Coupled Mode Theory
Derivation of CMT and Supermodes
Starting from the Helmholtz equation:
$$\left[\nabla^2 + n^2k_0^2\right]\psi = 0$$
The propagation along $z$ can be expressed as:
$$\partial_z^2\psi=\left[\nabla^2_\perp + \sum_{m}n_m^2k_0^2\right]\psi$$
The individual modes $u_m$ satisfy the eigenvalue equation:
$$\beta_m^2u_m(x,y)=\left[\nabla^2_\perp + n_m^2k_0^2\right]u_m(x,y)$$
We expand the total field as a sum of these modes:
$$\psi(x,y,z)=\sum_{m}a_m(z)u_m(x,y)e^{-i\beta_m z}=\sum_{m}A_m(z)u_m(x,y)$$
| Plugging into Eq(2) and using the Slowly Varying Envelope Approximation (SVEA), where $ | \partial_z a | \gg | \partial_z^2 a | $: |
$$\sum_{m}2i\beta_m\frac{da_m(z)}{dz}u_m e^{-i\beta_m z} = -\sum_{m}\Delta n^2 k_0^2 a_m u_m e^{-i\beta_m z}$$
Projecting using mode orthonormality $\int u_m^*u_p dx dy = \delta_{pm}$ yields:
$$i\frac{da_p(z)}{dz} = \sum_{m}\kappa_{pm} e^{i(\beta_p-\beta_m)z} a_m$$
where the coupling coefficient $\kappa_{pm}$ is defined as:
$$\kappa_{pm} = \frac{k_0^2}{2\beta_p}\int (n^2-n_m^2)u^*_p(x,y)u_m(x,y) dx dy$$
Using $A_m = a_m e^{-i\beta_m z}$, we can write this in matrix form:
$$i\frac{d\mathbf{A}}{dz}=\mathbf{H}\mathbf{A}$$
where $H_{pm} = \beta_p\delta_{pm} + \kappa_{pm}$. Therefore, the supermodes satisfy:
$$
\begin{aligned}
\mathbf{A} &= \mathbf{v}e^{-i\beta z} \\
\mathbf{H}\mathbf{v} &= \beta\mathbf{v} \\
\beta &\in \text{eig}(\mathbf{H})
\end{aligned}
$$