From 3a78748e8e4aacefbbc43fb7304fa50bbcad3864 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Fri, 11 Feb 2022 17:57:52 +0100
Subject: Expand knowledge base

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+---
+title: "SALT equation"
+firstLetter: "S"
+publishDate: 2022-02-07
+categories:
+- Physics
+- Optics
+
+date: 2022-01-20T22:01:48+01:00
+draft: false
+markup: pandoc
+---
+
+# SALT equation
+
+The **steady-state *ab initio* laser theory** (SALT) is
+a theoretical description of lasers, whose mode-centric approach
+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
+active atoms (or quantum dots) embedded in a passive linear background
+material with refractive index $c / v$.
+The system is affected by a driving [electric field](/know/concept/electric-field/)
+$\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$,
+such that the set of equations is:
+
+$$\begin{aligned}
+    - \mu_0 \pdv[2]{\vb{P}^{+}}{t}
+    &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{1}{v^2} \pdv[2]{\vb{E}^{+}}{t}
+    \\
+    \pdv{\vb{P}^{+}}{t}
+    &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}
+    - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D
+    \\
+    \pdv{D}{t}
+    &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
+\end{aligned}$$
+
+Where $\hbar \omega_0$ 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:
+
+$$\begin{aligned}
+    \vb{p}_0^{-}
+    \equiv q \matrixel{e}{\vu{x}}{g}
+    \qquad \qquad
+    \vb{p}_0^{+}
+    \equiv q \matrixel{g}{\vu{x}}{e}
+    = (\vb{p}_0^{-})^*
+\end{aligned}$$
+
+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.
+Note that this ansatz already suggests
+that the interactions between the modes are limited:
+
+$$\begin{aligned}
+    \vb{E}^{+}(\vb{x}, t)
+    = \sum_{n = 1}^N \Psi_n(\vb{x}) e^{- i \omega_n t}
+    \qquad \qquad
+    \vb{P}^{+}(\vb{x}, t)
+    = \sum_{n = 1}^N \vb{p}_n(\vb{x}) e^{- i \omega_n t}
+\end{aligned}$$
+
+Using the modes' linear independence to treat each term of the summation individually,
+the first two Maxwell-Bloch equations turn into, respectively:
+
+$$\begin{aligned}
+    \mu_0 \omega_n^2 \vb{p}_n
+    &= \nabla \cross \nabla \cross \Psi_n - \frac{1}{v^2} \omega_n^2 \Psi_n
+    \\
+    i \omega_n \vb{p}_n
+    &= \big( i \omega_0 + \gamma_\perp \big) \vb{p}_n
+    + \frac{i}{\hbar} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D
+\end{aligned}$$
+
+With being $\vb{p}_0^{+} \vb{p}_0^{-}$ a dyadic product.
+Isolating the latter equation for $\vb{p}_n$ gives us:
+
+$$\begin{aligned}
+    \vb{p}_n
+    &= \frac{\big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D}{\hbar \big((\omega_n - \omega_0) + i \gamma_\perp\big)}
+    = \frac{\gamma(\omega_n) D}{\hbar \gamma_\perp} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n
+\end{aligned}$$
+
+Where we have defined the Lorentzian gain curve $\gamma(\omega_n)$ as follows,
+which represents the laser's preferred frequencies for amplification:
+
+$$\begin{aligned}
+    \gamma(\omega_n)
+    \equiv \frac{\gamma_\perp}{(\omega_n - \omega_0) + i \gamma_\perp}
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    0
+    &= \bigg( \nabla \cross \nabla \cross - \, \omega_n^2 \frac{1}{v^2}
+    - \omega_n^2 \frac{\mu_0 \gamma(\omega_n) D}{\hbar \gamma_\perp} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg) \Psi_n 
+\end{aligned}$$
+
+To rewrite $D$, we turn to its (Maxwell-Bloch) equation of motion,
+making the crucial **stationary inversion approximation** $\pdv*{D}{t} = 0$:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
+\end{aligned}$$
+
+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 $D$:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \sum_{\nu, \mu}^\mathrm{active}
+    \bigg( \vb{p}_\nu^* \cdot \Psi_\mu e^{i (\omega_\nu - \omega_\mu) t}
+    - \vb{p}_\nu \cdot \Psi_\mu^* e^{i (\omega_\mu - \omega_\nu) t} \bigg)
+\end{aligned}$$
+
+Here, we make the [rotating wave approximation](/know/concept/rotating-wave-approximation/)
+to neglect all terms where $\nu \neq \mu$
+on the basis that they oscillate too quickly,
+leaving only $\nu = \mu$:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \sum_{\nu}^\mathrm{act.}
+    \bigg( \vb{p}_\nu^* \cdot \Psi_\nu - \vb{p}_\nu \cdot \Psi_\nu^* \bigg)
+\end{aligned}$$
+
+Inserting our earlier equation for $\vb{p}_n$
+and using the fact that $\vb{p}_0^{+} = (\vb{p}_0^{-})^*$ leads us to:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \bigg( \gamma^*(\omega_\nu) \big(\vb{p}_0^{+} \vb{p}_0^{-}\big)^* \!\cdot\! \Psi_\nu^* \cdot \Psi_\nu
+    - \gamma(\omega_\nu) \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \!\cdot\! \Psi_\nu \cdot \Psi_\nu^* \bigg)
+    \\
+    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \bigg( \gamma^*(\omega_\nu) \big(\vb{p}_0^{+} \cdot \Psi_\nu^*\big) \vb{p}_0^{-} \cdot \Psi_\nu
+    - \gamma(\omega_\nu) \big(\vb{p}_0^{-} \cdot \Psi_\nu\big) \vb{p}_0^{+} \cdot \Psi_\nu^* \bigg)
+    \\
+    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \Big( \gamma^*(\omega_\nu) - \gamma(\omega_\nu) \Big) \big|\vb{p}_0^{-} \cdot \Psi_\nu\big|^2
+\end{aligned}$$
+
+By putting the terms on a common denominator, it is easily shown that:
+
+$$\begin{aligned}
+    \gamma^*(\omega_\nu) - \gamma(\omega_\nu)
+    &= \frac{\gamma_\perp ((\omega_\nu - \omega_0) + i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
+    - \frac{\gamma_\perp ((\omega_\nu - \omega_0) - i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
+    \\
+    &= \frac{\gamma_\perp (i \gamma_\perp + i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
+    = i 2 \big|\gamma(\omega_\nu)\big|^2
+\end{aligned}$$
+
+Inserting this into our equation for $D$ gives the following expression:
+
+$$\begin{aligned}
+    D
+    &= D_0 - \frac{4 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2
+\end{aligned}$$
+
+We then properly isolate this for $D$ to get its final form, namely:
+
+$$\begin{aligned}
+    D
+    &= D_0 \bigg( 1 + \frac{4}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2 \bigg)^{-1}
+\end{aligned}$$
+
+Substituting this into the prototypical SALT equation from earlier
+yields the most general form of the **SALT equation**,
+upon which the theory is built:
+
+$$\begin{aligned}
+    \boxed{
+        0
+        = \bigg( \nabla \cross \nabla \cross
+        -\,\omega_n^2 \bigg[ \frac{1}{v^2(\vb{x})} + \frac{\mu_0 \gamma(\omega_n)}{\hbar \gamma_\perp}
+        \frac{D_0(\vb{x})}{1 + h(\vb{x})} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg] \bigg) \Psi_n(\vb{x})
+    }
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \boxed{
+        h(\vb{x})
+        \equiv \frac{4}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+        \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu(\vb{x})\Big|^2
+    }
+\end{aligned}$$
+
+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 $e_c$:
+
+$$\begin{aligned}
+    \tilde{\Psi}_n
+    \equiv \frac{\Psi_n}{e_c}
+    \qquad
+    e_c
+    \equiv \frac{\hbar \sqrt{\gamma_\parallel \gamma_\perp}}{2 |g|}
+    \qquad \qquad
+    \tilde{D}
+    \equiv \frac{D}{d_c}
+    \qquad
+    d_c
+    \equiv \frac{\varepsilon_0 \hbar \gamma_\perp}{|g|^2}
+\end{aligned}$$
+
+And then the SALT equation and hole burning function $h$ are reduced to the following,
+where the vacuum wavenumber $k_n = \omega_n / c$:
+
+$$\begin{aligned}
+    0
+    = \bigg( \nabla \cross \nabla \cross -\,k_n^2 \bigg[ \varepsilon_r
+    + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n
+    \qquad
+    h(\vb{x})
+    = \sum_{\nu}^\mathrm{act.} \Big|\gamma(c k_\nu) \tilde{\Psi}_\nu(\vb{x})\Big|^2
+\end{aligned}$$
+
+
+In addition, some papers only consider 1D or 2D *transverse magnetic* (TM) modes,
+in which case the fields are scalars. Using the vector identity
+
+$$\begin{aligned}
+    \nabla \cross \nabla \cross \Psi
+    = \nabla (\nabla \cdot \Psi) - \nabla^2 \Psi
+\end{aligned}$$
+
+Where $\nabla \cdot \Psi = 0$ thanks to [Gauss' law](/know/concept/maxwells-equations/),
+so we get an even further simplified SALT equation:
+
+$$\begin{aligned}
+    0
+    = \bigg( \nabla^2 +\,k_n^2 \bigg[ \varepsilon_r
+    + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n
+\end{aligned}$$
+
+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,
+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,
+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.
+
+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 $D_0$.
+
+
+## References
+1.  L. Ge, Y.D. Chong, A.D. Stone,
+    [Steady-state *ab initio* laser theory: generalizations and analytic results](http://dx.doi.org/10.1103/PhysRevA.82.063824),
+    2010, American Physical Society.
+
-- 
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