  Large scale atom interferometers promise unrivaled strain sensitivity to
midband (0.1 - 10 Hz) gravitational waves, and will probe a new parameter space
in the search for ultra-light scalar dark matter. These atom interferometers
require a momentum separation above 10^4 \hbar k between interferometer arms in
order to reach the target sensitivity. Prohibitively high optical intensity and
wavefront flatness requirements have thus far limited the maximum achievable
momentum splitting. We propose a scheme for optical cavity enhanced atom
interferometry, using circulating, spatially resolved pulses, and intracavity
frequency modulation to overcome these limitations and reach 10^4 \hbar k
momentum separation. We present parameters suitable for the experimental
realization of 10^4 \hbar k splitting in a 1 km interferometer using the 698 nm
clock transition in 87Sr, and describe performance enhancements in 10 m scale
devices operating on the 689 nm intercombination line in 87Sr. Although
technically challenging to implement, the laser and cloud requirements are
within the reach of upcoming cold-atom based interferometers. Our scheme
satisfies the most challenging requirements of these sensors and paves the way
for the next generation of high sensitivity, large momentum transfer atom
interferometers.
