  Optical properties of single-wall carbon nanotubes (SWCNTs) for light
polarized parallel to the nanotube axis have been extensively studied, whereas
their response to light polarized perpendicular to the nanotube axis has not
been well explored. Here, by using a macroscopic film of highly aligned
single-chirality (6,5) SWCNTs, we performed a systematic polarization-dependent
optical absorption spectroscopy study. In addition to the commonly observed
angular-momentum-conserving interband absorption of parallel-polarized light,
which generates $E_{11}$ and $E_{22}$ excitons, we observed a small but
unambiguous absorption peak whose intensity is maximum for
perpendicular-polarized light. We attribute this feature to the lowest-energy
cross-polarized interband absorption processes that change the angular momentum
along the nanotube axis by $\pm 1$, generating $E_{12}$ and $E_{21}$ excitons.
The energy difference between the $E_{12}$ and $E_{21}$ exciton peaks, expected
from asymmetry between the conduction and valence bands, was smaller than the
observed linewidth. Unlike previous observations of cross-polarized excitons in
polarization-dependent photoluminescence and circular dichroism spectroscopy
experiments, our direct observation using absorption spectroscopy allowed us to
quantitatively analyze this resonance. Specifically, we determined the energy
and oscillator strength of this resonance to be 1.54 and 0.05, respectively,
compared with the values for the $E_{11}$ exciton peak. These values, in
combination with comparison with theoretical calculations, in turn led to an
assessment of the environmental effect on the strength of Coulomb interactions
in this aligned single-chirality SWCNT film.
