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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 |