The trajectory of a single particle through a section of beam line elements can, to lowest order, be written in matrix form \[ \vec{X}_{\rm final} = M \;\vec{X}_{\rm initial} \]
where the matrix \(M\) corresponds to transport from location \(s_{\rm initial}\) to \(s_{\rm final}\) and \(\vec{X} = (x, x')^T\), \(x\) being the transverse coordinate and \(x' = dx/ds\). \(M\) itself may be decomposed into several individual matrices, each corresponding to transport through an individual bending or focusing element, or drift through a region absent of such elements:
\[ M = M_n \; M_{n-1} \cdots M_2 \; M_1. \]
Elsewhere1 we have investigated the propagation of the moments of particle distributions through such a system. Below we expand upon several representations of this motion in terms of previously introduced matrices.
From \[ \vec{X}_{\rm final} \cdot \vec{X}_{\rm final}^T = (M\vec{X}_{\rm initial})(M\vec{X}_{\rm initial})^T = M(\vec{X}_{\rm initial}\cdot \vec{X}_{\rm initial}^T) M^T = M \;\; \begin{pmatrix} x^2 & x'x \\ x'x &x'^2 \end{pmatrix}_{\rm initial} M^T \] we have seen that \[ \Sigma_{\rm final} = \begin{pmatrix} \langle x^2 \rangle & \langle x'x \rangle \\ \langle x'x \rangle & \langle x'^2 \rangle \end{pmatrix}_{\rm final} = M\; \Sigma_{\rm initial} \;M^T. \]
Re-writting the moments in the Sigma matrix in terms of Courant-Snyder parameters, \[ \alpha \equiv -\pi\langle x'x\rangle/\epsilon, ~~~~~~ \beta \equiv \pi\langle x^2\rangle/\epsilon, ~~~~~~~ \gamma \equiv \pi\langle x'^2\rangle/\epsilon, \] where \[ \epsilon = \pi\sqrt{\langle x^2\rangle \langle x'^2\rangle - \langle x'x\rangle^2} \] is the “rms emittance”, or rms phase space area of the particle distribution, then after noting that the emittance is conserved we arrive at \[ K_{\rm final} = M \;K_{\rm initial} \;M^T \] where \[ K \equiv \begin{pmatrix} \beta &-\alpha \\ -\alpha & \gamma \end{pmatrix}. \]
By writing down the solution to Hill’s equation as \(x(s) = \sqrt{\beta(s)}[a\cos\psi(s)+b\sin\psi(s)]\) where \(a\) and \(b\) are given by initial conditions, we found that the transport matrix \(M\) that propagates trajectories between two points \(s_1\) and \(s_2\) can be written in terms of the Courant-Snyder parameters at the two points and the phase advance between them:
\[ M = \begin{pmatrix} \sqrt{\frac{\beta_2}{\beta_1}}(\cos\Delta\psi + \alpha_1\sin\Delta\psi) & \sqrt{\beta_1\beta_2}\sin\Delta\psi \\ -\;\frac{(1+\alpha_1\alpha_2)\sin\Delta\psi +(\alpha_2-\alpha_1)\cos\Delta\psi}{\sqrt{\beta_1\beta_2}} & \sqrt{\frac{\beta_1}{\beta_2}}(\cos\Delta\psi - \alpha_2\sin\Delta\psi) \end{pmatrix} \]
Observe that the above form of the matrix \(M\) can be deconstructed into the multiplication of the following matrices: \[ M = U_2 \;R_{\Delta\psi}\;U_1^{-1} \] where \[ U \equiv \begin{pmatrix} \sqrt{\beta} & 0 \\ -\frac{\alpha}{\sqrt{\beta}} & \frac{1}{\sqrt{\beta}} \end{pmatrix}, ~~~~~~~~ R_\theta \equiv \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}. \]
This “Floquet” transformation suggests that we can describe the betatron motion as a pure rotation in the appropriate space given by \[ \vec{V}_2 = R_{\Delta\psi}\;\vec{V}_1 \] where \[ \vec{V} \equiv (U^{-1}\vec{X}) = \begin{pmatrix} \frac{x}{\sqrt{\beta}} \\ \frac{\alpha x + \beta x'}{\sqrt{\beta}} \end{pmatrix}. \]
We note that \(\det U = 1\) and \[ U^{-1} = \begin{pmatrix} \frac{1}{\sqrt{\beta}} & 0 \\ \frac{\alpha}{\sqrt{\beta}} & \sqrt{\beta} \end{pmatrix} \] and so, to summarize, the solution to the equation of motion can be factored as \[ \begin{pmatrix} x_2 \\ x'_2 \end{pmatrix} = \frac{1}{\sqrt{\beta_1\beta_2} } \begin{pmatrix} \beta_2 & 0 \\ -\alpha_2 & 1 \end{pmatrix} \begin{pmatrix} ~\cos\Delta\psi & ~\sin\Delta\psi \\ -\sin\Delta\psi & ~\cos\Delta\psi \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \alpha_1 & \beta_1 \end{pmatrix} \begin{pmatrix} x_1 \\ x'_1 \end{pmatrix}. \] If a new coordinate \(p=\beta x'+\alpha x\) is introduced, then the motion can also be described as \[ \begin{pmatrix} x_2 \\ p_2 \end{pmatrix} = \sqrt{\frac{\beta_2}{\beta_1} } \begin{pmatrix} ~\cos\Delta\psi & ~\sin\Delta\psi \\ -\sin\Delta\psi & ~\cos\Delta\psi \end{pmatrix} \begin{pmatrix} x_1 \\ p_1 \end{pmatrix}. \] Reiterating, the motion is a pure rotation in phase space when the Floquet variables are used: \[ \begin{pmatrix} x_2/\sqrt{\beta_2} \\ p_2/\sqrt{\beta_2} \end{pmatrix} = \begin{pmatrix} ~\cos\Delta\psi & ~\sin\Delta\psi \\ -\sin\Delta\psi & ~\cos\Delta\psi \end{pmatrix} \begin{pmatrix} x_1/\sqrt{\beta_1} \\ p_1/\sqrt{\beta_1} \end{pmatrix}, \] where the phase advance has the true meaning of a rotation angle.
Given the general transport matrix \(M\) between our two points of interest, it is often possible for it to be expressed in terms of periodic Courant-Snyder parameters and phase advance. In particular, if the trace of the matrix is less than two in absolute value, then there exist initial Courant-Snyder parameters \(\alpha_0\), \(\beta_0\), \(\gamma_0\) which will become reproduced after performing the transport transformation: \[ K_0 = M K_0 M^T . \] Obtaining the values of the periodic Courant-Snyder parameters is more straightforward when one notes that for \(\beta_2 = \beta_1 \equiv \beta_0\) and so forth, then the Courant-Snyder parameterization of \(M\) becomes
\[ M = \begin{pmatrix} \cos\mu + \alpha_0\sin\mu & \beta_0\sin\mu \\ -\gamma_0\sin\mu & \cos\mu - \alpha_0\sin\mu \end{pmatrix} \] where we remember the defnintion \(\gamma = (1+\alpha^2)/\beta\) and we denote the periodic phase advance through the system described by the matrix as \(\mu = \Delta\psi_{periodic}\).
Thus, if we have a numerical representation of \(M\) we can compute the periodic Courant-Snyder parameters via \[ \mu = \cos^{-1}[(M_{11}+M_{22})/2] \\ \alpha_0 = (M_{11}-M_{22})/(2\sin\mu) \\ \beta_0 = ~~M_{12}/\sin\mu \\ \gamma_0 = -M_{21}/\sin\mu \] where the sign of \(\sin\mu\) is equal to the sign of \(M_{12}\) (i.e., \(\beta >0\)). Even if \(M\) does not represent a periodic section of beam line or a ring, etc., it is possible to express \(M\) in terms of periodic Courant-Snyder functions so long as \(\mu\) is a real number.
Notice that the expression for \(M\) in terms of its periodic Courant-Snyder functions has the interesting form:
\[ M = \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \cos\mu + \begin{pmatrix} \alpha_0\ & \beta_0 \\ -\gamma_0 & -\alpha_0 \end{pmatrix}\sin\mu \\ = I\cos\mu + J_0\sin\mu \] where \[ J \equiv \begin{pmatrix} \alpha & \beta \\ -\gamma & -\alpha \end{pmatrix}. \]
Properties of \(J\) include: \(\det J = 1\), \(trace(J) = 0\), \(J^{-1} = -J\) and \(J^2 = -I\). This last property leads to the following notation: \[ M = I\cos\mu + J_0\sin\mu = e^{J_0\mu} \] which can be useful in certain calculations. For example, if \(M\) represents the passage of a particle about a storage ring, then in terms of the periodic values, \[ M = e^{J_02\pi\nu} \] where \(\nu\) is the betatron tune. The matrix for going twice around the ring is \(M^2 = e^{J_0\cdot4\pi\nu}\) and so forth. In general, akin to deMoivre’s Theorem, \[ M^n = (e^{J\mu})^n = e^{Jn\mu} = I\cos(n\mu)+J\sin(n\mu). \]
Just as values within the \(K\) matrix are transported according to \(K_2=MK_1M^T\), suppose that we have a periodic structure where the matrix from one point in a period to the corresponding point in the next downstream period is \(M_1\). Similarly, \(M_2\) corresponds to transport between two other similar points in the periodic sections. Finally, let \(M\) correspond to the transport matrix directly connecting points 1 and 2. Starting at point 1, \(M_2M\) is the same operation as \(MM_1\), or \[ M_2 = M \; M_1 \; M^{-1}. \] Hence it is easy to see that since \(M_i=I\cos\mu+J_i\sin\mu\), and since the trace of a prodcut of matrices (which yields \(\mu\)) is constant under cyclic permutations, then \[ J_2 = M J_1 M^{-1}. \]
As they appear so similar, it is natural to look for relationships between the \(J\) and \(K\) matrices. Upon inspection one can readily verify that \[ JK = S \] where \(S\) is the symplectic matrix, \[ S = \begin{pmatrix} ~~~0 & 1 \\ -1 & 0 \end{pmatrix}. \]
A matrix \(A\) is said to be symplectic if \(A^T S A = S\). Our matrix \(M\) is a symplectic matrix: \[ M^T S M = \begin{pmatrix} a & c \\ b & d \end{pmatrix} \begin{pmatrix} ~~~0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 0 & ad-bc \\ -(ad-bc) & 0 \end{pmatrix} = S \] since \(ad-bc = \det(M) = 1\).
Also, one can verify that
\[
K = SJ^T = -JS ~~~~~ K^T = -JS = K ~~~~~ K^{-1}=-SJ \\
J~=K^TS = KS~~~~~~~J^T = -SK ~~~~~~~~~~~~~ J^{-1}=-J~~~ \\
S^T = S^{-1} = -S ~~~~~~~~~~~~~ J^2 = S^2 = -I
\] Earlier we said that the periodic Courant-Snyder parameters when formed into the matrix \(K\) must satisfy \(K_0 = MK_0M^T\). If we start with the general result \(K_2 = M K_1 M^T\) and write the matrix \(M\) in terms of its periodic functions, then we have \[\begin{eqnarray}
K_2 &=& (Ic + J_0s)K_1(Ic+J_0s)^T \\
&=& (Ic + J_0s)K_1(Ic+J_0^Ts) \\
&=& K_1c^2 + J_0K_1cs + K_1J_0^Tcs + J_0K_1J_0^Ts^2
\end{eqnarray}\] where \(c \equiv \cos\mu\), \(s \equiv \sin\mu\).
We now suppose that we start with \(K_1 = K_0\), the periodic solutions. Then, \[\begin{eqnarray} K_2 &=& K_0c^2 + J_0K_0cs + K_0J_0^Tcs + J_0K_0J_0^Ts^2 \\ &=& K_0c^2 + (J_0K_0 - K_0(SK_0))cs - J_0K_0(S K_0)s^2 \\ &=& K_0c^2 + (J_0K_0 - (K_0S)K_0)cs - J_0(K_0S)K_0s^2 \\ &=& K_0c^2 + (J_0K_0 - J_0K_0)cs - J_0^2K_0s^2 \\ &=& K_0c^2 + K_0s^2 \\ &=& K_0 \end{eqnarray}\] and thus we end with the periodic solutions as expected.
The concept of momentum dispersion is typically introduced by noting that to lowest order in \(x\), \(x'\), and \(\delta\equiv \Delta p/p_0\), the solution to the equation of motion can be written as \[ x(s) = x_\beta(s) + D(s)\delta \] where \(x_\beta\) represents the solution for an “on-momentum” particle and \(x_p \equiv D\delta\) is the solution for the trajectory when \(x_\beta\) = 0; that is to say, from \[ x''(s,\delta) + K(s,\delta)x(s,\delta) = \frac{\delta}{\rho(s)}, \] the solution can be written generally in terms of a \(3\times 3\) matrix, \[ \begin{pmatrix} x_2 \\ x'_2 \\ \delta \end{pmatrix} = {\cal M} \begin{pmatrix} x_1 \\ x'_1 \\ \delta \end{pmatrix} = \begin{pmatrix} a & b & e \\ c & d & f \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x'_1 \\ \delta \end{pmatrix} \\ = {\cal M} \begin{pmatrix} x_{\beta 1} \\ x'_{\beta 1} \\ 0 \end{pmatrix} + {\cal M} \begin{pmatrix} D_1 \\ D'_1 \\ 1 \end{pmatrix}\delta. \]
For \(x_\beta=0\), the equation of motion of the dispersion function becomes \[ D''(s) + K(s)D(s) = \frac{1}{\rho(s)} \] where \(1/\rho\) is the local curvature generated by bending of the ideal trajectory, and the matrix solution for \(D\) therefore reduces to \[ \begin{pmatrix} D_2 \\ D'_2 \\ 1 \end{pmatrix} = \begin{pmatrix} a & b & e \\ c & d & f \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} D_1 \\ D'_1 \\ 1 \end{pmatrix}. \] Note that the sub-matrix \[ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] is the same transport matrix as we’ve been discussing (assuming \(K(s,\delta)\approx K(s)\) for small \(\delta\)) and the sub-vector \(\vec{E} = (e,f)^T\) comes from the dispersive elements present in the beam line segment.
In shorthand notation, \[ \begin{pmatrix} \vec{D}_2 \\ 1 \end{pmatrix} = {\cal M} \begin{pmatrix} \vec{D}_1 \\ 1 \end{pmatrix} = \begin{pmatrix} M & \vec{E} \\ \vec{0}^T & 1\end{pmatrix} \begin{pmatrix} \vec{D}_1 \\ 1 \end{pmatrix} \] with \(\vec{D} \equiv (D, D')^T\). We note that just as the matrix \(M\) can be written in terms of the Courant-Snyder parameters at the beginning and end of the beam line segment and the phase advance between the two points, the vector \(E\) can be written as \[ \vec{E} = \vec{D}_2 - M\vec{D}_1 \] and thus the entire matrix \({\cal M}\) can be parameterized in terms of initial and final lattice function values and the phase advance across the region.
Additionally, just as \(M\) can be written in terms of periodic Courant-Snyder parameters, \(\vec{E}\) can also: \[ \vec{E} = (I-M)\vec{D}_0 \] with \(\vec{D}_0\) being the periodic dispersion through the given region, if it exists.
So, any arbitrary \(3\times 3\) matrix describing transverse motion including off-momentum terms can be written in terms of periodic functions alone (assuming appropriate choice of \(\mu\) is permissible): \[ {\cal M} = \begin{pmatrix} e^{J_0\mu} & (I-e^{J_0\mu})\vec{D}_0 \\ \vec{0}^T & 1\end{pmatrix}. \]
See Ellipses tutorial.↩