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diff --git a/source/know/concept/coupled-mode-theory/index.md b/source/know/concept/coupled-mode-theory/index.md new file mode 100644 index 0000000..acb4710 --- /dev/null +++ b/source/know/concept/coupled-mode-theory/index.md @@ -0,0 +1,230 @@ +--- +title: "Coupled mode theory" +date: 2022-03-31 +categories: +- Physics +- Optics +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: + +$$\begin{aligned} + \dv{A}{t} + &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A +\end{aligned}$$ + +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: + +$$\begin{aligned} + A + = \frac{\alpha_\ell \tau_\ell}{2} S_\ell^\mathrm{in} + \qquad \implies \qquad + |\alpha_\ell|^2 |S_\ell^\mathrm{in}|^2 + = \frac{4}{\tau_\ell^2} |A|^2 +\end{aligned}$$ + +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 $\idv{|A|^2}{t}$ +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 $\alpha_\ell = \gamma_\ell$: + +$$\begin{aligned} + |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 + &= - \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big) \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big) + \\ + &\quad\; + S_\ell^\mathrm{in} \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big) + + (S_\ell^\mathrm{in})^* \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big) + \\ + &= - |\beta_\ell|^2 |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 + + \beta_\ell S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + \beta_\ell^* (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} + \\ + &\quad\; + S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* - \beta_\ell^* |S_\ell^\mathrm{in}|^2 + + (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} - \beta_\ell |S_\ell^\mathrm{in}|^2 + \\ + &= - (|\beta_\ell|^2 + \beta_\ell + \beta_\ell^*) |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 + \\ + &\quad\; + (1 - \beta_\ell) S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + (1 - \beta_\ell^*) (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} +\end{aligned}$$ + +This equation is only satisfied if $\beta_\ell = -1$. +Combined with $\alpha_\ell = \gamma_\ell = \sqrt{2 / \tau_\ell}$, +the **coupled-mode equations** take the following form: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \dv{A}{t} + &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A + - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A + + \sum_{\ell = 1}^N \sqrt{\frac{2}{\tau_\ell}} S_\ell^\mathrm{in} + \\ + S_\ell^\mathrm{out} + &= - S_\ell^\mathrm{in} + \sqrt{\frac{2}{\tau_\ell}} A + \end{aligned} + } +\end{aligned}$$ + +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. + |