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3.1 Transportation matrix

3.1.1 From equations of motion to matrix formalism

In the previous section, the resolution of the homogeneous equation of motion revealed, that in a none dispersive system, the coordinates

and

of a particle are entirely determinedd by initial conditions and the knowledge of principals solution. These functions,

and

, which depend only on the system, determine entirely the R matrix:

$\displaystyle R(s)=\begin{pmatrix} C(s) & S(s)\\ C'(s)& S'(s)\\ \end{pmatrix}$

(3.1)

In the case of

constant, the functions

and

are sine like functions, thus, easy to determine. This transformation matrix provides for any vector of input coordinates, the coordinates of the particle at any position is known by doing the simple multiplication:

$\displaystyle \begin{pmatrix}u(s)\\ u'(s)\\ \end{pmatrix}=\begin{pmatrix}C(s) & S(s)\\ C'(s)& S'(s)\\ \end{pmatrix}.\begin{pmatrix}u_0 \\ u'_0\\ \end{pmatrix}$

(3.2)

Most of the time, the matrix is used to determine coordinates at the otput of the element, so that the dependence in

is droped, remplaced by the length of the component, the R matrix being constant:

$\displaystyle X_{out} = R. X_{in}$

(3.3)

The 'R' matrix are called transport matrix. The matrix formalism offers the opportunity to follow any particle trajectory along a beam line by repeated matrix multiplications from element to element. If the beam line is made of

elements, including drifts, and if an R matrix can be associated with all elements(i.e.

on each element), a global transport matrix for the beam line can be easily defined: $R=R_n.R_{n-1}...R_2.R_1$ . This kind of method is used by some algorithm like TRACE3D[8], to perfom particle tracking.

3.1.2 Interpretation in term of optics

3.1.2.0.1 Case of a drift

According to figure 3.1, if the red line stands for the beam trajectory through a simple drift, we have the following result in two dimensions with the approximation of small angles:

$\displaystyle \begin{pmatrix} x_{out}\\ x'_{out}\\ \end{pmatrix}= \begin{pmat... ...\\ 0 & 1& \\ \end{pmatrix}\begin{pmatrix} x_{in} \\ x'_{in}\\ \end{pmatrix}$

In this example we recognize the matrix corresponding to the case

, developped in the previous chapter.

**Figure 3.1:** Illustration of the matrix formalism in optics - case of a drift. It appear clear tha for a linear trajectory (in red), stays constant, while the increase of is a function of the drift length .

3.1.2.0.2 Thin lens approximation

Principal solutions for

constant, have been given in the previous chapter. The corresponding transport matrix for an element of length

are:
For

(defocusing element)

$\displaystyle R_D= \begin{pmatrix}\cosh \sqrt{\vert k\vert} l & \frac{1}{\sqrt{... ...k\vert} \sinh \sqrt{\vert k\vert}l & \cosh \sqrt{\vert k\vert}l\\ \end{pmatrix}$

(3.4)

For

(focusing element)

$\displaystyle R_F= \begin{pmatrix}\cos \sqrt{k} l & \frac{1}{\sqrt{\vert k\vert}} \sin \sqrt{k}l \\ -\sqrt{k} \sin \sqrt{k}l & \cos \sqrt{k}l\\ \end{pmatrix}$

(3.5)

If the length of the element tend to 0 ( $l\rightarrow 0$ ), we have $\sin \sqrt{k}l \sim \sqrt{k}l$ , $\cos \sqrt{k}l \sim 1$ , and we can write the previous matrix in the thin lens approximation:

$\displaystyle R_{\frac{F}{D}}= \displaystyle\begin{pmatrix}\; \; 1 & 0\\ \pm \frac{1}{f}&1\\ \end{pmatrix}$

(3.6)

where we defined the focal length $f^{-1}=kl$ . One can recognize the matrix of a focusing/defocusing lens in the previous equation, revealing one again the analogy between optical and particle beams.

Emmanuel Branlard
http://emmanuel.branlard.free.fr/