Computer optimization of retarding lens systems for ESCA spectrometers

The performance of four-element electrostatic lenses as retarding systems between source and analyzer in ESCA spectrometers is calculated. The potential distribution in the lens is defined by an axial potential of the type (?(z) = V_o + Σ(V_i - V_{i- 1})/2 · tanh (ω/a_i (z - z_i)). For a given general shape of the lens and a given retardation ratio, the potentials of the two middle electrodes are fitted to give a paraxial image with a prescribed magnification at the exit slit of the lens system. The equipotential surfaces forming the electrodes are found by calculating the potential in an off-axis point, using the series expansion. All third-order geometrical and first-order chromatic aberrations of the lenses are calculated and used together with the second-order aberrations of the analyzer to calculate optimum dimensions of the lens elements and of the emittance-defining slits. A computer program, of which one part calculates the lens properties and one the properties of the entire system lens-analyzer, is described.Two lens systems are presented in some detail. The first one is intended for use with a hemispherical electrostatic analyzer. The angular acceptance is here defined by an aperture stop inside the lens. In this system, the image position and magnification can be kept constant for retardation ratios at least between 1:2 and 60:1, with moderate potentials on the middle electrodes. The second lens system is designed for a magnetic spectrometer of the π √2-type. Here, the central trajectory in the lens is slightly curved by the magnetic field, and the angular acceptance is defined by a baffle after the lens. This system is optimized for a constant retardation ratio of 5:1.