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Einzel Lens
Introduction
An Einzel lens is an electrostatic device used for focusing charged particle beams. It may be found in cathode ray tubes, ion beam and electron beam experiments, as well as in ion propulsion systems. This particular model consists of three axially aligned cylinders; the outer cylinders are grounded and the cylinder in the middle is held at a fixed voltage. The 3D electrostatic field is computed with the Electrostatics interface and the particle trajectories are computed using the Charged Particle Tracing interface.
Model Definition
The focusing capability of an Einzel lens depends on the initial particle energy, the voltage at each electrode, and the initial beam collimation (initial radius and transverse velocity of the charged particles).
The model geometry consists of three identical cylinders arranged on the same axis. The outer cylinders are grounded, and the middle cylinder has fixed voltage V0 = -10 kV. Note that this particular model uses cylinders, but other geometries are possible.
The electrons are released with an initial kinetic energy of 20 keV. The corresponding speed of the electrons is an appreciable fraction of the speed of light, so relativistic effects must be taken into account. The kinetic energy of a relativistic particle is
(1)
where c = 2.99792458 × 108 m/s is the speed of light in a vacuum, mr (SI unit: kg) is the particle rest mass, and
(2)
Substituting Equation 2 into Equation 1 and solving for the particle speed yields
(3)
For electrons, the rest mass is mr = me = 9.10938356 × 10-31 kg. Substituting mr = me and E = 20 keV into Equation 3 yields a velocity magnitude of about 0.27c, so the inclusion of the Relativistic correction term is justified.
This model uses two studies. First the electric potential is computed using a Stationary study. Then the corresponding electric field is used to exert an electric force on the model electrons using the Electric Force feature. The particle trajectories are computed using a Time Dependent study.
Results and Discussion
The equipotential surfaces surrounding the electrodes are shown in Figure 1. The electric potential and fringe fields in a cross section close to the electrodes are shown in Figure 2.
The electron trajectories are plotted as lines in Figure 3. The color is proportional to the particle kinetic energy and normalized to the initial kinetic energy of each particle. The particles decelerate as they approach the lens and then accelerate toward their initial speed as they pass through it. The nominal trajectory colored by hyperemittance, a common measurement the area occupied by a charged particle beam in transverse phase space, is shown in Figure 4.
Figure 1: Isosurfaces of electric potential in the vicinity of the electrodes.
Figure 2: Electric potential and fringe fields in the vicinity of the electrodes.
Figure 3: Electron trajectories in the Einzel lens. The color expression indicates the ratio of the particle kinetic energy to the initial kinetic energy.
Figure 4: Nominal beam trajectory in the Einzel lens. The color expression indicates the beam hyperemittance.
Application Library path: Particle_Tracing_Module/Charged_Particle_Tracing/einzel_lens
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select AC/DC>Electric Fields and Currents>Electrostatics (es).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Stationary.
6
Click  Done.
Global Definitions
Parameters 1
Load the model parameters from a file.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file einzel_lens_parameters.txt.
Geometry 1
Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.
1
In the Geometry toolbar, click  Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file einzel_lens_geom_sequence.mph.
3
In the Geometry toolbar, click  Build All.
4
Click the  Zoom Extents button in the Graphics toolbar.
5
Click the  Wireframe Rendering button in the Graphics toolbar.
Definitions
Create named selections for the electrodes. This will simplify the setup of the boundary conditions and mesh.
Ground Boundaries
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Ground Boundaries in the Label text field.
3
Select Domains 2 and 4 only.
4
Locate the Output Entities section. From the Output entities list, choose Adjacent boundaries.
Electrode Boundaries
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Electrode Boundaries in the Label text field.
3
Select Domain 3 only.
4
Locate the Output Entities section. From the Output entities list, choose Adjacent boundaries.
All Cylinder Surfaces
1
In the Definitions toolbar, click  Union.
2
In the Settings window for Union, type All Cylinder Surfaces in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click  Add.
5
In the Add dialog box, in the Selections to add list, choose Ground Boundaries and Electrode Boundaries.
6
Click OK.
Electrostatics (es)
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
2
In the Settings window for Electrostatics, locate the Domain Selection section.
3
Click  Clear Selection.
4
Select Domain 1 only. There is no need to solve for the electric potential in the other domains, which are inside the electrodes.
Charge Conservation 1
1
In the Model Builder window, under Component 1 (comp1)>Electrostatics (es) click Charge Conservation 1.
2
In the Settings window for Charge Conservation, locate the Constitutive Relation D-E section.
3
From the εr list, choose User defined. Keep the default value for the vacuum relative permittivity.
Grounded Vacuum Chamber Walls
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
In the Settings window for Ground, type Grounded Vacuum Chamber Walls in the Label text field.
3
Select Boundaries 3, 4, 8, and 13 only.
Grounded Cylinders
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
In the Settings window for Ground, type Grounded Cylinders in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Ground Boundaries.
Electric Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electric Potential.
2
In the Settings window for Electric Potential, locate the Boundary Selection section.
3
From the Selection list, choose Electrode Boundaries.
4
Locate the Electric Potential section. In the V0 text field, type V0.
Mesh 1
Since the focusing is accomplished via the fringe fields, it is important to solve for the electric potential near the cylinders accurately, so a finer mesh is used in that area.
Size 1
1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose All Cylinder Surfaces.
5
Locate the Element Size section. From the Predefined list, choose Fine.
Free Tetrahedral 1
1
In the Mesh toolbar, click  Free Tetrahedral.
2
In the Settings window for Free Tetrahedral, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Click  Clear Selection.
5
Select Domain 1 only.
6
Click  Build All.
Study 1
Step 1: Stationary
In the Home toolbar, click  Compute.
Results
Cut Plane: y=0
1
In the Model Builder window, expand the Results>Datasets node.
2
Right-click Datasets and choose Cut Plane.
3
In the Settings window for Cut Plane, type Cut Plane: y=0 in the Label text field.
4
Locate the Plane Data section. From the Plane list, choose xz-planes.
Equipotential Surfaces
1
In the Model Builder window, under Results click Electric Potential (es).
2
In the Settings window for 3D Plot Group, type Equipotential Surfaces in the Label text field.
3
Locate the Plot Settings section. Clear the Plot dataset edges check box.
Delete the default Multislice node.
Multislice 1
1
In the Model Builder window, expand the Equipotential Surfaces node.
2
Right-click Multislice 1 and choose Delete.
Contour 1
1
In the Model Builder window, right-click Equipotential Surfaces and choose Contour.
2
In the Settings window for Contour, locate the Data section.
3
From the Dataset list, choose Cut Plane: y=0.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Black.
6
Clear the Color legend check box.
Isosurface 1
1
Right-click Equipotential Surfaces and choose Isosurface.
2
In the Settings window for Isosurface, locate the Levels section.
3
In the Total levels text field, type 20.
Filter 1
Use a Filter to see a cross section of the equipotential surfaces. Otherwise, only the outermost surface is shown.
1
Right-click Isosurface 1 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type y>0.
Equipotential Surfaces
Next, color the electrode surfaces gray.
Surface 1
1
In the Model Builder window, right-click Equipotential Surfaces and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type 1.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Gray.
Selection 1
1
Right-click Surface 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
From the Selection list, choose All Cylinder Surfaces.
4
In the Equipotential Surfaces toolbar, click  Plot.
5
Click the  Go to Default View button in the Graphics toolbar. Compare the resulting plot to Figure 1.
Fringe Field
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Fringe Field in the Label text field.
Surface 1
Right-click Fringe Field and choose Surface.
Contour 1
1
In the Model Builder window, right-click Fringe Field and choose Contour.
2
In the Settings window for Contour, locate the Coloring and Style section.
3
From the Coloring list, choose Uniform.
4
From the Color list, choose Black.
5
Clear the Color legend check box.
Arrow Surface 1
1
Right-click Fringe Field and choose Arrow Surface.
2
In the Settings window for Arrow Surface, locate the Arrow Positioning section.
3
Find the x grid points subsection. From the Entry method list, choose Coordinates.
4
Click  Range.
5
In the Range dialog box, type 0.42 in the Start text field.
6
In the Step text field, type 0.0050.
7
In the Stop text field, type 0.6.
8
Click Add.
9
In the Settings window for Arrow Surface, locate the Arrow Positioning section.
10
Find the y grid points subsection. From the Entry method list, choose Coordinates.
11
Click  Range.
12
In the Range dialog box, type -0.095 in the Start text field.
13
In the Step text field, type 0.005.
14
In the Stop text field, type 0.095.
15
Click Add.
16
In the Settings window for Arrow Surface, locate the Coloring and Style section.
17
From the Color list, choose White.
18
Use the Zoom Box button to see the details around the electrodes more clearly.
Fringe Field
1
In the Model Builder window, click Fringe Field.
2
In the Fringe Field toolbar, click  Plot. Compare the resulting plot to Figure 2.
Now that the field has been computed, model the propagation of an electron beam through the Einzel lens.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select AC/DC>Particle Tracing>Charged Particle Tracing (cpt).
4
Find the Physics interfaces in study subsection. Clear the check box next to Study 1.
5
Click Add to Component 1 in the window toolbar.
6
In the Home toolbar, click  Add Physics to close the Add Physics window.
Charged Particle Tracing (cpt)
1
Click the  Zoom Extents button in the Graphics toolbar to see the entire geometry.
2
In the Settings window for Charged Particle Tracing, locate the Domain Selection section.
3
Click  Clear Selection.
4
Select Domain 1 only. When the electrode domains are removed from the physics interface selection, the default Wall condition applies to the adjacent boundaries and prevents electrons from passing through the excluded domains.
5
Locate the Particle Release and Propagation section. Select the Relativistic correction check box.
Electric Force 1
1
Right-click Component 1 (comp1)>Charged Particle Tracing (cpt) and choose Electric Force.
2
In the Settings window for Electric Force, locate the Electric Force section.
3
From the E list, choose Electric field (es/ccn1).
4
Select Domain 1 only.
5
Locate the Advanced Settings section. Select the Use piecewise polynomial recovery on field check box.
Particle Beam 1
1
In the Physics toolbar, click  Boundaries and choose Particle Beam.
2
Select Boundaries 5, 6, 9, and 10 only.
3
In the Settings window for Particle Beam, locate the Initial Transverse Velocity section.
4
In the εrms text field, type 5[um].
5
Locate the Initial Longitudinal Velocity section. In the E text field, type E0.
Add a Time Dependent study to compute the electron trajectories over time.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Time Dependent.
4
Find the Physics interfaces in study subsection. Clear the check box next to the Electrostatics (es) interface, which will not be solved for again.
5
Click Add Study in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Time Dependent
1
In the Settings window for Time Dependent, locate the Study Settings section.
2
In the Output times text field, type range(0,T/20,T*1.05).
Manually select the solution from the Stationary study in order to get the electric field computed by the Electrostatics interface.
3
Click to expand the Values of Dependent Variables section. Find the Values of variables not solved for subsection. From the Settings list, choose User controlled.
4
From the Method list, choose Solution.
5
From the Study list, choose Study 1, Stationary.
6
In the Home toolbar, click  Compute.
Results
Particle Trajectories 1
1
In the Model Builder window, expand the Particle Trajectories (cpt) node, then click Particle Trajectories 1.
2
In the Settings window for Particle Trajectories, locate the Coloring and Style section.
3
Find the Line style subsection. From the Type list, choose Line.
4
Find the Point style subsection. From the Type list, choose None.
Color Expression 1
1
In the Model Builder window, expand the Particle Trajectories 1 node, then click Color Expression 1.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type cpt.Ep/E0.
4
In the Particle Trajectories (cpt) toolbar, click  Plot. Compare the resulting plot to Figure 3.
Point Trajectories 1
1
In the Model Builder window, expand the Average Beam Position (cpt) node, then click Point Trajectories 1.
2
In the Settings window for Point Trajectories, locate the Coloring and Style section.
3
Find the Line style subsection. From the Type list, choose Tube.
4
In the Average Beam Position (cpt) toolbar, click  Plot. Compare the resulting plot to Figure 4.
Appendix — Geometry Instructions
Global Definitions
Parameters 1
Load the geometry parameters from a file.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file einzel_lens_geom_sequence_parameters.txt.
Geometry 1
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose zx-plane.
Work Plane 1 (wp1)>Plane Geometry
In the Model Builder window, click Plane Geometry.
Work Plane 1 (wp1)>Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type T_cyl.
4
In the Height text field, type L_cyl.
5
Locate the Position section. In the xw text field, type R_cyl.
6
In the yw text field, type d_lens.
Work Plane 1 (wp1)>Fillet 1 (fil1)
1
In the Work Plane toolbar, click  Fillet.
2
On the object r1, select Points 1–4 only.
3
In the Settings window for Fillet, locate the Radius section.
4
In the Radius text field, type R_cyl_fil.
Work Plane 1 (wp1)>Array 1 (arr1)
1
In the Work Plane toolbar, click  Transforms and choose Array.
2
Select the object fil1 only.
3
In the Settings window for Array, locate the Size section.
4
In the yw size text field, type 3.
5
Locate the Displacement section. In the yw text field, type L_cyl + cyl_sep.
Work Plane 1 (wp1)>Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type R_vac.
4
In the Height text field, type L_vac.
Work Plane 1 (wp1)>Point 1 (pt1)
1
In the Work Plane toolbar, click  Point.
2
In the Settings window for Point, locate the Point section.
3
In the xw text field, type initial_beam_radius.
Revolve 1 (rev1)
1
In the Model Builder window, right-click Geometry 1 and choose Revolve.
2
In the Settings window for Revolve, click  Build All Objects.