PSF reconstruction for Keck AO - Laboratory for Adaptive Optics

2 OTF CALCULATION 11whereV k (ρ) ===∑N c∑N c∑N c〈η k η k 〉 S ik S jk U ij (ρ) (29)k=1 i=1 j=1∑N ck=1〈η k η k 〉V k (ρ), (30)∫ ( ∑Nc)(dxi=1 S ∑Nc)ik [h i (x) − h i (x + ρ)]j=1 S jk [h j (x) − h j (x + ρ)] P (x)P (x + ρ)∫ (31)dx P (x)P (x + ρ)=∫ dx |h′k(x) − h ′ k (x + ρ)|2 NP (x)P (x + ρ)c∫ , and : h ′dx P (x)P (x + ρ)k (x) = ∑S ik h i (x), (32)or in vector notation h ′ = S T h is the transformed modal basis within which the covariance matrix 〈ηη T 〉 is diagonal.This requires the computation of only N c functions V k (ρ) (orV ii in the notation of Gendron et al.). To practicallycompute the V k functions we rewrite the numerator in terms of convolutions and use the Fourier transform to evaluatethem:∫V k (ρ) × (P ⋆P) = dx |h ′ k (x) − h′ k (x + ρ)|2 P (x)P (x + ρ) (33)∫∫= dx h ′2k (x)P (x)P (x + ρ)+ dx h ′2k (x + ρ)P (x + ρ)P (x) (34)∫− 2 dx h ′ k(x)P (x)h ′ k(x + ρ)P (x + ρ) (35)= 2 [ (h ′2k P ) ⋆P − (h′ k P ) ⋆ (h′ k P )] (36){ [= 2 F −1 F(h ′2k P ) ˜P ∗ −|F(h ′ k P )|2]} (37)where widetilde ˜P is shorthand for the Fourier transform, and the denominator is simply P⋆P= F −1 (| ˜P | 2 ).2.3 Non-stationary calculationAlthough it might have seemed computationally implausible at one time, the diagonalization introduced in 2.2together with Moore’s Law may render the stationarity approximation unnecessary. As we shall see later on, thefocal anisoplanatism structure function will in any case force us to calculate a non-stationary terms, so we mightas well get some practice starting right here. Recalling the original structure function expression and applying thediagonalization of the covariance matrix, we obtainwhereD ɛ‖ (x, ρ) ==H ′ k(x, ρ) =∑N c N c∑〈ɛ i (t)ɛ j (t)〉 [h i (x) − h i (x + ρ)][h j (x) − h j (x + ρ)] (38)i=1 j=1∑N ck=1〈η k η k 〉H ′ k(x, ρ), (39)∑N cS ik [h i (x) − h i (x + ρ)]∣∣i=12= |h ′ k(x) − h ′ k(x + ρ)| 2 ∑N c, and : h ′ k(x) = S ik h i (x). (41)And the OTF is computed explicitly as∫{}∑N c〈B(ρ/λ)〉 = dx P (x)P (x + ρ)exp −0.5 λ k |h ′ k(x) − h ′ k(x + ρ)| 2k=1i=1i=1(40)(42)