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diff --git a/content/know/concept/laser-rate-equations/index.pdc b/content/know/concept/laser-rate-equations/index.pdc new file mode 100644 index 0000000..d087035 --- /dev/null +++ b/content/know/concept/laser-rate-equations/index.pdc @@ -0,0 +1,330 @@ +--- +title: "Laser rate equations" +firstLetter: "L" +publishDate: 2022-03-16 +categories: +- Physics +- Optics +- Laser theory + +date: 2022-03-12T20:23:42+01:00 +draft: false +markup: pandoc +--- + +# Laser rate equations + +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 $D$: + +$$\begin{aligned} + - \mu_0 \pdv[2]{\vb{P}^{+}}{t} + &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{n^2}{c^2} \pdv[2]{\vb{E}^{+}}{t} + \\ + \pdv{\vb{P}^{+}}{t} + &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} + - \frac{i |g|^2}{\hbar} \vb{E}^{+} D + \\ + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) +\end{aligned}$$ + +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 $\vb{E}^{-} = (\vb{E}^{+})^*$. + +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 $\omega \approx \omega_0$: + +$$\begin{aligned} + \vb{E}^{+}(\vb{r}, t) + = \frac{1}{2} \vb{E}_0^{+}(\vb{r}, t) \: e^{-i \omega t} + \qquad \qquad + \vb{P}^{+}(\vb{r}, t) + = \frac{1}{2} \vb{P}_0^{+}(\vb{r}, t) \: e^{-i \omega t} +\end{aligned}$$ + +We insert this into the first MBE, +and assume that $\vb{E}_0^{+}$ and $\vb{P}_0^{+}$ +vary so slowly that their second-order derivatives are negligible, +i.e. $\pdv*[2]{\vb{E}_0^{+}\!}{t} \approx 0$ and $\pdv*[2]{\vb{P}_0^{+}\!}{t} \approx 0$, +giving: + +$$\begin{aligned} + \mu_0 \bigg( i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} \bigg) + = \nabla \cross \nabla \cross \vb{E}_0^{+} + - \frac{n^2}{c^2} \bigg( i 2 \omega \pdv{\vb{E}_0^{+}}{t} + \omega^2 \vb{E}_0^{+} \bigg) +\end{aligned}$$ + +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 +in which lasing will occur: + +$$\begin{aligned} + \nabla \cross \nabla \cross \vb{E}_0^{+} + = \frac{n^2}{c^2} \Omega^2 \vb{E}_0^{+} +\end{aligned}$$ + +For simplicity, we restrict ourselves to a single-mode laser, +where there is only one $\Omega$ and $\vb{E}_0^{+}$ to care about. +Substituting the above equation into the first MBE yields: + +$$\begin{aligned} + i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} + = \varepsilon_0 n^2 \bigg( (\Omega^2 - \omega^2) \vb{E}_0^{+} - i 2 \omega \pdv{\vb{E}_0^{+}}{t} \bigg) +\end{aligned}$$ + +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: + +$$\begin{aligned} + i 2 \pdv{\vb{P}_0^{+}}{t} + \omega \vb{P}_0^{+} + = \varepsilon_0 n^2 \bigg( 2 (\Omega - \omega) \vb{E}_0^{+} - i 2 \pdv{\vb{E}_0^{+}}{t} \bigg) +\end{aligned}$$ + +Moving on to the second MBE, +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 $\pdv*{\vb{P}_0^{+}\!}{t}$ 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 $\pdv*{\vb{P}_0^{+}\!}{t} \approx 0$, the second MBE becomes: + +$$\begin{aligned} + \vb{P}^{+} + = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 \!-\! \omega))} \vb{E}^{+} D + = \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D +\end{aligned}$$ + +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: + +$$\begin{aligned} + \gamma(\omega) + \equiv \frac{\gamma_\perp}{(\omega - \omega_0) + i \gamma_\perp} +\end{aligned}$$ + +Note that $\gamma(\omega)$ satisfies the following relation, +which will be useful to us later: + +$$\begin{aligned} + \gamma^*(\omega) - \gamma(\omega) + = \frac{\gamma_\perp (i \gamma_\perp + i \gamma_\perp)}{(\omega - \omega_0)^2 + \gamma_\perp^2} + = i 2 |\gamma(\omega)|^2 +\end{aligned}$$ + +Returning to the first MBE with $\pdv*{\vb{P}_0^{+}\!}{t} \approx 0$, +we substitute the above expression for $\vb{P}_0^{+}$: + +$$\begin{aligned} + \pdv{\vb{E}_0^{+}}{t} + &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{\omega}{2 \varepsilon_0 n^2} \vb{P}_0^{+} + \\ + &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \vb{E}_0^{+} D +\end{aligned}$$ + +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: + +$$\begin{aligned} + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) + \frac{i}{2 \hbar} + \Big( \frac{|g|^2 \gamma^*(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{-} D \cdot \vb{E}_0^{+} + - \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{+} D \cdot \vb{E}_0^{-} \Big) + \\ + &= \gamma_\parallel (D_0 - D) + \frac{i |g|^2}{2 \hbar^2 \gamma_\perp} \Big( \gamma^*(\omega) - \gamma(\omega) \Big) |\vb{E}|^2 D + \\ + &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D +\end{aligned}$$ + +This is the prototype of the first laser rate equation. +However, in order to have a practical set, +we need an equation for $|\vb{E}|^2$, +which we can obtain using the first MBE: + +$$\begin{aligned} + \pdv{|\vb{E}|^2}{t} + &= \vb{E}_0^{+} \pdv{\vb{E}_0^{-}}{t} + \vb{E}_0^{-} \pdv{\vb{E}_0^{+}}{t} + \\ + &= -i (\omega - \Omega^*) |\vb{E}|^2 - i \frac{|g|^2 \omega \gamma^*(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D + + i (\omega - \Omega) |\vb{E}|^2 + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D + \\ + &= i (\Omega^* - \Omega) |\vb{E}|^2 + + i \frac{|g|^2 \omega}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \Big(\gamma(\omega) - \gamma^*(\omega)\Big) |\vb{E}|^2 D + \\ + &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D +\end{aligned}$$ + +Where $\Im(\Omega) < 0$ represents the fact that the laser cavity is leaky. +We now have the **laser rate equations**, +although they are still in an unidiomatic form: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \pdv{|\vb{E}|^2}{t} + &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D + \\ + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D + \end{aligned} + } +\end{aligned}$$ + +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: + +$$\begin{aligned} + N_{p} + = \frac{U}{\hbar \omega} + = \frac{\varepsilon_0 n^2}{2 \hbar \omega} |\vb{E}|^2 +\end{aligned}$$ + +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. +The laser rate equations then take the following standard form: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \pdv{N_p}{t} + &= - \gamma_p N_p + G N_p N_e + \\ + \pdv{N_e}{t} + &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e + \end{aligned} + } +\end{aligned}$$ + +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 $R_\mathrm{pump}$ +are defined like so: + +$$\begin{aligned} + \gamma_p + = - 2 \Im(\Omega) + = \frac{Q}{\Re(\Omega)} + \qquad \quad + G + \equiv \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 + \qquad \quad + R_\mathrm{pump} + \equiv \gamma_\parallel D_0 +\end{aligned}$$ + +With $Q$ being the cavity mode's quality factor. +The nonlinear coupling term $G N_p N_e$ represents +[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 $t$: + +$$\begin{aligned} + 0 + &= - \gamma_p N_p + G N_p N_e + \\ + 0 + &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e +\end{aligned}$$ + +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: + +$$\begin{aligned} + N_e + = \frac{\gamma_p}{G} + \qquad \implies \qquad + \boxed{ + N_p + = \frac{1}{\gamma_p} \bigg( R_\mathrm{pump} - \frac{\gamma_e \gamma_p}{G} \bigg) + } +\end{aligned}$$ + +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 +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 $R_\mathrm{pump}$, +because all additional electrons are almost immediately +annihilated by stimulated emission. +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 $N_p$, +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. + +For this reason, the laser rate equations are typically presented +in a more empirical form, which "bookkeeps" the processes affecting $N_p$ and $N_e$. +Consider the following example: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \pdv{N_p}{t} + &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e + \\ + \pdv{N_e}{t} + &= R_\mathrm{pump} + \gamma_\mathrm{abs} N_p + - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e - G_\mathrm{stim} N_p N_e + \end{aligned} + } +\end{aligned}$$ + +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 +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 $N_e$, +so solutions can only be obtained numerically. + + + +## References +1. D. Meschede, + *Optics, light and lasers*, + Wiley. +2. L.A. Coldren, S.W. Corzine, M.L. Mašanović, + *Diode lasers and photonic integrated circuits*, 2nd edition, + Wiley. |