  Accurate and precise measurements of the Hubble constant are critical for
testing our current standard cosmological model and revealing possibly new
physics. With Hubble Space Telescope (HST) imaging, each strong gravitational
lens system with measured time delays can allow one to determine the Hubble
constant with an uncertainty of $\sim 7\%$. Since HST will not last forever, we
explore adaptive-optics (AO) imaging as an alternative that can provide higher
angular resolution than HST imaging but has a less stable point spread function
(PSF) due to atmospheric distortion. To make AO imaging useful for
time-delay-lens cosmography, we develop a method to extract the unknown PSF
directly from the imaging of strongly lensed quasars. In a blind test with two
mock data sets created with different PSFs, we are able to recover the
important cosmological parameters (time-delay distance, external shear, lens
mass profile slope, and total Einstein radius). Our analysis of the Keck AO
image of the strong lens system RXJ1131-1231 shows that the important
parameters for cosmography agree with those based on HST imaging and modeling
within 1-$\sigma$ uncertainties. Most importantly, the constraint on the model
time-delay distance by using AO imaging with $0.045"$resolution is tighter by
$\sim 50\%$ than the constraint of time-delay distance by using HST imaging
with $0.09"$when a power-law mass distribution for the lens system is adopted.
Our PSF reconstruction technique is generic and applicable to data sets that
have multiple nearby point sources, enabling scientific studies that require
high-precision models of the PSF.
