### Exploring the linearity of an Einzel lens

```2012
Simion project
Supervisors: Nick Tsoupas, Manolis Benis
By: Manos Zegkos, Çınar Bal and Fiona Hanton
Linearity of Einzel lens was investigated by:
Theoretical calculation of R transformation matrix.
Theoretical and experimental calculation of sigma matrix.
Comparison between input and output beam ellipse values.
Rotation of elliptical beam after passing through Einzel Lens.
Einzel Lens
Purpose of Einzel Lens:
To focus charge particles in flight path through manipulation by applied electric
fields.
Elliptical Beam

+
′

=1
a=1 mm b= 1o
Xo=displacement from central trajectory.
X’o = angle ray makes in horizontal plane w.r.t central trajectory
• A range of values
between 1 → -1 mm
was used.
Simion Set-up
• 22 particle groups were set-up.
• All particles located at x=-50 mm
with their own specific y coordinate to create ellipse.
• Particle type = electron
• Electrons were passed through
Einzel Lens consisting of 3 plates
of voltages: 0V, 110V and 0V.
• Output results were collected
and used for further analysis.
Simion Output Results
R Matrix Transformation
• The process of a charge particle through an Einzel Lens can be reduced to an R
matrix multiplication.
xout = Rxo
•
Considering only the first order transfer matrix!

′ =

* ′

•
This matrix is multiplied out, and to achieve R matrix values we assume:
x’o =0o and x’o =1o
•
From this we can extract R values and the determinate of R is found to be
0.95251
• │R│ = 1 in a perfect situation – Simion result is close But shows some
non-linearity.
Sigma Matrix
• From calculated R values σout matrix can be found!
Rσ RT =

σout =

Comparison:
Simion σout =
5.71
1.19
Theorectical σout =

x´
 22 ) ( s 0 )
x
 11 ( s 0 )
1.19
0.25
6.11 2.40
2.40 0.31
(x,x´)
slope 
 12
 11
An ellipse is transformed into another ellipse under linear
transformation and the area of the ellipse is conserved
Comparison between input and output values of elliptical beam
Angle of rotation ϑ = 11.7 o
Comparison between Simion and theoretical results.
Difference between theoretical and Simion output ellipse
Conclusions
 │R│ = 0. 953 which is close to expected value, though shows some nonlinearity due to aberrations.
 R matrix was used to find the theoretical output positions of the elliptical
beam.
 The discrepancy of the σ matrix as calculated further demonstrates the
nonlinearity of the Einzel lens.
 Input and output values of elliptical beam are compared .
 The rotation of beam after passing through Einzel Lens was found to be
ϑ=11.7o
Thank you
```