path: root/source/know/concept/coupled-mode-theory/index.md

---
title: "Coupled mode theory"
sort_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.