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Simulations of imperfect PRIMA fringe sensing units and calibration strategies

The PRIMA (Phase-Referenced Imaging and Microarcsecond Astrometry) instrument at ESO/VLTI is scheduled for commissioning in late 2008. It is designed for phased-referenced imaging and narrow-angle astrometry. The latter, which is the focus of this paper, may be used for exoplanet detection. A key PRIMA subsystem consists of two fringe sensing units. They employ polarized and dispersive optics to measure cross fluxes and differential phases in five narrow K band channels without the need of delay-line dithering. The differential phases are used to correct the differential delays, which are the primary observables used to determine relative proper motions, relative parallaxes, and planetary orbits. Real optical components are imperfect, which means that systematics will appear in the differential phases. In this paper, we 1) present a closed mathematical form for the differential phase, including small systematic offsets and random errors; 2) perform Monte Carlo simulations to understand how the small systematic offsets and random errors affect the differential phases; and 3) show that delay-line stepping can be used to eliminate the effects of small systematic offsets and random errors.