  We propose and demonstrate a new scheme for atom interferometry, using light
pulses inside an optical cavity as matter wave beamsplitters. The cavity
provides power enhancement, spatial filtering, and a precise beam geometry,
enabling new techniques such as low power beamsplitters ($<100 \mathrm{\mu
W}$), large momentum transfer beamsplitters with modest power, or new
self-aligned interferometer geometries utilizing the transverse modes of the
optical cavity. As a first demonstration, we obtain Ramsey-Raman fringes with
$>75\%$ contrast and measure the acceleration due to gravity, $\mathit{g}$, to
$60 \mathrm{\mu \mathit{g} / \sqrt{Hz}}$ resolution in a Mach-Zehnder geometry.
We use $>10^7$ cesium atoms in the compact mode volume ($600 \mathrm{\mu m}$
$1/e^2$ waist) of the cavity and show trapping of atoms in higher transverse
modes. This work paves the way toward compact, high sensitivity, multi-axis
interferometry.
