Study of the average short-exposure modulation transfer function in the presence of atmospheric turbulence

The ensemble average of the transfer function of an optical system that corrects wavefront tilt in real time is termed the average short-exposure modulation transfer function (SEMTF). Most of the previous studies regarding this topic neglect the influences of type of tilt. The difference between Zernike tilt (Z-tilt) and gradient tilt (G-tilt) is introduced in SEMTF. Fried-type analytical but less accurate forms that assume tilt is uncorrelated with residual phase and numerical but accurate double-integral forms that discard the above assumption have been derived. The double-integral forms can achieve both high accuracy and rapid computation speed without neglecting the location dependency of the tilt-corrected phase structure function. For the sake of generality, telescope aperture is set to be annular, and circular formulae can be obtained by setting the obstruction ratio to be zero.