  We present a new method, fan-beam modulation, for observing weak extended
x-ray sources with the Reuven Ramaty High-Energy Solar Spectroscopic Imager
(RHESSI). This space-based solar x-ray and gamma-ray telescope has much greater
sensitivity than previous experiments in the 3-25 keV range, but is normally
not well suited to detecting extended sources since their signal is not
modulated by RHESSI's rotating grids. When the spacecraft is offpointed from
the target source, however, the fan-beam modulation time-modulates the
transmission by shadowing resulting from exploiting the finite thickness of the
grids. In this paper we detail how the technique is implemented and verify its
consistency with sources with clear known signals that have occurred during
RHESSI offpointing: microflares and the Crab Nebula. In both cases the results
are consistent with previous and complementary measurements. Preliminary work
indicates that this new technique allows RHESSI to observe the integrated hard
x-ray spectrum of weak extended sources on the quiet Sun.
