PSF reconstruction for Keck AO - Laboratory for Adaptive Optics

5 FOURIER DOMAIN PSD MODELING 37wherewedefined=∑N l N l∑l=1 k=1〈〉˜ϕ † l (f,t) ˜ϕ k(f,t) Γ † l (f)Γ k(f), (178)∞∑Γ l (f) =1− sinc(f · v l t i )exp(2πif · v l t d ) g (ξ − g) n−1 exp(2πif · v l nt i ). (179)Separate turbulence layers are assumed to be uncorrelated, i.e. 〈ϕ l ϕ k 〉 = 〈|ϕ l | 2 〉δ kl , so this simplifies to the tubulencePSD (155) and removes one summation. Defining a = ξ − g and b l =2πf · v l t i , the summation over n in expression(179) can be written∑ ∞S l = a −1 a n e inb l(180)n=1n=1∑∞= a −1 a n (cos nb l + i sin nb l ), (181)n=1This can be evaluated as two Fourier series that have closed analytical forms:n∑r k cos kx = (1 − r cos x)(1 − rn cos nx)+r n+1 sin x sin nx1 − 2r cos x + r 2 , (182)k=0n∑k=1r k sin kx = r sin x(1 − rn cos nx) − (1 − r cos x)r n sin nx1 − 2r cos x + r 2 (183)These sums will convege as n →∞for any |r| < 1. Evaluating the asymptotic forms, substituting back into (181)and combining terms gives eventuallye ib l− aS l =1 − 2a cos b l + a 2 . (184)We can now jump to the final form of the PSD directly:wherewith a and b l defined as above.5.3.2 NoiseΦ sl (f) = 0.023N l(f 2 + f0 2) × ∑r −5/30l|Γ l (f)| 2 , (185)l=1Γ l (f) =1− sinc(f · v l t i )exp(2πif · v l t d ) ×g(e ib l− a)1 − 2a cos b l + a 2 (186)For WFS noise the function G n = R(ν n ), and the closed-loop PSD is given by〈Φ noise (f) = |F {s 0 (x,t)}| 2〉 (187)〈∣ {}∣ ∣∣∣∣ ∞∑∣∣∣∣ 2〉= F g (1 − g) n−1 R[ν n (x,t)](188)= g 2 ˜R† ˜Rn=1∞∑m=1 n=1∞∑(1 − g) m+n−2 〈˜ν m † (f,t) ˜ν n(f,t) 〉 . (189)Assuming spatially and temporally uncorrelated noise we have that 〈˜ν † m˜ν n〉= δmn Φ ν , the power spectrum of theinput noise ν. Defining a =(1− g) 2 we have thatΦ noise (f) =g 2 ˜R† ˜R Φν (f)∞∑a (n−1) . (190)n=1