Transverse Beam Emittance Measurement and Control

The beam emittance ∈xyz represents the volume of the beam occupied in the six dimensional phase space (x, x′, y, y′, φ, δ), where x and y are the transverse positions, x′ and y′ are the transverse angles, φ is the time-like variable representing the relative phase of the beam, and δ is the relative beam momentum error. Using the notation of the beam matrix Σbeam introduced in Chap. 1, the 6-dimensional emittance is $${\varepsilon _{xyz}} = \det \Sigma _{beam}^{xyz}.$$ Considering now only the horizontal plane, the corresponding 2-dimensional horizontal emittance is obtained from $${\varepsilon _x} = \sqrt {\left\langle {{x^2}} \right\rangle \left\langle {{{x'}^2}} \right\rangle- {{\left\langle {xx'} \right\rangle }^2}} ,$$ where the first moments have been subtracted, and the average (〈…〉) is taken over the distribution function of the beam; recall also (1.27–1.29). An analoguous expression holds for the vertical plane. For a coupled system, the general form of (4.1) must be taken.