= Solution
Atmospheric refractive-index fluctuations impose a changing phase screen on the pupil. For a <telescope> much larger than the <Fried parameter> $r_0$, long-exposure resolution is approximately $0.98\lambda/r_0$, rather than the <diffraction> scale $\lambda/D$. The scaling $r_0\propto\lambda^{6/5}$ makes the <seeing> width proportional to $\lambda^{-1/5}$ for fixed turbulence. An <atmospheric coherence time> sets the exposure or correction cadence needed to freeze the phase, while the <isoplanatic angle> sets the useful angular separation from a reference source. These are atmospheric quantities, not improvements automatically supplied by a larger <mirror>.
First reduce avoidable turbulence: choose a good high site, control <mirror> and enclosure <temperatures>, ventilate the dome, and observe when the <line of sight> crosses less turbulent air. Guiding removes <telescope> drift, and a fast tip-tilt correction removes image motion. Slow <active optics> corrects <mirror> figure and alignment; it cannot by itself follow all atmospheric phase fluctuations. These measures improve the input wavefront but do not guarantee diffraction-limited performance.
In <adaptive optics>, a <wavefront sensor> measures the reference-star wavefront, a reconstructor estimates its aberration, and a fast control loop commands a <deformable mirror> and tip-tilt element. A Shack-Hartmann sensor obtains local slopes from the displacement of focused lenslet spots. The science beam receives the corresponding optical-path correction. Residual phase <variance> limits the <Strehl ratio> approximately as $S\simeq e^{-\sigma_\phi^2}$ when the residual is small. Enough corrected spatial modes, adequate reference <photons> and a response faster than atmospheric changes are all needed. <infrared> correction is easier because a given path error has smaller phase in radians and the atmospheric coherence length is larger. A faint or distant <guide star> leaves larger errors and a core-plus-halo <point spread function>. The NACO manual documents these reference-star and anisoplanatic limitations.<NACO User Manual by European Southern Observatory>[§§3.1–3.2, printed p.13; §4.3, printed p.19: atmospheric coherence, real-time correction and anisoplanatism]{ref}
A <laser guide star> supplies a wavefront reference in fields lacking a bright suitable star, but a finite-altitude beacon samples a cone rather than the full stellar cylinder. It also does not by itself supply the absolute celestial tip-tilt reference; a natural source is still required for that measurement. Multiple <guide stars> and conjugate correction planes can improve wider-field compensation, with greater hardware and reconstruction demands. The <point spread function> can vary across the field and with time, so reference stars and telemetry are needed to calibrate <astrometry> and <photometry>; a sharper image alone is not a guarantee of accurate science.<Adaptive Optics by European Southern Observatory>[Laser guide stars: artificial reference sources, limited sky coverage and retained natural-star requirements]{ref}
Short-exposure methods provide another route. Exposures shorter than the <atmospheric coherence time> freeze <speckle patterns>. <speckle interferometry> uses calibrated Fourier statistics of many frames to recover diffraction-scale information; phase information requires methods such as <bispectrum> reconstruction, not just averaging the blurred frames. <lucky imaging> selects the best short frames, registers them and combines them. It sacrifices <photons> through selection and can be read-noise limited for faint targets, so it is most effective with enough reference <photons>. Ordinary shift-and-add mainly corrects motion; it does not remove every high-order phase error. Deconvolution using a measured <point spread function> can recover supported structure but amplifies noise and cannot reliably invent <spatial frequencies> absent from the observations. \b[Good-site engineering, fast wavefront correction and short-exposure reconstruction address different limitations and can be combined.]
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