Reflection matrix for Gaussian light beams in FEL oscillators

The transformations of Gaussian radiation beams caused by reflection off mirrors is an important issue for the operation of free-electron lasers as oscillators. The reflected radiation from a single incident Gaussian mode will contain other modes due to the finite size, the deflection of the beam, and mismatches in the curvature. A method for analytic computation of the reflection matrix is developed. The mirror surface that reflects spherical incoming wavefronts into spherical outgoing is determined analytically. The cross coupling among vector components of the radiation field, caused by the curvature of the mirror surface, is included