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Computationally efficient autoregressive method for generating phase screens with frozen flow and turbulence in optical simulations

Srikar Srinath, Lisa A. Poyneer, Alexander R. Rudy, and S. Mark Ammons

Open Access Open Access

Abstract

We present a sample-based, autoregressive (AR) method for the generation and time evolution of atmospheric phase screens that is computationally efficient and uses a single parameter per Fourier mode to vary the power contained in the frozen flow and stochastic components. We address limitations of Fourier-based methods such as screen periodicity and low spatial frequency power content. Comparisons of adaptive optics (AO) simulator performance when fed AR phase screens and translating phase screens reveal significantly elevated residual closed-loop temporal power for small increases in added stochastic content at each time step, thus displaying the importance of properly modeling atmospheric “boiling”. We present preliminary evidence that our model fits to AO telemetry are better reflections of real conditions than the pure frozen flow assumption.

© 2015 Optical Society of America

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Figures (6)
Fig. 1
Fig. 1 Flowcharts comparing the traditional frozen flow atmosphere generation process (left) and the autoregressive AR(1) process (right) for a 1-layer atmosphere (the process is repeated for each wind layer). The seeming simplicity of the frozen flow flowchart belies its memory requirements. In the autoregressive method, |α| ≤ 1 and this determines by how much prior atmospheric phase is attenuated. The phase of α carries information about wind velocity and direction in the layer as seen in Eq. 2.

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Fig. 2
Fig. 2 Spatial PSDs for datacubes comprising time series of autoregressive phase screens with |α| = 0.99 (left) and |α| = 0.999 (right). Overplotted on both is the theoretical Kolmogorov power spectrum derived from [12]. Power spectral density of each datacube follows the expected Kolmogorov slope (κ−11/3). The |α| = 0.99 datacube exhibits lower variance about the theoretical slope because of the higher magnitude scaled noise injected into successive frames.

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Fig. 3
Fig. 3 Temporal power spectral density comparison for two sets of autoregressive phase screen time series with |α| = 0.99 and |α| = 0.999 plotted over the full temporal frequency range (left) and zoomed in to lower frequencies (right) to show the lower power peak for the |α| = 0.99 case. Integrated power is the same for both cases.

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Fig. 4
Fig. 4 Radially averaged profile of science PSF image produced by the GPI AO simulator running in closed-loop for three realizations of the atmosphere: pure frozen flow (blue), autoregressive with |α| = 0.999 (black) and |α| = 0.99 (red). Intensity on the y-axis is normalized such that the peak of the ideal, unblocked PSF is 1. Inside the innermost 10-pixel radius, PSF intensity is suppressed by coronagraphy. Outside the corrected area at a radius of ∼ 100 pixels, the power in all PSFs converges. At lower orders, the AR atmospheres have greater residual power: > 2× for |α| = 0.999 and ~20× for |α| = 0.99). The scattered light at a spatial location in the PSF corresponds directly to a specific Fourier mode in the spatial PSD.

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Fig. 5
Fig. 5 Temporal power spectral comparison for open loop atmospheres (left) versus closed loop residuals (right) for AR atmospheres (|α| = 0.99 and |α| = 0.999, over a 22-second interval) and pure frozen flow (restricted to 4-second exposures by memory and computation limits). The AR atmospheres have much broader peaks than the frozen flow atmosphere realization in lower order modes beyond 5 Hz in the left plot, hence rejection is worse and residuals exhibit more power in the right hand figure. This manifests itself as a brighter PSF core outside the central obscuration in a science image. Dashed lines represent cumulative power from 0 Hz to ±512 Hz.

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Fig. 6
Fig. 6 Temporal PSD of Fourier mode k = 14,l = 36 for open-loop phase reconstructed from closed-loop residuals as measured by the Gemini Planet Imager (solid black line) with the overlaid model fit (solid red line) for peaks at DC (|α| = 0.995) and 10.2 Hz (|α| = 0.993).

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Equations (6)

Equations on this page are rendered with MathJax. Learn more.

P ∝ 2 π S N p r 0 − 5 / 6 ( f x 2 + f y 2 ) − 11 / 12
θ = − 2 π T ( f x v x + f y v y )
ϕ ˜ t = α ϕ ˜ t − 1 + 1 − | α 2 | P ω ˜ t
M = 2 b ( n + 1 ) N
E T F = | 1 1 + z − 2 C ( z ) | 2
P C L = E T F ( P ϕ + P N )
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