Quadrupole focusing and betatron motion

Take the reference beam along positive z and a positive charge with nearly constant reference momentum. Near the axis use a normal quadrupole field whose horizontal component is gradient times y and whose vertical component is gradient times x. This field has vanishing divergence and curl in the ideal current-free aperture. The transverse components of velocity cross field then have opposite signs, so a positive gradient restores horizontal displacement and increases vertical displacement.

Use path distance s as independent variable and define slopes as transverse derivatives with respect to s. In the paraxial approximation the longitudinal momentum dominates, allowing force to be converted into a slope derivative. A negative charge or reversed gradient exchanges the focusing planes. Real quadrupoles have finite length, higher multipoles and fringe fields; the linear equations describe a local small-amplitude approximation, not every trajectory that fits inside the pole gap.

When the magnet is short compared with the scale of transverse motion, integrate the horizontal force while holding position approximately fixed. The slope receives a kick proportional to negative position; the position itself is continuous across the ideal thin lens. A drift then changes position according to the new slope. Reversing those two operations generally changes the final position, which is why beam-line matrix products must state the order in which elements are encountered.

Write a column vector of position and slope. Each ideal lens and drift has determinant one at fixed reference momentum, reflecting area preservation in this uncoupled linear approximation. This property alone does not guarantee confinement: a map can preserve area while stretching one direction exponentially and squeezing another. Nor does it certify nonlinear dynamic aperture or account for momentum spread, which changes the quadrupole strength and introduces chromatic effects.

For a real two-by-two cell matrix with determinant one, eigenvalues solve a reciprocal-pair quadratic. A trace strictly between minus two and two gives complex unit-modulus eigenvalues and a nondegenerate stable linear cell. The cell phase advance follows from half the trace. At the endpoints, repeated unit eigenvalues need separate analysis and can produce secular growth, so equality is not automatically a robust operating point.

A simple equal-strength focusing–defocusing cell with equal intervening drift lengths has a trace reduced by the squared drift-to-focal-length ratio. Both transverse planes can satisfy the same strict stability condition even though each individual quadrupole defocuses one plane. Betatron motion in a ring is the corresponding oscillation about the closed orbit, with an envelope set by the lattice. Resonances, errors and collective forces require additional analysis beyond this ideal linear trace check.