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+---
+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.
+