The nature of the primary photoexcitations in semiconducting single-walled carbon nanotubes (S-SWCNTs) is of strong current interest. We have studied the emission spectra of S-SWCNTs and two different $pi$-conjugated polymers in solutions and films, and have also performed ultrafast pump-probe spectroscopy on these systems. The emission spectra relative to the absorption bands are very similar in S-SWCNTs and polymers, with redshifted photoluminescence in films showing exciton migration. The transient photoinduced absorptions (PAs) in SWCNTs and $pi$-conjugated polymers are also remarkably similar, with a low energy PA$_1$ and a higher energy PA$_2$ in all cases. Theoretical calculations of excited state absorptions within a correlated $pi$-electron Hamiltonian find the same excitonic energy spectrum for S-SWCNTs and $pi$-conjugated polymers, illustrating the universal features of quasi-one-dimensional excitons in carbon-based $pi$-conjugated systems. In both cases PA$_1$ is an excited state absorption from the optically allowed exciton to a two-photon exciton that occurs below the continuum band threshold. PA$_1$ therefore gives the lower limit of the binding energy of the lowest optical exciton. The binding energy of lowest exciton belonging to the widest S-SWCNTs with diameters $geq$ 1 nm in films is 0.3--0.4 eV, as determined by both experimental and theoretical methods.
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