108
An Introduction to Beam Physics
common particle optical elements. However, in general the motion is not
linear, and in many situations it is important to take into account various
contributions of the nonlinear effects.
Unfortunately, as straightforward as the determination of linear transfer
matrices is in many cases, the determination of nonlinear terms, especially
those of higher order, becomes exceedingly more difficult using paper and
pencil methods. In the following we will describe a general method that
in principle allows the recursive determination of aberrations of higher and
higher orders, but which in practice quickly succumbs to a rapid increase in
complexity for higher orders.
Let us assume we are given the system described by the following ordinary
differential equation (ODE)
d
ds
r =
f ( r, s) ,
which satisfies
f ( 0, s) = 0. We perform a Taylor expansion of the right hand
side. Because the system is origin preserving, the first contribution is linear,
and altogether we have
d
ds
r = ˆ
M (s) · r +
∞
j=2
N j ( r, s),
where the
N j are polynomials of exact order j, the coefficients of which may
depend on s.
The first step in obtaining a perturbative solution of the system is a linearization as in the previous sections. We have
d
ds
r = ˆ
M (s) · r.
For this system, we determine a system of n independent solutions l k (s) ,
k = 1, . . . , n, that satisfy the initial condition
l k (0) = (0, 0, . . . , 1
kth
, . . . 0, 0)
T .
We define the matrix
ˆ
L(s) =
l 1 (s) , l 2 (s) , . . . , l n (s)
,
and observe that the general solution of the linearized problem with initial
condition r i is then given by
r (s) = ˆ
L(s) · r i .
In practice, the determination of ˆ
L may be possible in closed form, depending on the structure of ˆ
M , or may have to rely on numerical integration. For
An Introduction to Beam Physics
common particle optical elements. However, in general the motion is not
linear, and in many situations it is important to take into account various
contributions of the nonlinear effects.
Unfortunately, as straightforward as the determination of linear transfer
matrices is in many cases, the determination of nonlinear terms, especially
those of higher order, becomes exceedingly more difficult using paper and
pencil methods. In the following we will describe a general method that
in principle allows the recursive determination of aberrations of higher and
higher orders, but which in practice quickly succumbs to a rapid increase in
complexity for higher orders.
Let us assume we are given the system described by the following ordinary
differential equation (ODE)
d
ds
r =
f ( r, s) ,
which satisfies
f ( 0, s) = 0. We perform a Taylor expansion of the right hand
side. Because the system is origin preserving, the first contribution is linear,
and altogether we have
d
ds
r = ˆ
M (s) · r +
∞
j=2
N j ( r, s),
where the
N j are polynomials of exact order j, the coefficients of which may
depend on s.
The first step in obtaining a perturbative solution of the system is a linearization as in the previous sections. We have
d
ds
r = ˆ
M (s) · r.
For this system, we determine a system of n independent solutions l k (s) ,
k = 1, . . . , n, that satisfy the initial condition
l k (0) = (0, 0, . . . , 1
kth
, . . . 0, 0)
T .
We define the matrix
ˆ
L(s) =
l 1 (s) , l 2 (s) , . . . , l n (s)
,
and observe that the general solution of the linearized problem with initial
condition r i is then given by
r (s) = ˆ
L(s) · r i .
In practice, the determination of ˆ
L may be possible in closed form, depending on the structure of ˆ
M , or may have to rely on numerical integration. For