A_j 
  \right) 
  \rho_{12}'
+ 
8 \imath f' \sum_j A_{j}  A_{j-1}^*
. w 
. e^{+\imath \left( k_j-k_{j-1}  \right) z - \imath \Delta t } \end{aligned}$$

$$\begin{aligned}
\partial_t w
&=&
- \gamma_1 \left( w - w_i \right)  
+  4 \imath f' \sum_j
\left[ 
  A_j^*A_{j+1}
  \rho_{12}' 
         . e^{+\imath \left( k_{j+1} -k_j \right) z - \imath \Delta t } 
-
  A_j^*A_{j-1}
  \rho_{12}'^* 
         . e^{+\imath \left( k_{j-1} -k_j \right) z + \imath \Delta t } 
\right]\end{aligned}$$

Here $\sigma$ is the number density of the atoms or molecules; $\gamma_1$, $\gamma_2$ are the population and polarization decay rates for the transition; $f'$ is the field-transition coupling constant; $g'$ is the stark shift coefficient; $\omega_j, k_j$ are the frequencies and wavevectors for the field components, $\Delta$ gives a rotating frame for $\rho_{12} \rightarrow \rho_{12}'$

Additionally, a Cauchy dispersion is applied to the propagation of the field components.

Because each of the field components has a time-history, this translates to a spectral width. In typical cases with roughly nanosecond pulse lengths, the bandwidth of each component will be tiny compared to the transition frequency, so there will be large uneventful gaps in the total spectrum. In contrast, if the pulse lengths drop to (say) roughly 100 femtoseconds, the bandwidths 