Apochromatic Twiss Parameters of Drift-Quadrupole Systems with Symmetries

In this article we continue the study of the apochromatic Twiss parameters of straight drift-quadrupole systems wit h special attention given to the properties of these paramete rs for beamlines with symmetries. INTRODUCTION A straight drift-quadrupole system can not be designed in such a way that a particle transport through it will not depend on the difference in particle energies and this dependence can not be removed even in first order with respect to the energy deviations. Nevertheless, the situatio n will change if instead of comparing the dynamics of individual particles one will compare the results of tracking of monoenergetic particle ensembles through the system or will look at chromatic distortions of the betatron function s appearing after their transport through the system. From this point of view, as it was proven in [1], for every driftquadrupole system which is not a pure drift space there exists an unique set of Twiss parameters (apochromatic Twiss parameters), which will be transported through that system without first order chromatic distortions 1. In this paper we continue the study of the apochromatic Twiss parameters of straight drift-quadrupole systems with special attenti o given to the properties of these parameters for beamlines with symmetries. TRANSFER MAPS AND APOCHROMATIC TWISS PARAMETERS Because we are interested in the lowest order chromatic effects and because the map of the drift-quadrupole system does not have second order geometric aberrations and transverse coupling terms, we will restrict our further con sideration to the motion in one degree of freedom, lets say, horizontal. As usual, we will take the longitudinal particl e position τ to be the independent variable and will use the variablesz = (x, px)⊤ for the description of the horizontal beam oscillations. Herex is the horizontal coordinate and px is the horizontal canonical momentum scaled with the constant kinetic momentum of the reference particle. To take into account energy dependence we will use the variable ε, which is proportional to the relative energy deviation, and we will treat this variable as a parameter. With the assumptions made and with the precision needed the horizontal mapM of the drift-quadrupole system can be represented through a Lie factorization as follows : M : =2 exp(: −(ε / 2) · Q(x, px) :) : M :, (1) ∗ vladimir.balandin@desy.de 1For a drift space the apochromatic Twiss parameters do not ex ist. whereM is 2 × 2 linear transfer matrix, the symbol =m denotes equality up to order m (inclusive) with respect to the variablesx, px andε when maps on both sides of (1) are applied to the phase space vector z, and Q(x, px) = c20 x + 2c11 xpx + c02 p2x. (2) As it was shown in [1], for every drift-quadrupole system which is not a pure drift space the quadratic form Q is negative-definite and the apochromatic Twiss parameters at the system entrance can be calculated using the coefficients of this quadratic form according to the following formulas βa = − c02 √ c20c02 − c211 , αa = − c11 √