Simulation of Time-dependent Karhunen-Loeve Phase Screens: an Ergodic Approach

TL;DR

This work develops an ergodic, time-dependent framework for simulating atmospheric phase screens by embedding conventional 2D Karhunen-Loève screens into a 3D isotropic volume and sampling them along a time axis. By formulating the KL evaluation in a 3D spherical domain with a von-Kármán Kolmogorov spectrum, it derives a Fourier-space eigenproblem based on a 3D Zernike basis, enabling consistent generation of phase-video realizations that respect the underlying turbulence statistics. The approach preserves the Kolmogorov exponent and outer-scale physics, provides practical guidance on outer-scale cutoffs via $\xi_L$, and offers ancillary tools (XML mode files, Maple/C++ utilities) to generate, tabulate, and read the KL modes for various telescope geometries. The method yields self-consistent, non-frozen, time-evolving phase screens suitable for ground-based astronomical simulations and instrument modeling.

Abstract

Time-dependent phase screens in ground-based astronomy are typically simulated in the so-called frozen-screen approximation by establishing a static phase screen on a large pupil and dragging an aperture equivalent to the size of the actual input pupil across this oversized phase screen. The speed of this motion sweeping through the large phase screen is equivalent to a wind speed that changes the phase screen as a function of time. The ergodic ansatz replaces this concept by constructing the structure function in a three-dimensional volume -- a sphere for reasons of computational efficiency -- , sampling phase screens by two-dimensional planar cuts through that volume, and dragging them along the surface normal at some speed which generates a video of a phase screen. This manuscript addresses the linear algebra of populating the three-dimensional volume with phase screens of the von-Karman model of atmospheric turbulence.

Figures (7)

• Figure 1: A 3-dimensional spherical domain outlined by three magenta circles and examples of two cylindrical subdomains in green and in blue with different sampling orientations. The circular cross sections of the cylinders (4 green plus 3 blue illustrated with time marks) are orthogonal to their long axes.
• Figure 2: The dominating (largest) eigenvalues $\lambda^2$ as a function of cutoff wavelength $\xi_L$.
• Figure 3: The dominating modes for even $n$ and $n'$ (therefore even $l$) in the Kolmogorov limit.
• Figure 4: The dominating modes for odd $n$ and $n'$ (therefore odd $l$) in the Kolmogorov limit.
• Figure 5: The dominating modes for odd $n$ and $n'$ comparing the Kolmogorov limit $\xi_L=0$ with an (inverse) outer scale $\xi_L=0.25$

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Remark 1
• Remark 2
• Definition 3
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Definition 4