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I am stuck on what I assume is a very basic rearranging of terms in Siegman's Lasers, Page 204. Here, the saturation of a laser medium is introduced. The change of the populations of two energy levels are given as

$$\frac{\text{d}N_1}{\text{d}t}=-\frac{\text{d}N_2}{\text{d}t}=-(W_{12}+w_{12})N_1+(W_{21}+w_{21})N_2$$

with $N=N_1+N_2$ and $\Delta N=N_1-N_2$

Now, as for the rearranging of terms:

The equations for $\text{d}N_1/\text{d}t$ and $\text{d}N_2/\text{d}t$ can be combined into a single rate equation

$$\frac{\text{d}}{\text{d}t}\Delta N=-(W_{12}+W_{21})\Delta N-(w_{12}+w_{21})\bigg(\Delta N-\frac{w_{21}-w_{12}}{w_{12}+w_{21}}N\bigg)$$

which is, if I didn't get anything wrong, just $$=N_1(-W_{12}-W_{21}-2w_{12})+N_2(W_{12}+W_{21}+2w_{21})$$

But with the information given in this section of the book, I just can't get it to agree with

$$\frac{\text{d}}{\text{d}t}\Delta N=\frac{\text{d}N_1}{\text{d}t}-\frac{\text{d}N_2}{\text{d}t}=2\cdot \frac{\text{d}N_1}{\text{d}t}=2\cdot(-\Delta N(W_{12}+W_{21}+w_{12}+w_{21}))$$

I hope I'm just stuck on something trivial.

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The thing I missed is the following trivial, yet important fact: Because the upwards- and downwards stimulated transition probabilities are the same $W_{12}\equiv W_{21}$,

\begin{align} \frac{\text{d}\Delta N}{\text{d}t}&=-(W_{12}+W_{21})\Delta N-(w_{12}+w_{21})\bigg(\Delta N-\frac{w_{21}-w_{12}}{w_{12}+w_{21}}N\bigg)\\ &=N_1(-W_{12}-W_{21}-2w_{12})+N_2(W_{12}+W_{21}+2w_{21}) \end{align}

becomes the expression given by Siegman

\begin{align} \frac{\text{d}N_1}{\text{d}t}-\frac{\text{d}N_2}{\text{d}t}&=2\bigg[-\big(W_{12}+w_{12}\big)N_1+\big(W_{21}+w_{21}\big)N_2\bigg]\\ \end{align}

From there on, the population recovery time $T_1$ can be introduced, along with the saturation.

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