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A symplectic matrix such as the normal mode matrix can be expressed in terms of the extended Twiss parameters. In 6 by 6 case, those are
\begin{pmatrix} \alpha_x & \beta_x & 0 & 0 & \zeta_x & \eta_x \\ 0 & \psi_x & 0 & 0 & \zeta '_x & \eta ' _x \\ r_1 & r_2 & \alpha_y & \beta _y & \zeta _y & \eta _y \\ r_3 & r_4 & 0 & \psi _y & \zeta '_y & \eta ' _y \\ 0 & 0 & 0 & 0 & \alpha _z & \beta _z \\ 0 & 0 & 0 & 0 & 0 & \psi _z \end{pmatrix} }}
\( \alpha _x,\alpha _y, \alpha _z, \beta_x, \beta_y, \beta_z \) are alphas and betas function in the usual sense, after a diagonalization to 2 by 2 submatrices. \( \psi_x,\psi_y,\psi_z \) are the rotation angle to set one the coordinate to parallel to the (X,Y,Z) axes. \( r_1,r_2,r_3,r_4 \) are the components of the x-y coupling matrix (see x-y-coupling). \( \eta_x,\eta ' _x, \eta_y, \eta ' _y \) are "dispersions" which decouples synchro-beta coupling terms together with \( \zeta _x,\zeta '_x,\zeta _y,\zeta '_y \). Those parameters should agree with what FFS calculates in the case of no synchro-beta couplings.
Let V denote the matrix to define the normal mode, i.e., \( \vec{X} = V \vec{x} \)
where \( \vec{X} = \{X, P_X, Y, P_Y, Z, P_Z \} \) and \( \vec{x} = \{x, p_x, y, p_y, z, \delta \} \) are normal and physical coordinates, respectively. The matrix V can be expressed as
\( V = P B R H \) ,
where
with
\( a^2 + det[H_x] + det[H_y] = 1 \) , \( b^2 + det[R] = 1 \) .
Symbols I, J2, Hx,y, r, Bx,y,z, Px,y,z above are 2 by 2 matrices:
,
,
,
,
.
Matrices Hx,y defines dispersions as
Hx,y are also expressed as
{{zetax, etax},
{zetapx, etapx}}
{zetay, etay},
{zetapy, etapy}} = R . Join[Hx, Hy] .
The transformation matrix from the physical coordinate {x,px,y,py} to the x-y decoupled coordinate {X,Px,Y,Py} is written as
R = {{mu I, J . Transpose[r] . J}, {r, mu I}},
with the submatrix r={{R1, R2},{R3, R4}}, where mu^2 + Det[r] = 1, I = {{1,0},{0,1}}, and J={{0, 1}, {-1, 0}}. Let T stand for the physical transfer matrix from location 1 to location 2, then the transformation in the decoupled coordinate is diagonalized as
R_2 . T . Inverse[R_1] = {{T_X ,0}, {0, T_Y}} .
The Twiss parameters are defined for the matrices T_X and T_Y.
The revolution matrix is expressed as
The matrix U are characterized by tunes of three modes.
The beam envelop matrix is expressed as
