4 The Correlating Shack–Hartmann Wavefront Sensor

The previous section summarized the vital role of the correlating SHWFS for solar adaptive optics. The principle of a correlating SHWFS is quite simple and is shown in Figure 14Watch/download Movie. The telescope aperture is sampled by an array of lenslets, which in turn forms an array of images of the object. In this case the object is solar granulation and typically 20 × 20 pixels are used to image a field of view of about 10” × 10” or less. The field can not be much larger to avoid averaging of wavefront information, in particular, from high turbulence layers. On the other hand, the FOV has to be large enough to contain a sufficient number of granules for the correlation algorithm to work in a robust manner (von der Lühe, 1983Jump To The Next Citation Point). As will be discussed in Section 6 depending on the severity of turbulence at high altitudes in the atmosphere and the zenith angle a WFS FOV of 10” × 10” can already severely limit the Strehl performance when compared to a point source WFS that is not subject to the directional averaging effect. The main challenge is to compute cross correlations in real-time between subaperture-images and a randomly selected subaperture-image, which serves as reference. The cross correlations are computed using:

∑ ∑ CC (Δ ⃗i ) = IM (⃗x) × IR(⃗x + Δ⃗i ), (15 )
where IM(⃗x ) is the subaperture image, IR(⃗x) is the reference image, and ⃗ Δi is the pixel shift between image and reference. The number of shifts between reference and image can be limited to just a few pixels in either direction, assuming the local tilts are small, i.e., the number of sums that have to be computed can be limited to a small number. Typically, computing the cross correlation on a pixel array of 5 × 5 pixels is sufficient, in particular once the control loop is closed. Alternatively the cross correlations can be computed using Fourier Transforms (FT) (von der Lühe et al., 1989Jump To The Next Citation Point). Using the FT approach may be of advantage on some processor platforms and provides the cross correlation for the entire FOV.

Variations of the classical cross-correlation algorithm have been proposed and are used in some solar AO systems (e.g., Shand et al., 1999Jump To The Next Citation Point; Scharmer et al., 2003Jump To The Next Citation Point). Computing the Square Difference Function or the Absolute Difference Function Squared may actually have slightly better performance (Löfdahl, 2010). The Absolute Difference Squared algorithm can be efficiently implemented on general purpose microprocessors using multimedia instruction set extensions (Shand et al., 1999Jump To The Next Citation Point).

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Figure 14: mpg-Movie (783 KB) Principle of correlating Shack–Hartmann wavefront sensor. Cross-correlation techniques are used to track the low contrast granulation images or any other extended object of sufficient contrast (Rimmele and Radick, 1998). The movie shows a time sequence of wavefront sensor camera images with 12 subapertures across the pupil of the DST. The cross-correlation functions of the subaperture images of granulation are shown on the right.

The full field cross correlations are shown in Figure 14Watch/download Movie, upper right. By locating the maximum of the cross correlation the displacement of the images with respect to the reference is determined, thereby measuring the local wavefront gradients or tilts. Image displacements are computed to subpixel precision by fitting a parabola to the correlation peak using and interpolating between pixels. Alternatively, a centroiding algorithm, commonly used for tracking point sources, can be used to track the correlation peaks since those closely resemble point sources. A tilt map is shown in the lower right corner of Figure 14Watch/download Movie. From the tilt vector map an estimate of the wavefront distortions is derived, i.e., the drive signals for the actuators of the deformable mirror using the same modal or zonal reconstruction schemes used for night-time AO systems (Hardy, 1998Jump To The Next Citation Point; Roddier, 1999Jump To The Next Citation Point; Tyson, 2011Jump To The Next Citation Point) are computed. Therefore, the main difference when compared to the night-time, SHWFS is the additional step required to compute the cross correlations, which adds significant computational expense.

Computing the cross correlations for a large number of subapertures requires not only substantial processing but also significant I/O capabilities. At the time when the first solar AO efforts were undertaken these capabilities were just not available. However, with the advances in the development of computer technology of recent years the processing power and I/O bandwidth are now readily available, for the most part as off-the-shelf products. The correlating Shack–Hartmann WFS is also of interest for tracking extended (elongated) “spots” produced by laser guide stars (e.g., Gratadour et al., 2010). It is interesting to note that images of the retina of the human eye with its cone structure look very similar to images of granulation, which in principle would make this wavefront sensor approach also interesting for vision science applications (Williams, 2000; Carroll et al., 2004). However, sufficient illumination of the retina is a problem and, hence, vision science AO systems project laser point sources onto the retina as wavefront sensing targets.

The subaperture size of Figure 14Watch/download Movie is 7 cm and diffraction at this small aperture limits the rms contrast of the granulation images to 1 – 3%, depending on the seeing conditions (Berkefeld and Soltau, 2010). This compares to typically 6 – 8% when imaged through the full aperture of the DST and an intrinsic contrast of about 13%. The low rms image contrast limits the sensitivity and ability to maintain lock of a correlating Shack–Hartmann wavefront sensor.

Potential alternatives to the correlating SHWFS

In the near future phase diversity (PD) might become an alternative to a correlating SHWFS, which currently appears to be the only viable choice for solar AO. Phase diversity has been used as a post-facto image reconstruction technique for solar high resolution imaging (Löfdahl and Scharmer, 1994Jump To The Next Citation Point). The implementation of real-time PD (Georges III et al., 2007; Warmuth et al., 2008) has made significant progress in recent years. Paxman et al. (2007Jump To The Next Citation Point) summarizes the current state and future prospects of real time PD and specifically compares performance and information content of PD and correlating SHWFS. Currently existing laboratory PD systems achieve 100 Hz frame rates, which is insufficient for solar AO applications. However, a number of speed-up factors may result in update rates for real time PD systems of 450 Hz to 5.4 kHz with a PD WFS sampling of 128 × 128 pixels or 1.6 to 20 kHz with sampling of 64 × 64 pixels and, thus, promises to provide a high performance alternative to the, by now, conventional correlating SHWFS approach.

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