.. _theory:

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,

.. math::
  H(z)\psi(x,y,z) = \lambda^2\psi(x,y,z)

| Here we have:
|    :math:`H` the Hamiltonian of the system
|    :math:`\psi` the amplitude of the electric field
|    :math:`\lambda` the eigenvalue of the problem

Applied to the optical mode propagation problem we retrieve the following relation:

.. math::
  \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:
|    :math:`k` the wave-number in of the light field
|    :math:`n(x,y)` the refractive index (RI) profile of the optical component
|    :math:`\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:

.. math::
   \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:

.. math::
  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.

.. math::
  \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

Where U is a vector array containing all the super-modes amplitudes.


Adiabatic criterion
-------------------

.. math::
  \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.