on the attenuation of waves propagating through floating broken ice of random thickness is extended to consider water of non-shallow depth. A theoretical model of broken floating ice is analysed using a multiple scales analysis to provide an explicit expression for the attenuation of waves as they propagate from a region of constant thickness ice into a semi-infinite region of ice whose thickness is a slowly-varying random function of distance. Theoretical predictions are shown to compare well to numerical simulations of scattering over long finite regions of ice of randomly-varying thickness computed from an approximate depth-averaged model derived under a mild-slope assumption. The theory predicts a low-frequency attenuation proportional to the eighth power of frequency and a roll-over effect at higher frequencies. The relationship between the results and field measurements are discussed.
Two-wavelength adaptive optics (AO) systems sense turbulence-induced wavefront distortions using an artificial beacon or natural guidestar at one wavelength, while correcting and possibly transmitting at another. Although most existing AO systems employ this methodology, the literature on atmospheric turbulence correction and AO system design generally focuses on performance at a single wavelength, neglecting the two-wavelength nature of the problem. In this paper, we undertake a rigorous study of the relevant wavefront errors necessary to quantify two-wavelength AO system performance. Since most AO systems employ separate tilt and higher-order correcting subsystems, our analysis mirrors this division, and we begin with higher-order wavefront errors. Utilizing Mellin transform techniques, we derive closed-form relations for the piston-removed and piston- and tilt-removed variances. The former is a measure of the total, residual wavefront error that a two-wavelength AO systems experiences; while the latter, quantifies the residual wavefront error due to higher-order aberrations. We then proceed to tilt or tracking errors and derive the two-wavelength Zernike- and gradient-tilt variances. Zernike tilt is the actual amount of tilt in the turbulent atmosphere; yet, most AO tracking subsystems measure gradient tilt. Consequently, we also derive the two-wavelength gradient-tilt, Zernike-tilt variance – also known as centroid anisoplanatism – to quantify this error. Lastly, we validate our analysis by performing two-wavelength wave-optics simulations and comparing the results to theory. We observe excellent agreement among the simulated results and our theoretical predictions. The analysis and findings presented in this paper will be useful in the characterization of existing, and the design of new, two-wavelength AO systems.
