Focus anisoplanatism effects on tip–tilt compensation for adaptive optics with use of a sodium laser beacon as a tracking reference

A Zernike decomposition of the pupil phase variance that is due to focus anisoplanatism for a sodium laser guide star is calculated, including the angular error variance associated with overall tilt. This quantity, or tilt focus anisoplanatism ${\sigma }_{\text{FA},tt}^2$, is necessary in evaluating the usefulness of new techniques developed to measure the wave-front tilt from a sodium laser guide star. The analytical expression of ${\sigma }_{\text{FA},tt}^2$ in the case of refractive-index fluctuation with a Kolmogorov power spectral density is given together with numerical results for two different $C_{n^{\prime }}^2(z)$ profiles. Numerical calculations show that tilt focus anisoplanatism may be a limitation when a sodium laser guide star is used to sense the overall wave-front tilt at visible wavelengths, especially for 8-m-class telescopes.