  Determination of optical absorption cross-section is always among the central
importance of understanding a material. However its realization on individual
nanostructures, such as carbon nanotubes, is experimentally challenging due to
the small extinction signal using conventional transmission measurements. Here
we develop a technique based on polarization manipulation to enhance the
sensitivity of single-nanotube absorption spectroscopy by two-orders of
magnitude. We systematically determine absorption cross-section over broad
spectral range at single-tube level for more than 50 chirality-defined
single-walled nanotubes. Our data reveals chirality-dependent one-dimensional
photo-physics through the behaviours of exciton oscillator strength and
lifetime. We also establish an empirical formula to predict absorption spectrum
of any nanotube, which provides the foundation to determine quantum
efficiencies in important photoluminescence and photovoltaic processes.
