  We study the internal structure of a non-Abelian vortex in color
superconductivity with respect to quark degrees of freedom. Stable non-Abelian
vortices appear in the Color-Flavor-Locked phase whose symmetry SU(3)_{c+L+R}
is further broken to SU(2)_{c+L+R} x U(1)_{c+L+R} at the vortex cores.
Microscopic structure of vortices at scales shorter than the coherence length
can be analyzed by the Bogoliubov-de Gennes (B-dG) equation (rather than the
Ginzburg-Landau equation). We obtain quark spectra from the B-dG equation by
treating the diquark gap having the vortex configuration as a background field.
We find that there are massless modes (zero modes) well-localized around a
vortex, in the triplet and singlet states of the unbroken symmetry
SU(2)_{c+L+R} x U(1)_{c+L+R}. The velocities v_i of the massless modes (i=t,s
for triplet and singlet) change at finite chemical potential \mu, and decrease
as \mu becomes large. Therefore, low energy excitations in the vicinity of the
vortices are effectively described by 1+1 dimensional massless fermions whose
velocities are reduced v_i<1.
