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-rw-r--r--source/know/concept/laser-rate-equations/index.md14
1 files changed, 8 insertions, 6 deletions
diff --git a/source/know/concept/laser-rate-equations/index.md b/source/know/concept/laser-rate-equations/index.md
index 1f42f73..c81f02b 100644
--- a/source/know/concept/laser-rate-equations/index.md
+++ b/source/know/concept/laser-rate-equations/index.md
@@ -97,14 +97,14 @@ $$\begin{aligned}
\end{aligned}$$
Typically, $$\gamma_\perp$$ is much larger than the rate of any other decay process,
-in which case $$\ipdv{}{\vb{P}0^{+}\!}{t}$$ is negligible compared to $$\gamma_\perp \vb{P}_0^{+}$$.
+in which case $$\ipdv{\vb{P}_0^{+}\!}{t}$$ is negligible compared to $$\gamma_\perp \vb{P}_0^{+}$$.
Effectively, this means that the polarization $$\vb{P}_0^{+}$$
near-instantly follows the electric field $$\vb{E}^{+}\!$$.
-Setting $$\ipdv{}{\vb{P}0^{+}\!}{t} \approx 0$$, the second MBE becomes:
+Setting $$\ipdv{\vb{P}_0^{+}\!}{t} \approx 0$$, the second MBE becomes:
$$\begin{aligned}
\vb{P}^{+}
- = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 \!-\! \omega))} \vb{E}^{+} D
+ = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 - \omega))} \vb{E}^{+} D
= \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D
\end{aligned}$$
@@ -137,7 +137,7 @@ $$\begin{aligned}
&= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \vb{E}_0^{+} D
\end{aligned}$$
-Next, we insert our ansatz for $$\vb{E}^{+}\!$$ and $$\vb{P}^{+}\!$$
+Next, we insert our ansatz for $$\vb{E}^{+}$$ and $$\vb{P}^{+}$$
into the third MBE, and rewrite $$\vb{P}_0^{+}$$ as above.
Using our identity for $$\gamma(\omega)$$,
and the fact that $$\vb{E}_0^{+} \cdot \vb{E}_0^{-} = |\vb{E}|^2$$, we find:
@@ -293,11 +293,13 @@ $$\begin{aligned}
\boxed{
\begin{aligned}
\pdv{N_p}{t}
- &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e
+ &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p
+ + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e
\\
\pdv{N_e}{t}
&= R_\mathrm{pump} + \gamma_\mathrm{abs} N_p
- - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e - G_\mathrm{stim} N_p N_e
+ - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e
+ - G_\mathrm{stim} N_p N_e
\end{aligned}
}
\end{aligned}$$