In this chapter I evaluate the performance of Lucky Imaging systems in a wide variety of conditions. The evaluations are based upon 10 nights of data taken during LuckyCam runs in 2000-2004; I participated in the June 2004 observing run (as well as the two 2005 LuckyCam runs).
The performance tests detailed here demonstrate that the prototype Lucky Imaging cameras give a very valuable increase in I-band resolution over a large area, with only modest signal loss due to frame selection. In section 4.1 I describe the point spread function improvements produced by Lucky Imaging and in section 4.2 I detail the performance produced in a variety of conditions. Section 4.3 addresses the effect on limiting magnitudes of the Lucky Imaging frame selection. Finally, section 4.4 discusses the guide star and hardware requirements.
In most observations presented here the camera is run approximately 2-3× too slowly to fully sample changes in the atmospheric turbulence and so these results cannot demonstrate properly the full potential of Lucky Imaging. Later chapters present data from the 2005–2006 Lucky Imaging system, which runs approximately 2× faster, and so more routinely achieves diffraction limited resolution. However, even the with slow camera used for the results the resolution gains are spectacular.
Figure 4.1 gives a typical example of the wide field PSF improvements obtained with Lucky Imaging, and examples of the general form of the Lucky Imaging PSF are given in fig. 4.2.
As with adaptive optics images, the radial shape takes the expected form for an image with partially compensated Kolmogorov turbulence (e.g.. Hardy (1998) and references therein) - a wide halo and a central compact core. As the selection of frames is made more stringent the fraction of light in the compact central core progressively increases.
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The Strehl ratio (commonly used as a high-resolution imaging performance measurement) is the peak value of a PSF divided by the theoretical diffraction-limited value. Figure 4.2 clearly shows that Lucky imaging offers a substantial Strehl ratio improvement over both autoguider and shift-and-add (effectively 100% selection) systems. Lucky Imaging routinely obtains I-band Strehl ratios in the range of 0.15-0.2 at high frame rates in good seeing on the 2.5m NOT.
It should be noted that the NOT is not designed to obtain diffraction limited images, with a specification to concentrate 80% of the light within 0.4 arcseconds* . Compared to a diffraction limit which concentrates approximately 80% of the light within 0.08 arcsecs, it is very likely that the performence of Lucky Imaging is limited by the telescope. Although on some occasions phase errors in the atmosphere can correct for telescope phase errors, this happens with reduced probability compared to a simple flat wavefront requirement. This subject has been treated in detail in Tubbs (2004); I here simply note that Lucky Imaging systems benefit from as high a telescope image quality as possible. Tests on the NTT telescope in Chile in June 2006, with a different telescope quality, are expected to quantify the effect of the telescope error budget.
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It is difficult to directly compare the Lucky Imaging performance to the standard imaging system on the NOT (which uses an autoguider), since it is not possible to take images simultaneously with the two systems. For all seeing values given in this chapter I apply a detailed simulation of the NOT autoguider (both the image analysis and the pointing system) to the LuckyCam data, effectively allowing a simultaneous test of the two systems. The simulation system is described in chapter 3.
The quoted seeings are measured from the FWHM of elliptical 2D Moffat profile fits to simulated autoguider images. They are thus measured at an effective wavelength of ~850nm and should be increased by approximately 10% to convert to a standard seeing quoted at 550nm. All other FWHMs detailed in this chapter are similarly measured from elliptical 2D Moffat profile fits.
The 1-D Moffat profile (eg. Trujillo et al. (2001) and references therein) is described by:
 | (4.1) |
where r is the radius from the peak, rs is a scale radius and β can be varied between 1 and ∞.
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The presented results are based on observations taken during 10 nights on five observing runs in May, June & July, 2000–2004. The 2.56m Nordic Optical Telescope on La Palma in the Canary Islands was used for all observations, which were principally made in several different fields in the cores of the Globular Clusters M3, M13 and M15. The 2001-2004 dataset is split into 42 short (2-3 minute) runs, totalling ~132,000 frames. All fields were observed in I-band.
I here investigate science-target performance - i.e. that for a star near the guide star. In each of the fields I chose a target star for PSF measurement at 1-3” separation from the guide star. The particular difficulties associated with using the guide star PSF are discussed in chapter 5.
The improvement in FWHM obtainable in different atmospheric conditions and with different percentage selections is shown in fig. 4.3. Much of the scatter in values is due to the range of distances of target stars from the reference star (section 4.2.2).
Figure 4.3 shows that under a wide range of conditions the resolution is improved by factors as large as ×4 in the most stringent selections in all of the 42 observations reduced. Less stringent selections give smaller improvements. There is an approximately linear correspondence between the autoguided long exposure seeing and the resolution attainable. This suggests that, at least over the ~200 second timescale of these observations, the standard measures of seeing are a reasonable guide to the atmospheric turbulence statistics. I thus adopt the seeing as the standard measure of the atmospheric turbulence strength in a particular run.
Figure 4.4 shows the effects of selecting differing fractions of images using empirical fits to the 12FPS data. Although limited by the slow frame rate, in 0.5 arcsec seeing the most stringently selected images are within ~0.06 arcsec of the 0.08 arcsec diffraction limit and Strehl ratios are >0.1.
The advantages of the full-frame field of view led the Lucky Imaging team to obtain most of our data in LuckyCam’s relatively slow 12 or 18 FPS modes. To obtain images with light reliably concentrated into bright single speckles, unblurred by image motion, the atmospheric coherence time must be oversampled. This would typically require > 40 FPS at the NOT. However, the image quality varies on every timescale slower than the coherence time and so image selection can always be expected to improve the resolution – even if very slow frame rates are used.
To acquire a high-quality frame there is a requirement for both an excellent point spread function and a stable atmosphere for longer than the frame integration time; this occurs with increasing probability as the frame rate is increased. As expected, the faster frame rates presented in fig. 4.3 on average give a greater resolution improvement than the slower rates. In good seeing the output resolution at fast frame rate is within 0.03 arcsec of the diffraction limit of the telescope, limited at least in part by LuckyCam’s slight PSF undersampling as well as the image quality of the telescope.
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Although the resolution can be adjusted by altering the number of frames selected, in poorer seeing the probability of a superb frame is greatly reduced. Empirically this appears to limit the LuckyCam resolution increase to a factor of ~3-4 when the seeing is poor.
The FWHM gives a good estimate of the system’s performance for resolution enhancement – i.e. the separation of two closely separated objects. However, for many observations (notably precision photometry) the degree of light concentration within the larger halo is also important. Figure 4.4 also details the Full Width at Half Enclosed Flux (FWHEF) performance. The FWHEF is the aperture size which contains half the light of the star and thus is a better measure for crowded observations where photometry is to be performed. The current LuckyCam improves the FWHEF by up to a factor of two, corresponding to concentrating half the light from a star into an area four times smaller than in seeing-limited images.
With a faster camera higher resolutions have been reached. Figure 4.5 shows an I-band image of the 0.8 arcsec binary ζ Boötis with a Strehl ratio of 0.26, taken in 0.42 arcsec seeing. The first Airy ring is clearly visible at a radius of ~0.1 arcsec, showing the point spread function is indeed diffraction-limited in width. This dataset was taken at 200Hz, approximately 5× faster than the coherence time of the atmosphere requires.
Chapters 6 and 7 detail the LuckyCam VLM binary survey, performed entirely at 30 FPS. Almost all of the 121 survey target were imaged with diffraction-limited cores and Strehl ratios up to 0.25.
Resolution enhancement systems such as LuckyCam provide useful turbulence correction only within some angular distance to the guide star. This distance is commonly quantified by the isoplanatic patch radius - the radius at which the Strehl ratio of stars is reduced by a factor of e-1 from those very close to the guide star. I here use this definition of the useful field size, although noting that the low frame rate wide-field data discussed here has generally smaller Strehl ratios (of order 0.1) than are usually used for this measurement.
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The difference in Strehl ratio between stars close to the guide star and those up to 25 arcsec off-axis (the 2001–2004 LuckyCam field limit) can be measured for several of the globular cluster fields in the dataset. I then calculate the isoplanatic patch radius from the point at which an empirical function fitted to the Strehl ratios drops to e-1 of its on-axis value. The models (fig. 4.6) suggest that in five fields of M15 taken over two nights in 2003 the isoplanatic patch size ranged from 17-30 arcsec in radius. However, it should be noted that the exact isoplanatic patch radii are extrapolated, as the isoplanatic patch width is larger than LuckyCam’s field of view in these observations. Selecting a larger fraction of the frames gives an improved patch size (up to 10 arcsec wider) as the degree of turbulence correction achieved on-axis is decreased along with the output resolution.
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It is clear that the atmospheric turbulence corrections produced even with a low frame rate camera have a remarkably large effective radius, in most cases larger than LuckyCam’s field of view. These results (from data taken in 2003) agree with earlier Lucky Imaging results presented in Tubbs et al. (2002), suggesting that the large patch size may be inherent to the technique over a wide range of conditions.
The isoplanatic patch for speckle imaging is expected to be proportional to r0, the standard atmospheric coherence length (Vernin & Munoz-Tunon (1994) and references therein). r0 is known to vary on relatively short timescales (fig. 4.7). If LuckyCam selects periods when r0 is larger (and not just periods when the turbulence induced phase errors randomly sum to a small value) it would be expected that those periods would have larger isoplanatic patches, as is observed.
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Although most observations presented in this chapter are taken in I-band, we have also tested Lucky Imaging at shorter wavelengths, down to B-band (430nm). Although diffraction-limited images are not obtained far blueward of I-band, very useful resolution improvements are still achieved.
As discussed in chapter 1, the probability of obtaining a diffraction-limited frame is:
![[ 2]
P ≈ 5.6 exp -0.1557(D ∕r0)](thesis68x.png) | (4.2) |
for Kolmogorov atmospheric turbulence, where D is the diameter of the telescope and r0 is the Fried seeing parameter. r0 scales as λ6∕5 (Fried, 1967) and thus the probability of obtaining a good frame decreases very rapidly as the wavelength becomes shorter. We would therefore not expect to acquire any diffraction limited images at B-band.
However, at short wavelengths the individual frames still show large variations in the underlying seeing (figure 4.7) and large image motion and so (as in poor seeing Lucky Imaging) much improved resolutions can be achieved even if the diffraction limit is not reached.
In figure 4.8 I present a three-colour Lucky Imaging image, where the blue corresponds to B-band, the green to SDSS r’ and the red to SDSS z’. The image thus spans the visible light wavelength range (and a little into the infrared). The red light is best concentrated but the B-band FWHM resolution is also improved by approximately a factor of two (accurate measurement of this is complicated by the undersampled pixel scale). The PSFs are also very stable across the FOV, further illustrating Lucky Imaging’s very large isoplanatic patch (or more probably in the case of B-band, isokinetic patch).
Although we have not yet acquired a large enough B-band dataset for detailed investigation of Lucky Imaging performance in the blue, these results are very encouraging for further investigations.
Figure 4.8 also shows the wide-field imaging capabilities of the June 2005 LuckyCam. By changing the pixel scale to 0.12 arcsec/pixel (such that even at I-band the diffraction limited PSF is severely undersampled) the LuckyCam FOV is increased to 1 arcminute, at the expense of some resolution. In poor seeing or at short wavelengths, such as figure 4.8, when a diffraction limited PSF is unlikely, this mode takes advantage of the expectation of lower resolution.
LuckyCam is also capable of narrow-band imaging, achieved simply by using a narrow-band filter. Narrow filters require accordingly brighter guide stars, but otherwise narrow-band imaging proceeds in exactly the same manner as broad-band observation.
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Insertion of the grism allows low-resolution spectroscopy with the spatial resolution offered by Lucky Imaging (see chapter 2). Some trial observations have been performed in this mode (eg. figure 4.9), which show both spectral features and spatially resolved spectra of binaries as close as 0.24 arcsecs.
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Since Lucky Imaging improves the PSF resolution, the detail visible in nebulosity is improved. However, it could be thought that the wide halo of Lucky Imaging PSFs would “wash-out” the improved detail when observing extended objects.
To test this, we preformed Lucky Imaging of the high-surface-brightness nebulosity around the Crab Pulsar (figure 4.10), in R-band and I-band. Comparison to the seeing-limited inset shows that the detail displayed in the nebulosity in both colours is indeed much improved by Lucky Imaging.
Every observation performed with LuckyCam comes with high-time-resolution data for free. In addition to use for atmospheric turbulence correction, the high-time-resolution capability offers a very interesting facility for observations of rapidly varying astrophysical objects, a capability that has only previously been available from specialised instruments (for example, ULTRACAM (Dhillon et al., 2002)).
The Crab pulsar has a period of ~1/30 sec with a double-peaked light curve. It is thus an ideal target with which to demonstrate LuckyCam’s high time resolution capabilities.
We observed the Crab pulsar in November 2005, during a short period with only small cloud extinction. We operated LuckyCam in its fastest frame mode, 10 ms/frame with a 536 × 100 pixels frame size.
The pulses are clearly visible in individual LuckyCam frames. The LuckyCam pulsar light curve is shown in figure 4.11, compared to a simple model used to accurately determine the period and phase of the pulsar. Using this model, each frame can be assigned a phase on the period-folded pulsar light curve. The frames can then be summed into phase-resolved bins. The resultant sequence of high signal-to-noise images is shown running down the left hand side of this page. Both pulses detailed in Percival et al. (1993) are clearly resolved with the expected flux ratios, and the times of no light output from the pulsar are also resolved.
Although the images to the left are each a simple sum of ~ 100 frames, with a larger set of frames Lucky Imaging can be performed on each phase-resolved bin. This would attain phase-resolved diffraction-limited imaging of faint objects in I-band at 10ms time resolution, a unprecedented ground-based capability for studies of transient phenomena.
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The 2.5m Hubble Space Telescope (HST) has the same diffraction-limited resolution as the 2.5m Nordic Optical Telescope. The Advanced Camera for Surveys (ACS) Wide Field Camera (Ford et al., 1998) has a plate scale of 0.05 arcsec per pixel, and is designed to achieve an 82% Strehl ratio. An HST ACS WFC image is therefore a good estimate of an ideal diffraction-limited image, and affords an excellent baseline with which to test the LuckyCam system.
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In figure 4.12 I show an ACS WFC camera field of the core of M3, compared to a 5% LuckyCam selected image of the same field. The two images are taken using very similar I-band filters (and the chip responses are much the same), although they unfortunately have very different exposure times – after image selection, the LuckyCam image has a 40× shorter exposure time (8 seconds).
Because of the much shorter exposure, the fainter objects visible in the HST exposure are not visible in the LuckyCam image. However, the FWHM resolution of the LuckyCam image is very similar to that of the HST image for all objects within a few arcseconds of the guide star. The fall-off in resolution due to the isoplanatic patch becomes visible compared to the HST image towards the upper right, at approximately 10 arcseconds radius from the guide star.
The most significant PSF difference between the LuckyCam and the HST images is the wide halo around the LuckyCam diffraction-limited cores. The halos limit the maximum small-radius contrast ratios that can be reached with LuckyCam compared to HST. However, as discussed in chapter 6, LuckyCam does reach very useful contrast ratios, for example enabling the detection of substellar companions around late M-dwarfs in only 100 seconds of observation.
Although not achieving the superb atmosphere-free imaging quality of the HST, LuckyCam reproduces the FWHM I-band resolution of HST in a very much cheaper and easier-to-construct system, even on a telescope not designed to give diffraction-limited images.
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An improvement in resolution allows smaller photometric apertures to be used. A smaller aperture contains fewer pixels and so less sky noise and (here negligible) detector noise, giving an increased SNR.
In figure 4.13 I compare modelled LuckyCam SNR performance with optimised apertures to measured results for an I=+19m star in the core of M15. The model assumes that the only sources of noise are photon shot noise from the sky and star (and L3CCD multiplication noise); the imaging resolution is modelled from the performance detailed in section 4.2.1. As for this star the noise in these images is background dominated, the SNR is measured by comparing the signal contained inside an aperture to the RMS noise in a nearby empty (background) aperture.
The 0.55 arcsec seeing model agrees well with the measured SNR in high-resolution observations but diverges by ~25% at lower resolutions. This is due to increased crowding (the fields are in the cores of globular clusters) leading to an increased background. This aside, the standard CCD photometric noise estimates model LuckyCam’s performance well, once the applicable aperture size changes are taken into account.
Although a 1% selection of frames would be expected to decrease the SNR by a factor of 
= 10
for a star in an empty field, the smaller aperture sizes that can be used reduce the decrease
to ~ 6.5× in both seeings in fig. 4.13 . At the 50-75% selection level the SNR matches
that available in a standard autoguided image, with resolution increased by a factor of
~2.
It is worth emphasising that Lucky Imaging’s ability to increase resolution by large factors even in poor seeing effectively increases the usable observation time at the telescope. For example, if a particular observation requires 0.5 arcsec seeing selecting a few percent of frames in 1.5 arcsec seeing will produce usable data in telescope time that would otherwise have been lost.
In this section I detail the two main requirements for a successful Lucky Imaging observation - a sufficiently bright guide star and a sufficiently fast camera.
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I simulated the effect of using a faint guide star by rescaling a bright (> 2000 photons / frame) star PSF to a specified flux, taking Poisson and L3CCD multiplication noise into account. As a check of the simulations’ accuracy I also performed the same measurements with some non-simulated faint guide stars.
Figure 4.14 shows, at a 10% selection, the reference star flux requirements at two seeings. The requirements are similar at all percentage selections.
At a guide star flux of 150 photons/frame (I ~ +16m) the resolution is reduced by 22% from that achieved with a very bright star. Because there appears to be a rapid falloff in resolution below that flux level I adopt 150 photons/frame as the minimum flux required for high resolution imaging.
The more spread out PSFs in poorer seeing require more photons to give an acceptable SNR in the PSF core. The real (non flux rescaled) guide stars give matching results, although with higher noise induced by anisoplanatism.
The ~150 photons/frame requirement can be understood in the following way. Even in poor seeing the best frames consist of a single speckle surrounded by a halo. If the speckle contains only a few tens of photons, Poisson shot-noise becomes a limiting factor for frame selection on the basis of Strehl ratios, thus degrading the output resolution.
If a field contains several stars which are individually fainter than the guide star limit, John Baldwin has found that it is possible to add the stars’ PSFs together to provide a useful estimate of a bright guide star PSF, which can then be used in the standard way for Lucky Imaging.
The isoplanatic patch and reference star flux requirements directly give LuckyCam’s achievable sky coverage. If the system is to achieve its most useful resolution gains an I ~ 16 reference star within 25” of our target is required.
At a galactic latitude of 60 - 70o approximately 7% of the sky can be used with chance-placed guide stars, based on averaged star counts from the USNO-B1.0 catalogue (Monet et al., 2003). 30% of the sky is accessible at a latitude of 20 - 30o while at lower galactic latitudes sky coverage is virtually complete.
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Freezing the effects of atmospheric turbulence requires exposures that are shorter than the atmospheric coherence time. One sufficiently fast (60Hz) trial dataset is presented in fig. 4.15, temporally rebinned to produce a range of effective effective frame rates. Although this is a single observation the performance at 12FPS and 30FPS is consistent with that found in the more numerous trials in previous sections of this chapter, and is thus probably not atypical.
Selection of frames (as opposed to simple shift-and-add) gives an improvement at all measured frame rates. At even 0.3 FPS the frame selection gives gains in FWHM of over 1.6×, with an estimated limiting reference star magnitude of I=+19m in dark time.
At ~30 FPS the atmospheric coherence time is well sampled in the best 10% of frames; there is no improvement with faster frame rates. The frames with poorer quality do, however, benefit from still faster rates. The periods of poor seeing have a relatively smaller atmospheric coherence time because there are more isophase patches per unit area to be swept over the telescope aperture by the wind. The more inclusive frame selections include these periods and so require a faster frame rate than when selecting only the very best frames.
