The Linearization of the Equations of Motion
93
our x and a are used frequently in the electron optics community; in particular,
the coordinate x is scaled to
x by a factor depending on the kinetic energy of
the particle. This leads to less change in the value of the position coordinate
x as we travel through a lens. This nonlinear transformation leads to the
approximation of
x=constant being better than that of x=constant, which in
turn leads to better estimates for focal length. There is a large amount of
work on this topic, but we forgo the details and refer to some of the literature
[56, 60, 67].
4.4.1.3 The Plate Lens
We now address a particular type of lens for which it is possible to derive
the transfer matrix analytically. We consider combinations of individual plates
placed perpendicular to the optical axis, each of which has a small hole in its
center through which the beam travels. Each of these plates represents a hard
edge fringe field as discussed above.
The plates are held at different potentials and form equipotential surfaces.
Assuming that the plates extend to infinity, the electric field between two
successive plates is that of a common plate capacitor, and it points in the
direction of the reference axis.
For any electrostatic lens it is desirable that the field far away vanishes.
Compared to the case of the hard edge fringe field, this requires the use of at
least two plates. Assuming the field between the plates to be E 0 and their
distance to be S, i.e.,
E s (s) =
E 0 for 0 ≤ s ≤ S
0
for s < 0, s > S
,
(4.13)
the potential at the second plate is V = −SE 0 , and we have the potential
function
V 0 (s) =
⎧
⎨
⎩
0
for s < 0
−sE 0 for 0 ≤ s ≤ S
−SE 0 for s > S
.
The effects at s = 0 and s = S are merely those of hard edge fringe fields as
discussed before. It is worthwhile to point out that if V > 0, which entails
that E 0 < 0, a particle of positive charge has lower energy at the second plate,
and thus is more susceptible to the transverse electric fields there. From the
discussion of hard edge fringe fields, we know that the transverse fields in
both places are of opposite sign but identical magnitude, thus the second
field, which happens to lead to focusing, has a more significant effect.
Using eqs. (4.11), we have the transfer matrices for the kicks at s = 0 and
s = S as follows. At s = 0, we have
V 0 (0) = 0, K(0) = K 0 , p s (0) = p 0 =
2mK 0 ,
(4.14)
93
our x and a are used frequently in the electron optics community; in particular,
the coordinate x is scaled to
x by a factor depending on the kinetic energy of
the particle. This leads to less change in the value of the position coordinate
x as we travel through a lens. This nonlinear transformation leads to the
approximation of
x=constant being better than that of x=constant, which in
turn leads to better estimates for focal length. There is a large amount of
work on this topic, but we forgo the details and refer to some of the literature
[56, 60, 67].
4.4.1.3 The Plate Lens
We now address a particular type of lens for which it is possible to derive
the transfer matrix analytically. We consider combinations of individual plates
placed perpendicular to the optical axis, each of which has a small hole in its
center through which the beam travels. Each of these plates represents a hard
edge fringe field as discussed above.
The plates are held at different potentials and form equipotential surfaces.
Assuming that the plates extend to infinity, the electric field between two
successive plates is that of a common plate capacitor, and it points in the
direction of the reference axis.
For any electrostatic lens it is desirable that the field far away vanishes.
Compared to the case of the hard edge fringe field, this requires the use of at
least two plates. Assuming the field between the plates to be E 0 and their
distance to be S, i.e.,
E s (s) =
E 0 for 0 ≤ s ≤ S
0
for s < 0, s > S
,
(4.13)
the potential at the second plate is V = −SE 0 , and we have the potential
function
V 0 (s) =
⎧
⎨
⎩
0
for s < 0
−sE 0 for 0 ≤ s ≤ S
−SE 0 for s > S
.
The effects at s = 0 and s = S are merely those of hard edge fringe fields as
discussed before. It is worthwhile to point out that if V > 0, which entails
that E 0 < 0, a particle of positive charge has lower energy at the second plate,
and thus is more susceptible to the transverse electric fields there. From the
discussion of hard edge fringe fields, we know that the transverse fields in
both places are of opposite sign but identical magnitude, thus the second
field, which happens to lead to focusing, has a more significant effect.
Using eqs. (4.11), we have the transfer matrices for the kicks at s = 0 and
s = S as follows. At s = 0, we have
V 0 (0) = 0, K(0) = K 0 , p s (0) = p 0 =
2mK 0 ,
(4.14)