Self-induced transparency of excitons and the dispersion law of steady-state exciton-photon pulses

We determine the dispersion law of the carrier wave of an exciton-photon pulse which propagates as a solitary wave. For short pulses, four branches $\Omega =\Omega$ ($\overrightarrow{\mathrm{K}}$) appear: a branch which goes through the intersection point $\hslash cK=E_o+\frac{{\hslash }^2{K}^2}{2M}$ of the uncoupled modes of the free exciton and the free light field, two polaritonlike branches and an excitonlike branch. The polaritonlike branches have a normal dispersion, whereas the branch which goes through the intersection point of the free modes has an anomalous dispersion for sufficiently short pulse width. This branch contains the solution of McCall and Hahn as a special case. For very short pulses the branch with anomalous dispersion continues into the two polaritonlike branches. For increasing pulse width a forbidden energy gap appears at $\overrightarrow{\mathrm{K}}=0\mathrm{}$. For long pulses, only the two polaritonlike solutions exist.