Angular correlation of Zernike polynomials for a laser guide star in adaptive optics

The residual wave-front error after correction by an adaptive optics system (AOS) with a laser guide star (LGS) as reference for the wave-front sensing is computed by using the wave-front expansion on the Zernike polynomials. The polynomial angular correlations are calculated for a spherical wave front coming from the LGS used to compensate for a plane wave front coming from a natural star and also for two arbitrary spherical wave fronts. The plot of decorrelation versus angular distance between the natural star and the LGS is much steeper for the sodium LGS than for the Rayleigh LGS. A $D/n$ dependence is found for the 50% correlation angle, where $D$ is the telescope diameter and $n$ is the polynomial radial degree. For focal anisoplanatism a modal $d_0$ parameter $[d_0^{\prime }(J)]$ is computed versus $J,$ the number of corrected polynomials. The Rayleigh LGS has the capacity for large-field-of-view observations but with poor image quality and a low-order AOS, while the sodium LGS is suitable for high-image-quality observations with a high-order AOS but at the price of a small field of view. The isoplanatic angle is computed for different $D$ and observation wavelengths. For $D$ $=8 \mathrm{m}$ and a sodium LGS, the wavelength must be larger than 1.6 μm in order to produce a $\mathrm{\lambda }/5$ wave-front error.