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author | Prefetch | 2022-03-31 21:27:48 +0200 |
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committer | Prefetch | 2022-03-31 21:27:48 +0200 |
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parent | eacd6f7bc1a4a048e1352b740dd3354e2a035106 (diff) |
Expand knowledge base
-rw-r--r-- | content/know/category/laser-theory.md | 9 | ||||
-rw-r--r-- | content/know/concept/coupled-mode-theory/index.pdc | 233 | ||||
-rw-r--r-- | content/know/concept/einstein-coefficients/index.pdc | 6 | ||||
-rw-r--r-- | content/know/concept/fabry-perot-cavity/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/laser-rate-equations/index.pdc | 330 | ||||
-rw-r--r-- | content/know/concept/maxwell-bloch-equations/index.pdc | 66 | ||||
-rw-r--r-- | content/know/concept/multi-photon-absorption/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/salt-equation/index.pdc | 5 |
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diff --git a/content/know/category/laser-theory.md b/content/know/category/laser-theory.md new file mode 100644 index 0000000..5437821 --- /dev/null +++ b/content/know/category/laser-theory.md @@ -0,0 +1,9 @@ +--- +title: "Laser theory" +firstLetter: "L" +date: 2022-03-16T20:36:17+01:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/concept/coupled-mode-theory/index.pdc b/content/know/concept/coupled-mode-theory/index.pdc new file mode 100644 index 0000000..5a44d6e --- /dev/null +++ b/content/know/concept/coupled-mode-theory/index.pdc @@ -0,0 +1,233 @@ +--- +title: "Coupled mode theory" +firstLetter: "C" +publishDate: 2022-03-31 +categories: +- Physics +- Optics + +date: 2022-03-12T20:22:21+01:00 +draft: false +markup: pandoc +--- + +# Coupled mode theory + +Given an optical resonator (e.g. a photonic crystal cavity), +consider one of its quasinormal modes +with frequency $\omega_0$ and decay rate $1 / \tau_0$. +Its complex amplitude $A$ is governed by: + +$$\begin{aligned} + \dv{A}{t} + &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A +\end{aligned}$$ + +We choose to normalize $A$ so that $|A(t)|^2$ +is the total energy inside the resonator at time $t$. + +Suppose that $N$ waveguides are now "connected" to this resonator, +meaning that the resonator mode $A$ and the outgoing waveguide mode $S_\ell^\mathrm{out}$ +overlap sufficiently for $A$ to leak into $S_\ell^\mathrm{out}$ at a rate $1 / \tau_\ell$. +Conversely, the incoming mode $S_\ell^\mathrm{in}$ brings energy to $A$. +Therefore, we can write up the following general set of equations: + +$$\begin{aligned} + \dv{A}{t} + &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A + - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A + \sum_{\ell = 1}^N \alpha_\ell S_\ell^\mathrm{in} + \\ + S_\ell^\mathrm{out} + &= \beta_\ell S_\ell^\mathrm{in} + \gamma_\ell A +\end{aligned}$$ + +Where $\alpha_\ell$ and $\gamma_\ell$ are unknown coupling constants, +and $\beta_\ell$ represents reflection. +We normalize $S_\ell^\mathrm{in}$ +so that $|S_\ell^\mathrm{in}(t)|^2$ is the power flowing towards $A$ at time $t$, +and likewise for $S_\ell^\mathrm{out}$. + +Note that we have made a subtle approximation here: +by adding new damping mechanisms, +we are in fact modifying $\omega_0$; +see the [harmonic oscillator](/know/concept/harmonic-oscillator/) for a demonstration. +However, the frequency shift is second-order in the decay rate, +so by assuming that all $\tau_\ell$ are large, +we only need to keep the first-order terms, as we did. +This is called **weak coupling**. + +If we also assume that $\tau_0$ is large +(its effect is already included in $\omega_0$), +then we can treat the decay mechanisms separately: +to analyze the decay into a certain waveguide $\ell$, +it is first-order accurate to neglect all other waveguides and $\tau_0$: + +$$\begin{aligned} + \dv{A}{t} + \approx \bigg( \!-\! i \omega_0 - \frac{1}{\tau_\ell} \bigg) A + \sum_{\ell' = 1}^N \alpha_\ell S_{\ell'}^\mathrm{in} +\end{aligned}$$ + +To determine $\gamma_\ell$, we use energy conservation. +If all $S_{\ell'}^\mathrm{in} = 0$, +then the energy in $A$ decays as: + +$$\begin{aligned} + \dv{|A|^2}{t} + &= \dv{A}{t} A^* + A \dv{A^*}{t} + \\ + &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_\ell} \bigg) |A|^2 + + \bigg( i \omega_0 - \frac{1}{\tau_\ell} \bigg) |A|^2 + \\ + &= - \frac{2}{\tau_\ell} |A|^2 +\end{aligned}$$ + +Since all other mechanisms are neglected, +all this energy must go into $S_\ell^\mathrm{out}$, meaning: + +$$\begin{aligned} + |S_\ell^\mathrm{out}|^2 + = - \dv{|A|^2}{t} + = \frac{2}{\tau_\ell} |A|^2 +\end{aligned}$$ + +Taking the square root, we clearly see that $|\gamma_\ell| = \sqrt{2 / \tau_\ell}$. +Because the phase of $S_\ell^\mathrm{out}$ is arbitrarily defined, +$\gamma_\ell$ need not be complex, so we choose $\gamma_\ell = \sqrt{2 / \tau_\ell}$. + +Next, to find $\alpha_\ell$, we exploit the time-reversal symmetry +of [Maxwell's equations](/know/concept/maxwells-equations/), +which govern the light in the resonator and the waveguides. +In the above calculation of $\gamma_\ell$, $A$ evolved as follows, +with the lost energy ending up in $S_\ell^\mathrm{out}$: + +$$\begin{aligned} + A(t) + = A e^{-i \omega_0 t - t / \tau_\ell} +\end{aligned}$$ + +After reversing time, $A$ evolves like so, +where we have taken the complex conjugate +to preserve the meanings of the symbols +$A$, $S_\ell^\mathrm{out}$, and $S_\ell^\mathrm{in}$: +$$\begin{aligned} + A(t) + = A e^{-i \omega_0 t + t / \tau_\ell} +\end{aligned}$$ + +We insert this expression for $A(t)$ into its original differential equation, yielding: + +$$\begin{aligned} + \dv{A}{t} + = \bigg( \!-\! i \omega_0 + \frac{1}{\tau_\ell} \bigg) A + = \bigg( \!-\! i \omega_0 - \frac{1}{\tau_\ell} \bigg) A + \alpha_\ell S_\ell^\mathrm{in} +\end{aligned}$$ + +Isolating this for $A$ leads us to the following power balance equation: + +$$\begin{aligned} + A + = \frac{\alpha_\ell \tau_\ell}{2} S_\ell^\mathrm{in} + \qquad \implies \qquad + |\alpha_\ell|^2 |S_\ell^\mathrm{in}|^2 + = \frac{4}{\tau_\ell^2} |A| +\end{aligned}$$ + +But thanks to energy conservation, +all power delivered by $S_\ell^\mathrm{in}$ ends up in $A$, so we know: + +$$\begin{aligned} + |S_\ell^\mathrm{in}|^2 + = \dv{|A|^2}{t} + = \frac{2}{\tau_\ell} |A|^2 +\end{aligned}$$ + +To reconcile the two equations above, +we need $|\alpha_\ell| = \sqrt{2 / \tau_\ell}$. +Discarding the phase thanks to our choice of $\gamma_\ell$, +we conclude that $\alpha_\ell = \sqrt{2 / \tau_\ell} = \gamma_\ell$. + +Finally, $\beta_\ell$ can also be determined using energy conservation. +Again using our weak coupling assumption, +if energy is only entering and leaving $A$ through waveguide $\ell$, we have: + +$$\begin{aligned} + |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 + = \dv{|A|^2}{t} +\end{aligned}$$ + +Meanwhile, using the differential equation for $A$, +we find the following relation: + +$$\begin{aligned} + \dv{|A|^2}{t} + &= \dv{A}{t} A^* + A \dv{A^*}{t} + \\ + &= - \frac{2}{\tau_\ell} |A|^2 + \alpha_\ell \Big( S_\ell^\mathrm{in} A^* + (S_\ell^\mathrm{in})^* A \Big) +\end{aligned}$$ + +By isolating both of the above relations for $\dv*{|A|^2}{t}$ +and equating them, we arrive at: + +$$\begin{aligned} + |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 + &= - \frac{2}{\tau_\ell} |A|^2 + \alpha_\ell \Big( S_\ell^\mathrm{in} A^* + (S_\ell^\mathrm{in})^* A \Big) +\end{aligned}$$ + +We insert the definition of $\gamma_\ell$ and $\beta_\ell$, +namely $\gamma_\ell A = S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in}$, +and use $\alpha_\ell = \gamma_\ell$: + +$$\begin{aligned} + |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 + &= - \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big) \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big) + \\ + &\quad\; + S_\ell^\mathrm{in} \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big) + + (S_\ell^\mathrm{in})^* \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big) + \\ + &= - |\beta_\ell|^2 |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 + + \beta_\ell S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + \beta_\ell^* (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} + \\ + &\quad\; + S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* - \beta_\ell^* |S_\ell^\mathrm{in}|^2 + + (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} - \beta_\ell |S_\ell^\mathrm{in}|^2 + \\ + &= - (|\beta_\ell|^2 + \beta_\ell + \beta_\ell^*) |S_\ell^\mathrm{in}|^2 + + (1 - \beta_\ell) S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + (1 - \beta_\ell^*) (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} +\end{aligned}$$ + +This equation is only satisfied if $\beta_\ell = -1$. +Combined with $\alpha_\ell = \gamma_\ell = \sqrt{2 / \tau_\ell}$, +the **coupled-mode equations** take the following form: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \dv{A}{t} + &= - i \omega_0 A - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A + + \sum_{\ell = 1}^N \sqrt{\frac{2}{\tau_\ell}} S_\ell^\mathrm{in} + \\ + S_\ell^\mathrm{out} + &= - S_\ell^\mathrm{in} + \sqrt{\frac{2}{\tau_\ell}} A + \end{aligned} + } +\end{aligned}$$ + +By connecting multiple resonators with waveguides, +optical networks can be created, +whose dynamics are described by these equations. + +The coupled-mode equations are extremely general, +since we have only used weak coupling, +conservation of energy, and time-reversal symmetry. +Even if the decay rates are quite large, +coupled mode theory still tends to give qualitatively correct answers. + + + +## References +1. H.A. Haus, + *Waves and fields in optoelectronics*, + 1984, Prentice-Hall. +2. J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, + *Photonic crystals: molding the flow of light*, + 2nd edition, Princeton. + diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc index b56af77..d50235f 100644 --- a/content/know/concept/einstein-coefficients/index.pdc +++ b/content/know/concept/einstein-coefficients/index.pdc @@ -5,9 +5,9 @@ publishDate: 2021-07-11 categories: - Physics - Optics -- Electromagnetism - Quantum mechanics - Two-level system +- Laser theory date: 2021-07-11T18:22:14+02:00 draft: false @@ -275,7 +275,7 @@ $$\begin{aligned} \boxed{ B_{21} = B_{12} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} \qquad - A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon \hbar c^3} + A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon_0 \hbar c^3} } \end{aligned}$$ @@ -332,7 +332,7 @@ $$\begin{aligned} \boxed{ B_{21} = B_{12} = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2} \qquad - A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon \hbar c^3} + A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon_0 \hbar c^3} } \end{aligned}$$ diff --git a/content/know/concept/fabry-perot-cavity/index.pdc b/content/know/concept/fabry-perot-cavity/index.pdc index 50b7c62..e4195d0 100644 --- a/content/know/concept/fabry-perot-cavity/index.pdc +++ b/content/know/concept/fabry-perot-cavity/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-09-18 categories: - Physics - Optics +- Laser theory date: 2021-09-18T00:42:59+02:00 draft: false diff --git a/content/know/concept/laser-rate-equations/index.pdc b/content/know/concept/laser-rate-equations/index.pdc new file mode 100644 index 0000000..d087035 --- /dev/null +++ b/content/know/concept/laser-rate-equations/index.pdc @@ -0,0 +1,330 @@ +--- +title: "Laser rate equations" +firstLetter: "L" +publishDate: 2022-03-16 +categories: +- Physics +- Optics +- Laser theory + +date: 2022-03-12T20:23:42+01:00 +draft: false +markup: pandoc +--- + +# Laser rate equations + +The [Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/) (MBEs) +give a fundamental description of light-matter interaction +for a two-level quantum system for the purposes of laser theory. +They govern the [electric field](/know/concept/electric-field/) $\vb{E}^{+}$, +the induced polarization $\vb{P}^{+}$, +and the total population inversion $D$: + +$$\begin{aligned} + - \mu_0 \pdv[2]{\vb{P}^{+}}{t} + &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{n^2}{c^2} \pdv[2]{\vb{E}^{+}}{t} + \\ + \pdv{\vb{P}^{+}}{t} + &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} + - \frac{i |g|^2}{\hbar} \vb{E}^{+} D + \\ + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) +\end{aligned}$$ + +Where $n$ is the background medium's refractive index, +$\omega_0$ the two-level system's gap resonance frequency, +$|g| \equiv |\matrixel{e}{\vu{x}}{g}|$ the transition dipole moment, +$\gamma_\perp$ and $\gamma_\parallel$ empirical decay rates, +and $D_0$ the equilibrium inversion. +Note that $\vb{E}^{-} = (\vb{E}^{+})^*$. + +Let us make the following ansatz, +where $\vb{E}_0^{+}$ and $\vb{P}_0^{+}$ are slowly-varying envelopes +of a plane wave with angular frequency $\omega \approx \omega_0$: + +$$\begin{aligned} + \vb{E}^{+}(\vb{r}, t) + = \frac{1}{2} \vb{E}_0^{+}(\vb{r}, t) \: e^{-i \omega t} + \qquad \qquad + \vb{P}^{+}(\vb{r}, t) + = \frac{1}{2} \vb{P}_0^{+}(\vb{r}, t) \: e^{-i \omega t} +\end{aligned}$$ + +We insert this into the first MBE, +and assume that $\vb{E}_0^{+}$ and $\vb{P}_0^{+}$ +vary so slowly that their second-order derivatives are negligible, +i.e. $\pdv*[2]{\vb{E}_0^{+}\!}{t} \approx 0$ and $\pdv*[2]{\vb{P}_0^{+}\!}{t} \approx 0$, +giving: + +$$\begin{aligned} + \mu_0 \bigg( i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} \bigg) + = \nabla \cross \nabla \cross \vb{E}_0^{+} + - \frac{n^2}{c^2} \bigg( i 2 \omega \pdv{\vb{E}_0^{+}}{t} + \omega^2 \vb{E}_0^{+} \bigg) +\end{aligned}$$ + +To get rid of the double curl, +consider the time-independent +[electromagnetic wave equation](/know/concept/electromagnetic-wave-equation/), +where $\Omega$ is an eigenfrequency of the optical cavity +in which lasing will occur: + +$$\begin{aligned} + \nabla \cross \nabla \cross \vb{E}_0^{+} + = \frac{n^2}{c^2} \Omega^2 \vb{E}_0^{+} +\end{aligned}$$ + +For simplicity, we restrict ourselves to a single-mode laser, +where there is only one $\Omega$ and $\vb{E}_0^{+}$ to care about. +Substituting the above equation into the first MBE yields: + +$$\begin{aligned} + i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} + = \varepsilon_0 n^2 \bigg( (\Omega^2 - \omega^2) \vb{E}_0^{+} - i 2 \omega \pdv{\vb{E}_0^{+}}{t} \bigg) +\end{aligned}$$ + +Where we used $1 / c^2 = \mu_0 \varepsilon_0$. +Assuming the light is more or less on-resonance $\omega \approx \Omega$, +we can approximate $\Omega^2 \!-\! \omega^2 \approx 2 \omega (\Omega \!-\! \omega)$, so: + +$$\begin{aligned} + i 2 \pdv{\vb{P}_0^{+}}{t} + \omega \vb{P}_0^{+} + = \varepsilon_0 n^2 \bigg( 2 (\Omega - \omega) \vb{E}_0^{+} - i 2 \pdv{\vb{E}_0^{+}}{t} \bigg) +\end{aligned}$$ + +Moving on to the second MBE, +inserting the ansatz $\vb{P}^{+} = \vb{P}_0^{+} e^{-i \omega t} / 2$ leads to: + +$$\begin{aligned} + \pdv{\vb{P}_0^{+}}{t} + = - \Big( \gamma_\perp + i (\omega_0 - \omega) \Big) \vb{P}_0^{+} - \frac{i |g|^2}{\hbar} \vb{E}_0^{+} D +\end{aligned}$$ + +Typically, $\gamma_\perp$ is much larger than the rate of any other decay process, +in which case $\pdv*{\vb{P}_0^{+}\!}{t}$ is negligible compared to $\gamma_\perp \vb{P}_0^{+}$. +Effectively, this means that the polarization $\vb{P}_0^{+}$ +near-instantly follows the electric field $\vb{E}^{+}\!$. +Setting $\pdv*{\vb{P}_0^{+}\!}{t} \approx 0$, the second MBE becomes: + +$$\begin{aligned} + \vb{P}^{+} + = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 \!-\! \omega))} \vb{E}^{+} D + = \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D +\end{aligned}$$ + +Where the Lorentzian gain curve $\gamma(\omega)$ +(which also appears in the [SALT equation](/know/concept/salt-equation/)) +represents a laser's preferred spectrum for amplification, +and is defined like so: + +$$\begin{aligned} + \gamma(\omega) + \equiv \frac{\gamma_\perp}{(\omega - \omega_0) + i \gamma_\perp} +\end{aligned}$$ + +Note that $\gamma(\omega)$ satisfies the following relation, +which will be useful to us later: + +$$\begin{aligned} + \gamma^*(\omega) - \gamma(\omega) + = \frac{\gamma_\perp (i \gamma_\perp + i \gamma_\perp)}{(\omega - \omega_0)^2 + \gamma_\perp^2} + = i 2 |\gamma(\omega)|^2 +\end{aligned}$$ + +Returning to the first MBE with $\pdv*{\vb{P}_0^{+}\!}{t} \approx 0$, +we substitute the above expression for $\vb{P}_0^{+}$: + +$$\begin{aligned} + \pdv{\vb{E}_0^{+}}{t} + &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{\omega}{2 \varepsilon_0 n^2} \vb{P}_0^{+} + \\ + &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \vb{E}_0^{+} D +\end{aligned}$$ + +Next, we insert our ansatz for $\vb{E}^{+}\!$ and $\vb{P}^{+}\!$ +into the third MBE, and rewrite $\vb{P}_0^{+}$ as above. +Using our identity for $\gamma(\omega)$, +and the fact that $\vb{E}_0^{+} \cdot \vb{E}_0^{-} = |\vb{E}|^2$, we find: + +$$\begin{aligned} + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) + \frac{i}{2 \hbar} + \Big( \frac{|g|^2 \gamma^*(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{-} D \cdot \vb{E}_0^{+} + - \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{+} D \cdot \vb{E}_0^{-} \Big) + \\ + &= \gamma_\parallel (D_0 - D) + \frac{i |g|^2}{2 \hbar^2 \gamma_\perp} \Big( \gamma^*(\omega) - \gamma(\omega) \Big) |\vb{E}|^2 D + \\ + &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D +\end{aligned}$$ + +This is the prototype of the first laser rate equation. +However, in order to have a practical set, +we need an equation for $|\vb{E}|^2$, +which we can obtain using the first MBE: + +$$\begin{aligned} + \pdv{|\vb{E}|^2}{t} + &= \vb{E}_0^{+} \pdv{\vb{E}_0^{-}}{t} + \vb{E}_0^{-} \pdv{\vb{E}_0^{+}}{t} + \\ + &= -i (\omega - \Omega^*) |\vb{E}|^2 - i \frac{|g|^2 \omega \gamma^*(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D + + i (\omega - \Omega) |\vb{E}|^2 + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D + \\ + &= i (\Omega^* - \Omega) |\vb{E}|^2 + + i \frac{|g|^2 \omega}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \Big(\gamma(\omega) - \gamma^*(\omega)\Big) |\vb{E}|^2 D + \\ + &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D +\end{aligned}$$ + +Where $\Im(\Omega) < 0$ represents the fact that the laser cavity is leaky. +We now have the **laser rate equations**, +although they are still in an unidiomatic form: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \pdv{|\vb{E}|^2}{t} + &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D + \\ + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D + \end{aligned} + } +\end{aligned}$$ + +To rewrite this, we replace $|\vb{E}|^2$ with the photon number $N_p$ as follows, +with $U = \varepsilon_0 n^2 |\vb{E}|^2 / 2$ being the energy density of the light: + +$$\begin{aligned} + N_{p} + = \frac{U}{\hbar \omega} + = \frac{\varepsilon_0 n^2}{2 \hbar \omega} |\vb{E}|^2 +\end{aligned}$$ + +Furthermore, consider the definition of the inversion $D$: +because a photon emission annihilates an electron-hole pair, +it reduces $D$ by $2$. +Since lasing is only possible for $D > 0$, +we can replace $D$ with the conduction band's electron population $N_e$, +which is reduced by $1$ whenever a photon is emitted. +The laser rate equations then take the following standard form: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \pdv{N_p}{t} + &= - \gamma_p N_p + G N_p N_e + \\ + \pdv{N_e}{t} + &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e + \end{aligned} + } +\end{aligned}$$ + +Where $\gamma_e$ is a redefinition of $\gamma_\parallel$ +depending on the electron decay processes, +and the photon loss rate $\gamma_p$, the gain $G$, +and the carrier supply rate $R_\mathrm{pump}$ +are defined like so: + +$$\begin{aligned} + \gamma_p + = - 2 \Im(\Omega) + = \frac{Q}{\Re(\Omega)} + \qquad \quad + G + \equiv \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 + \qquad \quad + R_\mathrm{pump} + \equiv \gamma_\parallel D_0 +\end{aligned}$$ + +With $Q$ being the cavity mode's quality factor. +The nonlinear coupling term $G N_p N_e$ represents +[stimulated emission](/know/concept/einstein-coefficients/), +which is the key to lasing. + +To understand the behaviour of a laser, +consider these equations in a steady state, +i.e. where $N_p$ and $N_e$ are constant in $t$: + +$$\begin{aligned} + 0 + &= - \gamma_p N_p + G N_p N_e + \\ + 0 + &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e +\end{aligned}$$ + +In addition to the trivial solution $N_p = 0$, +we can also have $N_p > 0$. +Isolating $N_p$'s equation for $N_e$ and inserting that into $N_e$'s equation, we find: + +$$\begin{aligned} + N_e + = \frac{\gamma_p}{G} + \qquad \implies \qquad + \boxed{ + N_p + = \frac{1}{\gamma_p} \bigg( R_\mathrm{pump} - \frac{\gamma_e \gamma_p}{G} \bigg) + } +\end{aligned}$$ + +The quantity $R_\mathrm{thr} \equiv \gamma_e \gamma_p / G$ is called the **lasing threshold**: +if $R_\mathrm{pump} \ge R_\mathrm{thr}$, the laser is active, +meaning that $N_p$ is big enough to cause +a "chain reaction" of stimulated emission +that consumes all surplus carriers to maintain a steady state. + +The point is that $N_e$ is independent of the electron supply $R_\mathrm{pump}$, +because all additional electrons are almost immediately +annihilated by stimulated emission. +Consequently $N_p$ increases linearly as $R_\mathrm{pump}$ is raised, +at a much steeper slope than would be possible below threshold. +The output of the cavity is proportional to $N_p$, +so the brightness is also linear. + +Unfortunately, by deriving the laser rate equations from the MBEs, +we lost some interesting and important effects, +most notably spontaneous emission, +which is needed for $N_p$ to grow if $R_\mathrm{pump}$ is below threshold. + +For this reason, the laser rate equations are typically presented +in a more empirical form, which "bookkeeps" the processes affecting $N_p$ and $N_e$. +Consider the following example: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \pdv{N_p}{t} + &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e + \\ + \pdv{N_e}{t} + &= R_\mathrm{pump} + \gamma_\mathrm{abs} N_p + - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e - G_\mathrm{stim} N_p N_e + \end{aligned} + } +\end{aligned}$$ + +Where $\gamma_\mathrm{out}$ represents the cavity's usable output, +$\gamma_\mathrm{abs}$ the medium's absorption, +$\gamma_\mathrm{loss}$ scattering losses, +$\gamma_\mathrm{spon}$ spontaneous emission, +$\gamma_\mathrm{n.r.}$ non-radiative electron-hole recombination, +and $\gamma_\mathrm{leak}$ the fact that +some carriers leak away before they can be used for emission. + +Unsurprisingly, this form is much harder to analyze, +but more accurately describes the dynamics inside a laser. +To make matters even worse, many of these decay rates depend on $N_p$ or $N_e$, +so solutions can only be obtained numerically. + + + +## References +1. D. Meschede, + *Optics, light and lasers*, + Wiley. +2. L.A. Coldren, S.W. Corzine, M.L. Mašanović, + *Diode lasers and photonic integrated circuits*, 2nd edition, + Wiley. diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc index e3a3680..2e0cdd9 100644 --- a/content/know/concept/maxwell-bloch-equations/index.pdc +++ b/content/know/concept/maxwell-bloch-equations/index.pdc @@ -7,6 +7,7 @@ categories: - Quantum mechanics - Two-level system - Electromagnetism +- Laser theory date: 2021-09-09T21:17:52+02:00 draft: false @@ -34,10 +35,10 @@ $\hat{H}_1$ is given by: $$\begin{aligned} \hat{H}_1(t) = - \hat{\vb{p}} \cdot \vb{E}(t) - \qquad \quad + \qquad \qquad \vu{p} \equiv q \vu{x} - \qquad \quad + \qquad \qquad \vb{E}(t) = \vb{E}_0 \cos\!(\omega t) \end{aligned}$$ @@ -72,7 +73,7 @@ Similarly, we define the transition dipole moment $\vb{p}_0^{-}$: $$\begin{aligned} \vb{p}_0^{-} \equiv q \matrixel{e}{\vu{x}}{g} - \qquad \quad + \qquad \qquad \vb{p}_0^{+} \equiv (\vb{p}_0^{-})^* = q \matrixel{g}{\vu{x}}{e} @@ -194,7 +195,7 @@ both decay with rate $\gamma_\perp$: $$\begin{aligned} \Big( \dv{\rho_{eg}}{t} \Big)_{\perp} = - \gamma_\perp \rho_{eg} - \qquad \quad + \qquad \qquad \Big( \dv{\rho_{ge}}{t} \Big)_{\perp} = - \gamma_\perp \rho_{ge} \end{aligned}$$ @@ -295,7 +296,7 @@ towards an equilbrium $d_0$: $$\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} = \gamma_\parallel (d_0 - d) - \qquad \quad + \qquad \qquad d_0 \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} \end{aligned}$$ @@ -367,37 +368,66 @@ Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$ together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields: $$\begin{aligned} + \nabla \cross \big( \nabla \cross \vb{E} \big) + = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t} +\end{aligned}$$ + +Where $\sigma$ is the active material's conductivity, if any; +almost all authors assume $\sigma = 0$. + +Recall that we are describing the dynamics of a two-level system. +In reality, such a system (e.g. a quantum dot) +is suspended in a passive background medium, +which reacts with a polarization $\vb{P}_\mathrm{med}$ +to the electric field $\vb{E}$. +If the medium is linear, i.e. $\vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E}$, +then: + +$$\begin{aligned} + \mu_0 \pdv[2]{\vb{P}}{t} + &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} + - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}_\mathrm{med}}{t} + \\ + &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} + - \mu_0 \pdv[2]{t} \Big( \varepsilon_0 \vb{E} + \varepsilon_0 \chi \vb{E} \Big) + \\ + &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} + - \mu_0 \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t} +\end{aligned}$$ + +Where $\varepsilon_r \equiv 1 + \chi_e$ is the medium's relative permittivity. +The speed of light $c^2 = 1 / (\mu_0 \varepsilon_0)$, +and the refractive index $n^2 = \mu_r \varepsilon_r$, +where $\mu_r = 1$ due to our assumption that $\vb{M} = 0$, so: + +$$\begin{aligned} \boxed{ - \nabla \cross \big( \nabla \cross \vb{E} \big) - = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t} + \mu_0 \pdv[2]{\vb{P}}{t} + = - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}}{t} } \end{aligned}$$ -Where $\sigma$ is the medium's conductivity, if any; -many authors assume $\sigma = 0$. -It is trivial to show that $\vb{E}$ and $\vb{P}$ -can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. - +$\vb{E}$ and $\vb{P}$ can trivially be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. It is also simple to convert $\vb{p}^{+}$ and $d$ -into the macroscopic polarization $\vb{P}^{+}$ and total inversion $D$ -by summing over the atoms: +into the macroscopic $\vb{P}^{+}$ and total $D$ +by summing over all two-level systems in the medium: $$\begin{aligned} \vb{P}^{+}(\vb{x}, t) - &= \sum_{n} \vb{p}^{+}_n \: \delta(\vb{x} - \vb{x}_n) + &= \sum_{\nu} \vb{p}^{+}_\nu \: \delta(\vb{x} - \vb{x}_\nu) \\ D(\vb{x}, t) - &= \sum_{n} d_n \: \delta(\vb{x} - \vb{x}_n) + &= \sum_{\nu} d_\nu \: \delta(\vb{x} - \vb{x}_\nu) \end{aligned}$$ We thus arrive at the **Maxwell-Bloch equations**, -which are relevant for laser theory: +which are the foundation of laser theory: $$\begin{aligned} \boxed{ \begin{aligned} \mu_0 \pdv[2]{\vb{P}^{+}}{t} - &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}^{+}}{t} + &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}^{+}}{t} \\ \pdv{\vb{P}^{+}}{t} &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} diff --git a/content/know/concept/multi-photon-absorption/index.pdc b/content/know/concept/multi-photon-absorption/index.pdc index b208cfe..a5f4ad7 100644 --- a/content/know/concept/multi-photon-absorption/index.pdc +++ b/content/know/concept/multi-photon-absorption/index.pdc @@ -4,6 +4,7 @@ firstLetter: "M" publishDate: 2022-01-30 categories: - Physics +- Optics - Quantum mechanics - Nonlinear optics - Perturbation diff --git a/content/know/concept/salt-equation/index.pdc b/content/know/concept/salt-equation/index.pdc index 2f2917b..6383469 100644 --- a/content/know/concept/salt-equation/index.pdc +++ b/content/know/concept/salt-equation/index.pdc @@ -5,6 +5,7 @@ publishDate: 2022-02-07 categories: - Physics - Optics +- Laser theory date: 2022-01-20T22:01:48+01:00 draft: false @@ -66,10 +67,10 @@ that the interactions between the modes are limited: $$\begin{aligned} \vb{E}^{+}(\vb{x}, t) - = \sum_{n = 1}^N \Psi_n(\vb{x}) e^{- i \omega_n t} + = \sum_{n = 1}^N \Psi_n(\vb{x}) \: e^{- i \omega_n t} \qquad \qquad \vb{P}^{+}(\vb{x}, t) - = \sum_{n = 1}^N \vb{p}_n(\vb{x}) e^{- i \omega_n t} + = \sum_{n = 1}^N \vb{p}_n(\vb{x}) \: e^{- i \omega_n t} \end{aligned}$$ Using the modes' linear independence to treat each term of the summation individually, |