Einzel lens

The equations of beamtracer.optimization.einzel, declared on the code that implements them and checked by the tests.

EinzelLens

classsource

The fixed settings of an einzel lens, and its problems.

ray

$$r'' + \frac{\phi'}{2\phi} r' + \frac{\phi''}{4\phi} r = 0, \qquad \phi(z) = \frac{E}{q} - V(z)$$

objective

$$\text{spot} = \sqrt{\tfrac12 \langle r^2 \rangle} \ \text{at the target plane}$$

constraints

$$\text{loss} = \max(0, 1 - T) - \text{max\_loss}, \qquad \text{surface\_field} = \frac{\max |E_\mathrm{surface}|}{\text{field\_limit}} - 1$$

transmission

$$T = \frac{I_\mathrm{target}}{\pi J R_b^2}$$

Assumes

• three coaxial tubes, the outer two grounded, the middle one at $V$, inside a grounded wall; a parallel beam launched upstream as rings
• without space charge (current 0) one field solve and trace is the solution, with $10^{-9}$ A as the weight of the moments
• a design that does not fit is a failed evaluation

References

• Reiser, Theory and Design of Charged Particle Beams, 2nd ed. (2008)
• Humphries, Principles of Charged Particle Acceleration (1986)
• Wollnik, Optics of Charged Particles (1987)

Built from

• BeamOptics
• thin_lens_focal_length
• paraxial_focus

paraxial_focus

functionsource

Return where a paraxial ray, parallel at the start, crosses the axis [m].

picht

$$R'' + \frac{3}{16}\left(\frac{\phi'}{\phi}\right)^2 R = 0, \qquad R = r\, \phi^{1/4}, \qquad R(z_0) = 1, \ R'(z_0) = 0$$

focus

$$z_f: R(z_f) = 0, \qquad \text{or } z_\mathrm{end} - R / R' \text{ for a ray still off the axis at the end}$$

Assumes

• a thick lens, paraxial and without aberrations; integrated by solve_ivp (rtol $10^{-9}$, at most one cell per step), the coefficient interpolated linearly
• $\infty$ for a diverging ray; the voltage that focuses on the target by Brent's method

thin_lens_focal_length

functionsource

Return the thin-lens focal length [m] of the on-axis potential [V].

thin lens

$$\frac{1}{f} = \frac{3}{16} \int \left(\frac{\phi'}{\phi}\right)^2 dz$$

discrete

$$\phi' \text{ by central differences}, \qquad \int \text{ by the trapezoidal rule on the mesh nodes}$$

voltage

$$\frac{1}{f(V)} = \frac{1}{x_\mathrm{target} - x_\mathrm{centre}} \quad \text{(Brent's method)}$$

Order

2

Assumes

• paraxial rays, a lens short compared with $f$, the same potential on both sides; $\phi > 0$ everywhere (NaN if an ion is stopped), $\infty$ without a lens
• the on-axis potential per volt from one vacuum solve at 1 kV on the middle tube
• the order is in the node spacing