Strong Focusing: Quadrupole FODO Lattice

Every modern particle accelerator, from hospital synchrotrons to the LHC, keeps its beam alive with alternating-gradient "strong focusing": a chain of quadrupole magnets that alternately focus and defocus the beam in each transverse plane, netting a stable focus in both. This simulator builds a real FODO lattice from focusing (QF) and defocusing (QD) quadrupoles separated by field-free drifts, propagates a ring of test particles through it using the exact linear transfer matrices used in real accelerator design codes, and renders the resulting betatron oscillation in 3D. Live readouts track the phase advance per cell, the lattice tune, and the stability condition — push the quadrupole strength or drift length too far and the beam envelope grows unbounded and strikes the vacuum chamber, exactly as it would in a real ring.