Fig. 3: Classical electron dynamics in a Lagrange waveguide.

a Stable trapping of an electron unfolding around \({{{{\rm{L}}}}}_{{{{\rm{A}}}}}\) for the potential landscapes depicted in Fig. 2a, b, as viewed within the co-rotating frame. The normalization factor \({{{{x}}}}_{0}\) here is taken to be 1 \({{{\upmu }}}{{{\rm{m}}}}\). b Spatial and velocity distributions associated with the injected electrons when the aperture at the input of the Lagrange waveguide has a radius of 10 \({{{\upmu }}}{{{\rm{m}}}}\) and is centered at \({{{{\rm{L}}}}}_{{{{\rm{A}}}}}\). c Histogram of electron spatial distribution after 30 cm in free space (\({{{{V}}}}_{0}=0\)). In this case, the electron beam diffracts to a spot size of ~1 mm. d Histogram of electron spatial distribution after 30 cm of propagation in a Lagrange waveguide when the repelling potential is \({{{{V}}}}_{0}=-2.15\,{{{\rm{kV}}}}\). In this latter scenario, the electron beam is stably guided with a mean spot size of ~65 μm. e Transverse trajectory (blue curve) of a high-speed electron (\({{{{v}}}}_{{{{e}}}}\) = 0.999 c) over 100 m. The yellow dashed line depicts the Lagrange point position, located at a radius of ~112 \({{{\upmu }}}{{{\rm{m}}}}\) from the center. The electron was positioned at a distance ~4 \({{{\upmu }}}{{{\rm{m}}}}\) away from the Lagrange point. f Electron beam transverse distribution histogram (for \({{{{v}}}}_{{{{e}}}}\) = 0.999 c) after 100 m of propagation in a Lagrange waveguide when the repelling potential is \({{{{V}}}}_{0}\) = \(-80\,{{{\rm{kV}}}}\). The electron beam is stably guided with a mean spot size of ~215 \({{{\upmu }}}{{{\rm{m}}}}\). The scaling bar (yellow) in (c, d, f) corresponds to 300 μm while that (green) in the inset of (d, f) to 200 \({{{\upmu }}}{{{\rm{m}}}}\).