at.beam improvement

[lfarv]

at.beam again ! The present version processes either fully coupled 6x6 sigma matrices, or fully decoupled (3 2x2 independent matrices). But it misses two common cases:

1. coupled h/v motion with zero bunch length and energy spread,
2. zero vertical emittance, but H-δ coupling always appear in dispersive regions.

In both cases, 6x6 analysis fails and at.beam falls back the the uncoupled case, which is wrong.

In this PR, we try in order the following cases:

1. 6x6 H-V-δ matrix
2. 4x4 H-V matrix
3. 4x4 H-δ matrix
4. fully decoupled

(The coupled V-δ case where ε_x=0 is not handled but looks unrealistic).

The following figures show the improvement. We consider a simplified radiation source: equilibrium emittance in horizontal and longitudinal planes, zero vertical emittance. The Σ-matrix is built with at.sigma_matrix using the twiss_in keyword argument, looking at a dispersive region of the ring. With at.beam we generate a particle distribution (1000 particles), and we rebuild the Σ-matrix from the particle distribution (at.sigma_matrix with the beam keyword argument).

Figure 1: master branch

The original projection of the Σ-matrix on the x-δ plane shows the correlation. The particle distribution has the right σ_x and σ_δ but not the right correlation, so the rebuilt Σ-matrix is wrong.

Figure 2: this branch

Now the particle distribution has the right correlation, and the re-built Σ is very close to the initial one, even with only 1000 particles.

Note: the figures above make use of a at.plot.plot_sigma() function which appear in another pull request

[lcarver]

i am convinced by your plots