A pattern of localization, called inhomogeneous localization, is found
for dipolar eigenmodes (surface plasmons or eigenstates of the corres
ponding Schrodinger equation) of fractal clusters. At any given freque
ncy, individual eigenmodes are dramatically different from each other,
their sizes vary in a wide range, and their internal geometry may be
topologically disconnected and singular at the small scale. These prop
erties differ principally from the results reported for vibrational ei
genmodes of fractals, which is attributed to the long-range interactio
n and non-Goldstonian nature of the polar modes.