Einzel Lens
© 2012 COMSOL. All
rights reserved.
Introduction
•
An
Einzel
lens is an electrostatic device used for focusing
charged particle beams.
•
The beam focusing depends on the initial particle energy, the
voltage on the
Einzel
lens and the initial beam collimation
(initial radius and transverse velocity of the charged particles).
•
The problem is almost axisymmetric: the electrostatic field is
axisymmetric but the particle velocity transverse to the beam
axis is random, which spoils the symmetry. Thus, the model is
solved in 3D.
•
The model uses COMSOL
Multiphysics
with the Particle
Tracing Module.
•
3 identical cylinders arranged on same axis
•
Outer cylinders are grounded
•
Middle cylinder has non

zero voltage
•
This particular model uses cylinders, but other geometries are
possible.
Geometry
𝑉
=
0
𝑉
=
0
𝑉
=
𝑉
0
≠
0
Focusing By Fringe Fields
Since the focusing is accomplished via the fringe fields, then it is important to model
this region accurately. This includes an accurate model of the cylinder edges (filleted)
and a dense mesh since the spatial variation of the field is highest near the edges.
Notes on COMSOL Model
•
The dimensions are listed in the Parameters table.
•
The geometry is constructed in a plane, which is then rotated
180
o
and mirrored.
•
Domains and boundaries are grouped in Selections (under
Definitions) to make the rest of the model clear.
•
The mesh is also constructed in the plane and then swept
180
o
in both directions.
•
The electrostatics problem is solved first. The particle
trajectories are computed in a separate study.
Initial Set Up
The first part is a 3D
e
lectrostatics calculation.
Parameters
Add a table of parameters under
Global Definitions.
Geometry
–
Work Plane
Draw a 2D cross

section of the geometry in a Work Plane. Later, the plane will
be revolved and mirrored to get the 3D geometry. The Work Plane is
𝑦
=
0
.
Geometry
–
Rectangle With Filleted Corners
Draw a Rectangle in the Work Plane, and
then Fillet all 4 corners.
Geometry
–
Array
Make copies of the filleted rectangle using a
1
×
3
array.
Geometry
–
Vacuum Chamber Rectangle
Add the large rectangle representing
the vacuum chamber, including an
inner box using “Layers”.
Geometry
–
Work Plane
Add a point that will eventually
trace out a circle for the beam inlet.
The finished 2D geometry.
Geometry
–
Revolve
Revolve all objects in
the Work Plane by
180
o
about its axis.
Geometry
–
Mirror
Mirror all of the revolved objects about the
xz

plane, whose normal is
0
,
1
,
0
.
Keep the original set of objects as well!
Selections
Add Selections to clearly group domains and boundaries for later use.
Right click on Definitions and choose “Selections > Explicit”.
Selections
–
Cylinder 1
1)
Click on “Wireframe Rendering” in the Graphics window.
2)
Click on “Go to ZX View” in the Graphics window.
3)
Click on “Select Box” in the Graphics window and drag the box around the cylinder
on the left. Right

click to add the domain numbers to the selection list.
4)
Rename “Explicit 1” to “Cylinder 1”.
Step 1
Step
2
Step
3
Selections
–
Cylinder 2
1)
Right

click on Definitions and choose
“Selections > Explicit”.
2)
Click on “Select Box” in the Graphics window and drag the box around the cylinder
in the middle. Right

click to add the domain numbers to the selection list.
3)
Rename “Explicit 2” to “Cylinder 2”.
Step
2
Selections
–
Cylinder 3
1)
Right

click on Definitions and choose
“Selections > Explicit”.
2)
Click on “Select Box” in the Graphics window and drag the box around the cylinder
on the right. Right

click to add the domain numbers to the selection list.
3)
Rename “Explicit 3” to “Cylinder 3”.
Step
2
Selections
–
Vacuum Domains
1)
Right

click on Definitions and choose
“Selections >
Complement”.
2)
Click on “Add” (the “+” sign) and choose all 3 cylinders.
3)
Rename “Complement 1” to “Vacuum domain”.
Selections
–
Boundaries
•
Define Selections to group the boundaries of the cylinders individually and
collectively.
•
Right

click on “Cylinder 1” and choose “Duplicate”.
•
Change “Output Entities” from “Selected domains” to “Adjacent boundaries”.
•
Rename “Cylinder 1.2” to “Cylinder 1
–
boundaries.
•
Repeat for the other 2 cylinders.
Selections
–
All cylinder boundaries
•
Right

click on Definitions and choose “Selections > Union”.
•
Under “Geometric Entity Level”, change Level from Domain to Boundary.
•
Press the Add button and select all of the cylinder boundaries defined previously.
•
Rename “Union 1” to “All cylinder boundaries”.
Electrostatics
•
Right

click on Electrostatics and choose
“Sort by Space Dimension”.
•
Under “Domain Selection”, change “All
domains” to “Vacuum domains”. (The
potential inside each cylinder is equal to
the value on the surface, so it does not
need to be computed.)
Electrostatics
–
Vacuum Properties
•
We could define a material called “Vacuum”, but only the permittivity is needed so
just set the value in “Charge Conservation 1”.
•
Click on “Charge Conservation 1” and under “Electric Field”, change “Relative
permittivity” from “From material” to “User defined”. Leave the value as 1.
Electrostatics
–
Boundary Conditions
•
The default boundary condition is “Zero Charge”, which means
∙
𝐷
=
∙
−
𝜖𝛻𝑉
=
0
, that is, the surface charge density is zero.
•
The “Zero Charge” boundary condition is fine for the ends of the vacuum chamber.
For the other surfaces, the potential will be fixed.
•
Right

click on “Boundaries” and choose Ground. Select all of the surfaces on the
sides of the vacuum chamber. (2, 3, 6, 8, 11, 13, 59, 61, 63, 111, 112, 113)
•
Rename “Ground 1” to “Ground
–
vacuum chamber walls”.
Electrostatics
–
Outer Cylinders
•
Right

click on “Boundaries” and choose “Ground”.
•
Under “Boundary Selection”, change “Manual” to “Cylinder 1

boundaries”.
•
Rename “Ground 2” to “Ground
–
Cylinder 1”.
•
Repeat for Cylinder 3.
Electrostatics
–
Middle Cylinder
•
Right

click on “Boundaries” and choose “Electric Potential”.
•
Under “Boundary Selection”, change “Manual” to “Cylinder 2

boundaries”.
•
Change “Electric potential” to “V0”, defined in the Parameters table above.
•
Rename “Electric Potential 1”
to “Electric Potential

Cylinder 2
”.
Mesh
–
Edge
•
The particle trajectories will be near the cylindrical axis and so construct a dense
mesh there.
•
Right

click on “Mesh 1” and choose “More Operations > Edge”.
•
Add the edges on the axis to the Selection list. (122, 125, 128)
•
Right

click on “Edge 1” and choose “Distribution”.
•
Change “Number of elements” from 5 to 120.
•
Press “Build Selected”.
Mesh
–
Free Triangular
•
Add boundaries 85, 87 and 89 to the Selection list.
•
Right

click on “Free Triangular 1” and choose “Size”.
•
In “Size 1”, clear boundaries 85 and 89 from the Selection
list, so that only boundary 87 remains.
•
Under “Element Size”, change “Predefined” from
“Normal” to “Extremely fine”.
•
Select “Custom” and change “Maximum element size” to
0.01.
•
Press “Build Selected”.
•
Right

click on “Mesh 1” and choose “More Operations > Free Triangular”.
Mesh
–
Swept
•
Right

click on “Mesh 1” and choose “Swept”.
•
Add the domains, source faces and destination
faces as shown on the left.
•
Right

click on “Swept 1” and choose “Distribution”.
•
Change “Number of elements” from 5 to 10.
•
Press “Build Selected”.
Mesh
–
Swept
•
Right

click on “Swept 1” and choose “Duplicate”.
•
In the newly created “Swept 2”, change the
domains to 1,3,5, which are the other half of the
vacuum chamber.
•
Press “Build All”.
•
Right click on “Study 1” and choose Compute!
Data Sets
–
Cut Plane: y=0
•
Create 3 new data sets to plot a cut

away view of the equipotential surfaces near
the
Einzel
lens
•
Right

click on “Data Sets” and choose “Cut Plane”
•
Under “Plane Data”, change Plane to “
xz

planes”
•
Rename “Cut Plane 1” to “Cut Plane: y=0”
Data Sets
–
Solution 1
–
behind lenses only
•
Under “Data Sets”, right

click on “Solution 1” and choose
Duplicate
•
Rename “Solution 2” to “Solution 1
–
behind lenses only”
•
Right

click on
“Solution 1
–
behind lenses only
” and choose
Selections
•
Change “Geometric entity level” to “Domain” and add
domain 3, which is one of the half cylinder sections near the
Einzel
lens
Data Sets
–
Solution
3
–
cylinder surfaces
•
Right

click on “Data Sets” and choose “Solution”
•
Rename “Solution 3” to “
Solution 3
–
cylinder
surfaces”
•
Right

click on
“Solution 3
–
cylinder surfaces
” and choose Selection
•
Change “Geometric entity level” to Boundary
•
Change Selection to “All cylinder boundaries”
Adding Views
•
Add a “View” for the upcoming plot so that the perspective and zoom level can
remain fixed, even after more model steps are added
•
Right

click on “Show” and check “Advanced Results Options”, which has the effect
of adding a new node called “Views” under Results
•
Right

click on “Results > Views” and choose “View 3D”.
•
In “View 3D 3”, uncheck “Show grid” and “Show axis orientation”.
Plots
–
Equipotential Surfaces
•
Rename “Electric Potential” to “Equipotential surfaces near
Einzel
lens”
•
Under “Plot Settings”, change the View to “View 3D 3” and uncheck the box “Plot
data set edges”
•
Right

click on
“Equipotential surfaces near
Einzel
lens
” and choose Contour. Also
add
Isosurface
and Surface.
•
The plot settings for each are shown on the next slide.
Plot Settings
–
Equipotential Surfaces
Contour
Isosurface
Surface
Equipotential Surfaces
•
Rotate and zoom

in to obtain a plot similar to the one below
2D Plot of Fringe Field
•
Plot the fringe field in the cut plane y=0.
•
Right

click on Results and choose “2D Plot Group”.
•
Rename “2D Plot Group 2” to “Fringe Field”.
•
Change “Data set” to “Cut Plane: y=0”.
•
Right

click on “Fringe Field” and choose “Surface”. Also add “Contour” and
“Arrow Surface”.
•
The plot settings are shown on the next slide.
Plot Settings for Fringe Field
Surface
Contour
Arrow Surface
Fringe Field
•
Zoom

in to obtain a plot of the fringe field similar to the one
below.
Add Particle Tracing
To add “Charged Particle Tracing” to the model:
1)
Right

click on “Model 1” and choose “Add Physics”.
2)
Choose “AC/DC > Charged Particle Tracing”.
3)
Choose “Time Dependent” and uncheck “Electrostatics” below.
Particle Tracing
–
Vacuum Domains
•
Once “Charged Particle Tracing” has been added to the model, right

click on it and
choose “Sort By Space Dimension”.
•
Change “Domain Selection” to “Vacuum domains”.
Particle Tracing
–
Electric Force
•
By default, there are no forces on the particles. Right

click on “Charged Particle
Tracing > Domains” and choose “Electric Force”.
•
The electric force only needs to be computed near the
Einzel
lens. (It is negligible
elsewhere.) Add only domains 3 and 4 to the selection list.
•
The electric field is computed in “Electrostatics”, so change “Electric field” from
“User defined” to “Electric field (
es
/ccn1)”.
•
The gradient of the electrostatic potential is not continuous across mesh element
boundaries, so then neither is the electric field. Check the box beside “Use
piecewise polynomial recovery on field” to apply smoothing.
Particle Tracing
–
Coulomb Repulsion
•
The particle density in the beam is very low so the Coulomb repulsion
between particles is negligible.
•
If the particle density were high enough so that the Coulomb repulsion
between particles is significant, then this effect could be added by right

clicking on Domains and choosing “Particle

Particle Interaction”.
•
“Particle

Particle Interaction” is not added to the model.
Particle Tracing
–
Random Function
•
Define a random function in the model under “Global Definitions”. This will be
used in the initial particle velocity components transverse to the beam axis.
•
Change the “Distribution” of random numbers to Normal.
•
The number of arguments is 1, which is the seed for the random number
generator.
Initial Conditions of the Particles
•
The initial conditions are the position and the velocity.
•
The initial position is on the small circle at the end of
the vacuum chamber. Right

click on Boundaries and
choose Inlet.
•
Select boundaries 57, 58, 84 and 86.
•
Add the initial velocity. The x

and y

components are
small and random using seeds “x/
initial_beam_radius
”
and “y/
initial_beam_radius
”, respectively.
•
With “Initial position = Mesh based” and “Refinement
factor = 1”, there will be 1 particle released on each
mesh element on the boundary.
Electrons
•
Click on “Global > Particle Properties 1” and verify that the particles are electrons.
•
The charge number is

1 for the electron, that is, the charge on the particle is
(charge number) x (charge on proton).
•
The mass is set to “
me_const
”, where “_
const
” signifies a built

in physical
constant.
•
To see a list of all built

in physical constants, click on “Help > Documentation”,
search with “_
const
” and click on “Physical Constants”.
Particle Tracing
–
Time Dependent Study
•
Click on “Study 2 > Step 1: Time Dependent”
and specify the times for the particle tracing
calculation.
•
Note that “T” is the time it would take a
particle to travel from one end of the vacuum
chamber to the other in the absence of any
external force. The end time of the study is
5% higher in case the particles slow down
near the
Einzel
lens. (“T” is defined in
“Global Definitions > Parameters”.)
•
Verify that only “Charged Particle Tracing” will
be computed, not “Electrostatics”.
•
To use the electrostatics solution from the
previous study, check the box beside “Values
of variables not solved for”, change “Method”
to “Solution” and change “Study” to “Study 1,
Stationary”.
•
Right

click on “Study 2” and choose Compute!
Create a New Plot View
•
To create a new plot view, right

click on “Results > Views” and choose “View 3D”.
•
Clear all of the boxes.
•
The angle and zoom level for the following plots can be set to this particular
view, “View 3D 5”. If we want to change the view while doing something else in
the Model above, the view for the plots using “View 3D 5” will not be changed.
Plot
–
Particle Trajectories
•
Change the plot settings for “Particle Trajectories (
cpt
)” to those below.
•
The default is to plot the particle position using a point at the final time. Change
this to a line to trace out the trajectory.
•
Change the color so that it is the ratio of the particle kinetic energy, “
cpt.Ep
”, to the
initial kinetic energy, “E0”.
Particle Trajectories and Energies
•
Modify the perspective to obtain a plot similar to the one below.
•
Focusing depends on the initial position and velocity as well as the charge

to

mass
ratio and the voltage on the
Einzel
lens.
What About Relativistic Effects?
The initial speed of the electrons is
~
0
.
3𝑐
. The relativistic kinetic energy is
𝛾
𝑐
2
, where
𝛾
=
1
1
−
𝑣
2
𝑐
2
≈
1
+
1
2
𝑣
2
𝑐
2
+
3
8
𝑣
4
𝑐
4
+
⋯
The first term in this expansion is
𝑐
2
, the rest mass
energy (which does not change in this example!), and the second term is
1
2
𝑣
2
, the
Galilean kinetic energy. The lowest order correction to the kinetic energy is
~
𝑣
4
𝑐
4
, so let’s
plot it!
•
Right

click on “Particle Trajectories (
cpt
)” and choose Duplicate.
•
Rename the new plot group and change “Color Expression” as shown below.
Relativistic effects < 1%, even at 0.3c!
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