  We develop a high-precision model for laser ranging interferometric (LRI)
observables of the GRACE Follow-On (GRACE-FO) mission. For this, we study the
propagation of an electromagnetic wave in the gravitational field in the
vicinity of an extended body, in the post-Newtonian approximation of the
general theory of relativity. We present a general relativistic model for the
phase of a plane wave that accounts for contributions of all the multipoles of
the gravitating body, its angular momentum, as well as the contribution of
tidal fields produced by external sources. We develop a new approach to model a
coherent signal transmission in the gravitational field of the solar system
that relies on a relativistic treatment of the phase. We use this approach to
describe high-precision interferometric measurements on GRACE-FO and formulate
the key LRI observables, namely the phase and phase rate of a coherent laser
link between the two spacecraft. We develop a relativistic model for the
LRI-enabled range between the two GRACE-FO spacecraft, accurate to less than 1
nm, and a high-precision model for the corresponding range rate, accurate to
better than 0.1 nm/s. We also formulate high-precision relativistic models for
the double one-way range (DOWR) and DOWR-enabled range rate observables
originally used on GRACE and now studied for interferometric measurements on
GRACE-FO. Our formulation justifies the basic assumptions behind the design of
the GRACE-FO mission and highlights the importance of achieving nearly circular
and nearly identical orbits for the GRACE-FO spacecraft.
