From aeacfca5aea5df7c107cf0c12e72ab5d496c96e1 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 3 Jan 2023 19:48:17 +0100
Subject: More improvements to knowledge base

---
 source/know/concept/salt-equation/index.md | 32 ++++++++++++++++++------------
 1 file changed, 19 insertions(+), 13 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 f5f085d..77f4755 100644
--- a/source/know/concept/salt-equation/index.md
+++ b/source/know/concept/salt-equation/index.md
@@ -52,7 +52,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 With $$q < 0$$ the electron charge, $$\vu{x}$$ the quantum position operator,
-and $$\Ket{g}$$ and $$\Ket{e}$$ respectively
+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 $$\Psi_n$$,
@@ -120,7 +120,7 @@ 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 $$D$$:
+and point out that only active lasing modes contribute to $$D$$:
 
 $$\begin{aligned}
     D
@@ -258,21 +258,27 @@ $$\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 $$D_0(\vb{x})$$,
+Given $$\varepsilon_r(\vb{x})$$ and $$D_0(\vb{x})$$,
+the challenge is to solve this eigenvalue problem for $$k_n$$ and $$\Psi_n$$,
 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.
+i.e. light is leaving the cavity.
+
+If $$\Imag(k_n) < 0$$, the $$n$$th mode's amplitude decays with time, so it acts as an LED:
+it emits photons without any significant light amplification taking place.
+Upon gradually increasing the pump $$D_0$$ in the active region,
+all $$\Imag(k_n)$$ become less negative,
+until one hits the real axis $$\Imag(k_n) = 0$$,
+at which point that mode starts lasing,
+i.e. the Light gets Amplified by [Stimulated Emission](/know/concept/einstein-coefficients/) (LASE).
 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 $$D_0$$.
+but above threshold it is nonlinear via $$h(\vb{x})$$.
+Then the amplitude of $$\Psi_n$$ gets adjusted
+such that its respective $$k_n$$ never leaves the real axis.
+Once a mode is lasing, hole burning makes it harder for any other modes to activate,
+since they modes must compete for the carrier supply $$D_0$$.
+
 
 
 ## References
-- 
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