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Steady-state space-charge optics without a mesh: trajectories as line charges.

A steady beam is a bundle of trajectories; the trajectory carrying the current dI (per unit depth, in 2D) at speed v is a curve of line charge dI / v per unit length. Its space-charge field is then a sum over the charged segments of all trajectories, which the tree code evaluates in O(N) once the segments are turned into sources, and the self-consistent beam is the fixed point “trace in the field of the segments, rebuild the segments”. Mesh codes (IBSimu, picnic’s BeamOptics) deposit the trajectories on a grid and solve Poisson’s equation instead. This example runs the gridless loop on the one 2D case with an exact answer, a uniform sheet beam expanding under its own space charge: inside a uniform slab the field is linear in y, so the beam stays uniform and every trajectory follows y(x) = y0 (1 + kappa x^2) with kappa = q I / (4 eps0 m v0^3), the planar analogue of the envelope equation. Units q = m = eps0 = v0 = 1, half-width 1, I = 0.01 (kappa = 1/400, so the beam doubles in width over 20 units).

Each segment is represented by Gauss-Legendre points carrying its charge dI * dt, so no new kernel is needed; the segments of a trajectory are smoothed over a fraction of the trajectory spacing (the “ribbon”, a Plummer softening), which is what regularises the field a trajectory sees from itself and from its neighbours. A beam that starts and stops has fringe fields at its ends that the infinite slab does not (about a / (pi s) of the slab field at distance s from an end); mesh codes remove them with Neumann planes at the domain ends, and the gridless code does the same by mirroring the segments at x = 0 and x = L (images reflections on each side), which is exact for the self-similar beam at the entrance, where its mirror image is the converging beam.

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

Figures

Section titled “Figures”

field error of 1, 2 and 3 Gauss-Legendre points per segment against the closed-form field of a uniform segment, against the distance over the segment length: the point representation is exponentially accurate beyond a segment length and only the near field needs the exact segment.

the converged trajectories, the envelope against the exact parabola, and the field-free straight rays the iteration starts from.

the change of the trajectories per iteration for several under-relaxation factors; for this beam plain substitution (omega = 1) converges fastest, and under-relaxation only slows it down.

the envelope’s deviation from the parabola along the beam for several trajectory counts (ribbon width 0.3 spacings) and for several numbers of mirror images: the trajectory count hardly matters, the images do, because a mirrored beam is still finite and the missing far slabs at distance s cost about a / (pi s) of the field; the deviation falls as the images extend the beam.

the same deviation against the ribbon width in trajectory spacings: a Plummer ribbon as wide as the spacing weakens the neighbours’ field (the softened field at one spacing is half the bare one, a deficit of about one trajectory’s worth in the whole slab), a narrow one leaves the discrete trajectories’ noise; the width is a numerical parameter of the method, as the cell size is for a mesh.

--quick uses fewer trajectories (a few minutes instead of ten); --smoke is the tiny configuration the test suite runs. Needs Plotly (pip install 'tree-d-3d[examples]').