6.3 Long exposure PSF estimation from AO telemetry

This section explains in some detail the shape of the typical AO corrected long exposure PSF already shown in Section 2. In particular, it will become clear why the PSF has two components – a diffraction limited core and a seeing limited halo. In order to derive an estimate for the solar AO corrected PSF (or equivalently the OTF) the AO structure function needs to be determined (Marino, 2007Jump To The Next Citation Point; Marino and Rimmele, 2010Jump To The Next Citation Point). The phase structure function produced by isotropic and uniform Kolmogorov turbulence depends on the separation parameter ρ only. This is no longer true once AO correction has been applied and the phase structure function now depends on separation βƒ—ρ and position βƒ—x in the pupil plane: D φ (βƒ—x,βƒ—ρ ) πœ–. It should be noted that the fact that the residual phase structure function no longer can be described with Kolmogorov statistics complicates post-facto image reconstruction of AO corrected images. Speckle interferometry algorithms have to be adapted to account for the field dependent AO correction (Wöger et al., 2008Jump To The Next Citation Point). Phase-diversity and MOMFBD, in principle, do not rely on atmospheric model information. However, as pointed out by Scharmer et al. (2010Jump To The Next Citation Point) in order to correct for stray-light from high order atmospheric aberrations that remain uncompensated by these methods, statistical estimation of the average effect of high order modes is required, which again relies on Kolmogorov statistics.

The AO system corrects a limited number of aberration modes. In the case of AO76 the system is typically able to correct between 65 – 80 Karhunen–Loève (KL) or Zernike modes. Higher order modes remain uncorrected. Hence, the AO system acts as an imperfect high pass filter. The corrected modes are not corrected perfectly, as was discussed in Section 6, and a residual variance remains for each corrected mode. The magnitude of the residual variance is mode dependent.

Closely following Veran et al. (1997Jump To The Next Citation Point), Marino (2007Jump To The Next Citation Point), and Marino and Rimmele (2010Jump To The Next Citation Point), the residual phase after correction can then be expressed as:

[ ] φ πœ–(βƒ—x,t) = β—Ÿφatmβˆ₯(βƒ—x,t)β—β—œ−-φm-(βƒ—x,t)β—ž+ φatm⊥(βƒ—x, t) φπœ–βˆ₯(βƒ—x,t) = φ πœ–βˆ₯(βƒ—x,t) + φatm⊥(βƒ—x,t), (27 )
where φ πœ–βˆ₯ is the residual phase of the corrected modes and φ atm ⊥ for the uncorrected, respectively. The phase structure function can then be written as:
¯D φπœ–(ρβƒ—) ≈ ¯Dφπœ–βˆ₯(βƒ—ρ) + ¯Dφatm⊥(βƒ—ρ), (28 )
where cross terms have been assumed to be negligible (Veran et al., 1997Jump To The Next Citation Point). D φatm⊥ is a function of βƒ—ρ only since the uncorrected atmospheric phase errors in a statistical sense (long exposure) are homogeneous and isotropic. Hence, D φatm ⊥ can be placed in front of the integral in Equation 11View Equation resulting in a long exposure OTF that can be written as the product of three independent terms:
OTFao (βƒ—ρβˆ•λ ) = OTF Ο•πœ–βˆ₯(βƒ—ρβˆ•λ) OTF Ο•πœ–⊥(βƒ—ρ βˆ•λ) OTFtel (βƒ—ρβˆ•λ ), (29 )
[ ] 1 OTF Ο•πœ–βˆ₯(βƒ—ρβˆ•λ) = exp − 2-¯Dφπœ–βˆ₯(βƒ—ρ) (30 ) (31 )
[ ] OTF (βƒ—ρβˆ•λ) = exp − 1D (βƒ—ρ) . (32 ) Ο•πœ–⊥ 2 φatm⊥

OTF Ο•πœ–βˆ₯ is the contribution from the residual variance of the corrected modes and can be derived from the residual wavefront errors measured by the WFS with the AO in closed loop. OTF Ο•πœ–⊥ derives from the high order modes that the AO system cannot correct. OTF tel is simply the diffraction limited telescope OTF. OTF Ο•πœ–⊥ is the main contributor to the seeing halo of the PSF, while the core of the PSF is formed by the other two terms.

φπœ–βˆ₯ can be extracted from the WFS measurements after accounting for various noise terms, such as the WFS noise described above and aliasing noise (Herrmann, 1981; Dai, 1996). A detailed description of this process and the calibration procedures involved is given by Veran et al. (1997) and Marino (2007Jump To The Next Citation Point).

The residual wavefront variance after correction φ πœ–βˆ₯ can be expressed as:

∑N φπœ–βˆ₯(βƒ—x,t) = [ai(t) − ki(t)]Ki(βƒ—x) i=1 N ∑ = πœ–i(t)Ki(βƒ—x), (33 ) i=1
where ai are the coefficients KL modes Ki of the uncorrected phase, ki are the KL coefficients of the applied correction, and the πœ–i are the noise free residual KL coefficients. The phase structure function of the residual corrected phase becomes:
| | D φ (βƒ—x,βƒ—ρ) = ⟨|φπœ–βˆ₯(βƒ—x,t) − φ πœ–βˆ₯(βƒ—x + βƒ—ρ,t)|2⟩ (34 ) πœ–βˆ₯
and using Equation (33View Equation):
∑N ∑N D (βƒ—x,βƒ—ρ) = βŸ¨πœ– πœ–βŸ© [K (βƒ—x) − K (βƒ—x + βƒ—ρ)][K (βƒ—x ) − K (βƒ—x + βƒ—ρ)]. (35 ) φπœ–βˆ₯ ij i i j j i=1 j=1

Due to the isotropy assumption mentioned above computing the mean phase structure function of the residuals averaged over the pupil is sufficient in this case:

∫ P(βƒ—x)P (βƒ—x + βƒ—ρ)D φ (βƒ—x, βƒ—ρ)dβƒ—x D¯φπœ–βˆ₯(βƒ—ρ ) = ----∫-------------πœ–βˆ₯---------, (36 ) P (βƒ—x)P (βƒ—x + βƒ—ρ)dβƒ—x
where P is the pupil function. Combining Equations (35View Equation) and (36View Equation) leads to:
∑N ∑N D¯φ (βƒ—ρ) = βŸ¨πœ–iπœ–j⟩Uij(βƒ—ρ), (37 ) πœ–βˆ₯ i=1 j=1
where the Uij functions are defined as:
∫ U (βƒ—ρ) = --P-(βƒ—x)P-(βƒ—x +-βƒ—ρ)[Ki(βƒ—x∫)-−-Ki-(βƒ—x +-βƒ—ρ)][Kj-(βƒ—x) −-Kj(βƒ—x-+-βƒ—ρ-)]d-βƒ—x. (38 ) ij P (βƒ—x)P (βƒ—x + βƒ—ρ)dβƒ—x

The important conclusion is that the information, which the AO telemetry has to deliver is the covariance of the noise free residual KL coefficients: βŸ¨πœ–iπœ–j⟩. The Uij functions can be pre-computed and stored. The computation of the covariance for the exposure interval can be performed within the AO system, which reduces the telemetry data rates and storage requirements significantly. The computation of the covariance is typically initiated and terminated by an external trigger signal from the science camera.

The term φatm ⊥ in Equation 32View Equation can be derived from the Kolmogorov model (Kolmogorov, 1941, 1991) if the Fried parameter r0 is known. The residual uncorrected phase after correction of N KL modes can be expressed in terms of KL modes:

∑N∞ φatm⊥(βƒ—x, t) = ai(t)Ki (βƒ—x). (39 ) i=N+1

Using the definition of the phase structure function the residual uncorrected phase structure function can be expressed as:

D φ⊥(βƒ—x, βƒ—ρ) = ⟨|φatm⊥ (βƒ—x,t) − φatm ⊥(βƒ—x + βƒ—ρ,t)|2⟩ N ∞ ∑ = ⟨aiaj⟩[Ki(βƒ—x) − Ki (βƒ—x + βƒ—ρ)][Kj (βƒ—x) − Kj(βƒ—x + βƒ—ρ )]. (40 ) i,j=N+1

The covariances ⟨aiaj⟩ for Zernike modes was given by Noll (1976) and scales with 5βˆ•3 (D βˆ•r0), where D is the aperture size of the telescope. A similar scaling law for the KL covariances can be derived. Only r0 needs to be determined, which can be estimated from the actuator commands. The modal variance of the DM actuator commands is compared in a least squares sense to the modal variance predicted by the Kolmogorov model in order to derive an estimation of the Fried parameter (r0).

Figure 24View Image summarizes the major steps of the PSF estimation method.

View Image

Figure 24: Schematic block diagram describing the method to estimate the long exposure PSF from solar AO loop data (from Marino, 2007Jump To The Next Citation Point).

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