The concept of using the atmospheric backscatter from a pulsed laser as an artificial guide star (AGS) for an adaptive-optics system in an astronomical telescope is analyzed. The extent to which such an AGS can be used to provide the information that is needed for adaptive-optics-system compensation of a wave front that is distorted by propagation through the atmosphere is studied. Attention is directed to the effect of focus anisoplanatism, the measurement error that is introduced when the probe light from the AGS at a finite range travels a different path than does the light from an astronomical object at a larger (infinite) range. Because of focus anisoplanatism there is a residual wave-front error that the wave-front-distortion sensor, relying on the AGS, is unable to sense and for which the adaptive-optics system is therefore unable to compensate. This residual wave-front error has a mean-square value that takes the form σφFA2 = (D/d0)5/3, where d0 is an aperture-diameter-sized quantity that measures the magnitude of the effect of focus anisoplanatism. The value of d0 depends on the vertical distribution of the optical strength of turbulence, the optical wavelength of the imaging system, the zenith angle, and the backscatter altitude. An expression is given for d0, and sample results for d0 are presented. Analytic results are also developed for the effect of focus anisoplanatism on the Strehl definition (or normalized antenna gain) of the imaging system. Numerical results are presented for the normalized antenna gain for a wide variety of backscatter altitudes and for several vertical distributions of the optical strength of turbulence. These results indicate that the diameter dependence of the normalized antenna gain is fully expressed as a function of D/d0. The peak achievable antenna gain is approximately equal to 40% of that of a diffraction-limited system with an aperture diameter that is equal to d0, and the peak is achieved with an actual aperture diameter in the range of (7/6)d0 to (9/6)d0.
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