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diff --git a/source/know/concept/coupled-mode-theory/index.md b/source/know/concept/coupled-mode-theory/index.md index ecae1bb..6a5ec1b 100644 --- a/source/know/concept/coupled-mode-theory/index.md +++ b/source/know/concept/coupled-mode-theory/index.md @@ -10,21 +10,21 @@ layout: "concept" 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: +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 $t$. +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 $A$. +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: $$\begin{aligned} @@ -36,35 +36,35 @@ $$\begin{aligned} &= \beta_\ell S_\ell^\mathrm{in} + \gamma_\ell A \end{aligned}$$ -Where $\alpha_\ell$ and $\gamma_\ell$ are unknown coupling constants, -and $\beta_\ell$ represents reflection. -We normalize $S_\ell^\mathrm{in}$ -so that $|S_\ell^\mathrm{in}(t)|^2$ is the power flowing towards $A$ at time $t$, -and likewise for $S_\ell^\mathrm{out}$. +Where $$\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}$$. Note that we have made a subtle approximation here: by adding new damping mechanisms, -we are in fact modifying $\omega_0$; +we are in fact modifying $$\omega_0$$; see the [harmonic oscillator](/know/concept/harmonic-oscillator/) for a demonstration. However, the frequency shift is second-order in the decay rate, -so by assuming that all $\tau_\ell$ are large, +so by assuming that all $$\tau_\ell$$ are large, we only need to keep the first-order terms, as we did. This is called **weak coupling**. -If we also assume that $\tau_0$ is large -(its effect is already included in $\omega_0$), +If we also assume that $$\tau_0$$ is large +(its effect is already included in $$\omega_0$$), then we can treat the decay mechanisms separately: -to analyze the decay into a certain waveguide $\ell$, -it is first-order accurate to neglect all other waveguides and $\tau_0$: +to analyze the decay into a certain waveguide $$\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}$$ -To determine $\gamma_\ell$, we use energy conservation. -If all $S_{\ell'}^\mathrm{in} = 0$, -then the energy in $A$ decays as: +To determine $$\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} @@ -77,7 +77,7 @@ $$\begin{aligned} \end{aligned}$$ Since all other mechanisms are neglected, -all this energy must go into $S_\ell^\mathrm{out}$, meaning: +all this energy must go into $$S_\ell^\mathrm{out}$$, meaning: $$\begin{aligned} |S_\ell^\mathrm{out}|^2 @@ -85,32 +85,32 @@ $$\begin{aligned} = \frac{2}{\tau_\ell} |A|^2 \end{aligned}$$ -Taking the square root, we clearly see that $|\gamma_\ell| = \sqrt{2 / \tau_\ell}$. -Because the phase of $S_\ell^\mathrm{out}$ is arbitrarily defined, -$\gamma_\ell$ need not be complex, so we choose $\gamma_\ell = \sqrt{2 / \tau_\ell}$. +Taking the square root, we clearly see that $$|\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}$$. -Next, to find $\alpha_\ell$, we exploit the time-reversal symmetry +Next, to find $$\alpha_\ell$$, we exploit the time-reversal symmetry of [Maxwell's equations](/know/concept/maxwells-equations/), which govern the light in the resonator and the waveguides. -In the above calculation of $\gamma_\ell$, $A$ evolved as follows, -with the lost energy ending up in $S_\ell^\mathrm{out}$: +In the above calculation of $$\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}$$ -After reversing time, $A$ evolves like so, +After reversing time, $$A$$ evolves like so, where we have taken the complex conjugate to preserve the meanings of the symbols -$A$, $S_\ell^\mathrm{out}$, and $S_\ell^\mathrm{in}$: +$$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}$$ -We insert this expression for $A(t)$ into its original differential equation, yielding: +We insert this expression for $$A(t)$$ into its original differential equation, yielding: $$\begin{aligned} \dv{A}{t} @@ -118,7 +118,7 @@ $$\begin{aligned} = \bigg( \!-\! i \omega_0 - \frac{1}{\tau_\ell} \bigg) A + \alpha_\ell S_\ell^\mathrm{in} \end{aligned}$$ -Isolating this for $A$ leads us to the following power balance equation: +Isolating this for $$A$$ leads us to the following power balance equation: $$\begin{aligned} A @@ -129,7 +129,7 @@ $$\begin{aligned} \end{aligned}$$ But thanks to energy conservation, -all power delivered by $S_\ell^\mathrm{in}$ ends up in $A$, so we know: +all power delivered by $$S_\ell^\mathrm{in}$$ ends up in $$A$$, so we know: $$\begin{aligned} |S_\ell^\mathrm{in}|^2 @@ -138,20 +138,20 @@ $$\begin{aligned} \end{aligned}$$ 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 $\alpha_\ell = \sqrt{2 / \tau_\ell} = \gamma_\ell$. +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. +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: +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 $A$, +Meanwhile, using the differential equation for $$A$$, we find the following relation: $$\begin{aligned} @@ -161,7 +161,7 @@ $$\begin{aligned} &= - \frac{2}{\tau_\ell} |A|^2 + \alpha_\ell \Big( S_\ell^\mathrm{in} A^* + (S_\ell^\mathrm{in})^* A \Big) \end{aligned}$$ -By isolating both of the above relations for $\idv{|A|^2}{t}$ +By isolating both of the above relations for $$\idv{|A|^2}{t}$$ and equating them, we arrive at: $$\begin{aligned} @@ -169,9 +169,9 @@ $$\begin{aligned} &= - \frac{2}{\tau_\ell} |A|^2 + \alpha_\ell \Big( S_\ell^\mathrm{in} A^* + (S_\ell^\mathrm{in})^* A \Big) \end{aligned}$$ -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 $\alpha_\ell = \gamma_\ell$: +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 $$\alpha_\ell = \gamma_\ell$$: $$\begin{aligned} |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 @@ -191,8 +191,8 @@ $$\begin{aligned} &\quad\; + (1 - \beta_\ell) S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + (1 - \beta_\ell^*) (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} \end{aligned}$$ -This equation is only satisfied if $\beta_\ell = -1$. -Combined with $\alpha_\ell = \gamma_\ell = \sqrt{2 / \tau_\ell}$, +This equation is only satisfied if $$\beta_\ell = -1$$. +Combined with $$\alpha_\ell = \gamma_\ell = \sqrt{2 / \tau_\ell}$$, the **coupled-mode equations** take the following form: $$\begin{aligned} |