Decoupling a transversely-coupled beam based on symplectic transformation theory and its application

Abstract

Transverse projected emittances of injected beams are strictly limited for synchrotrons with multi-loop injection. Horizontal and vertical emittances easily grow up when beam phase spaces in the xx′ and yy′ planes are coupled. Coupling in beam motion can be induced by tilted quadrupoles and solenoids. Various formalisms have been investigated to describe the beam coupling, which include the matrix and Hamiltonian formalism. In the matrix formalism, the transverse beam motion is parametrized by 4 × 4 beam matrices and the transfer matrices of the elements. Several methods or theories are used to obtain the decoupling matrix, including the equations solving, eigensystem analysis, geometrical method, Courant–Snyder theory and Hamiltonian perturbation theory. We propose an approach to obtain the decoupling matrix based on linear symplectic transformation theory. By adopting several consecutive tilted quadrupoles, the philosophy, as well as the numerical code for solving the decoupling and matching problem, are developed and checked for the general case. The transversely-coupled beam due to a 45∘-rotated radio frequency quadrupole (RFQ) can be decoupled in the downstream drift tube linac with only five tilted fixed-gradient permanent-magnet quadrupoles (PMQs), which is suggested to be the minimal number of PMQs for decoupling and matching. The transverse emittance growth is verified to be suppressed below 5% by multi-particle simulation. The error study shows that the transmission rate can reach 98% with a probability of 99%.