Turbulence has long been recognized as one of the most significant factors limiting the performance of optical systems operating in the presence of atmosphere. Atmospheric turbulence over vertical paths has been well characterized, both theoretically and experimentally. Much less is known about turbulence over long, horizontal paths. Perturbations of the wave-front phase can be measured with a Hartmann wave-front sensor (H-WFS). One can use these measurements to characterize atmospheric turbulence directly. Theoretical expressions for the slope structure function of the H-WFS measurements are derived and evaluated with the use of numerical quadrature. By concentrating on the slope structure function, we avoid the phase reconstruction step and use the slope measurements in a more direct fashion. The theoretical slope structure function is compared with estimated slope structure functions computed from H-WFS measurements collected in a series of experiments conducted by researchers at the U.S. Air Force’s Phillips Laboratory. These experiments involved H-WFS measurements over high-altitude (airborne) horizontal paths 20–200 km in length.
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