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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 $1 / \tau_0$ in isolation. Its complex amplitude $A$ then obeys:

$$\begin{aligned} \dv{A}{t} &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A \end{aligned}$$

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 $\idv{|A|^2}{t}$ 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.

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