Effect of spherical aberration on the emittance growth and frequency of oscillation of a beam crossing an RF gap

This paper presents the effect of spherical aberration on the transverse emittance growth and frequency of oscillation of a beam envelope inside an RF cavity. This paper is organized into two sections. In the first section, the coefficient of spherical aberration which arises due to third order terms of on-axis electric field component is discussed. An expression is derived for the growth of transverse emittance in an RF gap which includes the coupling between the phase spread of the beam and spherical aberration. second part, using reduced envelope equation for a laminar beam, effect of aberration on the invariant envelope solution is discussed. An expression is found using the Lindstedt–Poincare theory for solution of the envelope equation. The shift in frequency of oscillation of the beam envelope in the RF field is calculated.