Twiss

The Twiss parameters, also called Courant–Snyder parameters, are a set of three lattice functions α(s), β(s), and γ(s) that describe the linear optics of a charged-particle beam in the transverse plane. They characterize the phase-space ellipse of the beam, defined in each plane (x, x') by γ x^2 + 2 α x x' + β x'^2 = ε, where ε is the geometric emittance, an invariant under linear transport.

The three functions are not independent: γ = (1 + α^2)/β, and β γ − α^2 = 1. The moments of the beam relate to ε: ⟨x^2⟩ = ε β, ⟨x x'⟩ = − ε α, ⟨x'^2⟩ = ε γ. The rms beam sizes are σ_x = sqrt(ε β) and σ_{x'} = sqrt(ε γ), with covariances given by the cross-term.

Under linear optical transport, the Twiss parameters evolve with the path length s according to the focusing lattice. The beam matrix Σ transforms as Σ → M Σ M^T for a transfer matrix M; equivalently, the Twiss parameters evolve through the lattice according to the Courant–Snyder equations. The determinant condition β γ − α^2 = 1 ensures a conserved phase-space area ε.

Twiss parameters are fundamental in designing and matching beamlines, storage rings, and linear accelerators. They allow simple expressions for beam envelope and are used to optimize focusing, minimize beam size at a target, and describe optical functions independent of the full distribution.

History: The formalism originates in the Courant–Snyder treatment of linear optics in accelerators and is commonly referred to as Twiss parameters in many texts.