  This paper addresses a factorization method for imaging the support of a
wave-number-dependent source function from multi-frequency data measured at a
finite pair of symmetric receivers in opposite directions. The source function
is given by the inverse Fourier transform of a compactly supported
time-dependent source whose initial moment or terminal moment for radiating is
unknown. Using the multi-frequency far-field data at two opposite observation
directions, we provide a computational criterion for characterizing the
smallest strip containing the support and perpendicular to the directions. A
new parameter is incorporated into the design of test functions for indicating
the unknown moment. The data from a finite pair of opposite directions can be
used to recover the $\Theta$-convex polygon of the support. Uniqueness in
recovering the convex hull of the support is obtained as a by-product of our
analysis using all observation directions. Similar results are also discussed
with the multi-frequency near-field data from a finite pair of observation
positions in three dimensions. We further comment on possible extensions to
source functions with two disconnected supports. Extensive numerical tests in
both two and three dimensions are implemented to show effectiveness and
feasibility of the approach. The theoretical framework explored here should be
seen as the frequency-domain analysis for inverse source problems in the time
domain.
