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diff --git a/source/know/concept/laser-rate-equations/index.md b/source/know/concept/laser-rate-equations/index.md index 939a1a0..a84d274 100644 --- a/source/know/concept/laser-rate-equations/index.md +++ b/source/know/concept/laser-rate-equations/index.md @@ -12,9 +12,9 @@ layout: "concept" The [Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/) (MBEs) give a fundamental description of light-matter interaction for a two-level quantum system for the purposes of laser theory. -They govern the [electric field](/know/concept/electric-field/) $\vb{E}^{+}$, -the induced polarization $\vb{P}^{+}$, -and the total population inversion $D$: +They govern the [electric field](/know/concept/electric-field/) $$\vb{E}^{+}$$, +the induced polarization $$\vb{P}^{+}$$, +and the total population inversion $$D$$: $$\begin{aligned} - \mu_0 \pdvn{2}{\vb{P}^{+}}{t} @@ -28,16 +28,16 @@ $$\begin{aligned} &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) \end{aligned}$$ -Where $n$ is the background medium's refractive index, -$\omega_0$ the two-level system's gap resonance frequency, -$|g| \equiv |\matrixel{e}{\vu{x}}{g}|$ the transition dipole moment, -$\gamma_\perp$ and $\gamma_\parallel$ empirical decay rates, -and $D_0$ the equilibrium inversion. -Note that $\vb{E}^{-} = (\vb{E}^{+})^*$. +Where $$n$$ is the background medium's refractive index, +$$\omega_0$$ the two-level system's gap resonance frequency, +$$|g| \equiv |\matrixel{e}{\vu{x}}{g}|$$ the transition dipole moment, +$$\gamma_\perp$$ and $$\gamma_\parallel$$ empirical decay rates, +and $$D_0$$ the equilibrium inversion. +Note that $$\vb{E}^{-} = (\vb{E}^{+})^*$$. Let us make the following ansatz, -where $\vb{E}_0^{+}$ and $\vb{P}_0^{+}$ are slowly-varying envelopes -of a plane wave with angular frequency $\omega \approx \omega_0$: +where $$\vb{E}_0^{+}$$ and $$\vb{P}_0^{+}$$ are slowly-varying envelopes +of a plane wave with angular frequency $$\omega \approx \omega_0$$: $$\begin{aligned} \vb{E}^{+}(\vb{r}, t) @@ -48,9 +48,9 @@ $$\begin{aligned} \end{aligned}$$ We insert this into the first MBE, -and assume that $\vb{E}_0^{+}$ and $\vb{P}_0^{+}$ +and assume that $$\vb{E}_0^{+}$$ and $$\vb{P}_0^{+}$$ vary so slowly that their second-order derivatives are negligible, -i.e. $\ipdvn{2}{\vb{E}_0^{+}\!}{t} \approx 0$ and $\ipdvn{2}{\vb{P}_0^{+}\!}{t} \approx 0$, +i.e. $$\ipdvn{2}{\vb{E}_0^{+}\!}{t} \approx 0$$ and $$\ipdvn{2}{\vb{P}_0^{+}\!}{t} \approx 0$$, giving: $$\begin{aligned} @@ -62,7 +62,7 @@ $$\begin{aligned} To get rid of the double curl, consider the time-independent [electromagnetic wave equation](/know/concept/electromagnetic-wave-equation/), -where $\Omega$ is an eigenfrequency of the optical cavity +where $$\Omega$$ is an eigenfrequency of the optical cavity in which lasing will occur: $$\begin{aligned} @@ -71,7 +71,7 @@ $$\begin{aligned} \end{aligned}$$ For simplicity, we restrict ourselves to a single-mode laser, -where there is only one $\Omega$ and $\vb{E}_0^{+}$ to care about. +where there is only one $$\Omega$$ and $$\vb{E}_0^{+}$$ to care about. Substituting the above equation into the first MBE yields: $$\begin{aligned} @@ -79,9 +79,9 @@ $$\begin{aligned} = \varepsilon_0 n^2 \bigg( (\Omega^2 - \omega^2) \vb{E}_0^{+} - i 2 \omega \pdv{\vb{E}_0^{+}}{t} \bigg) \end{aligned}$$ -Where we used $1 / c^2 = \mu_0 \varepsilon_0$. -Assuming the light is more or less on-resonance $\omega \approx \Omega$, -we can approximate $\Omega^2 \!-\! \omega^2 \approx 2 \omega (\Omega \!-\! \omega)$, so: +Where we used $$1 / c^2 = \mu_0 \varepsilon_0$$. +Assuming the light is more or less on-resonance $$\omega \approx \Omega$$, +we can approximate $$\Omega^2 \!-\! \omega^2 \approx 2 \omega (\Omega \!-\! \omega)$$, so: $$\begin{aligned} i 2 \pdv{\vb{P}_0^{+}}{t} + \omega \vb{P}_0^{+} @@ -89,18 +89,18 @@ $$\begin{aligned} \end{aligned}$$ Moving on to the second MBE, -inserting the ansatz $\vb{P}^{+} = \vb{P}_0^{+} e^{-i \omega t} / 2$ leads to: +inserting the ansatz $$\vb{P}^{+} = \vb{P}_0^{+} e^{-i \omega t} / 2$$ leads to: $$\begin{aligned} \pdv{\vb{P}_0^{+}}{t} = - \Big( \gamma_\perp + i (\omega_0 - \omega) \Big) \vb{P}_0^{+} - \frac{i |g|^2}{\hbar} \vb{E}_0^{+} D \end{aligned}$$ -Typically, $\gamma_\perp$ is much larger than the rate of any other decay process, -in which case $\ipdv{}{\vb{P}0^{+}\!}{t}$ is negligible compared to $\gamma_\perp \vb{P}_0^{+}$. -Effectively, this means that the polarization $\vb{P}_0^{+}$ -near-instantly follows the electric field $\vb{E}^{+}\!$. -Setting $\ipdv{}{\vb{P}0^{+}\!}{t} \approx 0$, the second MBE becomes: +Typically, $$\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^{+}$$. +Effectively, this means that the polarization $$\vb{P}_0^{+}$$ +near-instantly follows the electric field $$\vb{E}^{+}\!$$. +Setting $$\ipdv{}{\vb{P}0^{+}\!}{t} \approx 0$$, the second MBE becomes: $$\begin{aligned} \vb{P}^{+} @@ -108,7 +108,7 @@ $$\begin{aligned} = \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D \end{aligned}$$ -Where the Lorentzian gain curve $\gamma(\omega)$ +Where the Lorentzian gain curve $$\gamma(\omega)$$ (which also appears in the [SALT equation](/know/concept/salt-equation/)) represents a laser's preferred spectrum for amplification, and is defined like so: @@ -118,7 +118,7 @@ $$\begin{aligned} \equiv \frac{\gamma_\perp}{(\omega - \omega_0) + i \gamma_\perp} \end{aligned}$$ -Note that $\gamma(\omega)$ satisfies the following relation, +Note that $$\gamma(\omega)$$ satisfies the following relation, which will be useful to us later: $$\begin{aligned} @@ -127,8 +127,8 @@ $$\begin{aligned} = i 2 |\gamma(\omega)|^2 \end{aligned}$$ -Returning to the first MBE with $\ipdv{\vb{P}_0^{+}}{t} \approx 0$, -we substitute the above expression for $\vb{P}_0^{+}$: +Returning to the first MBE with $$\ipdv{\vb{P}_0^{+}}{t} \approx 0$$, +we substitute the above expression for $$\vb{P}_0^{+}$$: $$\begin{aligned} \pdv{\vb{E}_0^{+}}{t} @@ -137,10 +137,10 @@ $$\begin{aligned} &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \vb{E}_0^{+} D \end{aligned}$$ -Next, we insert our ansatz for $\vb{E}^{+}\!$ and $\vb{P}^{+}\!$ -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: +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: $$\begin{aligned} \pdv{D}{t} @@ -155,7 +155,7 @@ $$\begin{aligned} This is the prototype of the first laser rate equation. However, in order to have a practical set, -we need an equation for $|\vb{E}|^2$, +we need an equation for $$|\vb{E}|^2$$, which we can obtain using the first MBE: $$\begin{aligned} @@ -171,7 +171,7 @@ $$\begin{aligned} &= 2 \Imag(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D \end{aligned}$$ -Where $\Imag(\Omega) < 0$ represents the fact that the laser cavity is leaky. +Where $$\Imag(\Omega) < 0$$ represents the fact that the laser cavity is leaky. We now have the **laser rate equations**, although they are still in an unidiomatic form: @@ -187,8 +187,8 @@ $$\begin{aligned} } \end{aligned}$$ -To rewrite this, we replace $|\vb{E}|^2$ with the photon number $N_p$ as follows, -with $U = \varepsilon_0 n^2 |\vb{E}|^2 / 2$ being the energy density of the light: +To rewrite this, we replace $$|\vb{E}|^2$$ with the photon number $$N_p$$ as follows, +with $$U = \varepsilon_0 n^2 |\vb{E}|^2 / 2$$ being the energy density of the light: $$\begin{aligned} N_{p} @@ -196,12 +196,12 @@ $$\begin{aligned} = \frac{\varepsilon_0 n^2}{2 \hbar \omega} |\vb{E}|^2 \end{aligned}$$ -Furthermore, consider the definition of the inversion $D$: +Furthermore, consider the definition of the inversion $$D$$: because a photon emission annihilates an electron-hole pair, -it reduces $D$ by $2$. -Since lasing is only possible for $D > 0$, -we can replace $D$ with the conduction band's electron population $N_e$, -which is reduced by $1$ whenever a photon is emitted. +it reduces $$D$$ by $$2$$. +Since lasing is only possible for $$D > 0$$, +we can replace $$D$$ with the conduction band's electron population $$N_e$$, +which is reduced by $$1$$ whenever a photon is emitted. The laser rate equations then take the following standard form: $$\begin{aligned} @@ -216,10 +216,10 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\gamma_e$ is a redefinition of $\gamma_\parallel$ +Where $$\gamma_e$$ is a redefinition of $$\gamma_\parallel$$ depending on the electron decay processes, -and the photon loss rate $\gamma_p$, the gain $G$, -and the carrier supply rate $R_\mathrm{pump}$ +and the photon loss rate $$\gamma_p$$, the gain $$G$$, +and the carrier supply rate $$R_\mathrm{pump}$$ are defined like so: $$\begin{aligned} @@ -234,14 +234,14 @@ $$\begin{aligned} \equiv \gamma_\parallel D_0 \end{aligned}$$ -With $Q$ being the cavity mode's quality factor. -The nonlinear coupling term $G N_p N_e$ represents +With $$Q$$ being the cavity mode's quality factor. +The nonlinear coupling term $$G N_p N_e$$ represents [stimulated emission](/know/concept/einstein-coefficients/), which is the key to lasing. To understand the behaviour of a laser, consider these equations in a steady state, -i.e. where $N_p$ and $N_e$ are constant in $t$: +i.e. where $$N_p$$ and $$N_e$$ are constant in $$t$$: $$\begin{aligned} 0 @@ -251,9 +251,9 @@ $$\begin{aligned} &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e \end{aligned}$$ -In addition to the trivial solution $N_p = 0$, -we can also have $N_p > 0$. -Isolating $N_p$'s equation for $N_e$ and inserting that into $N_e$'s equation, we find: +In addition to the trivial solution $$N_p = 0$$, +we can also have $$N_p > 0$$. +Isolating $$N_p$$'s equation for $$N_e$$ and inserting that into $$N_e$$'s equation, we find: $$\begin{aligned} N_e @@ -265,27 +265,27 @@ $$\begin{aligned} } \end{aligned}$$ -The quantity $R_\mathrm{thr} \equiv \gamma_e \gamma_p / G$ is called the **lasing threshold**: -if $R_\mathrm{pump} \ge R_\mathrm{thr}$, the laser is active, -meaning that $N_p$ is big enough to cause +The quantity $$R_\mathrm{thr} \equiv \gamma_e \gamma_p / G$$ is called the **lasing threshold**: +if $$R_\mathrm{pump} \ge R_\mathrm{thr}$$, the laser is active, +meaning that $$N_p$$ is big enough to cause a "chain reaction" of stimulated emission that consumes all surplus carriers to maintain a steady state. -The point is that $N_e$ is independent of the electron supply $R_\mathrm{pump}$, +The point is that $$N_e$$ is independent of the electron supply $$R_\mathrm{pump}$$, because all additional electrons are almost immediately annihilated by stimulated emission. -Consequently $N_p$ increases linearly as $R_\mathrm{pump}$ is raised, +Consequently $$N_p$$ increases linearly as $$R_\mathrm{pump}$$ is raised, at a much steeper slope than would be possible below threshold. -The output of the cavity is proportional to $N_p$, +The output of the cavity is proportional to $$N_p$$, so the brightness is also linear. Unfortunately, by deriving the laser rate equations from the MBEs, we lost some interesting and important effects, most notably spontaneous emission, -which is needed for $N_p$ to grow if $R_\mathrm{pump}$ is below threshold. +which is needed for $$N_p$$ to grow if $$R_\mathrm{pump}$$ is below threshold. For this reason, the laser rate equations are typically presented -in a more empirical form, which "bookkeeps" the processes affecting $N_p$ and $N_e$. +in a more empirical form, which "bookkeeps" the processes affecting $$N_p$$ and $$N_e$$. Consider the following example: $$\begin{aligned} @@ -301,17 +301,17 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\gamma_\mathrm{out}$ represents the cavity's usable output, -$\gamma_\mathrm{abs}$ the medium's absorption, -$\gamma_\mathrm{loss}$ scattering losses, -$\gamma_\mathrm{spon}$ spontaneous emission, -$\gamma_\mathrm{n.r.}$ non-radiative electron-hole recombination, -and $\gamma_\mathrm{leak}$ the fact that +Where $$\gamma_\mathrm{out}$$ represents the cavity's usable output, +$$\gamma_\mathrm{abs}$$ the medium's absorption, +$$\gamma_\mathrm{loss}$$ scattering losses, +$$\gamma_\mathrm{spon}$$ spontaneous emission, +$$\gamma_\mathrm{n.r.}$$ non-radiative electron-hole recombination, +and $$\gamma_\mathrm{leak}$$ the fact that some carriers leak away before they can be used for emission. Unsurprisingly, this form is much harder to analyze, but more accurately describes the dynamics inside a laser. -To make matters even worse, many of these decay rates depend on $N_p$ or $N_e$, +To make matters even worse, many of these decay rates depend on $$N_p$$ or $$N_e$$, so solutions can only be obtained numerically. |