PSF reconstruction for Keck AO - Laboratory for Adaptive Optics

4 ANISOPLANATISM 254 AnisoplanatismThis section was originally “Outer scale effects on anisoplanatism in adaptive optics,” R. Flicker, August 21, 2008.AbstractThe structure functions for angular and focal anisoplanatism in an astronomical adaptive optics system are derivedfor the case of a finite outer scale in a von Karman type turbulence model. The results derived are applicable tothe problem of point spread function (PSF) reconstruction and anisoplanatism PSF modeling. The analytical vonKarman model is compared with numerical simulations and with an analytical Kolmogorov anisoplanatism model.It is found that for 10-meter class telescopes, an outer scale smaller than 30 meters can have a non-negligible impacton the anisoplanatism PSF estimation, and under these conditions the von Karman structure function model shouldbe preferred.IntroductionAstronomical adaptive optics (AO) compensates for turbulence-induced wave-front aberrations in the direction ofa reference light source, most commonly a natural guide star (NGS). Angular anisoplanatism in a conventionalNGS AO system manifests at angular positions on the sky θ =(θ x ,θ y ) that are different from the position of theNGS, which we may take to define the zero position θ = 0 in the following. Wave-fronts propagating at an angleθ ≠ 0 through the atmosphere will acquire aberrations from turbulence that are different from those measuredand corrected by the AO system, which gives rise to the field-dependent wave-front error in the telescope pupilplane that is called anisoplanatism. This paper is cumulative with previous investigations into the properties ofanisoplanatism by deriving two new results, relating to the effects of a finite turbulence outer scale. Most adaptiveoptics analysis in astronomy relies on variants of the Kolmogorov turbulence model, which has been experimentallyverified by numerous observations, including [27, 7], but there is still much uncertainty regarding the nature of thepower spectrum at lower spatial frequencies, and the validity of the von Karman outer scale model. To date therehas been no routine outer scale monitoring implemented at astronomical observatories, which makes the outer scalea poorly known quantity. For the purpose of this paper, however, a simplistic model will adopted wherein the outerscale is independent of altitude and zenith angle. Both of these constraints can be relaxed and a more detailed modelincluded in the analysis, should more information on the outer scale become available.The current investigation is primarily motivated by a problem in point spread function (PSF) reconstruction,where explicit knowledge of the structure function and a viable way to compute the optical transfer function (OTF)is required. Pertinent reasons for astronomers for wanting to know the PSF include the potential for more accuratephotometry and astrometry, and improved deconvolution reliability when the PSF is better known, see e.g. [11,10, 26, 16, 33, 4]. Anisoplanatism in adaptive optics has been analyzed assuming Kolmogorov turbulence statisticsin many previous instances, including comprehensive studies by, e.g., Fried [18], Fried & Belsher [19], Sasiela [31]and Tyler [36]. These studies are chiefly concerned with calculating phase variances and Strehl ratios directly, ingeneral not yielding intermediate results that are suitable for the problem of PSF reconstruction. In the Kolmogorovturbulence model with an infinite outer scale, the angular anisoplanatism structure function was derived by Britton[6] (see also Sect. 4.1). Of interest to the topic of PSF reconstruction is to know within which parameter spaceof outer scale and telescope size the Kolmogorov model remains a good approximation, and beyond which a vonKarman model that incorporates a finite outer scale ought to be applied instead.4.1 Angular anisoplanatismDenote the optical phase of the wave-fronts integrated along the line of sight of the AO reference beam “a” and theobserving direction “b” by φ a (r,t)andφ b (r,t), where r =(x, y) is a spatial coordinate in the telescope pupil planeand φ b is inclined by an angle θ to φ a . The instantaneous anisoplanatism phase error φ Δ isand the anisoplanatism structure function is given byφ Δ (r,t)=φ a (r,t) − φ b (r,t), (102)D Δ (r 1 , r 2 , θ) = 〈|φ Δ (r 1 ,t) − φ Δ (r 2 ,t)| 2 〉 (103)