1. Phase space ellipse and its transformation by an Einzel lens
2012
Phase space ellipse and its transformation by an Einzel lens
Simion project
Supervisors: Nick Tsoupas, Manolis Benis
By: Manos Zegkos, Çınar Bal and Fiona Hanton

2. 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.

3. Einzel Lens Purpose of Einzel Lens:
To focus charge particles in flight path through manipulation by applied electric fields.

4. Elliptical Beam 𝒙𝒐 𝒂 𝟐 + 𝒙 ′ 𝒐 𝒃 𝟐 = 1
𝒙𝒐 𝒂 𝟐 + 𝒙 ′ 𝒐 𝒃 𝟐 = 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.

5. Simion Set-up 22 particle groups were set-up.
All particles located at x=-50 mm with their own specific y co-ordinate 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.

6. Simion Output Results

7. 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
│R│ = 1 in a perfect situation – Simion result is close But shows some non-linearity.
𝒙 𝒐𝒖𝒕 𝒙 𝒐𝒖𝒕 ′ = 𝑹 𝟏𝟏 𝑹 𝟏𝟐 𝑹 𝟐𝟏 𝑹 𝟐𝟐 * 𝒙 𝒐 𝒙 𝒐 ′

8. Sigma Matrix Rσ RT = σout = 𝝈 𝟏𝟏 𝝈 𝟏𝟐 𝝈 𝟐𝟏 𝝈 𝟐𝟐 Comparison:
From calculated R values σout matrix can be found!
Rσ RT = σout = 𝝈 𝟏𝟏 𝝈 𝟏𝟐 𝝈 𝟐𝟏 𝝈 𝟐𝟐 x´ x
(x,x´)
Comparison:
Simion σout =
Theorectical σout =
An ellipse is transformed into another ellipse under linear transformation and the area of the ellipse is conserved

9. Comparison between input and output values of elliptical beam
Angle of rotation ϑ = 11.7 o

10. Comparison between Simion and theoretical results.

11. Difference between theoretical and Simion output ellipse

12. Conclusions
│R│ = which is close to expected value, though shows some non-linearity 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

13. Thank you