Resolution Limits with Propagation Phase Errors

Resolution limits and corresponding optimum linear apertures are determined in the presence of phase errors. Let α(t) be the phase aberration at position t across the aperture; it is assumed that the random process α has a power law structure function, E{[(α(t)-α(τ)]2}= c|t-τ|n. Beam tilting caused by the phase error is "removed" (for each sample of α), then resolution formulas are developed. An approximate analysis is obtained in closed form and yields an optimum resolution proportional to c1/n for O < n < 2. The exact analysis is given for Gaussian α, and again the optimum resolution is proportional to c1/n. In applications n= 5/3 is of interest, and in the Gaussian case the best obtainable equivalent rectangle resolution is ρ λ)/2π (0.975)c3/5 radians with a corresponding optimum linear aperture of 14c-3/5. When long exposures are considered, imaging without removing beam tilting is of interest, and resolution is degraded by a factor of about 2.5 for a linear aperture. Alternatively, in some applications optimum focus as well as beam tilt should be considered, in this case resolution is improved by a factor of about 1.4 (again for n= 5/3). Finally, joint (tilt corrected) optimization over aperture length and taper is treated; however, as one might expect, the use of taper offers negligible resolution improvement.