Using a Matrix Approach in Nonlinear Beam Dynamics for Optimizing Beam Spot Size
Beam focusing is understood as the result of non-linear motion of a set of particles. As a result of this motion, we have the beam spot on the target. The set has a volume (the phase volume, or emittance). For a given brightness, the phase volume is proportional to the beam current and vice versa. The beam has an envelope surface. All particles of the beam are located inside of this surface, inside of this beam envelope. For the same phase volume (or beam current) the shape of the beam envelope can be different. We say the beam envelope is optimal if the spot size on the target has a minimum value for a given emittance. The essential feature of our optimization is the matrix approach for non-linear beam motion. In this approach we obtain and use analytical expressions for the matrizant (or transfer matrix) and for the envelope matrix. This matrix technique is known as the Matrizant method.