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The transverse transport of a charged-particle beam relative to a reference trajectory. Linear focusing gives betatron oscillations, described by Hill's equation and Courant-Snyder parameters; the beam envelope depends on its geometric emittance.

Quadrupole magnets provide transverse restoring forces proportional to displacement from the reference orbit. Their magnetic field increases linearly away from the axis, so an off-axis particle receives a kick that can bring its trajectory back toward the design orbit. A magnetic quadrupole focuses in one transverse plane and defocuses in the other; a single quadrupole cannot confine both planes.

An alternating-gradient focusing lattice alternates quadrupole orientations with drift spaces. With suitable strengths and separations, a focusing-defocusing sequence gives stable betatron oscillations in both planes. The focusing acts on transverse position, while the phase advance through the lattice converts an initial angular deviation into position and then a corrective kick. The beam is thus transported within a bounded envelope rather than allowed to drift into the pipe. Dipoles provide the main bending; the quadrupole lattice controls the small deviations.