Categories: Optics, Physics.

Coupled mode theory

Given an optical resonator (e.g. a photonic crystal cavity), consider one of its quasinormal modes with frequency ω0\omega_0 and decay rate 1/τ01 / \tau_0. Its complex amplitude AA is governed by:

dAdt=( ⁣ ⁣iω01τ0)A\begin{aligned} \dv{A}{t} &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A \end{aligned}

We choose to normalize AA so that A(t)2|A(t)|^2 is the total energy inside the resonator at time tt.

Suppose that NN waveguides are now “connected” to this resonator, meaning that the resonator mode AA and the outgoing waveguide mode SoutS_\ell^\mathrm{out} overlap sufficiently for AA to leak into SoutS_\ell^\mathrm{out} at a rate 1/τ1 / \tau_\ell. Conversely, the incoming mode SinS_\ell^\mathrm{in} brings energy to AA. Therefore, we can write up the following general set of equations:

dAdt=( ⁣ ⁣iω01τ0)A=1N1τA+=1NαSinSout=βSin+γA\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 α\alpha_\ell and γ\gamma_\ell are unknown coupling constants, and β\beta_\ell represents reflection. We normalize SinS_\ell^\mathrm{in} so that Sin(t)2|S_\ell^\mathrm{in}(t)|^2 is the power flowing towards AA at time tt, and likewise for SoutS_\ell^\mathrm{out}.

Note that we have made a subtle approximation here: by adding new damping mechanisms, we are in fact modifying ω0\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 τ\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 τ0\tau_0 is large (its effect is already included in ω0\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 τ0\tau_0:

dAdt( ⁣ ⁣iω01τ)A+=1NαSin\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 γ\gamma_\ell, we use energy conservation. If all Sin=0S_{\ell'}^\mathrm{in} = 0, then the energy in AA decays as:

dA2dt=dAdtA+AdAdt=( ⁣ ⁣iω01τ)A2+(iω01τ)A2=2τA2\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 SoutS_\ell^\mathrm{out}, meaning:

Sout2=dA2dt=2τA2\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 γ=2/τ|\gamma_\ell| = \sqrt{2 / \tau_\ell}. Because the phase of SoutS_\ell^\mathrm{out} is arbitrarily defined, γ\gamma_\ell need not be complex, so we choose γ=2/τ\gamma_\ell = \sqrt{2 / \tau_\ell}.

Next, to find α\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 γ\gamma_\ell, AA evolved as follows, with the lost energy ending up in SoutS_\ell^\mathrm{out}:

A(t)=Aeiω0tt/τ\begin{aligned} A(t) = A e^{-i \omega_0 t - t / \tau_\ell} \end{aligned}

After reversing time, AA evolves like so, where we have taken the complex conjugate to preserve the meanings of the symbols AA, SoutS_\ell^\mathrm{out}, and SinS_\ell^\mathrm{in}:

A(t)=Aeiω0t+t/τ\begin{aligned} A(t) = A e^{-i \omega_0 t + t / \tau_\ell} \end{aligned}

We insert this expression for A(t)A(t) into its original differential equation, yielding:

dAdt=( ⁣ ⁣iω0+1τ)A=( ⁣ ⁣iω01τ)A+αSin\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 AA leads us to the following power balance equation:

A=ατ2Sin    α2Sin2=4τ2A2\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 SinS_\ell^\mathrm{in} ends up in AA, so we know:

Sin2=dA2dt=2τA2\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 α=2/τ|\alpha_\ell| = \sqrt{2 / \tau_\ell}. Discarding the phase thanks to our choice of γ\gamma_\ell, we conclude that α=2/τ=γ\alpha_\ell = \sqrt{2 / \tau_\ell} = \gamma_\ell.

Finally, β\beta_\ell can also be determined using energy conservation. Again using our weak coupling assumption, if energy is only entering and leaving AA through waveguide \ell, we have:

Sin2Sout2=dA2dt\begin{aligned} |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2 = \dv{|A|^2}{t} \end{aligned}

Meanwhile, using the differential equation for AA, we find the following relation:

dA2dt=dAdtA+AdAdt=2τA2+α(SinA+(Sin)A)\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 dA2/dt\idv{|A|^2}{t} and equating them, we arrive at:

Sin2Sout2=2τA2+α(SinA+(Sin)A)\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 γ\gamma_\ell and β\beta_\ell, namely γA=SoutβSin\gamma_\ell A = S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in}, and use α=γ\alpha_\ell = \gamma_\ell:

Sin2Sout2=(SoutβSin)((Sout)β(Sin))  +Sin((Sout)β(Sin))+(Sin)(SoutβSin)=β2Sin2Sout2+βSin(Sout)+β(Sin)Sout  +Sin(Sout)βSin2+(Sin)SoutβSin2=(β2+β+β)Sin2Sout2  +(1β)Sin(Sout)+(1β)(Sin)Sout\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 β=1\beta_\ell = -1. Combined with α=γ=2/τ\alpha_\ell = \gamma_\ell = \sqrt{2 / \tau_\ell}, the coupled-mode equations take the following form:

dAdt=( ⁣ ⁣iω01τ0)A=1N1τA+=1N2τSinSout=Sin+2τA\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.

© Marcus R.A. Newman, CC BY-SA 4.0.
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