path: root/content/know/concept/coupled-mode-theory/index.pdc

---
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:

$$\begin{aligned}
    A
    = \frac{\alpha_\ell \tau_\ell}{2} S_\ell^\mathrm{in}
    \qquad \implies \qquad
    |\alpha_\ell|^2 |S_\ell^\mathrm{in}|^2
    = \frac{4}{\tau_\ell^2} |A|^2
\end{aligned}$$

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 $\dv*{|A|^2}{t}$
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 $\alpha_\ell = \gamma_\ell$:

$$\begin{aligned}
    |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
    &= - \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big) \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big)
    \\
    &\quad\; + S_\ell^\mathrm{in} \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big)
    + (S_\ell^\mathrm{in})^* \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big)
    \\
    &= - |\beta_\ell|^2 |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
    + \beta_\ell S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + \beta_\ell^* (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out}
    \\
    &\quad\; + S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* - \beta_\ell^* |S_\ell^\mathrm{in}|^2
    + (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} - \beta_\ell |S_\ell^\mathrm{in}|^2
    \\
    &= - (|\beta_\ell|^2 + \beta_\ell + \beta_\ell^*) |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
    \\
    &\quad\; + (1 - \beta_\ell) S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + (1 - \beta_\ell^*) (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out}
\end{aligned}$$

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.