For the purposes of this paper, Strehl ratios have been calculated from the ratio of peak to total fluxes in the PSF image, by comparing it to that calculated for the theoretical diffraction limited PSF (with a 3.50m mirror and 1.37m central obscuration). Unless stated, they take no account of where in the pixel the PSF is centred and hence may underestimate the actual Strehl. This error is rather variable: for example with 0.04 $^{\prime\prime}$ pixels a high Strehl measurement might be $50\pm5$ %, and with 0.08 $^{\prime\prime}$ pixels a lower Strehl measurement could be $15\pm5$ %.

For seeing of around 1 $^{\prime\prime}$ , K-band Strehls in excess of 60% can be reached for the brightest stars ( ${m_V}\mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$}\hss} \raise 2.0pt\hbox{$\mathchar''13C$}}8$ ), while values in the range $25-50\%$ can be attained for stars with ${m_V}\mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$}\hss} \raise 2.0pt\hbox{$\mathchar''13C$}}10$ .The performance achieved can be translated to other wavebands: we have achieved a J-band Strehl of 12% on SAO56114 (m_V=7.0) and a resolution better than 0.10 $^{\prime\prime}$ , close to the diffraction limit of 0.07 $^{\prime\prime}$ FWHM. Previous efforts can be summarised by an observation in July 1997 of the m_V=5 star 14Peg for which we achieved K-band strehl of only 20%, and so these new results represent a vast and speedy improvement, mainly due to upgrades in the hardware but also due to more careful alignment of the optical components and fine tuning of the modal reconstruction algorithms. Evidence for great advances in the field of AO generally can be found in the fact that such results are no more than is expected from other well established natural guide star systems such as Pueo (Rigaut et al. 1998), ADONIS (Le Mignant et al. 1999), and Hokupa'a (Close et al. 1999). However, it should be noted that operating in better seeing brings a huge benefit: a correction that results in a PSF with 25% strehl in 1 $^{\prime\prime}$ seeing would achieve 40% strehl in 0.7 $^{\prime\prime}$ seeing, which often occurs on Mauna Kea.

Even though the limiting magnitude is still lower than satisfactory ( ${m_V}\sim 12$ ) , the performance is very encouraging given that even if 32 (Zernike) modes are corrected perfectly, the residual wavefront error for the 3.5-m in 1 $^{\prime\prime}$ seeing in the K-band is $\sigma^2 = 0.43$ rad² (Noll 1976) giving a maximum theoretical Strehl ratio of only 65%; and this does not include bandwidth limitations, noise, or other residual static aberrations - which are discussed below. The disturbance rejection bandwidth is typically 1/12 of the sampling frequency, while the temporal timescale of the atmosphere at 2 $\mu$ m relates to a Greenwood frequency (Greenwood 1977) on the order of 10Hz. Thus for frame rates of 300Hz or more (e.g. for bright stars) the phase error due to bandwidth is small, while it becomes important at 100Hz. The optimal frame rate requires minimising temporal errors and those from noise; a further parameter enters as we have several lenslet arrays available. The limiting magnitude is 1-2 mags brighter than that calculated from throughput and photon noise, much of which was due to faulty electronics in the Shack-Hartmann sensor and has been corrected for the 1999 semesters. The last additional input of errors is from static aberrations; most of these are removed by adjusting the deformable mirror shape until a reference fibre appears as a near-perfect PSF on the science camera.

$\begin{figure} \includegraphics [width=7cm,clip]{H1405f1a.eps} \includegraphics [width=7cm,clip]{H1405f1b.eps}\end{figure}$

Figure 1: K-band surface plots of SAO36874 (m_V=5.9), observed with a pixel scale of 0.04 $^{\prime\prime}$ , and correcting 32 modes at a frame rate of 200Hz. Upper: a direct image. Lower: a deconvolved image (30 iterations of the Lucy algorithm); the PSF star (SAO56114, m_V=7.0) had a Strehl ratio of 45% and a FWHM of 0.13 $^{\prime\prime}$ . The two stars of SAO36874 are clearly resolved at a separation of 0.15 $^{\prime\prime}$ , only marginally greater than the K-band diffraction limit of the telescope

One particular example of ALFA's performance is the serendipitous discovery of a double star in SAO36784 as shown in Fig. 1, during testing of control parameters. This star is listed in the Washington Double Star Catalog (WDS, Worley & Douglass 1997) as the primary partner of a pair with separation of 20.5'' and magnitudes m_V=6.0 and 12.3. We have found that the primary itself is double, and almost certainly a true binary. Although we cannot rule out that it may be a projection effect, the probability of detecting any star (the cutoff is taken to be that the peak intensity of the companion is equal to the peak intensity of in the primary's first airy ring) this close, given the local stellar density, is $\sim 2 \ 10^{-8}$ .The separation of the two components is 0.15 $^{\prime\prime}$ , almost at the telescope's diffraction limit (0.13 $^{\prime\prime}$ FWHM), and they can clearly be discriminated after deconvolution. The observation shows the vast potential for studies of stars in multiple systems, for determining orbits and system stability, as well as for characterising spectroscopic and speckle binaries, by measuring broad-band colours or individual spectra. An example of the latter is given in the next section.

3 Best performance