  We propose a general theory on the standing waves (quasiparticle interference
pattern) caused by the scattering of surface states off step edges in
topological insulators, in which the extremal points on the constant energy
contour of surface band play the dominant role. Experimentally we image the
interference patterns on both Bi$_2$Te$_3$ and Bi$_2$Se$_3$ films by measuring
the local density of states using a scanning tunneling microscope. The observed
decay indices of the standing waves agree excellently with the theoretical
prediction: In Bi$_2$Se$_3$, only a single decay index of -3/2 exists; while in
Bi$_2$Te$_3$ with strongly warped surface band, it varies from -3/2 to -1/2 and
finally to -1 as the energy increases. The -1/2 decay indicates that the
suppression of backscattering due to time-reversal symmetry does not
necessarily lead to a spatial decay rate faster than that in the conventional
two-dimensional electron system. Our formalism can also explain the
characteristic scattering wave vectors of the standing wave caused by
non-magnetic impurities on Bi$_2$Te$_3$.
