PSF reconstruction for Keck AO - Laboratory for Adaptive Optics

2 OTF CALCULATION 9where angular brackets denote ensemble average. Expanding the square and rearranging terms allows us to relatethe structure function to the correlation function C φ (ρ) =〈φ(x,t)φ(x + ρ,t)〉 assuming homogeneityD φ (ρ) =2[C φ (0) − C φ (ρ)] , (9)which will be eminently useful. If we assume (A3) that ɛ ‖ and φ ⊥ are statistically uncorrelated, then the structurefunction D ɛ of the residual phase error isD ɛ (x, ρ) =D φ⊥ (x, ρ)+D ɛ‖ (x, ρ), (10)where D φ⊥ is the structure function of everything that the AO system does not attenuate (i.e. the fitting error), andis estimated separately (see Sect. 3.2). With the definition (7) we have thatD ɛ‖ (x, ρ) ==〈∣ ∣∣∣∣ ∑N c2〉ɛ i (t)[h i (x) − h i (x + ρ)]∣i=1∑N c N ci=1 j=1One may define an aperture-averaged structure function ¯D(ρ) according to(11)∑〈ɛ i (t)ɛ j (t)〉 [h i (x) − h i (x + ρ)][h j (x) − h j (x + ρ)]. (12)¯D φ (ρ) =∫dx Dφ (x, ρ)P (x)P (x + ρ)∫dx P (x)P (x + ρ), (13)where P (x) defines the aperture transmission function. Since the autocorrelation in the denominator is band-limited,the fraction must only be computed within the support of ∫ dx P (x)P (x + ρ). The homogenized structure function¯D ɛ‖ can be written more simply as∑N c∑N c¯D ɛ‖ (ρ) = 〈ɛ i (t)ɛ j (t)〉 U ij (ρ), (14)where the U ij functions are given byi=1 j=1U ij (ρ) =∫dx [hi (x) − h i (x + ρ)] [h j (x) − h j (x + ρ)] P (x)P (x + ρ)∫dx P (x)P (x + ρ). (15)The static U ij functions can be computed for a given set of mirror modes h i (x), while the covariance matrix 〈ɛ i (t)ɛ j (t)〉is computed from the AO telemetry stream of DM actuator commands and WFS measurements during the particularobservation (see Sect. 3.4).Optical transfer functionLet K = K(α, β) be the anisoplanatic PSF as a function of image and object space angular coordinates (α, β),resulting from the anisotropic complex field ψ = ψ(x, y) in the image and object space spatial coordinates (x, y). Forincoherent quasi-monochromatic sources it can be shown [5] that the instantaneous PSF in the far-field approximationis given by K(α, β) =|F[ψ(x, y)]| 2 ,whereF denotes the Fourier transform. Since the OTF is B = F(K), it can beshown by the Wiener-Khinchin theorem that the long-exposure OTF is∫〈B(u, v)〉 = dw 〈ψ(w, v)ψ ∗ (w + u, v)〉, (16)where (u, v, w) are angular frequency coordinates of the OTF domain, and the superscript asterisk ( ∗ ) indicatescomplex conjugate. Assuming the field amplitude to be uniform (A1) (and hence also isoplanatic), the complex fieldmay be described asψ(x, y) =P (x)exp[iφ(x, y)], (17)