The following sources of residual wavefront errors have to be considered:
The fitting error term is due to the limited number of actuators, which leads to an imperfect fit of the incoming wavefront by the DM and depends on the ration :
The coefficient is DM specific. For a continuous deformable mirror .
The aliasing error term is due to the limited spatial sampling of the wavefront by the wavefront sensor. Highorder modes can alias into loworder modes and, thus, contribute noise to those modes. The aliasing term is typically of order 30% of the fitting error variance:
A conventional adaptive optics system with the DM typically conjugated to the pupil plane provides optimal correction in one direction on the sky only. The wavefront sensor is measuring the wavefront aberrations typically for the center of the extended FOV. For field points offset from the center the light waves travel from different field directions and, hence, sample different turbulent volumes of the atmosphere. For that reason the wavefront sensor measurement and, thus, the AO correction becomes increasingly invalid as the separation from the lock center increases. For an extended object such as the Sun this effect can be and, in many cases, is a severe limitation. The error that is introduced when measuring the wavefront at an offaxis point can be derived from the phase structure function. For the condition the anisoplanatic error variance at angular distance can be expressed as (see Hardy, 1998; Quirrenbach, 2002):
The condition is typically valid for large aperture astronomical telescopes but may not hold for small aperture solar telescopes during excellent seeing conditions.
The isoplantic angle can be defined as the angular distance for which the anisoplanatic error variance is . A wavefront that has a variance is sometimes referred to as “flat”. With this definition the isoplanatic angle can be expressed as (see Hardy, 1998):
The anisoplanatism error then becomes simply:
The wavelength dependence of again derives from the wavelength dependence of r_{0} and is , i.e., the isoplanatic angle increases significantly towards infrared wavelengths.
With the above definition of the isoplanatic angle the residual wavefront variance amounts to 1 rad^{2} at an angular separation of , corresponding to a Strehl ratio of S = 0.37. The isoplanatic angle can be quite small in this case if for the high altitude turbulence is large. For current small aperture solar telescopes, however, (for the upper atmosphere) at a good site can be of order one. In this case the aberrations contributed by the upper atmosphere are dominated by low orders, which decorrelate less rapidly than the high order modes as we move away from the lock center. Similarly, if only a limited number of modes is corrected by the AO system the AO performance, in a relative sense, deteriorates less rapidly with angular separation because of the slower decorrelation of low order modes. Normalized modal correlation functions can be defined to quantify the decorrelation as a function of angular separation for individual modes, e.g., Zernikes (Valley and Wandzura, 1979; Fusco and Conan, 2004). If the correlation for a given mode falls below 0.5, adaptive correction for that mode degrades the phase as much as correcting it. Depending on the application and, in particular, the FOV requirements it may make sense to restrict the number of corrected modes in order to optimize correction over a larger FOV. The extreme case is tip/tilt correction only. The isoplanatic angle for tip/tilt, sometimes also referred to as isokinetic angle (Beckers, 1993a), can be many tens of arc seconds.
The size of the isoplanatic patch is determined by a height weighted () integral over . If strong turbulence is located high in the atmosphere the isoplanatic patch can become quite small as of the upper atmosphere becomes large. For example, at the DST site the jet stream occasionally moves far enough south and above the DST causing severe high altitude seeing. Although a meter is not installed at the DST the presence of high altitude seeing is usually quite easy to identify by assessing the real time video image and comparing the visual seeing to the readings of the Seykora seeing meter (Seykora, 1993). This seeing meter measures scintillation of an extended object (the solar disk) and, hence, provides a measure of the turbulence heavily weighted towards seeing layers close to the ground (Beckers, 1993b). Conditions where the video image indicates bad seeing but at the same time the Seykora meter reading indicates relatively good seeing are a clear indication of strong high altitude seeing.
In these conditions the dominant seeing layer produced by the jet stream shear layer is at about h = 10 km and the isoplanatic patch is only of order a few arcsec. This can be seen in Figure 19, left, which shows a long exposure image of solar granulation with the AO locked on the center of the field. The optimally corrected FOV is of order 5 – 10”, while the surrounding FOV is significantly blurred. The high zenith angle under which solar observations are generally performed to take advantage of the best seeing in the early morning hours adversely affects the isoplanatic patch size. Fortunately, these conditions are the exception and the jet stream usually passes north of the DST site (a reason for selecting sites in the south). In this case the seeing is generally dominated by ground layer turbulence caused by heating of the ground layer and the majority of the turbulence is located close to the telescope aperture. The isoplanatic patch for a small aperture telescope as the DST appears to be quite large under these conditions, which is shown in Figure 19, right.
This somewhat qualitative picture is based on observing experience and might be considered anecdotal. Nevertheless, this experience based knowledge can be helpful when trying to predict observing conditions in order to optimize observing programs ahead of time. More sophisticated prediction of seeing conditions use detailed and comprehensive atmospheric modeling (Vernin et al., 1998; Masciadri et al., 2001; Cherubini et al., 2008a,b, 2009).
The limiting effect of anisoplanatism can be pursued in more depth and more quantitatively with simulations that consider different turbulence profiles as well as the effect of aperture size (Marino and Rimmele, 2011). Results for current small aperture solar telescopes can be related to the observations shown in Figure 19.
Two atmospheric turbulence profiles are considered by Marino and Rimmele (2011) and are shown in Table 1. The first turbulence profile used represents realistic good, daytime conditions at the ATST site on Haleakala (see Rimmele et al., 2006c). The second model is built from measurements above Mt. Graham and Mt. Hopkins in Arizona (Milton et al., 2003; McKenna et al., 2003).
The Haleakala profile represents a case of relatively weak turbulence at high altitude with just 5% of the power above 6 km. If an total r_{0} of 10 cm is assumed this profile produces a layer r_{0} at an altitude of 13.5 km of 1.5 m. The Mt. Graham profile is measured at nighttime, i.e., the represented case has a much stronger higher atmosphere with 40% of the power above 6 km.
The intention here is not to compare sites but to illustrate the impact of the profile on the solar AO Strehl ratio and the isoplanatic angle. For each turbulence profile two seeing cases are modeled by setting the overall r_{0} to 10 cm (good seeing) and 20 cm (excellent seeing), respectively.
Haleakala

Mt. Graham


Height  Fraction of  Height  Fraction of 
(m)

total power 
(m)

total power 
0  0.715  200  0.34 
1852  0.232  2000  0.07 
6052  0.042  3400  0.19 
13552  0.011  6000  0.09 
7600  0.06  
13300  0.21  
16000  0.04  
In order to model an existing, small aperture solar telescope Marino and Rimmele (2011) (virtually) place the DST and its high order AO system on Haleakala and the AO76 performance in terms of Strehl is modeled as a function of field angle and with zenith angle and number of corrected modes as parameters. The adaptive optics performance is modeled using a large number of phase screens that obey Kolmogorov statistics at each of the turbulence layers. The fractional distribution of turbulent power and the corresponding layer r_{0} are shown in Table 1. The turbulence screens are projected onto the extended wavefront sensor with a FOV of 10” × 10”. This ensures that anisoplanatic effects are accurately taken into account also in modeling the extended field wavefront sensor measurement (see Section 6.1.6). Drive signals for the DM, which is modeled using realistic influence functions, are derived from the extended WFS FOV. The model assumes correction of 80 KL modes and also includes a realistic implementation of the servo loop.
Results are shown in Figures 20 and 21. Several interesting points can be made. The Strehl ratio at the lock point drops significantly for high zenith angle observations. Whereas the Strehl at the lockpoint would be independent of zenith angle (r_{0} is assumed to be the same for each zenith distance in this model calculation) if a pointsource WFS is used, the extended source WFS averages wavefront information from many different sky directions. In the extreme case of a very large WFS FOV a groundlayer AO is realized, i.e., the upper atmospheric turbulence is not corrected at all (see Section 9.3). This effect is more pronounced at high zenith angles and worse high altitude seeing due to the geometric projection of the turbulent phase screens. In addition to the high airmass solar AO therefore faces this additional disadvantage in achieving high Strehl in the early morning hours (zenith angle > 70°). Where a WFS FOV of 10” × 10” may be adequate for r_{0} of 20 cm Figures 20 and, in particular, Figure 21 indicate that for worse seeing conditions a smaller WFS FOV would be of advantage if high Strehl is the objective. For excellent seeing, reasonably low zenith angle and low altitude turbulence the isoplanatic patch size can be quite large. For the most ideal and, thus, rare case (Figure 20, r_{0} = 20 cm, vertical pointing) the Strehl dropps to 0.37 at an offaxis angle of about 30”; the isoplanatic patch size is 60”. Reasonably high Strehl is observed over an even larger FOV. It is also obvious that under less ideal conditions the isoplanatic patch size rapidly becomes smaller. Substantial high altitude turbulence leads to significantly reduced Strehl performance and a small isoplanatic patch as can be seen in Figure 21, right. A site with low high altitude turbulence is clearly preferable from the solar AO point of view and this was an important factor in the selection of the ATST site.
The equivalent and very important simulation results for future large aperture telescopes will be presented in Section 9 where it is demonstrated that anisoplanatism is even more of a challenge.
The bandwidth error is due to the limited correcting bandwidth of the AO system. The bandwidth error is proportional to the ratio between the frequency of the turbulence, quantified by the Greenwood frequency (Greenwood, 1977), and the bandwidth of the AO system (e.g., Hardy, 1998):
In the special case of a single turbulent layer moving at a speed , the Greenwood frequency can be written as (Hardy, 1998; Tyson, 2011):
As discussed in Section 2.3, closed loop bandwidths in excess of 100 Hz are required for solar AO systems.
Closed loop bandwidth should not be confused with sampling rate and using a meaningful definition of closed loop bandwidth is similarly important (Madec, 1999). A reasonable and conservative measure of closedloop bandwidth is given by the 0 dB error rejection crossover frequency, which for the DST AO76 system is about 120 Hz. As pointed out in Madec (1999) it is of critical importance to minimize compute and other latencies in order to obtain high closedloop bandwidth. Equation 21 provides an estimate of the bandwidth error. A more accurate value for can be computed if the closed loop error rejection transfer function can be measured or modeled and the PSD of the wavefront aberrations is known.
Often the incoming wavefront is decomposed into the well known Zernike modes (Noll, 1976). The Greenwood frequency depends on the radial mode number in the following way:
where is the radial degree of the mode, is the wind speed, and is the telescope aperture. This modal dependence of can be inferred from Figure 7, which shows modal PSDs for two Zernike modes. The Greenwood frequency for Z4 (astigmatism) is of order 5 Hz, while for Z21 is about 20 Hz. This means that the higher the order of correction, the more bandwidth will be required. Therefore, loworder systems require less bandwidth. Figure 7 also plots the corrected or residual wavefront error PSD, from which the residual wavefront error as a function of mode can be computed. This information can in principle be used to derive or optimize mode dependent servo gain factors as is done for some nighttime AO systems (Gendron and Léna, 1994, 1995). However, because of the low wavefront sensor noise of the solar correlating SHWFS the advantage of setting individual signaltonoise dependent gains for the modes may not be as convincing as it is for a photon starved nighttime WFS.
The WFS noise for a correlating Shack–Hartmann WFS has been studied by Michau et al. (1993, 2006). Michau et al. (1993) derived the following equation for the variance of the image position measurement by tracking the centerofgravity of the crosscorrelation peak:
where is the background noise variance, is the subimage size in pixels (typically 16 × 16 pixels), and is the total “energy” in the image, i.e., is the rms image contrast. Nyquist sampling of the image is assumed. A quantitative analysis of the wavefront sensor noise of a correlating Shack–Hartmann wavefront sensor was given by Poyneer (2003). Equation 24 indicates that the WFS S/N ratio is fundamentally given by the ratio of background noise to image contrast. It is of advantage to use more pixels per subaperture in order to reduce the WFS noise (Berkefeld, 2007).It is important to realize that in the case of a solar wavefront sensor a large number of photons are available in the wavefront sensor path since the WFS can work with broadband light (interference filters that can be several 100 A wide or simple color glass can be used). In praxis the number of photons that can be collected by the correlating SH wavefront sensor is limited by the well depth of the detector used. A typical well depth of a CCD is of order 50 ke. This means that the shot noise completely dominates the background noise even if relatively noisy CCD detectors or CMOS devices with readnoise levels of order 50 e are used. The low noise ( 1e) wavefront sensor detectors needed for tracking faint guide stars in the night sky are not needed for the solar application. However, because of the extended FOV ( 20 × 20 pixels/subaperture) the correlating SHWFS requires large format detectors and in order to achieve the required high bandwidths, the frame rates have to be very high (> 2 kHz). This means that detectors for the solar wavefront sensor typically need to have some sort of parallel readout. The signal to noise ratio might be improved by collecting more photons, which has to be achieved within an update rate time of typically 400 µs. A deep well detector or averaging multiple exposures of a extremely fast framing camera could potentially be used to achieve this and, thus, potentially allow smaller subapertures.
It should be noted that the wavefront sensor noise is object dependent in the sense that high signal to noise ratios can be achieved for high contrast objects, such as sunspots, whereas the S/N ratio for tracking the low contrast granulation is much lower. This is true, in particular, near the solar limb where the granulation contrast is even lower. Bright faculae near the limb can be used to lock the correlating SHWFS. The contrast of granulation imaged through a subaperture drops as the size of the subaperture becomes smaller. Granulation has a typical spatial scale of about 1”. This means that granulation will be “smeared out” by diffraction if a subaperture size much smaller than 10 cm is used. There is a “limiting image contrast” below which a correlating SHWFS will not work anymore. According to practical experience with existing systems the limiting contrast is between 1.5 – 2%. The limiting contrast can be compared to the “limiting magnitude” that exists for the photon starved nighttime AO. Because of the dependence of the wavefront sensor noise on image contrast and the limiting contrast the seeing at the site is extremely important for the performance of a solar AO systems. Matching a small Fried parameter at a bad site with a smaller and smaller subaperture size is not possible if granulation is to be used as tracking target.
One could consider substantially increasing the FOV of the subaperture images in order to be able to track on largescale intensity structures on the Sun. It has been demonstrated in the context of active optics wavefront sensing for large solar telescopes that a larger FOV can provide more robust tracking performance of the cross correlation algorithm (OwnerPetersen et al., 1993). However, in this case the wavefront sensor averages over many isoplanatic patches and essentially only the near ground turbulence can be corrected in this way. Although GroundLayer AO (GLAO) may be an attractive option for some solar applications (Rimmele, 2000; Rimmele et al., 2010c) the general conclusion is that scientific productivity of solar AO depends critically on the site performance.
Michau (2002) argued that Equation 24 overestimates the noise by a factor of two. Cain (2004) studied an image projection approach to the extended source wavefront sensor problem and found close agreement with the noise estimates from Equation 24 and his own wavefront sensor noise estimates. Figure 22 shows the subaperture tilt noise estimates based on Equation 24 as a function of number of photons collected by the wavefront sensor camera. Nyquist sampling (0.5”/pix) and 20 pixels across the subaperture are assumed. The number of photons collected during an exposure is only limited by the well depth of the CCD or CMOS device used. For example, the DST AO systems wavefront sensor camera uses a CMOS camera with well depth of 70 ke, i.e., the shot noise is 264 e compared to the camera read noise of about 50 e, leading to a total noise of 268 e. The different curves in Figure 22 are for subaperture contrasts of 0.015 (granulation, r_{0} = 5 cm), 0.025 (granulation, r_{0} > 10 cm), 0.05 (pore, r_{0} = 7 cm), and 0.1 (sunspot umbra, r_{0} = 7 cm) and, thus, give the range of typical observing targets and observing conditions considered in developing error budgets. It is obvious from Figure 22 that a detector with deep wells is desired. However, exceeding a well depth of about 40 ke results in insignificant gains in performance.
The DST AO system was used to verify the noise levels predicted by Equation 24. Openloop subaperture tilt PSDs with a sampling rate of 2500 Hz were collected. The example PSD of Figure 23 clearly shows a flat noise tail that, assuming white noise, can be used to directly determine the noise of the subaperture tilt measurements. The following subaperture tilt noise levels for different observing targets and seeing conditions were obtained:
The DST AO camera is operated at about 75% of its well depth or 52 ke for granulation. If a sunspot is tracked the number of photons collected per exposure can be as low as 30 ke. Comparing the measurements with the predicted noise levels one finds that Equation 24 seems to slightly overestimate the WFS noise. The WFS noise is propagated onto the actuator commands by the reconstruction process. The details of this noise propagation are again discussed in textbooks (Hardy, 1998; Roddier, 1999; Tyson, 2011). If B is the reconstruction matrix that converts wavefront sensor slope measurements into actuator commands, then the mean wavefront error over the aperture after the reconstruction is (Southwell, 1980):
The simplifying assumption has been made that the noise covariance matrix is diagonal, i.e., the noise is uncorrelated. The important point is that the error propagation coefficient is of order one or less.
The wavefront sensor noise due to anisoplanatism within the extended FOV of the SHWFS can be an important term that highly depends on seeing conditions and the size of the WFS FOV. Equation 24 was developed with the assumption that the correlated reference and live images are shifted copies of each other. However, reference and live image are taken at different times or from different subapertures. Distortion of the images due to anisoplanatism within the extended subimage is not considered. The subimage size of the reference and live image of the solar feature is typically between 5” × 5” to 10” × 10”. Depending on the seeing conditions and wavelength larger FOVs may contain several isoplanatic patches, which will compromise Strehl due to the already discussed field averaging effect. If the highest possible Strehl at or near the lock point is the objective a smaller WFS FOV is needed. Unfortunately, the field size can not be smaller than about 5” × 5” since a minimum number of granules are required within the FOV to obtain a distinct correlation peak that can provide an accurate tracking signal (von der Lühe, 1983).
It has been pointed out by Robert et al. (2006) and Védrenne et al. (2007) that anisoplanatism effects result in an additional noise term from the WFS. This noise term originates from a broadening of the PSF, which is an average over many different directions in the sky. Equivalently the crosscorrelation function is broadened, which lead to a less precise determination of the maximum position. Scintillation effects and crosscoupling between anisoplanatic and scintillation effects may produce yet another WFS noise terms of significant magnitude. Wöger and Rimmele (2009) performed a study to evaluate the effects of phase aberrations and scintillation within a subaperture of a correlating Shack–Hartmann wavefront sensor and compared the contribution of these effects to the wavefront sensor noise for both isoplanatic and anisoplanatic imaging. Realistic representations of the object, i.e., solar structure, and a close approximatation for a typical profile were used in these simulations. Wöger and Rimmele (2009) found that anisoplanatism in wavefront sensor subapertures can increase the measurement error of Equation 24 by 50% or more depending on the specifics of the WFS, such as pixel sampling and FOV. They also found that the effect of scintillation can be neglected in the solar AO case. The anisoplanatism can only be avoided by making the WFS FOV small enough to only cover one isoplanatic patch, which for practical purposes is not always possible.
The WFS and the science instrument in most cases do not share the entire optical path. In the simplest case the only difference is a beamsplitter. The science camera might be placed in the focal plane produced by the beam passing through the beamsplitter while the WFS is placed in the focal plan reflected off the beamsplitter. The aberrations introduced by the beamsplitter are different for transmitted and reflected beams and, in this example, the AO will correct what is introduced into the reflected beam and, in turn, add the aberration to the transmitted beam. In general, with more complex science instruments many more optical elements are noncommon. Since the instrument and WFS optical paths in the laboratory style environment typical for solar telescopes are through air, bench seeing adds to the noncommon path errors. In a well controlled lab environment local seeing can add 0.5 – 1 nm of wavefront error per meter optical path lengths (Biérent et al., 2008). Optical noncommon path errors can be calibrated out as described in Section 5.2.
Residual tip/tilt errors can severely lower the Strehl and degrade resolution. Residual image motion broadens the otherwise diffraction limited core of the PSF. Details can again be found in textbooks (Hardy, 1998). A separate tip/tilt error budget that considers most if not all terms discussed in this section should be developed, in particular, if a a separate tip/tilt system (e.g., correlation tracker (CT)) is used. The WFS noise of the CT sensor, the bandwidth and other terms may be different for the tip/tilt system.
Assuming statistical independence, which is usually a valid assumption in this case, the overall residual wavefront error can be computed as the RootSumSquared (RSS) of the individual error terms:
may, for example, include optical and seeing aberrations within the instrument, if not already accounted for in the noncommon path term.
Obviously the goal is to minimize the residual wavefront variance. This is a difficult task, in particular, for visible AO systems, which most solar AO systems are. At a wavelength of 500 nm the total allowable error is less than 50 nm if a Strehl of S = 0.7 is to be achieved. A Strehl of 0.7 in some textbooks is referred to a diffraction limited performance.
A careful error budget analysis during the design phase and careful attention to each of the error budget terms while building the AO system followed by a detailed evaluation of the actual performance are essential steps in ensuring that optimal performance is achieved. During the design a general goal of systems engineering is to produce a well balanced error budget, i.e., all terms should be of about equal magnitude. It does not pay to devote significant effort and expense to reduce one particular error term while other terms remain significantly larger.
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