We report the polarized optical absorption spectra of single-walled 4 Å carbon nanotubes arrayed in the channels of an AlPO 4–5 single crystal. When the light electric field (E) is polarized parallel to the tube direction (c), the spectra display a sharp peak at 1.37 eV, with two broadbands at 2.1 and 3.1 eV. In the E⊥c configuration, the tube is nearly transparent in the measured energy region 0.5–4.1 eV. The optical dipole selection rules are discussed, and the absorption bands are assigned to the dipole transitions between the Van Hove singularities. The measured absorption spectra agreed well with the ab initio calculations of band structure based on the local density function approximation. © 2001 The American Physical Society.