PSF reconstruction for Keck AO - Laboratory for Adaptive Optics

3 COMPONENT MODELING 17Figure 4: Schematic of AO temporal control loop (top) and a blowup of the WFS model (bottom). E and G are spatialmatrix mappings, independent of time, while S, R, C and D are temporal filters, generally matrix-valued but assumed in thisanalysis to be scalar filtersc = [0, 1.16136, 2.97422, −13.2381, 20.4395] (59)p 1 = a 0 + a 1 β + a 2 β 2 + a 3 β 3 (60)p 2 = b 0 + b 1 β + b 2 β 2 + b 3 β 3 (61)p 4 = c 0 + c 1 β + c 2 β 2 + c 3 β 3 (62)p 3 = (pp2 4 β − 1+p−p1 4 )− log p 4(63)where the a, b and c parameters were obtained (not by me) from a fitting procedure. This influence function is shownin the middle panels of Fig. 3, where I also include a simple bilinear influence function (rightmost panels of Fig. 3).3.4 Covariance matrix 〈ɛɛ T 〉Exact temporal modeling of the residual mode covariance matrix 〈ɛɛ T 〉 turns out to be problematic, and most PSFreconstruction implementations so far have applied varying levels of approximation to this aspect of the systemmodeling. Referring to the generic AO block diagram in Fig. 4, we can examine the loop dynamics in terms ofthe phase modal coefficients a (input turbulence), c (DM mode) and ɛ (residual error). The other variables are: s 0is an ideal noise-free and aliasing-free WFS measurement; n is the noise and r is the aliasing, and s is the actualWFS measurement, assuming the components add linearly; the u and y are internal variables used in defining thecompensator. The s and c vectors are the only ones which we have access to from AO telemetry data, and which wemust employ in order to estimate 〈ɛɛ T 〉.A subtlety to be aware of is that the low- and high-spatial-frequency modal sets φ ‖ and φ ⊥ were defined by theDM modal basis h i (x), and strictly speaking this need not be equivalent to the subsets of φ that are aliasing-freeand produce aliasing in the WFS, respectively. Hence, the aliasing r can not be said to have come solely from φ ⊥ ,since the DM influence functions themselves may produce a (very small) amount of aliasing. In practice, however,in a Fried-geometry Shack-Hartmann AO system, the modal basis of the DM influence functions and the basis thatis aliasing-free are very close to each other, and to a first-order approximation we can say that aliasing only arisesfrom φ ⊥ (and hence ɛ ⊥ ).To remind ourselves of the context, we are trying to calculate 〈ɛɛ T 〉 because it arises in the residual wavefront