Matching-Script-for-Twiss-Functions-Accelerator-Physics

This script will match twiss functions and magnetic elements for designing lattices. It uses linear matrix transport of Courant-Snyder parameters(3x3) linear matrix for matching drift lengths, quadrupole focusing strengths, solenoid focusing strengths as well as find the optimal twiss functions. It uses the following formalism:

$$ \begin{equation} \mathcal{M} = \begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} \\ M_{21} & M_{22} &M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ M_{41} & M_{42} & M_{43} & M_{44} \end{pmatrix} \end{equation} $$

The transformation of optic functions are given as:

$$ \begin{equation} \begin{pmatrix} \beta_{f} \\ \alpha_{f} \\ \gamma_{f} \end{pmatrix}=\begin{pmatrix} M_{11}^{2} & -2M_{11}M_{12} & M_{12}^{2} \\ -2M_{11}M_{21} & (M_{11}M_{22} + M_{12}M_{21}) & -2M_{12}M_{22} \\ M_{21}^{2} & -2M_{21}M_{22} & M_{22}^{2} \end{pmatrix}\cdot \begin{pmatrix} \beta_{i} \\ \alpha_{i} \\ \gamma_{i} \end{pmatrix} \end{equation} $$

Where, $i,f$ stands for initial and final state. Using this approach we can find the optimized magnetic strengths of the elements. Similarly for Dispersion function we have:

$$ \begin{equation} \begin{split} D_{xf} &= M_{11}D_{xi} + M_{12}D_{xi}^{'} + \rho(1-\cos(l/\rho)) + M_{13}D_{yi} + M_{14}D_{yi}^{'}\\ D_{xf}' &= M_{21}D_{xi} + M_{22}D_{xi}^{'} + \sin(l/\rho) + M_{23}D_{yi} + M_{24}D_{yi}^{'}\\ D_{yf} &= M_{31}D_{xi} + M_{32}D_{xi}^{'} + M_{33}D_{yi} + M_{34}D_{yi}^{'} \\ D_{yf}^{'} &= M_{41}D_{xi} + M_{42}D_{xi}^{'} + M_{43}D_{yi}^{'} \end{split} \end{equation} $$

Where, $\rho$ is bending radius and $l$ is dipole length. Since we know how dispersion gets transported, this script can also be used for matching dispersion. Where the vertical dispersion is also added for solenoid case.