diff options
Diffstat (limited to 'source/know/concept/laser-rate-equations/index.md')
| -rw-r--r-- | source/know/concept/laser-rate-equations/index.md | 14 |
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 @@ $$\begin{aligned} \boxed{ \begin{aligned} \pdv{N_p}{t} - &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e + &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e \\ \pdv{N_e}{t} &= R_\mathrm{pump} + \gamma_\mathrm{abs} N_p - - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e - G_\mathrm{stim} N_p N_e + - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e + - G_\mathrm{stim} N_p N_e \end{aligned} } \end{aligned}$$ |