Chapter 4
The Linearization of the Equations
of Motion
In order to develop a matrix theory of particle optics similar to the Gaussian
theory in glass optics, we have to linearize the equations of motion. This
procedure is rather similar to other linearizations in physics; in particular,
it is very similar to the study of so-called small oscillations in mechanics.
Since the solutions of linear systems depend linearly on the initial conditions,
indeed the resulting transfer maps will be linear as needed. It is worth noting
that, although a 6 × 6 matrix is required to describe the linear motion, only
blocks of 2 × 2 and 3 × 3 are needed for decoupled linear motion.
We begin the actual process of linearization with the linearization of the
fields, which corresponds to quadratic potentials in eq. (3.7). We begin our
discussion with the case in which the potentials on the reference orbit vanish,
which describes the situation of electric and magnetic multipoles as well as in
deflectors. The case of electric and magnetic lenses do require the presence of
potentials on axis, and they will be discussed in detail below.
In the electric case, let us assume that there is no potential on axis, i.e.,
a 0,0 = 0, and that in the midplane, we have
E x = −E x0 (1 + n e x) .
Because of the recursion relation for fields, eq. (3.8), we obtain an out-of-plane
expansion of
E y = E x0 (h + n e )y,
as well as an electrostatic potential
V (x, y) = E x0 x +
1
2
E x0 (n e x
2
− (h + n e ) y
2 ),
which is chosen in such a way as to vanish on the reference orbit.
In the magnetic case, let the midplane field be given by
B y = B y0 (1 + n b x).
Due to the recursion relation, we must then have
B x = B y0 n b y.
67
DOI:10.1201/b12074-4
The Linearization of the Equations
of Motion
In order to develop a matrix theory of particle optics similar to the Gaussian
theory in glass optics, we have to linearize the equations of motion. This
procedure is rather similar to other linearizations in physics; in particular,
it is very similar to the study of so-called small oscillations in mechanics.
Since the solutions of linear systems depend linearly on the initial conditions,
indeed the resulting transfer maps will be linear as needed. It is worth noting
that, although a 6 × 6 matrix is required to describe the linear motion, only
blocks of 2 × 2 and 3 × 3 are needed for decoupled linear motion.
We begin the actual process of linearization with the linearization of the
fields, which corresponds to quadratic potentials in eq. (3.7). We begin our
discussion with the case in which the potentials on the reference orbit vanish,
which describes the situation of electric and magnetic multipoles as well as in
deflectors. The case of electric and magnetic lenses do require the presence of
potentials on axis, and they will be discussed in detail below.
In the electric case, let us assume that there is no potential on axis, i.e.,
a 0,0 = 0, and that in the midplane, we have
E x = −E x0 (1 + n e x) .
Because of the recursion relation for fields, eq. (3.8), we obtain an out-of-plane
expansion of
E y = E x0 (h + n e )y,
as well as an electrostatic potential
V (x, y) = E x0 x +
1
2
E x0 (n e x
2
− (h + n e ) y
2 ),
which is chosen in such a way as to vanish on the reference orbit.
In the magnetic case, let the midplane field be given by
B y = B y0 (1 + n b x).
Due to the recursion relation, we must then have
B x = B y0 n b y.
67
DOI:10.1201/b12074-4