Categories: Optics, Physics.

# 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 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, 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.

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.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.