Reduction of degrees of freedom

[VForiel]

• [x] Show that some shifters can be removed
• [ ] Explore the idea of Romain Laugier about considering modes instead of independant shifters -> can it lead to a better convergeance method?

Discussed in https://github.com/Leirof/Tunable-Kernel-Nulling/discussions/31

^{Originally posted by **Leirof** March 4, 2024} Some degree of freedom seems redoundant, such as $P_1$ which can be removed and use the first input's phase as the reference phase. We can also consider removing $P_9$, $P_{10}$ and $P_{12}$ for the same reason. Does it work as expected? Can we also consider removing $P_3$, $P_5$ and $P_6$? 

[VForiel]

By adding the phase plots to the manual shift control interface and playing with it, I noticed that, for any initial perturbations, there is a way of optimizing the system using only P2, P3, P4, P6, P9, P10 and P12

Random initial state:

By playing with P2, P3 and P4 only, it is possible to optimize the bright output (and the first null output at the same time)

I can then correct the phases for the null output 2 and 3 by playing only with P6 (or alternatively with P7)

Finally, I can do the same for each pair of dark output by modifying only one of the two shifters before each cross recombiner

Unfortunately, this process require to know the phases at different points in the system, which is an information we don't have access to in practice. Thus, there is no straightforward implementation of this method.

Additionaly the current calibration algorithme seems very affected by the reduction of the degree of freedoms as it has more trouble to converge to a satisfying solution, and this solution seems often less good than when I let the algorithme play with all the degree of freedom.