PSF reconstruction for Keck AO - Laboratory for Adaptive Optics

4 ANISOPLANATISM 31Figure 10: Left: Normalized encircled energy of PSF in the analytical von Karman model of LGS anisoplanatism, comparingto the numerical von Karman simulation. Right: Simulation showing the effect of spatial filtering on LGS anisoplanatism.4.5 Spatial filteringIn a real AO system, anisoplanatism manifests only on the subspace of controlled modes. Applying the full powerspectrum of turbulence will overestimate the anisoplanatism error by counting the fitting error to the anisoplanatismerror budget. If the fitting error is small, this approximation might be acceptable. In order to project theanisoplanatism wave-front error onto controlled modes we rewrite Eq. (102) asφ Δ (r,t)=φ ‖a (r,t) − φ ‖b (r,t), (129)where the notation symbolizes that we have retained only the low-frequency (controlled) part φ ‖ of the decompositionφ = φ ‖ + φ ⊥ . This splitting of the phase into two orthogonal components in spatial frequency domain is commonin AO PSF modeling and PSF reconstruction methodology, see e.g. [29, 41, 25, 20]. In practice the domain of φ ‖ isdefined as the vector space spanned by the set of N a influence functions {h i (r)} Nai=1 of the DM. Defining w i as thepiston-removed influence function, we have the relationsφ ‖a (r,t) =a i (t) =∑N aa i (t)w i (r), (130)i=i∫dr w i (r)φ a (r,t), (131)w i (r) = P (r)[h i (r) − p i ], (132)∫dr P (r) hi (r)p i = ∫ , (133)dr P (r)where p i is the piston of each mode and {a i } Nai=1 is a set of expansion coefficients (i.e. actuator commands). Analogously,the off-axis beam φ b is projected onto the set {b i } Nai=1 . The covariance function now has the general form∑N a∑N a〈φ ‖a (r 1 )φ ‖b (r 2 )〉 = 〈a i b j 〉w i (r 1 )w j (r 2 ). (134)i=1 j=1The covariance matrix 〈a i b j 〉 is calculated from Eq. (110). When Eq. (130) is substituted into the expression forthe structure function, ten terms of the form above result, containing the two covariance matrices A ij = 〈a i a j 〉 and