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diff --git a/source/know/concept/salt-equation/index.md b/source/know/concept/salt-equation/index.md new file mode 100644 index 0000000..c938429 --- /dev/null +++ b/source/know/concept/salt-equation/index.md @@ -0,0 +1,281 @@ +--- +title: "SALT equation" +date: 2022-02-07 +categories: +- Physics +- Optics +- Laser theory +layout: "concept" +--- + +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: + +$$\begin{aligned} + - \mu_0 \pdvn{2}{\vb{P}^{+}}{t} + &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{1}{v^2} \pdvn{2}{\vb{E}^{+}}{t} + \\ + \pdv{\vb{P}^{+}}{t} + &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} + - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D + \\ + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) +\end{aligned}$$ + +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** $\ipdv{D}{t} = 0$: + +$$\begin{aligned} + D + &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) +\end{aligned}$$ + +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. + |