Planck intermediate results V. Pressure profiles of galaxy clusters from the Sunyaev-Zeldovich effect:(Corrigendum - vol 550, A131, 2013)

We have developed an approximate technique, based on the principles of multi-mode Gaussian optics, for studying the behaviour of shaped off-axis mirrors. We describe a mirror as an inclined phase-transforming plane, where the phase transformation across the plane is determined by the depth of the mirror as a function of position. The scattering matrix is calculated in the usual way by evaluating the overlap integrals over some surface for which the amplitudes and phases of the incoming and outgoing fields are known; because, however, the modes are not orthogonal over the surface of interest, a system of linear equations has to be solved. We demonstrate the “thin-mirror” technique by studying the behaviour of paraboloidal and ellipsoidal mirros, and we show how the performance of measured and approximate surfaces can be assessed.