Optimization of Flat to Round Transformers Using Adjoint Techniques

A continuous system of moment equations is introduced that models the transverse dynamics of a beam of charged particles as it passes through an arbitrary lattice of quadrupoles and solenoids in the presence of self-fields. Then, figures of merit are introduced specifying system characteristics to be optimized. The resulting model is used to optimize the parameters of the lattice elements of a flat to round transformer, as could be applied in relativistic electron cooling. Results are shown for a case of no space-charge and two cases with space-charge. The optimization is based on a gradient descent algorithm in which the gradient is calculated using adjoint methods that prove to be very computationally efficient. In the optimized configurations for each of the three cases, the beam in the solenoid is round and with uniform second moments.