From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/salt-equation/index.md | 92 +++++++++++++++--------------- 1 file changed, 46 insertions(+), 46 deletions(-) (limited to 'source/know/concept/salt-equation') diff --git a/source/know/concept/salt-equation/index.md b/source/know/concept/salt-equation/index.md index bfd69b9..f5f085d 100644 --- a/source/know/concept/salt-equation/index.md +++ b/source/know/concept/salt-equation/index.md @@ -15,11 +15,11 @@ makes it especially appropriate for microscopically small lasers. Consider the [Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/), governing the complex polarization -vector $\vb{P}^{+}$ and the scalar population inversion $D$ of a set of +vector $$\vb{P}^{+}$$ and the scalar population inversion $$D$$ of a set of active atoms (or quantum dots) embedded in a passive linear background -material with refractive index $c / v$. +material with refractive index $$c / v$$. The system is affected by a driving [electric field](/know/concept/electric-field/) -$\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$, +$$\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$$, such that the set of equations is: $$\begin{aligned} @@ -34,13 +34,13 @@ $$\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}$$ -Where $\hbar \omega_0$ is the band gap of the active atoms, -and $\gamma_\perp$ and $\gamma_\parallel$ are relaxation rates +Where $$\hbar \omega_0$$ is the band gap of the active atoms, +and $$\gamma_\perp$$ and $$\gamma_\parallel$$ are relaxation rates of the atoms' polarization and population inversion, respectively. -$D_0$ is the equilibrium inversion, i.e. the value of $D$ if there is no lasing. -Note that $D_0$ also represents the pump, -and both $D_0$ and $v$ depend on position $\vb{x}$. -Finally, the transition dipole matrix elements $\vb{p}_0^{-}$ and $\vb{p}_0^{+}$ are given by: +$$D_0$$ is the equilibrium inversion, i.e. the value of $$D$$ if there is no lasing. +Note that $$D_0$$ also represents the pump, +and both $$D_0$$ and $$v$$ depend on position $$\vb{x}$$. +Finally, the transition dipole matrix elements $$\vb{p}_0^{-}$$ and $$\vb{p}_0^{+}$$ are given by: $$\begin{aligned} \vb{p}_0^{-} @@ -51,12 +51,12 @@ $$\begin{aligned} = (\vb{p}_0^{-})^* \end{aligned}$$ -With $q < 0$ the electron charge, $\vu{x}$ the quantum position operator, -and $\Ket{g}$ and $\Ket{e}$ respectively +With $$q < 0$$ the electron charge, $$\vu{x}$$ the quantum position operator, +and $$\Ket{g}$$ and $$\Ket{e}$$ respectively the ground state and first excitation of the active atoms. -We start by assuming that the cavity has $N$ quasinormal modes $\Psi_n$, -each with a corresponding polarization $\vb{p}_n$ of the active matter. +We start by assuming that the cavity has $$N$$ quasinormal modes $$\Psi_n$$, +each with a corresponding polarization $$\vb{p}_n$$ of the active matter. Note that this ansatz already suggests that the interactions between the modes are limited: @@ -80,8 +80,8 @@ $$\begin{aligned} + \frac{i}{\hbar} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D \end{aligned}$$ -With being $\vb{p}_0^{+} \vb{p}_0^{-}$ a dyadic product. -Isolating the latter equation for $\vb{p}_n$ gives us: +With being $$\vb{p}_0^{+} \vb{p}_0^{-}$$ a dyadic product. +Isolating the latter equation for $$\vb{p}_n$$ gives us: $$\begin{aligned} \vb{p}_n @@ -89,7 +89,7 @@ $$\begin{aligned} = \frac{\gamma(\omega_n) D}{\hbar \gamma_\perp} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \end{aligned}$$ -Where we have defined the Lorentzian gain curve $\gamma(\omega_n)$ as follows, +Where we have defined the Lorentzian gain curve $$\gamma(\omega_n)$$ as follows, which represents the laser's preferred frequencies for amplification: $$\begin{aligned} @@ -97,10 +97,10 @@ $$\begin{aligned} \equiv \frac{\gamma_\perp}{(\omega_n - \omega_0) + i \gamma_\perp} \end{aligned}$$ -Inserting this expression for $\vb{p}_n$ +Inserting this expression for $$\vb{p}_n$$ into the first Maxwell-Bloch equation yields the prototypical form of the SALT equation, -where we still need to replace $D$ with known quantities: +where we still need to replace $$D$$ with known quantities: $$\begin{aligned} 0 @@ -108,8 +108,8 @@ $$\begin{aligned} - \omega_n^2 \frac{\mu_0 \gamma(\omega_n) D}{\hbar \gamma_\perp} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg) \Psi_n \end{aligned}$$ -To rewrite $D$, we turn to its (Maxwell-Bloch) equation of motion, -making the crucial **stationary inversion approximation** $\ipdv{D}{t} = 0$: +To rewrite $$D$$, we turn to its (Maxwell-Bloch) equation of motion, +making the crucial **stationary inversion approximation** $$\ipdv{D}{t} = 0$$: $$\begin{aligned} D @@ -119,8 +119,8 @@ $$\begin{aligned} This is the most aggressive approximation we will make: it removes all definite phase relations between modes, and effectively eliminates time as a variable. -We insert our ansatz for $\vb{E}^{+}$ and $\vb{P}^{+}$, -and point out that only excited lasing modes contribute to $D$: +We insert our ansatz for $$\vb{E}^{+}$$ and $$\vb{P}^{+}$$, +and point out that only excited lasing modes contribute to $$D$$: $$\begin{aligned} D @@ -130,9 +130,9 @@ $$\begin{aligned} \end{aligned}$$ Here, we make the [rotating wave approximation](/know/concept/rotating-wave-approximation/) -to neglect all terms where $\nu \neq \mu$ +to neglect all terms where $$\nu \neq \mu$$ on the basis that they oscillate too quickly, -leaving only $\nu = \mu$: +leaving only $$\nu = \mu$$: $$\begin{aligned} D @@ -140,8 +140,8 @@ $$\begin{aligned} \bigg( \vb{p}_\nu^* \cdot \Psi_\nu - \vb{p}_\nu \cdot \Psi_\nu^* \bigg) \end{aligned}$$ -Inserting our earlier equation for $\vb{p}_n$ -and using the fact that $\vb{p}_0^{+} = (\vb{p}_0^{-})^*$ leads us to: +Inserting our earlier equation for $$\vb{p}_n$$ +and using the fact that $$\vb{p}_0^{+} = (\vb{p}_0^{-})^*$$ leads us to: $$\begin{aligned} D @@ -168,7 +168,7 @@ $$\begin{aligned} = i 2 \big|\gamma(\omega_\nu)\big|^2 \end{aligned}$$ -Inserting this into our equation for $D$ gives the following expression: +Inserting this into our equation for $$D$$ gives the following expression: $$\begin{aligned} D @@ -176,7 +176,7 @@ $$\begin{aligned} \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2 \end{aligned}$$ -We then properly isolate this for $D$ to get its final form, namely: +We then properly isolate this for $$D$$ to get its final form, namely: $$\begin{aligned} D @@ -197,7 +197,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where we have defined **spatial hole burning** function $h(\vb{x})$ like so, +Where we have defined **spatial hole burning** function $$h(\vb{x})$$ like so, representing the depletion of the supply of charge carriers as they are consumed by the active lasing modes: @@ -209,10 +209,10 @@ $$\begin{aligned} } \end{aligned}$$ -Many authors assume that $\vb{p}_0^- \parallel \Psi_n$, -so that only its amplitude $|g|^2 \equiv \vb{p}_0^{+} \cdot \vb{p}_0^{-}$ matters. -In that case, they often non-dimensionalize $D$ and $\Psi_n$ -by dividing out the units $d_c$ and $e_c$: +Many authors assume that $$\vb{p}_0^- \parallel \Psi_n$$, +so that only its amplitude $$|g|^2 \equiv \vb{p}_0^{+} \cdot \vb{p}_0^{-}$$ matters. +In that case, they often non-dimensionalize $$D$$ and $$\Psi_n$$ +by dividing out the units $$d_c$$ and $$e_c$$: $$\begin{aligned} \tilde{\Psi}_n @@ -228,8 +228,8 @@ $$\begin{aligned} \equiv \frac{\varepsilon_0 \hbar \gamma_\perp}{|g|^2} \end{aligned}$$ -And then the SALT equation and hole burning function $h$ are reduced to the following, -where the vacuum wavenumber $k_n = \omega_n / c$: +And then the SALT equation and hole burning function $$h$$ are reduced to the following, +where the vacuum wavenumber $$k_n = \omega_n / c$$: $$\begin{aligned} 0 @@ -249,7 +249,7 @@ $$\begin{aligned} = \nabla (\nabla \cdot \Psi) - \nabla^2 \Psi \end{aligned}$$ -Where $\nabla \cdot \Psi = 0$ thanks to [Gauss' law](/know/concept/maxwells-equations/), +Where $$\nabla \cdot \Psi = 0$$ thanks to [Gauss' law](/know/concept/maxwells-equations/), so we get an even further simplified SALT equation: $$\begin{aligned} @@ -258,21 +258,21 @@ $$\begin{aligned} + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n \end{aligned}$$ -The challenge is to solve this equation for a given $\varepsilon_r(\vb{x})$ and $D_0(\vb{x})$, -with the boundary condition that $\Psi_n$ is a plane wave at infinity, +The challenge is to solve this equation for a given $$\varepsilon_r(\vb{x})$$ and $$D_0(\vb{x})$$, +with the boundary condition that $$\Psi_n$$ is a plane wave at infinity, i.e. that there is light leaving the cavity. -If $k_n$ has a negative imaginary part, then that mode is behaving as an LED. -Gradually increasing the pump $D_0$ in a chosen region -causes the $k_n$'s imaginary parts become less negative, +If $$k_n$$ has a negative imaginary part, then that mode is behaving as an LED. +Gradually increasing the pump $$D_0$$ in a chosen region +causes the $$k_n$$'s imaginary parts become less negative, until one of them hits the real axis, at which point that mode starts lasing. -After that, $D_0$ can be increased even further until some other $k_n$ become real. +After that, $$D_0$$ can be increased even further until some other $$k_n$$ become real. -Below threshold (i.e. before any mode is lasing), the problem is linear in $\Psi_n$, -but above threshold it is nonlinear, and the amplitude of $\Psi_n$ is adjusted -such that the corresponding $k_n$ never leaves the real axis. +Below threshold (i.e. before any mode is lasing), the problem is linear in $$\Psi_n$$, +but above threshold it is nonlinear, and the amplitude of $$\Psi_n$$ is adjusted +such that the corresponding $$k_n$$ never leaves the real axis. When any mode is lasing, hole burning makes it harder for other modes to activate, -since it effectively reduces the pump $D_0$. +since it effectively reduces the pump $$D_0$$. ## References -- cgit v1.2.3