and $c_2$ vary only slowly compared to the exponential terms.

[*Note: Separate from the oscillations that occur in my coupling parameters (but not in HPB), there is the issue of [*frequency dependence*]{} which applies to both HPB and my parameters.*]{}

For both HPB (their eqn.(15)), and my field-carrier based derivation (eqn.(\[eqn-single-A-alpha\])), the presence of the field carrier in the denominator is helpful. Since it can reasonably be assumed to be much greater than the inter(Raman)-level energy differences, the fractional terms will be correspondingly small, and so the $\alpha_{nj}$ ($\alpha_{nj}'$) parameters can be assumed to be independent of frequency (or at least nearly so).

This is not in any way obviously (or even partially) true for the no-carrier $E$ based (eqn.(\[eqn-single-E-alpha\]), $\alpha_{nj}$) parameters, which depend only on the difference in Raman levels – at best you might perhaps hope that the denominators were approximate multiples of each other.

### Near-zero difference

Here I assume that the forward and backward transitions (between levels 2 and 1) have nearly the same amplitude. It is not self-evidently true, but HPB must have made an equivalent assumption. In any case, since $\alpha_{12}'^* - \alpha_{21}'$ is the difference of similar terms, it will be smaller by 