Theoretical background
Super-modes
The 6-equation system for the electric field in a bounded environment can be reduced to
the Helmholtz equation as follows:
\[H(z)\psi(x,y,z) = \lambda^2\psi(x,y,z)\]
Here we have:
\(H\) the Hamiltonian of the system
\(\psi\) the amplitude of the electric field
\(\lambda\) the eigenvalue of the problem
Applied to the optical mode propagation problem we retrieve the following relation:
\[\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + k^2n^2(x,y) \right) \psi(x,y,z) = \lambda^2\psi(x,y,z)\]
Here we have:
\(k\) the wave-number in of the light field
\(n(x,y)\) the refractive index (RI) profile of the optical component
\(\psi\) the amplitude of the electric field
Couple mode theory
In order to study the behavior of the solutions (super-modes) we need to know,
how they couple together.
In the same RI configuration the modes or orthogonal, i.e:
\[\left< \psi_{l,m} \, | \psi_{l',m'} \, \right> = \delta_{l,l'} \,\, \delta_{m,m'}\]
However if the RI configuration is varying in the z-direction coupling can occur between modes.
This coupling is defined as follows:
\[C_{ij} = \frac{-i}{2}
\frac{k^2}{\sqrt{\lambda_i \lambda_2}}
\frac{1}{\Delta \lambda_{ij}}
\int_A \psi_i(r, \theta) \psi_j(r, \theta) \frac{d}{dz}n^2(r, \theta, z) dr d\theta\]
Mode propagation
Knowing all the coupling factors and propagation constant for each supermodes
at each z-axis slices we can compute the mode-propagation.
\[\begin{split}\frac{dU}{dz} = i \begin{bmatrix}
\lambda_1 & C_{12} & C_{13} \\
C_{21} & \lambda_2 & C_{23} \\
C_{31} & C_{32} & \lambda_3
\end{bmatrix} U\end{split}\]
Where U is a vector array containing all the super-modes amplitudes.
Adiabatic criterion
\[\widetilde{C}_{ij} (z) = C_{ij} . \left( \frac{1}{\rho} \frac{d\rho}{dz} \right)^{-1}\]
The respect of the adiabatic criterion ensure that modes wont be coupling together.