  Large-momentum-transfer~(LMT) atom interferometers using elastic Bragg
scattering on light waves are among the most precise quantum sensors to date.
To advance their accuracy from the mrad to the $\mu$rad regime, it is necessary
to understand the rich phenomenology of the Bragg interferometer, which differs
significantly from that of a standard two-mode interferometer. We develop an
analytic model for the interferometer signal and demonstrate its accuracy using
comprehensive numerical simulations. Our analytic treatment allows the
determination of the atomic projection noise limit of an LMT Bragg
interferometer, and provides the means to saturate this limit. It affords
accurate knowledge of the systematic phase errors as well as their suppression
by two orders of magnitude down to a few $\mu\mathrm{rad}$ using appropriate
light pulse parameters.
