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A slit (2D) einzel lens: both polarities focus, the decelerating one much more strongly.

A practical use of :class:tree_d_3d.BoundaryIntegralElectrodes, as opposed to examples/boundary_integral_electrodes.py’s closed-form validation cases: three slit electrodes on the beam axis, the outer two grounded and the centre one held at a lens voltage, act on a beam of test charges — the two-dimensional (slit, not round-aperture) analogue of the einzel lenses used in real ion-source LEBTs. Each electrode is a :class:simgeometry.Aperture2D, two plates of finite thickness, panelled by :func:tree_d_3d.conductors_from_shape with panels graded from short at the aperture tips (where the field varies fastest) to about the plate thickness far from the axis. Rays are traced through the static field by a plain RK4 push (ballistic: no self space charge, one lens voltage held fixed), the same approximation IBSimu and BeamOptics make for linear-optics design.

The textbook einzel (unipotential) lens converges a beam for either sign of the centre electrode’s voltage, but not equally: the decelerating polarity (for a positive test charge, a positive centre voltage) is considerably stronger than the accelerating one at the same voltage magnitude, because a decelerated particle spends longer in the lens field (see e.g. Hawkes & Kasper, Principles of Electron Optics, or Szilagyi, Electron and Ion Optics). This is usually stated for round apertures; this example checks that the slit lens reproduces the same behaviour: both polarities focus the beam at the voltages and beam energy tried here, the decelerating one at less than half the distance past the centre electrode (about 11 against 26 at |V| = 6, beam energy 12.5).

Source: examples/einzel_lens.py. Run it yourself with python examples/einzel_lens.py [--output examples/output] [--quick] (needs pip install "tree-d-3d[examples]").

Figures

Section titled “Figures”

the three slit electrodes and their potentials.

rays converging to a focus for the decelerating polarity.

the same rays for both polarities of equal magnitude — both converge, but the decelerating one focuses less than half as far past the centre electrode as the accelerating one, as for a round-aperture einzel lens.

the kinetic energy of the decelerating-case ray along its path, from well upstream of the lens to a symmetric point downstream (the two electrodes are not individually neutral, so the solved field has a net monopole and decays only as 1/r, the usual 2D electrostatics catch, so “upstream” and “downstream” need to be equidistant from the lens, not merely “before” and “after” it, for the potential — and so the kinetic energy — to return to its starting value).

the focus distance against the centre voltage, for the decelerating polarity only (the accelerating one needs several times the voltage-scan range’s beam energy before it is this strong, which starts to overlap the beam’s own KE): shorter (stronger focusing) at larger voltage, up to where the beam no longer has enough energy to cross the centre electrode’s potential barrier at all.

--quick uses fewer panels and voltage-scan points (under a minute instead of a few); --smoke is the tiny configuration the test suite runs. Needs Plotly (pip install 'tree-d-3d[examples]').